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+arXiv:2301.03318v1  [cs.LG]  9 Jan 2023
+ORIGINAL ARTICLE
+The Optimal Input-Independent Baseline for
+Binary Classification: The Dutch Draw
+Joris Pries1
+|
+Etienne van de Bijl1
+|
+Jan Klein1
+|
+Sandjai Bhulai2
+|
+Rob van der Mei1,2
+1Department of Stochastics, Centrum
+Wiskunde & Informatica, Amsterdam,
+North Holland, 1098 XG, Netherlands
+2Department of Mathematics, Vrije
+Universiteit, Amsterdam, North Holland,
+1081 HV, Netherlands
+Correspondence
+Joris Pries, Department of Stochastics,
+Centrum Wiskunde & Informatica,
+Amsterdam, North Holland, 1098 XG,
+Netherlands
+Email: [email protected]
+Funding information
+No additional funding
+Before any binary classification model is taken into practice,
+it is important to validate its performance on a proper test set.
+Without a frame of reference given by a baseline method, it
+is impossible to determine if a score is ‘good’ or ‘bad’. The
+goal of this paper is to examine all baseline methods that are
+independent of feature values and determine which model is
+the ‘best’ and why. By identifying which baseline models are
+optimal, a crucial selection decision in the evaluation process
+is simplified. We prove that the recently proposed Dutch
+Draw baseline is the best input-independent classifier (inde-
+pendent of feature values) for all positional-invariant mea-
+sures (independent of sequence order) assuming that the sam-
+ples are randomly shuffled. This means that the Dutch Draw
+baseline is the optimal baseline under these intuitive require-
+ments and should therefore be used in practice.
+KEYWORDS
+Baseline, binary classification, benchmark, evaluation, supervised
+learning
+1
+|
+INTRODUCTION
+A binary classification model is trying to answer the following question: Should the instance be labeled as zero or
+one? This question might seem simple, but there are many practical applications for binary classification, ranging from
+1
+
+2
+Joris Pries et al.
+predicting confirmed COVID-19 cases (Pirouz et al., 2020), detecting malicious intrusions (Li et al., 2018) to determin-
+ing if a runner is fatigued or not (Buckley et al., 2017). Whenever a classification model is developed for a practical
+application, it is important to validate the performance on a test set. However, a baseline is necessary to put the
+achieved performance in perspective. Without this frame of reference, only partial conclusions can be drawn from
+the results. An accuracy of 0.9 indicates that 90% of all predictions are correct. But it could be that the model actually
+did not learn anything and such a high accuracy can already be achieved by predicting only zeros. To put the perfor-
+mance in perspective, it should therefore be compared with some meaningful benchmark method, preferably with a
+state-of-the-art model.
+Nevertheless, many state-of-the-art methods are very problem-specific. They can rapidly change and often involve
+many fine-tuned parameters. Thus, as a necessary additional check in the development process, Van de Bijl et al.
+(2022) plead for a supplementary baseline that is general, simple, and informative. This baseline should test if the
+new model truly performs better than a simple model. It should be considered a major warning sign when a model
+is outperformed by e.g., a weighted coin flip. The binary classification model can use information about the feature
+values of a sample, yet it is outperformed by a model that does not even consider these values. Is the model then
+actually learning something productive?
+A theoretical approach for binary classificationis proposed in (Van de Bijl et al., 2022) based on Dutch Draw classifiers.
+Such a classifier draws uniformly at random (u.a.r.) a subset out of all samples, and labels these 1, and the rest 0. The
+size of the drawn subset is optimized to obtain the optimal expected performance, which is the Dutch Draw baseline.
+For most commonly used performance measures, a closed-form expression is given (Van de Bijl et al., 2022).
+However, there are infinitely many ways to devise a baseline method. We only investigate prediction models that do
+not take any information from the features into account, as this will result in a more general and simple baseline. We
+call these models input-independent. Irrespective of the input, the way that such a model predicts remains the same.
+Any newly developed model should at least beat the performance of these kinds of models, as an input-independent
+model cannot exploit patterns in the data to predict the labels more accurately. However, sometimes a model can get
+lucky by accidentally predicting the labels perfectly for a specific order of the labels. The order of the samples should
+not influence the ‘optimality’ of a model. This is why we introduce the notion of permutation-optimality. Furthermore,
+the order of the samples should not change the outcome of the performance measure (positional-invariant). This is
+not a strict condition, as most commonly used measures have this property. Under these restrictions, we prove that
+the Dutch Draw baseline is permutation-optimal out of all input-independent classifiers for any positional-invariant
+measure.
+To summarize, in this paper we:
+• determine natural requirements for a general, informative and simple baseline;
+• prove that the Dutch Draw baseline is the optimal baseline under these requirements.
+These contributions improve the evaluation process of any new binary classification method.
+The remainder of this paper is organized as follows. First, the necessary preliminaries and notations are discussed
+in Section 2. Next, in Section 3 we determine requirements for a general, simple and informative baseline. Further-
+more, we formally define what optimality entails under these requirements. In Section 4, an alternative definition
+
+Joris Pries et al.
+3
+for the Dutch Draw classifiers is given, which is necessary for the main proof. In Section 5, we prove that the Dutch
+Draw baseline is optimal. Finally, Section 6 summarizes the general findings and discusses possible future research
+opportunities.
+2
+|
+PRELIMINARIES
+Next, we introduce some concepts and notations to lay the foundation for the main proof. First, binary classifiers
+(Section 2.1) and performance measures (Section 2.2) for binary classification are discussed. Then, properties of
+permutations are examined in Section 2.3, which will play a crucial role in the proof of the main result.
+2.1
+|
+Binary classifiers
+To find a good baseline for a binary classification model, we first have to discuss what a binary classifier actually is. To
+this end, let X be the feature space (think e.g., �d ). Normally, a binary classifier is defined as a function h : X → {0, 1}
+that maps feature values to zero or one. However, this classifier only classifies one sample at a time. Instead, we are
+interested in classifiers that classify multiple samples simultaneously:
+hM : XM → {0, 1}M ,
+where M ∈ �>0 denotes the number of samples that are classified. This gives classifiers the ability to precisely predict
+k out of M samples positive. Note that a single sample classifier h can simply be extended to classify M samples
+simultaneously by applying the classifier for each sample individually:
+hM : (x1, . . . , xM ) ↦→ (h(x1), . . . , h(xM )) .
+Let HM = {hM : XM → {0, 1}M } be the set of all binary classifiers that classify M samples at the same time.
+|
+Example of a binary classifier
+An example of a binary classifier is a coin toss, where each sample is classified by throwing a coin and determining on
+which side it lands. Let θ ∈ [0, 1] be the probability that the coin lands head, and 1 − θ for tails. Assuming that head
+and tails are classified by 1 and 0 respectively, we get:
+hsingle
+coin (·) :=
+�
+1
+with probability θ,
+0
+with probability 1 − θ.
+Classifying M samples by repeatedly throwing coins can be achieved by:
+hcoin : (x1, . . . , xM ) ↦→
+�
+hsingle
+coin (x1), . . . , hsingle
+coin (xM )
+�
+.
+
+4
+Joris Pries et al.
+2.2
+|
+Performance measures for binary classification
+To assess the effectiveness of a binary classification model, it is necessary to choose a performance measure, which
+quantifies how much the predicted labels agree with the actual labels. Namely, each sample indexed by i has feature
+values xi ∈ X and a corresponding label yi ∈ {0, 1}. Let X := (x1 . . . xM ) ∈ XM be the combined feature values of
+M samples. Furthermore, let Y = (y1, . . . , yM ) denote the corresponding labels. A performance measure for binary
+classification is then defined as µ : {0, 1}M × {0, 1}M → �, where the first entry of µ is the predictions made by the
+classifier and the second entry is the corresponding labels. The performance of classifier hM can now be written as:
+µ(hM (X), Y).
+|
+Example of a performance measure
+An example of a performance measure for binary classification is accuracy (µacc). It is defined as the total number
+of correctly classified samples divided by the total number of samples. For any hM (X) = ( ˆy1, . . . , ˆyM ) ∈ {0, 1}M and
+Y = (y1, . . . , yM ) ∈ {0, 1}M , it holds that
+µacc (hM (X), Y) =
+�M
+i=1 1{ ˆyi =yi }
+M
+.
+|
+Undefined cases
+Some measures are undefined for specific combinations of hM (X) and Y. Take for example the true positive rate
+(Tharwat, 2021), which is the number of correctly predicted positives divided by the total number of actual positives.
+When there are no actual positives, the measure is ill-defined, as it divides by zero. Less obvious, the measure negative
+predictive value (Tharwat, 2021) is undefined when no negatives are predicted, as it is defined as the number of
+correctly predicted negatives divided by the total number of predicted negatives. Defining x
+0 := 0 for all x ∈ � will
+solve many undefined issues. However, this can make it desirable for a classifier to always predict labels that lead to
+a previously undefined measure in order to minimize the measure. Therefore, we redefine µ from now on for every
+ˆY, Y ∈ {0, 1}M to be equal to a constant Cundef, when µ( ˆY, Y) was undefined. We make a distinction for each objective
+(maximizing/minimizing). Let
+Cundef :=
+
+
+max ˆY,Y∈{0,1}M
+�
+µ( ˆY, Y)
+�
+if minimizing,
+min ˆY,Y∈{0,1}M
+�
+µ( ˆY, Y)
+�
+if maximizing.
+It is therefore always disadvantageous for a classifier to predict a previously undefined case. By defining Cundef in this
+way, we do not have to omit such classifiers from our analysis.
+2.3
+|
+Permutations
+To determine which binary classifier is considered to be the ‘best’, we define permutation-optimality in Section 3.3.3,
+which uses permutations to define ‘optimality’. In this section, we examine properties of permutations that are used
+in the main proof (see Section 5). A permutation is a bijective function from a set to itself (Dixon and Mortimer, 1996).
+This means that a permutation is not a reordered list; it is a function that determines where each element should be
+rearranged to.
+
+Joris Pries et al.
+5
+Let SM denote the set of all permutations of a set of size M , also called the symmetric group. More formally,
+SM :=
+�
+π : {1, . . . , M } → {1, . . . , M } s.t. {π (i ) }M
+i=1 = {1, . . . , M }
+�
+.
+|
+Example of symmetric group
+Using the Cauchy one-line notation (Cauchy, 1815), all possible permutations of three elements are given by
+�
+1
+2
+3
+�
+,
+�
+1
+3
+2
+�
+,
+�
+2
+1
+3
+�
+,
+�
+2
+3
+1
+�
+,
+�
+3
+1
+2
+�
+,
+�
+3
+2
+1
+�
+.
+The permutation
+�
+2
+3
+1
+�
+sends the first element to the second position, the second element to the third position
+and the third element to the first position.
+|
+Sample-wise permutations
+To apply permutations to a matrix, we discuss sample-wise permutations. For every M × K dimensional matrix X =
+(x1 . . . xM ), let Xπ denote the sample-wise permutation under π. Thus,
+Xπ := �xπ(1) . . . xπ(M )
+� ,
+with K ∈ �>0 the number of features. This means that the matrix X is reordered by row.
+|
+Properties of permutations
+Next, we outline some properties of SM that are used in the proof of the main result. SM is a group with the com-
+position of functions as group operator (denoted by ◦), thus the group axioms must hold (Dixon and Mortimer, 1996;
+Artin, 2011). This means that there exists an identity element id ∈ SM such that for all π ∈ SM :
+id ◦ π = π = π ◦ id.
+Furthermore, for every π ∈ SM , there exists a unique inverse element π−1 ∈ SM such that
+π ◦ π−1 = id = π−1 ◦ π.
+Thus, for each permutation, there exists an inverse permutation that reverses the change of order of the permutation,
+which is used in Section 5. As each inverse is unique and also contained in SM , it follows that
+{π ∈ SM } = {π−1 : π ∈ SM },
+(1)
+which means that the set of all permutations is the same as the set of all inverses of these permutations. Thus, taking
+
+6
+Joris Pries et al.
+an expectation over all permutations in SM is the same as taking the expectation over all inverse permutations of
+permutations in SM . This is used in the proof of the main result in Section 5.
+3
+|
+ESSENTIAL CONDITIONS
+To prove that the optimal Dutch Draw classifier yields the ‘optimal’ baseline, we first have to define ‘optimality’. When
+is a baseline considered to be optimal? To determine this, the following two questions must be answered: (1) which
+methods do we compare and (2) how do we compare them? To this end, we define the notion of input-independent
+classifiers, positional-invariant measures, and permutation-optimality.
+3.1
+|
+Input-independent classifier
+Any binary classifier can be used as a baseline. However, any good standardized baseline should be general, simple,
+and informative (Van de Bijl et al., 2022). Thus, it needs to be applicable to any domain, quick to train and clearly still
+beatable. To this end, we investigate all models that do not take any feature values into account, as they meet these
+three requirements. Without considering feature values, they can be applied to any domain. Furthermore, they do not
+require any training, because they cannot learn the relationship between the feature values and the corresponding
+labels. This makes them also clearly still beatable, as any newly developed model should leverage the information
+from the feature values to make better predictions.
+A binary classifier hM ∈ HM is called input-independent if for all feature spaces XM
+1 , XM
+2
+and for all feature values
+Xi ∈ XM
+1
+and Xj ∈ XM
+2
+it holds that hM (Xi ) and hM (Xj ) are identically distributed. In other words,
+hM (Xi ) d= hM (Xj ) d=: hM (·),
+where the notation of hM (·) is chosen to visualize that the classifier hM is not dependent on the input. By this
+definition, an input-independent classifier is not dependent on feature values or even the feature domains. Let
+Hi.i.
+M = {hM ∈ HM : hM is input-independent} be the set of all input-independent binary classifiers. A newly devel-
+oped model, that was optimized using the same performance measure, should always beat the performance of an
+input-independent model, as it gains information from the feature values. Otherwise, the model was not able to
+exploit this extra information to make better predictions.
+|
+Example of an input-independent classifier
+The coin flip (see Section 2.1) is by definition input-independent. The feature values have no influence on the proba-
+bility distribution of the coin. Thus, for any (x1, . . . , xM ) ∈ XM ,
+�
+hsingle
+coin (x1), . . . , hsingle
+coin (xM )
+�
+=
+�
+hsingle
+coin (·), . . . , hsingle
+coin (·)
+�
+.
+3.2
+|
+Positional-invariant measure
+To assess the performance of a method, a measure needs to be chosen. Reasonably, the order of the samples should
+not change the outcome of this measure. If a measure has this property, we call it positional-invariant. More formally, a
+
+Joris Pries et al.
+7
+measure µ is positional-invariant if for every permutation π ∈ SM and for all hM (X), Y ∈ {0, 1}M it holds that:
+µ (hM (X), Y) = µ(hM (X)π, Yπ).
+(2)
+This means that any reordering of the coupled predictedand actual labels does not affect the performance score.
+This is not a hard restriction, as most measures have this property. Note for example that the number of true
+positives (TP), true negatives (TN), false positives (FP), and false negatives (FN) are all positional-invariant. Most
+commonly used measures are a function of these four measures (Sokolova and Lapalme, 2009), making them also
+positional-invariant.
+|
+Example of a non-positional-invariant measure
+Nonetheless, it is possible to define measures that are not positional-invariant. For example, take the measure
+λ : {0, 1}M × {0, 1}M → �, (a = (a1, . . . , aM ), b) ↦→ a1,
+which is dependent on the first position of the prediction, as
+λ
+��
+0
+1
+�
+,
+�
+1
+0
+��
+= 0,
+λ
+��
+1
+0
+�
+,
+�
+0
+1
+��
+= 1.
+3.3
+|
+Defining optimality
+To find the ‘optimal’ baseline, it is first essential to specify what ‘optimality’ entails.
+3.3.1
+|
+Optimal classifier
+A binary classifier does not need to have a deterministic outcome. Thus, due to stochasticity, we consider a classifier to
+be optimal if it minimizes/maximizes the expected performance out of all considered binary classifiers (i.e., ˜HM ⊆ HM ).
+Whether optimization means minimization or maximization depends on the objective of the problem. So:
+hmin
+M
+∈ arg min
+hM ∈ ˜HM
+��hM (X) [µ(hM (X), Y)]� ,
+(3)
+hmax
+M
+∈ arg max
+hM ∈ ˜HM
+�
+�hM (X) [µ(hM (X), Y)]
+�
+.
+(4)
+For example, when the goal is to maximize the accuracy, then hmax
+M
+is an optimal baseline out of all other baselines in
+˜HM . Note that there could be multiple different optimal baselines.
+
+8
+Joris Pries et al.
+3.3.2
+|
+Trivial optimal solution
+However, this definition of ‘optimality’ leads to a trivial optimal solution, when we consider all input-independent
+classifiers ( ˜HM = Hi.i.
+M ). Take the deterministic classifier
+˜hmax
+M
+(·) := ˆYmax ∈ arg max
+ˆY∈{0,1}M
+µ( ˆY, Y),
+which always predicts a vector ˆYmax that maximizes the measure µ. Note that ˜hmax
+M
+is clearly input-independent (see
+Section 3.1), thus ˜hmax
+M
+∈ Hi.i.
+M . Furthermore, it holds that
+max
+hM ∈ ˜HM
+��hM (X) [µ(hM (X), Y)]� ≤
+max
+ˆY∈{0,1}M µ( ˆY, Y)
+= �˜hmax
+M
+(·)
+�
+µ( ˜hmax
+M
+(·), Y)
+�
+.
+In other words, the expected performance of ˜hmax
+M
+is always higher or equal compared to any other classifier. Thus,
+˜hmax
+M
+is considered to be optimal (see Equation (4)). The same holds for minimization with
+˜hmin
+M (·) := ˆYmin ∈ arg min
+ˆY∈{0,1}M
+µ( ˆY, Y).
+Essentially, a perfect prediction can always be made by an input-independent classifier, using the actual labels and the
+performance measure. Consider for example the commonly used performance measure: accuracy, which is maximized
+if the prediction ˆY = Y. A classifier ˜hmax
+M
+that always predicts Y, is thus optimal for these given labels. This shows that
+an extension to the definition of ‘optimality’ should be considered.
+3.3.3
+|
+Permutation-optimality
+The optimal property (see Equations (3) and (4)) is not very insightful when we consider all deterministic classifiers,
+as the perfect prediction is always made by one of them. Similarly, a broken clock gives the correct time twice
+a day, but should not be used to determine the time. Therefore, we introduce a new optimality condition called
+permutation-optimality.
+It is often assumed that the test set is randomly shuffled. Therefore, we introducethe notion of permutation-optimality.
+Instead of being optimal for the distinct order that the feature values and corresponding labels are given in, now all per-
+mutations of the samples are considered. A classifier is permutation-optimal if it minimizes/maximizes the expected
+performance for a random permutation of the test set out of all considered binary classifiers ( ˜HM ). Thus,
+hmin
+M
+∈ arg min
+hM ∈ ˜HM
+��π∼U(SM )
+��hM (Xπ ) [µ(hM (Xπ), Yπ)]�� ,
+(5)
+hmax
+M
+∈ arg max
+hM ∈ ˜HM
+�
+�π∼U(SM )
+�
+�hM (Xπ ) [µ(hM (Xπ), Yπ)]
+��
+.
+(6)
+
+Joris Pries et al.
+9
+4
+|
+DUTCH DRAW CLASSIFIER
+A Dutch Draw classifier is defined in (Van de Bijl et al., 2022) for θ ∈ [0, 1], as
+σθ (X) := (1E (i ))i∈{1,...M } with E ⊆ {1, . . . M }
+drawn u.a.r. such that |E | = ⌊M · θ⌉.
+(7)
+In other words, the classifier draws u.a.r. a subset E of size ⌊M · θ⌉ out of all samples, which it then labels as 1, while
+the rest is labeled 0. In this section, we introduce an alternative definition, that is used in the main proof, and show
+that all Dutch Draw classifiers are input-independent.
+4.1
+|
+Alternative definition
+Insteadof the definition in Equation (7), we introduce an alternative definition for the Dutch Draw classifiers to simplify
+the proof of the main result. Given a binary vector (y1, . . . , yM ) ∈ {0, 1}M of length M , note that the number of ones
+it contains can be counted by taking the sum �M
+i=1 yi . Next, we define sets of binary vectors (of the same length) that
+contain the same number of ones. For all j ∈ {0, . . . , M }, define
+Yj :=
+�
+ˆY = (y1, . . . , yM ) ∈ {0, 1}M s.t.
+M
+�
+i=1
+yi = j
+�
+.
+(8)
+In other words, Yj contains all binary vectors of length M with exactly j ones and M −j zeros. For example, for M = 4
+it holds that
+Y0 = {(0, 0, 0, 0) },
+Y1 = {(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0) },
+Y2 = {(0, 0, 1, 1), (0, 1, 0, 1), (0, 1, 1, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 1, 0, 0) },
+Y3 = {(0, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0) },
+Y4 = {(1, 1, 1, 1) }.
+A Dutch Draw classifier selects u.a.r. E out of M samples and labels these as one, and the rest zero. Note that this
+is the same as taking u.a.r. a vector from YE . To simplify notation, let U(A) denote the uniform distribution over a
+finite set A. Thus, when X ∼ U(A) it must hold that �(X = a) =
+1
+|A| for each a ∈ A. Now, a Dutch Draw classifier σθ
+can be rewritten as
+σθ (X) := ˆY with ˆY ∼ U �Y⌊M ·θ⌉
+� .
+(9)
+Put differently, a Dutch Draw classifier σθ chooses u.a.r. a vector with exactly ⌊M · θ⌉ ones as prediction out of all
+vectors with ⌊M · θ⌉ ones (Y⌊M ·θ⌉). This alternative definition simplifies the proof of the main result.
+
+10
+Joris Pries et al.
+4.2
+|
+Input-independence
+Next, we discuss why all Dutch Draw classifiers are input-independent (see Section 3.1). Note that a Dutch Draw
+classifier σθ is independent of feature values, as it is only dependent on θ and M , see Equation (9). In other words, any
+Dutch Draw classifier is by definition input-independent. Instead of σθ (X), we can therefore write σθ (·). To conclude,
+for every θ ∈ [0, 1] it holds that σθ (·) ∈ Hi.i.
+M , which is the set of all input-independent binary classifiers.
+4.3
+|
+Optimal Dutch Draw classifier
+The optimal Dutch Draw classifier σθopt is determined by minimizing/maximizing the expected performance for the
+parameter θ out of all allowed parameter values Θ (Van de Bijl et al., 2022). Note that some measures are undefined
+for certain predictions, thus Θ is not always equal to [0, 1]. Take e.g., the measure precision (Tharwat, 2021), which is
+defined as the number of true positives divided by the total number of predicted positives. Therefore, if no positives
+are predicted, the measure becomes undefined (division by zero). By adapting each measure according to Section 2.2,
+all undefined cases are resolved and Θ = [0, 1] always holds.
+Using the alternative definition of the Dutch Draw classifier (see Equation (9)), we obtain:
+θ∗
+min ∈ arg min
+θ∈[0,1]
+�
+� ˆY∼U
+�
+Y⌊M ·θ⌉
+� �
+µ( ˆY, Y)
+��
+,
+(10)
+θ∗
+max ∈ arg max
+θ∈[0,1]
+�
+� ˆY∼U
+�
+Y⌊M ·θ⌉
+� �
+µ( ˆY, Y)
+��
+.
+(11)
+Depending on the objective, either σθ∗
+min or σθ∗max is an optimal Dutch Draw classifier.
+5
+|
+THEOREM AND PROOF
+After defining input-independence (Section3.1), positional-invariance (Section3.2), permutation-optimality(Section3.3.3),
+and introducing an alternative formulation for the Dutch Draw classifier, all ingredients for the following theorem are
+present.
+Theorem 1 (Main result) The optimal Dutch Draw classifier σθopt is permutation-optimal out of all input-independent clas-
+sifiers (Hi.i.
+M ), for any positional-invariant measure µ. In other words:
+σθ∗
+min ∈ arg min
+hM ∈Hi .i .
+M
+��π∼U(SM )
+��hM (Xπ ) [µ(hM (Xπ), Yπ)]�� ,
+(12)
+σθ∗max ∈ arg max
+hM ∈Hi .i .
+M
+��π∼U(SM )
+��hM (Xπ ) [µ(hM (Xπ), Yπ)]�� .
+(13)
+This means that the optimal Dutch Draw classifier is the best general, simple, and informative baseline.
+Proof Let hM ∈ Hi.i.
+M be an input-independent classifier and let µ be a positional-invariant measure, the classifier is
+permutation-optimal if it minimizes/maximizes the expected performance under a random permutation of the test
+set out of all input-independent classifiers (see Equations (5) and (6)).
+For any input-independent classifier hM , it holds that
+
+Joris Pries et al.
+11
+�hM (Xπ ) [µ(hM (Xπ), Yπ)] = �hM (·) [µ(hM (·), Yπ)] .
+(14)
+The input Xπ is not relevant for the classification, and can thus be omitted.
+In total, there are 2M unique possible predictions in {0, 1}M . Denote these distinct vectors by ˆY1, . . . , ˆY2M such that
+�2M
+i=1 ˆYi = {0, 1}M . Next, the expectation in Equation (14) can be written out by:
+�hM (·) [µ(hM (·), Yπ)] =
+2M
+�
+i=1
+�(hM (·) = ˆYi) · µ( ˆYi, Yπ).
+(15)
+As we need to proof permutation-optimality, we have to take the expectation of Equation (15) over all permutations.
+Using linearity of expectation gives:
+�π∼U(SM )
+
+2M
+�
+i=1
+�(hM (·) = ˆYi) · µ( ˆYi, Yπ)
+
+=
+2M
+�
+i=1
+�(hM (·) = ˆYi) · �π∼U(SM )
+�
+µ( ˆYi, Yπ)
+�
+.
+(16)
+Instead of taking the expectation of a sum, we now take the sum of expectations.
+The measure µ is positional-invariant, thus using Equation (2) gives
+µ( ˆYi, Yπ) = µ(( ˆYi)π−1, (Yπ)π−1) = µ(( ˆYi)π−1, Y).
+(17)
+Applying a permutation does not change a positional-invariant measure µ. In this case, we apply the inverse permuta-
+tion π−1 to retrieve Y.
+Because of Equation (17), it therefore also holds that
+�π∼U(SM )
+�
+µ( ˆYi, Yπ)
+�
+= �π∼U(SM )
+�
+µ(( ˆYi)π−1, Y)
+�
+.
+(18)
+Equation (1) shows that the set of all inverse permutations is the same as the set of all permutations. Given that the
+permutations are drawn u.a.r., taking the expectation over all the inverse permutations is the same as taking the expec-
+tation over all permutations. When permutation π is drawn u.a.r., it namely holds that �(π = s) = �(π = s−1) =
+1
+|SM |
+for all s ∈ SM . Therefore,
+�π∼U(SM )
+�
+µ(( ˆYi)π−1, Y)
+�
+=
+�
+s∈SM
+�
+µ(( ˆYi)s−1, Y) · �(π = s)
+�
+=
+�
+s∈SM
+�
+µ(( ˆYi)s−1, Y) · �(π = s−1)
+�
+= �π∼U(SM )
+�
+µ(( ˆYi)π, Y)
+�
+.
+(19)
+
+12
+Joris Pries et al.
+Thus, π−1 can be replaced with π in Equation (18).
+Recall that Yj is the set of all binary vectors of length M with j ones (see Equation (8)). Furthermore, note that applying
+a u.a.r. chosen permutation π ∈ SM on ˆYi ∈ Yj is the same as selecting u.a.r. ˆY ∈ Yj as outcome, because for every
+ˆY⋆ ∈ Yj it holds that
+�
+�
+( ˆYi)π = ˆY⋆
+�
+=
+1
+|Yj | with π ∼ U (SM ) ,
+and
+�
+�
+ˆY = ˆY⋆
+�
+=
+1
+|Yj | with ˆY ∼ U �Yj
+� .
+Therefore, we can rewrite the expectation �π∼U(SM ) [·] over all permutations into an expectation over a u.a.r. drawn
+vector with the same number of ones, by
+�π∼U(SM )
+�
+µ(( ˆYi)π, Y)
+�
+= � ˆY∼U
+�
+Yj
+�
+: ˆYi∈Yj
+�
+µ( ˆY, Y)
+�
+.
+(20)
+Using Equations (18), (19), and (20) in combination with Equation (16) gives
+2M
+�
+i=1
+�(hM (·) = ˆYi) · �π∼U(SM )
+�
+µ( ˆYi, Yπ)
+�
+=
+2M
+�
+i=1
+�(hM (·) = ˆYi) · � ˆY∼U
+�
+Yj
+�
+: ˆYi∈Yj
+�
+µ( ˆY, Y)
+�
+.
+We have now eliminated all permutations from the equation. Note that the expectation in the right-hand side is the
+same for each ˆYi ∈ Yj . In other words, the expectation is the same for two vectors, when they have the same number
+of ones. Grouping the vectors with the same number of ones, gives
+2M
+�
+i=1
+�(hM (·) = ˆYi) · � ˆY∼U
+�
+Yj
+�
+: ˆYi∈Yj
+�
+µ( ˆY, Y)
+�
+=
+M
+�
+j =0
+�(hM (·) ∈ Yj ) · � ˆY∼U
+�
+Yj
+� �
+µ( ˆY, Y)
+�
+.
+Instead of summing over all possible binary vectors ˆYi ∈ {0, 1}M , all vectors with the same number of ones are grouped
+together, as they have the same expectation. All probability mass of the grouped vectors is also added up. Note, that
+it is thus only relevant for a classifier in which group Yj the prediction hM (·) belongs.
+For any j ∈ {0, . . . , M } it holds that � ˆY∼U
+�
+Yj
+� �
+µ( ˆY, Y)
+�
+is bounded by minimizing/maximizing over all possible values
+
+Joris Pries et al.
+13
+of j . Thus,
+� ˆY∼U
+�
+Yj
+� �
+µ( ˆY, Y)
+�
+≥
+min
+j ′∈{0,...,M } � ˆY∼U
+�
+Yj ′
+� �
+µ( ˆY, Y)
+�
+,
+(21)
+� ˆY∼U
+�
+Yj
+� �
+µ( ˆY, Y)
+�
+≤
+max
+j ′∈{0,...,M } � ˆY∼U
+�
+Yj ′
+� �
+µ( ˆY, Y)
+�
+.
+(22)
+Observe that �M
+j =0 �(hM (·) ∈ Yj ) = 1 and �(hM (·) ∈ Yj ) ≥ 0 hold for each j , therefore it follows using Equations (21)
+and (22) that
+M
+�
+j =0
+�(hM (·) ∈ Yj ) · � ˆY∼U
+�
+Yj
+� �
+µ( ˆY, Y)
+�
+≥
+min
+j ′∈{0,...,M } � ˆY∼U
+�
+Yj ′
+� �
+µ( ˆY, Y)
+�
+,
+M
+�
+j =0
+�(hM (·) ∈ Yj ) · � ˆY∼U
+�
+Yj
+� �
+µ( ˆY, Y)
+�
+≤
+max
+j ′∈{0,...,M } � ˆY∼U
+�
+Yj ′
+� �
+µ( ˆY, Y)
+�
+.
+Consequently, we have found a lower and upper bound for Equations (12) and (13), respectively. Namely,
+min
+hM ∈Hi .i .
+M
+�
+�π∼U(SM )
+�
+�hM (Xπ ) [µ(hM (Xπ), Yπ)]
+��
+≥
+min
+j ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+�µ( ˆY, Y)
+�
+,
+(23)
+max
+hM ∈Hi .i .
+M
+��π∼U(SM )
+��hM (Xπ ) [µ(hM (Xπ), Yπ)]�� ≤
+max
+j ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+�µ( ˆY, Y)
+�
+.
+(24)
+Equality only holds for any classifierhM ∈ Hi.i.
+M , when all probability mass is given to arg minj ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+�µ( ˆY, Y)
+�
+and arg maxj ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+�µ( ˆY, Y)
+�
+, respectively. In other words, the minimum can only be attained if
+�
+jmin∈arg minj ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+� µ( ˆY,Y)
+� �(hM (·) ∈ Yjmin) = 1,
+(25)
+and the maximum only if
+�
+jmax∈arg maxj ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+� µ( ˆY,Y)
+� �(hM (·) ∈ Yjmax) = 1.
+(26)
+A classifier hM ∈ Hi.i.
+M can therefore only attain the minimum/maximum if all predictions belong to a group Yj or
+possibly multiple groups that all minimize/maximize the expectation (depending on the objective).
+
+14
+Joris Pries et al.
+Remember that the Dutch Draw selects the optimal classifier based on Equations (10) and (11),which leads to
+⌊M · θ∗
+min⌉ ∈ arg min
+j ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+� �
+µ( ˆY, Y)
+��
+,
+⌊M · θ∗
+max⌉ ∈ arg max
+j ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+� �
+µ( ˆY, Y)
+��
+.
+Combining this with the alternative definition of the Dutch Draw (Equation (9)) directly gives that
+�
+jmin∈arg minj ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+� µ( ˆY,Y)
+� �(σθ∗
+min (·) ∈ Yjmin) = 1,
+�
+jmax∈arg maxj ∈{0,...,M }
+�
+� ˆY∼U
+�
+Yj
+� µ( ˆY,Y)
+� �(σθ∗max (·) ∈ Yjmax) = 1.
+This shows in combination with Equations (25) and (26) that the optimal Dutch Draw classifier actually attains the
+bound given in Equations (23) and (24). It now follows that,
+σθ∗
+min ∈ arg min
+hM ∈Hi .i .
+M
+��π∼U(SM )
+��hM (Xπ ) [µ(hM (Xπ), Yπ)]�� ,
+σθ∗max ∈ arg max
+hM ∈Hi .i .
+M
+��π∼U(SM )
+��hM (Xπ ) [µ(hM (Xπ), Yπ)]�� .
+Thus, we can conclude that the optimal Dutch Draw classifier attains the minimum/maximum expected performance
+and is therefore permutation-optimal for all input-independent classifiers with a positional-invariant measure.
+6
+|
+DISCUSSION AND CONCLUSION
+A baseline is crucial to assess the performance of a prediction model. However, there are infinitely many ways to
+devise a baseline method. As a necessary check in the development process, Van de Bijl et al. (2022) plead for a
+supplementary baseline that is general, simple, and informative. In this paper, we have therefore examined all base-
+lines that are independent of feature values, which makes them general and relatively simple. Additionally, these
+baselines are also informative, as it should be considered a major warning sign when a newly developed model is out-
+performed by a model that does not take any feature values into account. In this paper, we have shown that, out of
+all input-independent binary classifiers, the Dutch Draw baseline is permutation-optimal for any positional-invariant
+measure. Our findings improve the evaluation process of any new binary classification method, as we have proven
+that the Dutch Draw baseline is ideal to gauge the performance score of a newly developed model.
+Next, we discuss two points that could be considered an ‘unfair’ advantage for the Dutch Draw baseline. First of all,
+we have considered in this paper classifiers that predict M labels simultaneously. This gives classifiers a potential
+advantage over classifying each sample sequentially, as e.g., exactly k out of M samples can be labeled positive. This
+can only be done sequentially when a classifier is allowed to track previous predictions or to change based on the
+
+Joris Pries et al.
+15
+number of classifications it has made. Even with this advantage, we believe that all input-independent models still
+remain clearly beatable by a newly developed model.
+Secondly, the Dutch Draw baseline can be derived for most commonly used measures without any additional knowl-
+edge about the number of positive labels P . Nonetheless, it was shown in (Van de Bijl et al., 2022) that the Dutch
+Draw baseline can only be calculated for the measure accuracy when it is known if P ≥ M /2 holds. If the distribution
+of the training set is the same as the test set, the training set can be used to determine whether P ≥ M /2 is likely
+to hold. Furthermore, a domain expert could estimate whether it is likely that a dataset contains more positives than
+negatives. Take for example a cybersecurity dataset, where there are often significantly less harmful instances and
+more normal instances (Wheelus et al., 2018). There are thus many ways to estimate if P ≥ M /2 holds. Nevertheless,
+even if the Dutch Draw baseline uses this information (only for the accuracy), we believe that any newly developed
+model should still beat the Dutch Draw baseline, as it does not use any feature values to improve prediction.
+Finally, we address future researchopportunities. In this paper, we have only considered binary classification. A natural
+extension would be to also consider multiclass classification (Grandini et al., 2020). Is a strategy similar to the Dutch
+Draw optimal in this case? Can a closed-form expression of the optimal baseline be derived? We believe that the three
+introduced properties (namely, input-independent, positional-invariant, and permutation-optimal) are still relevant for
+the multiclass case. This could help identify what kind of classifier is considered to be optimal. Van de Bijl et al. (2022)
+stated that the Dutch Draw baseline could be used to scale existing measures. This paper provides more motivation
+to scale measures with the Dutch Draw baseline and not by using any other input-independent classifier. Yet, it could
+still be investigated how each measure should be scaled in order to maximize the explainability behind a performance
+score.
+Disclosure statement
+The authors have no relevant financial or non-financial interests to disclose.
+Funding
+No funding was received for conducting this study.
+Availability of data
+No datasets were used in this research.
+references
+Artin, M. (2011) Algebra. Prentice Hall, 2nd edn.
+van de Bijl, E., Klein, J., Pries, J., Bhulai, S., Hoogendoorn, M. and van der Mei, R. (2022) The dutch draw: Constructing a
+universal baseline for binary prediction models. URL: https://arxiv.org/abs/2203.13084.
+Buckley, C., O’Reilly, M., Whelan, D., Farrell, A. V., Clark, L., Longo, V., Gilchrist, M. and Caulfield, B. (2017) Binary classification
+of running fatigue using a single inertial measurement unit. In 2017 IEEE 14th International Conference on Wearable and
+Implantable Body Sensor Networks (BSN), 197–201.
+
+16
+Joris Pries et al.
+Cauchy, A.-L. (1815) Mémoire sur le nombre des valeurs qu’une fonction peut acquérir lorsqu’on y permute de toutes les
+maniéres possibles les quantités qu’elle renferme. Journal de l’École polytechnique.
+Dixon, J. D. and Mortimer, B. (1996) Permutation groups, vol. 163. Springer Science & Business Media.
+Grandini, M., Bagli, E. and Visani, G. (2020) Metrics for multi-class classification: an overview.
+Li, L., Yu, Y., Bai, S., Hou, Y. and Chen, X. (2018) An effective two-step intrusion detection approach based on binary classifi-
+cation and k -nn. IEEE Access, 6, 12060–12073.
+Pirouz, B., Shaffiee Haghshenas, S., Shaffiee Haghshenas, S. and Piro, P. (2020) Investigating a serious challenge
+in the sustainable development process:
+Analysis of confirmed cases of covid-19 (new type of coronavirus)
+through a binary classification using artificial intelligence and regression analysis.
+Sustainability, 12.
+URL:
+https://www.mdpi.com/2071-1050/12/6/2427.
+Sokolova, M. and Lapalme, G. (2009) A systematic analysis of performance measures for classification tasks. Information Pro-
+cessing & Management, 45, 427–437. URL: https://www.sciencedirect.com/science/article/pii/S0306457309000259.
+Tharwat, A. (2021) Classification assessment methods.
+Applied Computing and Informatics, 17, 168–192.
+URL:
+https://doi.org/10.1016/j.aci.2018.08.003.
+Wheelus, C., Bou-Harb, E. and Zhu, X. (2018) Tackling class imbalance in cyber security datasets. In 2018 IEEE International
+Conference on Information Reuse and Integration (IRI), 229–232.
+

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+FullStop: Punctuation and Segmentation Prediction for
+Dutch with Transformers
+Vincent Vandeghinste∗
+[email protected]
+Oliver Guhr†
+[email protected]
+∗Instituut voor de Nederlandse Taal, Leiden, the Netherlands and Centre for Computational Linguistics,
+Leuven.AI, KU Leuven, Belgium
+†Hochschule f¨ur Technik und Wirtschaft, Dresden, Germany
+Abstract
+When applying automated speech recognition (ASR) for Belgian Dutch (Van Dyck et al. 2021),
+the output consists of an unsegmented stream of words, without any punctuation. A next step is
+to perform segmentation and insert punctuation, making the ASR output more readable and easy
+to manually correct. As far as we know there is no publicly available punctuation insertion system
+for Dutch that functions at a usable level.
+The model we present here is an extension of the models of Guhr et al. (2021) for Dutch and is
+made publicly available.1 We trained a sequence classification model, based on the Dutch language
+model RobBERT (Delobelle et al. 2020). For every word in the input sequence, the models predicts
+a punctuation marker that follows the word. We have also extended a multilingual model, for cases
+where the language is unknown or where code switching applies.
+When performing the task of segmentation, the application of the best models onto out of
+domain test data, a sliding window of 200 words of the ASR output stream is sent to the classifier,
+and segmentation is applied when the system predicts a segmenting punctuation sign with a ratio
+above threshold. Results show to be much better than a machine translation baseline approach.
+1. Introduction
+Language is primarily a spoken medium, as every human society has a fully functioning spoken
+language, and until some hundred years ago, only relatively few societies had a written language,
+accessible to only a small class of people (Aronoff 2007).
+In order to study properties of language performance, linguists can use corpora, and in order
+to study properties of spoken language, speech corpora are often used. A written transcript of the
+speech used in these corpora, aligned at the word or sentence/utterance level increases the usability
+of these corpora as they facilitate search and analysis of the contents.
+Manual transcription of speech corpora is a very costly process and is therefore often unavailable.
+Automated speech recognition (ASR) can provide a cheap, albeit imperfect, solution, by providing
+a rough transcript. Manual transcription can then be redefined as a post-editing process on the
+output of the ASR system. For Belgian Dutch spoken data we can use the relatively recent ASR
+system developed by Van Dyck et al. (2021).
+As shown in Figure 1, incoming Belgian Dutch speech is recognized by an ASR system, resulting
+in a transcript consisting of a stream of words with time stamps (not shown in Figure 1), but
+without any segmentation (into sentences or utterances) or punctuation. While this already makes
+it possible to search for the occurrence of specific words in the speech, the streams have a low
+readability because of the lack of segmentation.
+The FullStop model for Dutch provides a punctuation and segmentation prediction system, that
+consists of two steps:
+1. https://huggingface.co/oliverguhr/
+arXiv:2301.03319v1  [cs.CL]  9 Jan 2023
+
+Figure 1: An incoming sound is recognized with an ASR system, resulting in a stream of words. The
+FullStop approach, presented in this paper, segments this stream of words in segments, by predicting
+punctuation.
+1. A sliding window of 200 words slides over the input text that needs to be segmented. Per
+window the system checks how often segmenting punctuation is predicted.
+If this relative
+frequency is above a threshold θ, then the predicted punctuation is accepted and segmentation
+is applied as well. Increasing θ will result in a higher precision at the price of a lower recall.
+We have two experimental conditions with respect to the set of segmenting punctuation S:
+only the full stop (S = {.}), and the full stop and the question mark (S = {., ?}).
+2. This step takes as input the 200 words from the previous step and segments it in batches
+of up to 512 tokens. This is necessary since the used transformer-based models are limited
+to 512 tokens input length. The model predicts for every token whether it is followed by a
+punctuation sign that is part of P = {:-,?.0}, where 0 indicates that the classifier predicts that
+no punctuation follows.
+Together, these two steps apply the classifier of step 2 onto every sliding window of 200 words,
+implying that we get a maximum of 200 punctuation predictions per word, depending on in how
+many sliding windows the word appears. The ratio of predictions of a certain punctuation should
+be above θ before it is accepted.
+One type of speech corpora that particularly motivates this study is the preprocessing of mul-
+timedia corpora, consisting of video and speech. Video is becoming an increasingly popular means
+of communication, with 300 hours of video being uploaded to YouTube every minute,2 and TikTok
+becoming ever more popular. It therefore makes sense that the creation of video corpora becomes
+more important. Audio transcription of these video corpora is often the first step, and the usage
+of ASR helps in the transcription process. There are two video corpora that motivate the current
+research:
+1. The Spoken Academic Belgian Dutch corpus, and
+2. https://fortunelords.com/youtube-statistics/
+
+Motivation
+Stream of words
+ASR
+kijk om je heen alles beweegt alles draait zo komen wij ter
+: wereld de zon de maan de planeten en de sterren kijken
+: toe en wij staan in het midden onze plaats maar Nicolaas
+Copernicus kwam en stelde dat de Zon in het midden staat
+en dat wij om haar heen draaien net als de andere planeten
+FullStop
+Segmented and punctated text
+kijk om je heen .
+: alles beweegt .
+: alles draait .
+ zo komen wij ter wereld .
+de zon de maan de planeten en de sterren kijken toe en wij staan in het midden .
+onze plaats .
+maar Nicolaas Copernicus kwam en stelde dat de Zon in het midden staat en dat
+wij om haar heen draaien net als de andere planeten ..2. The Belgian Federal COVID-19 Sign language (BeCoS) corpus.
+Spoken Academic Belgian Dutch (SABeD)
+The first corpus is the Spoken Academic Belgian
+Dutch (SABeD) corpus,3 which is currently under development. The SABeD project is an interdis-
+ciplinary research project which develops a corpus of spoken academic Belgian Dutch consisting of
+at least 200 lectures.
+Lectures are typical of higher education.
+In lectures students learn new course content in a
+language register they are not familiar with, viz. academic Dutch. The SABeD project will
+• compile a corpus of spoken academic Belgian Dutch;
+• investigate the effectiveness of ASR for automatic transcription of spoken texts;
+• create a word frequency list of spoken academic Belgian Dutch and
+• develop a vocabulary test of spoken academic Belgian Dutch.
+The corpus is mainly based on video lectures of academic teaching to first-year bachelor students.
+These videos are, as a positive consequence of the COVID-19 pandemic, abundantly available. The
+lectures contain content in a language register students are not familiar with, viz. academic dutch.
+The corpus will serve as a source to develop representative study and test material on academic
+language. Furthermore, it can also serve as a source for linguistic research and a tool to optimize
+the language policy of higher education institutions. It will allow us to create study material and
+tests for international students and will be an important tool for researchers, language support and
+policy makers.
+An additional aim is the improvement of Belgian Dutch ASR through the creation of a spoken
+Dutch corpus with manually transcribed (or corrected) speech. Based on these manual transcriptions
+the ASR system of Van Dyck et al. (2021) will be retrained to improve fully automated transcriptions
+at a later stage so the corpus can be expanded efficiently. Once ready, the corpus will be made freely
+available for research through the Dutch Language Institute and the CLARIN Virtual Language
+Observatory.4
+In order to speed up the transcription process, the first step consists of applying ASR. As a
+second step, human transcribers edit and correct the ASR output. A tool that provides manual
+transcription functionality is ELAN (Wittenburg et al. 2006). ELAN is an annotation tool for audio
+and video recordings and supports the creation of multiple tiers of annotation. We integrate the ASR
+output into ELAN by converting the automatically recognized words and associated time stamps
+into an ELAN tier. The human editor can easily adapt the content of the tier, but the unit of
+annotation in the tier is the word, as this is the unit of output of the ASR system. This is not the
+most convenient unit for manual transcription, due to the fact that ASR errors often lead to errors
+over the word boundaries. Human editors would have to make changes over several units and adapt
+unit boundaries (to keep the alignment with the audio/video), which is quite a time-consuming task
+in ELAN. A much more convenient unit to work with is the sentence or utterance level. The model
+described in this paper provides a way to segment the ASR output stream into appropriate segments,
+usable for human annotation.
+Within the SABeD corpus we are not going to provide manually corrected transcripts for the
+entire videos, but limit these to the first 25 minutes and the last 5 minutes, to keep the data set
+balanced, irrespective of the length of the videos. The rest of the video corpus will be released with
+fully automatic transcripts (and fully automatic segmentation).
+The Belgian Federal COVID-19 Sign language (BeCoS) corpus
+The second corpus that
+motivates the described punctuation and segmentation approach is the BeCoS corpus, the Belgian
+federal COVID-19 Sign language video corpus. This corpus is extensively described in Vandeghinste
+3. https://www.arts.kuleuven.be/ling/language-education-society/projects/sabed
+4. https://vlo.clarin.eu
+
+et al. (2022), and is developed within the SignON project.5 SignON is a user-centric and community-
+driven project that aims to facilitate the exchange of information among Deaf, hard of hearing and
+hearing individuals across Europe, targeting the Irish, British, Dutch, Flemish and Spanish sign as
+well as the English, Irish, Dutch and Spanish spoken languages.
+One of the bottlenecks for developing machine translation (MT) systems between sign languages
+and spoken/written languages is the lack of parallel data. The BeCoS corpus addresses this issue.
+It consists of 220 press conferences of the Belgian Federal Government concerning the COVID-19
+pandemic, totalling 178 hours of speech. These press conferences were live interpreted into sign
+language: when speech was in Dutch, the sign language was Vlaamse GebarenTaal (VGT, Flemish
+Sign Language), when speech was in French, the sign language was Langue des Signes de Belgique
+Francophone (LSFB, Belgian Francophone Sign Language). The Dutch-VGT part of the data can
+serve as a parallel corpus for training a machine translation engine from VGT to Dutch or vice versa.
+The speech in this corpus is unscripted, and manual transcription is currently unattainable.
+As MT engines commonly are trained on parallel data at the sentence level, the corpus requires a
+sentence-like segmentation, which is attained by the models proposed in this paper.
+Segmenting transcriptions generated by ASR systems has an application in voice user interfaces
+as well.
+We developed the first FullStop model to segment multi sentence user statements into
+single statements. The goal was to process the sentences of users’ utterances individually. This is
+important as typical text classification models can only classify the users’ intention, e.g a command,
+reliably if the input text does not contain multiple intentions.
+***
+Section 2 presents related work. Section 3 describes the data sets we used and how they were
+processed, section 4 presents the models, and section 5 first describes an experimental evaluation as
+a classifier, then performs a qualitative discussion on some system output, and ends with showing
+results on full stop prediction on out of domain data, using the sliding window on continuous text
+streams. The final section, section 6 concludes the paper.
+2. Related Work
+P˘ai¸s and Tufi¸s (2022) provides an extensive survey of what they call punctuation restoration, making
+a distinction between methods that only use lexical features and methods that include audio specific
+features. As we work on the recognition output of the ASR, we do not consider the latter.
+Within the methods with lexical features, there are the early rule-based approaches and boot-
+strapping approaches that extract rules from large corpora, such as Petasis et al. (2001). There are
+also the n-gram approaches, such as Stolcke and Shriberg (1996) for sentence boundary detection.
+Conditional random fields are used by Lu and Ng (2010) amongst others. Character-level recurrent
+neural networks have been used by Susanto et al. (2016). Tilk and Alum¨ae (2016) approach the
+punctuation restoration problem as a bidirectional recurrent neural network with attention model.
+Guhr et al. (2021) lists other sources of punctuation/segmentation research. Attia et al. (2014)
+constitutes a rather traditional approach to spelling and punctuation correction for Arabic. Classifi-
+cation is carried out with Support Vector Machines and Conditional Random Field (CRF) classifiers,
+using part-of-speech and morphological information, and obtains an F1-score of 0.56, with the CRF
+classifier and a window size of five tokens.
+Che et al. (2016) experiments with different neural network architectures, using pretrained GloVe
+embeddings (Pennington et al. 2014) as inputs. It evaluates its models on ASR transcripts of TED
+5. https://www.signon-project.eu/
+
+talks, predicting commas, periods, and question marks. Its best result in this 4-class classification
+is an F1 -score of 0.54.
+Sunkara et al. (2020) works in the clinical domain on the output of medical ASR systems. It
+jointly models punctuation and truecasing by predicting a punctuation sequence and then the case of
+each input word. It uses a pretrained transformer model in combination with subword embeddings
+to overcome lexical sparsity in the medical domain. It carries out a fine-tuning step on medical
+data and a task adaptation step, randomly masking punctuation marks, before training the actual
+model. Predicting full stops and commas, it achieves F1 -scores of 0.81 (for commas) and 0.92 (for
+full stops) with Bio-BERT (Lee et al. 2019), which was trained on biomedical corpora.
+Previous work on multilingual punctuation prediction is described in Li and Lin (2020) and
+Guerreiro et al. (2021).
+Vandeghinste et al. (2018) models punctuation prediction in the context of speech translation, but
+also investigates a monolingual approach for Dutch, modelling punctuation prediction as a machine
+translation problem, in which the source language is the text without punctuation, and the target
+language is the text with punctuation. It shows that a neural MT approach that uses LSTM cells
+works much better than a language modelling approach using LSTMs, scoring on in domain data for
+the punctuation set P = {., ?! :; ()/−} an F1 of 0.82. We have used this approach, but now using a
+transformer model, as a baseline in the experiments in section 5.3. A statistical MT approach using
+Moses (Koehn et al. 2007) is shown to work at least equally well, with F1 = 0.83.
+In 2021 there was the shared task in Sentence End and Punctuation Prediction in NLG Text
+(SEPP-NLG) (Tuggener and Aghaebrahimian 2021),6 which consisted of two subtasks:
+1. Fully unpunctuated sentences - full stop detection: Given the textual content of an utterance
+where the full stops are fully removed, correctly detect the end of sentences by placing a full
+stop in appropriate positions.
+2. Fully unpunctuated sentences - full punctuation marks: Given the textual content of an utter-
+ance where all punctuation marks are fully removed, correctly predict all punctuation marks.
+Guhr et al. (2021) modelled this task as a token-wise prediction and examined several language
+models based on the transformer architecture. They trained two separate models for the two tasks
+and submitted their results for all four languages of the shared task, reaching state-of-the-art F-
+scores.
+They advocated transfer learning for solving the task and showed that the multilingual
+transformer models yielded better results than monolingual models. It is this approach that is taken
+in the current paper, and which is applied and evaluated on Dutch.
+For the SEPP-NLG task Guhr et al. (2021) also evaluated a CRF based model and found that
+this approach was outperformed by transformer based models. A GRU based model was submitted
+by (Masiello-Ruiz et al. 2021) for the shared task. This model scored 10 to 20 percent points lower
+F1 scores than the best transformer based models. For these reasons we did not consider RNN based
+models or more classical ML approaches for this work.
+3. Data
+To finetune our models we experimented with two different data sets: Europarl (Koehn 2005) and
+SoNaR (Oostdijk et al. 2013).
+Europarl data set (EP)
+contains transcribed plenary sessions of the European Parliament. For
+our models we used the Europarl v8 data, to be analogous with the other languages in the model.
+The text was extracted from the data downloads from OPUS (Tiedemann 2012).
+6. https://sites.google.com/view/sentence-segmentation
+
+...
+doos
+0
+0
+van
+0
+0
+pandora
+0
+0
+zouden
+0
+0
+openen
+0
+.
+hoe
+0
+0
+...
+op
+0
+0
+de
+0
+0
+volgende
+0
+0
+vraag
+0
+:
+kunnen
+0
+0
+...
+Table 1: Sepp format for text ... doos van pandora zouden openen. hoe ... op de volgende vraag:
+kunnen ...
+SoNaR data set
+contains texts from different genres and domains in standard Dutch that have
+been written after 1954. The data was obtained from the SoNaR website.7 We found that SoNaR
+contains a number of artefacts like HTML code that can lead to issues when processing this data
+set.
+For out of domain evaluation, as described in sections 5.2 and 5.3, we make use of the OpenSub-
+titles data8 for Dutch, as made available on OPUS (Lison and Tiedemann 2016). OpenSubtitles is
+a large database of movie and TV subtitles. We chose this data as it mainly contains translations
+of spoken language.
+Data Preprocessing
+Data was split at the sentence level and tokenized at the word level using
+the Moses corpus preprocessing tools included in the Moses3.0 distribution (Koehn et al. 2007).
+All data was truecased, as the ASR output is also truecased. The data was then converted into a
+tab-separated format, where the first column contains the word, the second column contains a 0 if
+the word is not followed by a full stop and a 1 if the word is followed by a full stop (to allow for
+binary classification as sentence segmentation), and the third column contains a 0 if the word is not
+followed by punctuation but contains the punctuation sign if it is followed by it. An example is given
+in Table 1. This format is consistent with the format used in the shared task on Sentence End and
+Punctuation Prediction in NLG Text (SEPP-NLG 2021) held at SwissText (Swiss Text Analytics
+Conference) in 2021.
+For both data sets we split the data into 75% training data and 25% test data. For the SoNaR
+data set we needed to downsample the training data to 1 GB, due to limited computing resources.
+4. The Models
+Transformers (Vaswani et al. 2017) and combining transformers with transfer learning (Devlin et al.
+2019) have led to performance gains for many different NLP tasks.
+The first model we present here is an extension for Dutch of the models of Guhr et al. (2021).
+We trained a token classification model, based on the Dutch language model RobBERT (Delobelle
+et al. 2020). We also trained a Dutch model based on BERTje (de Vries et al. 2019), but found
+that RobBERT slightly outperformed BERTje with a 0.75% better F1 score. For every token in
+7. http://hdl.handle.net/10032/tm-a2-h5
+8. http://www.opensubtitles.org/
+
+the input sequence, the model predicts a punctuation marker that follows the token. The model is
+trained to predict punctuation marks of the set P = {: −, ?.0}, with 0 indicating that the word is
+not followed by a marker. We finetuned several variants of this model on the two different datasets.
+These variants are shown in Table 2.
+Transformer models can only process sequences of a fixed length, typically 512 tokens. Therefore
+we implemented a sliding window approach to process the documents in our data set, which are
+typically longer than 512 tokens. The simplest method to achieve that is by splitting the text into
+chunks of 200 words before processing. The number of 200 words was chosen empirically to account
+for the fact that words get tokenized into more than one token (subword tokenization). With this
+method, it is important to leave some headroom since some words get decomposed into multiple
+tokens. This problem is more prominent in languages that allow compound words, such as Dutch.
+We choose to train a multilingual model for this work as well. A multilingual model can simplify
+the data processing in mixed language scenarios. In our previous work (Guhr et al. 2021) we found
+that multilingual models perform on par and in some cases, like Italian, significantly better than
+monolingual models. We wanted to investigate if this is the case for the Dutch language as well
+and trained a model on Dutch, English, German, French, and Italian texts. For the multilingual
+model, we used ”xlm-roberta-base” (Conneau et al. 2020). We previously evaluated a list of current
+language models and gained the best results using XLM Roberta in multilingual settings.
+We choose to use the same hyper-parameters that we evaluated in our previous work.
+This
+hyper-parameters search included the optimiser algorithm, learning rate, random initialisation seed.
+All models were trained for 3 epochs using Adafactor (Shazeer and Stern 2018) and a learning rate
+of 4e−5 with a batch size of 8 and 16 as the seed. Furthermore we used 16-bit-precision training to
+improve training and inference efficiency.
+As explained in section 3, for our experiments we used two different data sets: Europarl (Koehn
+2005) and SoNaR (Oostdijk et al. 2013).
+We trained one model for each data set, as well as
+a multilingual model on Dutch, English, German, French, and Italian sentences from the Europarl
+data set. For this model we used about 400 MB of data per language. Lastly we tested a combination
+of all available data from the SoNaR and Europarl data set. For this model the Dutch data set was
+downsampled to be on par with the other languages and contains 200 MB of Europarl and 200 MB
+of SoNaR data. Table 2 provides an overview of the trained models.
+Model Name
+Data Set
+Base Model
+Monolingual Europarl
+Nl EuroParl
+RobBERT
+Monolingual SoNaR
+Nl SoNaR
+RobBERT
+Multilingual EP
+Nl, En, Fr, De, It Europarl
+xlm-roberta-base
+Multilingual EP+SoNaR
+Nl, En, Fr, De, It Europarl + Nl SoNaR
+xlm-roberta-base
+Table 2: The list of data set and model combinations that we evaluated for this work.
+The models described here are made available on HuggingFace.9 We also provide a high level
+software library to simplify the usage of the models.10
+5. Experimental Results
+In section 5.1 we describe the evaluations of different variants of the model, evaluated as a multi-
+class classifier. Section5.2 shows some qualitative evaluation. Section 5.3 describes how the model
+performs on out-of-domain data.
+9. https://huggingface.co/oliverguhr/
+10. https://github.com/oliverguhr/deepmultilingualpunctuation
+
+5.1 Evaluation as a classifier
+Table 3 compares the per class F1 scores for each model on the test set of the corresponding data
+set it was trained on. Tables with the precision and recall values are presented in the Appendix.
+The overall macro and micro averaged F1 scores are in the same range for every model. There are
+more pronounced differences for certain classes. Question marks and full stops for models including
+SoNaR data have 10 to 13 percentage points lower scores than models trained on Europarl without
+SoNaR. We assume the reason for this is that the SoNaR data set contains more diverse data and
+more noise (e.g. HTML code) than Europarl and is, therefore, harder to learn for the model.
+Label/Model
+EP
+SoNaR
+Multilingual EP
+Multilingual EP + SoNaR
+0
+0.994
+0.986
+0.994
+0.987
+.
+0.961
+0.855
+0.959
+0.854
+,
+0.811
+0.721
+0.813
+0.723
+?
+0.849
+0.687
+0.817
+0.671
+-
+0.462
+0.723
+0.464
+0.613
+:
+0.655
+0.697
+0.657
+0.709
+macro F1
+0.789
+0.778
+0.784
+0.760
+micro F1
+0.983
+0.964
+0.983
+0.965
+Table 3:
+Per class F1 scores for the FullStop models on the Dutch test data sets. For detailed
+evaluation results of each model and per class precision and recall metrics please see the appendix.
+Table 3 also shows that the multilingual Europarl model improved the F1 scores in every class
+over its monolingual version. Part of this improvements can be explained by the fact that XLM-
+Roberta base uses more than twice as many parameters than RobBERT. Note that it is not possible
+to compare the performance of the other models directly, since models using the SoNaR data set in
+Table 3 where trained and tested on different data sets. Therefore we compared the models using
+an out of domain data set in section 5.3.
+Figure 2 shows a confusion matrix for the Dutch language for every trained model.
+Models
+trained on Europarl tend to confuse dashes with commas. SoNaR based models predict 14% to 18%
+of the colons and question marks as 0 or no punctuation mark. All models predict 10% to 25% of
+the colons and question marks with full stops. This is to be expected, as colons are more stylistic
+markers and there are no strict usage rules. Overall the Dutch language results are in line with
+English, French, German and Italian language predictions from our previous work.
+Furthermore we compared the performance of the different languages that both multilingual
+model where trained on, in Tables 4 and 5. We think these models are useful in scenarios where
+users mix languages or the source language is unknown, for example in social media posts. The
+evaluation results of the multilingual Europarl model (Table 4) are comparable between all five
+languages. We see an overall drop in Dutch language performance for the multilingual model using
+both Europarl and SoNaR data in Table 5. This is to be expected, as the SoNaR data set is more
+diverse. The results of the other four languages remain the same for both models.
+For efficiency reasons we choose to train XLM-Roberta base instead of large models. Comparing
+the results from Tables 4 and 5 with the finetuned multilingual model from our previous work, we
+estimate that a size ”large” model could improve the macro F1 by 5%. However, XLM-Roberta
+large models use more than twice as many parameters as base models, with 550 million parameters
+compared to 270 million.
+
+(a) Monolingual Europarl
+,
+-
+.
+0
+:
+?
+Predicted label
+,
+-
+.
+0
+:
+?
+True label
+0.8
+0.01
+0.018
+0.17
+0.0037 0.00098
+0.39
+0.34
+0.056
+0.2
+0.014
+0.0013
+0.028
+0.0011
+0.95
+0.012
+0.0054
+0.0027
+0.0049 0.00015 0.00031
+0.99
+7e-05
+3e-05
+0.088
+0.0076
+0.25
+0.06
+0.59
+0.0023
+0.037
+0.0016
+0.17
+0.039
+0.0024
+0.75
+0.2
+0.4
+0.6
+0.8
+(b) Monolingual SoNaR
+,
+-
+.
+0
+:
+?
+Predicted label
+,
+-
+.
+0
+:
+?
+True label
+0.69
+0.0026
+0.086
+0.21
+0.0082
+0.0053
+0.11
+0.63
+0.046
+0.2
+0.012
+0.0025
+0.042
+0.00039
+0.85
+0.088
+0.0079
+0.0094
+0.0056 0.00033 0.0036
+0.99
+0.00086 0.0004
+0.041
+0.0043
+0.14
+0.15
+0.66
+0.0072
+0.04
+0.00052
+0.16
+0.14
+0.011
+0.64
+0.2
+0.4
+0.6
+0.8
+(c) Multilingual EP
+,
+-
+.
+0
+:
+?
+Predicted label
+,
+-
+.
+0
+:
+?
+True label
+0.81
+0.0093
+0.016
+0.16
+0.0036 0.00093
+0.39
+0.36
+0.052
+0.18
+0.016
+0.0012
+0.025
+0.00081
+0.96
+0.01
+0.0049
+0.0021
+0.0048 0.00011 0.00026
+0.99
+7.1e-05 2.5e-05
+0.079
+0.0073
+0.25
+0.052
+0.61
+0.0027
+0.026
+0.001
+0.17
+0.032
+0.0019
+0.77
+0.2
+0.4
+0.6
+0.8
+(d) Multilingual EP+SoNaR
+,
+-
+.
+0
+:
+?
+Predicted label
+,
+-
+.
+0
+:
+?
+True label
+0.73
+0.0063
+0.048
+0.2
+0.0082
+0.0059
+0.19
+0.53
+0.038
+0.23
+0.015
+0.0019
+0.074
+0.00095
+0.81
+0.093
+0.0078
+0.011
+0.0055 0.00036 0.0025
+0.99
+0.00046 0.0004
+0.044
+0.0076
+0.11
+0.14
+0.69
+0.0074
+0.068
+0.0022
+0.12
+0.18
+0.015
+0.61
+0.2
+0.4
+0.6
+0.8
+Figure 2: Confusion matrices for Dutch language for the FullStop models. Note that all values are
+rounded.
+Label
+EN
+DE
+FR
+IT
+NL
+0
+0.990
+0.996
+0.991
+0.988
+0.994
+.
+0.924
+0.951
+0.921
+0.917
+0.959
+,
+0.798
+0.937
+0.811
+0.778
+0.813
+?
+0.825
+0.829
+0.800
+0.736
+0.817
+-
+0.345
+0.384
+0.353
+0.344
+0.464
+:
+0.535
+0.608
+0.578
+0.544
+0.657
+macro F1
+0.736
+0.784
+0.742
+0.718
+0.784
+micro F1
+0.975
+0.987
+0.977
+0.972
+0.983
+Table 4: Per class F1 scores of the multilingual Europarl model. Tested on English, German, French,
+Italian and Dutch language on the test data set.
+5.2 Qualitative Evaluation
+To better understand the capabilities and the limitations of the model, we qualitatively discuss some
+examples, presented in Table 6. The examples are selected from the OpenSubtitles corpus (Lison
+and Tiedemann 2016).
+
+Label
+EN
+DE
+FR
+IT
+NL
+0
+0.990
+0.996
+0.991
+0.988
+0.987
+.
+0.924
+0.950
+0.921
+0.917
+0.854
+,
+0.797
+0.937
+0.810
+0.778
+0.723
+?
+0.823
+0.826
+0.802
+0.731
+0.671
+-
+0.349
+0.380
+0.359
+0.348
+0.613
+:
+0.533
+0.606
+0.576
+0.541
+0.709
+macro F1
+0.736
+0.783
+0.743
+0.717
+0.760
+micro F1
+0.975
+0.987
+0.977
+0.972
+0.965
+Table 5:
+Per class F1 scores of the multilingual Europarl + SoNaR model. Tested on English,
+German, French, Italian and Dutch language data on the test set.
+The results that are shown are those of a single input of the words to the classifier, so there is
+no effect of θ in Table 6.
+We can see that the Gold strings of the examples contain different punctuation signs, belonging
+to the set P = {., ?}.They are tokenized at the word level and truecased. The Input strings simulate
+the stream of words as coming from an ASR system, without any punctuation. The Prediction
+strings show the output of the classification model, not in SEPP format, but in string format. In
+the examples we have marked the inserted punctuation in the Prediction in green where they are
+correct and in red when there is a mismatch (be it a deletion, substitution or insertion) between
+the Prediction and the Gold version. For ease of reference we have indexed the predicted or omitted
+punctuation with a subscript.
+The first example shows the stream of words that was also presented in Figure 1, but now
+somewhat longer. Prediction 1 is a comma where the gold standard has a full stop. This is counted
+as a substitution error, but it is not an implausible prediction and could be correct. The next 5
+predictions, containing comas and full stops, are correct.
+Then our model misses a full stop at
+prediction 7, which may be explained by the next sentence starting with the conjunction en. Then
+two more correct predictions and an omission of a comma (10), which can be attributed to the next
+word being en again. As Dutch has no Oxford comma rule, you would not necessarily expect a
+comma here. In (11) the system substitutes a full stop with a comma. A comma would be plausible
+at this position. (12) is a correct full stop prediction. In (13) and (14) question marks should have
+been predicted, but there is nothing in the word order that indicates that these are questions, so
+this information would have to come from the intonation, which is not available to our model. Then
+the system misses the final full stop (15). This may be due to the lack of context.
+The second example starts with an insertion (1) of a comma. This could be considered correct if
+we would put the apposition hier between commas, as is done often. The next comma, ending the
+apposition is correctly predicted. (3) and (4) omit commas at the start of a relative phrase, which
+is not uncommon. (5), (6) and (7) are correctly predicted full stops and commas. (8) is an omitted
+comma before en. (9) is a correctly predicted full stop before an en. (10) omits a comma before a
+subordinate clause. (11) correctly predicts the final full stop.
+From these samples we can see that the system makes several real mistakes, but that most of
+the differences between the Gold version and the Predicted version can be attributed to the fact
+that there are often are no strict punctuation rules, and that the output of the system could be
+argumented for. It would be interesting to see how humans would add punctuation purely based on
+the input text and what the inter-annotator agreement would be, but such an exercise is outside the
+scope of this paper.
+
+Example 1
+Gold
+kijk om je heen . alles beweegt , alles draait . zo komen wij ter wereld . de zon ,
+de maan , de planeten en de sterren kijken toe . en wij staan in het midden .
+onze plaats . maar Nicolaas Copernicus kwam en stelde dat de Zon in het midden
+staat , en dat wij om haar heen draaien . net als de andere planeten . een Aarde
+die beweegt ? maar daar zien en voelen we toch niets van ? dat was 1543 .
+Input
+kijk om je heen alles beweegt alles draait zo komen wij ter wereld de zon
+de maan de planeten en de sterren kijken toe en wij staan in het midden
+onze plaats maar Nicolaas Copernicus kwam en stelde dat de Zon in het midden
+staat en dat wij om haar heen draaien net als de andere planeten een Aarde
+die beweegt maar daar zien en voelen we toch niets van dat was 1543
+Prediction
+kijk om je heen ,1 alles beweegt ,2 alles draait .3 zo komen wij ter wereld .4 de zon ,5
+de maan ,6 de planeten en de sterren kijken toe
+7 en wij staan in het midden .8
+onze plaats .9 maar Nicolaas Copernicus kwam en stelde dat de Zon in het midden
+staat
+10 en dat wij om haar heen draaien ,11 net als de andere planeten .12 een Aarde
+die beweegt ,13 maar daar zien en voelen we toch niets van .14 dat was 1543
+15
+Example 2
+Gold
+en waar we nu zitten hier , dat is bij een fotografische kijker , die heel veel gelijkenis
+vertoont met de kijker , die werd gebruikt door David Gill . zo ’ n kijker moet dus in
+staat zijn om foto ’ s te nemen . maar als je foto ’ s neemt met die kijker , moet je
+natuurlijk ook een oogje houden op het stukje hemel waar hij op gericht is , en zorgen
+dat de kijker heel nauwkeurig de dagelijkse beweging van de hemel volgt . en daarom
+is zo ’ n kijker zo gebouwd , dat hij een gedeelte heeft waar de fotografische plaat zich
+bevindt .
+Input
+en waar we nu zitten hier dat is bij een fotografische kijker die heel veel gelijkenis
+vertoont met de kijker die werd gebruikt door David Gill zo ’ n kijker moet dus in
+staat zijn om foto ’ s te nemen maar als je foto ’ s neemt met die kijker moet je
+natuurlijk ook een oogje houden op het stukje hemel waar hij op gericht is en zorgen
+dat de kijker heel nauwkeurig de dagelijkse beweging van de hemel volgt en daarom
+is zo ’ n kijker zo gebouwd dat hij een gedeelte heeft waar de fotografische plaat zich
+bevindt
+Prediction
+en waar we nu zitten ,1 hier ,2 dat is bij een fotografische kijker
+3 die heel veel gelijkenis
+vertoont met de kijker
+4 die werd gebruikt door David Gill .5 zo ’ n kijker moet dus in
+staat zijn om foto ’ s te nemen .6 maar als je foto ’ s neemt met die kijker ,7 moet je
+natuurlijk ook een oogje houden op het stukje hemel waar hij op gericht is
+8 en zorgen
+dat de kijker heel nauwkeurig de dagelijkse beweging van de hemel volgt .9 en daarom
+is zo ’ n kijker zo gebouwd
+10 dat hij een gedeelte heeft waar de fotografische plaat zich
+bevindt .11
+Table 6: We generated these examples with the FullStop SoNaR model.
+5.3 Experiments on full stop prediction on out of domain data
+In this section we describe an evaluation on out of domain test data, i.e. test data coming from
+OpenSubtitles, as described in section 3.
+We test segmentation in two variants: with segementation set S = {.} and with segementation
+set S = {.?}.
+Baseline
+As a baseline we tested a machine translation approach, similar to Vandeghinste et al.
+(2018), in which we consider texts with all punctuation and segmentation removed as the source
+language and the punctuated version as the target language.
+We trained an OpenNMT (Klein et al. 2017) transformer model on a randomly resegmented
+version of the SoNaR corpus complemented with the Corpus Spoken Dutch (Oostdijk et al. 2002).
+
+Resegmenation was random, but distributed normally with an average of 14 tokens and standard
+deviation of 3 tokens. These values are based on the average length and stdev of sentences in De
+Standaard, according to Vandeghinste and Bult´e (2019). In order to create the training data for the
+MT system, we removed all punctuation in the source side, and kept the original punctuation in the
+target side.
+In our best model and parameter setting, such an approach reached a segmentation prediction
+F1 score of 42%, which seems not good enough for practical usage. The low F1 score is mostly due
+to low recall, as shown in table 7.
+Evaluation Procedure
+Evaluation was performed on 1000 sentences from OpenSubtitles, and
+applied with a sliding window of size 200. For every sequence of 200 words we checked where the
+system inserted an element of S. We accept a segmentation if the element of S is predicted in more
+than θ of all cases. We tried different θ values, but a θ = 0.1 setting seemed to provide the best
+results.
+Results
+Table 7 shows the results for the different models we trained. It is clear that the current
+models present a major improvement over the baseline MT apprach. The best F1-score for this
+evaluation seems to be the monolingual model trained on SoNaR only. This may be due to the
+fact that SoNaR contains different registers, amongst which some more colloquial forms of Dutch,
+that may be closer to the subtitles register than the data from Europarl. Note that the multilingual
+models also have good scores and an even higher precision than the monolingual models. There is
+also a clear improvement when using S = {.?} as set of segmenters over just using S = {.}.
+Model
+S
+Precision
+Recall
+F1-score
+Baseline ONMT
+{.}
+0.7187
+0.2986
+0.4219
+FullStop EP
+{.}
+0.8939
+0.7609
+0.8221
+FullStop SoNaR
+{.}
+0.8734
+0.8750
+0.8742
+FullStop Multilingual
+{.}
+0.9010
+0.7719
+0.8314
+FullStop Multilingual EP+SoNaR
+{.}
+0.9021
+0.7915
+0.8424
+FullStop EP
+{.?}
+0.8962
+0.8193
+0.8561
+FullStop SoNaR
+{.?}
+0.8749
+0.9380
+0.9053
+FullStop Multilingual
+{.?}
+0.9013
+0.8330
+0.8658
+FullStop Multilingal EP+Sonar
+{.?}
+0.9040
+0.8504
+0.8764
+Table 7: Evaluation results on out of domain data
+At this point, we are interested in the significance levels: do the models differ significantly or
+not, and what is the 95% confidence interval?
+Multiple testfiles
+In order to determine whether the F-scores between the S = {.?} condition
+and the S = {.} condition, or the scores for different θ values differ significantly, we have tested the
+Condition
+Characteristics
+95% Conf.
+Model
+θ
+S
+Median
+Average
+Std dev
+Lo
+Hi
+A
+SoNaR
+0.1
+{.}
+0.7975
+0.7785
+0.0818
+0.5918
+0.8567
+B
+SoNaR
+0.2
+{.}
+0.7933
+0.7746
+0.0820
+0.5882
+0.8540
+C
+SoNaR
+0.3
+{.}
+0.7892
+0.7705
+0.0821
+0.5841
+0.8505
+D
+SoNaR
+0.1
+{.?}
+0.8812
+0.8611
+0.0839
+0.6802
+0.9308
+Table 8: Characteristics of the distribution of F1 scores over 10000 test files
+
+Figure 3: Distribution of F1 scores over 10000 test files. The blue curve shows the distribution
+of using {.} as a segmentation marker. The red curve shows the distribution when using {.?} as
+segmentation markers.
+SoNaR model, which was the best scoring model in Table 7 on 10000 test sets of each 1000 sentences.
+These test sets were created by splitting up the OpenSubtitle file into sections of 1000 lines each.
+Per condition, we have evaluated these 10000 test sets, ranked the F1 scores and taken the score
+at rank 251 and at rank 9750 as the values of the 95% confidence interval.
+Table 8 lists the main characteristics of the F1 scores for the different conditions that were tested
+on the 10000 evaluation files.
+The difference between Condition A and Condition D has a p-value of 0.0981, so it is significant
+at the p < .10 level.
+Difference between Condition B and D is not significant (p = 0.100801).
+Difference between C and D has a p = 0.091266.
+The effect of the θ parameter is not significant, but the effect of adding the question mark to S
+is mildly significant at the p < .10 level.
+Figure 3 presents a visualisation of the distribution of the F1 scores for the two different S
+conditions, which shows clearly that S = {.?} scores better than only S = {.}.
+6. Conclusions
+We have presented several models that perform punctuation prediction and evaluated them in dif-
+ferent settings. We have made various models specifically for Dutch, but have also extended the
+multilingual model from Guhr et al. (2021) with Dutch. The models use transfer learning from large
+pretrained models and are finetuned as per token classifiers.
+The models are evaluated as classifiers, reaching a similar accuracy as for the other languages,
+and they have also been tested on out-of-domain data, on which they present a great improvement
+over a baseline MT model.
+The best models are publicly available through Huggingface.11. The use of the model to actually
+predict segmentation on a stream of text, as used in the out-of-domain evaluation is made available
+on Github.12
+For future work, it would make sense to predict different sets of tokens. As for now, we have
+taken the set of tokens as defined in the shared task. Training a classifier just for the prediction
+of segmentation, punctuation signs in set S = {.?!} or for the prediction of all non-alphanumeric
+characters would be possible. The model could also be extended to predict the segmentation tokens
+11. https://huggingface.co/oliverguhr
+12. https://github.com/VincentCCL/Segment_FullStop
+
+Distribution of F-scores
+25.00%
+F0.1
+ F0.1?
+20.00%
+15.00%
+10.00%
+5.00%
+0.00%
+0.60
+0.70
+0.80
+0.90
+1.00and the true case of every word in the sequence. This is a common use case in processing the output
+of automatic speech recognition systems.
+Another line of work would be to make a lighter version of the model, with fewer parameters,
+through knowledge distillation which is quicker in inference.
+All in all, we can conclude that the model we present provides a usable and practical way of
+inserting punctuation into streams of words, and therefore turning a sequence of words into a text.
+The SoNaR model came out as the best model and will therefore serve this purpose in further
+processing of the SABeD corpus and has been used in processing of the BeCoS corpus.
+7. Acknowledgements
+Work in this paper is partly financed by the SignON project.13 This project has received funding from
+the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement
+No. 101017255. The SABeD project is funded by KU Leuven Internal Funding, Research Project
+3H200610.
+Oliver Guhr has been funded by the European Social Fund (ESF), SAB grant number 100339497
+and the European Regional Development Funds (ERDF) (ERDF-100346119).
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+conf.org/proceedings/lrec2006/pdf/ pdf.pdf.
+Appendix
+In this appendix we present more detailed classification evaluation results, including precision and
+recall.
+class
+precision
+recall
+F1-score
+samples
+0
+0.992584
+0.994595
+0.993588
+9627605
+.
+0.960450
+0.962452
+0.961450
+433554
+,
+0.816974
+0.804882
+0.810883
+379759
+?
+0.871368
+0.826812
+0.848506
+13494
+-
+0.619905
+0.367690
+0.461591
+27341
+:
+0.718636
+0.602076
+0.655212
+18305
+accuracy
+0.983874
+10500058
+macro avg
+0.829986
+0.759751
+0.788538
+10500058
+weighted avg
+0.983302
+0.983874
+0.983492
+10500058
+Table 9: Monolingual Europarl model tested on Nl EuroParl data.
+class
+precision
+recall
+F1-score
+samples
+0
+0.982554
+0.989277
+0.985904
+73926815
+.
+0.858432
+0.852403
+0.855407
+4941897
+,
+0.754981
+0.689276
+0.720634
+3127454
+?
+0.732037
+0.646400
+0.686558
+410416
+-
+0.849020
+0.629105
+0.722703
+331849
+:
+0.740604
+0.659131
+0.697497
+590946
+accuracy
+0.964436
+83329377
+macro avg
+0.819604
+0.744266
+0.778117
+83329377
+weighted avg
+0.963170
+0.964436
+0.963641
+83329377
+Table 10: Monolingual SoNaR model tested on Nl SoNaR.
+
+class
+precision
+recall
+F1-score
+samples
+0
+0.992625
+0.994700
+0.993662
+9627605
+.
+0.960790
+0.956852
+0.958817
+433554
+,
+0.815222
+0.810991
+0.813101
+379759
+?
+0.867011
+0.772047
+0.816778
+13494
+-
+0.657312
+0.358070
+0.463597
+27341
+:
+0.708049
+0.613166
+0.657201
+18305
+accuracy
+0.983884
+10500058
+macro avg
+0.833501
+0.750971
+0.783859
+10500058
+weighted avg
+0.983364
+0.983884
+0.983499
+10500058
+Table 11: Multilingual EP model tested on Nl Europarl data
+class
+precision
+recall
+F1-score
+samples
+0
+0.983286
+0.990781
+0.987020
+8982463
+.
+0.900062
+0.812584
+0.854089
+588253
+,
+0.713272
+0.732957
+0.722980
+356718
+?
+0.739526
+0.614814
+0.671428
+59631
+-
+0.727932
+0.529030
+0.612744
+32828
+:
+0.725112
+0.694275
+0.709358
+49708
+accuracy
+0.966042
+10069601
+macro avg
+0.798198
+0.729074
+0.759603
+10069601
+weighted avg
+0.965309
+0.966042
+0.965441
+10069601
+Table 12: Multilingual EP+SoNaR tested on Nl, Europarl + Nl SoNaR data.
+

09E1T4oBgHgl3EQf5AUq/content/tmp_files/2301.03506v1.pdf.txt

@@ -0,0 +1,738 @@
+This article has been published in Nature Physics: Vila-Costa, A., Gonzalez-Silveira, M., 
+Rodríguez-Tinoco, C. et al. Emergence of equilibrated liquid regions within the glass. Nat. Phys. 
+(2022). https://doi.org/10.1038/s41567-022-01791-w 
+ 
+Experimental evidence of a crossover between cooperative relaxation and liquid growth 
+dynamics  
+Ana Vila-Costa, Marta Gonzalez-Silveira$, Cristian Rodríguez-Tinoco, Marta Rodríguez-López, 
+Javier Rodríguez-Viejo$  
+Departament de Física. Facultat de Ciències, Universitat Autònoma de Barcelona, 08193, 
+Bellaterra, Spain 
+Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, 
+Bellaterra, 08193, Barcelona, Spain 
+Abstract: 
+In stark contrast with the conventional understanding of the glass transition, where the 
+transition from glass to liquid appears as a dynamic process where atoms/molecules 
+cooperatively relax into the equilibrium phase, we experimentally show that the nature of the 
+glass transition depends at a given temperature on the ratio between the relaxation time of the 
+glass, 𝜏𝑔𝑙𝑎𝑠𝑠, taken as its transformation time, and the alpha relaxation time, 𝜏𝛼. Although the 
+relaxation of liquid-cooled glasses is not totally synchronous, due to the existence of a 
+distribution of relaxation times, there has been no clear observation of phase separation. 
+However, at temperatures at which 𝜏𝑔𝑙𝑎𝑠𝑠/𝜏𝛼 is large, high mobility regions nucleate into the 
+liquid phase that subsequently grow by dynamic facilitation before – or while - cooperative glass 
+relaxation sets into play. On the contrary, at temperatures associated to smaller 𝜏𝑔𝑙𝑎𝑠𝑠/𝜏𝛼 the 
+glass transition proceeds by cooperative relaxation dynamics all-across the material. This 
+behavior is independent of the experimental procedure or protocol to produce the glass. 
+ 
+$Corresponding authors: [email protected], [email protected] 
+ 
+ 
+ 
+
+ 
+ 
+ 
+ 
+Understanding the physics of glass formation upon cooling a liquid and its converse effect, the 
+transition to a supercooled liquid upon heating the glass, is still a challenge despite the intense 
+experimental and theoretical research of the last 100 years 1. The main difficulty stems from the 
+apparent disagreement between the insignificant changes of structure and the concomitant 
+huge variations of the dynamics across the glass transition. Whether this transition is at its core 
+a thermodynamic phenomenon with a hidden phase transition below the experimentally 
+observed at the glass transition temperature, Tg, or it is a dynamic one related to the kinetic 
+arrest of atoms/molecules at Tg is still an unsolved question2. A methodical experimental study 
+of the glass transition is also challenging, due to the limited time scale range we can 
+experimentally access.  
+Glasses can be seen as a mosaic of regions each one characterized by a different relaxation time, 
+resulting from the original distribution of relaxation times in the liquid state3. Since glasses are 
+out-of-equilibrium systems, they will evolve towards equilibrium, and therefore towards a new 
+distribution of relaxation times, when the glass is annealed at a certain temperature.  Whether 
+the relaxation times will become slower or faster will be determined by the annealing 
+temperature being, respectively, below (ageing) or above (rejuvenation4,5, anti-aging6 or 
+devitrification) the limiting fictive temperature of the glass, Tf 7. This temperature is directly 
+related to the thermodynamic stability of the glass and can be calculated as the temperature at 
+which the state variable (enthalpy or density) has the same value for the glass and for the 
+extrapolated equilibrium liquid 8,9. While it has been recognized that glasses do not show a 
+unique, macroscopic Tf, but rather a microscopic dispersion of a mosaic of fictive temperatures4, 
+it is a useful parameter to globally define the stability of a glass. Each of these mosaic regions 
+conforming the glass will relax according to its mobility and the external temperature, 
+producing, globally, a stretching relaxation signature with an exponent beta related to the 
+distribution of relaxation times 3. Mean field models like the Tool-Narayanaswamy-Moynihan 
+(TNM) are based on this approximation to simulate the evolution of glasses under a specific 
+thermal history 10–12. However, this model is too simple since these regions are not isolated. One 
+also has to consider mobility transport, as predicted by the kinetic facilitation models, that 
+pushes/induces the relaxation in adjacent zones13,14. Considering both the relaxation of the 
+
+different regions and the mobility transport, the glass is expected to relax progressively towards 
+the equilibrium liquid, in a cooperative and practically homogeneous way. This type of relaxation 
+is the one expected in liquid-cooled glasses and models based on this approach fit well enough 
+the available experimental data 15. However, recent works based on theoretical developments 
+and simulations introduce one more mechanism for the transition from glass to liquid at 
+temperatures above the limiting fictive temperature of the glass. Wolynes et al., based on the 
+combination of Random First Order Theory (RFOT)16,17 and Mode Coupling Theory (MCT)18, 
+expose that we must consider the formation of entropy drops in the glass, small mobile glassy 
+regions that by statistical fluctuations relax fully into the equilibrated liquid and once relaxed, 
+propagate the relaxation into the less mobile adjacent regions via a kinetic facilitation process4.  
+This relaxation would spread as a flame, accelerating the transition of the glass into the liquid. 
+The spreading of this equilibrated liquid into the adjacent regions would be faster than the time 
+required for each of the individual regions to relax, dominating the transition of the whole glass 
+into the liquid. Other authors have explored this alternative view of the glass transition by 
+performing different type of simulations. In this way, Douglass and Harrowell, use the facilitated 
+kinetic Ising model to study the relaxation of the glass into the supercooled liquid and the 
+existence of high mobility regions is introduced as inherent dynamic heterogeneities 19. 
+Gutiérrez and Garrahan use local excitations to initiate the transition in the kinetically 
+constrained model when simulating ultrastable glasses 20. Lulli et al. introduce equilibrated 
+higher temperature regions in a model based on a distinguishable-particle lattice, where the 
+authors follow the spatial profiles of particle displacement and their interactions to study the 
+relaxation of the glass 21.  Jack and Berthier use a triangular plaquette model based on spin 
+variables considering simple interactions to reproduce the relaxation of glasses equilibrated at 
+different temperatures (i.e. of different stability)22. They find that the transition in stable glasses 
+takes place via the nucleation and growth of equilibrated liquid drops, the same transformation 
+mechanism proposed by Wolynes et al4. Moreover, using the Avrami formalism23 they 
+successfully reproduce the dynamics of the transition. On the contrary, they find that glasses 
+equilibrated at higher temperatures, relax via a relaxation with a broad range of relaxation 
+times, as expected for this type of glasses. Lastly, Fullerton and Berthier 24 used the Swap Monte 
+Carlo approach to generate in-silico glasses of ultra-high stability. The transition of this type of 
+glasses into the liquid state takes place via the formation of liquid patches that grow until 
+consuming the static glass matrix, following again an Avrami-like kinetics. In all these models, 
+the requirement to observe these differentiated liquid regions is generally a big contrast in 
+mobility between the glass and the equilibrium liquid at that temperature. This can be an 
+explanation as why these liquid drops have not been observed experimentally in liquid-cooled 
+
+glasses up to now, although according to Guiselin et al. 25, even close to Tg, fast equilibrated 
+regions would form at the tail of the distribution of relaxation times, as shown in a recent 
+simulation study where equilibrated configurations of a supercooled liquid with 𝜏𝛼 ≈ 100 𝑠 
+where produced by a Swap Monte Carlo algorithm.  
+This theoretical framework, with a clear phase separation between the liquid and the remaining 
+untransformed glass, has been experimentally observed in thin films of vapor-deposited 
+ultrastable glasses, where the free surface exhibits larger mobility, and, hence, acts as a seed 
+plane for a liquid front propagation following the kinetic facilitation concept 26–28. The same has 
+been possible for the bulk glass transition in ultrastable glasses either by measuring very thick 
+films (several micrometers 29) or by arresting the mobility of the surface 30 and avoiding in this 
+way the formation of a propagating front 5,31. Ultrastable glasses are prepared by means of 
+Physical Vapor Deposition (PVD), which allows the obtention of glasses with outstanding kinetic 
+and thermodynamic stability (i.e. higher transformation temperatures and very low fictive 
+temperatures, respectively) 32–35, only attainable after hundreds or thousands of years of ageing 
+of a conventional glass 27, defined as a glass obtained by cooling the liquid at q=-10 K/min. The 
+requirements to prepare amorphous solids  with such unique properties are a sufficiently slow 
+deposition rate and a deposition temperature high enough to allow the exploration of low 
+energy equilibrium states 27,36,37. Experimentally, it has been found that this temperature is 
+around 0.85Tg for most organic molecules38 for deposition rates around 0.1-0.2 nm/s, being Tg 
+the glass transition temperature of the conventional glass. For these ultrastable glasses, the ratio 
+between the average relaxation time of the glass and that of an equilibrated liquid drop at 
+temperatures close to Tg, (𝜏𝑔𝑙𝑎𝑠𝑠/𝜏𝛼), is expected to be very large, several orders of magnitude. 
+It has been previously shown that 𝜏𝑔𝑙𝑎𝑠𝑠(T) can be understood as the time required to 
+completely transform the glass into liquid at a given temperature 39,40 . We note that at Tf, 
+𝜏𝑔𝑙𝑎𝑠𝑠
+𝜏𝛼
+≈ 1. When an ultrastable glass is heated up to temperatures far above its fictive 
+temperature, the emergence and growth of these liquid drops is apparently much faster than 
+the relaxation of the glass and, therefore would dominate the transition of the ultrastable glass 
+into the liquid. This is indeed what has been observed by Rodríguez-Viejo and coworkers, where 
+ultrastable vapor deposited TPD (N,N′-Bis(3-methylphenyl)-N,N′-diphenylbenzidine) thin film 
+glasses capped with TCTA (Tris(4-carbazoyl-9-ylphenyl)amine) were shown to transform into the 
+liquid by the nucleation and growth of equilibrated liquid regions when submitted to isothermal 
+treatments above the conventional glass transition temperature5.  
+
+However, according to models and simulations, a glass with low stability, i.e one cooled from 
+the liquid at standard rates, could also experience the bulk transition into the equilibrated 
+supercooled liquid by forming localized liquid droplets at spots with short relaxation times 
+provided there is sufficient mobility contrast with the adjacent regions 4,21. Although there is not 
+yet experimental evidence of this behavior, the measurement strategy would be to shift the 
+transition to the supercooled liquid to temperatures much higher than its limiting fictive 
+temperature, so the contrast in mobility between the liquid drops and the adjacent glass could 
+be high enough for the formation and propagation of these liquid regions to be faster than the 
+intrinsic relaxation of the surrounding glass. This assumption was already verified for surface 
+front transformation, where experiments by Sadtchenko et al.41 and Rodriguez-Tinoco et al.26 
+showed that, at sufficient high heating rates and, hence, by sufficiently shifting the 
+devitrification temperature of the glass, a liquid-cooled glass would also transform via front 
+propagation.  
+In this article, we demonstrate experimentally that the bulk glass transition in liquid-cooled 
+glasses can also take place via the formation of localized liquid regions instead of a cooperative 
+relaxation under specific experimental conditions. We also observe a transition between these 
+two mechanisms depending on the transformation temperature and the initial stability of the 
+glass. We study the glass transition in both vapor-deposited and liquid-cooled glasses of 
+different stabilities by carrying isothermal treatments above Tf. We show how all studied glasses 
+can rejuvenate via the formation and growth of liquid regions given enough contrast in mobility 
+between the glass and the equilibrated liquid, which in our case, translates into performing the 
+annealing at high temperature above the crossover between 𝜏𝑔𝑙𝑎𝑠𝑠 and 𝜏𝑐𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟 = (180 ±
+ 60)𝜏𝛼. When the transformation takes place below this crossover it develops through gradual 
+softening, that is via a progressive relaxation of the whole glass due to the small contrast in 
+mobility between nearby regions.  
+Results and discussion 
+We prepare TPD glasses (Tg=333 K on heating/cooling at 10 K/min) from the liquid state and 
+from the vapor-phase to test the presence of phase separation during the transition to the super 
+cooled liquid (SCL) depending on the stability of the glass and on the isothermal treatment above 
+Tf. The glasses we analyze are: i) vapor deposited glasses grown at two different deposition 
+temperatures: Tdep=285 K (0.85Tg), for the ultrastable glass (Tf=292 K) and Tdep=330 K (0.99Tg), a 
+glass that grows in equilibrium with the supercooled liquid and has Tf=330 K. We cap both sides 
+of the TPD glasses with TCTA (Tg= 428 K) to inhibit the formation of a liquid front at 
+
+surfaces/interfaces 30,31; ii) liquid-cooled glass: we deposit in the liquid state, 5 K above Tg 
+(Tdep=338K), we then cool it down to room temperature and then we age the resulting glass at 
+Tann=319 K (Tg-14 K) for 96 h until it is completely equilibrated at that temperature (Tf=319K). We 
+cap it afterwards with TCTA to avoid the formation of the liquid front on the surface during the 
+final upscan (see figure 1). Since the sample is directly deposited in the liquid state well above 
+Tg, this approach allows us to generalize the observation of phase separation during the glass to 
+liquid transition to both vapor-deposited and liquid-cooled glasses. The samples are directly 
+deposited onto the membrane of a nanocalorimeter which allows us to apply the thermal 
+protocol shown in figure 1 right after growth without breaking vacuum. After deposition the 
+glasses are taken to a certain temperature above the Tf of that specific glass and remain there 
+for different times. During this annealing treatment, the glass partially rejuvenates. We then 
+cool down the sample at approximately -500 K/s and perform a subsequent fast heating 
+(β≈3.5x104 K/s) scan with the nanocalorimeter.  The heat capacity traces of the final fast upscan 
+are therefore representative of the thermal history of the glass after the isothermal treatment. 
+The fast cooling/fast heating attained with our custom-made nanocalorimetric chips is a key 
+point to resolve the heat capacity overshoots of glassy zones with different stabilities. That is, if 
+some liquid regions are formed during the previous isotherm, they will become glass regions of 
+very low stability because of the fast cooling rate employed5. In this case, two glass transition 
+signatures will be observed during heating, one at lower temperature for the very low stability 
+glass and one at higher temperature, corresponding to the untransformed (or partially relaxed) 
+glass during the isotherm.  
+ 
+Figure 1. a) Thermal treatment performed on the different samples. After sample preparation, 
+samples are annealed at Tann for different times. After cooling them down at ~-500 K/s, heat 
+capacity data is recorded at a heating rate of 3.5x104 K/s. Different samples under study: b) 
+ultrastable vapor deposited glass grown at Tdep=285 K (0.85Tg); c) low stability vapor deposited 
+
+CH3
+HaC
+Temperature
+a)
+TCTA
+TPD
+heating
+scan
+Annealing
+b)
+Ultrastableglass
+C
+Lowstableglass
+3.5x104 K/s
+Tdep=285K
+Tdep=330K
+Tg= 333 K
+sample
+prepa-
+fast
+ration
+cooling
+d)
+~500K/s
+Cooled from the liguid +
+Agedglass
+aged 96h at 319 K
+Timeglass grown at Tdep=330 K (0.99Tg); d) glass obtained after depositing 5 K above Tg, cooled down 
+to 319 K and aged there until complete stabilization (96h).  
+ 
+Figure 2 shows the specific heat curves for a TPD glass deposited at Tdep=330 K (Tf=330 K) after 
+isothermal treatments at 341K (Tg+8K) and 347 K (Tg+14K) for different times. The as-deposited 
+glass, grown in equilibrium with the supercooled liquid at Tdep, is comparable to a glass obtained 
+by cooling the liquid at a cooling rate around 1 K/min. The transformation of this glass shows 
+remarkable differences depending on the annealing temperature. Annealing at Tann=341 K 
+results in a shift of the glass transition peak towards lower temperatures as time increases, 
+indicating a progressive relaxation of the glass towards the equilibrium liquid at that specific 
+annealing temperature. On the contrary, at Tann=347 K we see both the relaxation of the as-
+deposited glass, by the shift of the original peak to lower temperatures, and the formation of 
+distinct liquid regions in the glass that are manifested through the apparition of a second peak 
+at lower temperature5 identified by a green arrow in Figure 2a and a cartoon representing the 
+formation and growth of the liquid drops. We would like to emphasize the relevance of what is 
+shown in figure 2a. The same glass can experience different transformation mechanisms during 
+the glass transition depending on the temperature at which the transition takes place. 
+Remarkably, even that both temperatures differ just by 6 K, the contrast in mobility between 
+liquid and glass is apparently different enough at these two temperatures to show distinct 
+outputs when the transformation is partially fulfilled.  
+ 
+ 
+Figure 2. a) Specific heat traces obtained at a scanning rate of 3x104 K/s after different annealing 
+treatments of samples deposited at 0.99Tg. The annealing temperatures and times are indicated 
+in the legend. The endothermic peak that appears at lower temperatures is an indication of the 
+formation of liquid patches in the glass during the annealing treatment. b) curves corresponding 
+to the alpha relaxation time of the liquid (blue) 42, the relaxation time of the glass (green), which 
+
+10
+4,0-
+a)
+Tdep=330K
+Tdep=330K
+b)
+Tann=341K
+Tann=347K
+Specific Heat (J/g/K)
+3,5
+ad
+106
+0.5s
+ad
+1s
+3,0
+3s
+10s
+5s
+2,5-
+30s
+10
+2
+2,0
+6
+1,5
+10
+1,0-
+0
+320
+340
+360
+380
+400
+420
+370380390400410
+440
+370380390400410
+Temperature (K)
+Temperature(K)
+Temperature (K)depends on the limiting fictive temperature of the glass (Tf=330 K) and has been calculated 
+according to Rodriguez-Tinoco et al39 (green), the line corresponding to the crossover relaxation 
+time calculated as 𝜏𝑔𝑙𝑎𝑠𝑠 = (180 ±  60)𝜏𝛼 (grey). In orange, the temperature regions for which 
+a cooperative relaxation mechanism is expected for the glass transition. In green, the 
+temperature region where we expect to find the nucleation and growth of liquid regions during 
+the glass transition. The cartoons represent the two transformation mechanisms.  
+ 
+ 
+We propose the appearance of equilibrated regions can be rationalized from the ratio between 
+the relaxation time of the glass and the alpha relaxation time of the liquid (𝜏𝑔𝑙𝑎𝑠𝑠/𝜏𝛼) at the 
+temperature of the isothermal treatment. If this ratio is large, the nucleation and growth of 
+liquid drops dominates the transition5 since on average the relaxation of the glass is slower than 
+the rate at which the liquid emerges and consumes the glass. On the contrary, small ratios are 
+an indication that a cooperative heterogenous dynamics across the sample is fast enough to be 
+the main active mechanism during the transition. To further analyze this assumption and provide 
+a numeric estimation, we represent in figure 2b the experimental transformation times of the 
+glass deposited at Tdep=0.99Tg as a function of temperature (green circles). A detailed 
+explanation on how to calculate these times can be found in the methods section. In the same 
+graph, we have also plotted the alpha relaxation time (blue line) and the 
+relaxation/transformation time of the glass (green line), calculated as explained elsewhere39 and 
+briefly addressed at the methods section. In this particular case we see a change of mechanism 
+in a very small range of temperatures, so we assume that the crossover temperature between 
+the two regimes, gradual softening (orange colored) and formation of distinctive liquid regions 
+(green colored), should be at a temperature between these two. For the annealing at 341 K, 
+𝜏𝑔𝑙𝑎𝑠𝑠/𝜏𝛼 ≈ 60, while at Tann=347 K, 𝜏𝑔𝑙𝑎𝑠𝑠/𝜏𝛼 ≈ 300. As a rough estimation, we take the 
+average between these two values as the crossover ratio. Figure 2b shows as a grey line this 
+crossover relaxation time, 𝜏𝑔𝑙𝑎𝑠𝑠,𝑐𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟 = (180 ±  60)𝜏𝛼. We use this value as a predictor to 
+identify the temperature required to observe the formation of isolated liquid regions during the 
+glass transition for a specific glass irrespective of the initial state of the glass. The transformation 
+time at 341 K falls to the left of the crossover line, while the one at 347 K will be at the right side, 
+showing at each temperature different transformation mechanisms.  
+The temperature of the crossover is somewhat ill-defined because of the difficulty to establish 
+in this region a clear difference between both mechanisms in the heat capacity traces and 
+therefore we represent it by a broad white-graded area (see figure 2b). The cartoons (together 
+
+with background color) in figure 2b clearly illustrate the impact of the ratio 
+𝜏𝑔𝑙𝑎𝑠𝑠
+𝜏𝛼   on the 
+transformation, and schematically represent the transformation at both sides, showing regions 
+of liquids drops at the right (darker blue) within a glassy matrix (softer blue), while in the left 
+region the transformation is spatially less resolved.  
+ 
+ 
+ 
+Figure 3. a) Specific heat traces after different annealing treatments of samples deposited at 
+1.05Tg, fast cooled and aged at 319K for 4 days. The annealing temperatures and times are 
+indicated in the legend. Left pannel: The overlap between the curve corresponding to a sample 
+deposited at 328K and the one annealed for 5h at 328 K indicates that after this time, the glass 
+has fully transformed and reached equilibrium. b) curves corresponding to the alpha relaxation 
+time of the liquid (blue) 42, the relaxation time of the glass (purple), and the line corresponding 
+to the crossover relaxation time calculated as 𝜏𝑔𝑙𝑎𝑠𝑠,𝑐𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟 = (180 ±  60)𝜏𝛼 (grey). Violet 
+circles correspond to experimental data. 
+ 
+A similar behavior has been observed in a glass cooled from the liquid. Figure 3a shows the 
+specific heat traces of a glass deposited 5 K above Tg (338 K), cooled down and aged at 319 K for 
+96 h. This treatment led to full equilibration of the glass at this temperature, i.e. Tf=319 K, before 
+performing the isothermal treatment at 347 K (Tg+14 K) and at Tg-5 K (328 K). At both 
+temperatures the glass is expected to evolve towards the supercooled liquid, which means 
+towards faster relaxation times 7. As can be seen in the calorimetric traces, during the annealing 
+at 347 K, distinguishable liquid regions are formed as part of the transition of the glass into the 
+supercooled liquid. On the contrary, when the annealing is performed at 328 K, there is a 
+continuous shift of the glass transition overshoot, with no distinguishable fast-cooled glassy 
+regions, and the transition takes place exclusively via a cooperative relaxation process (gradual 
+softening). In figure 3b we represent the relaxation times of the liquid and the glass (for a glass 
+
+Tdep=338K&aged96hat319K
+10
+Tdep=338K&aged96hat319K
+laxation/transformationtime
+Tann=328K
+Tann=347K
+8
+Specific heat (J/K/g)
+as-aged
+as-aged
+45min
+18s
+2h
+30s
+6
+5h
+1min30s
+Tdep=328K
+6
+a)
+b)
+Rel
+320
+340
+360380400420440
+360
+380
+400
+420360
+380
+400
+420
+Temperature (K)
+Temperature (K)
+Temperature (K)with Tf=319K), the transformation times and the crossover relaxation time as in figure 2b. As can 
+be seen in the figure, the transformation times at 328 K (cooperative heterogenous dynamics 
+on the whole sample) and 347 K (localized heterogenous dynamics + facilitation) fall respectively 
+at left and right of the crossover line (𝜏𝑔𝑙𝑎𝑠𝑠,��𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟 = (180 ±  60)𝜏𝛼), consistent with the 
+transition mechanism observed during the annealing treatments (figure 3a) and providing 
+solidity to the assumption that this crossover curve is independent of the characteristics of the 
+glass. According to our estimation, for this particular glass, since its stability is high (Tf=319 K), 
+the annealing temperatures required to exclusively see the cooperative relaxation would be 
+below 338 K. In fact, the higher the stability of the glass, the lower the crossover temperature. 
+This is even clearer in the case of the ultrastable glass. Figure 4a shows the calorimetric traces 
+for a sample vapor-deposited at Tdep=285 K (0.85 Tg), which is the temperature at which the 
+maximum kinetic and thermodynamic stability is attained 31,43, after annealing at 347 K (Tg+14 
+K) for different times. At this temperature the ratio 𝜏𝑔𝑙𝑎𝑠𝑠/𝜏𝛼 ≈ 2 ∙ 106 and the transformation 
+is clearly dominated by the nucleation and growth of liquid patches in the glass 5. Looking at a 
+similar scheme than in figures 2b and 3b, one can see in the inset of figure 4b that we would 
+need to attain T<300 K (33 K below Tg) with unreachable transformation times above 1020𝜏𝛼 to 
+transform the glass by the cooperative gradual softening.  
+ 
+ 
+Figure 4. a) Specific heat traces after different annealing treatments of samples deposited at 
+0.85Tg (285 K) and annealed at 347 K.  The annealing times are indicated in the legend. b) curves 
+corresponding to the alpha relaxation time of the liquid (blue) 42, the relaxation time of the glass 
+(black), and the line corresponding to the crossover relaxation time calculated as 𝜏𝑔𝑙𝑎𝑠𝑠 =
+(180 ±  60)𝜏𝛼 (grey). The inset shows the same curves but in a lower temperature range, so the 
+crossover point can be depicted.   
+ 
+
+elaxation/transformationtime (s)
+15
+b)
+1042
+ad
+10
+Tdep=285K
+a
+30min
+Tann=347K
+1032
+1h30min
+6
+10
+10
+3h
+1022
+4h
+5h
+1012
+290295300305
+4
+6h30min
+Temperature (K)
+8h15min
+2
+10
+Rel
+380
+400
+420
+460
+320
+340
+360
+380
+400
+360
+440
+420
+440
+Temperature (K)
+Temperature (K)The crossover line could be rationalized by considering the intrinsic dynamical heterogeneity of 
+the supercooled liquid state. Even though the characteristic time of the relaxation of a liquid is 
+monitored via a single value of structural relaxation for each temperature, the true relaxation 
+proceeds via non-stretched exponential decay of dynamic correlations in spontaneous density 
+fluctuations. The relaxation dynamics of the SCL is typically assessed by dielectric spectroscopy, 
+where this relaxation is manifested as a peak in the complex part of the permittivity of the 
+sample (dielectric loss). While the maximum of this peak is regarded as the (main) relaxation 
+time of the system, its width is strongly influenced by the dynamical heterogeneities in the SCL. 
+As a consequence, even after a time τα, parts of the SCL are still not relaxed, corresponding to 
+the low frequency side of the dielectric loss, which for most glass-formers, but in particular for 
+TPD, spreads along two orders of magnitude above the maximum of the peak (alpha relaxation 
+value)44. Interestingly, this value is consistent with the observation of the crossover time of 
+(180 ±  60)𝜏𝛼. Under this view, we could interpret that if the relaxation time of the glass is 
+inside the distribution of relaxation times of the liquid, the whole system relaxes as a SCL would 
+do, i.e., via heterogeneous relaxation mechanism with no nucleation of liquid clusters. 
+The relaxation time of the glass at the crossover temperature may have been the major handicap 
+for observing experimentally these liquid regions during the glass transition. The lower the 
+stability of the glass, the higher the crossover temperature and, therefore, the faster the 
+transformation time of the glass. In the case of the glass deposited at 0.99Tg, for instance, the 
+crossover occurs for transformation times of the order of a few seconds. For a conventional glass 
+cooled at -10 K/min with 𝜏𝛼 ≈ 100 𝑠 at Tg ‘nucleation and growth’ behavior would be identified 
+only at temperatures above 359 K, with transformation times around several ms. Since 
+conventional calorimetry is unable to work in these short time scales, most of the experiments 
+up to now were limited to temperatures near Tg (at the left side of the crossover) and therefore, 
+only the cooperative relaxation mechanism had been observed so far. Now with the general 
+availability of fast scanning calorimetry short time scales are within reach and we challenge 
+other experimental groups to carry out measurements far from equilibrium to test the 
+occurrence of similar behavior in other glassy systems.  
+Our work provides conclusive experimental evidence to test theories of the glass transition and 
+is compatible with models that include heterogeneous dynamics and dynamic facilitation and 
+predict the formation of distinct liquid regions when there exists a large contrast between the 
+mobility of the equilibrated liquid and the glass. We have also shown that the ratio of 𝜏𝑔𝑙𝑎𝑠𝑠/𝜏𝛼 
+at a given temperature appears to be a good indicator to predict the mechanism of the 
+transformation. Importantly, the existence of the two regimes identified in this work is 
+
+independent of the experimental procedure or protocol to produce the glass but their 
+exploration may require the use of ultrafast experimental techniques to identify the liquid 
+regions. 
+ 
+Acknowledgements 
+JRV 
+and 
+MGS 
+acknowledge 
+Grant 
+MAT2016-79759-R 
+funded 
+by 
+MCIN/AEI/ 
+10.13039/501100011033 and “ERDF A way of making Europe”, and Grant PID2020-117409RB-
+I00 funded by MCIN/AEI/ 10.13039/501100011033. CRT is a Serra Húnter Fellow. The ICN2 was 
+funded by the CERCA programme / Generalitat de Catalunya. The ICN2 was supported by the 
+Severo Ochoa Centres of Excellence Programme, funded by the Spanish Research Agency (AEI, 
+Grant no. SEV-2017-0706). All the authors acknowledge L. Abad and IMB-CNM for the 
+fabrication of the nanocalorimeters. 
+ 
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+51. 
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+Technol. 102, 171 (1997). 
+ 
+ 
+Methods: 
+TCTA, with a Tg 91 K above the one of TPD (Tg=333 K), is a good candidate to arrest the mobility 
+of the TPD surfaces, as has been shown previously 31. So, in order to access the bulk 
+transformation of the TPD glass, a 13 nm TCTA layer has been deposited at the bottom and the 
+top of the TPD layers, with thicknesses around 65 nm.  The TPD/TCTA sandwich has been 
+deposited on the active zone of the membrane based calorimetric chips by means of physical 
+vapor deposition. The deposition chamber has a N2 trap to improve the vacuum and to act as a 
+heat sink for the fast cooling of the samples. The base pressure is of the order of 10-8 mbar. Two 
+evaporators and two shutters allow the sequential evaporation of the two materials without 
+breaking the vacuum. The thickness is controlled by a previously calibrated quartz crystal 
+monitor. The temperature control during deposition and annealing treatment is performed with 
+the calorimetric chips, by providing a specific intensity, which will heat the Pt circuit on the 
+membrane. Heat scans are performed by introducing a short pulse of intensity in the chips, 
+which will induce a constant heating ramp. In this work, pulses of 35 mA have been used, which 
+
+result in heating rates of around 3.5·104 K/s. Data of the resistivity of the chip as a function of 
+time can be processed to obtain the heat capacity as function of temperature. More information 
+about the technique can be found in 45,46 . Specific heat data for TPD has been obtained first 
+subtracting the heat capacity contribution of glassy TCTA to the heat capacity curves and dividing 
+them by the TPD mass.  
+In order to calculate the relaxation/transformation time of the glass we use two different 
+approaches as explained elsewhere 39. In the first approach, we employ the expression 𝜏1𝛽1 =
+𝜏2𝛽2
+47 to obtain the value of the glass relaxation time from the heating rate of the experiment, 
+assigning this value to the onset temperature of the glass transition. As reference we use a value 
+for the relaxation time of the glass of 100 s for a heating rate of 10 K/min 48,49 . In the second 
+approach, we calculate the transformation time from the width of the transformation peak, and 
+the corresponding value of the heating rate, assigning it to the temperature at the maximum of 
+the transformation peak: ttrans(Tmax) = ΔT/β, where ΔT is the width of the transformation peak 
+and β the value of the heating rate during the transformation. Both approaches yield 
+comparable results. Previous works have already considered this equivalence 39,50. The 
+transformation/relaxation time curve as function of temperature corresponds to the equation 
+𝜏𝑔 = 𝜏𝑔0exp [
+𝜉(𝑇𝑓
+′)𝑇0
+(𝑇−𝑇0)]  , which is a generalization of the well-known Vogel-Fulcher-Tamman, VFT, 
+equation 51, aimed at describing the dynamics of supercooled liquids and glasses with different 
+thermal stability. In this equation all the parameters have an analogous meaning as in VFT 
+equation. In this case, however, D has been substituted by a linear function of the limiting fictive 
+temperature of the glass, 𝜉(𝑇𝑓
+′) = 𝐴𝑇𝑓
+′ + 𝐵. In a supercooled liquid the fictive temperature Tf = T 
+at all temperatures, from the definition of Tf. More information of the validity of this equation 
+can be found in 39.  
+ 
+

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@@ -0,0 +1,1521 @@
+Evaluating the Transferability of Machine-Learned Force Fields
+for Material Property Modeling
+Shaswat Mohantya,1, SangHyuk Yoob,1, Keonwook Kangb, Wei Cai∗,a
+aDepartment of Mechanical Engineering, Stanford University, CA 94305-4040, USA
+bSchool of Mechanical Engineering, Yonsei University, Seoul 03722, Republic of Korea
+Abstract
+Machine-learned force fields have generated significant interest in recent years as a tool
+for molecular dynamics (MD) simulations, with the aim of developing accurate and
+efficient models that can replace classical interatomic potentials. However, before these
+models can be confidently applied to materials simulations, they must be thoroughly
+tested and validated. The existing tests on the radial distribution function and mean-
+squared displacements are insufficient in assessing the transferability of these models.
+Here we present a more comprehensive set of benchmarking tests for evaluating the
+transferability of machine-learned force fields. We use a graph neural network (GNN)-
+based force field coupled with the OpenMM package to carry out MD simulations
+for Argon as a test case. Our tests include computational X-ray photon correlation
+spectroscopy (XPCS) signals, which capture the density fluctuation at various length
+scales in the liquid phase, as well as phonon density-of-state in the solid phase and the
+liquid-solid phase transition behavior. Our results show that the model can accurately
+capture the behavior of the solid phase only when the configurations from the solid
+phase are included in the training dataset. This underscores the importance of appro-
+priately selecting the training data set when developing machine-learned force fields.
+The tests presented in this work provide a necessary foundation for the development
+and application of machine-learned force fields for materials simulations.
+Key words:
+Graph Neural Network, Machine-learned Force Field, X-ray photon
+correlation spectroscopy, Optical contrast, Phonon Density of States
+Highlights
+• Development of a suite of benchmarking tests, including X-ray photon correlation
+spectroscopy for machine-learned force fields (code available).
+∗Corresponding author
+Email address: [email protected] (Wei Cai)
+1Equal contribution
+Preprint submitted to Computer Physics Communications
+January 11, 2023
+arXiv:2301.03729v1  [cs.LG]  10 Jan 2023
+
+• Phonon density of states and melting point calculations are necessary tests for
+the solid phase.
+• Training data should include both solid and liquid configurations to accurately
+model material behavior.
+1. Introduction
+Molecular dynamics (MD) is a powerful tool for studying the equilibrium and trans-
+port properties of many-body systems with applications in diverse fields such as ma-
+terials science [1, 2, 3], polymer chemistry [4, 5, 6], biochemistry [7] and medical sci-
+ence [8, 9, 10]. In MD simulations, the motion of the atoms is driven by interatomic
+forces, which are evaluated using interatomic potentials. However, there are limitations
+to traditional molecular dynamics simulations, including a limited length scale, a lim-
+ited time scale, and insufficient force field accuracy. In recent years, machine learning
+techniques have been used to address these limitations, with the goal of accelerating
+molecular dynamics simulations and improving force field accuracy. Machine learning
+has been used to predict atomic trajectories [11, 12] with the ultimate aim of bridging
+the timescale gap between MD simulations and experiments [13, 14]. However, these
+techniques often require retraining the model for different system sizes, making it chal-
+lenging to apply to large systems, which are often necessary for property predictions.
+As a result, there is significant interest in developing scalable machine-learning methods
+for MD simulations.
+Graph-based deep learning techniques have proven effective in developing accurate
+and scalable force fields that can be trained on data from small simulation cells and
+then applied to larger systems. Graph neural networks (GNN) can accurately predict
+atomic forces with a computational cost that scales linearly with the number of atoms,
+making them well-suited for large-scale simulations. Recently, GNN-based force fields
+have been used to predict MD trajectories using approaches such as SchNet [15], and
+DeepMD [11, 16, 17]. These force fields, when trained on high-quality ab initio data,
+can perform large-scale simulations at an accuracy comparable to that of ab initio model
+but only at a small fraction of the cost [18, 19, 20]. However, before GNN force fields
+(or any machine-learned force fields) can be confidently deployed, their transferability
+to configurations beyond the training data set must be established. Currently, most
+existing benchmark tests on machine-learned force fields are limited to the radial distri-
+bution function and self-diffusivity in the liquid phase. As we show below, these tests
+are insufficient for establishing their reliability for materials modeling.
+To this end, we present a series of benchmarking tests, comparing the predictions
+of GNN-based force fields [21] to those of the original model that produces the training
+data. As an example, the original model is the classical Lennard-Jones interatomic
+potential for Argon. The purpose of choosing such a simple model as a benchmark in
+this work is to provide a baseline to evaluate the transferability of machine-learned force
+fields without incurring significant computational costs. Using such a simple model, a
+2
+
+large amount of data can be easily generated for training and comparison purposes. The
+findings from this study pave the road for future tests for machine-learned force fields
+fitted to ab initio data which are much more expensive to generate. In addition to the
+radial distribution function and self-diffusivity, we compare the computational X-ray
+photon correlation spectroscopy (XPCS) signals [22], which capture density fluctuations
+at different length scales in the liquid phase. We further show that a model trained
+exclusively on liquid configurations fails to accurately capture the vibrational frequency
+distribution in the solid phase. This deficiency is fixed only after the model is trained
+on configurations sampled from both the liquid and solid phases. These findings raise
+a concern about the transferability of the GNN-based force field and underscore the
+importance of ensuring that the training data adequately cover the areas of interest in
+the configurations space.
+The paper is structured as follows. In Section 2 we discuss the preparation of training
+data, the GNN model details, and the hyperparameters used for training the model.
+Additionally, we present the metrics that will be used to analyze the performance of
+the GNN-MD simulations against the MD simulations using the original model. In
+Section 3, we present the results of these benchmark tests. We summarize our findings
+in Section 4.
+2. Model details
+GNN-based models provide a linearly scalable force field that has the potential to
+be used in ab initio molecular dynamics (AIMD) simulations. However, verifying a
+model against direct ab initio simulations are computationally expensive, especially for
+larger systems. To address this issue, we will focus on conducting benchmarking tests
+on a GNN force field that is trained on forces generated using the classical Lennard-
+Jones interatomic potential. This will allow us to analyze the model’s performance
+without the added computational cost of using ab initio data. We assume that the
+transferability of the GNN model can be assessed when trained on forces from either
+classical interatomic potentials or ab initio data, with comparable results.
+2.1. Preparation of reference data
+We prepared a 256-atom configuration of Argon (Ar), described by the Lennard-
+Jones (LJ) interatomic potential used in our earlier work [22], by using LAMMPS [23].
+VLJ(r) = 4ϵ
+��σ
+r
+�12
+−
+�σ
+r
+�6�
+,
+(1)
+where ϵ = 0.0103 eV, σ = 3.40 ˚A. The density of the system used is ρ = 0.858 amu/˚A3
+(0.844 in LJ unit system). The cut-off radius of the neighbor list is 8.5 ˚A (2.5 in LJ
+unit system). Periodic Boundary Conditions (PBC) were applied in all three directions.
+The initial positions of atoms were generated according to a perfect face-centered cu-
+bic (FCC) lattice and the velocities were assigned random values corresponding to a
+temperature of 10 K. To reach the state of thermal equilibrium, the Nos-Hoover NPT
+3
+
+ensemble and the NVT ensemble were applied sequentially. The equilibration simula-
+tion was performed for about 2.156 ns, with a timestep of 10.78 fs. Simulations under
+both ensembles used a thermostat with a collision frequency of 0.02 /ps, a chain length
+of 5, and 5 MTK [24] loops were used. For pressure control under the NPT ensemble, a
+barostat with a collision frequency of 0.2 /ps was used. After implementing this proto-
+col of initial equilibration, the thermostat temperature (NVT simulation) was increased
+with a linear ramp from 10 K to 105 K over the 2.156 ns trajectory, during which
+atomic configurations were saved every 100 timesteps. This same process is repeated
+ten times with different initialization of velocities to generate a total of 20000 configu-
+rations. Additionally, the configuration at the end of the initial equilibration protocol
+is used as the initial state to run the analysis trajectory by using LAMMPS for the MD
+results and the Atomic Simulation Environment [25] and OpenMM [26] script for the
+GNN-MD simulation, respectively.
+2.2. GNN model
+Our benchmarking tests are carried out on the GNN-based force field following the
+GNN structure and training procedure of Li et al. [21]. For completeness, we present
+a brief description of the GNN model in this section. For the GNN-based force pre-
+dictor, the atomic configurations are first converted into graphs that encode the local
+atomic environment. Here, each atom acts as a node, and the vectors towards neigh-
+boring atoms within a cut-off radius act as edges in the graph. The node information
+includes the atomic position and force vector. The edge information includes the dis-
+tance between neighboring atoms and the unit vector toward the neighbors. A brief
+description of each step in the GNN force field prediction is enlisted below. A more
+detailed description of the model can be found in [21].
+• Encoding layer
+The feature v(l)
+i
+is an array of size h and represents the encoded node feature
+at the lth message-passing layer. The reference node encoding, v(0)
+i , is created
+by passing the atomic species information si through an encoding multi-layer
+perceptron (MLP), eN, such that v(0)
+i
+= eN(si). The feature e(l)
+ij is also an array
+of size h represents the encoded edge feature at the lth message-passing layer. The
+reference edge feature is also created through an MLP, eE, where e(l)
+ij = eE(qij, dij).
+Here, for a given pair of atoms[i,j], the unit vector and the distance between
+them are denoted by qij and dij, respectively. The superscript l is the index of
+the message-passing layer, which varies from 0 to n and denotes how many layers
+the features have passed through. At the encoding step, dij is represented by the
+radial basis functions, following the implementation of Schnet [15].
+• Message passing layer
+The encoded features, v(l)
+i
+and e(l)
+ij , are passed through the message passing layer
+m(l)
+j→i. The equation of the message-passing layer is as follows,
+m(l)
+j→i = Φ(l) �
+v(l−1)
+j
++ e(l−1)
+ij
++ v(l−1)
+i
+�
+⊙ v(l−1)
+j
+,
+∀j ∈ N(i),
+(2)
+4
+
+where Φ(l) denotes a MLP with the activation function being the Gaussian Error
+Linear Units (GELU) function [27]. As an input of Φ(l), the edge features and
+node features are added up as a vector sum since their dimensions are the same.
+⊙ refers to elemental multiplication, and j is the index of the set of neighbors
+N(i) of the ith node.
+Each message from j-th atom aggregates into M (l)
+i
+and the new node feature v(l)
+i
+is computed by the following equations.
+M (l)
+i
+=
+�
+∀j∈N(i)
+m(l)
+j→i,
+(3)
+v(l)
+i
+= Θ(l) �
+v(l−1)
+i
++ M (l)
+i
+�
++ v(l−1)
+i
+.
+(4)
+Here, Θ(l) is the node update MLP function at the lth layer. Eq. (4) shows that
+the node features are updated recursively. However, the edge features are not
+updated recursively [21]. Instead, the edge features at every level are computed
+directly from the reference edge encoding through an MLP function, A(l), i.e.,
+e(l)
+ij = A(l)(e(0)
+ij ).
+• Decoding layer
+At the last step of the neural network, the updated node feature is decoded into
+the interatomic forces fi by executing a forward pass through the decoding MLP.
+Although we present our benchmarking tests on a model trained on pair potential, the
+GNN model is not constrained to a force field derived from a pair potential form. The
+GNN architecture used here is also capable of learning a force field derived from ab
+initio data [21].
+2.3. Details of the training
+We set the number of encoding features and latent features as h = 128 and used
+n = 4 message-passing layers in the GNN. The cut-off radius for converting atomic
+configuration to graph is 7.5 ˚A, which is 1 ˚A smaller than that in the LJ potential.
+The effects of the cutoff radius will be studied in the future. The loss function for the
+training contains the L1 distance between the reference force fi and the predicted force
+ˆfi, as well as a penalty function, as follows.
+L = 1
+N
+N
+�
+i=1
+���fi − ˆfi
+���
+1 + λ
+�����
+1
+N
+N
+�
+i=1
+ˆfi
+�����
+1
+,
+(5)
+where N is the number of atoms in the reference data. The force on the center of mass
+from an interatomic potential is expected to be zero. However, the GNN model does not
+compute the forces from the derivative of a potential function, we are not guaranteed a
+zero force on the center of mass. We minimize the magnitude of the total force on the
+entire system to prevent divergent dynamics of the center of mass by adding a penalty
+5
+
+function to the loss function. The regularization parameter, λ, is set to 0.01 during
+the training. As described earlier, we used 20,000 atomic configurations as our training
+dataset. The dataset is shuffled and split into the train and test set with a 90:10 ratio.
+During training, we carry out standard normalization of the atomic forces such that
+the mean atomic force is zero and the variance is unity. We used PyTorch 1.11.0,
+PyTorch-lightning 1.6.3, DGL 0.8.1 and Scikit-learn 0.24.2 packages to train
+and test the model. We use the Adam optimizer with an exponential learning rate
+scheme (from 3 × 10−4 to 1 × 10−7) during the training of the force field. The total
+number of epochs is set to 30 and we choose the trained model parameter (weights and
+biases) at the last epoch as the force calculator for our GNN-MD simulation.
+2.4. Metrics for model performance
+Here we list the static and dynamic metrics that will be used to compare the per-
+formance of the GNN-MD simulations against the MD simulations. To examine the
+structure of the liquid phase we will study the pair distribution function. To examine
+the dynamics of the liquid phase, we evaluate the self-diffusivity and carry out the
+computational XPCS analysis on the MD trajectory. The relations to compute the
+properties such as the radial distribution function, the structure factor, and the com-
+putational XPCS signal, are described only briefly in A.2 and A.3 (see [22] for more
+details), since the focus of this paper is to establish the accuracy of the GNN-MD in
+capturing the dynamics and structure of the simulation. To evaluate the model perfor-
+mance in the solid phase, we compute the phonon density of states (PDOS), and the
+melting point by using the interface method. Further details on the key equations we
+solve to obtain the numerical results are given in A.
+3. Numerical Results
+The MD and GNN-MD simulations discussed in this section are carried out on a
+256-atom and a 4000-atom system to test the scalability and transferability of the GNN
+force field which was trained on 256-atom configurations. The training samples for the
+256 and 4000 atoms system are obtained from MD simulations using LAMMPS [23]. While
+in the previous work [21] only liquid configurations are included in the training data,
+here we train two models: Model A is trained only using liquid configurations and
+Model B is trained using both liquid and solid configurations (see Section 2.1 for the
+procedure of preparing these configurations). The results presented in Sections 3.1, 3.2,
+and 3.3, corresponds to Model B. However, Model A gives similar results for the tests
+in these sections; hence they are omitted here for brevity. On the other hand, Model A
+and Model B give different results for some of the tests involving the solid phase, and
+both will be presented in Section 3.4.
+The initial configuration for the testing MD and GNN-MD simulations are prepared
+using the initial relaxation protocol. The final trajectory for the analysis is then run
+using the same NVT ensemble as in the final relaxation step.
+However, the total
+simulation time extends to 539 ps with a timestep of 10.78 fs. The simulation frames
+6
+
+are stored at 107.8 fs intervals for our analysis. In comparison to MD simulations using
+the LJ potential, the GNN-MD is 101 times slower, because the atomic forces from the
+LJ potential can be computed very quickly. We reiterate that the focus of this work is
+to use the LJ potential as an example to test the transferability of the GNN force field
+in capturing the static and dynamic properties of the material system that it is trained
+on.
+3.1. Accuracy of force prediction
+We test the atomic forces predicted atomic forces against the calculated atomic
+forces from the Lennard-Jones potential for 1,000 sampled configurations of our 256-
+atom simulation. These configurations are different from the ones used for the training
+of the GNN force field. We see that the predicted forces are well correlated with the
+actual forces across all atoms and all configurations, as shown in Fig. 1(a) (parity plot).
+The coefficient of determination, R2, is almost 1.0 (1−R2 ≈ 10−4) which shows how well
+correlated the predicted force is to the actual forces computed from LAMMPS by using
+the Lennard-Jones force field. The generalizability of the force field to a 4000-atom
+system is captured by the parity plot shown in Fig. 1(b), where the quality of the force
+prediction remains qualitatively the same.
+(a)
+(b)
+(c)
+Figure 1: Parity plot for the predicted forces and the actual forces for the (a) 256 atom and (b) 4000
+atom system. (c) Distribution of the absolute error of the predicted forces for the 256 atom and 4000
+atom system over the configurations of analysis.
+Figure 1(c) shows the distribution of the absolute error of the predicted forces, for the
+256-atom (distribution of atomic force components over 1,000 configurations) and the
+4000-atom (distribution of atomic force components over 100 configurations) system.
+The standard deviation of the force error is 0.15 meV/˚A. We see that the absolute error
+distribution is the same irrespective of the system size, which indicates the error of the
+trained model and is independent of the size of the system. This establishes the quality
+of the trained model consistent with previous report [21].
+3.2. Structure of liquid phase
+Here we examine the orientation-averaged radial distribution function, g(r), as de-
+fined in A.2. We average the g(r) over 100 configurations that are sampled every 5.34 ps
+7
+
+0.3
+0.2
+A
+0.1
+(eV
+0
+-0.1
+- - R2 = 1.0
+F
+-0.2
+-0.3
+-0.4
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+F
+(eV/A)
+pred0.4
+0.3
+0.2
+A
+0.1
+(eV
+ref
+-0.1
+- - R² = 1.0
+F
+-0.2
+-0.3
+-0.4
+-0.4
+-0.2
+0
+0.2
+0.4
+F
+(eV /A)
+pred0.03
+256 atom
+0.025
+4000 atom
+0.02
+Probability
+0.015
+0.01
+0.005
+0
+-0.5
+0
+0.5
+Absolute Error
+(meV/A)from our simulation. Figure 2(a) shows that the g(r) obtained from the MD and GNN-
+MD simulation are in close agreement for the (small) system it is trained on, confirming
+the previous observation [21]. We further find that g(r) between the MD and GNN-MD
+simulation agree well with each other even for the larger simulation cell containing 4000
+atoms, as shown in Fig. 2(b). We also observe that the orientation-averaged structure
+factor, S(q) (not shown here), which can be obtained from the Fourier transform of g(r),
+agrees between MD and GNN-MD simulations, for both the 256-atom and 4000-atom
+systems.
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+0
+0.5
+1
+1.5
+2
+2.5
+3
+(a)
+0
+1
+2
+3
+4
+0
+0.5
+1
+1.5
+2
+2.5
+3
+(b)
+Figure 2: The g(r) over 100 configurations for the (a) 256 atoms and (b) 4000 atoms MD and GNN-MD
+simulation.
+3.3. Dynamics of liquid phase
+Given that atomic trajectories in MD simulations are chaotic, small differences in
+atomic forces can lead to large divergences in the atomic trajectories at a later time.
+Therefore, we don’t expect the atomic trajectories in MD and GNN-MD to agree well
+with each other. However, for the GNN-based force field to be useful, the statistical
+properties of the atomic trajectories need to be consistent with those of the original
+MD model. A commonly tested statistical property is the self-diffusivity obtained from
+the mean-squared displacement (MSD) as described in A. Figure 1 shows that the MSD
+predicted by the GNN model is consistent with that from the original MD model. For
+the 256-atom system, the MSD from GNN-MD agrees very well with the MD model
+for the first 150 ps, after which the difference grows somewhat larger, as shown in
+Fig. 3(a). For the 4000-atom system, the MSD from GNN-MD agrees very well with
+the MD model for the first 300 ps, after which the difference also grows somewhat
+larger, as shown in Fig. 3(b).
+8
+
+0
+100
+200
+300
+400
+500
+600
+0
+200
+400
+600
+800
+1000
+1200
+1400
+(a)
+0
+100
+200
+300
+400
+500
+600
+-200
+0
+200
+400
+600
+800
+1000
+1200
+1400
+(b)
+Figure 3: The mean-squared displacement (MSD) for the (a) 256 atoms and the (b) 4000 atoms from
+the MD and GNN-MD simulation over the 539 ps trajectory at 100 K.
+The MSD curves from the entire (∼ 539 ps) trajectories are fitted to straight
+(dashed) lines to obtain the predicted self-diffusivity D (see Fig. 3). The agreement
+between MD and GNN-MD is within 10% for the 256-atom system, and within 5% for
+the 4000-atom system, at the temperature of 100 K. The level of agreement for the
+4000-atom system is maintained at all temperatures from 95 K to 110 K, and appears
+to improve with increasing temperature, as shown in Table 1.
+Temperature
+D (µm2/s)
+MD
+GNN-MD
+Relative Error (%)
+95 K
+3254.70
+3096.14
+4.87
+100 K
+3673.08
+3802.89
+3.53
+110 K
+5010.12
+4979.92
+0.60
+Table 1: The self-diffusivity calculated from the MSD for the 4000 atom system at different tem-
+peratures using the MD and GNN-MD simulation. The MSD for all temperatures can be found in
+Fig. 11.
+In addition to the bulk diffusivity, we also test the GNN-based force field by carrying
+out the XPCS analysis.
+This allows us to quantify the density fluctuations arising
+from atomic motion at different length scales. Figure 4 shows two computed XPCS
+speckle patterns from snapshots of the MD and GNN-MD trajectories, respectively.
+The speckle patterns are computed for all saved configurations (∼ 107.8 fs apart) in
+the atomic trajectories so that the time correlation at each pixel can be obtained.
+9
+
+-2.21
+0    
+2.21 
+-2.21
+0    
+2.21 
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+(a)
+-2.21
+0    
+2.21 
+-2.21
+0    
+2.21 
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+(b)
+Figure 4: The speckle patterns of scattered intensity I(q) on a spherical slice (81 × 81 pixels) in the
+q-space for (a) MD and (b) GNN-MD simulation.
+Figure 5 shows that the decay of the time correlation of the speckle averaged over
+all wave-vectors close to a given q = |q|, agrees well between MD and GNN-MD, for
+both the 256-atom system and the 4000-atom system. If the system dynamics is purely
+diffusive, then g2(q, τ) should decay exponentially.
+0
+50
+100
+150
+200
+250
+300
+350
+1
+1.2
+1.4
+1.6
+1.8
+2
+(a)
+0
+50
+100
+150
+200
+250
+300
+350
+1
+1.2
+1.4
+1.6
+1.8
+2
+(b)
+Figure 5: The g2(q, τ) at q = 1.844 ± 0.029 ˚A−1 over the first 3.5574 ps for the (a) 256 atoms and the
+(b) 4000 atoms MD and GNN-MD simulation at 100 K.
+Figure 5 compares the time correlation for q = 1.844 ± 0.029 ˚A−1 (probing the
+length scale corresponding to the first nearest neighbor). We have also carried out
+the same computation for all wave-vector magnitudes from q = 0.461 ± 0.029 ˚A−1
+to q = 1.844 ± 0.029 ˚A−1 and observed close agreement between MD and GNN-MD
+10
+
+models. In all cases, we see that the g2(q, t) decays as a single exponential after the
+initial sub-diffusive dynamics associated with caging effects [22], as is expected for liquid
+Ar.
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+2
+4
+6
+8
+10
+12
+14
+(a)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+2
+4
+6
+8
+10
+12
+14
+(b)
+Figure 6: The Γ(q) as a function of q2 for the (a) 256 atoms and the (b) 4000 atoms MD and GNN-MD
+simulation at 100 K.
+From g2(q, τ), we extract the rate of exponential decay Γ(q) in the long time limit,
+which reveals the dispersion relation in the liquid [22]. Figure 6 shows that the Γ(q)
+function obtained from MD and GNN-MD are in good agreement. Interestingly, the
+agreement appears better in the 4000-atom system than in the 256-atom system on
+which the GNN-base force field is trained. Additionally, we found that the dispersion
+relation from the GNN-MD simulation agrees well with MD simulation results for all
+temperatures of 95 K, 100 K, and 110 K (see Fig. 13).
+0
+5
+10
+15
+20
+25
+0
+0.2
+0.4
+0.6
+0.8
+1
+(a)
+0
+5
+10
+15
+20
+25
+0
+0.2
+0.4
+0.6
+0.8
+1
+(b)
+Figure 7: The optical contrast β∆(q) as a function of X-ray pulse width (∆τ) for the (a) 256 atoms
+and the (b) 4000 atoms MD and GNN-MD simulation for |q| = 1.844 ± 0.029 ˚A−1 at 100 K.
+11
+
+In addition to the dynamics obtained from the XPCS speckle fluctuations, we also
+test the statistical distribution of the X-ray speckles by evaluating the optical contrast in
+the q-range |q| = 1.844± 0.029 ˚A−1 for varying pulse widths, ∆τ, of the incident X-ray.
+Figure 7 shows that the optical contrast obtained from MD and GNN-MD models are
+in close agreement. The optical contrast β∆(q) is expected to decrease with increasing
+pulse width, ∆τ, where the time at which β∆(q) decreases to half of its maximum value
+gives an estimate of the correlation time. Table 2 lists the correlation time obtained
+at three temperatures, where the estimates from MD and GNN-MD models agree well
+with each other (time resolution of 0.11 ps).
+Temperature
+Correlation time (ps)
+MD
+GNN-MD
+Relative Error (%)
+95 K
+2.16
+2.25
+∼ 4.17
+100 K
+2.05
+2.05
+∼ 0.0
+110 K
+1.83
+1.83
+∼ 0.0
+Table 2: The correlation time (time taken for β∆(q) to drop to 0.5) for the 4000 atom system at
+different temperatures using the MD and GNN-MD simulation. The β∆(q) as a function of ∆τ for all
+temperatures can be found in Fig. 14.
+3.4. Solid phase and melting
+In order to test the model performance in the solid phase, we performed MD and
+GNN-MD simulations of a face-centered cubic (FCC) lattice at 20 K, which is below
+the melting temperature. The resulting pair distribution function g(r) captures the
+amplitude of thermal vibration in the solid phase. Figure 8 shows that the GNN-MD
+captures the g(r) of the solid phase accurately even for Model A, which is the GNN-
+based force field trained exclusively on liquid configurations. Similar results for the
+predicted g(r) of the solid phase are observed using Model B (not shown).
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+2
+4
+6
+8
+10
+Figure 8: The predicted pair distribution function g(r) of the solid phase at 20 K using Model A,
+which is trained only using liquid configurations.
+To further test the capability of the model to capture solid phase configurations, we
+estimate the melting point of the system by using the solid-liquid interface method [28].
+12
+
+The solid half is initialized as an FCC lattice and is equilibrated at 20 K, whereas,
+the liquid half is equilibrated at 100 K to yield the initial configuration, as shown in
+Fig. 9(a).
+(a)
+20
+40
+60
+80
+100
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+(b)
+Figure 9: (a) The solid-liquid interface used for the melting point estimation simulation and (b) the
+interface velocity at different temperatures for Model A and Model B.
+The solid-liquid interface is maintained at 100 K, 61.36 K, and 20 K across three
+different simulations. The interface will move into the liquid or solid phase depending
+on whether the temperature is below or above the melting point. To quantify these
+transitions we estimate the velocity of the solid-liquid interface. Our results show a
+linear dependence of the interface velocity with the simulation temperature, as shown
+in Fig. 9(b). Here the melting point corresponds to the temperature where the interface
+velocity is zero. We obtain the melting point from the regular MD simulation to be
+55.2 K, whereas the melting point from the GNN-MD is: 56.4 K from Model A and
+55.8 K from Model B. (The differences here are below the error bar ∼ 1.34 K.) This
+demonstrates that the GNN force field is able to capture the solid-liquid phase transition
+with sufficient accuracy. It is somewhat surprising that this level of agreement can be
+reached even for Model A, which is trained on liquid configurations only.
+To see whether Model A can capture all the properties of the solid phase without
+being trained on any solid configurations, we performed further tests. First, we examine
+the distribution of the eigenfrequencies of a perfect FCC crystal. The eigenfrequencies
+can be obtained by first diagonalizing the Hessian matrix, H, and then dividing the
+eigenvalues by the atomic mass m and taking the square root. The distribution of the
+eigenfrequencies provides an estimate of the phonon density of states (PDOS) in the
+crystal, as shown in Fig 10. For this analysis, we use a simulation cell with 2048 atoms,
+such that the H is defined over 6144 degrees of freedom. The details of the calculations
+13
+
+can be found in A.4.
+(a)
+(b)
+Figure 10: The phonon density of states obtained from Lennard-Jones force field computation and the
+GNN force field trained on: (a) exclusively liquid configurations (Model A), and (b) solid and liquid
+configurations (Model B).
+The Hessian matrix computed using the Lennard-Jones potential (HLJ) is perfectly
+symmetric whereas the one computed from the GNN force field ( ˜HGNN) is only ap-
+proximately symmetrical. Here we use the ∼ sign to indicate that the matrix is not
+exactly symmetrical. Before computing the eigenfrequencies we need to symmetrize
+the Hessian matrix by Hs
+GNN = 1
+2
+�
+˜HGNN + ˜HT
+GNN
+�
+. Figure 10(a) shows that the PDOS
+computed from GNN Model A (trained on liquid configurations only) exhibits an ap-
+preciable difference from the reference LJ model. The agreement with the reference LJ
+model becomes significantly improved for GNN Model B (trained on both liquid and
+solid configurations), as shown in Fig. 10(b).
+The lack of perfect symmetry in the Hessian matrix is a type of error in the GNN-
+based force field considered here, in which the atomic forces are not obtained from the
+partial derivatives of a potential energy function. To quantify this error, we define a
+symmetricity measure S for any square matrix (see A.4) such that S = 1 if the matrix
+is perfectly symmetric.
+Configuration
+Training Data
+Liquid (Model A)
+Solid and Liquid (Model B)
+Solid
+0.785
+0.949
+Liquid
+0.984
+0.984
+Table 3: The symmetricity, S( ˜HGNN), of the Hessian matrix of GNN Model A (trained on liquid
+configuration only) and Model B (trained on both liquid and solid configurations) when tested on a
+solid configuration (at ∼ 20 K) and a liquid configuration (at ∼ 100 K).
+Table 3 shows the symmetricity of the Hessian matrix from Model A and Model
+14
+
+0.1
+MD
+GNN - MD
+0.08
+ (a.u.)
+0.06
+PDOS
+0.04
+0.02
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Vo (THz)0.1
+MD
+GNN - MD
+0.08
+ (a.u.)
+0.06
+PDOS
+0.04
+0.02
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Vo
+(THz)B evaluated on a solid (perfect FCC crystal) and on a randomly chosen liquid con-
+figuration. The symmetricity on the liquid configuration is 0.984 for both Model A
+and Model B, which is considered acceptable in this study. The symmetricity on the
+solid configuration is only 0.785 for Model A, which is correlated with its poor pre-
+diction of the PDOS of the perfect crystal (Fig. 10(a)). On the other hand, after the
+solid configurations are included in the training set, Model B gives a symmetricity of
+0.949 for the solid configuration, which is correlated with the improved prediction of
+the PDOS of the perfect crystal (Fig. 10(b)). Therefore, our results demonstrate the
+need for a comprehensive set of tests to evaluate the transferability of machine-learned
+force fields, and the importance of including both liquid and solid configurations in the
+training dataset.
+4. Conclusions
+This paper discusses the comprehensive lists of tests needed to check the transfer-
+ability of a neural network force field. We carry out our benchmarking tests on the
+GNN-based force field presented in [21]. Our analysis is based on a reference system
+of liquid Argon by modeling it as a Lennard-Jones fluid. The typical tests for a force
+field include validating the radial distribution function and mean-squared displacement.
+However, in this paper, we present additional tests to validate the dynamics from the
+GNN force field by carrying out the computational XPCS analysis. These tests are
+carried out at different system sizes and temperatures to further establish the transfer-
+ability of the GNN force field.
+In addition to testing the liquid behavior, we present a few solid-phase tests such
+as melting point estimation, which are not usually validated for machine-learned force
+fields. Notably, most solid and liquid phase tests were passed by the force field trained
+on just liquid configurations, however, estimating the phonon density of states required
+solid configurations in the training dataset. This form of dataset engineering is neces-
+sary for the force field to be able to capture both solid and liquid behavior. In summary,
+these benchmarking tests provide a more comprehensive check on the transferability of
+the GNN-based force fields.
+On having established the transferability of the GNN force field, efforts can now
+be made to train the model from AIMD calculations for systems that are not well
+modeled by traditional MD force fields. This resulting GNN-AIMD promises the force
+field accuracy of AIMD and can be used for larger atomic systems which is not possible
+using traditional AIMD (due to computational expense) and traditional MD (due to
+the inaccuracy in the force field).
+Conflict of Interest
+The authors declare no competing interests.
+15
+
+Data Access
+All data is available in the main text and the Supplementary appendices. Further
+information about the computation can be obtained on request from the corresponding
+author. The library for the testbed of machine-learned force fields can be found at
+TB-MLFF. The code on the XPCS and XSVS analysis is obtained from the C-XPCS
+library.
+Acknowledgements
+S.M. and W.C. acknowledge support from the Precourt Pioneering Project of Stan-
+ford University. S.Y. was supported by Korea Institute for Advancement of Technology
+(KIAT) grant funded by the Korea Government (MOTIE) (P0017304, Human Resource
+Development Program for Industrial Innovation). K.K. acknowledges support from the
+National Research Foundation of Korea (NRF) grant funded by the Korean government
+(MSIT) (NRF- 2022R1A2C2011266).
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+[31] E Jakeman.
+Photon correlation.
+In Photon correlation and light beating spec-
+troscopy, pages 75–149. Springer, 1974.
+19
+
+A. Analysis Details
+A.1. Mean-squared Displacement
+Mean squared displacement (MSD) is a metric used in statistical mechanics to mea-
+sure the diffusive motion of the material system being simulated. The MSD is the most
+commonly used metric to measure the spatial deviation as a function of time. The
+MSD is an ensemble average and is computed using the position (r ∈ Rd where d is
+the dimensionality of the position),
+MSD(τ) = ⟨(r(τ) − r(0))2⟩ = 1
+N
+N
+�
+i=1
+(ri(τ) − ri(0))2.
+(6)
+For a purely diffusive motion, such as the dynamics of Brownian motion, the MSD is
+related to the diffusivity, D, by,
+MSD(τ) = 2dDτ.
+(7)
+For the liquid simulations, the MSD varies linearly with time after the initial slow
+dynamics (attributed to the sub-diffusive motion brought about by the caging effect).
+As a consequence, the MSD is analyzed for t > 1 ps.
+A.2. Basics of structural analysis
+An important structural measure of a collection of atoms is the pair distribution
+function, g(r, t), defined by,
+g(r, t) =
+1
+Nρ0
+�
+i
+�
+j
+j̸=i
+δ(r − (ri(t) − rj(t))),
+(8)
+where ρ0 is the bulk atomic density of the sample, N is the total number atoms, and
+ri (and rj) is the atomic position in the simulation box. Although the convention is
+to exclude the correlation between the atom and itself (i.e. i = j) from the definition
+of g(r, t), in the following it is more convenient to introduce an alternative definition,
+˜g(r, t), where such a constraint is removed.
+˜g(r, t) =
+1
+Nρ0
+�
+i
+�
+j
+δ(r − (ri(t) − rj(t))) = g(r, t) + 1
+ρ0
+δ(r)
+(9)
+When all atoms are of the same type, the Fourier transform of ρ0 ˜g(r, t) is the structure
+factor, ˜S(q, t),
+˜S(q, t) = 1
+N
+�
+i
+�
+j
+e−iq·[ri(t)−rj(t)],
+(10)
+where q is the scattered wave-vector. When the sample contains atoms of different
+types, the definition of the structure factor is generalized to the following
+S(q, t) =
+1
+�
+j fj(q)2
+�
+i
+�
+j
+fi(q)fj(q) e��iq·[ri(t)−rj(t)].
+(11)
+20
+
+Here fi(q) is the X-ray atomic form factor of atom i. The X-ray atomic form factor
+can be obtained by the Fourier transform of the electron density field for a given type
+of atom. For many elements, the atomic form factor can be well parameterized by a
+sum of Gaussians [29].
+A.3. Basics of XPCS and scattering statistics
+We then consider the time correlation function of the intensity of the XPCS speckles
+that we observe. The normalized auto-correlation function helps in studying the dy-
+namics of the simulation. The auto-correlation is denoted by g2(τ), since it is a second
+order correlation and is given by,
+g2(q, τ) = ⟨I(q, t)I(q, t + τ)⟩/⟨I⟩2
+(12)
+I(q, τ) ∝ S(q, τ)
+(13)
+Another metric that helps us in studying the dynamic characteristics for the simu-
+lation is the intermediate scattering function, f(q, τ), described as,
+f(q, τ) = 1
+N
+��
+i̸=j
+e(iq(ri(0)−rj(τ))
+�
+t
+(14)
+For a simple diffusive process following Brownian motion, the intermediate scatter-
+ing function reduces to a single exponential, F(q, τ) = f(q,τ)
+f(q,0) = e−Γτ, where Γ = Dq2
+[30].
+Since the time auto-correlation (g2(q, τ)) is related to the f(q, τ) by the Siegert
+relation [31], it can also be represented as a single exponential under the assumption of
+Brownian dynamics,
+g2(q, τ) = 1 + β(q)|F(q, τ)|2
+= 1 + β(q) exp[−2Γ(q)τ]
+(15)
+g2(q, τ) − 1 = G(q, τ) ∝ exp[−2Γ(q)τ]
+(16)
+where Γ(q) is the relaxation rate and the factor of 2 arises from the Siegert relation
+which relates the intensity auto-correlation function to the electric field correlation
+function [31].
+Furthermore, the auto-correlation of the speckle at q is denoted by g2(q, τ) and it
+is non-dimensionalized by the square of its mean at τ = 0, as given below for infinites-
+imally short pulses,
+g2(q, τ) = ⟨I(q, t) I(q, t + τ)⟩t
+⟨I(q, t)⟩2
+t
+= ⟨S(q, t) S(q, t + τ)⟩t
+⟨S(q, t)⟩2
+t
+,
+(17)
+where ⟨·⟩t represents the time-averaged value of the enclosed entity. Experimentally,
+the X-ray pulses always have a finite duration. Hence the intensity should be replaced
+21
+
+by the time-averaged X-ray intensity, I∆(q), over the exposure duration of ∆τ,
+I∆(q) =
+� t+∆τ
+t
+I(q, t) dt.
+(18)
+In this case,
+g2(q, τ) = ⟨I∆(q, t) I∆(q, t + τ)⟩t
+⟨I∆(q, t)⟩2
+t
+.
+(19)
+For a X-ray speckle pattern obtained from a single X-ray pulse, the optical contrast
+β(q) is defined by the variance of the intensity divided by the square of its mean [22].
+While β(q) is given by the scattering intensity distribution I(q) from an infinitesimally
+short pulse, we denote the optical contrast from an X-ray pulse of finite duration ∆t as
+β∆(q) where,
+β∆(q) = ⟨I∆(q)2⟩q − ⟨I∆(q)⟩2
+q
+⟨I∆(q)⟩2
+q
+,
+(20)
+where ⟨·⟩q represents the average over all detector pixels that satisfy q − dq/2 ≤ |q| <
+q + dq/2, for a small dq. For an ergodic and isotropic system, the distribution of pixel
+intensity at around a particular q is the same in q-space and t, so that
+⟨I∆(q)⟩q ≈ ⟨I∆(q)⟩t,
+⟨I∆(q)2⟩q ≈ ⟨I∆(q)2⟩t.
+(21)
+This means that we can also define an optical contrast β0(q) from the time variation
+of the intensity at a single q, i.e,
+β0(q) = g2(q, τ = 0) − 1.
+(22)
+For an ergodic and isotropic system, β∆(q) and β0(q) should be equal. The detailed
+discussion on the XPCS relations and the statistics of X-ray speckles, the computational
+algorithm and its implementation can be found in [22].
+A.4. Hessian matrix and phonon density of states
+To estimate the phonon density of states (PDOS) we examine the eigenvalues of the
+Hessian matrix, H, of our relaxed system. The H is defined as,
+Hij =
+∂
+∂xj
+�∂U
+∂xi
+�
+,
+(23)
+= − ∂fi
+∂xj
+,
+(24)
+where U is the potential energy and fi is the atomic force component corresponding to
+the ith degree of freedom (i goes from 1 to 3N). We can approximate this derivative
+numerically by a central difference scheme,
+Hij = fi(xj − ∆x) − fi(xj + ∆x)
+2∆x
+,
+(25)
+22
+
+where ∆x = 3.405 × 10−6 ˚A(10−6 in LJ unit). If λn represents the eigenvalues of H
+then the eigenfrequencies νn can be obtained from
+νn = 1
+2π
+�
+λn
+m .
+(26)
+The distribution of the νn gives us the PDOS in arbitrary units. The Hessian matrix
+H is expected to be symmetric for a conservative force field that is derived from an
+interatomic potential. Consistency or symmetricity requires,
+∂fi
+∂xj
+= ∂fj
+∂xi
+.
+(27)
+The Hessian matrix computed using the Lennard-Jones potential (HLJ) is perfectly
+symmetric whereas the one computed from the GNN force field ( ˜HGNN) is only nearly
+symmetric. We define a measure for symmetricity, S, for a matrix A as
+S(A) = ||As||2 − ||Aa||2
+||As||2 + ||Aa||2
+,
+(28)
+where,
+As = A + AT
+2
+,
+(29)
+Aa = A − AT
+2
+.
+(30)
+The symmetricity is such that −1 ≤ S(A) ≤ 1, where S(A) = 1 for a perfectly sym-
+metric matrix and S(A) = −1 for a perfectly anti-symmetric matrix.
+23
+
+B. Analysis at different temperatures
+In this appendix, we show the results corroborating our findings that have been
+presented in main text. We present simulation results corresponding to the 4000 atoms
+case which is run at 95 K, 100 K, and 110 K.
+0
+100
+200
+300
+400
+500
+600
+0
+200
+400
+600
+800
+1000
+1200
+1400
+(a)
+0
+100
+200
+300
+400
+500
+600
+-200
+0
+200
+400
+600
+800
+1000
+1200
+1400
+(b)
+0
+100
+200
+300
+400
+500
+600
+-200
+0
+200
+400
+600
+800
+1000
+1200
+(c)
+0
+100
+200
+300
+400
+500
+600
+0
+500
+1000
+1500
+2000
+(d)
+Figure 11: The mean-squared displacement (MSD) for the (a) 256 atoms at 100 K and the 4000 atoms
+at (b) 100 K, (c) 95 K, and (d) 110 K, MD and GNN-MD simulation over the 539 ps trajectory.
+We see that the MSD is in close agreement between the MD and GNN-MD simula-
+tions, as shown in Fig. 11, irrespective of the temperature at which the simulations are
+carried out.
+24
+
+0
+50
+100
+150
+200
+250
+300
+350
+1
+1.2
+1.4
+1.6
+1.8
+2
+(a)
+0
+50
+100
+150
+200
+250
+300
+350
+1
+1.2
+1.4
+1.6
+1.8
+2
+(b)
+0
+50
+100
+150
+200
+250
+300
+350
+1
+1.2
+1.4
+1.6
+1.8
+2
+(c)
+0
+50
+100
+150
+200
+250
+300
+1
+1.2
+1.4
+1.6
+1.8
+2
+(d)
+Figure 12: The g2(q, τ) at q = 1.844 ± 0.029 ˚A−1 over the first ∼ 3.56 ps for the (a) 256 atoms (25
+tracked atoms) at 100 K and the 4000 atoms (45 tracked atoms) at (b) 100 K, (c) 95 K and (d) 110
+K, MD and GNN-MD simulation.
+We observe that the decay in g2(q, τ) between the MD and the GNN-MD simulation
+is also in agreement at all three temperatures.
+25
+
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+2
+4
+6
+8
+10
+12
+14
+(a)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+2
+4
+6
+8
+10
+12
+14
+(b)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+600
+800
+1000
+1200
+1400
+1600
+1800
+(c)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+2
+4
+6
+8
+10
+12
+14
+(d)
+Figure 13: The Γ(q) as a function of q2 for the (a) 256 atoms at 100 K and the 4000 atoms at (b) 100
+K, (c) 95 K, and (d) 110 K, MD and GNN-MD simulation.
+We examine the Γ(q) as a function of q2 for different temperatures and see a very
+close agreement between the MD and GNN-MD simulations.
+26
+
+0
+5
+10
+15
+20
+25
+0
+0.2
+0.4
+0.6
+0.8
+1
+(a)
+0
+5
+10
+15
+20
+25
+0
+0.2
+0.4
+0.6
+0.8
+1
+(b)
+0
+5
+10
+15
+20
+25
+0
+0.2
+0.4
+0.6
+0.8
+1
+(c)
+0
+5
+10
+15
+20
+25
+0
+0.2
+0.4
+0.6
+0.8
+1
+(d)
+Figure 14: The β∆(q) as a function of ∆τ for the (a) 256 atoms at 100 K and the 4000 atoms at (b)
+100 K, (c) 95 K and (d) 110 K, MD and GNN-MD simulation for |q| = 1.844 ± 0.029 ˚A−1.
+In addition to the previous analysis, we also examine the XSVS analysis by plot-
+ting the optical contrast, β∆(q), as a function of the exposure time ∆τ. The decay in
+the optical contrast is also in agreement irrespective of the temperature of the simula-
+tion which shows that the GNN force field approximates the Lennard-Jones potential
+sufficiently to capture the dynamics on liquid phase.
+27
+

19E1T4oBgHgl3EQfRwPe/content/tmp_files/2301.03058v1.pdf.txt

@@ -0,0 +1,304 @@
+arXiv:2301.03058v1  [math.AG]  8 Jan 2023
+GROMOV ELLIPTICITY AND SUBELLIPTICITY
+SHULIM KALIMAN AND MIKHAIL ZAIDENBERG
+Abstract. We establish the equivalence of Gromov ellipticity and
+subellipticity in the algebraic category.
+Contents
+Introduction
+1
+0.1.
+Ellipticity versus subellipticity
+1
+0.2.
+Examples of elliptic varieties
+3
+0.3.
+Ellipticity versus rationality
+3
+1.
+Composing and extending sprays
+3
+1.1.
+Composition of sprays
+4
+1.2.
+Extension lemma
+5
+1.3.
+Proof of the main theorem
+6
+References
+6
+Introduction
+0.1. Ellipticity versus subellipticity. We are working over an alge-
+braically closed field K of characteristic zero; and An = An
+K stands for
+the affine n-space over K. All varieties and vector bundles in this note
+are algebraic; a variety is always reduced and irreducible. Abusing the
+language, in this paper the terms spray and (sub)ellipticity stand for
+algebraic spray resp. algebraic (sub)ellipticity.
+The notion of ellipticity was introduced, both in analytic and al-
+gebraic setting, in [Gro89, Sect. 0.5 and 3.5.A]. In the Localization
+Lemma (see [Gro89, Sect. 3.5.B]) Gromov actually considers a local
+version of ellipticity. The subellipticity (implicitly present in [Gro89,
+Sec. 3.5]) was introduced in [For02]; see [For17, Sect. 5.1] and [For20]
+for a historical account. Recall these notions.
+A spray of rank r over a smooth algebraic variety X is a triple
+(E, p, s) consisting of a vector bundle p: E → X of rank r and a
+Date: 06.01.2023.
+1
+
+2
+SHULIM KALIMAN AND MIKHAIL ZAIDENBERG
+morphism s: E → X such that s|Z = p|Z where Z ⊆ E stands for
+the zero section of p. This spray is dominating at x ∈ X if the restric-
+tion s|Ex : Ex → X to the fiber Ex = p−1(x) is dominant at the origin
+0x = Z ∩ Ex of the vector space Ex. The variety X is called elliptic if
+it admits a spray (E, p, s) which is dominating at each point x ∈ X.
+A local spray (E, p, s) with values in X at a point x ∈ X consists of
+a vector bundle p: E → U over a neighborhood U of x and a morphism
+s: E → X such that s|ZU = p|ZU where ZU stands for the zero section
+of p: E → U. One says that X is locally elliptic if for any x ∈ X there
+is a local spray in a neighborhood of x dominating at x. The variety
+X is called subelliptic if it admits a family of sprays (Ei, pi, si) defined
+over the whole X which is dominating at each point x ∈ X, that is,
+TxX =
+n
+�
+i=1
+dsi(T0i,xEi,x)
+∀x ∈ X.
+Thus, an elliptic variety is both locally elliptic and subelliptic. In this
+note we establish the converse implications.
+Theorem 0.1. For a smooth algebraic variety X the following are
+equivalent:
+• X is elliptic;
+• X is subelliptic;
+• X is locally elliptic.
+Theorem 0.1 answers a question in [For17, p. 230]. The equivalence
+of the first two properties is well known for homogeneous spaces of
+a linear algebraic group, see [For20, Proposition 6.7]. It is unknown
+however whether this equivalence also holds in the analytic category;
+cf. [Kus20].
+Comparing Theorem 0.1 with [LT17, Theorem 1] and [For20, Corol-
+lary 6.6] we deduce the following results, see [LT17] and [For20] for the
+terminology.
+Corollary 0.2. For a smooth algebraic manifold X the following con-
+ditions are equivalent:
+(a) X is elliptic;
+(b) X is locally elliptic;
+(c) X is subelliptic;
+(d) X satisfies Gromov’s condition aEll1;
+(e) X satisfies the algebraic homotopy Runge principle.
+See [For20, Sec. 6.2] for relations of (d) and (e) to approximation,
+interpolation and Oka-Grauert h-Principle in the algebraic setting.
+
+GROMOV ELLIPTICITY AND SUBELLIPTICITY
+3
+As another immediate corollary we mention the following version of
+[Kus22, Corollary 1.5], see also [For20, Corollary 6.26].
+Corollary 0.3. The universal cover of a smooth (sub)elliptic variety
+is an elliptic algebraic variety.
+0.2. Examples of elliptic varieties. Recall that a variety X of di-
+mension ≥ 2 is flexible if through any smooth point x ∈ X pass one-
+dimensional orbits of Ga-actions on X whose velocity vectors spend
+the tangent space TxX, see [AFKKZ13]. It is known that every smooth
+flexible variety is elliptic, see [Gro89, 0.5.B] and [For17, Proposition
+5.6.22(C)]; cf. also [AFKKZ13, Appendix]. There are numerous ex-
+amples of flexible varieties; see e.g. [AFKKZ13] and survey articles
+[CPPZ21] and [Arz22].
+The complement of a closed subset Y of codimension ≥ 2 in a flexible
+smooth quasi-affine variety X of dimension ≥ 2 is again flexible, see
+[FKZ16, Theorem 1.1]. In particular, X \ Y is elliptic.
+Recall that an algebraic variety X of dimension n belongs to class
+A0 if it can be covered by a finite number of copies of An. It belongs
+to class A if it is the complement of a closed subvariety of codimension
+at least 2 in a variety of class A0. Any variety of class A is subelliptic,
+see [For17, Proposition 6.4.5]. By Theorem 0.1 it is elliptic.
+For example, the Grassmann manifold X = G(k, n) of k-dimensional
+subspaces in An belongs to class A0. Hence X \ Y is elliptic for any
+closed subset Y ⊆ X of codimention ≥ 2, see [For17, Corollary 5.6.18(D)].
+This answers a question in [ibid].
+Furthermore, a variety of class A blown up along a smooth closed
+subvariety is subelliptic. The same concerns a smooth locally stably
+flexible variety, see [LT17], [KKT18] and [For17, Theorem 6.4.8]. By
+Theorem 0.1 all these varieties are elliptic. See also [Gro89, Sec. 3.4(F)]
+for further potential examples.
+0.3. Ellipticity versus rationality. By definition, any smooth (sub)elliptic
+variety is dominated by an affine space, hence is unirational and ratio-
+nally connected. Notice, however, that there are examples of flexible
+and so, elliptic smooth affine varieties that are not stably rational, see
+[Pop11, Example 1.22]. Gromov asked in [Gro89] whether any ratio-
+nal projective variety is elliptic. The same question can be asked for
+unirational projective varieties, see [BKK13].
+1. Composing and extending sprays
+In this section X stands for a smooth algebraic variety. We develop
+here several technical tools for the proof of Theorem 0.1.
+
+4
+SHULIM KALIMAN AND MIKHAIL ZAIDENBERG
+1.1. Composition of sprays. The composition (E1∗E2, p1∗p2, s1∗s2)
+of two given sprays (E1, p1, s1) and (E2, p2, s2) over X is defined via1
+E1 ∗ E2 = {(e1, e2) ∈ E1 × E2 | e1 ∈ p−1(X), s1(e1) = p2(e2)} = s∗
+1(E2),
+p1 ∗ p2(e1, e2) = p1(e1),
+s1 ∗ s2(e1, e2) = s2(e2),
+see [Gro89, 1.3.B]. One considers also the iterated composition
+(E, p, s) = (E1 ∗ . . . ∗ Em, p1 ∗ . . . ∗ pm, s1 ∗ . . . ∗ sm)
+of a sequence of m sprays (Ei, pi, si) over X, see [ibid] or [For17, Def-
+inition 6.3.5]. Clearly, p: E → X is a fiber bundle whose general fiber
+is isomorphic to an affine space AN viewed as a variety. In general,
+p: E → X does not admit a vector bundle structure. However, it ad-
+mits a natural section σ: X → E (which plays a role of zero section)
+such that s|σ(X) = p|σ(X), see [For17, Definition 6.3.5].
+If pi : Ei → X are trivial vector bundles for all i, then the composition
+(E, p, s) is a spray over X with a trivial vector bundle p: E → X, see
+[For17, Lemma 6.3.1]. Furthermore, we have the following fact.
+Proposition 1.1. Consider the composition (E, p, s) = (E1 ∗ E2, p1 ∗
+p2, s1 ∗ s2) of sprays (E1, p1, s1) and (E2, p2, s2) over X. If (E2, p2, s2)
+is of rank 1, then (E, p, s) is a spray over X.
+Proof. By the preceding it suffices to show that p: E → X is a vector
+bundle. The pullback via the projection p1 : E1 → X yields an isomor-
+phism Pic(X) ∼= Pic(E1), see [Mag75, Theorem 5]. Therefore, the line
+bundle s∗
+1(E2) → E1 is isomorphic to p∗
+1(L1) for a line bundle L1 → X.
+Choose an affine open cover {Ui} of X which is trivializing simulta-
+neously for p1: E1 → X and for L1 → X and consider the cylinders
+Yi := p−1
+1 (Ui) ≃ Ui × Ar and p∗
+1(L|Ui) ≃ Ui × Ar × A1. Up to these triv-
+ializations the composition p∗
+1(L1|Ui) → Ui coincides with the standard
+projection pr1: Ui × Ar+1 → Ui. The transition function ϕi,j between
+p∗
+1(L1|Ui) → Ui and p∗
+1(L1|Uj) → Uj over Ui ∩ Uj is of the form
+ϕi,j : (x, v, t) �→ (x, fi,j(x, v), gi,j(x, v)t),
+(x, v, t) ∈ (Ui ∩Uj)×Ar ×A1
+where
+fi,j ∈ GL(r, O(Ui ∩ Uj))
+and
+gi,j ∈ O∗(Yi ∩ Yj) = p∗
+1(O∗(Ui ∩ Uj)),
+that is, the functions gi,j(x, v) = gi,j(x) do not depend on v ∈ Ar.
+Finally ϕi,j ∈ GL(r+1, O(Ui∩Uj)), which proves that p = p1∗p2 : E →
+X is a vector bundle.
+□
+1Abusing notation, we do not distinguish between a vector bundle and its total
+space.
+
+GROMOV ELLIPTICITY AND SUBELLIPTICITY
+5
+Corollary 1.2. The composition (E, p, s) of r rank 1 sprays (Li, qi, si)
+over X is a spray of rank r over X. If the family (Li, qi, si) is domi-
+nating, then also the spray (E, p, s) is.
+Proof. The first assertion is immediate from Proposition 1.1; see [For17,
+Lemma 6.3.6] for the second.
+□
+1.2. Extension lemma. The following lemma and its proof follow
+closely Gromov’s Localization Lemma and its proof, see [Gro89, 3.5.B].
+Lemma 1.3. Let U ⊆ X be a dense open subset and (E, p, s) be a local
+spray on U with values in X dominating at a point x ∈ U. Then there
+exists a spray ( ˜E, ˜p, ˜s) on X dominating at x.
+Proof. Shrinking U we may suppose that
+• U = X \ supp(D) where D is a reduced effective divisor on X,
+and
+• p: E = U × Ar → U is a trivial vector bundle on U.
+We extend p: E → U to a trivial vector bundle p′: E′ = X × Ar → X
+on X. Let ˜p: ˜E = E′ ⊗ OX(−nD) → X where n ∈ N. Thus, ˜E is a
+direct sum of r samples of the line bundle OX(−nD). We have
+Hom(OX(−nD), OX) = Hom(OX, OX(nD)) = H0(X, OX(nD)).
+Pick a section σ ∈ H0(X, OX(D)) with div(σ) = D. Then σn defines a
+homomorphism of line bundles OX(−nD) → OX identical on X which
+vanishes to order n on ˜p−1(supp(D)) and restricts to an isomorphism
+over U. The direct sum of these homomorphisms yields a homomor-
+phism of vector bundles ψ: ˜E → E′ identical on X with similar prop-
+erties. Extend s to a rational map s′: E′ ��� X. Then for all n ≫ 1
+the composition ˜s = s′ ◦ ψ: ˜E → X is a morphism such that s| ˜Z = ˜p| ˜Z
+where ˜Z is the zero section of ˜p: ˜E → X. The resulting spray ( ˜E, ˜p, ˜s)
+on X is clearly dominating at x.
+□
+The following corollary is immediate. It is implicitly present in the
+proof of the Localization Lemma in [Gro89, 3.5.B], cf. also [Gro89,
+3.5.B′] and [For17, Proposition 6.4.2].
+Corollary 1.4. If X is locally elliptic, then it is subelliptic.
+Next we deduce the following useful fact.
+Lemma 1.5. Any subelliptic smooth variety X admits a dominating
+family of sprays of rank 1.
+Proof. Let (Ei, pi, si) be a dominating family of sprays over X. For a
+point x ∈ X we can find a neighborhood U such that pi|U : Ei|U → U
+
+6
+SHULIM KALIMAN AND MIKHAIL ZAIDENBERG
+is a trivial bundle of rank, say ri. Choose a decomposition Ei|U =
+�ri
+j=1 Li,j where pi,j = pi|Li,j : Li,j → U are trivial line bundles. Let-
+ting si,j = s|Li,j : Li,j → X yields a family of local rank 1 sprays
+(Li,j, pi,j, si,j)j=1,...,ri on U. It is easily seen that this family is domi-
+nating at x. Shrinking U and proceeding in the same way as in the
+proof of Lemma 1.3 we extend the latter family to a family of rank 1
+sprays (˜Li,j, ˜pi,j, ˜si,j)j=1,...,ri on the whole X which is still dominating
+at x. In this way we can construct a family (˜Li,j, ˜pi,j, ˜si,j)i,j which is
+dominating at each point of X.
+□
+1.3. Proof of the main theorem. In view of Corollary 1.4 the follow-
+ing proposition ends the proof of Theorem 0.1 from the Introduction.
+Proposition 1.6. Let X be a smooth algebraic variety. If X is subel-
+liptic, then it is elliptic.
+Proof. By Corollary 1.2 it suffices to construct a dominating family of
+rank 1 sprays on X. However, Lemma 1.5 provides such a family.
+□
+References
+[Arz22] I. Arzhantsev, Automorphisms of algebraic varieties and infinite transitiv-
+ity, arXiv:2212.13616 (2022).
+[AFKKZ13] I. V. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, and
+M. Zaidenberg, Flexible varieties and automorphism groups, Duke Math. J. 162:4
+(2013), 767–823.
+[BKK13] F. Bogomolov, I. Karzhemanov, and K. Kuyumzhiyan, Unirationality and
+existence of infinitely transitive models. In: Birational Geometry, Rational Curves,
+and Arithmetic, pp. 77–92. Springer, 2013.
+[CPPZ21] I. Cheltsov, J. Park, Yu. Prokhorov, and M. Zaidenberg, Cylinders in
+Fano varieties. EMS Surv. Math. Sci. 8, 39–105 (2021).
+[FKZ16] H. Flenner, S. Kaliman, and M. Zaidenberg, A Gromov-Winkelmann type
+theorem for flexible varieties. J. Eur. Math. Soc. (JEMS) 18:11 (2016), 2483–2510.
+[For02] F. Forstneric, The Oka principle for sections of subelliptic submersions.
+Math. Z. 241:3 (2002), 527–551.
+[For17] F. Forstneric, Stein manifolds and holomorphic mappings. The homotopy
+principle in complex analysis. Second edition. Springer-Verlag, Berlin-Heidelberg,
+2017.
+[For20] F. Forstneric, Recent developments on Oka manifolds. arXiv:2006.07888
+(2020).
+[Gro89] M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles,
+J. Amer. Math. Soc. 2 (1989), 851–897. MR1001851. DOI10.2307/1990897.
+[KKT18] S. Kaliman, F. Kutzschebauch, and T. T. Truong, On subelliptic mani-
+folds. Israel J. Math. 228:1 (2018), 229–247.
+[Kus20] Y. Kusakabe. Oka complements of countable sets and nonelliptic Oka man-
+ifolds. Proc. Amer. Math. Soc. 148:3 (2020), 1233–1238.
+
+GROMOV ELLIPTICITY AND SUBELLIPTICITY
+7
+[Kus22] Y.
+Kusakabe.
+On
+the
+fundamental
+groups
+of
+subelliptic
+varieties.
+arXiv:2212.07085 (2022).
+[LT17] F. L´arusson and T. T. Truong, Algebraic subellipticity and dominability of
+blow-ups of affine spaces. Documenta Mathematica 22 (2017), 151–163.
+[Mag75] A. Magid, The Picard sequence of a fibration. Proc. Amer. Maths. Soc.
+53:1 (1975), 37–40.
+[Pop11] V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite auto-
+morphism groups of algebraic varieties. In: Affine algebraic geometry, 289–311.
+CRM Proc. Lecture Notes 54, Amer. Math. Soc., Providence, RI, 2011. Zbl
+1242.14044, MR 2768646.
+University of Miami, Department of Mathematics, Coral Gables, FL
+33124, USA
+Email address: [email protected]
+Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France
+Email address: [email protected]
+

19E1T4oBgHgl3EQfRwPe/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf,len=362
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='03058v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='AG]  8 Jan 2023  GROMOV ELLIPTICITY AND SUBELLIPTICITY  SHULIM KALIMAN AND MIKHAIL ZAIDENBERG  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' We establish the equivalence of Gromov ellipticity and  subellipticity in the algebraic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Contents  Introduction  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Ellipticity versus subellipticity  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Examples of elliptic varieties  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Ellipticity versus rationality  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Composing and extending sprays  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Composition of sprays  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Extension lemma  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Proof of the main theorem  6  References  6  Introduction  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Ellipticity versus subellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' We are working over an alge-  braically closed field K of characteristic zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' and An = An  K stands for  the affine n-space over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' All varieties and vector bundles in this note  are algebraic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' a variety is always reduced and irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Abusing the  language, in this paper the terms spray and (sub)ellipticity stand for  algebraic spray resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' algebraic (sub)ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  The notion of ellipticity was introduced, both in analytic and al-  gebraic setting, in [Gro89, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' In the Localization  Lemma (see [Gro89, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='B]) Gromov actually considers a local  version of ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The subellipticity (implicitly present in [Gro89,  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5]) was introduced in [For02];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' see [For17, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1] and [For20]  for a historical account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Recall these notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  A spray of rank r over a smooth algebraic variety X is a triple  (E, p, s) consisting of a vector bundle p: E → X of rank r and a  Date: 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  1  2  SHULIM KALIMAN AND MIKHAIL ZAIDENBERG  morphism s: E → X such that s|Z = p|Z where Z ⊆ E stands for  the zero section of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' This spray is dominating at x ∈ X if the restric-  tion s|Ex : Ex → X to the fiber Ex = p−1(x) is dominant at the origin  0x = Z ∩ Ex of the vector space Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The variety X is called elliptic if  it admits a spray (E, p, s) which is dominating at each point x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  A local spray (E, p, s) with values in X at a point x ∈ X consists of  a vector bundle p: E → U over a neighborhood U of x and a morphism  s: E → X such that s|ZU = p|ZU where ZU stands for the zero section  of p: E → U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' One says that X is locally elliptic if for any x ∈ X there  is a local spray in a neighborhood of x dominating at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The variety  X is called subelliptic if it admits a family of sprays (Ei, pi, si) defined  over the whole X which is dominating at each point x ∈ X, that is,  TxX =  n  �  i=1  dsi(T0i,xEi,x)  ∀x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Thus, an elliptic variety is both locally elliptic and subelliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' In this  note we establish the converse implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' For a smooth algebraic variety X the following are  equivalent:  X is elliptic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  X is subelliptic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  X is locally elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1 answers a question in [For17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' 230].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The equivalence  of the first two properties is well known for homogeneous spaces of  a linear algebraic group, see [For20, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' It is unknown  however whether this equivalence also holds in the analytic category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' [Kus20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Comparing Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1 with [LT17, Theorem 1] and [For20, Corol-  lary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='6] we deduce the following results, see [LT17] and [For20] for the  terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Corollary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' For a smooth algebraic manifold X the following con-  ditions are equivalent:  (a) X is elliptic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  (b) X is locally elliptic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  (c) X is subelliptic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  (d) X satisfies Gromov’s condition aEll1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  (e) X satisfies the algebraic homotopy Runge principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  See [For20, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2] for relations of (d) and (e) to approximation,  interpolation and Oka-Grauert h-Principle in the algebraic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  GROMOV ELLIPTICITY AND SUBELLIPTICITY  3  As another immediate corollary we mention the following version of  [Kus22, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5], see also [For20, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Corollary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The universal cover of a smooth (sub)elliptic variety  is an elliptic algebraic variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Examples of elliptic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Recall that a variety X of di-  mension ≥ 2 is flexible if through any smooth point x ∈ X pass one-  dimensional orbits of Ga-actions on X whose velocity vectors spend  the tangent space TxX, see [AFKKZ13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' It is known that every smooth  flexible variety is elliptic, see [Gro89, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='B] and [For17, Proposition  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='22(C)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' also [AFKKZ13, Appendix].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' There are numerous ex-  amples of flexible varieties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' [AFKKZ13] and survey articles  [CPPZ21] and [Arz22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  The complement of a closed subset Y of codimension ≥ 2 in a flexible  smooth quasi-affine variety X of dimension ≥ 2 is again flexible, see  [FKZ16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' In particular, X \\ Y is elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Recall that an algebraic variety X of dimension n belongs to class  A0 if it can be covered by a finite number of copies of An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' It belongs  to class A if it is the complement of a closed subvariety of codimension  at least 2 in a variety of class A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Any variety of class A is subelliptic,  see [For17, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' By Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1 it is elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  For example, the Grassmann manifold X = G(k, n) of k-dimensional  subspaces in An belongs to class A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Hence X \\ Y is elliptic for any  closed subset Y ⊆ X of codimention ≥ 2, see [For17, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='18(D)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  This answers a question in [ibid].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Furthermore, a variety of class A blown up along a smooth closed  subvariety is subelliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The same concerns a smooth locally stably  flexible variety, see [LT17], [KKT18] and [For17, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' By  Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1 all these varieties are elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' See also [Gro89, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='4(F)]  for further potential examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Ellipticity versus rationality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' By definition, any smooth (sub)elliptic  variety is dominated by an affine space, hence is unirational and ratio-  nally connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Notice, however, that there are examples of flexible  and so, elliptic smooth affine varieties that are not stably rational, see  [Pop11, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Gromov asked in [Gro89] whether any ratio-  nal projective variety is elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The same question can be asked for  unirational projective varieties, see [BKK13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Composing and extending sprays  In this section X stands for a smooth algebraic variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' We develop  here several technical tools for the proof of Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  4  SHULIM KALIMAN AND MIKHAIL ZAIDENBERG  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Composition of sprays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The composition (E1∗E2, p1∗p2, s1∗s2)  of two given sprays (E1, p1, s1) and (E2, p2, s2) over X is defined via1  E1 ∗ E2 = {(e1, e2) ∈ E1 × E2 | e1 ∈ p−1(X), s1(e1) = p2(e2)} = s∗  1(E2),  p1 ∗ p2(e1, e2) = p1(e1),  s1 ∗ s2(e1, e2) = s2(e2),  see [Gro89, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' One considers also the iterated composition  (E, p, s) = (E1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' ∗ Em, p1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' ∗ pm, s1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' ∗ sm)  of a sequence of m sprays (Ei, pi, si) over X, see [ibid] or [For17, Def-  inition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Clearly, p: E → X is a fiber bundle whose general fiber  is isomorphic to an affine space AN viewed as a variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' In general,  p: E → X does not admit a vector bundle structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' However, it ad-  mits a natural section σ: X → E (which plays a role of zero section)  such that s|σ(X) = p|σ(X), see [For17, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  If pi : Ei → X are trivial vector bundles for all i, then the composition  (E, p, s) is a spray over X with a trivial vector bundle p: E → X, see  [For17, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Furthermore, we have the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Consider the composition (E, p, s) = (E1 ∗ E2, p1 ∗  p2, s1 ∗ s2) of sprays (E1, p1, s1) and (E2, p2, s2) over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' If (E2, p2, s2)  is of rank 1, then (E, p, s) is a spray over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' By the preceding it suffices to show that p: E → X is a vector  bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The pullback via the projection p1 : E1 → X yields an isomor-  phism Pic(X) ∼= Pic(E1), see [Mag75, Theorem 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Therefore, the line  bundle s∗  1(E2) → E1 is isomorphic to p∗  1(L1) for a line bundle L1 → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Choose an affine open cover {Ui} of X which is trivializing simulta-  neously for p1: E1 → X and for L1 → X and consider the cylinders  Yi := p−1  1 (Ui) ≃ Ui × Ar and p∗  1(L|Ui) ≃ Ui × Ar × A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Up to these triv-  ializations the composition p∗  1(L1|Ui) → Ui coincides with the standard  projection pr1: Ui × Ar+1 → Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The transition function ϕi,j between  p∗  1(L1|Ui) → Ui and p∗  1(L1|Uj) → Uj over Ui ∩ Uj is of the form  ϕi,j : (x, v, t) �→ (x, fi,j(x, v), gi,j(x, v)t),  (x, v, t) ∈ (Ui ∩Uj)×Ar ×A1  where  fi,j ∈ GL(r, O(Ui ∩ Uj))  and  gi,j ∈ O∗(Yi ∩ Yj) = p∗  1(O∗(Ui ∩ Uj)),  that is, the functions gi,j(x, v) = gi,j(x) do not depend on v ∈ Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Finally ϕi,j ∈ GL(r+1, O(Ui∩Uj)), which proves that p = p1∗p2 : E →  X is a vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  □  1Abusing notation, we do not distinguish between a vector bundle and its total  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  GROMOV ELLIPTICITY AND SUBELLIPTICITY  5  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The composition (E, p, s) of r rank 1 sprays (Li, qi, si)  over X is a spray of rank r over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' If the family (Li, qi, si) is domi-  nating, then also the spray (E, p, s) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The first assertion is immediate from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' see [For17,  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='6] for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  □  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Extension lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The following lemma and its proof follow  closely Gromov’s Localization Lemma and its proof, see [Gro89, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Let U ⊆ X be a dense open subset and (E, p, s) be a local  spray on U with values in X dominating at a point x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Then there  exists a spray ( ˜E, ˜p, ˜s) on X dominating at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Shrinking U we may suppose that  U = X \\ supp(D) where D is a reduced effective divisor on X,  and  p: E = U × Ar → U is a trivial vector bundle on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  We extend p: E → U to a trivial vector bundle p′: E′ = X × Ar → X  on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Let ˜p: ˜E = E′ ⊗ OX(−nD) → X where n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Thus, ˜E is a  direct sum of r samples of the line bundle OX(−nD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' We have  Hom(OX(−nD), OX) = Hom(OX, OX(nD)) = H0(X, OX(nD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Pick a section σ ∈ H0(X, OX(D)) with div(σ) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Then σn defines a  homomorphism of line bundles OX(−nD) → OX identical on X which  vanishes to order n on ˜p−1(supp(D)) and restricts to an isomorphism  over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The direct sum of these homomorphisms yields a homomor-  phism of vector bundles ψ: ˜E → E′ identical on X with similar prop-  erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Extend s to a rational map s′: E′ ��� X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Then for all n ≫ 1  the composition ˜s = s′ ◦ ψ: ˜E → X is a morphism such that s| ˜Z = ˜p| ˜Z  where ˜Z is the zero section of ˜p: ˜E → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' The resulting spray ( ˜E, ˜p, ˜s)  on X is clearly dominating at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  □  The following corollary is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' It is implicitly present in the  proof of the Localization Lemma in [Gro89, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='B], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' also [Gro89,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='B′] and [For17, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' If X is locally elliptic, then it is subelliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Next we deduce the following useful fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Any subelliptic smooth variety X admits a dominating  family of sprays of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Let (Ei, pi, si) be a dominating family of sprays over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' For a  point x ∈ X we can find a neighborhood U such that pi|U : Ei|U → U  6  SHULIM KALIMAN AND MIKHAIL ZAIDENBERG  is a trivial bundle of rank, say ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Choose a decomposition Ei|U =  �ri  j=1 Li,j where pi,j = pi|Li,j : Li,j → U are trivial line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Let-  ting si,j = s|Li,j : Li,j → X yields a family of local rank 1 sprays  (Li,j, pi,j, si,j)j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=',ri on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' It is easily seen that this family is domi-  nating at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Shrinking U and proceeding in the same way as in the  proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3 we extend the latter family to a family of rank 1  sprays (˜Li,j, ˜pi,j, ˜si,j)j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=',ri on the whole X which is still dominating  at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' In this way we can construct a family (˜Li,j, ˜pi,j, ˜si,j)i,j which is  dominating at each point of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  □  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' In view of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='4 the follow-  ing proposition ends the proof of Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='1 from the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Let X be a smooth algebraic variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' If X is subel-  liptic, then it is elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='2 it suffices to construct a dominating family of  rank 1 sprays on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' However, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='5 provides such a family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  □  References  [Arz22] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Arzhantsev, Automorphisms of algebraic varieties and infinite transitiv-  ity, arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='13616 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  [AFKKZ13] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Arzhantsev, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Flenner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Kaliman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Kutzschebauch, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Zaidenberg, Flexible varieties and automorphism groups, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' 162:4  (2013), 767–823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  [BKK13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Bogomolov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Karzhemanov, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Kuyumzhiyan, Unirationality and  existence of infinitely transitive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' In: Birational Geometry, Rational Curves,  and Arithmetic, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' 77–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content=' Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
+page_content='  [CPPZ21] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQfRwPe/content/2301.03058v1.pdf'}
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39FQT4oBgHgl3EQfHTWW/content/tmp_files/2301.13248v1.pdf.txt

@@ -0,0 +1,1071 @@
+A	platform	for	trapped	cryogenic	electrons,	anions	and		
+cations	for	fundamental	physics	and	chemical	studies	
+ 
+L.	O.	A.	Azevedo1,	R.	J.	S.	Costa1,	W.	Wolff1,	A.	N.	Oliveira2,	R.L.	Sacramento1,	D.M.	Silveira1,	C.	L.	Cesar1*		
+		
+1Instituto	de	Física,	Universidade	Federal	do	Rio	de	Janeiro,	Rio	de	Janeiro	21941–909,	Brazil	
+2INMETRO,	Caxias,	219999,	Brazil		
+	
+Abstract:		
+Cryogenic	cations,	electrons	and	anions	are	ubiquitous	in	space,	participate	on	star	
+formation	chemistry	and	are	relevant	to	studies	on	the	origin	of	molecular	biology	
+homochirality.	We	report	on	a	system	to	directly	generate	and	trap	these	species	in	
+the	laboratory.	Laser	ablation	of	a	solid	target	(LiH)	facing	a	sublimating	Ne	matrix	
+generates	cryogenic	electrons,	anions,	and	cations.	Axial	energy	distributions	(of	𝐞!,	
+𝐇±	and	𝐋𝐢±)	peaked	at	0–25	meV	are	obtained	in	a	Penning	trap	at	90	mT	and	0.5	eV	
+barrier.	Anions	can	be	guided	and	turned	neutral	with	low	recoil	energy	by	near–
+threshold	 photodetachment.	 An	 immediate	 prospect	 for	 this	𝐇! 	source	 is	 to	 load	
+hydrogen	 atoms	 into	 the	 ALPHA	 antihydrogen	 trap	 at	 CERN	 towards	 direct	
+spectroscopic	comparison	of	both	conjugated	species	beyond	13	significant	figures.	
+The	production	is	scalable	and	adaptable	to	different	species	including	deuterium	and	
+tritium,	relevant	for	neutrino	mass	and	fusion	research.	
+	
+Introduction	
+	
+Sources	 and	 traps	 of	 cold	 negative	 and	 positive	 species	 are	 needed	 to	 study	 the	 low-
+temperature	species	themselves	and	the	reactions	between	trapped	charged	particles	and	
+neutral	species1.	Crucial	requirements	on	both,	sources	and	traps,	represent	a	significant	
+challenge	due	to	specific	limitations	and	conditions	on	a	variety	of	setups.		Here,	we	report	
+source	and	trap	with	innovative	methods	with	an	integrated	mass	discriminator,	describing	
+the	elements	and	quantitatively	assessing	their	performance.	Many	of	the	innovations	and	
+findings	are	of	general	interest,	as	briefly	presented	for	diverse	applications.	
+	
+Measurements	 of	 the	 1S–2S	 transition	 frequency	 in	 antihydrogen	 ( H& )	 by	 the	 ALPHA	
+collaboration	at	12	significant	figures2,3	entered	uncharted	territory	in	the	comparison	of	
+charge	conjugated	species,	a	test	of	the	Charge–Parity–Time	(CPT)	symmetry.	We	foresee	
+the	 need	 to	 perform	 laser	 spectroscopy	 of	 H	 in	 the	 same	 trap4	 as	H&	to	 achieve	 aimed	
+precisions	 of	 15	 significant	 figures	 in	 search	 of	 explanations	 for	 the	 matter–antimatter	
+asymmetry	in	the	Universe.	In	the	same	trap	and	reference	frame,	both	species	could	be	
+studied	under	the	same	conditions	enabling	better	control	over	systematic	effects,	such	as	
+trap	magnetic	fields	and	laser	power	causing	AC	Stark	shift	–.	The	H&	research	program	also	
+involves	 probing	 the	 gravitational	 acceleration5,6.	 Antihydrogen	 is	 produced	 –	 from	 its	
+constituents,	antiproton	and	positron	–	and	studied	in	challenging	conditions,	such	as	an	
+exquisite	ultra–high–vacuum	(UHV)	environment	to	avoid	annihilation	to	background	gas.	
+
+Thus,	it	is	not	always	possible	to	readily	use	normal	matter	techniques	to	load	the	antimatter	
+trap	with	matter	species.	For	example,	H	has	been	trapped	and	subjected	to	high–precision	
+laser	spectroscopy	at	MIT7	and	Amsterdan8	in	a	trap	that	required	a	superfluid	liquid	helium	
+covered	 cell,	 conditions	 incompatible	 with	 the	 H&	 trap	 at	 CERN.	 Therefore,	 innovative	
+techniques	are	required	for	loading	H	in	the	H&	trap	and	the	developments	presented	here	
+can	be	readily	adapted	as	a	solution.		
+	
+The	samples	of	H!	produced	by	our	system	are	within	the	temperature	of	the	antiproton	and	
+positron	samples	used	by	the	ALPHA	experiment	for	H&		synthesis.	The	H!	can	be	produced	
+and	trapped	adjacent	to	the	main	apparatus,	and	after	the	UHV	condition	is	regained,	the	
+anions	will	be	guided	into	the	composed,	Penning	and	magnetic,	ALPHA	trap.	They	can	be	
+further	cooled	by	evaporative	cooling	by	themselves	or	with	pre–cooled	electrons.	Then,	a	
+laser	pulse,	with	photon	energy	near	the	photodetachment	threshold9	of	H!(0.754	eV),	will	
+neutralize	the	anions	imparting	low	recoil	energy.	For	example,	a	laser	at	1575	nm	will	leave	
+0.2	K	of	recoil	energy	–	less	than	the	typical	temperature	or	energy	dispersion	of	the	ion	
+sample	–	to	the	neutral	H.	The	fraction	of	resulting	atoms	with	energy	below	0.5	K	will	remain	
+trapped	in	the	superposed	magnetic	trap.	
+	
+Moreover,	the	developments	with	H&	research	have	spurred	a	renewed	interest	in	hydrogen	
+trapping	and	spectroscopy	by	many	groups.	New	techniques	to	produce	cold	hydrogen10-13	
+are	 being	 investigated	 with	 interests	 ranging	 from	 wavefunction	 in	 gravity	 quantum	
+reflection14,	to	scientific	metrology	–	in	tests	of	Quantum	Electrodynamics,	proton	radius	
+puzzle,	and	search	for	variation	of	fundamental	constants	–	and	to	produce	larger	Bose–
+Einstein	condensates.	The	system	presented	offers	an	alternative,	study	model,	or	a	proof–
+of–principle	for	some	of	these	studies.	For	example,	the	generation	of	cold	H	from	H!	is	the	
+charge	conjugated	process	from	one	proposed	experiment15,16	for	measuring	gravity	with	H&.		
+	
+The	present	demonstration,	starting	from	laser	ablation	of	LiH	and	generating	H!,	Li!	and	
+Li#H$
+! 	does	not	show,	a	priori,	a	specificity	that	would	prevent	it	from	being	applicable	to	
+other	simple	species,	such	as	D!	and	T!.		An	example	of	such	a	specificity	is	the	case	of	the	
+MIT	and	Amsterdan	H	traps	that	could	not	trap17	D	because	of	its	slightly	higher	binding	
+energy	to	liquid	helium.	A	proposal	for	neutrino	mass	measurement18,19	requires	quasi–
+trapped	tritium	for	the	experiment	to	be	performed	in	a	magnetic	field.		Many	low–energy	
+atomic	 processes	 involving	 deuterium	 are	 relevant	 in	 fusion	 research20	 for	 a	 copious	
+production	of	D!.	The	source	here	described,	adapted	for	T!	and	D!,	might	find	use	for	these	
+studies.	
+	
+Lastly,	the	interaction	of	low	energy	ions	and	electrons	with	neutral	atoms	and	molecules	is	
+important	 from	 astrophysics	 to	 the	 origin	 of	 biological	 molecules21-23.	 It	 would	 thus	 be	
+desirable	to	have	a	platform	to	generate	various	cryogenic	species	in	a	direct	way.	Producing	
+
+and	trapping	slow	anions	is	not	trivial.	Stray	or	contact	electric	fields	can	easily	generate	
+thousands	of	kelvins,	since	an	energy	of	only	26	meV	is	equivalent	to	300	K	in	temperature	
+scale.	Difference	of	work	functions	in	metals	can	reach	1	V.	Patch	or	domain	potentials	within	
+an	electrode24	can	reach	250	mV.	Anions	are	typically	created	at	energies25	of	many	eV	or	
+keV.	Various	methods	have	been	employed	for	further	cooling	ionized	species:	–	entrainment	
+in	buffer	gas26,27;	–	resistive	cooling28;	–	direct	sympathetically	cooling29	and	LC	mediated	
+sympathetic	cooling30;	–	direct	laser	cooling	for	species,	such	as	cations31	and	proposed	for	
+molecular	anions32,	with	nearly	closed	optical	transitions.		
+	
+At	the	ALPHA	experiment,	where	the	H&	trapping	rate	depends	strongly	on	a	low	positron	
+cloud	temperature,	the	final	cooling	to	~20	K	relies	on	evaporative	cooling33.	In	summary,	
+there	are	many	different	techniques	being	explored	and	developed	to	produce	cold	charged	
+samples	due	to	their	scientific	relevance.	Here,	we	report	on	a	platform	to	directly	generate:	
+(i)	 electrons	 with	 a	 perspective	 for	 a	 cryogenic	 polarized	 electron	 beam;	 (ii)	 simple	
+molecular	ions	at	temperatures	compatible	with	interstellar	media	chemistry;	(iii)	hydrogen	
+anions	suitable	for	antihydrogen	research	in	a	method	adaptable	to	deuterium	and	tritium	
+research.	It	may	also	serve	as	an	attractive	initial	step	towards	samples	in	the	cold	and	
+ultracold	regime.	
+ 
+Experimental	Setup	and	Procedures	
+	
+The	 method	 is	 based	 on	 the	 Matrix	 Isolation	 Sublimation	 (MISu)	 technique34–37	 and	
+variations.	In	MISu,	beams	of	atoms	and	molecules	can	be	produced	either	in	a	fast	pulse	(in	
+~10–100	 µs)	 or	 in	 a	 quasi–continuous	 beam	 depending	 on	 the	 sublimation	 regime.	 The	
+implantation	of	different	atoms	into	the	rare–gas	matrix,	typically	via	laser	ablation	of	solid	
+precursor	such	as	pellets	of	LiH,	Ca,	Cr,	graphite,	can	lead	to	the	formation	of	molecules	
+within	the	matrix.	As	previously	reported37,	we	produced	LinCam	molecules	from	alternating	
+laser	ablation	of	a	Li	and	a	Ca	pellet.	Since	electrons	should	not	attach	to	the	Ne	atoms,	it	was	
+initially	thought	that	implanting	electrons	together	with	atoms	and	molecules	would	lead	to	
+the	formation	of	different	anionic	species	in	the	matrix.	As	a	faster	variation,	we	found	that	
+the	 laser	 ablated	 species	 impinging	 onto	 the	 Ne	 matrix	 already	 generate	 low–energy	
+electrons	and	atomic	or	molecular	ions.	
+		
+The	sapphire	substrate,	upon	which	we	deposit	the	neon	matrix,	is	coated	on	both	sides	with	
+NiCr	resistive	film.	The	back	film	is	used	as	a	resistor	to	apply	a	sublimating	heat	pulse	while	
+the	front	film	serves	as	a	mirror	and	provides	a	way	to	bias	the	sapphire	with	a	positive	or	
+negative	voltage.	This	bias	can	be	used	to	repel	or	attract	the	species	to	the	matrix	during	the	
+ablation	process	and/or	during	the	sublimation	phase.		
+	
+
+In	figure	1	we	present	the	basic	setup	used	in	this	study.	A	two–stage	closed	cycle	cryostat	
+with	1	W	of	cooling	power	at	4	K	is	used.	An	UHV	environment	can	be	achieved	below	6.9	K,	
+where	the	saturated	vapor	pressure	of	Ne	is	predicted	to	be	~7.5x10-10	Torr	and	falls	12	
+orders	of	magnitude	at	4	K	if	one	had	a	vacuum	tight	enclosure.	This	UHV	condition	is	not	
+met	during	the	matrix	growth	or	sublimation,	and	it	takes	milliseconds	for	the	Ne	gas	to	
+cryopump	 back	 to	 the	 walls.	 	 Due	 to	 a	 poor	 thermal	 link	 in	 the	 present	 setup,	 the	 trap	
+electrodes	region	only	reaches	~6.5	K	while	the	main	chamber	is	at	~4	K.	The	sapphire	
+substrate	is	thermally	anchored	to	the	“4K	plate”	through	a	designed	thermal	link	that	allows	
+it	to	reach	3.2	K	and	be	heated	for	the	sublimation	process	without	warming	the	whole	
+experimental	cell.	The	Ne	gas	delivery	tube	is	brought	into	the	cryostat	with	a	first	thermal	
+link	at	the	1st	stage	(40	K	nominal)	and	its	exit	is	above	16	K,	to	avoid	plugging,	and	directed	
+towards	the	sapphire.	The	laser	reflections	on	the	interfaces,	vacuum–matrix	and	matrix–
+sapphire,	cause	interference	fringes	that	allow	monitoring	the	matrix	thickness	down	to	the	
+atomic	monolayer	as	well	as	performing	Doppler–sensitive	spectroscopy	of	neutral	Li	atoms	
+as	they	sublimate	from	the	matrix	into	vacuum.	
+
+	
+Figure 1. a) A diagram of the basic setup is shown. The sapphire (“Sapph”) where the Ne matrix is deposited, as the gas is delivered 
+through the tube, and the inner chamber are thermally anchored at the 2nd stage 4 K cryohead surrounded by a “40K” black–
+body shield. The spectroscopy laser monitors the matrix thickness and the Li atoms absorption. A pulsed laser at 532 or 1064 nm 
+(in dashed green) promotes ablation from a LiH pellet imparting atoms, molecules, electrons, and ions onto the Ne matrix. Two 
+magnets, “Mag.Sap.” and “Trap coil”, for guiding (dotted blue arrows) and trapping are placed around the sapphire and the trap 
+region. An aperture avoids excess Ne gas towards the cell bottom and trap region.  The Penning–Malmberg trap uses six ring 
+electrodes (E0–E5), glued externally to a grounded copper tube that reaches 6.5 K, and is followed by a channeltron CEM detector. 
+The CEM is loosely thermally anchored to the “40K” stage. b) Axial profiles of the magnetic field (red line) and a configuration, 
+for cations, of the potentials for a dump at E4 (black solid line going in time to the lowered dotted curves) for a trap initially 
+centered around E3.  
+ 
+During	 or	 after	 the	 matrix	 growth	 a	 single,	 or	 multiple,	 laser	 ablation	 pulses	 onto	 the	
+appropriate	solid	precursor	–	mainly	LiH	during	this	study	–	release	and	implant	neutral	and	
+Laser 670.8nm
+Ablation Laser 
+532nm or 1064 nm
+lens
+E0
+E1
+E2 E3 E4
+E5
+CEM
+Vacuum Isolation
+40K Shield
+Trap coil
+a)
+4K Shield
+E0
+E1
+E2
+E3
+E4
+E5
+V(z)
+B(z)
+50
+100
+150
+200
+- 2
+0
+2
+4
+- 25
+0
+25
+50
+75
+Position z (mm)
+V(z)
+(V)
+B(z)
+(mT)
+b)
+T~4K 
+Sapph
+Ne Tube
+LiH
+Mag.Sap.
+Apert.
+
+charged	particles	into	the	matrix	or	reflect	from	it.	The	matrix	can	be	sublimated	at	different	
+times	with	respect	to	the	ablation	pulses.	We	have	employed	many	variations	on	these	timing	
+parameters	and	laser	pulse	energies.		For	ablation	we	mostly	employed	a	doubled	Nd:YAG	at	
+532	nm,	with	a	few	mJ	in	5	ns	pulses.	This	flashlamp	pumped	laser	became	unreliable	and	
+we	switched	to	a	diode	laser	pumped	Nd:YAG	at	1064	nm	with	up	to	1.5	mJ	using	a	tighter	
+focusing	for	similar	fluences.	The	front	sapphire	terminal	can	be	monitored	for	the	charge	
+deposited	into	(or	released	from)	the	matrix,	acting	as	a	capacitor,	during	the	ablation	(or	
+sublimation)	process.	Through	laser	ablation,	enough	charge	can	be	deposited	to	achieve	
+many	volts	of	space	charge	or	to	thermally	explode	the	matrix.	Both	the	ablation	process	and	
+space	charge	variations	can	affect	the	results.	We	monitored	the	matrix	size,	the	deposited	
+charge,	and	the	sapphire	temperature	change	due	to	the	ablation	and	sublimation.	Adjusting	
+these	parameters	and	the	very	critical	timing	for	switching	the	trapping	electrodes	potentials,	
+we	have	conducted	hundreds	of	experimental	cycles	in	a	robust	manner.	
+		
+A	dry	superconducting	coil	(“Mag.	Sap.”	in	Fig.	1)	placed	around	the	sapphire	generates	a	
+field	of	~41	mT	in	a	configuration	to	guide	the	particles	into	the	trap	region.	The	Penning–
+Malmberg	trap,	in	the	horizontal	direction,	is	composed	of	six	similar	electrodes,	named	E0–
+E5,	of	17.4	mm	of	inner	diameter	and	20	mm	length	each.	The	magnetic	field	at	the	trap	is	
+generated	by	an	external	resistive	coil	(“Trap	coil”	in	Fig.	1)	which	produces	92	mT	on	axis,	
+around	E3–E4,	at	100	A.	Such	a	short	coil	was	dictated	by	space	constraints	and	resulted	in	
+a	 highly	 non-uniform	 field.	 This	 leads	 to	 magnetic	 mirroring	 reflections	 if	 we	 dump	 the	
+particles	from	electrodes	E2–E3.	Due	to	heating,	the	coil	can	only	be	left	energized	for	~1.5	
+second,	limiting	the	trapping	time.	The	resulting	field	from	the	two	coils	generates	a	guiding	
+magnetic	field	that	leads	the	particles	from	the	source	making	a	turn	into	the	trap.	Along	the	
+guiding	lines,	the	field	reach	values	as	low	as	1.5	mT,	which	is	enough	for	guiding	low	energy	
+and	low	mass	species.		
+	
+For	detection	of	the	charged	species,	we	use	a	Channel	Electron	Multiplier	(CEM),	a	Magnum	
+Channeltron	from	Photonis	Inc,	to	the	right	of	electrode	E5	in	Fig.1.	We	set	the	voltages	in	
+the	CEM	to	attract	either	negative	or	positive	charges	before	each	experiment.	For	cations,	
+the	cone	of	the	CEM	is	set	at	-2	kV	while	the	anode	is	grounded	via	a	load	resistor	of	1	kΩ.	
+For	anions,	the	cone	is	set	at	+0.4	kV	while	the	anode	is	connected	to	+2.4	kV	via	a	1.2	MΩ	
+resistor.	In	both	configurations	the	output	is	capacitively	coupled	out	of	the	cryostat	and	into	
+a	radio–frequency	pre–amplifier	which	is	directly	recorded	by	an	oscilloscope	at	40	GSa/s,	
+with	 up	 to	 5	 ms	 acquisition,	 and	 analysed	 by	 a	 peak	 detection	 routine	 above	 a	 certain	
+threshold.	To	prevent	excess	black–body	heat	coming	from	the	CEM	into	the	trap	we	partially	
+anchor	 the	 wires	 to	 the	 CEM	 at	 the	 40	 K	 heat	 shield,	 at	 the	 known	 expense	 of	 a	 higher	
+detector's	resistance	resulting	in	longer	recharging	times.	With	a	high	detection	rate,	we	had	
+to	deal	with	saturation	of	the	cold	detector.	Particularly,	using	a	fast	dump	configuration	for	
+
+a	time–of–flight	mass	discrimination	(ToF)	for	negative	particles,	the	first	coming	electron	
+signal	can	saturate	the	CEM	that	ends	up	detecting	very	few	H!	afterwards.	
+	
+Experimental	Results	
+	
+Initial	experiments	involved	using	a	homemade	time–of–flight	mass	spectrometer	(ToF–MS)	
+to	detect	the	different	species	emanating	from	the	sapphire	(see	figure	1).	The	details	of	this	
+ToF–MS	can	be	found	in	ref37.	The	region	where	the	ions	are	extracted	is	large	and	the	ToF–
+MS	resolution	is	poor,	but	sufficient	to	discriminate	electrons	from	H!	and	Li!.	A	typical	
+mass	 spectrum	 of	 the	 raw	 data	 and	 histogram	 is	 shown	 in	 figure	 2.	 The	 extraction	
+accelerating	potential	was	switched	on	at	50	µs	after	ablation.	The	electrons	are	detected	
+immediately	after	the	extraction	pulse,	followed	by	the	heavier	species	with	clear	presence	
+of	H!,	Li!	and	molecular	anions.	As	expected	from	the	ablation	of	LiH	we	generated	both	Li	
+and	H	ionized	species,	with	both	signs	of	charge,	as	well	as	neutral	atoms,	and	molecules.	
+	
+	
+Figure 2. Initial ToF–MS results, in the absence of the trap. The accelerating electric field is switched on at t=50 µs after the 
+ablation laser pulse onto the sublimating Ne matrix. The electrons appear immediately, followed by the heavier species. The top 
+graph (a) shows the data collected from the oscilloscope in a single realization. The red dots are peaks identified by the LabView 
+routine. The bottom histogram (b), with mass identification up to Li! or LiH!and showing heavier masses, is an accumulation of 
+35 runs. The fraction of counts for different species depends on timing. 
+ 
+In	a	MISu	procedure	the	trapping	experiment	is	as	follows:	(i)	the	Ne	matrix	is	grown	and	
+laser	ablation	generates	and	implants	atoms,	molecules,	ions	and	electrons	into	the	matrix;	
+(ii)	both	magnets	are	energized	and	the	Ne	matrix	is	sublimated	with	a	heat	pulse	starting	at	
+time	t=0,	sometimes	the	last	electrode	E5	is	energized	from	the	beginning;	(iii)	at	time	t1	the	
+entrance	trap	electrode	(E2,	for	example),	or	both	E2	and	E5,	is	switched	on	and	hold	the	
+(a)
+0
+1.0
+2.0
+3.0
+Signal (V)
+(b)
+e-
+H-
+Li-/LiH-
+50
+60
+70
+80
+90
+100
+200
+400
+600
+800
+Time (μs)
+Counts
+
+trapped	ions;	(iv)	the	trapping	potentials	are	manipulated	and	then	lowered	at	different	
+times	and	rates	and	the	ions	detected	in	the	CEM.		
+	
+The	 raw	 data	 for	 cations	 of	 a	 trapping	 sequence	 is	 shown	 in	 figure	 3.	 During	 the	 slow	
+sublimation	(heat	pulse	starts	at	t=0),	the	spectroscopy	laser	(purple	trace)	continuously	
+scanning	 (at	 2	 kHz)	 around	 the	 D2	 resonance	 in	 Li	 (670.776	 nm)	 records	 the	 atoms’	
+absorption	after	they	are	released	from	the	matrix.	With	E5	=	5	V	during	sublimation,	the	
+energy	 barrier	 prevents	 any	 cation	 from	 reaching	 the	 CEM.	 At	 t=1080	 µs	 the	 entrance	
+electrode	E2	is	switched,	during	20	µs,	to	5	V	closing	the	trap.	At	time	t=3500	µs	the	exit	
+electrode	E5	is	linearly	brought,	during	800	µs,	to	0	V	and	the	ions	are	detected	in	the	CEM	
+as	shown	in	the	blue	histogram.	
+	
+Figure 3. Trapping of cations using the MISu procedure. A matrix is grown while 6 ablation pulses deposit atoms and charged 
+particles into it. At t=0, a current is applied to the sapphire’s NiCr film resistor causing a slow sublimation of the matrix. The laser 
+transmission is shown in purple and presents information on the matrix thickness and atomic absorption. The E5 electrode (black 
+trace) is set at 5 V since t=0. The E2 electrode (red dashed trace) is switched to 5 V at t=1080 µs trapping particles between E2 
+and E5. At t=3500 µs, E5 is linearly brought to 0 V and cations are allowed to scape towards the CEM. The cations count is shown 
+in the blue histogram. The appearance of signal at the end of, and after, the ramp is compatible with very low energy ions, despite 
+a small propagation delay of the ions to the detector. The small signal of ions at t~1100 µs results from the energy gained by the 
+switching of E2 that pushes ions over the E5 barrier. 
+	
+Several	studies	were	performed	using	the	MISu	procedure	described	above,	with	positive	
+and	negative	charges.	The	critical	optimization	parameters	include	the	times	between	the	
+beginning	of	the	sublimation	pulse,	the	ablation	pulse,	and	the	trap	closing.	Figure	4	shows	
+results	for	cations	done	lowering	the	trap	barrier	(electrode	E4)	at	a	much	lower	rate	than	
+Laser signal
+Arbitrary Vertical Scale
+Electrode 2 Potential (V)
+Electrode 5 Potential (V)
+Histogram Counts
+0
+1000
+2000
+3000
+4000
+5000
+0
+2
+4
+6
+8
+10
+12
+0
+20
+40
+60
+80
+100
+Time (μs)
+Electrodes Voltage (V)
+Counts
+
+in	figure	3,	taking	2500	µs	to	go	from	1	V	to	0	V.	In	this	case,	the	propagation	delay	to	the	
+CEM	is	of	smaller	relevance	during	the	ramp	down.	An	energy	distribution,	incorporating	the	
+counts	after	the	ramp	down	into	the	first	bin,	is	shown	as	an	inset	of	the	figure.	Notice	that	
+time	and	energy	are	not	linear	near	the	end	of	the	ramp	as	there	is	a	small	potential	at	the	
+central	(E3)	electrode	region.	The	peak	of	the	distribution	is	in	the	first	bin	(0	to	25	meV)	
+which	accounts	for	32%	of	the	total	counts	shown.	Among	the	various	manipulations	we	
+included	squeezing	the	trap	into	a	single	electrode	to	rid	the	trap	of	species	with	masses	
+higher	than	10.	
+	
+	
+Figure 4. Results with cations (H" and Li") from the accumulation of 5 runs. A matrix (~1 µm) was grown over 20 s with 40 
+ablation pulses. The sublimation heat pulse started at t=0 and the Li atoms were seen (not shown in the graph) sublimating from 
+~4.5 – 8 ms. An extra ablation pulse is given at t=6140 µs and E2 rose to 5 V from 6200 – 6250 µs closing the trap between E2–
+E5. The sample was later squeezed between E2–E4 and E4 was brought to 1 V. The histogram (in blue) from the final slow linear 
+dump (from 8000–10500 µs) of E4 from 1 V is shown above in solid black. In the inset the time distribution is mapped into an 
+energy distribution with the counts after the ramp down being incorporated in the first bin (0 – 25 meV).  
+	
+The	result	of	the	studies	above	with	negative	charges	showed	that	electrons	dominated	the	
+counts.	To	improve	the	anions’	yield,	we	developed	a	variation	on	MISu	which	consists	of	
+applying	a	single	ablation	pulse	during	the	sublimation	of	the	Ne	matrix.	In	figure	5,	the	
+results	of	such	an	experiment	for	negative	charged	particles	is	shown.	In	this	case	the	trap	
+was	manipulated	as	in	figure	4	above,	going	from	an	initial	trap	in	E2–E5	to	a	squeezed	E2–
+E4	and	then	E4	slowly	dumped	from	-2	V	while	E2	was	kept	at	-5	V.	The	time	distribution	
+can	be	folded	into	an	energy	distribution	shown	in	the	inset	with	bins	of	25	meV.		
+	
+0
+100
+200
+300
+400
+500
+600
+0
+200
+400
+600
+800
+1000
+1200
+Energy (meV)
+Counts
+Electrode 4 Voltage
+Histogram Counts
+8000
+9000
+10000
+11000
+0
+50
+100
+150
+200
+250
+0
+200
+400
+600
+800
+1000
+Time (μs)
+Counts
+Barrier Voltage (mV)
+
+ 
+Figure 5. Histogram (in blue) from final dump of electrons and anions (about 1/4 e!and 3/4 H!) from a trap between E2–E4 after 
+similar manipulations as in figure 4. This data represents the accumulation of 21 runs at a reduced laser energy. The matrix was 
+80 nm thick. The sublimation heat pulse starts at t=0 and a single ablation pulse is given at t=6140 µs and E2 rises to -5 V from 
+6200 – 6250 µs closing the trap between E2–E5. The sample is later squeezed between E2–E4 and E4 is brought to -2 V. The 
+histogram from the final slow linear dump (from 8000–10500 µs) of E4 from -2 V is shown above. The inset shows the 
+corresponding energy distribution, with a nearly exponential decay shape, with the counts after the ramp down being 
+incorporated in the first bin (0 – 25 meV).  
+ 
+ 
+To	determine	which	species	were	trapped,	we	employed	a	ToF	discrimination	in	the	small	
+space	available,	far	from	the	ideal	Wiley–McLaren38	condition.	By	trapping	the	particles	in	a	
+single	electrode,	e.g.,	E2–E4	with	E2	at	±5	V	and	E4	at	±1	V,	we	can	quickly	lower	E4	to	±0.2	
+V	while	raising	E3	to	±0.4	V	(where	the	±	applies	to	positive	or	negative	particles)	imposing	
+a	wiggling	acceleration	region	from	E2	to	E4	(see	Methods	for	details).	The	particles	are	
+accelerated	towards	the	channeltron	and	have	some	short	length	of	free	flight.	The	condition	
+is	not	ideal	but	enough	to	discriminate	e!,	H!	and	Li!/LiH!,	though	not	Li!	from	LiH!.	Due	
+to	the	short	length	of	the	trap	coil	we	cannot	employ	this	technique	at	electrodes	farther	
+from	the	detector.	The	result	of	this	ToF	discriminator	is	shown	in	figure	6,	using	the	same	
+matrix	and	ablation	conditions	as	in	figure	5.	The	ToF	discrimination	shows	electrons	and	
+hydrogen	anions	as	majority,	followed	by	a	small	fraction	of	Li!	or	LiH!	for	those	conditions.	
+For	 this	 procedure	 we	 had	 to	 considerably	 decrease	 the	 ablation	 laser	 energy	 to	 avoid	
+saturating	 the	 detector	 with	 the	 e! 	signal.	 Details	 on	 the	 simulation	 are	 presented	 in	
+Methods	but	it	is	relevant	to	mention	that	it	is	a	simple	1D	simulation	along	the	axis;	that	it	
+does	not	consider	the	large	finite	time	for	the	“quick	ramp	down”,	which	typically	has	a	1.2	
+µs	time	decay;	and	that	it	uses	an	energy	distribution	that	would	reproduce	figure	5.	
+ 
+0
+200
+400
+600
+800
+0
+50
+100
+150
+200
+250
+300
+Energy (meV)
+Counts
+Electrode 4 Voltage
+Histogram Counts
+8000
+9000
+10000
+11000
+0
+50
+100
+150
+200
+0
+500
+1000
+1500
+2000
+Time (μs)
+Counts
+Barrier Voltage (-mV)
+
+ 
+ 
+Figure 6. ToF discrimination for negative species showing e! , H!  and Li!	(or LiH! ) in the same trapping conditions and 
+manipulations as in figure 5 but now doing a quick ramp down from the trap with E1= -10 V, E2 = -5 V, E3=0 V, E4= -2 V to an 
+acceleration ramp with E1=-10 V, E2= -5 V, E3 = -0.2 V, E4= -0.1V and E5 = 0 V. The experimental data from an accumulation of 
+21 runs is shown in blue while a simulated histogram for the species identified is shown with a red envelope. In this figure, t=0 is 
+the beginning of the quick ramp down, taken as instantaneous in the simulation. The ablation laser energy was considerably 
+decreased to avoid saturation of the CEM with the e! signal. See text for more discussion on the “quick ramp down” and Methods 
+for the simulation. 
+	
+Due	to	some	saturation	of	the	CEM	with	the	electron	signal	in	the	ToF,	we	employed	another	
+method	to	get	a	ratio	of	H!	to	e!	under	these	conditions.	The	electrons	arrive	at	the	trap	
+region	almost	immediately	after	the	ablation	in	this	variation	while	the	H!‘s	take	about	100	
+µs.	By	measuring	that	the	electrons	trapped	number	would	not	change	from	60	to	90	µs,	after	
+ablation	and	before	the	arrival	time	of	the	H!,	we	can	compare	the	trapped	distribution	for	
+slow	ramps	for	samples	comprised	only	of	electrons	to	samples	trapped	at	the	best	time	for	
+H!.	The	result	is	shown	in	figure	7.	The	ratio	of	the	areas	(~3.6)	represents	that,	at	its	optimal	
+timing,	 H! 	is	 trapped	 2.6	 times	 more	 efficiently	 than	 e! 	and	 with	 a	 similar	 energy	
+distribution.	
+ 
+e-
+H-
+Li-
+0
+10
+20
+30
+40
+50
+0
+50
+100
+150
+200
+Time (μs)
+Counts
+
+ 
+Figure 7. Comparison of slow dumps for a sample trapped at earlier times (60–90 µs after ablation) comprised only of e!‘s (in 
+red), to sample comprised of e! and H! (in blue) trapped at the proper H! trapping time 100 µs after ablation. In these 
+conditions, the ratio of integrated counts results in 2.6 times more H!‘s than e!‘s trapped at the optimized H! trapping time. 
+The energy distributions are similar. 
+	
+Another interesting manipulation, able to discriminate masses, is to make the trap unstable to 
+certain species by changing the electrostatic potentials and the magnetic field. For a harmonic 
+Penning trap to be stable39 the cyclotron frequency, 𝜔% = 𝑞
+&
+$,  and the axial frequency, 
+𝜔'	~	1/√𝑚 , should relate as 	𝜔% >	√2	𝜔'. The condition relates the confining potential shape 
+(in 𝜔'), the magnetic field (B) and the particle mass (m) and charge (q). Squeezing and deepening 
+the electrostatic potentials – alternating -10 V and +10 V in electrodes E3, E4 and E5 – we can 
+easily make the trap unstable for species other than	e! and H! even at the maximum current of 
+100 A in the trap magnet. Under this quasi-harmonic trap configuration even H!  becomes 
+unstable under ~55 A and we performed a study on the trapped particles number as function of 
+the current. Due to a hardware limitation, we could only set the trap magnet current just before 
+sublimation and could not separate the guiding and initial trap loading processes from the trap 
+instabilities. To keep the magnetic field guiding lines configuration unchanged, the sapphire 
+magnet current was changed in the same proportion as the trap magnet current. In figure 8 the 
+result of this study is shown.  
+ 
+The data clearly shows a decrease of the number trapped as the current decreases until a cutoff 
+near 55 A where H! becomes unstable and the residual counts at lower currents are due to e!.  
+Two models attempt to capture the qualitative behavior. A first one, in purple, supposes a simple 
+axial 1D model for ions already in a harmonic trap resulting in a step function separating the 
+regions of stability and instability. The other consider the entrance aperture transmission of the 
+Electrode 4 voltage
+Counts with Η- and e-
+Counts with only e-
+20000
+21000
+22000
+23000
+24000
+25000
+0
+10
+20
+30
+40
+0
+100
+200
+300
+400
+500
+Time (μs)
+Counts
+Barrier Voltage (-mV)
+
+trap, and trap stability also considering the finite electrodes diameters. At the entrance aperture 
+the magnetic field is small, at a ratio of ~1/13 of that in the trap center, and the radius of 
+curvature of the cyclotron motion may exceed the entrance opening. Inside the trap one may 
+have instability or simply a large cyclotron plus magnetron motion that gets to the electrode walls 
+and lead to losses. Monte-Carlo simulations, using an axial energy distribution like the 
+exponential one in figure 5 inset (despite different conditions and manipulations for this data set 
+to that of figure 5) and different perpendicular energy distribution lead to the other curves in 
+figure 8. The details on the models are presented in Methods. 
+ 
+The good agreement of the last model, without adjustable parameters other than the arbitrary 
+scaling number for the total counts at 100 A, suggests that this method may be further developed 
+as a diagnostic tool for the radial density distribution – requiring validating studies with an 
+imaging detector, or employing an axially displaced photodetachment laser beam, and higher 
+data statistics – or the transverse energy distribution. These are two identified parameters 
+affecting the curves. 
+ 
+ 
+Figure 8. Trap loading and survival of H! in the trap due to a varying magnetic field. The data is shown as grey dots, representing 
+individual realizations, and the blue dots are the average of the grey dots values for the corresponding current. Two models are 
+shown. The purple line represents a simple axial 1D model concerning only the (harmonic) axial and cyclotron frequencies relation 
+for stability. The red lines guide the eyes through the results of a Monte-Carlo model considering the entrance aperture of trap 
+region, at magnetic fields much lower (~1/13) than in the trap, and the survival of the particles in the trap given its diameter for 
+different radial energy distributions. The axial energy distribution is taken as an exponential decay as in figure 5 and the transverse 
+energy distributions are taken as exponential decays but with different energy scales. In the continuous, dashed, and dotted red 
+lines the transverse energy scale is 1/12, 2, or 20 times the axial energy scale, respectively.   Magnetic mirroring effects are taken 
+into effect in the energy rescaling between the trap entrance aperture to the trap center. The models do not include a full particle 
+tracing but only consider the particles at the entrance aperture and the trap center.  See text and Methods for further discussion. 
+↘
+55 A
+20
+40
+60
+80
+100
+0
+500
+1000
+1500
+Trap Coil Current (A)
+Counts
+
+ 
+The results shown in figures 2–8 above are just a subset of the possibilities with this technique. 
+For instance, we produced trapped H! from ablation on TiH2 and performed a few runs with a 
+H2, instead of Ne, matrix. In the present trap we can make all atomic masses above 7 (Li), or 
+higher, to be unstable just by squeezing and deepening the potentials, thus making the axial 
+frequency to go over the stability condition compared to the cyclotron frequency. Light particles, 
+like electrons can be expelled from the trap by a fast (~1 µs, in the case of e!) opening–and–
+closing of the trap barrier, but this required adding some extra complexity to our hardware and 
+software and we performed just a few validating tests. Therefore, one could select a range of 
+masses to keep in the trap for studies such as molecular formation. Splitting the ablation laser, 
+one could simultaneously ablate different targets to produce molecules already in the collision 
+with the sublimating plume or one could press a sintered composite material with the different 
+precursors into a single ablation target. We have also produced a cryogenic beam of electrons 
+from electrostatic sublimation, i.e., we can switch on and off a beam of electrons, and 
+subsequently trap them, just by applying a potential to the sapphire substrate without 
+sublimating a matrix that was previously implanted via laser ablation.  
+ 
+Discussion		
+	
+In a next generation of the apparatus, with a longer trap magnet – and more uniform field – we 
+will be able to achieve much better mass discrimination and more relevant: stack samples in a 
+very simple manner. Even if we keep only 6 electrodes, we could trap between E0–E3, then move 
+the sample into E3–E5, trap again in E0–E3, then accumulate in E3–E5, and repeat the process. 
+Using a MISu variation we were able to trap over one thousand H!‘s after a single ablation pulse. 
+With a diode pumped laser operating at high frequency and higher magnetic fields the system 
+can be scaled up by orders of magnitude to large amounts of trapped species. Trapping heavier 
+atomic and molecular ion species at large numbers will also require higher magnetic fields.  
+ 
+This trap holding time was limited by the trap coil’s heating at ~1 s time scale. In a test, we held 
+a sample up to 1.5 s obtaining ~1000 counts in final dump of that realization. In a next realization 
+we plan to use a superconducting coil which, together with a better thermal anchor in the 
+trapping region, should enable long holding times and thus to achieve thermalization and 
+perform evaporative cooling of the trapped samples achieving a few kelvins of temperature.  
+ 
+The system is very rich and has many adjustable parameters that affect the outcome. Despite 
+that, we were able to achieve a robust operation. Moreover, the system still holds many 
+intriguing and open questions. Why we trap only few H! ‘s in the typical MISu procedure. Is the 
+H! not penetrating the matrix, or being stripped of its extra electron as it enters the solid Ne, or 
+being ionized under the matrix’s high electric fields? Using a variation, we were able to 
+
+circumvent the problem and produce large amounts of H!, but it would be important to better 
+understand this process. We also observed production of these charged species using a H2 matrix, 
+instead of a Ne matrix, but were not able to do a full exploration on different parameters.  
+ 
+We want to explore the conditions to produce ions at the typical atomic temperatures of kelvins. 
+The MISu technique generates an intense Li beam at a few kelvins, and below, despite Li having 
+a positive (expelling) energy40 of 0.25 eV (~2900 K) in Ne. This is a strong argument why we 
+expect to produce ions coming from the matrix at these low temperatures as well, even with 
+space charge effects with the ions. The trap closing procedure itself imparts large energy to a 
+fraction of the ions under the rising electrodes potentials and this is clearly seen in figure 3. If 
+these high energy ions would exchange some energy, the “high energy” from our samples could 
+be coming from the dynamic trapping procedure (one gets 1.6 eV average energy from a trap 
+switching to 5 eV, supposing a uniformly distributed flux of ions over the electrodes region). In 
+the contrary direction, the ratio of magnetic fields of 41 mT below the sapphire, where collisions 
+with the Ne gas may not be relevant anymore, to 92 mT in the trap gives some artificial cooling 
+of the measured axial energy since the transverse energy for particles following the field lines 
+will scale with the magnetic field at the expense of the axial energy. We plan to use laser 
+spectroscopy on suitable cations, such as Ca+ to study the real temperature of the ions’ sample 
+and see in which step of the procedure the sample is gaining energy. At one earlier construction, 
+with different sapphire attaching plates and without all the tools and the trap, we saw time 
+delaying evidence of being able to control and produce a charged sample at speeds comparable 
+to the atomic speeds.  
+ 
+In	the	present	case	we	still	have	a	large	amount	of	Ne	and	ions	of	the	opposite	charge	that	
+come	to	the	direction	of	the	trap.	This	condition	can	be	mitigated.	As	for	the	opposite	charges,	
+or	stacking,	we	could	easily	setup	the	system	with	different	ablation	targets	and	build	a	
+nested	Penning	trap	configuration	for	positive	and	negative	species	trapped	side–by–side	
+ready	to	study	molecular	recombination,	as	it	is	done	with	H&	formation	by	mixing	p9‘s	into	a	
+e(cloud.		
+	
+With the electrostatic sublimation of electrons from the matrix, we speculate on producing a low 
+energy spin–polarized beam of electrons. If the matrix is at 3 K and would be subjected to a 4 T 
+field, a spin thermalized electron sample could be 80% polarized. Such a simple low–energy 
+polarized electrons’ source could be very attractive to biochemical experiments, especially with 
+respect to the issue of biology homochirality41,42, and to fundamental quantum collisional 
+processes.43 
+	
+
+We	believe	the	data	is	quite	a	strong	proof–of–principle	for	what	is	needed	to	load	H	atoms	
+in	the	H&	ALPHA	experiment	trap.	The	system	here	presented	is	at	a	more	advanced	stage	
+than	other	alternatives44,45	proposed.	As	for	applications	with	large	amounts	of	ions,	such	as	
+with	 tritium	 or	 deuterium,	 we	 believe	 the	 system	 is	 scalable.	 The	 use	 of	 higher	 power	
+ablation	 lasers	 and	 at	 high	 rates,	 together	 with	 higher	 guiding	 magnetic	 fields,	 would	
+improve	the	numbers	by	orders	of	magnitude.			
+ 
+Conclusions	and	Prospects		
+	
+We have demonstrated a platform to produce cryogenic (below 25 meV) beams or trapped 
+samples of electrons, anions, and cations. The system was first designed to produce H! towards 
+loading the H& trap in the ALPHA experiment at CERN and we find it meets the necessary criteria 
+of numbers, energy range, and UHV conditions after trapping and waiting the matrix gas to 
+cryopump, for this application. The system should work equally well for T!/T, perhaps suitable 
+for a neutrino mass experiment, and for D!/D for studies relevant to fusion research. Trapped 
+anions and cations, or in a beam, can be used to study chemistry at astrophysical conditions, such 
+as in star formation. The system can generate an electrically controlled electron beam which we 
+speculate might be suitable for spin polarization with applications towards studies of chirality 
+with relevance to biochemistry. Based on the present data and previous studies on MISu, the 
+system is quite versatile and general in the sense of being able to produce different species of 
+atomic and molecular neutrals, anions, and cations. 
+ 
+Acknowledgements This work was supported by: CNPq, FAPERJ, and RENAFAE (Brazilian agencies). We thank Profs. 
+Massimo Xi Liu, from USP-São Carlos, and Vitoria Barthem, from UFRJ, for depositing the NiCr resistive films, first 
+and second batches respectively. 
+ 
+Author Contributions The initial conception was proposed by CLC. All the authors participated in the design and 
+implementation at various stages. LOAA, primarily, and RJSC were responsible for the experimental runs. Data 
+processing was performed by LOAA. The initial manuscript was written by CLC and edited and improved by WW, 
+DMS, RLS, and LOAA.  
+ 
+Reprints and permissions information is available online at www.nature.com/reprints. Readers are welcome to 
+comment on the online version of the paper. Correspondence and requests for materials should be addressed to 
+CLC or LOAA ([email protected] , [email protected]).  
+
+ 
+Data availability statement The datasets generated during and/or analysed during the current study are available 
+from CLC or LOAA ([email protected] , [email protected]) on reasonable request. 
+ 
+Competing	financial	interests		The	authors	declare	no	competing	financial	interests.		
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+ 
+Methods / Supplementary Material:  
+ 
+Time-of-Flight mass discrimination (ToF-MD) of trapped species 
+ 
+Once the ions are trapped, we have implemented a potential ramp to discriminate the masses. 
+A Wiley-McLaren38 configuration would require a free-flight region twice as long as the 
+accelerating region. Within our trap space constrain, we could switch from a trap configuration 
+with potentials (in V) 𝐸2 = 5, 𝐸3 = 0, 𝐸4 = 1 and 𝐸5 = 0  to the configuration of 𝐸2 = 5, 𝐸3 =
+0.2, 𝐸4 = 0.1 and 𝐸5 = 0 (or 𝐸2 = 5, 𝐸3 = 0.4, 𝐸4 = 0.2 and 𝐸5 = 0) as shown in figure 8 
+below. 
+ 
+For initial estimates of this ToF-MD functionality we used a simple analytical function that 
+approximately simulate each electrode potential and added their values to represent different 
+trap configurations. The simplest function, with just 3 adjustable parameters, is given by: 
+ 
+𝑉(𝑧) =		∑
+𝑓)	𝐴#		GErf K
+*'(('!'!")	
+.'
+L + Erf K
+*'!('!'!")	
+.'
+LN
+/
+#01
+, 
+
+ 
+where  𝐴#, 𝑧1#  represent the applied voltage amplitude and the center position of the 𝑛23  
+electrode. The geometrical factor 𝑓) 	≈ 0.47, 𝛿𝑧	 ≈ 7	mm, and 𝑑𝑧	 ≈ 11	mm is the length of the 
+electrode (10 mm) plus the distance between electrodes (1 mm). From the above expression the 
+electric field can be analytically obtained. 
+ 
+The configuration for the ToF-MD, with precise potentials, is shown in figure 9 below. Notice that 
+the electric field is not ideally uniform in the acceleration region but shows wiggling, and the free-
+flight region is small compared to the acceleration one.  
+ 
+ 
+Figure 9. ToF mass discrimination potentials (depicted for the case of anions). The dashed curve is the initial trap around E3 (E2=5 
+V, E3=0, E4=1 V, E5=0) before the potentials are switched to the ToF configuration (E2=5 V, E3=0.2 V, E4=0.1 V, E5=0) shown in 
+solid line.  The negative potential at z > 200 mm is due to the CEM. 
+ 
+The simulation considered a 1D system, along the trap axes, concerning only the electrostatic 
+field produced by the electrodes and CEM detector and not the magnetic field. According to the 
+energy distribution in figure 6, we place ions at the returning points in the trap, in both ends, 
+given its energy and let each ion evolve inside the trap for a random time between 0 and half of 
+the period of axial oscillation for that energy. This new state of each ion is the corresponding 
+initial state for the second part of the simulation where the trap is switched instantaneously to 
+the ToF configuration. The actual driving hardware takes about 1.2 µs to reach 90% of the set 
+value and this would cause extra spreading in the detection time distribution, but it was not 
+considered in the simulation. The results are sufficient for discriminating our species of interest 
+for this work. 
+E2
+E3
+E4
+E5
+100
+120
+140
+160
+180
+200
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Position z (mm)
+V(z)
+(-V)
+
+ 
+For the ToF-MD simulations shown in figure 6 we used an interpolation for the electrodes’ voltage 
+configuration, 𝑉456!78(𝑧), given by a finite element package solver instead of the analytical 
+potential described above. The time-of-flight is given by the usual integration 𝑡456!78 =
+∫
+9
+: #
+${<!=	>%&'()*(')}
+'+
+'!
+𝑑𝑧, where 𝑧1 is the position at the time of switch,  𝑧@ is a position near the 
+CEM – where particles at rest would take less than 0.2 µs to reach the CEM  – and 𝜖 is the total 
+energy the particle has immediately after the switch, i.e., its initial kinetic energy (when it was in 
+the trap) plus the potential energy in ToF configuration: 𝑞	𝑉456!78(𝑧1). 
+ 
+Magnetic field variation study 
+ 
+To analyze the stability of 	H!  and the losses by collisions in the trap entrance as we change the 
+trapping/guiding magnetic field, we repeated for different magnetic field conditions the 
+following steps: a matrix of about ~370 nm is grown; the currents on the magnets are set to the 
+desired value (at maximum, 100 A in the trap magnet and 2.5 A in the sapphire coil); during a 
+slow sublimation, a single laser ablation hits the LiH target; the trap loading potential, with the 
+electrode configuration (in V) E0=0, E1=0, E2=-5, E3=0, E4=0, E5=-5, is closed 100 µs after the 
+ablation; the sample is adiabatically squeezed by changing the electrodes configuration (in V) to 
+E0=0, E1=0, E2=0, E3=-10, E4=10, E5=-10 and waiting for 1 ms; particles with energies above 
+~700 meV are released by changing the electrode configuration (in V) to E0=0, E1=0, E2=0, E3=-
+10, E4=0, E5=-1; after 500 µs the remaining trapped particles are slowly released and detected. 
+The data is shown in figure 8. 
+ 
+Some assumptions were made for constructing the models (1D and 3D) to qualitatively describe 
+the survival of	H! after this process. A uniform magnetic field along all the trap volume was 
+considered, as the trap is just one electrode wide, taking the mean value of field in the region. 
+We considered the trap to be harmonic for energies < 700 meV since the axial oscillation 
+frequency changes less than 2% within this energy range, hence the equations for the stability 
+inside the trap are exact. The simple axial 1D model only considered the stability criteria of the 
+relation of the cyclotron and axial frequencies. For the 3D models, we considered an axial (z) 
+exponential decaying energy distribution as measured in Figure 5. The transverse energy 
+distribution was also considered exponential but with different decay values and it was initially 
+mapped only on 𝑣A. The initial position of the anions in the xy-plane inside the trap  (𝑥1) imitates 
+a uniforme surface distribution but mapped only in 𝑥 with probability proportional to 𝑥.  
+ 
+The axial and transverse energies are drawn for a particle inside the trap. Considering the 
+transverse energy adiabatic propagation in magnetic field – where the transverse energy scales 
+with magnetic field at the expense of axial energy – we map both the transverse and axial 
+energies to the entrance (𝜖2,CD and  𝜖',CD) of the trap region (just left of E0). If  
+<,,./
+<0,./ >
+&1234
+&./ 	≈
+12.85, where 𝐵EFGH	and 𝐵CD are the magnetic fields at the trap and at the entrance, the particle 
+
+is considered in the Monte-Carlo since it will not reflect magnetically. If 𝜖2,CD > 172	meV the 
+cyclotron radius is greater then the entrance radius and we consider the particle lost at the 
+entrance. If the particle enters the trap, given the drawn initial position (𝑥1) and transverse 
+energy inside the trap (𝜖2,EFGH, supposed initially along 𝑦), we calculate the maximum radius46 of 
+the particle as the sum of the magnetron and cyclotron radii for this particle for each current 
+considered. If this maximum radius, for each current, is larger than the electrode radius this 
+particle is lost in the trap. Finally, if the current is bellow ~55	𝐴, the trap is unstable and no H! 
+would be trapped.  
+ 
+The shapes of the curves resulting from these simulations have a dependence with the transverse 
+energy distribution (see Figure 8). Figure 10 shows the histograms of distributions for the 
+particles that survived the losses considered for the trap at 100 A consisting of the transverse 
+and axial energies (𝜖2,EFGH and  𝜖',EFGH) distributions and initial position (𝑥1)  considering three 
+different initial distributions for the transverse energy.  
+ 
+ 
+Figure 10. Histograms from the Monte-Carlo Simulations of the surviving particles inside the trap for a trap coil current of 100 A 
+for the 3D model with different exponential energy scales for the transverse energy distribution: (a) 2 times greater than in the 
+axial distribution, (b) 12 times smaller and (c) 20 times greater. In all three cases an exponential decaying axial energy distribution 
+was assumed in the beginning, but some low energy particles are magnetically reflected and thus considered not trapped. Notice 
+(a,I)
+0
+200
+400
+600
+800
+1000
+0
+5000
+10000
+15000
+20000
+Energy (meV)
+Transverse Energy Inside Trap
+(a,II)
+0
+100 200 300 400 500 600 700
+0
+2000
+4000
+6000
+8000
+10000
+12000
+Axial energy (meV)
+Axial Energy Inside Trap
+(a,III)
+0
+1
+2
+3
+4
+5
+6
+0
+2000
+4000
+6000
+8000
+10000
+12000
+14000
+Initial Position (mm)
+Initial Position (x axis)
+(b,I)
+0
+20
+40
+60
+80
+100
+0
+10000
+20000
+30000
+40000
+50000
+Energy (meV)
+(b,II)
+0
+100 200 300 400 500 600 700
+0
+5000
+10000
+15000
+20000
+Axial energy (meV)
+(b,III)
+0
+1
+2
+3
+4
+5
+6
+0
+5000
+10000
+15000
+20000
+25000
+Initial Position (mm)
+(c,I)
+0
+500
+1000
+1500
+2000
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+Energy (meV)
+(c,II)
+0
+100 200 300 400 500 600 700
+0
+1000
+2000
+3000
+4000
+5000
+6000
+Axial energy (meV)
+(c,III)
+0
+1
+2
+3
+4
+5
+6
+0
+1000
+2000
+3000
+4000
+5000
+6000
+Initial Position (mm)
+
+the very large loss at the highest transverse energy distribution from the total of 150000 particles, simulated at each case. See 
+text for discussion. 
+ 
+46 M Kretzschmar, Particle motion in a Penning trap, Eur. J. Phys. 12 240 (1991) 
+ 
+ 
+

59FAT4oBgHgl3EQfnB39/content/tmp_files/2301.08627v1.pdf.txt

@@ -0,0 +1,892 @@
+arXiv:2301.08627v1  [cs.CY]  20 Jan 2023
+“This Applies to the Real World”: Student Perspectives on
+Integrating Ethics into a Computer Science Assignment
+Julie Jarzemsky
+[email protected]
+University of Colorado Boulder
+Joshua Paup
+[email protected]
+University of Colorado Boulder
+Casey Fiesler
+casey.fi[email protected]
+University of Colorado Boulder
+ABSTRACT
+There is a growing movement in undergraduate computer science
+(CS) programs to embed ethics across CS classes rather than re-
+lying solely on standalone ethics courses. One strategy is creat-
+ing assignments that encourage students to reflect on ethical is-
+sues inherent to the code they write. Building off prior work that
+has surveyed students after doing such assignments in class, we
+conducted focus groups with students who reviewed a new intro-
+ductory ethics-based CS assignment. In this experience report, we
+present a case study describing our process of designing an ethics-
+based assignment and proposing the assignment to students for
+feedback. Participants in our focus groups not only shared feed-
+back on the assignment, but also on the integration of ethics into
+coding assignments in general, revealing the benefits and challenges
+of this work from a student perspective. We also generated novel
+ethics-oriented assignment concepts alongside students. Deriving
+from tech controversies that participants felt most affected by, we
+created a bank of ideas as a starting point for further curriculum
+development.
+CCS CONCEPTS
+• Social and professional topics;
+KEYWORDS
+ethics, introductoryprogramming, CS1, social impact, assignments,
+university, undergraduate, content, focus groups, content modera-
+tion
+ACM Reference Format:
+Julie Jarzemsky, Joshua Paup, and Casey Fiesler. 2023. “This Applies to
+the Real World”: Student Perspectives on Integrating Ethics into a Com-
+puter Science Assignment. In Proceedings of the 54th ACM Technical Sym-
+posium on Computer Science Education V. 1 (SIGCSE 2023), March 15–18, 2023,
+Toronto, ON, Canada. ACM, New York, NY, USA, 7 pages. https://doi.org/10.1145/3545945.3569846
+1
+INTRODUCTION
+Rising awareness of the harm and injustice that technology can
+inflict upon people and society has led to calls for increased cri-
+tique and interrogation of current practices within the tech indus-
+try [3, 8, 25], as well as proposals to expand the inclusion of con-
+cepts such as ethics, social impact, and justice within computer
+science and other technology-related education programs [11, 21,
+This work is licensed under a Creative Commons Attribu-
+tion International 4.0 License.
+SIGCSE 2023, March 15–18, 2023, Toronto, ON, Canada.
+© 2023 Copyright held by the owner/author(s).
+ACM ISBN 978-1-4503-9431-4/23/03.
+https://doi.org/10.1145/3545945.3569846
+22, 31]. Though in the U.S., the Accreditation Board for Engineer-
+ing and Technology requires computer science curricula to edu-
+cate about the impacts of computing [1]. Solutions for accomplish-
+ing this vary. For example, many undergraduate CS programs sat-
+isfy this requirement with standalone ethics courses [16], which
+though an important component of ethics education [11], have the
+drawback of being potentially disconnected from technical con-
+tent.
+In addition to these standalone ethics classes, there is also now a
+growing movement within universities to embed ethics across the
+entire CS curriculum [2, 6, 12], with one common strategy being in-
+corporating ethical and social context into technical assignments.
+Recent examples from prior work have situated assignments in the
+context of ethical dilemmas such as ad personalization, college ad-
+missions algorithms, bias in criminal justice systems, and privacy
+[6, 9, 10, 18, 26]. These efforts also go beyond learning about ethics
+in general to instilling a responsibility within CS students to con-
+sider the impacts of the code they write. This is a step towards
+what Amy Ko calls a “critical literacy of computing,” beyond “just
+an ethics requirement for CS majors...recasting computing itself
+in moral, ethical, and social terms” [21].
+To contribute to these efforts, we created a programming as-
+signment for an introductory CS class centered on the topic of
+automated content moderation. After creating the initial version
+of this assignment, we conducted focus groups with students at
+our university to gather detailed feedback on the assignment and
+the ethical discussions it prompted. The focus groups enabled us
+to expand on previous work that surveyed introductory comput-
+ing students on their attitudes towards ethics based assignments
+[10], in order to gather more detailed feedback that both provided
+guidance for iterating on the assignment, and ideas that will help
+guide the integration of ethics into CS curriculum in a way that
+centers student engagement.
+2
+CREATING THE ASSIGNMENT
+We began our process by reviewing literature and existing ethics-
+based assignments, discussing assignment concepts with colleagues,
+and creating a list of social issues that the assignment could focus
+on. We ultimately decided to focus on automated content moder-
+ation, given its abundance of both related technical skills and eth-
+ical considerations, as well as the authors’ own observations of
+discussions in a standalone ethics class that were highly engaging
+to students. We noted that much of the literature about the tech-
+nical challenges of automated content moderation involve issues
+such as hate speech detection [28, 32], but due to the obvious po-
+tential for harm we did not want to ask students to write code that
+directly involved hate speech. We instead considered the shape of
+
+the problem in terms of automated language detection and bad ac-
+tors, and created a hypothetical problem: writing and reasoning
+about a content moderation system for a cat-run social media plat-
+form. The following context was provided for the assignment:
+Catter is a social media platform built by and for cats.
+The cats’ platformhas recently been getting spammed
+by dogs, so they have decided to remove all mentions
+of dogs from their platform entirely. However, cats
+are not great at programming. They need your help
+in removing all of the dog content from their plat-
+form.
+We used Python for our assignment since this is a common lan-
+guage for intro-level courses. The assignment covers file input and
+output, parsing text using regular expressions, functions, loops,
+conditionals, and using a natural language processing library (Python’s
+NLTK library).
+The assignment asks students to write several iterations of a
+content moderator for Catter. First, students must simply detect
+whether a string has the word “dog” in it, scan a file to remove all
+posts with occurrences of “dog,” and write the filtered content to
+a new file. In the next section, the dogs in our hypothetical sce-
+nario learn to bypass this filter, and the students are asked to up-
+date their system to detect a list of words, including slang such
+as “doggos” and “dawg.” They are asked to read in a file that con-
+tains this list, then use their previous code to remove all posts with
+those words. Finally, the dogs start posting negative comments
+about cats, bringing in the need for the NLTK sentiment analyzer.
+The last part requires the students to remove any posts containing
+the word “cat” that the analyzer marks as having a negative sen-
+timent. These different steps are designed to illustrate challenges
+a real developer might face when moderating content with code:
+wrestling with users who “outsmart” the algorithm so their con-
+tent gets posted–regardless of whether they are bad actors or are
+themselves wrestling with systematic errors in the system.
+In order to encourage students to consider the nuances within
+content moderation beyond this hypothetical scenario, and to ex-
+amine the decisions they made in their own code, we followed the
+coding exercise with a reflection section. In this part of the assign-
+ment, students watch Maarten Sap’s talk on Social Bias Frames,
+which shows how binary keyword matching, like the students were
+required to code, can miss sexist, racist or otherwise hateful speech
+[29]. The assignment asks for written responses to questions that
+prompt students to critique the way that the code they wrote mod-
+erates content. We chose to include written reflections because pre-
+vious work suggests that integrating reflections is critical in en-
+couraging students to consider their ethical own responsibility in
+creating technology [4]. The reflection questions also broaden the
+discussion, asking students to provide a news article related to the
+topic of content moderation. Our aim here was to expand on real
+world connections, which has been shown to have a positive effect
+on student engagement [13, 23], including increasing participation
+amongst underrepresented groups in computing [19].
+3
+EVALUATION METHODS
+To evaluate the design of the assignment, we created a plan for 90-
+minute focus groups with 3-5 participants each, all of whom had
+already taken an intro programming course. These groups partici-
+pated in the following tasks:
+(1) Pre-questionnaire about content moderation
+(2) Review of coding portion of assignment
+(3) Group discussion and individual surveys on coding portion:
+(a) Reactions to the assignment
+(b) How they would complete it
+(c) Any changes they would suggest
+(4) Review of reflection portion
+(5) Group discussion and individual surveys on reflection por-
+tion
+(6) Brainstorm new assignment ideas centered around ethical
+dilemmas
+(a) Generated list of tech controversies
+(b) Discussed how these controversies tie in with CS topics
+One key goal for the group discussion and questionnaires was
+to gauge whether and how this assignment made participants con-
+sider ethical dilemmas within content moderation, through the cod-
+ing alone or with the supplementary reflection. For the final brain-
+storming session where the students thought about how we might
+integrate ethics into programming assignments more broadly, we
+drew from our own assignment creation process, starting with list-
+ing social issues the assignment could be based on. We created the
+list by asking participants “What controversies or impacts of tech-
+nology have you either heard about or been personally affected
+by?” which they responded to by posting on a shared Google Jam-
+board. Afterwards, we asked participants to choose one contro-
+versy that impacts them or other students the most. Finally, we
+presented the students with a list of CS topics and asked how they
+might tie in with the list of tech controversies they created.
+We ran one pilot focus group with members of our lab, and af-
+ter the study was approved by the university’s institutional review
+board, began recruiting participants, looking for anyone who had
+taken an introductory programming course at our large, predom-
+inantly white, public university in the United States. The recruit-
+ment materials, which were shared with computer science profes-
+sors and through flyers on campus, mentioned only that partici-
+pants would be evaluating CS curriculum. We did not mention that
+the assignment involved ethics to reduce selection bias of students
+who had a specific interest in ethics.
+We ultimately had a total of 16 individual participants, with 2-5
+participants per group. Our participants were majority white, with
+a roughly even split between men and women. The focus groups
+included the following participants:
+• Group 1 (P1-3): a freshman, sophomore, and junior, all CS
+majors
+• Group 2 (P4-7): a freshman CS major, two juniors with dual
+majors in CS and applied math and economics, and a grad-
+uate student CS major
+• Group 3 (P8-10): a junior and sophomore information sci-
+ence majors, and a senior with dual majors in CS and me-
+chanical engineering
+• Group 4 (P11-12): two graduate student CS majors
+• Group 5 (P13-16): a freshman aerospace engineering major,
+a senior CS major, a graduate student information science
+major,and a graduate student CS major
+
+We adapted the plan as we ran the groups, shortening the sec-
+tion about how the assignment would be solved in exchange for
+more time for reflecting on how the assignment prompted ethical
+reasoning and on integrating ethics in general.
+Following the focus groups, we conducted a thematic analysis
+of the transcripts [7]. Two of the authors highlighted anything of
+note in the transcripts and created a list of themes from these high-
+lights. We then performed a second round of coding for those spe-
+cific themes and finally we wrote memos about each theme, which
+we synthesized into our findings and discussion. During this pro-
+cess, all authors met to discuss and iterate on themes. Our findings
+below represent the synthesis of these themes, including represen-
+tative quotes.
+4
+FINDINGS
+From the focus groups, we gained perspective on the strengths and
+weaknesses of the assignment and for student opinions integrating
+ethics in general. We also created a bank of new CS assignment
+ideas integrated with ethical topics that the students in our groups
+cared about.
+4.1
+Strengths of the assignment
+Using a hypothetical scenario within the assignment was success-
+ful from our observations and from our participants’ perspectives.
+Starting with the scenario of a social media site for cats neutralized
+the subject of content moderation, which can include politically
+or emotionally charged subjects such as misinformation and hate
+speech. However, the hypothetical still promoted discussion on a
+topic that can be uncomfortable for students, as well as educators,
+to talk about, by allowing them to bring up these issues themselves.
+Participants responded in the questionnaires that the pretend sce-
+nario was helpful for easing into reflection on content moderation
+before jumping into controversial real world issues. The groups
+readily connected the fake scenario to real issues, with the benefit
+that students conjured up these issues organically, instead of the
+assignment feeding them specific situations.
+The assignment brought to mind real-world issues in content
+moderation, even prior to reviewing Sap’s video and the reflection
+questions. When asked if the coding piece made participants think
+differently about content moderation, some responded that it did,
+bringing up issues like censorship, Facebook filtering out content,
+bias in automated systems, and misinformation. In group discus-
+sion, when asked what the implications of the cats’ moderator are,
+participants immediately tied the hypothetical situation to reality.
+Many thought the content moderator resembled “echo chambers”
+within real social media platforms, which limit content to only
+include opinions the user agrees with. Another theme discussed
+across groups was over-moderation of content. One participant
+gave the example of YouTube removing benign comments, which
+negatively affects both the content creators and the commenters.
+They also reasoned about the fairness of moderation systems de-
+fined by companies, with P16 reflecting that removal of content is
+“not within the users’ power, the power is in the platform,” and an-
+other participant responding that “the gatekeepers are often a small
+group of executives. There aren’t many regulations that companies
+need to follow.”
+The assignment caused P14 to reconsider third-party tools she
+used in other courses, saying that the assignment “shows you that
+you might be given a function, or a tool, but how much do you just
+put your blind trust in it? In my class, we were given a lot of like
+functions that we would have to put together into our program but
+that we didn’t write ourselves. But [it] makes you wonder ‘how was
+this written? How does it work?’ even if it’s supposed to be a black
+box type thing...you can’t just go and completely put your trust in
+like the sentiment analyzer or the AI program that’s supposed to be
+moderating it.” For them, the assignment spurred critical ethical
+thinking about their coding.
+All groups were able to create a feasible plan for completing
+the code within a short time-frame. Most participants agreed that
+the assignment would be appropriate for an intro-level course, but
+some gauged its difficulty as mid or even high-level. The next sec-
+tion outlines modifications we could make to the assignment to
+address the difficulty level, as well as other improvements partici-
+pants suggested.
+4.2
+Suggestions for improving the assignment
+To reduce the difficulty of the assignment to an introductory level,
+participants suggested alternatives to regular expressions for the
+keyword matching or ways to provide more support, such as code
+examples using regular expressions for students to work off of.
+Participants also thought installing the NLTK library and under-
+standing sentiment analysis would be difficult for intro-level stu-
+dents. The sentiment analysis requirement could be removed from
+the coding section but be kept as a discussion topic within the re-
+flection. Depending on the difficulty level and other aspects of the
+course, making these kinds of adjustments could be critical; prior
+work has noted that for programming assignments that integrate
+ethics, when the students are struggling too much with the techni-
+cal content, they focus less on the contextual aspects [10].
+Participants also offered ideas for additions to the coding re-
+quirements. P10 proposed requiring students to give each other
+example phrases that could break their partner’s moderator, and
+then build something to handle those phrases. This method, which
+is known as “Build it, Break it, Fix it” within cybersecurity com-
+petitions, has been used in other ethics-oriented technical assign-
+ments [15]. P16 recommended creating a display of phrases that
+were moderated out, and creating a tagging system to categorize
+moderated phrases. Research on empowering real creators over
+moderation of their content has shown demand for features like
+this [17]. These ideas could be added to a follow-up assignment or
+to modify the existing assignment for a higher-level course.
+Participants also provided constructive feedback about how stu-
+dents given this assignment would reflect on the ethics of content
+moderation. We observed from our analysis that discussion was
+heavily biased towards the negative effects of content moderation
+with little discussion of cases where it could be beneficial. One
+topic rarely mentioned within the focus groups was the impact of
+false negatives, when harmful phrases pass through a moderator
+undetected. This may be partially due to participants’ pre-existing
+beliefs about content moderation, but the scenario in the assign-
+ment could be adjusted to show how a lack of moderation can
+have negative impacts just as over-moderation can. For example,
+
+the assignment could include examples of posts that are reason-
+able to moderate out, rather than the cats’ current strategy leaning
+towards censorship. P15 recommended incorporating fake facts or
+news that dogs have spread, like providing false information about
+a vet clinic to steer cats away from getting health care. Based on
+this feedback we were able to make a number of improvements on
+the assignment which we will cover in more detail after the find-
+ings.
+4.3
+Opinions on integrating ethics
+Aside from gathering feedback on this specific assignment through
+the focus groups, we also aimed to gauge students’ opinions of in-
+tegrating ethical concepts in general. Their reactions to including
+an ethical reflection within this assignment were mostly positive
+(though we acknowledge the possibility of response bias within the
+study, given the topic). All but two participants responded in the
+questionnaire that they think Sap’s video and the reflection belong
+within a CS class. They thought it would prompt students to think
+about the direct impacts of technology they are contributing to,
+something “many students up until this assignment may never have
+been challenged about,” according to one participant. They stressed
+the importance of reflecting on impacts, one writing: “Implications
+of algorithms and ethics should be part of CS curriculum. People
+creating the problems should also be fixing them or at least know
+about them.”
+Participants also mentioned the potential of the real world con-
+text to spark interest in more advanced topics. It “makes students
+think more about applications out in the real world...maybe if peo-
+ple are interested, you could point them to the class for natural lan-
+guage processing, [for] down the road,” one remarked. Another re-
+sponded that “contextualizing” the assignment “provides extra mo-
+tivation/understanding and maybe career insight,” suggesting that
+including a context could motivate students to continue pursuing
+careers within technology.
+However, participants did raise concerns about the integration
+of these topics into technical assignments. Some expressed that
+they would rather study ethics outside of CS curriculum, and prefer
+it be in a standalone course. “If I am expecting and wanting a class
+to be technical in nature, I would likely be unhappy writing a paper
+since it is not what I signed up for,” P6 explained. Others mentioned
+they would see the reflection portion of the assignment as a “grade
+bump.” P8, who had previously experienced ethics content in an
+economics course, said they did not have to reason deeply about
+conflicts to complete the assignment. It is a challenge to determine
+how to grade whether a student has really thought through the im-
+pacts of the technology in question. Provoking meaningful think-
+ing while not adding too much additional work is a fine line to
+tread.
+Students are already working hard to learn the technical skills of
+programming, so adding extra work could become a burden rather
+than an opportunity to reflect and learn. For our assignment specif-
+ically, building a fair moderation system seemed too difficult to ac-
+complish for some participants. Many expressed that real-world
+content moderation systems are so flawed that we are better off
+with no moderation at all. This presents a challenge for integrat-
+ing ethics: how can we incorporate ethics without only focusing
+on discussions of harmful technologies, but also create solutions
+to these difficult problems?
+Another concern a small number of participants raised was that
+ethics-oriented assignments carry the risk of inappropriately im-
+posing a particular set of values on students. “Make the students
+think about this on their own, rather than trying to influence their
+opinions,” suggested one student. Another raised concerns about
+conformity; that students would side with the opinion that the ma-
+jority of the class was expressing rather than speak their own mind.
+Ethics can be a thorny subject, and uncharted territory for some
+CS instructors, but the effort to incorporate them is worth it. We
+explain suggestions for facing these barriers in the discussion.
+4.4
+New assignment ideas
+One other product of the groups was ideas for new ethics-based
+assignments. Collaborating with students proved to be a fruitful
+method for creating assignment ideas. Participants generated a list
+of tech controversies they felt most affected by, then connected
+these controversies with assignment ideas and corresponding con-
+cepts covered in CS curriculum. Several of these assignment ideas
+from our participants can be found in Table 1.
+One group agreed that the worst of the tech controversies was
+addiction to social media and “doom-scrolling,” when a user gets
+stuck endlessly scrolling through content. They proposed a sorting
+algorithm assignment, P15 saying “you are recommendedcontent on
+TikTok based off of how relevant it is to you, but also how popular
+it is, right? So...how do you decide what is sorted to the top?” An-
+other group suggested an assignment to create a user interface or
+experience to either increase or decrease screen time.
+Others situated sorting algorithms in the context of search re-
+sults for news, and how this could influence a users’ opinions over
+time. P8 suggested graph algorithms for determining relevance,
+drawing from an example she had seen in another course, where
+she determined sort order based on how many connections a node
+had to other nodes in a graph.
+Another idea came from a blood type matching algorithm P7
+was required to write for an algorithms class she had taken. She
+thought it could be expanded to have ethical reasoning about or-
+gan donor matching. The algorithm could “take into account how
+long someone’s been on the organ waiting list, how fresh the organ is,
+and then assign it based on age,” according to this participant. We
+could see a scenario where students create different versions of the
+system and run tests for various groups, discussing the trade-offs
+in fairness for each system.
+Another common controversy was data collection and privacy.
+Participants and the first author discussed one assignment idea to
+develop an object-oriented design or a database representing user
+data that highlights how much information platforms gather on
+their users. The assignment could incorporate discussions of how
+engineers manage data, especially in cases where it needs to be
+erased for privacy reasons. Participants also mentioned the pro-
+cess of data extraction. P2 remarked that a system “could have your
+favorite TV show be known just by what you’ve been clicking...your
+race, gender, and name.” Perhaps students could develop and reason
+about systems that infer information about a user based on pieces
+of data they have collected. Targeted ads also came up, a subject
+
+Table 1: The first column represents a controversy in the
+tech industry that our participants identified, followed by
+an assignment idea that could be developed around that con-
+troversy in the second column, and the assignment’s corre-
+sponding CS topic in the third column.
+Context
+Assignment Idea
+CS Topic
+Algorithms for
+organ donor
+matching
+Sort recipients using differ-
+ent algorithms and discuss
+trade-offs between them
+Sorting algo-
+rithms
+Addiction to
+social media
+Create a program that de-
+cides how social media
+posts are ordered
+Conditionals,
+trees, graphs
+Addiction to
+social media
+Design UI/UX that increases
+or decreases screen time
+UI/UX design
+Dark patterns
+on the web
+Design a system to make it
+difficult or easy to unsub-
+scribe from an email list or
+opt out of cookies
+UI/UX design
+Influence of
+search engines
+on what news
+stories users
+see
+Use various search or sort
+algorithms to create a list
+of ordered results of news
+stories
+Sorting algo-
+rithms, search al-
+gorithms, object-
+oriented pro-
+gramming
+Misuse of user
+data and right
+to be forgotten
+Compare data storage meth-
+ods and the complexities of
+keeping data private and
+ensuring complete deletion
+Memory alloca-
+tion and man-
+agement, object-
+oriented pro-
+gramming
+User data and
+privacy
+Write a program that infers
+information about a user
+based on existing user data
+Conditionals,
+variable
+Usability and
+inclusivity in
+web forms
+Compare from different
+users’ perspectives how in-
+clusive different web forms
+are in terms of race, gender,
+sexuality
+UI/UX design,
+personas
+Mental health
+impacts of so-
+cial media apps
+Create an image filter that
+changes the sentiment of
+the image
+Image process-
+ing, could use
+machine learn-
+ing
+which has already been incorporated into CS assignments by other
+researchers [10].
+5
+IMPROVEMENTS TO THE ASSIGNMENT
+In considering both the specific feedback from participants and
+their reflections about this type of assignment in general, we iter-
+ated on our original assignment. After observing the participants
+think through how they would complete the coding portion of the
+assignment, we modified the wording of the instructions to make
+them easier to read. Instead of using a paragraph of text to lay out
+instructions, we detailed out steps in a list and made it more clear
+what each function in the skeleton code should do. Because many
+participants noted the NLTK toolkit as the most challenging part
+of the assignment, and that it would be useful to have more back-
+ground information, we added a more detailed explanation of the
+output and a link to a guide on using this library. We also removed
+a question from the reflection section about comparing binary to
+structured toxic speech detection that some participants found dif-
+ficult to answer given the 3-minute video.
+We also received suggestions for extending the assignment, and
+included these in the resources we are providing to instructors who
+want to use the assignment. These extensions included adding a
+tagging system to moderate posts, allowing users to input certain
+words they would like to moderate, creating a display of posts that
+were flagged for removal, and including a pair programming activ-
+ity where students would try and break each others’ moderation
+systems, then modify code to solve for those cases.
+In addition to the changes we implemented, we have ideas for
+other adaptations to the assignment. The feedback indicated the
+need to balance out the scenario so that students understand the
+potential benefits of content moderation. The participants gave us
+specific suggestions on how we might do this: by including exam-
+ples where the dogs were posting misinformation that could be
+detrimental to the cats’ lives, for example. For the reflection, the ex-
+ample detailed in the video is offensive to women. Students could
+be asked to provide examples of how other groups are affected by
+false negatives or positives in content moderation. Scholarly works
+about bias against marginalized groups in speech toxicity detec-
+tion systems could also be discussed [14, 27, 30]. Instructors us-
+ing this assignment could consider using in-class discussion rather
+than written responses for the reflection piece of the assignment.
+This would address students’ aversion to being required to do writ-
+ten work within a technical assignment. We also observed benefits
+in the focus group of having a group discussion. The group discus-
+sion allowed participants to hear each others’ viewpoints instead
+of reflecting individually. Opting for a discussion would also take
+out the need for class staff to grade written responses.
+6
+DISCUSSION
+We believe CS educators can empower students to become more
+proactive in considering social impacts of their work. Embedding
+ethics topics and discussions throughout computer science course-
+work can help accomplish this, and has other positive side effects.
+Including ethics-based assignments can bolstercultural competency
+when it brings up issues such as racial or gender bias in AI, prepar-
+ing students to be more inclusive and equitable when they enter
+an industry known for the opposite [33]. Previous work also shows
+that including a real-world context can increase retention of under-
+represented groups in computing [19]. Our participants reported
+our assignment to be more interesting and memorable than a stan-
+dard programming assignment, reinforcing findings from previous
+work that integrating ethics improves engagement [10].
+Despite these benefits, we also recognize that embedding ethics
+can be daunting. In this section, we give recommendations in creat-
+ing ethics-based assignments. We explain why gathering feedback
+was useful for us, give strategies for tackling controversial subject
+matter, and measure the trade-offs of hypothetical vs. real scenar-
+ios for sensitive subjects.
+Through our focus groups, we tested the effectiveness of embed-
+ding ethics within our new CS assignment. The participants’ feed-
+back positively reinforced our methods of embedding ethics and
+gave us ideas for improving the assignment. The feedback helped
+
+us gauge whether including a reflection in addition to embedding
+the coding situation in an ethical scenario was necessary. We found
+that the reflection was worth including.
+Prior to the reflection part of our assignment, few participants
+acknowledged the potential benefits of content moderation or con-
+sidered the impacts of false negatives, when content that should be
+removed passes through a moderation system undetected. The dis-
+cussions in the focus groups revealed that the scenario in our cod-
+ing portion highlighted the impacts of false positives, when benign
+content is flagged for removal, and failed to draw attention to false
+negatives. This may also be due to participants’ experiences with
+content moderation outside of the focus groups, since both false
+positives and false negatives in content moderation often dispro-
+portionately impact people from marginalized groups, such as peo-
+ple of color [14, 24]. Participants in our groups, which reflected the
+population of our predominantly white university, may have never
+personally experienced the negative effects of false negatives. In a
+study of an ethics-related assignment, Klassen and Fiesler found
+that when speculating about ethics in the classroom, students and
+instructors may fail to consider perspectives outside of their own;
+as an instructor in their study said, “People tend to lean on their
+own experiences pretty heavily in speculation, and don’t, unless
+they’re very carefully prompted, consider broader context” [20].
+Including a reflection is an opportunity to “carefully prompt” stu-
+dents to consider other perspectives.
+In the focus groups we were also able to test out using a hy-
+pothetical situation, which we found to be effective. However, in-
+structors do need to be careful about their transitions to real issues
+from the hypothetical. In our case, at least one participant found
+the transition from the cats’ platform into more charged topics like
+racist hate speech “too sudden.” There is also the risk that students
+will fail to bridge the gap between the hypothetical situation and
+the severity of the ethical impact in real life. While we did not ob-
+serve this to be the case for our assignment, we believe it’s impor-
+tant to ensure that students are not getting a watered-down rep-
+resentation of the ethical issue being addressed by an assignment.
+We found the benefits of using a hypothetical scenario worth these
+extra considerations, because this approach removed barriers in
+considering ethics for those averse to controversial discussions.
+Running the focus groups provided us with valuable feedback
+and we recommend gathering feedback to anyone creating ethics-
+based CS assignments. Without hearing from our participants, we
+would have remained unaware of the blind spots we had to issues
+with our assignment. Some of the improvements listed above came
+directly from students in our groups. We got suggestions for con-
+fronting barriers they perceived students would have with ethics-
+based content. We also generated a wealth of ideas for more ethics-
+oriented assignments, spending only 30 minutes or less in each of
+the ���ve focus groups. The tactic we used to facilitate brainstorm-
+ing, starting from tech controversies and then drawing connec-
+tions to CS topics, flowed more easily than simply asking students
+if they had ideas for ethics-based CS assignments. The engagement
+in the brainstorming activity was also encouraging; participants
+were active in generating ideas and applying them to CS topics.
+Once we had student feedback showing their concerns with
+ethics-based content, we could create some mitigation strategies.
+Some participants shared concerns about ethics-oriented assign-
+ments imposing a particular belief onto them about a particular
+technology. One article on integrating ethics states that “a good
+technology ethics course teaches students how to think, not what
+to think, about their role in the development and deployment of
+technology.” [5]. Consider the whole picture of any ethical context
+you integrate and keep questions posed to students open-ended.
+We made edits along these lines to our focus group script after pi-
+loting it with a group from our lab. In place of “Do you think that
+the moderation system you implemented is harmful?” we asked
+“What are the implications of the moderation system?”.
+Finally, we strongly encourage CS educators to utilize the re-
+sources around them when designing similar assignments. Our
+content moderator assignment materials are publicly available.1
+There are also many open source ethics-based CS assignments cre-
+ated by other educators, such as those developed as part of Mozilla’s
+Responsible Computing Challenge [2]. We also invite educators to
+use the ideas generated by our participants to create new assign-
+ments that fit into their courses. Educators looking to test their as-
+signments could use teaching assistants to pilot assignments, there-
+fore making them collaborators in the process as well, in place of
+running more time-consuming focus groups. Providing in-depth
+feedback could also be presented as an extra-credit opportunity
+for students. Incorporating ethics may seem daunting, but can be
+accomplished, as others have suggested, by starting small and col-
+laborating between disciplines, adapting existing assignments to
+include an ethics-oriented context [10].
+7
+CONCLUSION
+Rather than relying on a standalone course for teaching the ethi-
+cal implications of the technology students will produce, there is
+a growing movement to embed ethics throughout the entire com-
+puter science curriculum. In this experience report, we demoed an
+assignment to five small groups of participants that reflected on the
+design and implications of a content moderation system for a cat-
+driven social media platform. Discussions with these participants
+revealed students’ opinions about integrating ethics into course-
+work, feedback on the assignment itself, and ideas for future as-
+signments. With these findings in mind, we recommend that CS
+educators creating ethics-based assignments make use of student
+or TA feedback to improve assignments, use strategies to limit im-
+position of biases on the ethical dilemma, consider using hypothet-
+ical scenarios in their assignments, and embrace resources around
+them. The benefits of integrating ethics into CS curriculum, includ-
+ing better student engagement as well as preparing them to create
+technology with societal impacts in mind, are worth the effort of
+creating ethics-based assignments.
+8
+ACKNOWLEDGMENTS
+This research was supported by Omidyar Network as well as the
+Responsible Computer Science Challenge (with funding from Mozilla,
+Omidyar Network, Schmidt Futures, and Craig Newmark Philan-
+thropies). We would like to thank our participants and the Inter-
+net Rules Lab for their feedback on this project, particularly Ella
+Sarder, Johnny Sreenan, and Camryn Kelley.
+1https://www.internetruleslab.com/ethicsbased-computer-science-assignments#content-mod
+
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+

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+Data-Driven Model Identification via
+Hyperparameter Optimization for the
+Autonomous Racing System
+Hyunki Seong1, Chanyoung Chung2∗, and David Hyunchul Shim1
+Abstract— In this letter, we propose a model identifica-
+tion method via hyperparameter optimization (MIHO). Our
+method is able to identify the parameters of the paramet-
+ric models in a data-driven manner. We utilize MIHO for
+the dynamics parameters of the AV-21, the full-scaled au-
+tonomous race vehicle, and integrate them into our model-
+based planning and control systems. In experiments, the
+models with the optimized parameters demonstrate the
+generalization ability of the vehicle dynamics model. We
+further conduct extensive field tests to validate our model-
+based system. The tests show that our race systems lever-
+age the learned model dynamics well and successfully
+perform obstacle avoidance and high-speed driving over
+200km/h at the Indianapolis Motor Speedway and Las Vegas
+Motor Speedway. The source code for MIHO and videos of
+the tests are available at https://github.com/hynkis/MIHO.
+Index Terms— Model identification, hyperparameter opti-
+mization, autonomous vehicle
+I. INTRODUCTION
+U
+nderstanding the system model is essential for robotic
+applications, especially high-speed safety-critical au-
+tonomous systems. Unlike low-speed driving, various dynam-
+ics elements such as chassis, tires, or engines become cru-
+cial to implement high-speed autonomy. Model-based optimal
+control [1], [2] is well-suited for handling those factors and is
+widely used to design dynamics system control. By leveraging
+physics-based parametric dynamics models, It optimizes driv-
+ing maneuvers with respect to a designed objective function
+and enables safe and reliable control system design.
+Despite the success of the model-based approach in robotics,
+the model-based algorithm has two fundamental challenges:
+model fidelity and tractability. The performance of model-
+based approaches relies heavily on the accuracy of the model.
+However, identifying accurate models is often laborious or in-
+tractable because of their large search space and non-linearity.
+This work was supported by Institute of Information communica-
+tions Technology Planning Evaluation (IITP) grant funded by the Korea
+government(MSIT) (2021-0-00029, Development of Indoor Autonomous
+Drone for Performing Multiple Missions Based on Artificial Intelligence)
+∗Corresponding author
+1 The authors are with the School of Electrical Engineering, Korea
+Advanced Institute of Science and Technology, Daejeon, South Korea.
+(email: [email protected];[email protected])
+2 Chanyoung Chung is with the JPL Science Division, NASA, Califor-
+nia, USA. (email: [email protected])
+Besides model accuracy, models also need to be computa-
+tionally feasible considering the real-time control applications.
+High-fidelity but highly complex models are difficult to inte-
+grate into real-time safety-critical driving systems.
+To tackle those challenges, conventional approaches, includ-
+ing the Prediction Error Method, are used to identify model
+parameters [3]. However, those methods often require the
+model structure to be linear or in a specific mathematical
+form, which might not be feasible for the design of the
+real-time autonomous driving system. On the other hand,
+in several recent works, data-driven approaches using neural
+networks, Gaussian processes, or Bayesian methods have been
+actively employed for nonlinear system dynamics modeling
+and have shown promising results [4]. In [5], they proposed
+a simple neural network to replace a single-track vehicle
+model and used it to generate feedforward control signals.
+Similarly, in [6], they designed Deep Neural Networks (DNN)
+as a model approximator to identify the vehicle dynamics
+model in an end-to-end learning fashion. However, while DNN
+is an efficient way to approximate nonlinear systems, it is
+difficult to integrate with non-learning model-based methods,
+which are reliable in real-world applications. Furthermore, it
+is challenging to ensure the validity of the DNN model in
+unseen driving scenarios without large-scale field tests.
+In this letter, we propose a data-driven model identifica-
+tion method via hyperparameter optimization (MIHO) for a
+high-speed autonomous driving system. Our key idea is to
+leverage a data-driven parameter optimization approach from
+machine learning to identify physics-based parametric models
+without any structural model requirement. To this end, we
+adopt a novel hyperparameter optimization (HPO) method
+that has an efficient exploration and exploitation strategy.
+Using the proposed method, we estimate the parameters of
+the integrable parametric dynamics models for a full-scaled
+autonomous racecar system, Dallara AV-21 (Fig. 1), at the
+Indy Autonomous Challenge (IAC) [7]. We validate our
+proposed approach by integrating identified models into the
+high-speed autonomous system and conducting extensive field
+experiments, including over 200km/h autonomous driving
+and obstacle avoidance scenarios in the Indianapolis Motor
+Speedway (IMS) and Las Vegas Motor Speedway (LVMS).
+In summary, our technical contributions are as follows:
+• We propose a data-driven model identification method via
+hyperparameter optimization.
+arXiv:2301.01470v1  [cs.RO]  4 Jan 2023
+
+• We design model-based planning and control systems
+incorporating the learned vehicle dynamics models.
+• We integrate the systems with learned model parameters
+into the full-scaled autonomous race vehicle and exten-
+sively validate them during the IAC.
+II. MODEL IDENTIFICATION VIA HYPERPARAMETER
+OPTIMIZATION
+The more accurately the system dynamics are described, the
+greater the nonlinearity and number of parameters required
+for the dynamics model. Therefore, an efficient parameter
+estimation approach is necessary to find the parameter con-
+figuration of such a complex model. In this letter, we propose
+a model identification method via hyperparameter optimization
+(MIHO) to learn the optimal model parameter configuration by
+a data-driven approach. Hyperparameter optimization (HPO) is
+the problem of selecting an optimal hyperparameter configura-
+tion required for neural network training in the machine learn-
+ing field [8]. A hyperparameter is a parameter that controls
+the training process. HPO optimizes the hyperparameter con-
+figuration by evaluating the performance of the configuration
+during the model training process. Since one course of neural
+network training requires a substantial time, HPO focuses
+on the balanced exploration and exploitation strategy for the
+efficient optimal hyperparameter selection. [8]. Motivated by
+the balanced strategy, we design MIHO by adopting the HPO
+to the model identification problem. First, we regard a model
+parameter configuration p as a set of hyperparameters of a
+nonlinear dynamics model f. Then, we identify the parameter
+configuration by evaluating the following objective function
+inspired by the standard supervised learning problem:
+L =
+1
+|D|
+�
+(x,y)∈D
+∥y − f(x; p)∥2,
+(1)
+where x, y denote the sampled input and output data of the
+model f from a given dataset D. By minimizing this learning
+objective, we find an optimized model parameter configuration
+p∗ that has the minimum model error with the observed model
+output y. The model f has no limitation on its formula or form.
+Thus, our method is able to be used for arbitrary parametric
+models, such as a combination of polynomial or mathematical
+terms, as well as analytic physics-based models.
+We implement MIHO incorporating a bandit-based HPO
+algorithm, Hyperband [9], as summarized in Algorithm 1. It is
+Algorithm 1 MIHO Algorithm based on Hyperband
+Input: R, η, D
+1: smax ← ⌊logη(R)⌋, B = (smax + 1)R
+2: for s ∈ {smax, smax − 1, ..., 0} do
+3:
+n = ⌈ B
+R
+ηs
+(s+1)⌉, r = Rη−s
+4:
+P = get model param config(n)
+5:
+for j ∈ {0, ..., s} do
+6:
+nj = ⌊nη−j⌋, rj = rηj
+7:
+L = {eval with mutation(p, rj, D) : p ∈ P}
+8:
+P = select top k config(P, L, ⌊nj/η⌋)
+Output: Optimized parameters p∗ with the smallest loss.
+Fig. 1.
+Overview of our autonomous driving system in the AV-21. Our
+learned model parameters are embedded in the planning and control
+modules that are covered in this letter (highlighted in blue). Several input
+variables are omitted for clarity.
+a variation of a random search algorithm with explore-exploit
+theory to find the optimal hyperparameter configuration based
+on an evaluation loss. The algorithm needs two arguments:
+R, the maximum amount of resource (e.g., the number of
+evaluation iterations) that can be allocated to a single config-
+uration, and η, a value that determines the proportion of the
+discarded configurations. The two arguments derive smax + 1
+combinations (called ”brackets” in [9]) of the values n and r,
+which enables various ratios of exploration and exploitation
+for finding the optimal parameter configuration. Hyperband
+compares the evaluation loss of each sampled configuration
+and allocates more resources to the configurations with lower
+evaluation losses, excluding the configurations with higher
+losses. It repeats the sampling and exclusion processes until
+the last configuration remains to obtain the optimal set of
+hyperparameters. To adjust the HPO algorithm to the model
+parameter optimization, we add the Gaussian mutation [10]
+during the evaluation to explore the new neighbor parameters
+that might have less model loss. Unlike the original HPO,
+which only allocates more resources ri, our approach, MIHO,
+adds noise perturbation at the selected parameter configuration
+p after the resource allocation as:
+pmut = p + σ ⊙ ϵ,
+ϵ ∼ N(0, I),
+(2)
+where σ is the standard deviation of the exploration noise that
+is annealed over the course of the evaluation [11]. We define
+the following three functions for the HPO process in MIHO:
+• get model param config(n): a function that returns a set
+of n i.i.d model parameter configurations from the normal
+distribution pre-defined over the configuration space.
+• eval with mutation(p, rj, D): a function that receives a
+parameter configuration p, an allocated resource rj, and
+a dataset D as arguments. Using the dataset, this function
+evaluates an initial configuration and mutates it for the
+allocated rj iterations by Eq. 2. If a mutated configuration
+pmut has a less loss than the initial one, the function
+replaces p with pmut. It returns the final loss after
+spending the allocated resources.
+• select top k config(P, L, k): a function that receives a
+set of hyperparameter configurations P with their corre-
+sponding evaluation losses L and returns the top k high-
+performing configurations (here, k = ⌊nj/η⌋).
+
+Planning
+Control
+Full-scale Racecar
+Reference
+Lateral
+Trajectory
+Control
+K
+Tt,Tb
+Velocity
+Vx,des
+Longitudinal
+Throttle / Brake
+Planner
+Control
+(axr)
+ControlIII. VEHICLE DYNAMICS MODEL
+A. Tire Dynamics Model
+Tire dynamics is one of the factors that significantly affect
+the nonlinearity of driving dynamics. Especially the lateral
+tire model is crucial to design stable path-tracking control in
+high-speed driving. The tire model [12] can be described as a
+function of the slip angle αi, slip ratio ρx,i, inclination angle
+θi, tire load Fz,i, and current velocity vx,i, which has a lateral
+tire force F ∗
+y,i of each tire (i ∈ {LF, LR, RF, RR}) as,
+F ∗
+y,i = ftire(αi, ρx,i, θi, Fz,i, vx,i).
+(3)
+Although the model has high fidelity with various dynamics
+perspectives, it has low suitability for designing the controller
+of high-speed driving, which requires real-time performance.
+Therefore, we first define a tire model with dimension-
+reductionthat can be applied to model-based control design
+within an acceptable complexity. We then optimize the model’s
+parameter configuration to represent the overall tire charac-
+teristic of a given dataset using our MIHO algorithm. We
+follow the Pacejka tire model [13] to define the tire dynamics.
+While the prior work neglect offsets, we formulate a tire model
+Fy,i = ft,i(αi; pt,i) containing offset parameters Sx,i, Sy,i
+to describe the asymmetric tire characteristic determined to
+maximize cornering performance on an oval track:
+Fy,i = Di sin(Ci arctan(Bi(αi + Sx,i))) + Sy,i,
+(4)
+where
+the
+tire
+model
+parameter
+configuration
+pt,i
+=
+{Bi, Ci, Di, Sx,i, Sy,i} is identified by minimizing the follow-
+ing tire model objective with a given dataset Dt,i as
+Lt,i =
+1
+|Dt,i|
+�
+(αi,F ∗
+y,i)∈Dt,i
+∥F ∗
+y,i − ft,i(α; pt,i)∥2.
+(5)
+B. Engine Torque Model
+The powertrain system of our racecar consists of an internal
+combustion engine, transmission, and wheels. The racecar is a
+rear-wheel-drive vehicle whose traction force Fx,r is generated
+by engine-based driveline dynamics. We model the equation
+of the longitudinal dynamics [14] as follows:
+max = Fx,r − Cdv2
+x − Cr,
+(6)
+where m is the vehicle mass, vx is the longitudinal velocity,
+Cd denotes the drag coefficient, and Cr denotes the rolling
+resistance. Following a prior work [15], the traction force can
+be expressed as:
+Fx,r = max,r = Teηtigi0
+Rw
+,
+(7)
+where ax,r denotes the traction acceleration, ηt denotes the
+efficiency of the transmission, ig, i0 denote the transmission
+ratio of the current gear and final reducer, and Rw denotes
+the wheel radius. Te = fe(we, τt) is the engine torque map
+in terms of the engine speed we and throttle command τt.
+Due to the high complexity of the engine characteristic, the
+torque map is expressed as an experimental lookup table
+based on engine torque curves of specific throttle opening
+commands. Although the engine torque model can be obtained
+by the engine dynamometer testing [16], it could suffer from
+modeling error because the dynamometer testing is done in a
+static environment without longitudinal experiment. Therefore,
+we build the engine torque map that integrates the dyno data
+with learned torque curves based on our data-driven model
+identification approach. We express an engine torque curve
+Te,τt = fτt(we; pτt) of a throttle command τt as a 3rd order
+polynomial function of the engine speed we as:
+Te,τt = pτt,0 + pτt,1we + pτt,2w2
+e + pτt,3w3
+e,
+(8)
+where pτt = {pτt,0, pτt,1, pτt,2, pτt,3} is the torque model
+parameter configuration. Using the resultant traction acceler-
+ations ax,r while driving, the engine torque output T ∗
+e,τt is
+obtained by Eq. 7. Then, pτt is learned by minimizing the
+following engine model objective with a given dataset Dτt:
+Lτt =
+1
+|Dτt|
+�
+(we,T ∗
+e,τt)∈Dτt
+∥T ∗
+e,τt − fτt(we; pτt)∥2.
+(9)
+To stabilize the learning process, we normalize the engine
+speed in the range [0,1] with the maximum engine speed.
+IV. MODEL-BASED PLANNING AND CONTROL
+In this session, we introduce a model-based planning and
+control algorithm that uses the learned model parameters. We
+exploit the learned tire parameters to design a dynamics-aware
+velocity planning and model-based lateral controller (Fig. 1).
+We also integrate the engine dyno data with the learned engine
+torque model to construct an engine lookup table.
+A. Dynamics-aware Velocity Planning
+During high-speed racing, cornering with high velocity
+generates lateral acceleration at the roll axis, which causes
+significant load transfer on each wheel. Since the tire load
+governs the maximum performance of the tire, a model-based
+velocity strategy accounting for the real-time wheel load is
+necessary to maximize the tire performance without losing
+tire grip. We introduce a dynamics-aware velocity planning
+algorithm that derives the velocity plans with maximum tire
+performance based on the learned tire dynamics. We first
+compute the real-time vertical tire load Fz,i affected by the
+lateral load transfer ∆Wf [17]. The diagram of the load
+transfer at the roll axis is illustrated in the left of Fig. 2. The
+load transfer is computed by the roll couple Croll = ms ˙vyha,
+where ms is the sprung mass, ˙vy is the lateral acceleration,
+and ha is the roll height. As the learned tire model describes
+the characteristic for the nominal tire load ¯Fz,i, we compute
+the maximum lateral force of each tire F max
+y,i
+in terms of the
+tire load ratio with peak value of the tire model as:
+F max
+y,i
+= µFz,i
+¯Fz,i
+F peak
+y,i
+,
+(10)
+where µ is a tire performance factor to control the confidence
+and maximum performance of the tire model, Fz,i
+¯
+Fz,i is the tire
+load ratio. The maximum lateral acceleration is determined by
+the following lateral motion dynamics [13]:
+ay,max = 1
+m(F max
+y,r
++ F max
+y,f cos(δ) − mvx ˙ψ),
+(11)
+
+Fig. 2.
+Left: Lateral load transfer generation by the lateral acceleration
+at the roll axis. Right: Overall diagram of the vehicle model.
+where δ is the steering angle and ˙ψ is the yaw rate. Then a
+desired maximum velocity vx,des is planned according to the
+curvature κ of a reference path from a planning module [18]:
+vx,des =
+�
+ay,max/κ.
+(12)
+B. Throttle and Brake Control
+The planned desired velocity is fed to a feedback control
+module [19] to compute the traction force. However, as shown
+in Fig 1, another low-level controller to transform the traction
+force to the throttle command is required to control the racecar
+with nonlinear driveline dynamics. We design the throttle and
+brake control system following [20]. Fig. 3 shows the details of
+the low-level control system. We exploit the integrated engine
+torque map to convert the desired engine torque Te,des to the
+desired throttle command τt,des. As the torque map is built as
+a lookup table, we search the desired throttle with respect to
+a given engine speed and desired torque. The inverse brake
+model is a module to convert the braking force to the brake
+pedal command, which is activated if Fx,r is negative. The
+brake model is also attained by our proposed MIHO, but
+details are omitted to conserve space.
+C. Model-based Path Tracking Control
+We follow the lateral vehicle dynamics of [14] illustrated
+in the right of Fig 2. The lateral model is derived from the
+objective of tracking a reference trajectory. We implement path
+tracking control by stabilizing a velocity-dependent chassis
+model in terms of the error state variables ξ and control u.
+ξ = [ey, ˙ey, eψ, ˙eψ]T ,
+u = δ,
+(13)
+where ey, eψ denote the position and orientation error with
+respect to a given path trajectory. The lateral model contains
+tire-related model parameters such as the cornering stiffnesses
+of the front and rear tires Cα,f, Cα,r. Since the lateral dynam-
+ics is obtained from the bicycle model, the front and rear tire
+models are optimized according to Eq 4, but the sum of the left
+and right tire forces is used as the model output. The cornering
+stiffnesses then can be approximated as follows [14]:
+Cαf ≈ Bf × Cf × Df,
+Cαr ≈ Br × Cr × Dr.
+(14)
+Based on the lateral vehicle model, we design the Linear
+Quadratic Regulator with the following optimization problem:
+min
+u
+� ∞
+0
+(ξT Qξ + uT Ru)dt,
+(15)
+where Q, R denote gain matrices for LQR. For the real-time
+control performance, we compute the state feedback optimal
+LQR gains over piecewise velocity intervals offline [21].
+Fig. 3.
+Throttle and brake control system.
+V. EVALUATION
+A. Analysis for Model Identification
+1) Tire Dynamics Model: Fig. 4 illustrates the learned tire
+models with datasets provided as the tire property files (*.tir).
+The files contain tire force and moment characteristics with
+high fidelity [22]. For model identification, we sampled 3000
+data for each tire using the property files in various tire state
+conditions such as tire load, camber angle, slip angle, and
+slip ratio. As illustrated in the left and middle of Fig. 4, the
+learned models show good fitness to the tire characteristic
+distribution. We further investigated the tire model with the
+driving data collected during track racing. The right of Fig.
+4 illustrates the front tire model of the single-track bicycle
+dynamics learned by the provided tire property data comparing
+it with the driving data that is not used for learning. The
+learned model shows the generalization ability for the overall
+data distribution represented by blue dots. However, since we
+obtained the model by offline optimization and focused on
+the representativeness of data, the model needs more accuracy
+in some edge cases near the peaks of the lateral force. To
+handle those cases, an online parameter optimization can be
+used by parallelizing the HPO process in MIHO, and we will
+implement it in future work.
+2) Engine Torque Model: Fig. 5 illustrates the learned engine
+torque curves and integrated engine map. The data for the
+engine map was provided by engine dynamometer testing.
+For better reliability, we incorporated our data-driven engine
+torque models with the dyno data, especially for the throttle
+pedals 5, 15, and 20%, where the dynamometer had shown in-
+sufficient accuracy of torque measurements. The result shows
+that the learned torque curves are able to represent the change
+of the maximum torque according to the throttle commands.
+Moreover, the learned models also fit the torque curves that
+change with nonlinearity in terms of engine speed. We inte-
+grated the learned torque models with the provided dyno data
+and interpolated the torque data to construct an engine lookup
+table. The blue area on the right of Fig 5 shows the interpolated
+region by the learned torque curves. Our vehicle utilized the
+learned region in racing scenarios such as pit-in/out, obstacle
+avoidance, and driving within 100km/h (Fig. 7).
+B. Control Performance in Indy Autonomous Challenge
+Our model-based planning and control algorithms were
+deployed in the full-scale racecar platform. Moreover, we
+extensively validated our learned model parameter-based al-
+gorithms in the real-world race tracks, IMS and LVMS. The
+algorithms successfully performed various race scenarios, such
+
+eu
+msi.
+H
+yf
+Reference
+V,
+F.
+xr
+lf
+Z
++AW,
+Tf
+AW
+X
+(a)
+(b)Te,des
+Throttle
+Inverse
+Wheel
+Transmission
+command
+Engine Map
+(tt)
+Gear number
+Engine speed
+(G)
+(we)
+Brake
+Inverse
+command
+Brake Model
+(tb)Fig. 4.
+Learned tire models with the provided tire data (Left: left-front and right-front, Middle: left-rear and right-rear). Right: Learned front tire
+dynamics of the single-track bicycle model and the distribution of the collected data on the track.
+Fig. 5.
+Learned engine torque curves and the integrated engine map.
+as obstacle avoidance and high-speed autonomous driving over
+200km/h on the race tracks (Fig. 6).
+1) Obstacle Avoidance in IMS: Fig. 7 shows the quantitative
+results of the obstacle avoidance mission. In the mission,
+obstacles are located before the first-corner section, where the
+velocity plan is critical for avoiding collision while keeping
+close to the racing line. For the sake of safety, we set
+the tire performance factor µ as 0.7. Our dynamics-aware
+velocity planner was able to allow the racecar to maximize
+the velocity while regulating the lateral acceleration within the
+learned maximum tire performance during the rapid avoidance
+maneuvers. The obstacle avoidance was initiated in high-speed
+driving at 100km/h, and steering commands were computed
+up to -10 degrees to follow a generated collision-free reference
+trajectory. The sharp steering command could cause significant
+lateral acceleration higher than 9.0m/s2 in our vehicle, which
+might cause critical tire grip loss. However, our tire model-
+based velocity planner inferred the allowable maximum lateral
+acceleration based on the real-time tire load. As a result, it was
+able to plan safe desired velocities within lateral acceleration
+allowance capable of preserving the tire grip performance.
+2) High-Speed Autonomous Driving in LVMS: Furthermore,
+we extensively validated the control performance based on the
+optimized model parameters at the Tri-Oval Superspeedway,
+LVMS. Fig. 8 illustrates the quantitative results of the lateral
+and longitudinal control while our vehicle raced more than
+nine laps (23km). Our path-tracking algorithm shows robust
+control performance leveraging the learned tire parameters.
+The largest position and orientation errors were 0.6m and
+−2.2 degrees, respectively. In addition, the AV-21 succeeded
+high-speed autonomous driving at above 144km/h (with a top
+speed of 217.4km/h), where the dynamic scenario had yet to
+be visited and adjusted before this track experiment. These
+results demonstrate that MIHO can optimize and provide
+appropriate prior dynamics models offline for the design of
+model-based control before deployment. However, the charac-
+teristic of vehicle dynamics changed and affected the control
+performance over high-speed driving. As shown in the bottom
+of Fig. 8, the tire temperatures were increased after reaching
+the unseen velocity range. In addition, after visiting the range
+of over 144km/h, our low-level controller computed throttle
+commands of over 50% with the engine range consisting only
+of the provided dyno data. Those factors might have an effect
+on the velocity error in the velocity range [42, 52]m/s of Fig
+8. Nevertheless, the control system can be improved with an
+extended data-driven engine map for throttle control at high-
+speed. We also point out that our method has the potential
+to be processed online by parallelizing the HPO process [9],
+which enables the method to identify the model in real-time
+during deployment. The online MIHO could be incorporated
+with the offline model optimization introduced in this work,
+and we leave it as an important future work.
+VI. CONCLUSION
+We present MIHO, a data-driven model identification
+method
+via
+hyperparameter
+optimization.
+Our
+approach
+showed the ability to optimize the parameters of the dy-
+namics models, such as the tire models and engine torque
+curves. Furthermore, the model-based planning and control
+system with the learned model parameters demonstrated stable
+performance in the real-world track environments, IMS and
+LVMS. In future works, we will implement the online HPO
+method and integrate it with the offline method of this work
+to iteratively infer the changing parameters of the vehicle
+dynamics while on track.
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+
+Lateral Tire Model (LF, LR)
+Lateral Tire Model (RF, RR)
+Lateral Tire Model (Front)
+6
+Data (LF)
+Data (RF)
+Driving data
+Data (LR)
+Data (RR)
+Learned (Front)
+Learned (LF)
+Learned (RF)
+4
+Learned (LR)
+Learned (RR)
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+2
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+-0.3
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+Engine speed [RPM]
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+2.5
+Data (interpolated):
+by learned models
+Learned Engine Torque Model
+by dyno data
+Data:
+Tt,5
+Tt,15
+Tt,20
+Learned model
+J Tt,15Fig. 6.
+Left: Team KAIST’s successful obstacle avoidance at IMS. The bottom illustrates the point cloud data and traveled trajectory during
+avoidance. Right: Our AV-21 drove more than nine laps (23km) at the Tri-Oval Superspeedway of LVMS.
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+
+Counter-clockwise
+forward
+-200 -
+Top speed
+reached
+(217.4 km/h)
+-400
+[m] 
+ 009-
+Obstacles
+-800 -
+Driving direction
+Traveled trajectory
+0
+500
+X [m]Velocities (Vx,des, Vx [m/s])
+3525050
+x.des
+280
+290
+300
+310
+320
+Lateral Accelerations (ay,max, ay,abs [m/s2])
+10
+8
+max
+6
+ ay,abs
+4
+2
+0
+280
+290
+300
+310
+320
+Steering Angle (s [degree])
+5
+0
+-5 
+-10
+280
+290
+300
+310
+320
+Time [sec]
+Obstacle avoidanceErrors (ey[m], e[degree])
+1
+0
+-1
+500
+600
+700
+800
+900
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+50
+40
+30
+20
+10
+0
+500
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+800
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+Throttle & Brake Commands (Tt, Tb [-])
+0.6
+0.67 (67 % Throttle)
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+500
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+Tire Temperature(tRR, tRF [C])
+80
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8NAzT4oBgHgl3EQfgfyv/content/tmp_files/load_file.txt

@@ -0,0 +1,441 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf,len=440
+page_content='Data-Driven Model Identification via  Hyperparameter Optimization for the  Autonomous Racing System  Hyunki Seong1, Chanyoung Chung2∗, and David Hyunchul Shim1  Abstract— In this letter, we propose a model identifica-  tion method via hyperparameter optimization (MIHO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Our  method is able to identify the parameters of the paramet-  ric models in a data-driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We utilize MIHO for  the dynamics parameters of the AV-21, the full-scaled au-  tonomous race vehicle, and integrate them into our model-  based planning and control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' In experiments, the  models with the optimized parameters demonstrate the  generalization ability of the vehicle dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We  further conduct extensive field tests to validate our model-  based system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The tests show that our race systems lever-  age the learned model dynamics well and successfully  perform obstacle avoidance and high-speed driving over  200km/h at the Indianapolis Motor Speedway and Las Vegas  Motor Speedway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The source code for MIHO and videos of  the tests are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='com/hynkis/MIHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Index Terms— Model identification, hyperparameter opti-  mization, autonomous vehicle  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' INTRODUCTION  U  nderstanding the system model is essential for robotic  applications, especially high-speed safety-critical au-  tonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Unlike low-speed driving, various dynam-  ics elements such as chassis, tires, or engines become cru-  cial to implement high-speed autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Model-based optimal  control [1], [2] is well-suited for handling those factors and is  widely used to design dynamics system control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' By leveraging  physics-based parametric dynamics models, It optimizes driv-  ing maneuvers with respect to a designed objective function  and enables safe and reliable control system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Despite the success of the model-based approach in robotics,  the model-based algorithm has two fundamental challenges:  model fidelity and tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The performance of model-  based approaches relies heavily on the accuracy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  However, identifying accurate models is often laborious or in-  tractable because of their large search space and non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  This work was supported by Institute of Information communica-  tions Technology Planning Evaluation (IITP) grant funded by the Korea  government(MSIT) (2021-0-00029, Development of Indoor Autonomous  Drone for Performing Multiple Missions Based on Artificial Intelligence)  ∗Corresponding author  1 The authors are with the School of Electrical Engineering, Korea  Advanced Institute of Science and Technology, Daejeon, South Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  (email: hynkis@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='geninfty@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='kr)  2 Chanyoung Chung is with the JPL Science Division, NASA, Califor-  nia, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' (email: chanyoung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='chung@jpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='gov)  Besides model accuracy, models also need to be computa-  tionally feasible considering the real-time control applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  High-fidelity but highly complex models are difficult to inte-  grate into real-time safety-critical driving systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  To tackle those challenges, conventional approaches, includ-  ing the Prediction Error Method, are used to identify model  parameters [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' However, those methods often require the  model structure to be linear or in a specific mathematical  form, which might not be feasible for the design of the  real-time autonomous driving system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' On the other hand,  in several recent works, data-driven approaches using neural  networks, Gaussian processes, or Bayesian methods have been  actively employed for nonlinear system dynamics modeling  and have shown promising results [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' In [5], they proposed  a simple neural network to replace a single-track vehicle  model and used it to generate feedforward control signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Similarly, in [6], they designed Deep Neural Networks (DNN)  as a model approximator to identify the vehicle dynamics  model in an end-to-end learning fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' However, while DNN  is an efficient way to approximate nonlinear systems, it is  difficult to integrate with non-learning model-based methods,  which are reliable in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Furthermore, it  is challenging to ensure the validity of the DNN model in  unseen driving scenarios without large-scale field tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  In this letter, we propose a data-driven model identifica-  tion method via hyperparameter optimization (MIHO) for a  high-speed autonomous driving system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Our key idea is to  leverage a data-driven parameter optimization approach from  machine learning to identify physics-based parametric models  without any structural model requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' To this end, we  adopt a novel hyperparameter optimization (HPO) method  that has an efficient exploration and exploitation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Using the proposed method, we estimate the parameters of  the integrable parametric dynamics models for a full-scaled  autonomous racecar system, Dallara AV-21 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 1), at the  Indy Autonomous Challenge (IAC) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We validate our  proposed approach by integrating identified models into the  high-speed autonomous system and conducting extensive field  experiments, including over 200km/h autonomous driving  and obstacle avoidance scenarios in the Indianapolis Motor  Speedway (IMS) and Las Vegas Motor Speedway (LVMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  In summary, our technical contributions are as follows:  We propose a data-driven model identification method via  hyperparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='01470v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='RO]  4 Jan 2023  We design model-based planning and control systems  incorporating the learned vehicle dynamics models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  We integrate the systems with learned model parameters  into the full-scaled autonomous race vehicle and exten-  sively validate them during the IAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' MODEL IDENTIFICATION VIA HYPERPARAMETER  OPTIMIZATION  The more accurately the system dynamics are described, the  greater the nonlinearity and number of parameters required  for the dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Therefore, an efficient parameter  estimation approach is necessary to find the parameter con-  figuration of such a complex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' In this letter, we propose  a model identification method via hyperparameter optimization  (MIHO) to learn the optimal model parameter configuration by  a data-driven approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Hyperparameter optimization (HPO) is  the problem of selecting an optimal hyperparameter configura-  tion required for neural network training in the machine learn-  ing field [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' A hyperparameter is a parameter that controls  the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' HPO optimizes the hyperparameter con-  figuration by evaluating the performance of the configuration  during the model training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Since one course of neural  network training requires a substantial time, HPO focuses  on the balanced exploration and exploitation strategy for the  efficient optimal hyperparameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Motivated by  the balanced strategy, we design MIHO by adopting the HPO  to the model identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' First, we regard a model  parameter configuration p as a set of hyperparameters of a  nonlinear dynamics model f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Then, we identify the parameter  configuration by evaluating the following objective function  inspired by the standard supervised learning problem:  L =  1  |D|  �  (x,y)∈D  ∥y − f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' p)∥2,  (1)  where x, y denote the sampled input and output data of the  model f from a given dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' By minimizing this learning  objective, we find an optimized model parameter configuration  p∗ that has the minimum model error with the observed model  output y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The model f has no limitation on its formula or form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Thus, our method is able to be used for arbitrary parametric  models, such as a combination of polynomial or mathematical  terms, as well as analytic physics-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  We implement MIHO incorporating a bandit-based HPO  algorithm, Hyperband [9], as summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' It is  Algorithm 1 MIHO Algorithm based on Hyperband  Input: R, η, D  1: smax ← ⌊logη(R)⌋, B = (smax + 1)R  2: for s ∈ {smax, smax − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=', 0} do  3:  n = ⌈ B  R  ηs  (s+1)⌉, r = Rη−s  4:  P = get model param config(n)  5:  for j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=', s} do  6:  nj = ⌊nη−j⌋, rj = rηj  7:  L = {eval with mutation(p, rj, D) : p ∈ P}  8:  P = select top k config(P, L, ⌊nj/η⌋)  Output: Optimized parameters p∗ with the smallest loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Overview of our autonomous driving system in the AV-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Our  learned model parameters are embedded in the planning and control  modules that are covered in this letter (highlighted in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Several input  variables are omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  a variation of a random search algorithm with explore-exploit  theory to find the optimal hyperparameter configuration based  on an evaluation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The algorithm needs two arguments:  R, the maximum amount of resource (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=', the number of  evaluation iterations) that can be allocated to a single config-  uration, and η, a value that determines the proportion of the  discarded configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The two arguments derive smax + 1  combinations (called ”brackets” in [9]) of the values n and r,  which enables various ratios of exploration and exploitation  for finding the optimal parameter configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Hyperband  compares the evaluation loss of each sampled configuration  and allocates more resources to the configurations with lower  evaluation losses, excluding the configurations with higher  losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' It repeats the sampling and exclusion processes until  the last configuration remains to obtain the optimal set of  hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' To adjust the HPO algorithm to the model  parameter optimization, we add the Gaussian mutation [10]  during the evaluation to explore the new neighbor parameters  that might have less model loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Unlike the original HPO,  which only allocates more resources ri, our approach, MIHO,  adds noise perturbation at the selected parameter configuration  p after the resource allocation as:  pmut = p + σ ⊙ ϵ,  ϵ ∼ N(0, I),  (2)  where σ is the standard deviation of the exploration noise that  is annealed over the course of the evaluation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We define  the following three functions for the HPO process in MIHO:  get model param config(n): a function that returns a set  of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='d model parameter configurations from the normal  distribution pre-defined over the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  eval with mutation(p, rj, D): a function that receives a  parameter configuration p, an allocated resource rj, and  a dataset D as arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Using the dataset, this function  evaluates an initial configuration and mutates it for the  allocated rj iterations by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' If a mutated configuration  pmut has a less loss than the initial one, the function  replaces p with pmut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' It returns the final loss after  spending the allocated resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  select top k config(P, L, k): a function that receives a  set of hyperparameter configurations P with their corre-  sponding evaluation losses L and returns the top k high-  performing configurations (here, k = ⌊nj/η⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Planning  Control  Full-scale Racecar  Reference  Lateral  Trajectory  Control  K  Tt,Tb  Velocity  Vx,des  Longitudinal  Throttle / Brake  Planner  Control  (axr)  ControlIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' VEHICLE DYNAMICS MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Tire Dynamics Model  Tire dynamics is one of the factors that significantly affect  the nonlinearity of driving dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Especially the lateral  tire model is crucial to design stable path-tracking control in  high-speed driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The tire model [12] can be described as a  function of the slip angle αi, slip ratio ρx,i, inclination angle  θi, tire load Fz,i, and current velocity vx,i, which has a lateral  tire force F ∗  y,i of each tire (i ∈ {LF, LR, RF, RR}) as,  F ∗  y,i = ftire(αi, ρx,i, θi, Fz,i, vx,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  (3)  Although the model has high fidelity with various dynamics  perspectives, it has low suitability for designing the controller  of high-speed driving, which requires real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Therefore, we first define a tire model with dimension-  reductionthat can be applied to model-based control design  within an acceptable complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We then optimize the model’s  parameter configuration to represent the overall tire charac-  teristic of a given dataset using our MIHO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We  follow the Pacejka tire model [13] to define the tire dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  While the prior work neglect offsets, we formulate a tire model  Fy,i = ft,i(αi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' pt,i) containing offset parameters Sx,i, Sy,i  to describe the asymmetric tire characteristic determined to  maximize cornering performance on an oval track:  Fy,i = Di sin(Ci arctan(Bi(αi + Sx,i))) + Sy,i,  (4)  where  the  tire  model  parameter  configuration  pt,i  =  {Bi, Ci, Di, Sx,i, Sy,i} is identified by minimizing the follow-  ing tire model objective with a given dataset Dt,i as  Lt,i =  1  |Dt,i|  �  (αi,F ∗  y,i)∈Dt,i  ∥F ∗  y,i − ft,i(α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' pt,i)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  (5)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Engine Torque Model  The powertrain system of our racecar consists of an internal  combustion engine, transmission, and wheels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The racecar is a  rear-wheel-drive vehicle whose traction force Fx,r is generated  by engine-based driveline dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We model the equation  of the longitudinal dynamics [14] as follows:  max = Fx,r − Cdv2  x − Cr,  (6)  where m is the vehicle mass, vx is the longitudinal velocity,  Cd denotes the drag coefficient, and Cr denotes the rolling  resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Following a prior work [15], the traction force can  be expressed as:  Fx,r = max,r = Teηtigi0  Rw  ,  (7)  where ax,r denotes the traction acceleration, ηt denotes the  efficiency of the transmission, ig, i0 denote the transmission  ratio of the current gear and final reducer, and Rw denotes  the wheel radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Te = fe(we, τt) is the engine torque map  in terms of the engine speed we and throttle command τt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Due to the high complexity of the engine characteristic, the  torque map is expressed as an experimental lookup table  based on engine torque curves of specific throttle opening  commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Although the engine torque model can be obtained  by the engine dynamometer testing [16], it could suffer from  modeling error because the dynamometer testing is done in a  static environment without longitudinal experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Therefore,  we build the engine torque map that integrates the dyno data  with learned torque curves based on our data-driven model  identification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We express an engine torque curve  Te,τt = fτt(we;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' pτt) of a throttle command τt as a 3rd order  polynomial function of the engine speed we as:  Te,τt = pτt,0 + pτt,1we + pτt,2w2  e + pτt,3w3  e,  (8)  where pτt = {pτt,0, pτt,1, pτt,2, pτt,3} is the torque model  parameter configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Using the resultant traction acceler-  ations ax,r while driving, the engine torque output T ∗  e,τt is  obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Then, pτt is learned by minimizing the  following engine model objective with a given dataset Dτt:  Lτt =  1  |Dτt|  �  (we,T ∗  e,τt)∈Dτt  ∥T ∗  e,τt − fτt(we;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' pτt)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  (9)  To stabilize the learning process, we normalize the engine  speed in the range [0,1] with the maximum engine speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' MODEL-BASED PLANNING AND CONTROL  In this session, we introduce a model-based planning and  control algorithm that uses the learned model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We  exploit the learned tire parameters to design a dynamics-aware  velocity planning and model-based lateral controller (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  We also integrate the engine dyno data with the learned engine  torque model to construct an engine lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Dynamics-aware Velocity Planning  During high-speed racing, cornering with high velocity  generates lateral acceleration at the roll axis, which causes  significant load transfer on each wheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Since the tire load  governs the maximum performance of the tire, a model-based  velocity strategy accounting for the real-time wheel load is  necessary to maximize the tire performance without losing  tire grip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We introduce a dynamics-aware velocity planning  algorithm that derives the velocity plans with maximum tire  performance based on the learned tire dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We first  compute the real-time vertical tire load Fz,i affected by the  lateral load transfer ∆Wf [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The diagram of the load  transfer at the roll axis is illustrated in the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The  load transfer is computed by the roll couple Croll = ms ˙vyha,  where ms is the sprung mass, ˙vy is the lateral acceleration,  and ha is the roll height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' As the learned tire model describes  the characteristic for the nominal tire load ¯Fz,i, we compute  the maximum lateral force of each tire F max  y,i  in terms of the  tire load ratio with peak value of the tire model as:  F max  y,i  = µFz,i  ¯Fz,i  F peak  y,i  ,  (10)  where µ is a tire performance factor to control the confidence  and maximum performance of the tire model, Fz,i  ¯  Fz,i is the tire  load ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The maximum lateral acceleration is determined by  the following lateral motion dynamics [13]:  ay,max = 1  m(F max  y,r  + F max  y,f cos(δ) − mvx ˙ψ),  (11)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Left: Lateral load transfer generation by the lateral acceleration  at the roll axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Right: Overall diagram of the vehicle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  where δ is the steering angle and ˙ψ is the yaw rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Then a  desired maximum velocity vx,des is planned according to the  curvature κ of a reference path from a planning module [18]:  vx,des =  �  ay,max/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  (12)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Throttle and Brake Control  The planned desired velocity is fed to a feedback control  module [19] to compute the traction force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' However, as shown  in Fig 1, another low-level controller to transform the traction  force to the throttle command is required to control the racecar  with nonlinear driveline dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We design the throttle and  brake control system following [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 3 shows the details of  the low-level control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We exploit the integrated engine  torque map to convert the desired engine torque Te,des to the  desired throttle command τt,des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' As the torque map is built as  a lookup table, we search the desired throttle with respect to  a given engine speed and desired torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The inverse brake  model is a module to convert the braking force to the brake  pedal command, which is activated if Fx,r is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The  brake model is also attained by our proposed MIHO, but  details are omitted to conserve space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Model-based Path Tracking Control  We follow the lateral vehicle dynamics of [14] illustrated  in the right of Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The lateral model is derived from the  objective of tracking a reference trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We implement path  tracking control by stabilizing a velocity-dependent chassis  model in terms of the error state variables ξ and control u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  ξ = [ey, ˙ey, eψ, ˙eψ]T ,  u = δ,  (13)  where ey, eψ denote the position and orientation error with  respect to a given path trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The lateral model contains  tire-related model parameters such as the cornering stiffnesses  of the front and rear tires Cα,f, Cα,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Since the lateral dynam-  ics is obtained from the bicycle model, the front and rear tire  models are optimized according to Eq 4, but the sum of the left  and right tire forces is used as the model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The cornering  stiffnesses then can be approximated as follows [14]:  Cαf ≈ Bf × Cf × Df,  Cαr ≈ Br × Cr × Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  (14)  Based on the lateral vehicle model, we design the Linear  Quadratic Regulator with the following optimization problem:  min  u  � ∞  0  (ξT Qξ + uT Ru)dt,  (15)  where Q, R denote gain matrices for LQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' For the real-time  control performance, we compute the state feedback optimal  LQR gains over piecewise velocity intervals offline [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Throttle and brake control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' EVALUATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Analysis for Model Identification  1) Tire Dynamics Model: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 4 illustrates the learned tire  models with datasets provided as the tire property files (*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='tir).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  The files contain tire force and moment characteristics with  high fidelity [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' For model identification, we sampled 3000  data for each tire using the property files in various tire state  conditions such as tire load, camber angle, slip angle, and  slip ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' As illustrated in the left and middle of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 4, the  learned models show good fitness to the tire characteristic  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We further investigated the tire model with the  driving data collected during track racing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  4 illustrates the front tire model of the single-track bicycle  dynamics learned by the provided tire property data comparing  it with the driving data that is not used for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The  learned model shows the generalization ability for the overall  data distribution represented by blue dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' However, since we  obtained the model by offline optimization and focused on  the representativeness of data, the model needs more accuracy  in some edge cases near the peaks of the lateral force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' To  handle those cases, an online parameter optimization can be  used by parallelizing the HPO process in MIHO, and we will  implement it in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  2) Engine Torque Model: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 5 illustrates the learned engine  torque curves and integrated engine map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The data for the  engine map was provided by engine dynamometer testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  For better reliability, we incorporated our data-driven engine  torque models with the dyno data, especially for the throttle  pedals 5, 15, and 20%, where the dynamometer had shown in-  sufficient accuracy of torque measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The result shows  that the learned torque curves are able to represent the change  of the maximum torque according to the throttle commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Moreover, the learned models also fit the torque curves that  change with nonlinearity in terms of engine speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We inte-  grated the learned torque models with the provided dyno data  and interpolated the torque data to construct an engine lookup  table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The blue area on the right of Fig 5 shows the interpolated  region by the learned torque curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Our vehicle utilized the  learned region in racing scenarios such as pit-in/out, obstacle  avoidance, and driving within 100km/h (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Control Performance in Indy Autonomous Challenge  Our model-based planning and control algorithms were  deployed in the full-scale racecar platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Moreover, we  extensively validated our learned model parameter-based al-  gorithms in the real-world race tracks, IMS and LVMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The  algorithms successfully performed various race scenarios, such  eu  msi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  H  yf  Reference  V,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  xr  lf  Z  +AW,  Tf  AW  X  (a)  (b)Te,des  Throttle  Inverse  Wheel  Transmission  command  Engine Map  (tt)  Gear number  Engine speed  (G)  (we)  Brake  Inverse  command  Brake Model  (tb)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Learned tire models with the provided tire data (Left: left-front and right-front, Middle: left-rear and right-rear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Right: Learned front tire  dynamics of the single-track bicycle model and the distribution of the collected data on the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Learned engine torque curves and the integrated engine map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  as obstacle avoidance and high-speed autonomous driving over  200km/h on the race tracks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  1) Obstacle Avoidance in IMS: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 7 shows the quantitative  results of the obstacle avoidance mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' In the mission,  obstacles are located before the first-corner section, where the  velocity plan is critical for avoiding collision while keeping  close to the racing line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' For the sake of safety, we set  the tire performance factor µ as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Our dynamics-aware  velocity planner was able to allow the racecar to maximize  the velocity while regulating the lateral acceleration within the  learned maximum tire performance during the rapid avoidance  maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The obstacle avoidance was initiated in high-speed  driving at 100km/h, and steering commands were computed  up to -10 degrees to follow a generated collision-free reference  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The sharp steering command could cause significant  lateral acceleration higher than 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='0m/s2 in our vehicle, which  might cause critical tire grip loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' However, our tire model-  based velocity planner inferred the allowable maximum lateral  acceleration based on the real-time tire load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' As a result, it was  able to plan safe desired velocities within lateral acceleration  allowance capable of preserving the tire grip performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  2) High-Speed Autonomous Driving in LVMS: Furthermore,  we extensively validated the control performance based on the  optimized model parameters at the Tri-Oval Superspeedway,  LVMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 8 illustrates the quantitative results of the lateral  and longitudinal control while our vehicle raced more than  nine laps (23km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Our path-tracking algorithm shows robust  control performance leveraging the learned tire parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  The largest position and orientation errors were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='6m and  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='2 degrees, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' In addition, the AV-21 succeeded  high-speed autonomous driving at above 144km/h (with a top  speed of 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='4km/h), where the dynamic scenario had yet to  be visited and adjusted before this track experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' These  results demonstrate that MIHO can optimize and provide  appropriate prior dynamics models offline for the design of  model-based control before deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' However, the charac-  teristic of vehicle dynamics changed and affected the control  performance over high-speed driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' As shown in the bottom  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 8, the tire temperatures were increased after reaching  the unseen velocity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' In addition, after visiting the range  of over 144km/h, our low-level controller computed throttle  commands of over 50% with the engine range consisting only  of the provided dyno data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Those factors might have an effect  on the velocity error in the velocity range [42, 52]m/s of Fig  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Nevertheless, the control system can be improved with an  extended data-driven engine map for throttle control at high-  speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' We also point out that our method has the potential  to be processed online by parallelizing the HPO process [9],  which enables the method to identify the model in real-time  during deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The online MIHO could be incorporated  with the offline model optimization introduced in this work,  and we leave it as an important future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' CONCLUSION  We present MIHO, a data-driven model identification  method  via  hyperparameter  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Our  approach  showed the ability to optimize the parameters of the dy-  namics models, such as the tire models and engine torque  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Furthermore, the model-based planning and control  system with the learned model parameters demonstrated stable  performance in the real-world track environments, IMS and  LVMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' In future works, we will implement the online HPO  method and integrate it with the offline method of this work  to iteratively infer the changing parameters of the vehicle  dynamics while on track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  REFERENCES  [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Lewis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Vrabie, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Syrmos, Optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' John Wiley  & Sons, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Jung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Seong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Finazzi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Shim, “Game-theoretic  model predictive control with data-driven identification of vehicle model  for head-to-head autonomous racing,” arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='04094,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Lateral Tire Model (LF, LR)  Lateral Tire Model (RF, RR)  Lateral Tire Model (Front)  6  Data (LF)  Data (RF)  Driving data  Data (LR)  Data (RR)  Learned (Front)  Learned (LF)  Learned (RF)  4  Learned (LR)  Learned (RR)  Lateral Tire Force [kN]  Lateral Tire Force [kN]  [kN]  2  0  0  2  2  4  4  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='3  Side Slip Angle [rad]  Side Slip Angle [rad]  Side Slip Angle [rad]Engine torque [N /m]  80  0  300  60  200  100  7000  100  6000  ¥3  5000  400  00  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='5  Integrated  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='5  3500  3000  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='0  Engine Torque Map  2500  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='5  Engine speed [RPM]  2000  1500  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='0  1000  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='5  Data (interpolated):  by learned models  Learned Engine Torque Model  by dyno data  Data:  Tt,5  Tt,15  Tt,20  Learned model  J Tt,15Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Left: Team KAIST’s successful obstacle avoidance at IMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' The bottom illustrates the point cloud data and traveled trajectory during  avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Right: Our AV-21 drove more than nine laps (23km) at the Tri-Oval Superspeedway of LVMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
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+page_content='  [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content=' Schmeitz, “Mf-tyre/mf-swift,” Delft: TNO, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='  Counter-clockwise  forward  200 -  Top speed  reached  (217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='4 km/h)  400  [m]  009-  Obstacles  800 -  Driving direction  Traveled trajectory  0  500  X [m]Velocities (Vx,des, Vx [m/s])  3525050  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='des  280  290  300  310  320  Lateral Accelerations (ay,max, ay,abs [m/s2])  10  8  max  6  ay,abs  4  2  0  280  290  300  310  320  Steering Angle (s [degree])  5  0  5  10  280  290  300  310  320  Time [sec]  Obstacle avoidanceErrors (ey[m], e[degree])  1  0  1  500  600  700  800  900  Velocities (Vx,des, Vx [m/s])  60  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='39 m/s (217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='4 km/h)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='des  50  40  30  20  10  0  500  600  700  800  900  Throttle & Brake Commands (Tt, Tb [-])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='67 (67 % Throttle)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}
+page_content='1  0  500  600  700  800  900  Tire Temperature(tRR, tRF [C])  80  RR  60  40  20  0  20  500  600  700  800  900  Time [sec]  Unseen dynamic scenario' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfgfyv/content/2301.01470v1.pdf'}

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+Measuring the maximally allowed polarization states of the isotropic stochastic
+gravitational wave background with the ground-based detectors
+Hidetoshi Omiya∗ and Naoki Seto†
+Department of Physics, Kyoto University, Kyoto 606-8502, Japan
+(Dated: January 5, 2023)
+We discuss the polarizational study of isotropic gravitational wave backgrounds with the second
+generation detector network, paying special attention to the impacts of adding LIGO-India. The
+backgrounds can be characterized by at most five spectral components (three parity-even ones
+and two parity-odd ones).
+They can be algebraically decomposed through the difference of the
+corresponding overlap reduction functions defined for the individual spectra. We newly identify two
+interesting relations between the overlap reduction functions, and these relations generally hamper
+the algebraic decomposition in the low frequency regime f ≲ 30Hz. We also find that LIGO-India
+can significantly improve the network sensitives to the odd spectral components.
+I.
+INTRODUCTION
+A stochastic gravitational wave background is one
+of the primary targets of gravitational wave detectors.
+There exist a large number of theoretical predictions for
+generation processes such as an inflationary expansion
+[1–4], a phase transition [5, 6], and distant unresolved bi-
+naries [7, 8] (for other sources, see [9, 10]). Many of these
+backgrounds were generated in strong gravity regimes or
+high energy states and could be a good probe for physics
+in an extreme environment. Note also that these back-
+grounds are expected to be highly isotropic.
+In General Relativity (GR), we only have the two ten-
+sor degrees of freedom, the + and × modes. In contrast,
+some alternative theories of gravity predict additional po-
+larization modes; the two vector (x and y) modes and
+the two scalar (b and l) modes [11]. Therefore, through
+a polarization study of background, we might detect a
+signature of modification to GR [12, 13] (see [14–19] for
+studies on the polarization of gravitational waves from
+compact binary). Furthermore, even if GR is not mod-
+ified at present, a parity violation process in the early
+universe could generate an asymmetry between right- and
+left-handed polarization patterns [20–27].
+The cross-correlation analysis is an efficient method for
+detecting a weak gravitational wave background [28–30].
+By taking products of data streams of noise-independent
+detector pairs, we can gradually improve the sensitivity
+to a background by increasing observational time. When
+the gravitational wave frequency is much longer than the
+arm lengths of detectors [31], the two scalar modes are
+observationally non-separable, and we can generally mea-
+sure the five background spectra IT , IV , IS, WT , and WV .
+The three spectra IT , IV , and IS represent the total in-
+tensity of the tensor, vector, and scalar modes.
+The
+remaining two spectra WT and WV correspond to the
+Stokes “V” parameters which probe the degrees of circu-
+lar polarization of the tensor and vector modes. In this
+∗ [email protected]
+† [email protected]
+paper, we utilize W for the “V” parameter to avoid con-
+fusion with the vector modes. Since the spectra IT , IV ,
+and IS transform as parity even quantities and the WT
+and WV transform as parity odd quantities, we refer to
+the former three as parity even spectra and the later two
+as parity odd spectra.
+At the correlation analysis, we can measure linear com-
+binations of the five spectra with the five coefficients
+known as the overlap reduction functions (ORFs). The
+ORFs characterize the sensitivities to the corresponding
+spectra and depend on gravitational wave frequency as
+well as the relative configuration of the two pairwise de-
+tectors. We apply the parity even/odd classification also
+to the five ORFs.
+For probing the existence of the anomalous polariza-
+tion spectra IV , IS, WT , and WV , we desire to clean the
+contribution from the standard spectrum IT (see also [32]
+for a maximum likelihood analysis). In addition, we pre-
+fer to break down the four anomalous modes and measure
+them separately. Our strategy in this paper is to utilize
+the difference between the five ORFs and algebraically
+decompose the five spectra by taking appropriate linear
+combinations of the correlation products from multiple
+pairs (originally proposed in [33]). We mainly study the
+prospects of this algebraic scheme with the second gener-
+ation detector network. We pay special attention to the
+impacts of adding LIGO-India as the fifth detector.
+In the middle of our study, we newly identify two de-
+generate relations between the ORFs. The first one is
+for the three even ORFs, and the second one is for the
+two odd ORFs. These two relations generally limit the
+performance of the algebraic decomposition in the low
+frequency regime f ≲ 30Hz. On the other hand, LIGO-
+India can largely mitigate the damage associated with
+the degeneracy for the even ORFs, because of its rela-
+tively remote location from the two other LIGO detec-
+tors. Furthermore, LIGO-India can significantly improve
+the sensitivities to the odd spectra.
+This paper is organized as follows. In Sec. II, we re-
+view the polarization decomposition of an isotropic back-
+ground and present the analytical expressions for the as-
+sociated ORFs. We also explain our two new findings
+arXiv:2301.01489v1  [astro-ph.CO]  4 Jan 2023
+
+2
+with respect to the ORFs. In Sec. III, we concretely study
+the geometry of the second generation terrestrial detec-
+tor network, including LIGO-India. In Sec. IV, we review
+the correlation analysis, primarily focusing on the evalu-
+ation of the signal-to-noise ratio. In Sec. V, we explain
+the algebraic decomposition scheme for multiple spectral
+components. In Secs. VI and VII, we apply this scheme
+to the second generation ground-based detector network.
+We discuss how the sensitivity depends on the target po-
+larization spectra and the network combinations. Finally,
+in Sec. VIII, we summarize our paper.
+II.
+BASIC QUANTITIES
+Following our preceding work [31] on formal aspects,
+we first review the basic ingredients for the correlated
+signals of stochastic backgrounds with ground-based de-
+tectors. Since our universe is highly isotropic and homo-
+geneous, the monopole components of the backgrounds
+are assumed to be our primary target. In addition, be-
+cause the observed speed of gravitational wave vg is close
+to the speed of light c, we set vg = c.
+In Sec. II A, we describe the polarization states of
+the isotropic backgrounds and introduce the five relevant
+spectra IT , IV , IS, WT , and WV . In Sec. II B, we discuss
+the ORFs which characterize the correlated response of
+pairwise detectors to the backgrounds. In Sec. II C, we
+give analytic expressions of the ORFs for the ground-
+based detectors. In Secs. II D and II E, we discuss their
+asymptotic behaviors. In Secs. II F and II G, we report
+our two new findings on the ORFs.
+A.
+Polarization states of a stochastic gravitational
+wave background
+We start with the plane wave decomposition of the
+metric perturbation hij generated by the gravitational
+waves
+hij(t, x) =
+�
+P
+�
+df
+�
+dΩ
+× ˜hP (f, Ω)eP,ij(Ω)e−2πif(t−Ω·x/c) ,
+(1)
+where Ω is the unit vector for the propagation direction,
+normalized by
+�
+dΩ = 4π. Here, eP (P = +, ×, x, y, b
+and l) represent the polarization tensors given by
+e+ = m ⊗ m − n ⊗ n ,
+e× = m ⊗ n + n ⊗ m ,
+ex = Ω ⊗ m + m ⊗ Ω ,
+ey = Ω ⊗ n + n ⊗ Ω ,
+eb =
+√
+3(m ⊗ m + n ⊗ n) ,
+el =
+√
+3(Ω ⊗ Ω)
+(2)
+with the orthonormal vectors m and n in addition to Ω
+(see [34] for geometrical interpretation of these modes).
+Note that our definitions for eb and el are different from
+the conventional one such as used in [13] (see also Ap-
+pendix in [35]). They are written by the standard polar
+coordinates (θ, φ) as
+Ω =
+�
+�
+sin θ cos φ
+sin θ sin φ
+cos θ
+�
+� ,
+(3)
+m =
+�
+�
+cos θ cos φ
+cos θ sin φ
+− sin θ
+�
+� ,
+(4)
+n =
+�
+�
+− sin φ
+cos φ
+0
+�
+� .
+(5)
+In Eq. (2), the labels P = +, × correspond to the ten-
+sor (T) modes, P = x, y to the vector (V ) modes, and
+P = b, l to the scalar (S) modes. Note that GR predicts
+only the tensor modes. However, numerous alternative
+theories of gravity allow the existence of the remaining
+V and S modes.
+For a stochastic background, the expansion coefficients
+˜hP can be regarded as random quantities.
+Their sta-
+tistical properties are specified by the power spectrum
+matrix ⟨˜hP (f, Ω)˜h∗
+P ′(f ′, Ω)⟩ with no correlation between
+T, V and S modes for statistically isotropic backgrounds
+[31]. In the case of the tensor modes (P, P ′ = +, ×), the
+matrix can be written in terms of the Stokes parameters
+as [36]
+⟨˜hP (f, Ω)˜h∗
+P ′(f ′, Ω′)⟩ =1
+2δΩΩ′δ(f − f ′)
+×
+�
+IT + QT
+UT − iWT
+UT + iWT
+IT − QT
+�
+P P ′
+.
+(6)
+In the standard literature of polarization, the chiral
+asymmetry is usually denoted as the Stokes “V ” param-
+eter. In this paper, we apply the notation V to repre-
+sent the vector modes, and use W for the chiral asymme-
+try. Note that the combinations QT ± iUT do not have
+isotropic components, as understood from their transfor-
+mation properties [31, 36]. We thus drop them hereafter.
+In Eq.
+(6), we use the coefficients ˜hP (f, Ω) for the
+linear polarization bases (e+, e×). However, the physi-
+cal meaning of the W parameter becomes transparent by
+introducing the circular (right- and left-handed) polar-
+ization bases given by
+eT
+R =
+1
+√
+2 (e+ + ie×) ,
+eT
+L =
+1
+√
+2 (e+ − ie×) ,
+(7)
+with the corresponding coefficients
+˜hT
+R(f, Ω) =
+1
+√
+2
+�
+˜h+(f, Ω) − i˜h×(f, Ω)
+�
+,
+(8)
+˜hT
+L(f, Ω) =
+1
+√
+2
+�
+˜h+(f, Ω) + i˜h×(f, Ω)
+�
+.
+(9)
+
+3
+We then have
+IT = ⟨˜hT
+R˜hT ∗
+R ⟩ + ⟨˜hT
+L˜hT ∗
+L ⟩ ,
+(10)
+WT = ⟨˜hT
+R˜hT ∗
+R ⟩ − ⟨˜hT
+L˜hT ∗
+L ⟩ ,
+(11)
+omitting apparent delta functions.
+These expressions
+show that the spectra IT and WT characterize the to-
+tal and asymmetry of the amplitudes of the right- and
+left-handed polarization patterns of the tensor modes.
+Since the parity transformation interchanges the right-
+and left-handed waves, we resultantly have I′
+T = IT and
+W ′
+T = −WT (′ representing parity transformed quanti-
+ties).
+For the vector modes, we can repeat almost the same
+arguments as Eqs. (6)-(9) and obtain
+IV = ⟨˜hV
+R˜hV ∗
+R ⟩ + ⟨˜hV
+L ˜hV ∗
+L ⟩ ,
+(12)
+WV = ⟨˜hV
+R˜hV ∗
+R ⟩ − ⟨˜hV
+L ˜hV ∗
+L ⟩
+(13)
+with the correspondences I′
+V = IV and W ′
+V = −WV for
+the parity transformation.
+For the scalar modes (P, P ′ = b, l), considering their
+potential correlation, we can generally put
+⟨˜hP (f, Ω)˜h∗
+P ′(f ′, Ω′)⟩ =1
+2δΩΩ′δ(f − f ′)
+×
+�
+Ib
+CS
+C∗
+S
+Il
+�
+P P ′ .
+(14)
+described by the four real parameters in the power spec-
+tra. In reality, as long as the low frequency approxima-
+tion is valid (f ≪ (2πL/c)−1, L: the arm length), only
+the combination
+IS ≡ 1
+2(Ib + Il − CS − C∗
+S) ,
+(15)
+appears in the correlation analysis [31]. Therefore, in the
+following, we keep only IS for the scalar modes. Because
+of its spin-0 nature, we also have I′
+S = IS for the parity
+transformation.
+Up to now, we see that an isotropic background is
+characterized by the five quantities IT , IV , IS, WT , and
+WV . Here, we introduce another commonly used repre-
+sentation for the magnitudes of these spectra. In GR,
+the amplitude IT (f) can be simply related to the energy
+density of the background. More specifically, with the
+Hubble parameter H0, we have the relation
+ΩIT
+GW (f) =
+�32π3
+3H2
+0
+�
+f 3IT (f)
+(16)
+for the energy density of the background per logarithmic
+frequency (normalized by critical density of universe) [28,
+29]. In a modified theory of gravity, the relation (16) for
+the energy density might be invalid [37]. However, we
+are not directly interested in the actual energy density
+of the backgrounds, and thus continue to use Eq. (16) as
+the definition of ΩIT
+GW (f). Similarly, we use the effective
+energy densities
+ΩIV
+GW (f) ≡
+�32π3
+3H2
+0
+�
+f 3IV (f) ,
+(17)
+ΩIS
+GW (f) ≡
+�32π3
+3H2
+0
+�
+f 3IS(f) ,
+(18)
+ΩWT
+GW (f) ≡
+�32π3
+3H2
+0
+�
+f 3WT (f) ,
+(19)
+ΩWV
+GW (f) ≡
+�32π3
+3H2
+0
+�
+f 3WV (f) .
+(20)
+If the left handed modes dominate the right handed ones,
+we have ΩWT
+GW (f) < 0 (and ΩWV
+GW (f) < 0).
+B.
+Correlation Analysis
+Now we discuss how to detect the five spectral com-
+ponents by using multiple interferometers.
+In the low
+frequency regime (f ≪ (2πL/c)−1), the response of an
+interferometer A (at the position xA) can be modeled
+as [31]
+hA(f) = Dij
+A˜hij(f, xA) ,
+(21)
+with the beam pattern function
+DA = uA ⊗ uA − vA ⊗ vA
+2
+.
+(22)
+Here, ˜hij(f, xA) is the metric perturbation of the back-
+ground at the detector, and the two unit vectors uA and
+vA represent the two arm directions of the detector.
+By correlating data streams of multiple detectors, we
+can statistically amplify the background signals relative
+to the detector noises (closely discussed in Sec. IV). We
+denote the correlation product of two detector A and B
+by
+CAB(f) ≡ ⟨hA(f)h∗
+B(f)⟩
+(23)
+(again omitting delta functions).
+Leaving only the
+monopole components of the background, we obtain
+CAB(f) = 4π
+5
+�
+�
+�
+P =T,V,S
+γIP IP +
+�
+P =T,V
+γWP WP
+�
+� .
+(24)
+Here, γIP and γWP are the ORFs which characterize the
+correlated response of two detectors to the relevant com-
+ponents of an isotropic background.
+They are written
+as
+γIP
+AB(f) ≡ DA,ijDB,klΓIP
+ijkl ,
+(25)
+γWP
+AB (f) ≡ DA,ijDB,klΓWP
+ijkl ,
+(26)
+
+4
+β
+σA
+σB
+A
+B
+FIG. 1. The relative geometry of the ground-based detector
+pair A and B. The two detectors are on the same great circle
+and their detector planes are tangential to the earth sphere.
+The opening angle β is measured from the center of the Earth.
+The angles σA and σB correspond to the orientations of the
+bisectors of the two arms (dotted line) measured counter clock
+wisely relative to the great circle.
+with the angular integrals
+ΓIT
+ijkl = 5
+8π
+�
+dΩ(e+,ije+,kl + e×,ije×,kl)eiyΩ· ˆ
+d ,
+(27)
+ΓIV
+ijkl = 5
+8π
+�
+dΩ(ex,ijex,kl + ey,ijey,kl)eiyΩ· ˆ
+d ,
+(28)
+ΓIS
+ijkl = 5
+8π
+�
+dΩ(eb,ijeb,kl + el,ijel,kl)eiyΩ· ˆ
+d ,
+(29)
+ΓWT
+ijkl = − 5i
+8π
+�
+dΩ(e+,ije×,kl − e×,ije+,kl)eiyΩ· ˆ
+d , (30)
+ΓWV
+ijkl = − 5i
+8π
+�
+dΩ(ex,ijey,kl − ey,ijex,kl)eiyΩ· ˆ
+d .
+(31)
+Here, we put d ≡ |xA − xB|, ˆd ≡ (xA − xB)/d and
+y = 2πfd/c.
+As already in Eq. (24), we will use the label P for the
+polarization modes P = (T, V, S), extending it from the
+original patterns P = (+, ×, x, y, b, l). In addition, we
+introduce the label Q to represent all the ��ve spectral
+modes (IT , IV , IS, WT , WV ) of interest.
+For notational
+simplicity, we also omit the labels for the detectors in
+obvious cases.
+C.
+ORFs for ground-based detectors
+Now we focus on the ground-based detectors that are
+assumed to be tangential to the Earth sphere of the ra-
+dius RE = 6400km. As shown in Fig. 1, the relative
+geometry of two interferometers A and B are fully char-
+acterized by three angles β, σA and σB (following the
+convention in [28]). The angle β represents the opening
+angle between the two detectors, measured from the cen-
+ter of the Earth, and we have d = 2RE sin(β/2). Mean-
+while, the angle σA shows the orientation of the bisector
+of the two arms of the detector A measured counterclock-
+wise relative to the great circle joining the two detectors.
+The angle σB is defined similarly. Below, instead of σA
+and σB, we use the angles ∆ and δ
+∆ ≡ σA + σB
+2
+,
+δ ≡ σA − σB
+2
+,
+(32)
+following the standard convention.
+The close expressions of the ORFs are presented in [31]
+as
+γIP = ΘP
+∆(y, β) cos 4∆ + ΘP
+δ (y, β) cos 4δ ,
+(P = T, V, S) ,
+(33)
+γWP = ΞP (y, β) sin 4∆ ,
+(P = T, V ) .
+(34)
+Here the angles δ and ∆ appear only in the forms
+cos 4δ, cos 4∆, and sin 4∆, reflecting certain symmetries
+[38]. The coefficients ΞP , ΘP
+∆, and ΘP
+δ are given by
+ΘT
+∆(y, β) = − sin4
+�β
+2
+�
+j0(y)
+− 5
+56(−9 + 8 cos β + cos 2β)j2(y)
+−
+1
+896(169 + 108 cos β + 3 cos 2β)j4(y) , (35)
+ΘV
+∆(y, β) = − sin4
+�β
+2
+�
+j0(y)
++
+5
+112(−9 + 8 cos β + cos 2β)j2(y)
++
+1
+224(169 + 108 cos β + 3 cos 2β)j4(y) , (36)
+ΘS
+∆(y, β) = − sin4
+�β
+2
+�
+j0(y)
++ 5
+56(−9 + 8 cos β + cos 2β)j2(y)
+−
+3
+448(169 + 108 cos β + 3 cos 2β)j4(y) , (37)
+ΘT
+δ (y, β) = cos4
+�β
+2
+� �
+j0(y) + 5
+7j2(y) +
+3
+112j4(y)
+�
+,
+(38)
+ΘV
+δ (y, β) = cos4
+�β
+2
+� �
+j0(y) − 5
+14j2(y) − 3
+28j4(y)
+�
+,
+(39)
+ΘS
+δ (y, β) = cos4
+�β
+2
+� �
+j0(y) − 5
+7j2(y) + 9
+56j4(y)
+�
+,
+(40)
+ΞT (y, β) = sin
+�β
+2
+� �
+(1 − cos β)j1(y) − 7 + 3 cos β
+8
+j3(y)
+�
+,
+(41)
+ΞV (y, β) = 1
+2 sin
+�β
+2
+� �
+(1 − cos β)j1(y) + 7 + 3 cos β
+2
+j3(y)
+�
+(42)
+with the spherical Bessel functions jn(y).
+
+5
+D.
+Asymptotic Behaviors at y → ∞
+In this subsection, we briefly discuss the asymptotic
+profiles of the ORFs at y → ∞, based on Eqs. (33)-(42).
+For the spherical Bessel functions, at large y, we have
+the following correspondences
+j2l(y) ∝ sin y
+y
+,
+j2l+1(y) ∝ cos y
+y
+,
+(43)
+Then, we can put
+γIP → CIP
+sin y
+y
+,
+γWP → CWP
+cos y
+y
+(44)
+with the coefficients CIP and CWP presented shortly.
+Roughly speaking, these relations show the phase offset of
+∼ π/2 (as in the combination of sin y and cos y), depend-
+ing on the two parity types of the background spectra IP
+and W P .
+We can readily evaluate the coefficients CIP and CWP
+as follows;
+CIT =
+5
+128
+�
+8 cos4
+�β
+2
+�
+cos 4δ
+− (cos 2β − 28 cos β + 35) cos 4∆
+�
+,
+(45)
+CIV = 5
+8 cos2
+�β
+2
+� �
+2 cos2
+�β
+2
+�
+cos 4δ
+− (cos β − 3) cos 4∆
+�
+,
+(46)
+CIS = 15
+8 cos4
+�β
+2
+�
+(cos 4δ − cos 4∆) ,
+(47)
+CWT = − 5
+16
+�
+− sin
+�3β
+2
+�
++ 7 sin
+�β
+2
+��
+sin 4∆ , (48)
+CWV = 5
+2 sin
+�β
+2
+�
+cos2
+�β
+2
+�
+sin 4∆ .
+(49)
+We have CWT · CWV ≤ 0. Notice that CWT and CWV
+vanish at β = 0. Two detectors on a plane are appar-
+ently mirror symmetric, and thus blind to the parity odd
+polarizations.
+E.
+Asymptotic Behaviors at y → 0
+At the opposite limit, y → 0, we have
+γIT,V,S(y) → 2DA,ijDij
+B
+(50)
+= − sin4
+�β
+2
+�
+cos 4∆ + cos4
+�β
+2
+�
+cos 4δ ,
+(51)
+γWT (y) → 2 sin3
+�β
+2
+�
+y sin 4∆ ,
+(52)
+γWV (y) → sin3
+�β
+2
+�
+y sin 4∆ .
+(53)
+The first expression shows the degeneracy of the parity
+even ORFs. Meanwhile, the parity odd ORFs vanish at
+y → 0, due to the parity symmetry of a network at the
+same place with d = 0 [31]. Thus, a network becomes
+blind to the parity odd polarizations for small y.
+F.
+Trinity degeneracy of even ORFs at the
+sub-leading order O(y2)
+At
+the
+sub-leading
+order
+O(y2)
+(or
+equivalently
+O(f 2)), we can easily confirm a cancellation for the three
+even ORFs and have
+γIT (y) − 4γIV (y) + 3γIS(y) = O(y4) .
+(54)
+This trinity degeneracy will later play an important role
+in the spectral decomposition of the three even spectra.
+G.
+Degeneracy of odd ORFs at 13Hz
+For detectors on the Earth, we can put y = ζ sin(β/2)
+with ζ ≡ 4πREf/c. In Fig. 2, we present a contour plot
+for the ratio between the odd ORFs
+γWT
+γWV
+= ΞT (y, β)
+ΞV (y, β) ≡ Θ(ζ, β).
+(55)
+At the left end, we can see the limit limζ→0 Θ(ζ, β) = 2
+following from Eqs. (52) and (53).
+Surprisingly, the function Θ depends very weakly on β
+around ζ = 3.57, as shown with the almost vertical con-
+tour Θ = 1.26 in Fig. 2. Indeed, along this contour, the
+variation of ζ is within ±0.01. Later, we will find that the
+odd spectral decomposition practically collapses around
+ζ = 3.57, corresponding to f = 13Hz for the Earth’s
+radius RE = 6400km.
+This anathematic frequency is
+intrinsic to ground-based detectors.
+In space, we might realize a detector network com-
+posed by multiple LISA-like units orbiting around the
+Sun [39, 40] (see also [41]).
+For their typical orbital
+configuration, we need at least three separated units
+for fully decomposing the five polarization spectra, and
+these units contact with a virtual sphere of radius 1.15
+a.u. [35, 42, 43] (see also [44]). In this case, the anathe-
+matic frequency becomes 0.57mHz.
+III.
+SECOND GENERATION DETECTOR
+NETWORK
+From now on, we mainly discuss the ground-based de-
+tector networks composed by the following five second
+generation interferometers; LIGO-Handford (H), LIGO-
+India (I), KAGRA (K), LIGO-Livingston (L) and Virgo
+(V). We present their basic angular parameters in Ta-
+ble I.
+
+6
+0
+1
+2
+3
+4
+5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+ζ
+β
+0
+1.00
+1.26
+1.50
+1.75
+1.99
+FIG. 2. The contour plot for the ratio Θ(ζ, β). We have the
+limit limζ→0 Θ = 2 and almost vertical contour line around
+ζ = 3.575 (corresponding to 13Hz for ground-based detec-
+tors).
+TABLE I. The latitudes, longitudes, and orientations of the
+five ground-based detectors in units of degree. The angle α is
+the orientation angle of the bisector of the two arms measured
+from the local east at each detectora.
+detector
+latitude
+longitude
+α
+KAGRA(K)
+36.41
+137.31
+74.60
+LIGO-I(I)
+19.61
+77.03
+162.62
+LIGO-H(H)
+46.45
+-119.41
+171.00
+LIGO-L(L)
+30.56
+-90.8
+242.17
+Virgo(V)
+43.63
+10.50
+115.57
+a https://git.ligo.org/
+From these five interferometers, we can make 5C2 = 10
+pairs and introduce the abstract index u to represent
+these ten pairs {HI, HK, · · · , LV}. Their relative geomet-
+rical parameters are presented in Table II.
+Since
+each
+pair
+has
+the
+five
+ORFs
+γQ
+u (f)
+(Q; IT , IV , IS, WT , WV ),
+the total number of ORFs
+is 50. In Fig. 3, we present all of them at a clip. Later,
+we will come to deal with the sums of their products
+such as �
+u γQ
+u (f)γQ′
+u (f), and the collective behaviours
+of these large number of ORFs would be important
+there.
+As explained in Sec. II.E, at f = 0, we have the de-
+generacies γIT
+u
+= γIV
+u
+= γIS
+u
+and γWT
+u
+= γWV
+u
+=
+0.
+In Fig. 3, we can easily identify the three conspicuous
+curves starting from γQ
+u ≃ −0.9 at f = 0. These are the
+even ORFs of the HL pair. This pair is designed to have
+a large overlap with cos 4δ ≃ −1. In Fig. 4, its five ORFs
+are presented, showing loose oscillation patterns due to
+the small separation angle β. The small angle β also sup-
+presses the amplitudes of the odd ORFs, in contrast to
+the even ones (see Sec. II D and II E).
+Meanwhile, the HI and IL pairs have large separation
+KI
+KH
+KL
+KV
+IH
+IL
+IV
+HL
+HV
+LV
+0
+50
+100
+150
+-1.0
+-0.5
+0.0
+0.5
+f[Hz]
+Even ORFs
+KI
+KH
+KL
+KV
+IH
+IL
+IV
+HL
+HV
+LV
+0
+50
+100
+150
+-1.0
+-0.5
+0.0
+0.5
+f[Hz]
+Odd ORFs
+FIG. 3. All the 50 ORFs formed from the five interfeometers
+H, I, K, L and V. In the upper panel, the three curves starting
+from −0.9 correspond to the HL pair.
+angles β and thus provide relatively large value
+y ∝ f sin(β/2)
+(56)
+for a given frequency f. This will help us to use the higher
+order correction terms of the variables y (e.g., breaking
+the spectral degeneracy). Together with the preferred rel-
+ative orientation |sin 4∆|∼ 1, these pairs also have good
+sensitivities to the odd parity spectra WT and WV .
+As examples of typical pairs, in Fig. 5, we show the
+ORFs of the LV-pair. In the bottom panel, we compare
+the asymptotic profiles discussed in Sec. II.D. At f ≳
+80Hz (y ≳ 4π), they show reasonable agreements with
+the original curves. Accordingly, in the upper panel, we
+can see the phase offset ∼ π/2 between the odd and even
+ORFs there.
+In Table III, we present the asymptotic
+coefficients CQ for the ten pairs.
+IV.
+CORRELATION ANALYSIS WITH
+GROUND-BASED DETECTORS
+Up to this point, we only considered the response of
+detectors to stochastic backgrounds. In reality, the data
+streams of the detectors are contaminated by the detec-
+tor noises. As we see below, the correlation analysis is
+a powerful framework to coherently amplify the back-
+ground signals relative to the noises [28, 29].
+Under the existence of the detector noises, the outputs
+
+7
+TABLE II. (Upper right) The opening angle β (in units of degree) of the detector pairs, measured from the center of the Earth.
+(Lower left) The values of (cos 4δ, cos 4∆, sin 4∆).
+KAGRA
+LIGO-I
+LIGO-H
+LIGO-L
+Virgo
+KAGRA
+*
+54.89
+72.37
+99.27
+86.52
+LIGO-I
+(-0.41,0.63,0.78)
+*
+112.28
+128.47
+59.79
+LIGO-H
+(0.99,-0.34,0.94)
+(0.75,0.47,-0.88)
+*
+27.22
+79.62
+LIGO-L
+(-1.00,0.19,-0.98)
+(-0.80,-0.06,1.00)
+(-1.00,-0.40,-0.91)
+*
+76.76
+Virgo
+(-0.60,0.87,0.50)
+(-0.99,0.14,-0.99)
+(-0.43,-0.80,-0.60)
+(-0.31,0.86,-0.50)
+*
+TABLE III. The expansion coefficients (CIT , CIV ,CIS) (upper right) and (CWT , CWV ) (lower left).
+KAGRA
+LIGO-I
+LIGO-H
+LIGO-L
+Virgo
+KAGRA
+*
+(-0.54,0.43,-1.22)
+(0.48,0.15,1.06)
+(-0.34,-0.06,-0.39)
+(-1.1,0.64,-0.77)
+LIGO-I
+(-0.54,0.70)
+*
+(-0.80,0.40,0.05)
+(0.11,-0.06,-0.05)
+(-0.29,-0.53,-1.20)
+LIGO-H
+(-0.93,0.90)
+(1.55,-0.57)
+*
+(-0.11,-1.62,-1.00)
+(0.86,-1.02,0.24)
+LIGO-L
+(1.48,-0.78)
+(-2.04,0.42)
+(0.28,-0.51)
+*
+(-0.97,0.77,-0.83)
+Virgo
+(-0.63,0.45)
+(0.77,-0.93)
+(0.68,-0.57)
+(0.54,-0.48)
+*
+Tensor even
+Vector even
+Scalar even
+Tensor odd
+Vector odd
+0
+50
+100
+150
+200
+250
+300
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+f[Hz]
+ORFs
+FIG. 4.
+All the five ORFs of the HL pair with y
+=
+6.3(f/100Hz). The solid lines correspond to the even ORFs.
+The dashed lines show the odd ones.
+of two detectors A and B can be modeled as
+sA(f) = hA(f) + nA(f) ,
+sB(f) = hB(f) + nB(f) .
+(57)
+Here, hA,B are the signals from stochastic backgrounds
+(see Eq. (21)) and nA,B are the detector noises. In this
+paper, we assume the noises nA,B to be stationary, Gaus-
+sian, and mutually independent. In addition, the signals
+are assumed to be much smaller than the noises, namely
+|hA,B|≪ |nA,B| (the weak signal condition). Then the
+covariance of the detector noises is given by
+⟨nA(f)n∗
+B(f ′)⟩ = δAB
+2 NA(f)δ(f − f ′) ,
+(58)
+where NA is the noise power spectrum.
+As a preparation of the correlation analysis, let us take
+the product of the two outputs of pairwise detectors (u =
+AB) as (again omitting the delta functions)
+µu(f) ≡ Re[sA(f)s∗
+B(f)] .
+(59)
+Here we extracted the real part. This is because, we know
+the following relation
+⟨sA(f)s∗
+B(f)⟩ = ⟨hA(f)hB(f)⟩ + ⟨hA(f)nB(f)⟩
+(60)
++ ⟨nA(f)hB(f)⟩ + ⟨nA(f)nB(f)⟩]
+= ⟨hA(f)hB(f)⟩
+(61)
+= Cu(f) ∈ Real
+(62)
+for the expectation value (using the statistical indepen-
+dence between hA,B and nA,B). As we see shortly, this
+projection can reduce the associated noise level [45].
+The variance can be calculated similarly. Under the
+weak signal condition (|hA(f)|≪ |nA(f)|), we have
+Nu(f) = ⟨µ2
+u⟩ − ⟨µu⟩2 ∼ ⟨µ2
+u⟩
+=1
+4 ⟨(sAs∗
+B + s∗
+AsB)(f)(sAs∗
+B + s∗
+AsB)(f)⟩
+∼1
+2 ⟨nA(f)n∗
+B(f)n∗
+A(f)nB(f)⟩
+=1
+8NA(f)NB(f)
+(63)
+with
+�
+Nu(f) ≫ ⟨µu(f)⟩. Note that we have the addi-
+tional factor 2−1 due to the real projection (59).
+The basic idea of the correlation analysis is to coher-
+ently amplify the background signal relative to the noise,
+by using a large number of Fourier modes, after a long
+observational time. We now explain this by deriving Eqs.
+(66) and (67).
+To deal with the frequency dependence, we first divide
+the Fourier modes into N bins (B1, B2, · · · , Bρ, · · · , BN)
+characterized by the central frequencies fρ and a fixed
+width δf [45]. We take δf to be much smaller than fρ,
+such that involved quantities (e.g. IP (f), W P (f), and
+γIP ,WP (f)) are nearly the same in each bin. Meanwhile,
+we also set the width δf to be much larger than the
+frequency resolution T −1
+obs determined by the observation
+
+8
+Tensor even
+Vector even
+Scalar even
+Tensor odd
+Vector odd
+0
+50
+100
+150
+200
+-0.2
+-0.1
+0.0
+0.1
+0.2
+f[Hz]
+ORFs
+Even
+CIT
+Sin (y)
+y
+Odd
+CWT
+Cos (y)
+y
+0
+50
+100
+150
+200
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+f[Hz]
+ORF
+Tensor
+FIG. 5. The ORFs of the LV pair with y = 16.65(f/100Hz).
+(Top) All the five ORFs. The solid lines correspond the even
+ORFs.
+The dashed lines are the odd ones.
+(Bottom) The
+ORFs for the two tensor modes IT and WT . The solid ones
+are the original expressions, and the dotted lines show their
+asymptotic profiles Eq. (44) with the coefficients (CIT , CWT )
+given in Table III.
+time Tobs (i.e. the number of the modes Tobsδf ≫ 1 in
+each bin).
+Now, we sum up the product µu in each bin as
+µρ
+u =
+�
+f∈Bρ
+Re[sA(f)sB(f)∗]
+≃
+�
+f∈Bρ
+Re[hA(f)hB(f)∗ + nA(f)nB(f)∗]
+(64)
+≃ ⟨µρ
+u⟩ +
+�
+f∈Bρ
+Re[nA(f)nB(f)∗] .
+(65)
+In Eq. (64), the first term comes from the background
+and can be coherently amplified.
+On the other hand,
+the second term is due to the noises and is not amplified
+because of its incoherence.
+Let us calculate the expectation value and the variance
+of the compressed estimator µρ
+u.
+From Eqs.
+(23) and
+(24), the expectation value ⟨µρ
+u⟩ is given by
+⟨µρ
+u⟩ =
+�
+f∈Bρ
+Re[⟨hA(f)hB(f)∗⟩]
+∼8π
+5 Tobsδf
+�
+�
+�
+P =T,V,S
+γIP
+u (fρ)IP (fρ)
++
+�
+P =T,V
+γWP
+u
+(fc)WP (fc)
+�
+� .
+(66)
+The variance is given by the second term in Eq. (64) as
+N ρ
+u = ⟨µρ2
+u ⟩ − ⟨µρ
+u⟩2
+∼1
+2
+�
+f
+�
+f ′
+⟨nA(f)n∗
+A(f ′)⟩ ⟨nB(f)n∗
+B(f ′)⟩
+∼Tobsδf
+8
+NA(fρ)NB(fρ)
+(67)
+with no noise correlation between different pairs (e.g.,
+between HK and HL). The last line is obtained by sub-
+stitution of Eq. (58). These expressions show that ex-
+pectation value ⟨µρ
+u⟩ is proportional to the number of
+the Fourier modes Tobsδf but the variance
+√
+N ρ
+u is pro-
+portional to √Tobsδf. For Tobsδf ≫ 1, the background
+signal is relatively amplified to the noise, as expected.
+Combining Eqs. (66) and (67), we obtain the SNR of
+each bin as
+SNRρ2
+u = ⟨µρ
+u⟩2
+N ρ
+u
+∼ 2
+�16π
+5
+�2
+Tobsδf
+NA(fρ)NB(fρ)
+�
+�
+�
+P =T,V,S
+γIP
+u (fρ)IP (fρ)
++
+�
+P =T,V
+γWP
+u
+(fρ)WP (fρ)
+�
+�
+2
+.
+(68)
+Quadratically summing up all the frequency bin, we ob-
+tain total SNR for the detector pair u as
+SNR2
+u =
+�
+ρ
+SNRρ2
+u
+= 2Tobs
+�16π
+5
+�2
+×
+�
+df
+��
+P =T,V,S γP
+u IP + �
+T,V γWP
+u
+WP
+�2
+NA(f)NB(f)
+.
+(69)
+In this paper, we assume that all detectors have the
+noise spectrum NAL identical to the design sensitivity of
+the advanced LIGO [46] (see Fig. 6 for NAL(f)). Con-
+sidering the current status of the LVK-network, this as-
+sumption looks unrealistic. However, it is virtually diffi-
+cult for a largely less sensitive detector to make an effec-
+tive contribution to the network, and we expect that our
+
+9
+5
+10
+50
+100
+500
+1000
+5000
+1.×10-24
+1.×10-23
+1.×10-22
+1.×10-21
+1.×10-20
+f[Hz]
+NALIGO ( f ) [Hz-1/2]
+FIG. 6. Noise power spectrum of advanced LIGO, taken from
+[46]. The spike around 9Hz is due to the resonance of the
+anti-vibration components.
+assumption will eventually become a reasonable approx-
+imation.
+Note that it is, in principle, straightforward
+to taking into account the difference between detector
+noise spectra for the rest of this paper.
+For simplic-
+ity, unless otherwise stated, we also assume flat spectra
+ΩQ
+GW (f) ∝ f 0 for the injected backgrounds.
+In the upper right panel of Table IV, we present SNRu
+for ΩIT
+GW = 10−8, setting other four spectra at zero. Sim-
+ilarly, in the lower left part, we show SNRu only with
+the non-vanishing compoent ΩWT
+GW = 10−8 (ignoring the
+physical requirement |ΩWT
+GW |≤ ΩIT
+GW ). In the ten detec-
+tor pairs, the HL pair has the best sensitivity to IT , but
+the worst sensitivity to WT .
+This is due to the small
+separation angle β = 27◦ of the HL pair, as pointed out
+earlier in Sec. II D. In contrast, the IL pair has the worst
+sensitivity to IT but the best sensitivity to WT with the
+largest separation angle β = 128◦.
+For a background purely made by IT , we have the net-
+work sensitivity
+SNR2
+IT = 2Tobs
+�16π
+5
+�2 �
+df
+�
+u
+�
+γIT
+u
+�2 I2
+T
+N 2
+AL(f)
+.
+(70)
+For the HIKLV network and a flat spectrum, we numer-
+ically have
+SNRIT = 19.0
+�
+ΩIT
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+(≡ SNR0) ,
+(71)
+which gives the maximum sensitivity to IT achieved by
+the five detectors. In Eq. (71), we introduced the nota-
+tion SNR0 in order to use this result as a reference value
+in our study below.
+V.
+SEPARATION OF THE FIVE COMPONENTS
+As shown in Eq. (66), the expectation value of a single
+segment ⟨µρ
+u(f)⟩ is given as the linear combination of
+the five spectra Q = {IT , IV , IS, WT , WV }. For testing
+alternative gravity theories, we would like to handle them
+separately.
+Such a method has been discussed in the
+literature (IT and WT in [33, 36], and IT , IV , and IS in
+[47]). Its basic strategy is to take the appropriate linear
+combinations of the cross correlation signals µρ
+u(f) and
+algebraically isolate the background spectra.
+To this end, we need at least 5 detector pairs. This
+can be satisfied by 4 or more detectors, which provide 6
+or more pairs (not equal to 5). Thus the spectral decom-
+position is actually an overdetermined problem.
+Our first objective in this section is to present a simple
+expression for the signal-to noise ratios SNRρ
+Q after the
+algebraic spectral decomposition. However, as outlined
+in Sec.
+VA, under the orthodox approach, we have a
+technical difficulty at deriving the simplified expression
+SNRρ
+Q. Thus, basically following the arguments in Ref.
+[36], we provide the desired expression that is not proven
+in a precise mathematical sense.
+A.
+Orthodox Approach
+As an example, let us consider the five detector net-
+work with 10 data set µρ
+u (u = 1, · · · , 10). Each segment
+contains the five polarization spectra as in Eq. (66). Us-
+ing the difference between the ORFs, we can isolate a
+specific spectrum Q (e.g., IV ) by algebraically cancelling
+other four spectra (e.g., IT , IS, WT , WV ). Then we ob-
+tain the six linear combinations of the original data µρ
+u.
+In contrast to the original ten data µρ
+u, the resultant
+six combinations have correlated detector noises. We can
+newly generate six noise orthogonal combinations, as a
+standard eigenvalue decomposition for the 6 × 6 noise
+matrix. Then, quadratically adding the six orthogonal
+elements, we obtain the network SNR for the target spec-
+trum Q. We can formally put
+(SNRρ
+Q)2 = 2Tobsδf
+�16π
+5
+�2 Q(f)2XQ(f)
+N 2
+AL(f)
+.
+(72)
+Here the factor XQ(f) is given by the 50 ORFs, and can
+be effectively regarded as the square of a compiled ORF.
+Unfortunately, following the above line of argument,
+we could not analytically obtain the simplified symmet-
+rical form for the factor XQ(f) even with Mathematica.
+B.
+Alternative Approach
+In Ref.
+[36], a convenient construction scheme was
+deduced for the factor XQ, on the basis of the likeli-
+hood study for the multiple spectra (closely related to
+the Fisher matrix analyses). Here we concisely provide
+their final expression (see [36] for detail).
+
+10
+TABLE IV. The upper right corresponds to SNRAB for ΩIT
+GW = 10−8 setting other four spectra at zero. The lower left is only
+with ΩWV
+GW = 10−8.
+KAGRA
+LIGO-I
+LIGO-H
+LIGO-L
+Virgo
+KAGRA
+*
+2.16
+2.42
+1.38
+4.47
+LIGO-I
+2.32
+*
+2.79
+0.34
+3.27
+LIGO-H
+3.67
+5.07
+*
+15.4
+3.51
+LIGO-L
+5.09
+6.33
+1.04
+*
+3.83
+VIRGO
+2.28
+3.22
+2.56
+2.07
+*
+We first compose a 5 × 5 matrix F as
+F QQ′ ≡
+np
+�
+u=1
+γQ
+u γQ′
+u
+.
+(73)
+Next, we take its inverse matrix
+Σ ≡ F −1 .
+(74)
+Then, we presume the following relation for the factor
+XQ
+XQ =
+1
+ΣQQ(f).
+(75)
+Below, we mention some circumstance evidences for its
+validity.
+For decomposing only two spectra (e.g., IT and WT ),
+we can analytically confirmed that this relation is ac-
+tually true for an arbitrary number of detectors.
+For
+the five spectral decomposition with ten detector pairs,
+we numerically generated the 50 ORFs randomly in the
+range [−1, 1] and evaluated the both sides of Eq. (75)
+with Mathematica.
+We repeated this experiments for
+many times and confirmed equality within numerical ac-
+curacy. Note that, with Mathematica, we need much less
+computational resources at numerical evaluation than at
+corresponding symbolic processing.
+We hereafter use relation (75) and put
+(SNRρ
+Q)2 = δf
+f ZQ(f)
+�
+ΩQ
+GW (f)
+10−8
+�2 �Tobs
+3yr
+�
+,
+(76)
+where we defined (e.g., with Eqs. (17) and (72))
+ZQ(f) ≡ 3.7 × 10−82XQ(f)
+� f
+1Hz
+�−5 �NAL(f)
+1Hz−1
+�−2
+.
+(77)
+Here we used H0 = 70km s−1 Mpc−1.
+This function
+ZQ(f) shows the contribution of background signals from
+various frequencies.
+After the frequency integral, we obtain
+(SNRQ)2 =
+� ∞
+0
+df
+f ZQ(f)
+�
+ΩQ
+GW (f)
+10−8
+�2 �Tobs
+3yr
+�
+. (78)
+VI.
+STATISTICAL LOSS ASSOCIATED WITH
+THE MODE SEPARATION
+We now examine the matrices F and Σ, in particular
+the role of their off-diagonal elements.
+A.
+Reduction Factors
+For simplicity, we first deal with the two component
+analysis with the spectra IT and Q′ (Q′ = IV , IS or WT ).
+The 2 × 2 matrix F is given by
+F =
+� �np
+u=1 γIT
+u γIT
+u
+�np
+u=1 γIT
+u γQ′
+u
+�np
+u=1 γIT
+u γQ′
+u
+�np
+u=1 γQ′
+u γQ′
+u
+�
+,
+(79)
+and we have
+XIT =
+1
+ΣIT IT
+= (1 − R2
+IT Q′)
+np
+�
+u=1
+γIT
+u γIT
+u
+,
+(80)
+XQ′ =
+1
+ΣQ′Q′ = (1 − R2
+IT Q′)
+np
+�
+u=1
+γQ′
+u γQ′
+u
+.
+(81)
+Here we defined the coefficient RIT Q′ by
+RIT Q′ ≡
+�np
+u=1 γIT
+u γQ′
+u
+�
+�nt
+u=1
+�
+γIT
+u
+�2�
+�np
+u=1
+�
+γQ′
+u
+�2 .
+(82)
+From Cauchy-Schwartz inequality, we have |RIT Q′|≤ 1
+with equality only for two parallel vectors {γIT
+u } and
+{γQ′
+u }. The coefficient RIT Q′ represents the correlation
+between the two spectra and reduces SNRs after the spec-
+tral decomposition through the factor (1 − R2
+IT Q′) (see
+Eqs. (72) and (80)). This factor shows the statistical
+loss associated with the decomposition.
+So far, we discussed two component decomposition.
+When the number nQ of the target spectral components
+is larger than two (nQ > 2), we can similarly define the
+reduction factor 1 − R2
+Qi (i = 1, · · · , nQ) by
+1 − R2
+Qi =
+XQi(f)
+�
+u γQi
+u γQi
+u
+=
+1
+(F −1)QiQiF QiQi .
+(83)
+Note that we omitted the subscripts other than the com-
+ponent of interest for the notational simplicity.
+If the
+
+11
+HIKLV
+HKLV
+HLV
+0
+50
+100
+150
+200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+f[Hz]
+1-RIT IV
+2
+HIKLV
+HKLV
+HLV
+0
+50
+100
+150
+200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+f[Hz]
+1-RIT IS
+2
+FIG. 7. The reduction factors 1−R2
+IT IV (upper) and 1−R2
+IT IS
+(lower) respectively for the two component analyses {IT , IV }
+and {IT , IS}.
+vectors {γQi
+u } (i = 1, · · · , nQ) are close to linearly depen-
+dent, the matrix F becomes nearly singular, and we could
+have |(F −1)QiQiF QiQi|≫ 1, resulting in a large signal
+loss 1 − R2
+Qi ≪ 1. In this relation, our two new findings
+in Sec. II could play interesting roles, as explained in the
+next subsection.
+B.
+Numerical Results
+In Fig. 7, we show the reduction factor (1 − R2
+IT ) for
+the two component models {IT , IV } (upper) and {IT , IS}
+(lower). As shown in Eq. (51), we have the degeneracy
+limf→0{γIT
+u } = {γIV
+u } = {γIS
+u } and need the sub-leading
+correction O(f 2) to decompose the two spectra. We thus
+have a significant suppression (1 − R2
+IT ) ≲ 0.1 at f ≲
+10Hz.
+In Fig. 8, we examined the hypothetical case for de-
+composing the two odd spectra {WT , WV }. Their ORFs
+are parallel at the low frequency limit and nearly parallel
+around the anathematic frequency 13Hz. We thus have
+the siginificant signal reduction below 13Hz, as in Fig. 8.
+Next we move to examine the decomposition of more
+than two spectra nQ > 2. In Fig. 9, we show the reduc-
+HIKLV
+HKLV
+HLV
+0
+50
+100
+150
+200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+f[Hz]
+1-RWT WV
+2
+FIG. 8. The reduction factors 1−R2
+WT WV for the hypothetical
+two component analysis {WT , WV }.
+ITIVIS(HIKLV)
+ITIVIS(HKLV)
+ITIVIS(HLV)
+ITIVISWTWV(HIKLV)
+50
+100
+150
+200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+f[Hz]
+1-RIT
+2
+FIG. 9. The reduction factors 1 − R2
+TT for the three and five
+component analyses.
+HIKLV
+HKLV
+HLV
+0
+50
+100
+150
+200
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+f[Hz]
+1-RIT WT
+2
+FIG. 10. The reduction factors 1 − R2
+IT WT for the two com-
+ponent analysis {IT , WT }.
+
+12
+tion factor (1−R2
+IT ) at separating the three even spectra
+{IT , IV , IS}. In contrast to the two component cases in
+Fig. 7, the strong suppression 1 − R2
+IT ≲ 0.1 continues
+up to 20Hz. This is because the trinity degeneracy (54)
+works still at the sub-leading order O(f 2). We thus need
+the higher corrections O(f 4) to isolate the three spectra.
+In fact, even for a detector network not tangential to a
+sphere, we still have det F = O(f 4) for the 3 × 3 matrix
+F of the three even spectra {IT , IV , IS}.
+Note that the LIGO-India plays a key role for the usage
+of the higher order terms O(f 4) (or more appropriately
+O(y4) for the perturbative expansion). In Fig. 9, we can
+clearly see the resulting improvement around 20-40Hz.
+Here the mechanism around Eq. (56) works efficiently,
+in particular, with the HI and IL pairs.
+In Fig. 10, we present the result for the two tensorial
+spectra {IT , WT }. Since their ORFs {γIT
+u } and {γWT
+u
+}
+are generally not parallel, the reduction is not significant.
+If we use the HIKLV pair, the reduction factor is no less
+than 0.8.
+VII.
+SIGNAL TO NOISE RATIO
+A.
+Results for the HIKLV Network
+Now we discuss the signal-to-noise ratios SNRQ after
+the spectral decomposition and the associated frequency
+profiles ZQ(f) defined in Eq. (77). We start with the
+results for the HIKLV network and flat spectra ΩQ
+GW =
+const.
+In Fig. 11, we show the profile ZIT (f) for IT . The
+sharp dip around 10Hz is caused by the noise spike in
+Fig.
+5.
+The uppermost blue line shows the result for
+the simplest case only with IT (no reduction factor). Its
+peak is around 25Hz with the integrated value SNRIT
+(see Eq. (71))
+SNR0 = 19.0
+�
+ΩIT
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+.
+(84)
+We use this expression to normalize the signals SNRQ
+in different settings, as presented in Tables V and VI.
+In Fig. 11, the four lines other than the blue one show
+the profiles ZIT (f) after decomposing multiple spectra.
+Their fractional differences from the blue lines represent
+the corresponding reduction factor (1 − R2
+IT ).
+At the decomposition of IT and WT (dashed orange
+line in Fig. 12), the statistical loss is inconspicuous with
+the total value SNRIT /SNR0 = 0.99 (see Table V). How-
+ever, we need to pay a significant cost to isolate the three
+even spectra IT , IV and IS. The total signal decreases
+down to SNRIT /SNR0 = 0.47 and the peak of the profile
+ZIT (f) moves up to ∼ 40Hz.
+In Figure 12 and Table VI, we show the results for the
+odd tensor spectrum WT . Similarly to Fig. 11, we can
+isolate it from IT with almost no loss of the integrated
+signal SNRWT (see Table VI). When we separate WT and
+IT only
+ITWT
+ITIV
+ITIVIS
+ITIVISWTWV
+5
+10
+50
+100
+1. × 10-12
+1. × 10-9
+1. × 10-6
+0.000
+1
+1000
+f[Hz]
+ZIT(f)
+FIG. 11.
+The factor ZIT (f) showing the signal strength
+defined in Eq. (77) for the HIKLV network. The blue and
+orange curves are nearly overlapped. The ratio between the
+blue and other curves corresponds to the reduction factor 1−
+R2
+IT due to the signal correlation. The sharp dip around 9Hz
+is due to the noise spectrum NAL(f).
+WT only
+ITWT
+WTWV
+ITIVISWTWV
+5
+10
+50
+100
+1. × 10-8
+1. × 10-4
+1
+f[Hz]
+ZWT(f)
+FIG. 12. The factor ZWT (f) showing the signal strength de-
+fined in Eq. (77) for the HIKLV network. The green and the
+red curves have the sharp dips around 13Hz.
+WV , the anathematic frequency 13Hz clearly appears,
+as shown by the green and red lines, and the function
+ZWT (f) is significantly suppressed below ∼ 20Hz.
+In
+contrast to ZIT (f), the peak of the profile ZWT (f) stays
+around 25Hz.
+B.
+LIGO-India
+Next we discuss the impacts of adding LIGO-India to
+the detector network. As shown in Table V, for the single
+spectral search IT , LIGO-India increases the total signal
+SNRIT by only 3%. However, together with KAGRA, it
+makes a notable contribution to improve the sensitivity
+to the odd spectra WT (see Table VI). In addition, as
+explained earlier, LIGO-India also helps us to use the
+higher order terms O(f 4) for decomposing the three even
+spectra. For the five spectral search, we can double both
+
+13
+HIKLV
+HKLV
+HILV
+5
+10
+50
+100
+1. × 10-13
+1. × 10-8
+0.000
+100.000
+f[Hz]
+ZIT(f)
+HIKLV
+HKLV
+HILV
+5
+10
+50
+100
+1. × 10-11
+1. × 10-8
+1. × 10-5
+0.01
+10
+f[Hz]
+ZWT(f)
+FIG. 13. The top and bottom panels respectively show ZIT (f)
+and ZIT (f) for three networks. The dotted lines in the upper
+panel are proportional to f 2 and f −2.
+SNRIT and SNRWT by adding the LIGO-India detector.
+C.
+Power-law Models
+So far, we have assumed that the background has flat
+spectra ΩQ
+GW = const. Now, we briefly discuss a power-
+law form ΩQ
+GW ∝ f α in the frequency regime in interest.
+The integrated signal SNRQ in Eq. (78) has the dom-
+inant contribution around the frequency where the func-
+tion ZQ(f) is tangential to a curve f −2α. As deduced
+from Fig. 13, for the five spectral decompositions with
+the HIKLV network, the tangential frequencies are 40Hz
+for IT and 25Hz for WT , as long as the index α is in
+the range [−1, 1]. Therefore, the total signals are very
+roughly given as
+SNRIT ∼ 19 × 0.4
+�
+ΩIT
+GW (40Hz)
+10−8
+� �Tobs
+3yr
+�1/2
+(85)
+SNRWT ∼ 19 × 0.39
+�
+ΩWT
+GW (25Hz)
+10−8
+� �Tobs
+3yr
+�1/2
+.(86)
+TABLE V. Ratio SNRIT /SNR0 after the spectral isolation.
+All five components of LHV is missing since LHV has only
+three independent detector pairs. We assumed a flat spectrum
+ΩIT
+GW = const.
+background components
+KILHV
+KLHV
+LHV
+IT only
+1
+0.97
+0.91
+IT , WT
+0.99
+0.96
+0.91
+IT , IV
+0.63
+0.57
+0.47
+IT , IS
+0.89
+0.82
+0.74
+IT , IV , IS
+0.47
+0.33
+0.22
+All five
+0.40
+0.20
+*
+TABLE VI. Ratio SNRWT /SNR0 after the spectral isolation.
+We assume flat spectra and omit the factor ΩWT
+GW /ΩIT
+GW for
+simplicity.
+background components
+KILHV
+KLHV
+LHV
+WT only
+0.62
+0.40
+0.18
+WT , IT
+0.62
+0.39
+0.18
+WT , WV
+0.43
+0.20
+0.06
+All five
+0.39
+0.17
+*
+VIII.
+SUMMARY
+In this paper, we studied the prospects for the polar-
+izational study of isotropic stochastic gravitational wave
+backgrounds by correlating second generation detectors.
+In the long-wave approximation, the backgrounds are
+generally characterized by the five spectra IT,V,S and
+WT,V . The modes other than IT can appear in modi-
+fied theories of gravity.
+For correlation analysis, the ORFs play key roles. In
+this paper, we newly identified two simple relations be-
+hind them. The first one is the trinity degeneracy (54)
+between the three even ORFs at the sub-leading order
+O(f 2). The second one is the degeneracy between the
+two odd ORFs around the specific frequency 13Hz.
+For each detector pair, the correlation product is given
+as a linear combination of the five spectra. To closely
+examine theories of gravitation, we desire to separate
+the five spectra clearly.
+We thus examined their alge-
+braic decomposition using the difference between the in-
+volved ORFs. Here we generally need to handle an over-
+determined problem. By extending an analytic frame-
+work in the literature, we derived the formal expression
+(78) for the optimal SNRs after the spectral decomposi-
+tion.
+Then, assuming an identical noise curve for the five
+detectors and flat background spectra, we discussed the
+statistical loss of sensitivities accompanied by the decom-
+position. This loss is closely related to the off-diagonal
+elements of the matrix F QQ′ ∝ �np
+i=1 γQ
+u γQ′
+u .
+In this context, our two findings are quite useful for
+following the singular behaviors at the decomposition.
+On the one hand, when simultaneously dealing with the
+
+14
+three even spectra, due to the higher order degeneracy of
+their ORFs, we have a large signal reduction below 20Hz,
+unlike the two spectral decomposition (such as IT − IV
+and IT − IS). On the other hand, it is very difficult to
+separate the two odd spectra below ∼ 20Hz, including
+the anathematic frequency 13Hz.
+Given the structure
+of the covariance matrix F, these limitations will also
+appear in the likelihood or Fisher matrix analyses.
+We also discussed the advantage of adding the LIGO-
+India detector to the ground-based detector network. As
+shown in Tables V and VI, it can largely increase the
+sensitivities to the odd spectra and will also help us to
+decompose multiple spectra. Here the HI and LI pairs
+are particularly useful with the large separation angles
+β.
+In this paper, we have mainly considered the sec-
+ond generation ground-based detectors.
+However, our
+method is general enough to be straightforwardly applied
+to the third generation ground-based detectors (such as
+ET [48] and CE [49], see also
+[50]) and partially to
+space borne detectors (LISA [39], TAIJI [40], and Tian-
+Qin [41]). The former will cover a lower frequency regime
+than that of the second generation ones and will be more
+severely affected by the limitations associated with our
+two findings.
+ACKNOWLEDGMENTS
+We would like to thank M. Ando and S. Bose for valu-
+able comments. This work is supported by JSPS Kakenhi
+Grant-in-Aid for Scientific Research (Nos. 17H06358 and
+19K03870). HO is supported by Grant-in-Aid for JSPS
+Fellows JP22J14159.
+Appendix A: optimal SNR for the ground-based
+detectors
+Assuming the flat spectrum of the background and us-
+ing Eq. (78), we can evaluate the SNR for each spectra
+after the decomposition. As a reference, we provide nu-
+merical results for the five spectral components. For the
+HKLV-network, we obtain
+SNRIT = 3.94
+�
+ΩIT
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+(A1)
+SNRIV = 2.75
+�
+ΩIV
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+(A2)
+SNRIS = 6.81
+�
+ΩIS
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+(A3)
+SNRWT = 3.14
+�
+ΩWT
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+(A4)
+SNRWV = 4.07
+�
+ΩWV
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+.
+(A5)
+For the HIKLV-network, we have
+SNRIT = 7.53
+�
+ΩIT
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+(A6)
+SNRIV = 6.14
+�
+ΩIV
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+(A7)
+SNRIS = 9.74
+�
+ΩIS
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+(A8)
+SNRWT = 7.50
+�
+ΩWT
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+(A9)
+SNRWV = 8.27
+�
+ΩWV
+GW
+10−8
+� �Tobs
+3yr
+�1/2
+.
+(A10)
+[1] A. A. Starobinsky, Spectrum of relict gravitational radi-
+ation and the early state of the universe, JETP Lett. 30,
+682 (1979).
+[2] R. Easther, J. T. Giblin, and E. A. Lim, Gravitational
+wave production at the end of inflation, Phys. Rev. Lett.
+99, 221301 (2007).
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+

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+Supervised Acoustic Embeddings And Their Transferability Across
+Languages
+Sreepratha Ram
+UAE University
+[email protected]
+Hanan Aldarmaki
+MBZUAI
+[email protected]
+Abstract
+In speech recognition, it is essential to model
+the phonetic content of the input signal while
+discarding irrelevant factors such as speaker
+variations and noise, which is challenging in
+low-resource settings.
+Self-supervised pre-
+training has been proposed as a way to
+improve both supervised and unsupervised
+speech recognition, including frame-level fea-
+ture representations and Acoustic Word Em-
+beddings (AWE) for variable-length segments.
+However, self-supervised models alone cannot
+learn perfect separation of the linguistic con-
+tent as they are trained to optimize indirect ob-
+jectives. In this work, we experiment with dif-
+ferent pre-trained self-supervised features as
+input to AWE models and show that they work
+best within a supervised framework. Models
+trained on English can be transferred to other
+languages with no adaptation and outperform
+self-supervised models trained solely on the
+target languages.
+Keywords— Unsupervised ASR, Transfer Learn-
+ing, Acoustic Word Embeddings
+1
+Introduction
+With supervised speech recognition systems get-
+ting more robust and accurate due to the avail-
+ability of large amounts of labeled data and com-
+putational power (Gulati et al., 2020; Baevski
+et al., 2020b), more attention is now given to low-
+resource languages for which training data are lim-
+ited or non-existent (Aldarmaki et al., 2022). Un-
+supervised pre-training using unlabeled speech can
+be leveraged to improve both supervised and un-
+supervised models; for instance, speech represen-
+tations pre-trained on large amounts of unlabeled
+speech from multiple languages have been shown
+to improve ASR performance for low-resource lan-
+guages (Kawakami et al., 2020; Conneau et al.,
+2020).
+While most supervised ASR models operate at
+the level of phones, word-level segmental ASR
+where variable-length segments are modeled and
+embedded into fixed-dimensional vectors have also
+been explored with relative success (Abdel-Hamid
+et al., 2013; He and Fosler-Lussier, 2015). In a
+similar vein, Acoustic Word Embeddings (AWEs)
+have been proposed as a way to efficiently compare
+variable-length speech segments in low-resource
+settings (Peng et al., 2020; Kamper et al., 2020).
+Unlike written words, spoken words naturally con-
+tain speaker and phonetic variability that makes
+them more difficult to model in a latent space with-
+out supervision. Self-supervised pre-training and
+cross-lingual transfer are two possible approaches
+to make unsupervised models more robust to non-
+linguistic variations in the input signal.
+In this work, we investigate the performance of
+self-supervised training of AWE models versus su-
+pervised training with zero-shot cross-lingual trans-
+fer. We experiment with different types of acoustic
+features and measure their performance separately
+and within the AWE models. While we find that
+pre-trained acoustic features improve the perfor-
+mance of self-supervised AWE models to some
+extent, a larger improvement can be achieved when
+the AWE models are trained in a supervised manner
+using small amount of labeled data from a different
+language. This zero-shot cross-lingual transfer is
+observed consistently across different languages,
+and particularly with the use of pre-trained feature
+representations. Our results suggest that supervised
+training with zero-shot cross-lingual transfer is a
+more effective approach for low-resource speech
+models compared with purely self-supervised train-
+ing1.
+2
+Background & Related Work
+Spoken language is often modeled using short
+fixed-length frames of 10 to 30 ms duration, which
+1We provide python training and evaluation scripts
+for replicating our experiments:
+https://github.com/h-
+aldarmaki/acoustic_embeddings
+arXiv:2301.01020v1  [cs.CL]  3 Jan 2023
+
+results in variable-length word segments. Dynamic
+Time Warping (DTW) is an early technique that
+uses dynamic programming to compare variable-
+length segments by finding optimal frame-wise
+alignment. DTW is rather inefficient, which mo-
+tivates embedding variable-length segments into
+vectors of fixed size that can be compared using
+more efficient metrics such as cosine or Euclidean
+distance (Levin et al., 2013). Different types of
+Acoustic Word Embeddings (AWE) have been pro-
+posed. As these techniques are generally meant for
+low-resource languages, they are typically trained
+in a self-supervised manner, most commonly us-
+ing an auto-encoder network with reconstruction
+loss (Chung et al., 2016; Holzenberger et al., 2018).
+Compared with direct comparison via DTW, these
+AWEs generally result in similar or slightly supe-
+rior performance while being far more efficient
+(Holzenberger et al., 2018). Peng et al. (2020)
+describes an alternative training strategy using cor-
+respondence auto-encoders, which relies on word
+pairs extracted via unsupervised spoken term dis-
+covery, and further improvements can be achieved
+using contrastive learning and multi-lingual adap-
+tation (Jacobs et al., 2021).
+The above models use static acoustic features
+(e.g. MFCCs) as input. van Staden and Kam-
+per (2021) shows that using pre-trained features
+like CPC (van den Oord et al., 2018) improves the
+performance of unsupervised AWE models. Pre-
+trained features have been repeatedly shown to im-
+prove performance in supervised downstream tasks
+(Yang et al., 2021). In addition, pre-trained features
+have been shown to transfer across languages. For
+instance, a modified version of CPC (MCPC) is de-
+scribed in Riviere et al. (2020), which demonstrates
+that pre-training these features on Egnlish results
+in improved phone classification accuracy for other
+languages. Other types of pre-trained features, such
+as wav2vec 2.0 (Baevski et al., 2020a) have been
+shown to improve both supervised and unsuper-
+vised ASR performance (Baevski et al., 2021), and
+multi-lingual training of these features (i.e. XLSR-
+53) can lead to improvements across many lan-
+guages compared to monolingual pre-training (Con-
+neau et al., 2020).
+3
+Objectives & Methodology
+The objective of this study is to investigate the effec-
+tiveness and trasnsferability of pre-trained acoustic
+features when used as input to acoustic word em-
+beddings. To that end, we compare self-supervised
+AWEs trained directly on the target languages ver-
+sus zero-shot cross-lingual transfer of supervised
+AWEs trained on a different source language. To
+our knowledge, the combination of pre-trained fea-
+tures with AWE models has not been fully investi-
+gated; most AWE models are trained with stan-
+dard acoustic features like MFCCs, while self-
+supervised features are typically evaluated within
+supervised models fine-tuned for the target lan-
+guages. Furthermore, zero-shot cross-lingual trans-
+fer of supervised AWEs has not been the focus of
+previous works in this area, which mainly focused
+on improving self-supervised AWEs.
+For the purpose of this evaluation, we use a rela-
+tively simple architecture for the embedding model
+and we fix the hyper-parameters based on prelimi-
+nary validation results for English self-supervised
+AWEs2. We do not do any further tuning of the
+self-supervised or the supervised models. We use
+English as the source language, and evaluate zero-
+shot transfer on four other languages: French, Ger-
+man, Spanish, and Arabic, with the latter used as
+a challenge set since it contains more variability
+and noise. No labeled data were used for the target
+languages with the exception of word boundaries
+which were obtained via force alignment. We eval-
+uate mainly using minimal-pair ABX error rates
+to measure phonetic discriminability and speaker
+invariance. We also cluster the embedded words
+and measure how often different occurrences of the
+same words end up in the same cluster.
+4
+Experimental Settings
+4.1
+Model Architecture
+Our AWE model consists of a multi-layer bidirec-
+tional LSTM encoder, followed by a uni-directional
+LSTM decoder, similar to Chung et al. (2016) and
+(Holzenberger et al., 2018). The encoder takes a se-
+quence of T acoustic features representing one spo-
+ken word. The forward and backward states of the
+last hidden layer of the encoder are concatenated
+and used as an embedding of the given word, call
+it hT . The decoder generates the target sequence
+one step at a time, conditioned on hT and the out-
+put at the previous time step, similar to Chung and
+2We observed that self-supervised models were very sen-
+sitive to the choice of architecture and hyper-parameters, so
+we fixed these in favor of self-supervised models. As shown
+in later sections, we still got better results with the supervised
+models, which shows that they are more robust and easier to
+optimize on top of being more effective.
+
+Glass (2018). In the self-supervised setting, the
+target sequence is the same as the input sequence,
+so the model is trained as an auto-encoder with
+MSE loss. In the supervised setting, the target is a
+sequence of phonemes representing the input word,
+and the model is trained by minimizing the nega-
+tive log-likelihood. We used 2-layer networks with
+100 hidden units for most models, which results
+in embeddings of size 200. We also used dropout
+with probability 0.3 on the input features, similar to
+the denoising networks used in Chung et al. (2016).
+More details of the parameters and training process
+can be found in the Appendix.
+4.2
+Feature Extraction
+For easier reproduciblity, we used the s3prl toolkit3
+for extracting all features. We used the pre-trained
+s3prl upstream models; among the many pretrained
+self supervised speech representations available,
+modified CPC, Wav2Wec2 and XLSR-53 were cho-
+sen based on superior DTW-based ABX scores4.
+All pre-trained models, with the exception of
+XLSR, have been exclusively pre-trained on En-
+glish data. XLSR-53 was pre-trained on unlabeled
+speech from 53 languages, including all target lan-
+guages in our experiments. As observed by other
+researchers (Bartelds et al., 2022), the performance
+of features extracted from transformer-based mod-
+els is largely dependent on the choice of layer; we
+used the last hidden layer for modified CPC, the
+second to last hidden layer for Wav2Vec2 and the
+central hidden layer (layer 12) for XLSR-53. Aver-
+aging all layers gave reasonable results, but these
+choices led to the best performance. For MFCC fea-
+tures, we also used the s3prl implementation, which
+includes 13 static features as well as dynamic delta
+and delta-delta coefficients.
+4.3
+Data
+We used the Librispeech (Panayotov et al., 2015)
+and Multilingual Librispeech (Pratap et al., 2020)
+datasets for English (en), French (fr), German (de),
+and Spanish (es). We used the dev sets for train-
+ing, and test sets for evaluation (dev-clean and test-
+clean for English). We obtained the word bound-
+aries automatically by forced alignment. For Ara-
+bic (ar), we used the dev and test sets of MGB2
+3https://github.com/s3prl/s3prl
+4We did experiment with other features like APC, VQ-
+APC, and VQ-Wav2Vec, and got similar or inferior perfor-
+mance to MCPC and Wav2Vec2. We opted to omit these for
+brevity.
+(Ali et al., 2016). This dataset is expected to be
+more challenging as it contains a diversity of di-
+alects as well as various noise conditions. See the
+Appendix for more details on the datasets and the
+word alignment process.
+4.4
+Evaluation Scheme
+We constructed Minimal-Pair ABX tasks, as de-
+scribed in Schatz et al. (2013). ABX tasks are
+typically used to measure phoneme discrimination
+in zero-resource settings, and they consist of two
+segments, A and B, that differ by a minimal con-
+trast (e.g. one phoneme difference), and a third
+segment X that matches either A or B. A distance
+measure such as DTW or cosine is used to find
+the closest match. We used two variants of this
+task: within-speaker ABX, where all three words
+are spoken by the same speaker, and cross-speaker
+ABX, where X is spoken by a different speaker.
+We automatically extracted the words from each
+test set; we selected A and B by finding word pairs
+that have the same length5 and Levenshtein edit
+distance of 1 or 2, which roughly corresponds to a
+difference of one or two phonemes most of the time.
+For Arabic, the dataset did not have speaker ids, so
+all three words could be from different speakers.
+In addition, due to the lower quality of the sound
+recordings and the presence of noise in this dataset,
+the word alignment quality is much lower than the
+other languages, so the automatic process resulted
+in many invalid segments. To have a more reliable
+test set for Arabic, we manually checked the valid-
+ity of the extracted words and kept 954 validated
+word pairs for evaluation.
+We also used clustering for complementary eval-
+uation. We clustered the embeddings using K-
+Means with K being the number of unique words
+in the test set. We calculated the accuracy of clus-
+tering as the percentage of words that match their
+cluster label, which is the word id of the majority of
+segments in each cluster. This allows us to measure
+if the embeddings of the same words are similar
+enough to be clustered together.
+5
+Results
+Table 1 shows ABX error rates using the input
+features directly (with DTW as distance metric),
+self-supervised AWEs trained on each language,
+5Since automatic word alignments tend to be inaccurate
+around the boundaries, we only used words that have at least
+five characters.
+
+en
+fr
+de
+es
+ar
+within
+across
+within
+across
+within
+across
+within
+across
+across
+Using DTW
+MFCC
+9.98
+19.85
+12.59
+24.82
+11.46
+25.03
+11.83
+25.27
+40.98
+wav2vec
+8.51
+11.15
+9.95
+15.61
+9.01
+15.08
+10.13
+15.46
+37.42
+MCPC
+7.80
+11.74
+9.96
+17.09
+9.22
+15.85
+11.14
+18.03
+38.99
+XLSR-53
+9.45
+13.72
+10.97
+16.77
+10.89
+15.76
+14.31
+19.31
+40.15
+Self-Supervised AWEs in each language
+MFCC
+12.30
+19.12
+16.63
+25.06
+16.71
+25.99
+16.58
+25.21
+43.92
+wav2vec
+6.63
+9.27
+9.94
+13.19
+10.56
+15.09
+12.25
+15.23
+38.16
+MCPC
+7.66
+9.53
+11.64
+16.19
+10.24
+15.30
+13.25
+16.29
+41.09
+XLSR-53
+10.61
+12.19
+13.72
+16.19
+12.59
+15.73
+17.10
+20.14
+37.00
+Supervised AWEs trained on English
+MFCC
+3.83
+4.57
+10.77
+15.32
+9.16
+13.06
+11.49
+16.56
+38.15
+wav2vec
+1.38
+1.14
+6.59
+9.44
+4.98
+7.32
+7.32
+10.12
+34.80
+MCPC
+2.49
+2.51
+8.13
+12.23
+6.66
+10.68
+9.89
+13.99
+39.20
+XLSR-53
+0.93
+0.79
+4.12
+5.71
+1.92
+2.83
+5.05
+6.21
+31.76
+Table 1: ABX error rates (%) within and across speakers for each language
+en
+fr
+de
+es
+ar
+Self-Supervised AWEs for each language
+MFCC
+47.3
+48.4
+45.0
+54.3
+30.9
+wav2vec
+66.1
+59.9
+59.5
+67.5
+35.9
+MCPC
+57.2
+53.6
+53.9
+60.4
+33.5
+XLSR-53
+52.0
+52.2
+54.9
+55.0
+31.0
+Supervised AWEs trained on English
+MFCC
+68.5
+52.7
+56.7
+61.1
+33.4
+wav2vec
+82.3
+64.8
+69.3
+71.1
+38.8
+MCPC
+74.5
+56.4
+62.7
+66.2
+35.6
+XLSR-53
+84.3
+69.1
+78.1
+75.8
+41.8
+Table 2: K-Means Clustering Accuracy (%)
+and supervised AWEs trained on English. Cosine
+similarity is the metric used in the latter two set-
+tings. Confirming previous results (Riviere et al.,
+2020), we do observe that pre-trained acoustic fea-
+tures like Modified CPC and Wav2Vec2, which are
+trained exclusively on English unlabeled speech,
+transfer well across languages. These pre-trained
+features consistently outperformed MFCC features
+for all languages, particularly in cross-speaker eval-
+uation. Unsurprisingly, the English language has
+the best ABX scores overall simply because the
+pre-trained features used are all trained on English.
+The results for self-supervised AWE models are
+mixed, but generally they are in the same range
+as DTW performance, which also conforms with
+previously published results (Holzenberger et al.,
+2018).
+With supervised training, we see significant re-
+duction in errors rates for all languages. The lowest
+error rates are achieved on the English test set, as
+expected. More notably, the largest reduction in
+error rates is achieved with the XLSR features. It is
+also interesting to note that XLSR features were not
+impressive in the self-supervised setting compared
+with other features; Wav2Vec2 and MCPC, which
+were trained on English only, gave better results
+in the self-supervised framework for all test lan-
+guages. The advantage of using these cross-lingual
+features was only evident in the supervised and
+transfer learning setting, where they consistently
+outperformed all other features. For Arabic, the
+error rates are higher overall due to the nature of
+the dataset, but we still observe the lowest error
+rate in the transfer learning setting.
+Finally, we see in table 2 that the clustering ac-
+curacy results are consistent with the ABX results,
+where supervised models trained on English con-
+sistently gave higher accuracy compared with self-
+supervised models trained on the target languages.
+6
+Conclusions
+Our results demonstrate the superior effectiveness
+of zero-shot transfer learning of acoustic word em-
+beddings compared with self-supervised training
+in the target languages. This is particularly use-
+ful for low-resource languages for which data may
+not be available for supervised or self-supervised
+
+training. The mechanism of this transfer is mainly
+through the reduction in speaker variability which
+is far easier to achieve via supervised training. In
+addition, supervised training makes the most out
+of pre-trained features, where we see further re-
+duction in error rates that far exceed the reduction
+observed in self-supervised settings. The presence
+of noise naturally results in larger error rates; fur-
+ther investigations are needed to demonstrate the
+transferability of noise robustness in a similar man-
+ner.
+Acknowledgement
+This work was supported by grant no. 31T139
+at United Arab Emirates University and partially
+funded under UAEU-ZU Joint Research Grant
+G00003715 (Fund No.: 12T034) through Emirates
+Center for Mobility Research.
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+Workshop (SLT), pages 927–934. IEEE.
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+arXiv preprint
+arXiv:2105.01051.
+A
+Appendix
+A.1
+Dataset Details
+A.2
+Model Architecture &
+Hyper-Parameters
+The architecture described in section 4.1 was mod-
+eled after other acoustic word embedding mod-
+els (Chung et al., 2016; Chung and Glass, 2018;
+Holzenberger et al., 2018) with slight variations
+Dataset
+test
+dev
+English
+52,576
+54,402
+French
+90,958
+83,560
+German
+121,713
+122,903
+Spanish
+88,417
+87,417
+Arabic
+62,745
+57,532
+Table 3: Total number of words in each dataset
+in details. We found that this particular configura-
+tion worked best across different acoustic features,
+whereas other choices gave mixed results. For ex-
+ample, using GRUs instead of LSTMs worked well
+with pre-trained features but was worse for MFCCs.
+The decoding process described in Holzenberger
+et al. (2018), where positional encodings are used
+instead of previous outputs also resulted in infe-
+rior performance. We also found that using teacher
+forcing instead of the model’s previous output as
+input to the decoder hurt the performance. Finally,
+using two layers was crucial to get results in line
+with DTW perforamnce for most self-supervised
+models. The only exception is the self-supervised
+model with XLSR features which resulted in un-
+stable training with 2 layers. We found it to work
+much better with a single layer network and slightly
+larger embedding size. Generally, larger embed-
+dings sizes improved performance to some extent,
+but the improvements were smaller beyond the
+values that we have chosen; furthermore, using
+smaller sizes is more advantageous in terms of
+computational efficiency. We did not perform any
+hyper-parameter tuning for the target languages
+since we are working within the premise of low-
+resource settings where validation data may not be
+available.
+Table 4 shows the number of parameters for each
+model. Since the decoder is only used for training
+and can be discarded after that, we only show the
+number of encoder parameters.
+A.3
+Training Details
+The supervised models were trained with NLL loss,
+and the training targets are sequences of phonemes
+obtained using the Phonemizer package 6 (Bernard
+and Titeux, 2021). This choice seemed more sen-
+sible at first, but we found that using sequences
+of characters instead of phonemes worked equally
+well.
+The model was implemented using PyTorch and
+6https://github.com/bootphon/phonemizer
+
+Model
+input
+hidden
+no.of parameters
+Self-Supervised
+MFCC
+39
+100
+354,400
+Wav2Vec
+768
+100
+937,600
+MCPC
+256
+100
+528,000
+XLSR-53
+1024
+250
+1,411,200
+Supervised
+MFCC
+39
+100
+354,400
+Wav2Vec
+768
+100
+937,600
+MCPC
+256
+100
+528,000
+XLSR-53
+1024
+100
+1,142,400
+Table 4: Input size, hidden layer size, and total number
+of encoder parameters for each model.
+trained on NVIDIA K80 GPU as provided in AWS
+p2.xlarge instances. For optimization, we found
+that adam optimizer worked for all features except
+MFCCs, for which SGD with cyclical or step learn-
+ing rate schedule was more stable.
+Table 3 shows the number of words in each
+dataset.
+The word alignments were obtained
+via force alignment using The Montreal Forced
+Aligner7 (McAuliffe et al., 2017) for English, Ger-
+man, French, and Spanish. The Montreal aligner
+uses an ASR engine, and since these datasets are
+relatively clean, the alignments are generally accu-
+rate. For Arabic, the best option was the aeneas
+toolkit8, which relies on a TTS engine to align the
+synthesized words with the actual audio segments.
+We used Amazon Polly TTS for higher quality, but
+overall the alignments were not as accurate as the
+other datasets, which we believe is due to the low
+quality of the recordings, presence of noise, and
+high variability in accents. The low clustering accu-
+racy could be partially attributed to the inaccurate
+labeling of the segments as a result of this. For
+ABX evaluation on the Arabic set, we manually
+filtered the segments that had somewhat accurate
+boundaries; the chosen pairs still contained high
+level of noise conditions, such as background mu-
+sic and interfering speech.
+7https://github.com/MontrealCorpusTools/Montreal-
+Forced-Aligner
+8www.readbeyond.it/aeneas
+

AdAzT4oBgHgl3EQfF_tg/content/tmp_files/load_file.txt

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+page_content='Supervised Acoustic Embeddings And Their Transferability Across  Languages  Sreepratha Ram  UAE University  sree_ram@uaeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='ae  Hanan Aldarmaki  MBZUAI  hanan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='aldarmaki@mbzuai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='ae  Abstract  In speech recognition, it is essential to model  the phonetic content of the input signal while  discarding irrelevant factors such as speaker  variations and noise, which is challenging in  low-resource settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Self-supervised pre-  training has been proposed as a way to  improve both supervised and unsupervised  speech recognition, including frame-level fea-  ture representations and Acoustic Word Em-  beddings (AWE) for variable-length segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  However, self-supervised models alone cannot  learn perfect separation of the linguistic con-  tent as they are trained to optimize indirect ob-  jectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In this work, we experiment with dif-  ferent pre-trained self-supervised features as  input to AWE models and show that they work  best within a supervised framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Models  trained on English can be transferred to other  languages with no adaptation and outperform  self-supervised models trained solely on the  target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Keywords— Unsupervised ASR, Transfer Learn-  ing, Acoustic Word Embeddings  1  Introduction  With supervised speech recognition systems get-  ting more robust and accurate due to the avail-  ability of large amounts of labeled data and com-  putational power (Gulati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Baevski  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2020b), more attention is now given to low-  resource languages for which training data are lim-  ited or non-existent (Aldarmaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Un-  supervised pre-training using unlabeled speech can  be leveraged to improve both supervised and un-  supervised models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' for instance, speech represen-  tations pre-trained on large amounts of unlabeled  speech from multiple languages have been shown  to improve ASR performance for low-resource lan-  guages (Kawakami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=',  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  While most supervised ASR models operate at  the level of phones, word-level segmental ASR  where variable-length segments are modeled and  embedded into fixed-dimensional vectors have also  been explored with relative success (Abdel-Hamid  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' He and Fosler-Lussier, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In a  similar vein, Acoustic Word Embeddings (AWEs)  have been proposed as a way to efficiently compare  variable-length speech segments in low-resource  settings (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Kamper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Unlike written words, spoken words naturally con-  tain speaker and phonetic variability that makes  them more difficult to model in a latent space with-  out supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Self-supervised pre-training and  cross-lingual transfer are two possible approaches  to make unsupervised models more robust to non-  linguistic variations in the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  In this work, we investigate the performance of  self-supervised training of AWE models versus su-  pervised training with zero-shot cross-lingual trans-  fer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We experiment with different types of acoustic  features and measure their performance separately  and within the AWE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' While we find that  pre-trained acoustic features improve the perfor-  mance of self-supervised AWE models to some  extent, a larger improvement can be achieved when  the AWE models are trained in a supervised manner  using small amount of labeled data from a different  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' This zero-shot cross-lingual transfer is  observed consistently across different languages,  and particularly with the use of pre-trained feature  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Our results suggest that supervised  training with zero-shot cross-lingual transfer is a  more effective approach for low-resource speech  models compared with purely self-supervised train-  ing1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  2  Background & Related Work  Spoken language is often modeled using short  fixed-length frames of 10 to 30 ms duration, which  1We provide python training and evaluation scripts  for replicating our experiments:  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='com/h-  aldarmaki/acoustic_embeddings  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='01020v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='CL]  3 Jan 2023  results in variable-length word segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Dynamic  Time Warping (DTW) is an early technique that  uses dynamic programming to compare variable-  length segments by finding optimal frame-wise  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' DTW is rather inefficient, which mo-  tivates embedding variable-length segments into  vectors of fixed size that can be compared using  more efficient metrics such as cosine or Euclidean  distance (Levin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Different types of  Acoustic Word Embeddings (AWE) have been pro-  posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' As these techniques are generally meant for  low-resource languages, they are typically trained  in a self-supervised manner, most commonly us-  ing an auto-encoder network with reconstruction  loss (Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Holzenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Compared with direct comparison via DTW, these  AWEs generally result in similar or slightly supe-  rior performance while being far more efficient  (Holzenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' (2020)  describes an alternative training strategy using cor-  respondence auto-encoders, which relies on word  pairs extracted via unsupervised spoken term dis-  covery, and further improvements can be achieved  using contrastive learning and multi-lingual adap-  tation (Jacobs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  The above models use static acoustic features  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' MFCCs) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' van Staden and Kam-  per (2021) shows that using pre-trained features  like CPC (van den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2018) improves the  performance of unsupervised AWE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Pre-  trained features have been repeatedly shown to im-  prove performance in supervised downstream tasks  (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In addition, pre-trained features  have been shown to transfer across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' For  instance, a modified version of CPC (MCPC) is de-  scribed in Riviere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' (2020), which demonstrates  that pre-training these features on Egnlish results  in improved phone classification accuracy for other  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Other types of pre-trained features, such  as wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='0 (Baevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2020a) have been  shown to improve both supervised and unsuper-  vised ASR performance (Baevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2021), and  multi-lingual training of these features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' XLSR-  53) can lead to improvements across many lan-  guages compared to monolingual pre-training (Con-  neau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  3  Objectives & Methodology  The objective of this study is to investigate the effec-  tiveness and trasnsferability of pre-trained acoustic  features when used as input to acoustic word em-  beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' To that end, we compare self-supervised  AWEs trained directly on the target languages ver-  sus zero-shot cross-lingual transfer of supervised  AWEs trained on a different source language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' To  our knowledge, the combination of pre-trained fea-  tures with AWE models has not been fully investi-  gated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' most AWE models are trained with stan-  dard acoustic features like MFCCs, while self-  supervised features are typically evaluated within  supervised models fine-tuned for the target lan-  guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Furthermore, zero-shot cross-lingual trans-  fer of supervised AWEs has not been the focus of  previous works in this area, which mainly focused  on improving self-supervised AWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  For the purpose of this evaluation, we use a rela-  tively simple architecture for the embedding model  and we fix the hyper-parameters based on prelimi-  nary validation results for English self-supervised  AWEs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We do not do any further tuning of the  self-supervised or the supervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We use  English as the source language, and evaluate zero-  shot transfer on four other languages: French, Ger-  man, Spanish, and Arabic, with the latter used as  a challenge set since it contains more variability  and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' No labeled data were used for the target  languages with the exception of word boundaries  which were obtained via force alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We eval-  uate mainly using minimal-pair ABX error rates  to measure phonetic discriminability and speaker  invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We also cluster the embedded words  and measure how often different occurrences of the  same words end up in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  4  Experimental Settings  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='1  Model Architecture  Our AWE model consists of a multi-layer bidirec-  tional LSTM encoder, followed by a uni-directional  LSTM decoder, similar to Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' (2016) and  (Holzenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The encoder takes a se-  quence of T acoustic features representing one spo-  ken word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The forward and backward states of the  last hidden layer of the encoder are concatenated  and used as an embedding of the given word, call  it hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The decoder generates the target sequence  one step at a time, conditioned on hT and the out-  put at the previous time step, similar to Chung and  2We observed that self-supervised models were very sen-  sitive to the choice of architecture and hyper-parameters, so  we fixed these in favor of self-supervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' As shown  in later sections, we still got better results with the supervised  models, which shows that they are more robust and easier to  optimize on top of being more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Glass (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In the self-supervised setting, the  target sequence is the same as the input sequence,  so the model is trained as an auto-encoder with  MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In the supervised setting, the target is a  sequence of phonemes representing the input word,  and the model is trained by minimizing the nega-  tive log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We used 2-layer networks with  100 hidden units for most models, which results  in embeddings of size 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We also used dropout  with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='3 on the input features, similar to  the denoising networks used in Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  More details of the parameters and training process  can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='2  Feature Extraction  For easier reproduciblity, we used the s3prl toolkit3  for extracting all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We used the pre-trained  s3prl upstream models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' among the many pretrained  self supervised speech representations available,  modified CPC, Wav2Wec2 and XLSR-53 were cho-  sen based on superior DTW-based ABX scores4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  All pre-trained models, with the exception of  XLSR, have been exclusively pre-trained on En-  glish data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' XLSR-53 was pre-trained on unlabeled  speech from 53 languages, including all target lan-  guages in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' As observed by other  researchers (Bartelds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2022), the performance  of features extracted from transformer-based mod-  els is largely dependent on the choice of layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' we  used the last hidden layer for modified CPC, the  second to last hidden layer for Wav2Vec2 and the  central hidden layer (layer 12) for XLSR-53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Aver-  aging all layers gave reasonable results, but these  choices led to the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' For MFCC fea-  tures, we also used the s3prl implementation, which  includes 13 static features as well as dynamic delta  and delta-delta coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='3  Data  We used the Librispeech (Panayotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2015)  and Multilingual Librispeech (Pratap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2020)  datasets for English (en), French (fr), German (de),  and Spanish (es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We used the dev sets for train-  ing, and test sets for evaluation (dev-clean and test-  clean for English).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We obtained the word bound-  aries automatically by forced alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' For Ara-  bic (ar), we used the dev and test sets of MGB2  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='com/s3prl/s3prl  4We did experiment with other features like APC, VQ-  APC, and VQ-Wav2Vec, and got similar or inferior perfor-  mance to MCPC and Wav2Vec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We opted to omit these for  brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  (Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' This dataset is expected to be  more challenging as it contains a diversity of di-  alects as well as various noise conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' See the  Appendix for more details on the datasets and the  word alignment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='4  Evaluation Scheme  We constructed Minimal-Pair ABX tasks, as de-  scribed in Schatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' ABX tasks are  typically used to measure phoneme discrimination  in zero-resource settings, and they consist of two  segments, A and B, that differ by a minimal con-  trast (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' one phoneme difference), and a third  segment X that matches either A or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' A distance  measure such as DTW or cosine is used to find  the closest match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We used two variants of this  task: within-speaker ABX, where all three words  are spoken by the same speaker, and cross-speaker  ABX, where X is spoken by a different speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  We automatically extracted the words from each  test set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' we selected A and B by finding word pairs  that have the same length5 and Levenshtein edit  distance of 1 or 2, which roughly corresponds to a  difference of one or two phonemes most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  For Arabic, the dataset did not have speaker ids, so  all three words could be from different speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  In addition, due to the lower quality of the sound  recordings and the presence of noise in this dataset,  the word alignment quality is much lower than the  other languages, so the automatic process resulted  in many invalid segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' To have a more reliable  test set for Arabic, we manually checked the valid-  ity of the extracted words and kept 954 validated  word pairs for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  We also used clustering for complementary eval-  uation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We clustered the embeddings using K-  Means with K being the number of unique words  in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We calculated the accuracy of clus-  tering as the percentage of words that match their  cluster label, which is the word id of the majority of  segments in each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' This allows us to measure  if the embeddings of the same words are similar  enough to be clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  5  Results  Table 1 shows ABX error rates using the input  features directly (with DTW as distance metric),  self-supervised AWEs trained on each language,  5Since automatic word alignments tend to be inaccurate  around the boundaries, we only used words that have at least  five characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  en  fr  de  es  ar  within  across  within  across  within  across  within  across  across  Using DTW  MFCC  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
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+page_content='9  MCPC  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='2  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='6  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='9  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='4  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='5  XLSR-53  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='0  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='2  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='9  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='0  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='0  Supervised AWEs trained on English  MFCC  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='5  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='7  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='7  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='1  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='4  wav2vec  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='3  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='8  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='3  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='1  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='8  MCPC  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='5  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='4  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='7  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='2  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='6  XLSR-53  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='3  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='1  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='1  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='8  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='8  Table 2: K-Means Clustering Accuracy (%)  and supervised AWEs trained on English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Cosine  similarity is the metric used in the latter two set-  tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Confirming previous results (Riviere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=',  2020), we do observe that pre-trained acoustic fea-  tures like Modified CPC and Wav2Vec2, which are  trained exclusively on English unlabeled speech,  transfer well across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' These pre-trained  features consistently outperformed MFCC features  for all languages, particularly in cross-speaker eval-  uation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Unsurprisingly, the English language has  the best ABX scores overall simply because the  pre-trained features used are all trained on English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  The results for self-supervised AWE models are  mixed, but generally they are in the same range  as DTW performance, which also conforms with  previously published results (Holzenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=',  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  With supervised training, we see significant re-  duction in errors rates for all languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The lowest  error rates are achieved on the English test set, as  expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' More notably, the largest reduction in  error rates is achieved with the XLSR features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' It is  also interesting to note that XLSR features were not  impressive in the self-supervised setting compared  with other features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Wav2Vec2 and MCPC, which  were trained on English only, gave better results  in the self-supervised framework for all test lan-  guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The advantage of using these cross-lingual  features was only evident in the supervised and  transfer learning setting, where they consistently  outperformed all other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' For Arabic, the  error rates are higher overall due to the nature of  the dataset, but we still observe the lowest error  rate in the transfer learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Finally, we see in table 2 that the clustering ac-  curacy results are consistent with the ABX results,  where supervised models trained on English con-  sistently gave higher accuracy compared with self-  supervised models trained on the target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  6  Conclusions  Our results demonstrate the superior effectiveness  of zero-shot transfer learning of acoustic word em-  beddings compared with self-supervised training  in the target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' This is particularly use-  ful for low-resource languages for which data may  not be available for supervised or self-supervised  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The mechanism of this transfer is mainly  through the reduction in speaker variability which  is far easier to achieve via supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In  addition, supervised training makes the most out  of pre-trained features, where we see further re-  duction in error rates that far exceed the reduction  observed in self-supervised settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The presence  of noise naturally results in larger error rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' fur-  ther investigations are needed to demonstrate the  transferability of noise robustness in a similar man-  ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Acknowledgement  This work was supported by grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 31T139  at United Arab Emirates University and partially  funded under UAEU-ZU Joint Research Grant  G00003715 (Fund No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' : 12T034) through Emirates  Center for Mobility Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
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+page_content=' wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
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+page_content=' Advances in Neural Information Processing  Systems, 33:12449–12460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
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+page_content=' Journal of Open Source Software,  6(68):3958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Speech2vec: A  sequence-to-sequence framework for learning word  embeddings from speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
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+page_content=' Interspeech 2018,  pages 811–815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Yu-An Chung, Chao-Chung Wu, Chia-Hao Shen,  Hung-Yi Lee, and Lin-Shan Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Audio  word2vec: Unsupervised learning of audio segment  representations using sequence-to-sequence autoen-  coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Interspeech 2016, pages 765–769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Unsupervised cross-lingual representation learn-  ing  for  speech  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  arXiv  preprint  arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Conformer: Convolution-augmented transformer for  speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Interspeech 2020, pages  5036–5040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Yanzhang He and Eric Fosler-Lussier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Segmen-  tal conditional random fields with deep neural net-  works as acoustic models for first-pass word recog-  nition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In Sixteenth Annual Conference of the Inter-  national Speech Communication Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Nils Holzenberger, Mingxing Du, Julien Karadayi,  Rachid Riad, and Emmanuel Dupoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Learn-  ing word embeddings: Unsupervised methods for  fixed-size representations of variable-length speech  segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Interspeech 2018, pages 2683–  2687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Christiaan Jacobs, Yevgen Matusevych, and Herman  Kamper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Acoustic word embeddings for  zero-resource languages using self-supervised con-  trastive learning and multilingual adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  In  2021 IEEE Spoken Language Technology Workshop  (SLT), pages 919–926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Herman Kamper, Yevgen Matusevych, and Sharon  Goldwater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Multilingual acoustic word em-  bedding models for processing zero-resource lan-  guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In ICASSP 2020-2020 IEEE International  Conference on Acoustics, Speech and Signal Pro-  cessing (ICASSP), pages 6414–6418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Kazuya Kawakami, Luyu Wang, Chris Dyer, Phil Blun-  som, and Aaron van den Oord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Learning  robust and multilingual speech representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In  Findings of the Association for Computational Lin-  guistics: EMNLP 2020, pages 1182–1192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Keith Levin, Katharine Henry, Aren Jansen, and Karen  Livescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Fixed-dimensional acoustic embed-  dings of variable-length segments in low-resource  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  In 2013 IEEE Workshop on Automatic  Speech Recognition and Understanding, pages 410–  415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Michael McAuliffe, Michaela Socolof, Sarah Mi-  huc, Michael Wagner, and Morgan Sonderegger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Montreal forced aligner:  Trainable text-  speech alignment using kaldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Interspeech  2017, pages 498–502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Vassil Panayotov, Guoguo Chen, Daniel Povey, and  Sanjeev Khudanpur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Librispeech: an asr cor-  pus based on public domain audio books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In 2015  IEEE international conference on acoustics, speech  and signal processing (ICASSP), pages 5206–5210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Puyuan Peng, Herman Kamper, and Karen Livescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' A correspondence variational autoencoder for  unsupervised acoustic word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Advances  in neural information processing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Vineel Pratap, Qiantong Xu, Anuroop Sriram, Gabriel  Synnaeve, and Ronan Collobert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  MLS: A  large-scale multilingual dataset for speech research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Morgane Riviere, Armand Joulin, Pierre-Emmanuel  Mazaré, and Emmanuel Dupoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Unsuper-  vised pretraining transfers well across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  In ICASSP 2020-2020 IEEE International Confer-  ence on Acoustics, Speech and Signal Processing  (ICASSP), pages 7414–7418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Thomas Schatz, Vijayaditya Peddinti, Francis Bach,  Aren Jansen, Hynek Hermansky, and Emmanuel  Dupoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Evaluating speech features with  the minimal-pair abx task: Analysis of the classi-  cal mfc/plp pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In INTERSPEECH 2013: 14th  Annual Conference of the International Speech Com-  munication Association, pages 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Aaron van den Oord, Yazhe Li, and Oriol Vinyals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Representation learning with contrastive pre-  dictive coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' arXiv e-prints, pages arXiv–1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Lisa van Staden and Herman Kamper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' A compar-  ison of self-supervised speech representations as in-  put features for unsupervised acoustic word embed-  dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' In 2021 IEEE Spoken Language Technology  Workshop (SLT), pages 927–934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Shu-wen Yang, Po-Han Chi, Yung-Sung Chuang,  Cheng-I Jeff Lai, Kushal Lakhotia, Yist Y Lin,  Andy T Liu, Jiatong Shi, Xuankai Chang, Guan-  Ting Lin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Superb: Speech processing  universal performance benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  arXiv preprint  arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='01051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  A  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='1  Dataset Details  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='2  Model Architecture &  Hyper-Parameters  The architecture described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='1 was mod-  eled after other acoustic word embedding mod-  els (Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Chung and Glass, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Holzenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2018) with slight variations  Dataset  test  dev  English  52,576  54,402  French  90,958  83,560  German  121,713  122,903  Spanish  88,417  87,417  Arabic  62,745  57,532  Table 3: Total number of words in each dataset  in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We found that this particular configura-  tion worked best across different acoustic features,  whereas other choices gave mixed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' For ex-  ample, using GRUs instead of LSTMs worked well  with pre-trained features but was worse for MFCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  The decoding process described in Holzenberger  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' (2018), where positional encodings are used  instead of previous outputs also resulted in infe-  rior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We also found that using teacher  forcing instead of the model’s previous output as  input to the decoder hurt the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Finally,  using two layers was crucial to get results in line  with DTW perforamnce for most self-supervised  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The only exception is the self-supervised  model with XLSR features which resulted in un-  stable training with 2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We found it to work  much better with a single layer network and slightly  larger embedding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Generally, larger embed-  dings sizes improved performance to some extent,  but the improvements were smaller beyond the  values that we have chosen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' furthermore, using  smaller sizes is more advantageous in terms of  computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' We did not perform any  hyper-parameter tuning for the target languages  since we are working within the premise of low-  resource settings where validation data may not be  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Table 4 shows the number of parameters for each  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' Since the decoder is only used for training  and can be discarded after that, we only show the  number of encoder parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='3  Training Details  The supervised models were trained with NLL loss,  and the training targets are sequences of phonemes  obtained using the Phonemizer package 6 (Bernard  and Titeux, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' This choice seemed more sen-  sible at first, but we found that using sequences  of characters instead of phonemes worked equally  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  The model was implemented using PyTorch and  6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='com/bootphon/phonemizer  Model  input  hidden  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='of parameters  Self-Supervised  MFCC  39  100  354,400  Wav2Vec  768  100  937,600  MCPC  256  100  528,000  XLSR-53  1024  250  1,411,200  Supervised  MFCC  39  100  354,400  Wav2Vec  768  100  937,600  MCPC  256  100  528,000  XLSR-53  1024  100  1,142,400  Table 4: Input size, hidden layer size, and total number  of encoder parameters for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  trained on NVIDIA K80 GPU as provided in AWS  p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='xlarge instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' For optimization, we found  that adam optimizer worked for all features except  MFCCs, for which SGD with cyclical or step learn-  ing rate schedule was more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  Table 3 shows the number of words in each  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  The word alignments were obtained  via force alignment using The Montreal Forced  Aligner7 (McAuliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=', 2017) for English, Ger-  man, French, and Spanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The Montreal aligner  uses an ASR engine, and since these datasets are  relatively clean, the alignments are generally accu-  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' For Arabic, the best option was the aeneas  toolkit8, which relies on a TTS engine to align the  synthesized words with the actual audio segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  We used Amazon Polly TTS for higher quality, but  overall the alignments were not as accurate as the  other datasets, which we believe is due to the low  quality of the recordings, presence of noise, and  high variability in accents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' The low clustering accu-  racy could be partially attributed to the inaccurate  labeling of the segments as a result of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' For  ABX evaluation on the Arabic set, we manually  filtered the segments that had somewhat accurate  boundaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content=' the chosen pairs still contained high  level of noise conditions, such as background mu-  sic and interfering speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='  7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='com/MontrealCorpusTools/Montreal-  Forced-Aligner  8www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='readbeyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}
+page_content='it/aeneas' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf'}

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+arXiv:2301.02887v1  [math.DG]  7 Jan 2023
+ORBIFOLDS AND MANIFOLD QUOTIENTS
+WITH UPPER CURVATURE BOUNDS
+CHRISTIAN LANGE
+Abstract. We characterize Riemannian orbifolds with an upper curvature bound in the
+Alexandrov sense as reflectofolds, i.e. Riemannian orbifolds all of whose local groups are
+generated by reflections, with the same upper bound on the sectional curvature. Combined
+with a result by Lytchak–Thorbergsson this implies that a quotient of a Riemannian manifold
+by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov
+sense if and only if it is a reflectofold.
+1. Introduction
+Let M be a Riemannian manifold and let G be a closed group of isometries of M. Then the
+quotient space M/G is a metric space whose metric properties are often related to properties
+of the action in an interesting way, see e.g. [GLLM22, LT10]. Usually, this quotient is not a
+Riemannian manifold, but an Alexandrov space with curvature locally bounded from below.
+Nevertheless, it is stratified by Riemannian manifolds and the so-called principle stratum is
+open and dense [AB15]. Lytchak and Thorbergsson have shown that the sectional curvature
+of this principle stratum is bounded from above in the neighborhood of a point if and only
+if this neighborhood in M/G is a Riemannian orbifold, i.e. a metric space which is locally
+isometric to the quotient of a Riemannian manifold by an isometric action of a finite group
+[LT10, Theorem 1.1]. However, the curvature of such a quotient is in general still locally
+unbounded from above in the Alexandrov sense. For instance, the quotient of R2 by a finite
+cyclic group of rotations around the origin is isometric to the Euclidean cone over a circle of
+radius 2π/k for some k > 1 and exhibits infinite positive curvature at the tip of the cone.
+Infinitesimally, the only exceptions of this phenomenon one can think of are quotients of Rn
+by finite reflection groups [Hu90]. In this case the quotient is isometric to a Weyl chamber
+of the corresponding reflection group and thus flat. Globally, these examples correspond to
+so-called reflectofolds, i.e. Riemannian orbifolds all of whose local groups are reflection groups
+[Da11]. In particular, reflectofolds are Riemannian manifolds in their interior. In fact, metric
+spaces with two-sided curvature bounds are always Riemannian manifolds (perhaps of low
+regularity) in their interior [BN93], cf. [BBI01, Theorem 10.10.13]. Here we confirm that
+for the whole quotient space no other examples than reflectofolds locally have a two-sided
+curvature bound.
+Theorem 1.1. The curvature of a Riemannian orbifold is locally bounded from above in the
+Alexandrov sense if and only if it is a reflectofold.
+In this case, locally, the curvature is
+bounded from above by k in the Alexandrov sense if and only if the sectional curvature is
+bounded from above by k.
+1
+
+2
+C. LANGE
+In the manifold case this statement is due to Alexandrov [Al51] based on earlier work by
+Cartan [Ca28]. Our proof of Theorem 1.1 works by induction on the dimension and uses the
+fact that tangent spaces of metric spaces with curvature bounded from above are CAT(0),
+see Theorem 2.2. The main step is to prove Lemma 2.3 which characterizes CAT(0) spaces
+among quotient spaces Rn/Γ for finite subgroups Γ < O(n).
+Namely, such a quotient is
+CAT(0) if and only if Γ is generated by reflections, in which case the quotient is isometric
+to a Weyl chamber of Γ if Γ is nontrivial. The strategy of the proof is related to the one in
+[La16, Section 4] and [La19, Lemma 4.17].
+Based on the result by Lytchak–Thorbergsson [LT10, Theorem 1.1] we state the following
+corollary.
+Corollary 1.2. Let M be a Riemannian manifold and let G be a closed group of isometeries
+of M. Let p ∈ M be a point with isotropy Gp. Set ¯p = Gp ∈ M/G. Then the following are
+equivalent:
+(i) The curvature of M/G is bounded from above in a neighborhood of ¯p in the Alexandrov
+sense.
+(ii) A neighborhood of ¯p in M/G is a reflectofold.
+(iii) The action of Gp on TpM is polar and and the corresponding polar group Π is gen-
+erated by reflections.
+Here the action of Gp on TpM is called polar if there exists a subspace Σ ⊆ TpM that
+meets all orbits of the Gp-action and meets them always orthogonally. The polar group Π is
+defined to be the quotient of the subgroup that leaves Σ invariant modulo the subgroup that
+fixes Σ pointwise. It acts naturally on Σ. In particular, condition (iii) is satisfied, if Gp is
+connected [GZ12, Proposition 1.4] and if Gp is Coxeter polar like the isotropy representations
+of a symmetric space. For more details on polar action we refer to the survey [GZ12].
+2. Quotients with upper curvature bounds
+2.1. Spaces with upper curvature bounds. We say that a metric space X is D-geodesic
+for some D > 0 if all points of distance less than Dk are connected by a geodesic. Let Dk be
+the diameter of the complete model plane of constant curvature k. We say that X is CAT(k)
+if it is Dk-geodesic and all geodesic triangles in X of perimeter less than 2Dk satisfy the
+CAT(k) comparison condition, see [BH99, II.1]. We say that X has curvature bounded from
+above by k in the Alexandrov sense if it is locally CAT(k). This definition is motivated by
+the following result of Alexandrov. A proof can also be found in [BH99, Theorem 1A.6].
+Theorem 2.1 (Alexandrov, [Al51]). A Riemannian manifold has curvature bounded from
+above by k in the Alexandrov sense if and only if its sectional curvature is bounded from above
+by k.
+For a metric space X with curvature bounded from above in the Alexandrov sense we
+denote the (completion of the) space of directions at a point p ∈ X by ΣpX and the tangent
+cone of X at p, which is isometric to the Euclidean cone over the space of directions ΣpX, by
+TpX, cf. [BH99, II 3]. A proof of the following statement, first outlined by Kleiner and Leeb
+[KL97], can be found in [BH99, II 3.19].
+Theorem 2.2 (Nikolaev, [N95]). If a metric space has curvature bounded from above in the
+Alexandrov sense, then all spaces of directions are CAT(1) and all tangent spaces are CAT(0).
+
+ORBIFOLDS AND MANIFOLD QUOTIENTS WITH UPPER CURVATURE BOUNDS
+3
+2.2. Riemannian orbifolds. A Riemannian n-orbifold O can be defined as a length space
+which is locally isometric to the quotient of a Riemannian n-manifold by an isometric action
+of a finite group [La20]. The isotropy group of the preimage of a point p ∈ O in such a
+manifold chart under the finite group action is uniquely defined up to conjugation in O(n),
+and it is called the local group of O at p. The set of points with trivial local group is called
+the regular part of O. From the local model it is easy to see that the regular part is open
+and dense. Hence, if the curvature of a Riemannian orbifold is bounded from above by k in
+the Alexandrov sense, then the sectional curvature of its regular part must satisify the same
+curvature bound by Theorem 2.1. Moreover, since the regular part is dense, the sectional
+curvature bound must be satisfied everywhere.
+2.3. Proof of Theorem 1.1 and Corollary 1.2. The space of directions of a Riemannian
+orbifold O at a point p ∈ O with local group Γp is isometric to Sn−1/Γp, where Sn−1 denotes
+the unit sphere in Rn, and the tangent cone of O at p is isometric to Rn/Γp. Therefore, the
+statement in Theorem 1.1 that local groups are generated by reflections if the curvature is
+bounded from above in the Alexandrov sense is a consequence of the following lemma.
+Lemma 2.3. Let Γ < O(n) be a finite subgroup such that Rn/Γ is CAT(0).
+Then Γ is
+generated by reflections.
+Proof. We prove the claim by induction on n. For n = 1 the claim is obvious.
+Assume that the claim holds for some n ∈ N and let Γ < O(n + 1) be a finite subgroup
+such that Rn+1/Γ is CAT(0). Then Σ0(Rn+1/Γ) = Sn/Γ is CAT(1) by Theorem 2.2. Hence,
+for any v ∈ Sn we have that Tv(Sn/Γ) = TvSn/Γv = Rn/Γv is CAT(0) and so for any v ∈ Sn
+the isotropy group Γv is generated by reflections by our induction assumption.
+We set Γ′ = ⟨Γv | v ∈ Sn⟩. By construction Γ′ is a normal subgroup of Γ generated by
+reflections. The induced action of Γ/Γ′ on Sn/Γ′ is isometric and free, see [La19, Lemma 4.16].
+We first suppose that Γ′ is nontrivial. In this case Rn+1/Γ′ is isometric to a Weyl chamber
+[Hu90] and so the quotient Sn/Γ′ is contractible. This implies Γ = Γ′ since a nontrivial finite
+group cannot act freely on a finite dimensional complex (although it can act without fixed
+points [FR59]). Alternatively, we can also argue geometrically as follows. The group Γ leaves
+the subspace
+V = Fix(Γ′) = {v ∈ Rn | gv = v for all g ∈ Γ′} ∼= Rk
+and its orthogonal complement invariant. Since Γ′ is nontrivial we have that k > 0. Then ∆ =
+Sn−k/Γ′ is a (strictly convex) spherical simplex [Hu90]. The join splitting Sn = Sk−1 ∗ Sn−k
+induces a splitting Sn/Γ′ = Sk−1 ∗ ∆ of its Γ′-quotient whose subspaces Sk−1 and ∆ are
+invariant under the Γ/Γ′ action. Since the barycenter of ∆ is fixed by Γ/Γ′, we again conclude
+that Γ = Γ′.
+Hence, we can assume that Γ′ is trivial. In this case Γ acts freely on Sn. Therefore, for any
+nontrivial g ∈ Γ there exists a periodic geodesic on Sn which together with its orientiation
+is preserved by g. Suppose that there is such a nontrivial g. Then a corresponding invariant
+geodesic projects onto a periodic geodesic of Sn/Γ of length < 2π, again because the action of
+Γ on Sn is free. This contradicts the fact that Sn/Γ is CAT(1), see [AKP19, Proposition 2.2.7],
+and so Γ has to be trivial as well in this case. The claim follows.
+□
+This completes the proof of the if direction of Theorem 1.1.
+The proof of the only if
+direction is based on the fact that the quotient Rn/Γ of Rn by a reflection group Γ < O(n)
+
+4
+C. LANGE
+is isometric to a Weyl chamber in Rn [Hu90]. If a finite group Γp acts isometrically on a
+Riemannian manifold M fixing a point p ∈ M such that the induced action of Γp on TpM is
+generated by reflections, then a small r-neighborhood of the image of p in M/Γp is isometric to
+the image under the exponential map expp : TpM → M of the intersection of a corresponding
+Weyl chamber with a small r-neighborhood of 0 ∈ TpM. The claim now follows from the
+observation that for sufficiently small r > 0 this image is a convex subset of M, which is an
+easy exercise in Riemannian geometry.
+Proof of Corollary 1.2. To prove Corollary 1.2 we first observe that an upper curvature bound
+in the Alexandrov sense for a neighborhood U in M/G implies an upper bound on the sectional
+curvature of the intersection of U with the principle stratum of M/G by Theorem 2.1. In
+this case U is a Riemannian orbifold by [LT10, Theorem 1.1] and hence a reflectofold by
+Theorem 1.1. The reverse implication also follows from Theorem 1.1.
+The other equivalence follows from [LT10, Theorem 1.1] together with the observation that
+the quotient Σ/Π of a section Σ of the polar action of Gp on TpM modulo the polar group Π
+is isometric to the tangent cone of the orbifold M/G at p
+□
+Acknowledgements.
+The author thanks Claudio Gorodoski for discussions about polar
+actions and the members of the LMU geometry seminar for useful questions and comments.
+References
+[AKP19]
+S. Alexander, V. Kapovitch and A. Petrunin, An invitation to Alexandrov geometry. CAT(0)
+spaces. SpringerBriefs in Mathematics. Springer, Cham, 2019. xii+88 pp.
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+tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag,
+Berlin, 1999. xxii+643 pp.
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+unit sphere. J. Eur. Math. Soc. (2022), DOI 10.4171/JEMS/1272.
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+ematics, vol. 29, Cambridge University Press, Cambridge, 1990.
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+
+ORBIFOLDS AND MANIFOLD QUOTIENTS WITH UPPER CURVATURE BOUNDS
+5
+[La16]
+C. Lange, Characterization of finite groups generated by reflections and rotations. J. Topol. 9
+(2016), no. 4, 1109–1129.
+[La19]
+C. Lange, When is the underlying space of an orbifold a manifold? Trans. Amer. Math. Soc. 372
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+C. Lange, Orbifolds from a metric viewpoint. Geom. Dedicata, 209 (2020), 43–57.
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+Ludwig-Maximilians-Universität München, Mathematisches Institut
+Theresienst. 39, 80333 München, Germany
+Email address: [email protected], [email protected]
+

DtE1T4oBgHgl3EQfEQNL/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='02887v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='DG]  7 Jan 2023  ORBIFOLDS AND MANIFOLD QUOTIENTS  WITH UPPER CURVATURE BOUNDS  CHRISTIAN LANGE  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' We characterize Riemannian orbifolds with an upper curvature bound in the  Alexandrov sense as reflectofolds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Riemannian orbifolds all of whose local groups are  generated by reflections, with the same upper bound on the sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Combined  with a result by Lytchak–Thorbergsson this implies that a quotient of a Riemannian manifold  by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov  sense if and only if it is a reflectofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Introduction  Let M be a Riemannian manifold and let G be a closed group of isometries of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Then the  quotient space M/G is a metric space whose metric properties are often related to properties  of the action in an interesting way, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' [GLLM22, LT10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Usually, this quotient is not a  Riemannian manifold, but an Alexandrov space with curvature locally bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Nevertheless, it is stratified by Riemannian manifolds and the so-called principle stratum is  open and dense [AB15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Lytchak and Thorbergsson have shown that the sectional curvature  of this principle stratum is bounded from above in the neighborhood of a point if and only  if this neighborhood in M/G is a Riemannian orbifold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' a metric space which is locally  isometric to the quotient of a Riemannian manifold by an isometric action of a finite group  [LT10, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' However, the curvature of such a quotient is in general still locally  unbounded from above in the Alexandrov sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' For instance, the quotient of R2 by a finite  cyclic group of rotations around the origin is isometric to the Euclidean cone over a circle of  radius 2π/k for some k > 1 and exhibits infinite positive curvature at the tip of the cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Infinitesimally, the only exceptions of this phenomenon one can think of are quotients of Rn  by finite reflection groups [Hu90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' In this case the quotient is isometric to a Weyl chamber  of the corresponding reflection group and thus flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Globally, these examples correspond to  so-called reflectofolds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Riemannian orbifolds all of whose local groups are reflection groups  [Da11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' In particular, reflectofolds are Riemannian manifolds in their interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' In fact, metric  spaces with two-sided curvature bounds are always Riemannian manifolds (perhaps of low  regularity) in their interior [BN93], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' [BBI01, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Here we confirm that  for the whole quotient space no other examples than reflectofolds locally have a two-sided  curvature bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The curvature of a Riemannian orbifold is locally bounded from above in the  Alexandrov sense if and only if it is a reflectofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  In this case, locally, the curvature is  bounded from above by k in the Alexandrov sense if and only if the sectional curvature is  bounded from above by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  1  2  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' LANGE  In the manifold case this statement is due to Alexandrov [Al51] based on earlier work by  Cartan [Ca28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1 works by induction on the dimension and uses the  fact that tangent spaces of metric spaces with curvature bounded from above are CAT(0),  see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The main step is to prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='3 which characterizes CAT(0) spaces  among quotient spaces Rn/Γ for finite subgroups Γ < O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Namely, such a quotient is  CAT(0) if and only if Γ is generated by reflections, in which case the quotient is isometric  to a Weyl chamber of Γ if Γ is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The strategy of the proof is related to the one in  [La16, Section 4] and [La19, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Based on the result by Lytchak–Thorbergsson [LT10, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1] we state the following  corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Let M be a Riemannian manifold and let G be a closed group of isometeries  of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Let p ∈ M be a point with isotropy Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Set ¯p = Gp ∈ M/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Then the following are  equivalent:  (i) The curvature of M/G is bounded from above in a neighborhood of ¯p in the Alexandrov  sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  (ii) A neighborhood of ¯p in M/G is a reflectofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  (iii) The action of Gp on TpM is polar and and the corresponding polar group Π is gen-  erated by reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Here the action of Gp on TpM is called polar if there exists a subspace Σ ⊆ TpM that  meets all orbits of the Gp-action and meets them always orthogonally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The polar group Π is  defined to be the quotient of the subgroup that leaves Σ invariant modulo the subgroup that  fixes Σ pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' It acts naturally on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' In particular, condition (iii) is satisfied, if Gp is  connected [GZ12, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='4] and if Gp is Coxeter polar like the isotropy representations  of a symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' For more details on polar action we refer to the survey [GZ12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Quotients with upper curvature bounds  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Spaces with upper curvature bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' We say that a metric space X is D-geodesic  for some D > 0 if all points of distance less than Dk are connected by a geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Let Dk be  the diameter of the complete model plane of constant curvature k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' We say that X is CAT(k)  if it is Dk-geodesic and all geodesic triangles in X of perimeter less than 2Dk satisfy the  CAT(k) comparison condition, see [BH99, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' We say that X has curvature bounded from  above by k in the Alexandrov sense if it is locally CAT(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' This definition is motivated by  the following result of Alexandrov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' A proof can also be found in [BH99, Theorem 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1 (Alexandrov, [Al51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' A Riemannian manifold has curvature bounded from  above by k in the Alexandrov sense if and only if its sectional curvature is bounded from above  by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  For a metric space X with curvature bounded from above in the Alexandrov sense we  denote the (completion of the) space of directions at a point p ∈ X by ΣpX and the tangent  cone of X at p, which is isometric to the Euclidean cone over the space of directions ΣpX, by  TpX, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' [BH99, II 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' A proof of the following statement, first outlined by Kleiner and Leeb  [KL97], can be found in [BH99, II 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='2 (Nikolaev, [N95]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' If a metric space has curvature bounded from above in the  Alexandrov sense, then all spaces of directions are CAT(1) and all tangent spaces are CAT(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  ORBIFOLDS AND MANIFOLD QUOTIENTS WITH UPPER CURVATURE BOUNDS  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Riemannian orbifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' A Riemannian n-orbifold O can be defined as a length space  which is locally isometric to the quotient of a Riemannian n-manifold by an isometric action  of a finite group [La20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The isotropy group of the preimage of a point p ∈ O in such a  manifold chart under the finite group action is uniquely defined up to conjugation in O(n),  and it is called the local group of O at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The set of points with trivial local group is called  the regular part of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' From the local model it is easy to see that the regular part is open  and dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Hence, if the curvature of a Riemannian orbifold is bounded from above by k in  the Alexandrov sense, then the sectional curvature of its regular part must satisify the same  curvature bound by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Moreover, since the regular part is dense, the sectional  curvature bound must be satisfied everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The space of directions of a Riemannian  orbifold O at a point p ∈ O with local group Γp is isometric to Sn−1/Γp, where Sn−1 denotes  the unit sphere in Rn, and the tangent cone of O at p is isometric to Rn/Γp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Therefore, the  statement in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1 that local groups are generated by reflections if the curvature is  bounded from above in the Alexandrov sense is a consequence of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Let Γ < O(n) be a finite subgroup such that Rn/Γ is CAT(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Then Γ is  generated by reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' We prove the claim by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' For n = 1 the claim is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Assume that the claim holds for some n ∈ N and let Γ < O(n + 1) be a finite subgroup  such that Rn+1/Γ is CAT(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Then Σ0(Rn+1/Γ) = Sn/Γ is CAT(1) by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Hence,  for any v ∈ Sn we have that Tv(Sn/Γ) = TvSn/Γv = Rn/Γv is CAT(0) and so for any v ∈ Sn  the isotropy group Γv is generated by reflections by our induction assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  We set Γ′ = ⟨Γv | v ∈ Sn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' By construction Γ′ is a normal subgroup of Γ generated by  reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The induced action of Γ/Γ′ on Sn/Γ′ is isometric and free, see [La19, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  We first suppose that Γ′ is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' In this case Rn+1/Γ′ is isometric to a Weyl chamber  [Hu90] and so the quotient Sn/Γ′ is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' This implies Γ = Γ′ since a nontrivial finite  group cannot act freely on a finite dimensional complex (although it can act without fixed  points [FR59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Alternatively, we can also argue geometrically as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The group Γ leaves  the subspace  V = Fix(Γ′) = {v ∈ Rn | gv = v for all g ∈ Γ′} ∼= Rk  and its orthogonal complement invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Since Γ′ is nontrivial we have that k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Then ∆ =  Sn−k/Γ′ is a (strictly convex) spherical simplex [Hu90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The join splitting Sn = Sk−1 ∗ Sn−k  induces a splitting Sn/Γ′ = Sk−1 ∗ ∆ of its Γ′-quotient whose subspaces Sk−1 and ∆ are  invariant under the Γ/Γ′ action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Since the barycenter of ∆ is fixed by Γ/Γ′, we again conclude  that Γ = Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Hence, we can assume that Γ′ is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' In this case Γ acts freely on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Therefore, for any  nontrivial g ∈ Γ there exists a periodic geodesic on Sn which together with its orientiation  is preserved by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Suppose that there is such a nontrivial g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Then a corresponding invariant  geodesic projects onto a periodic geodesic of Sn/Γ of length < 2π, again because the action of  Γ on Sn is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' This contradicts the fact that Sn/Γ is CAT(1), see [AKP19, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='7],  and so Γ has to be trivial as well in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  □  This completes the proof of the if direction of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  The proof of the only if  direction is based on the fact that the quotient Rn/Γ of Rn by a reflection group Γ < O(n)  4  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' LANGE  is isometric to a Weyl chamber in Rn [Hu90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' If a finite group Γp acts isometrically on a  Riemannian manifold M fixing a point p ∈ M such that the induced action of Γp on TpM is  generated by reflections, then a small r-neighborhood of the image of p in M/Γp is isometric to  the image under the exponential map expp : TpM → M of the intersection of a corresponding  Weyl chamber with a small r-neighborhood of 0 ∈ TpM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The claim now follows from the  observation that for sufficiently small r > 0 this image is a convex subset of M, which is an  easy exercise in Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' To prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='2 we first observe that an upper curvature bound  in the Alexandrov sense for a neighborhood U in M/G implies an upper bound on the sectional  curvature of the intersection of U with the principle stratum of M/G by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' In  this case U is a Riemannian orbifold by [LT10, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1] and hence a reflectofold by  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' The reverse implication also follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  The other equivalence follows from [LT10, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='1] together with the observation that  the quotient Σ/Π of a section Σ of the polar action of Gp on TpM modulo the polar group Π  is isometric to the tangent cone of the orbifold M/G at p  □  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='  The author thanks Claudio Gorodoski for discussions about polar  actions and the members of the LMU geometry seminar for useful questions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
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+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
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+page_content=' 39, 80333 München, Germany  Email address: lange@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='de, clange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='math@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQfEQNL/content/2301.02887v1.pdf'}

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+arXiv:2301.01736v1  [math.DS]  4 Jan 2023
+Non-singular actions of infinite-dimensional groups and
+polymorphisms
+Yury A.Neretin1,2
+Let Z be a probabilistic measure space with measure ζ, R× be the group of positive
+reals, let t be the coordinate on R×. A polymorphism of Z is a measure π on Z × Z × R×
+such that for any measurable A, B ⊂ Z we have π(A × Z × R×) = ζ(A) and
+�
+t dπ(z, u, t)
+over Z ×B ×R× is ζ(B). The set of all polymorphisms has a natural semigroup structure,
+the group of all non-singular transformations is dense in this semigroup. We discuss a
+problem of closure in polymorphisms for certain types of infinite dimensional (’large’)
+groups and show that a non-singular action of a group generates a representation of its
+train (category of double cosets) by polymorphisms.
+Let Z = (Z, ζ) be a Lebesgue probabilistic measure space with a non-atomic
+measure. It is well-known that the group of all measure preserving transforma-
+tions of Z has a natural compactification, which is a metrizable semigroup with
+separately continuous multiplication (this arises to E. Hopf, [6], 1954). Its points
+(’polymorphisms’, the term from [36]) are Borel measures π on Z × Z, whose push-
+forwards under projections of Z × Z to the first and second factors coincide with
+ζ, for details, see, e.g. [10], [36], [37], [13], Sect. VIII.4. Such objects play differ-
+ent roles. They are a kind of multivalued transformations of measure spaces and
+have the same rights as usual transformations, see, e.g., [10], [36]. They can be
+intertwiners between dynamical systems (joinings), see, e.g., [31], [9], [4], [32].
+Closures of groups in polymorphisms is an interesting and non-trivial topic,
+starting from the group Z, see, e.g, [9], [5], [34]. The subject of the present note
+are closures of actions of infinite-dimensional groups. The first result of this kind
+was obtained by Nelson [11], who showed that the closure of the orthogonal group
+O(∞) acting on the space with Gaussian measure is the semigroup of all contractive
+operators.
+Similar constructions related to non-singular transformations are well known in
+ergodic theory, see [10], [3]. The matter of author’s interest are unitary representa-
+tions related to non-singular actions, a definition of polymorphisms relevant to this
+situation was proposed in [12].
+1. Polymorphisms. We consider only Lebesgue measure spaces, see [30], i.e.,
+spaces, which are equivalent to a union of some interval in R and an at most count-
+able collection of atoms having non-zero measure. For a space Z = (Z, ζ) with
+a continuous (non-atomic) measure ζ we denote by Ams(Z) (an abbreviation of
+’automorphisms of a measure space’) the group of all measure preserving transfor-
+mations, by Gms(Z) the group of all non-singular transformations; in both cases
+transformations are defined up sets of zero measure. Recall that a transformation
+g : Z → Z is non-singular if g and g−1 send sets of zero measure to sets of zero
+measure.
+Denote by R× the multiplicative group of positive reals, denote by t the coordi-
+nate on it. Let (X, ξ), (Y, υ) be probabilistic Lebesgue spaces.
+1Supported by the grant FWF P31591.
+2This note is an extended Section 2 of preprint https://arxiv.org/abs/2202.12978.
+1
+
+2
+Definition 1. ([12]) A polymorphism p : X ։ Y is a measure on X ×Y ×R× such
+that
+1. The pushforward of p under the projection X × Y × R× → X is the measure
+ξ.
+2. The pushforward of the measure t · p under the projection to Y coincides with
+measure υ.
+Denote by Pol(X, Y ) the space of all polymorphisms X ։ Y .
+Example. Let q ∈ Gms(X, ξ). Consider the map X → X × X × R× defined by
+x �→
+�
+x, q(x), q′(x)
+�
+,
+where q′ is the Radon–Nikodym derivative of q.
+Then the pushforward of the
+measure ξ under this map is a polymorphism X ։ X.
+⊠
+Topology.
+Let pj, p ∈ Pol(X, Y ).
+Consider measurable subsets A ⊂ X,
+B ⊂ Y .
+Restrict p to A × B × R× and take its pushforward under the map
+A × B × R× → R×. Denote the resulting measure on R× by p[A × B]. We say that
+a sequence pj ∈ Pol(M, N) converges to p, if for any measurable subsets A ⊂ M,
+B ⊂ N we have weak convergences of measures
+pj[A × B] → p[A × B],
+t · pj[A × B] → t · p[A × B].
+It is easy to show (see [18], Theorem 5.3) that the group Gms(M) is dense in
+Pol(M).
+Remark. Spaces Pol(X, Y ) are not compact. I do not know, is it reasonable to
+compactify them.
+⊠
+Continuous polymorphisms. Denote by M▽(R×) the set of all positive finite
+Borel measures σ on R× such that t · σ is finite. This set is a semigroup, a product
+is a convolution ∗ of measures on the group R×. Clearly, a measurable function
+(x, y) → sx,y from X × Y to M▽ determines a certain measure s on X × Y × R×.
+Namely, for measurable sets A ⊂ X, B ⊂ Y , C ⊂ R× we assume
+s(A × B × R×) =
+�
+A×B
+sx,y(C) dξ(x) dυ(y).
+We also must assume that
+�
+X
+sx,y(R×) dυ(y) = 1,
+�
+Y
+�
+R× t dsx,y(t) dξ(x) = 1,
+under this condition we get an element of Pol(X, Y ).
+Let continuous polymorphisms u ∈ Pol(X, Y ), v ∈ Pol(Y, Z) be determined by
+functions (x, y) �→ ux,y, (y, z) �→ vy,z. Then their product w = v ⋄ u ∈ Pol(X, Z) is
+determined by the function
+wx,z =
+�
+Y
+ux,y ∗ vy,z dυ(y).
+This is a formula for a product of integral operators, where the usual multiplication
+is replaced by a convolution.
+Product of polymorphisms. It is easy to see, that continuous polymorphisms
+are dense in Pol(X, Y ). It can be shown that the multiplication ⋄ extends to a
+separately continuous map
+Pol(X, Y ) × Pol(Y, Z) → Pol(X, Z)
+
+3
+and this operation is associative (see [18], Theorem 5.5, Theorem 5.9), i.e., for any
+Lebesgue spaces X, Y , Z, U and for any
+p ∈ Pol(X, Y ),
+q ∈ Pol(Y, Z),
+r ∈ Pol(Z, U),
+we have (r ⋄ q) ⋄ p = r ⋄ (q ⋄ p).
+So we get a category Pol whose objects are Lebesgue probabilistic measure spaces
+and morphisms are polymorphisms.
+Remarks. a) For other ways to define the multiplication of polymorphisms, see
+[18], however it seems that the definition by continuity is most convenient.
+b) Informally, polymorphisms are ’maps’ X → Y , which spread points of X
+along Y and the Radon–Nikodym derivative also is spread.
+⊠
+Involution. Finally, we define an involution on the category Pol. For p ∈
+Pol(X, Y ) we consider its pushforward p′ under the map (x, y, t) �→ (y, x, t−1) and
+define p⋆ ∈ Pol(X, Y ) as the measure t · p′. Then
+(q ⋄ p)⋆ = p⋆ ⋄ q⋆.
+Description of closures of some infinite-dimensional groups in polymorphisms
+were obtained in [14], [17], [20]. The purpose of this paper is to formulate a general
+statement on this topic.
+2. Multiplicativity. M-subgroups and M-families. Let G be a separable
+topological group (not necessary metrizable). Let ρ be a unitary representation of
+G in a Hilbert space H (we consider only separable Hilbert spaces). For any closed
+subgroup K ⊂ G we denote by HK the subspace of K-fixed vectors, by P K the
+orthogonal projection to HK. For two subgroups K, L ⊂ G we define operators
+�ρK,L(g) : HL → HK
+by
+�ρK,L(g) := P Kρ(g)
+���
+HL.
+It is easy to see that
+p ∈ K, q ∈ L
+=⇒
+�ρK,L(pgq) = �ρK,L(g).
+So �ρK,L(·) actually depends on a double coset g := K · g · L. Denote the space of
+all double cosets by K\G/L. Next, we define a weaker equivalence on G,
+g ∼ g′ ⇔ �ρK,L(g) = �ρK,L(g′), for all unitary representations ρ.
+Denote by [K\G/L] the set of all equivalence classes.
+Definition 2. A subgroup K ⊂ G is an M-subgroup in G (or a pair (G, K) satisfy
+multiplicativity) if for each g, h ∈ [K\G/K] there is an element g ◦ h ∈ [K\G/K]
+such that for any unitary representation ρ of G we have
+ρK,K(g) ρK,K(h) = ρK,K(g ◦ h).
+Since a product of operators is associative, the ◦-product in [K\G/K] also is
+associative.
+Remarks. 1) In all known cases there is an explicit sequence qj ∈ K such that
+for any unitary representation ρ of G we have a weak operator convergence of ρ(qj)
+to P K (existence of such sequences is usual for topological groups independently of
+M-property); moreover, usually ◦-product can be defined as limj→∞ K · gqjh · K in
+
+4
+the quotient topological space [K\G/K], where g, h are representatives of double
+cosets.
+2) Our definition does not exclude trivial situations. For instance, it can hap-
+pened that G has no nontrivial unitary representations. Then any K ⊂ G is an M-
+subgroup, a semigroup [K\G/K] exists and consists of one element. If G = SL(n, R)
+and K is a noncompact subgroup, then for any unitary representation we have
+HK = 0 and again we come to a one-point semigroup. Also, M-property automat-
+ically holds if K is a normal subgroup in G. Then K\G/K is the quotient group
+G/K.
+3) Let G be a locally compact group and K be a compact subgroup.
+Then
+measures on K\G/K are in one-to-one correspondence with measures on G invari-
+ant with respect to left and right shifts by elements of K, such measures form a
+convolution semigroup. Semigroups of this type are widely used in representation
+theory as tools or objects of interest in their own right (as Hecke(–Iwahori) algebras
+or affine Hecke(–Iwahori) algebras). But sets K\G/K have no natural semigroup
+structure.
+⊠
+Next, let we have a subgroup K = K0 ⊂ G and a family of subgroups Kα ⊂ K,
+α ranges in some set A. Let us denote
+HKα =: Hα,
+P Kα =: Pα,
+�ρKα,Kβ(·) =: �ρα,β(·).
+Definition 3. A family {Kα} is an M-family (or a family satisfying to multiplica-
+tivity), if for each α, β, γ ∈ A for any g ∈ [Kα\G/Kβ], h ∈ [Kα\G/Kβ] there is
+an element g ◦ h such that for any unitary representation ρ of G we have
+�ρα,β(g) �ρβ,α(h) = �ρα,γ(g ◦ h).
+Since a product of operators is associative, ◦-product also is associative. So we
+get a category (train Tr(G, {Kα}) of pair (G, K)). Its objects are α ∈ A and sets
+of morphisms are
+Mor(β, α) = [Kα\G/Kβ].
+3. Examples of M-families.
+1⋆. Infinite-dimensional real classical groups. Let G = GL(∞, R) be the group of
+invertible real infinite matrices g such that g − 1 has only finite number of nonzero
+elements. Let K = O(∞) be the subgroup consisting of orthogonal matrices. Let
+Kα ⊂ K be the stabilizer of first α elements of the basis. Then {K}α ⊂ G is an
+M-family.
+Variant 1⋆
++ (which is de facto almost equivalent to the previous case). Let G be
+the group of invertible operators in real ℓ2, which can be represented in the form
+U(1 + T ), where U is an orthogonal operator and T is a Hilbert–Schmidt operator.
+Subgroup Kα are defined in the same way. See [27], [29], [28], [13], Sect. IX.3-4,
+[15], [16].
+2⋆. Infinite symmetric groups. Denote by S∞ the group of finitely supported
+permutations of N, by S∞ the group of all permutations. Let G be the product of
+n copies of S∞, where n ⩾ 1. Let K be the diagonal, so K ≃ S∞. Let {Kα} be
+point-wise stabilizers of sets {1, . . . , α}.
+Variant 2⋆
++. Consider the product of n copies of S∞. Let G consist of tuples
+(g1, . . . , gn) such that gig−1
+j
+∈ S∞ for all i, j.
+Subgroups {Kα} are point-wise
+stabilizers of sets {1, . . . , α}.
+
+5
+Variant 2⋆
+++. Let G, K be as in the first case. Let Ω be the set of all subsets
+⊂ N consisting of odd numbers. For ω ∈ Ω we define Kω as the point-wise stabilizer
+of the set ω. See [26], [28], [19].
+3⋆. Infinite-dimensional matrix groups over finite fields. Let Fq be a finite field,
+let V be the space of sequences (x1, x2, . . . ) such that all but a finite number of
+elements are zero. Let G be the group of all linear transformations of V . Let K = G
+and Kα be the stabilizer of the first α elements of the basis. See [28], [21].
+4⋆. Groups of transformations of measure spaces. Let Z be a probabilistic non-
+atomic measure space. Let G = Gms(Z), K = Ams(Z). For any finite partition h
+of Z we consider the subgroup Kh ⊂ K consisting of transformations preserving this
+partition. In this case we get the category of all polymorphisms of finite measure
+spaces. See [12], [13], Sect. VIII.4, Chapter X, [14].
+5⋆. Groups acting on trees and R-trees. Consider a tree T (a graph without
+cycles) such that each vertex is contained in a countable family of edges. Let G be
+the group of its automorphisms, K be a stabilizer of a vertex, say w. Subgroups Kα
+are stabilizers of finite subtrees containing w. See [25], [13], Sect. VIII.6 (Remarks),
+[23].
+6⋆. Oligomorphic groups. Let G = K be the group of automorphisms of the
+Rado graph (see, e.g., [2]). Subgroups KI are stabilizers of finite subgraphs I. See
+[35], [1].
+Remarks. 1) Examples 1⋆–6⋆ are representatives of families of similar pairs
+G ⊃ K (see references); known families are relatively wide in the cases 1⋆–2⋆ and
+relatively small in the case 3⋆. In all cases there are explicit descriptions of train
+categories, usually we come to non-standard algebraic structures.
+2) First examples of M-subgroups were discovered by Ismagilov [7], [8]. They
+correspond to actions of groups on trees and R-trees (but he did not observed a
+presence of trees).
+3) For the case of the Rado graph objects of the train are finite graphs and
+morphisms I → J are isomorphisms of complete subgraphs A ⊂ I to complete
+subgraphs B ⊂ J. The authors of [1] use another language, but our statement
+easily follows from [35], [1].
+On the other hand, the group of order preserving
+bijections of Q is oligomorphic, but it has no train.
+4) For representatives of types 1⋆, 2⋆, 4⋆ for cases discussed until 1996 ([27], [26],
+[12], [13]) sets [Kα\G/Kβ] coincide with Kα\G/Kβ or with maximal Hausdorff
+quotients of G with respect to equivalence g ≃ pgq, where p ∈ Kα, q ∈ Kβ; I do
+not know such statements for more general cases. On the other hand, for cases 3⋆,
+6⋆ spaces [Kα\G/Kβ] and Kα\G/Kβ can be essentially different (as it was firstly
+shown in [28]).
+4. The statement of the paper. Let G be a separable topological group,
+{Kα} be an M-family of subgroups. Consider an action of G by nonsingular trans-
+formations τ(g) of a Lebesgue probabilistic space (Z, ζ). Let K act by measure
+preserving transformations.
+For each α denote by (Zα, ζα) the quotient of (Z, ζ) by the action of the group
+Kα. Namely, consider the sigma-algebra Σα of all Kα-invariant measurable subsets3
+3A measurable subset A ⊂ Z is invariant if the symmetric differences A △ g(A) have zero
+measure for all elements g of the group.
+
+6
+in Z. Also consider the space Hα consisting of Kα-fixed functions in L2(Z, ζ). For
+each A ∈ Σα we assign its indicator function IA(z). Since L2 is separable, we can
+choose a dense countable subset IAj in the space of all functions IA, where A ranges
+in Σα. Consider the sigma-algebra Σ′
+α generated by Aj. According Rohlin [30], it
+determines a measurable partition of Z. Denote by Zα the quotient space with
+respect to this partition. We have a canonical map Z → Zα and therefore we get a
+measure, say ζα, on Zα.
+Let (X, ξ), (Y, υ) be probabilistic measure spaces, let π : X → Y be a map
+such that for each measurable B ⊂ Y we have ξ(π−1(B)) = υ(B). We define a
+polymorphism l[π] : X ։ Y as the image of ξ under the map X → X × Y × R×
+given by x �→ (x, π(x), 1), see [18], Subsect. 3.10.
+Since we have a canonical map Z → Zα, we have a canonical polymorphism
+lα : Z ։ Zα.
+Theorem 4. Let G be a separable topological group, {Kα}α∈A be an M-family
+of subgroups. Consider an action of G by nonsingular transformations τ(g) of a
+Lebesgue probabilistic space Z. Let K act by measure preserving transformations.
+For any α, β ∈ A we define
+Tα,β(g) = l∗
+ατ(g)lβ ∈ Pol(Zα, Zβ).
+Then T is a functor from the train Tr(G, {Kα}α∈A) to the category of polymor-
+phisms; i.e., Tα,β(g) depends only on a double coset [Kα\G/Kβ] containing g and
+for any g ∈ [Kα\G/Kβ], h ∈ [Kβ\G/Kγ] we have
+Tα,β(g) ⋄ Tβ,γ(h) = Tα,γ(h ◦ g).
+We need some preliminaries.
+5. Mellin–Markov transform of polymorphisms. For a Lebesgue space Z
+denote by L∞−(Z) the space of bounded measurable functions on Z equipped with
+the following convergence: ϕj → ϕ if essential supremums of |ϕj| are uniformly
+bounded and the sequence ϕj converges to ϕ in measure, see [18],
+Let r+is ∈ C ranges in the strip Π : 0 ⩽ r ⩽ 1. For p ∈ Pol(X, Y ) and r+is ∈ Π,
+we define a bilinear form Br+is : L∞−(X) × L∞−(Y ) → C by
+Br+is[p](ϕ, ψ) =
+��
+X×Y ×R×
+ϕ(x) ψ(y) tr+is dp(x, y, t).
+We define operators
+Tr+is(p) : L1/r(Y ) → L1/r(X)
+(Mellin–Markov transform of p) by the duality
+(1)
+�
+X
+ϕ(x) · Tr+is(p)ψ(x) dξ(x) = Br+is[p](ϕ, ψ)
+for any ϕ ∈ L∞−(X), ψ ∈ L∞−(Y ). Then ∥Tr+is(p)∥L1/r ⩽ 1 for r > 0. Also
+T0+is(p) is a continuous operator L∞−(Y ) → L∞−(X), see [18], Theorem 6.3.
+Example. Let g ∈ Gms(Z) be considered as a polymorphism. Then
+(2)
+Tr+is(g)ϕ(z) = ϕ(g(z)) g′(z)r+is.
+
+7
+A polymorphism p is uniquely determined by its Mellin–Markov transform ([18],
+Theorem 6.12). For any p ∈ Pol(M, N), q ∈ Pol(N, K) we have (see [18], Theorem
+6.14)
+Tr+is(q) Tr+is(p) = Tr+is(p ⋄ q).
+This transform is holomorphic in r + is in the following sense: for any bounded
+measurable functions ϕ, ψ the matrix element r + is �→ Br+is(ϕ, ψ) is holomorphic
+in the strip 0 < r < 1 and continuous in the closed strip 0 ⩽ r ⩽ 1.
+Pointwise convergence of forms
+Br+is[pj](ϕ, ψ) → Br+is[p](ϕ, ψ)
+in the closed strip Π implies convergence of polymorphisms pj → p ([18], Theorem
+6.14).
+6.
+Proof of Theorem 4.
+For α ∈ A we denote by L2(Z)α the subspace
+consisting of Kα-fixed functions in L2(Z).
+First, we notice that Mellin–Markov transforms T1/2+is[lα] are usual Markov
+operators (see, e.g, [18], Subsect. 3.9) and they do not depend on the parameter s.
+The operator T1/2+is[l∗
+αlα] is the operator of orthogonal projection Pα : L2(Z) →
+L2(Z)α or equivalently, the operator of conditional expectation, see [18], Subsect.
+3.10. We denote
+(3)
+mα := l∗
+αlα.
+The operator T1/2+is[lα] : L2(Zα) → L2(Z) is an operator of isometric embed-
+ding, its image coincides with L2(Z)α. The operator
+T1/2+is[l∗
+α]
+���
+L2(Z)α : L2(Z)α → L2(Zα)
+is unitary and
+T1/2+is[l∗
+α]
+���
+(L2(Z)α)⊥ = 0.
+Second, we slightly reformulate the M-property. In notation of Sect. 2, we define
+operators �ρα,β(g) in H by
+�ρα,β(g) = Pαρ(g)Pβ.
+These operators have the following block form
+�ρα,β(g) =
+��ρα,β(g)
+0
+0
+0
+�
+: Hβ ⊕ H⊥
+β → Hα ⊕ H⊥
+α .
+Clearly, these operators depend only on double cosets and the M-property can be
+written in the form
+(4)
+�ρα,β(g) �ρβ,γ(h) = �ρα,γ(g ◦ h).
+In the same way we define polymorphisms
+�Tα,β(g) := mατ(g)mβ ∈ Pol(Z, Z).
+The conclusion of the theorem can be reformulated in the form
+(5)
+�Tα,β(g) ⋄ �Tβ,γ(h) = �Tα,γ(h ◦ g).
+
+8
+Applying the Mellin-Markov transform to both sides of the hypothetic equality
+(5) we get an equivalent equality
+(6)
+�Tr+is(mα) Tr+is(g) �Tr+is(mβ) Tr+is(h) �Tr+is(mγ) =
+= �Tr+is(mα) Tr+is(h ◦ g) �Tr+is(mγ),
+where operators Tr+is(g) are given by (2), and g ◦ h denotes an arbitrary represen-
+tative of the product of double cosets. Setting r = 1/2 we come to a hypothetic
+equality of operators in L2(Z):
+Pα T1/2+is(g) Pβ T1/2+is(h)Pγ = Pα T1/2+is(h ◦ g) Pγ.
+But this is the identity (4). So (6) holds on the line r = 1/2. By holomorphy of
+the Mellin–Markov transform, this holds in the whole strip Π. This implies (5).
+7.
+Closures of groups in polymorphisms.
+Let G, {Kα}, τ(·) be as in
+Theorem 4. Assume that for any Kα there is a sequence qj ∈ Kα such that for any
+unitary representation π of Kα the sequence π(qj) converges to the projector Pα
+in the weak operator topology. Applying the Mellin–Markov transform we observe
+that mα := l∗
+αlα ∈ Pol(Z, Z) is contained in the closure of Kα in polymorphisms.
+So all elements
+mα τ(g) mβ ∈ Pol(Z, Z)
+are contained in the closure of G in Pol(Z, Z).
+The following statement allows to obtain estimates of the closure of G in poly-
+morphisms if we know explicit description of the action of the train (as in [14],
+[17]).
+Proposition 5. Let G, {Kα}, τ(·) be as above. Assume that there is a chain of
+subgroups Kγ1 ⊃ Kγ2 ⊃ . . . such that for any unitary representation of G in any
+Hilbert space H the subspace ∪Hγj is dense4. Then a polymorphism r ∈ Pol(Z, Z)
+is contained in the closure of G if and only if for any γj the element mγj r mγj is
+contained in the closure of G for all j.
+Proof. Only ⇐ needs in proof. Let us show that r is the limit of mγj r mγj. It
+is sufficient to prove a pointwise convergence of corresponding forms Br+is, i.e.,
+Br+is[m∗
+αrm∗
+β](ϕ, ψ) → Br+is[r](ϕ, ψ)
+Applying the Mellin–Markov transform, we get
+Br+is[m∗
+αrm∗
+β](ϕ, ψ) = Br+is[r](Pγjϕ, Pγjψ).
+So we must verify convergence
+(7)
+Br+is[r](Pγjϕ, Pγjψ) → Br+is[r](ϕ, ψ),
+where ϕ, ψ are bounded functions. The sequences Pγjϕ, Pγjψ are bounded martin-
+gales, so they converge a.s and in the L1-sense (see, e.g., [33], Theorems VIII.4.1-2).
+By the density of ∪L2(Z)γj, our sequences converge to ϕ, ψ in L2, and therefore
+we have a.s. convergences Pγjϕ → ϕ, Pγjψ → ψ.
+Formula (7) contains integrations over measures trr on Z × Z × R×, where
+0 ⩽ r ⩽ 1.
+By the definition of polymorphisms, these measures are finite and
+their pushforwards to two copies of Z are absolutely continuous with respect to ζ.
+Therefore the sequence Pγjϕ ∈ L∞− converges a.s. as a sequence on a measure
+4Examples in Sect. 3 satisfy this condition except 2⋆
+++.
+
+9
+space (Z ×Z ×R×, trr). Now (7) follows from the Lebesgue dominated convergence
+theorem.
+□
+8. Some questions. The main question is examinations of explicit non-singular
+actions. This note is abstractionist and we add some abstractionist remarks.
+1) Apparently, there is a version of Theorem 4 for spaces with σ-finite measures.
+But this requires some modification of a notion of polymorphisms.
+2) According [24], any non-singular action of S∞ is equivalent to a measure
+preserving action. So for K = S∞ we can omit the condition ’K acts by measure
+preserving transformations’. Apparently, this case is not unique.
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+

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+Inertial effects on rectification and diffusion of active Brownian particles in an
+asymmetric channel
+Narender Khatri1, ∗ and Raymond Kapral1, †
+1Chemical Physics Theory Group, Department of Chemistry,
+University of Toronto, Toronto, Ontario M5S 3H6, Canada
+(Dated: January 10, 2023)
+Micro- and nano-swimmers moving in a fluid solvent confined by structures that produce entropic
+barriers are often described by overdamped active Brownian particle dynamics, where viscous effects
+are large and inertia plays no role.
+However, inertial effects should be considered for confined
+swimmers moving in media where viscous effects are no longer dominant. Here, we study how inertia
+affects the rectification and diffusion of self-propelled particles in a two-dimensional asymmetric
+channel.
+We show that most of the particles accumulate at the channel walls as the masses of
+the particles increase. Furthermore, the average particle velocity has a maximum as a function of
+the mass, indicating that particles with an optimal mass M ∗
+op can be sorted from a mixture with
+particles of other masses. In particular, we find that the effective diffusion coefficient exhibits an
+enhanced diffusion peak as a function of the mass, which is a signature of the accumulation of most
+of the particles at the channel walls. The dependence of M ∗
+op on the rotational diffusion rate, self-
+propulsion force, aspect ratio of the channel, and active torque is also determined. The results of
+this study could stimulate the development of strategies for controlling the diffusion of self-propelled
+particles in entropic ratchet systems.
+I.
+INTRODUCTION
+Many biological microorganisms, as well as artificial
+active particles, take free energy from their environments
+and convert it under nonequilibrium conditions into per-
+sistent motion. The mechanisms that underlie such ac-
+tive motion and the dynamical properties of these sys-
+tems are diverse and have been studied extensively [1–
+15].
+For the most part, the biological and synthetic ac-
+tive agents mentioned above have micrometer or sub-
+micrometer dimensions and move in viscous environ-
+ments under conditions where inertia does not play an
+important role.
+In such circumstances, the active dy-
+namics is often described by the overdamped Langevin or
+continuum models that neglect inertia. Inertia cannot al-
+ways be neglected, and an increasing body of research [16]
+considers the effects of inertia on active particle motion
+and describes the new phenomena that arise as a result
+of its inclusion. While the systems where inertial effects
+are important are diverse, some examples include sys-
+tems that support a temperature gradient across coexist-
+ing phases [17], vibrobots [18–20], active particle motion
+in low-density media, such as gases [21], plasmas [22–
+24], superfluids [25], and active aerosols [26], etc. Such
+inertia-dominated active particles are termed micro- and
+nano-flyers rather than swimmers [16].
+The rectification of artificial active particles in con-
+fined environments in the absence of external forces has
+attracted interest [27–32]. Geometrical confinement con-
+trols the volume of phase space that is accessible to the
+∗ Corresponding author: [email protected]
+† Corresponding author: [email protected]
+active particles, resulting in entropic barriers that signif-
+icantly influence their transport properties [33–37]. As
+well, confined environments possessing spatial ratchet
+asymmetry give rise to an entropic ratchet potential
+that can induce active directed transport in the system
+[27, 31].
+We investigate the underdamped dynamics of self-
+propelled particles confined by a two-dimensional asym-
+metric channel.
+We use a minimal underdamped
+Langevin model for the dynamics of the self-propelled
+particles that accounts for inertia.
+The collisional dy-
+namics of particles with the channel walls are modeled by
+sliding-reflecting boundary conditions [27, 31, 38]. We fo-
+cus on how inertial effects influence the rectification and
+diffusion of active particles in the asymmetric channel.
+The article is organized as follows: in Sec. II, we in-
+troduce the underdamped Langevin model used to de-
+scribe the dynamics of the active particles in the two-
+dimensional asymmetric channel.
+Section III discusses
+the effects of inertia on the spatial distribution of par-
+ticles, while Sec. IV presents results on the rectification
+and effective diffusion in the channel. The main conclu-
+sions of the article are given in Sec. V.
+II.
+MODEL
+The system is confined to a two-dimensional asym-
+metric channel with periodicity L (see Fig. 1), and, as
+in other studies [16], a minimal underdamped Langevin
+model is used to describe the active dynamics under con-
+ditions where inertia is important. The coupled Langevin
+equations for an active particle with position r = (x, y),
+orientation ˆn = (cos θ, sin θ), mass M, and moment of
+arXiv:2301.02902v1  [cond-mat.soft]  7 Jan 2023
+
+2
+FIG. 1.
+Schematic illustration of an active (self-propelled)
+Brownian particle of mass M and moment of inertia I con-
+fined in a two-dimensional triangular channel with periodicity
+L. The active force F0 = F0ˆn, angle θ, active torque T0, local
+width of the channel 2 w(x), and local length of a cell of the
+channel ∆(y) are indicated. The shape of the channel struc-
+ture is prescribed by Eq. (2). The particle cannot penetrate
+through the channel walls, which are considered rigid; how-
+ever, the particle is free to rotate and slide along the walls.
+inertia I read
+M ¨r(t) = −γt ˙r(t) + F0ˆn(t) + γt
+�
+2Dt ξ(t),
+I ¨θ(t) = −γr ˙θ(t) + T0 + γr
+�
+2Dr ζ(t).
+(1)
+Here γt and γr are the translational and rotational fric-
+tion coefficients, Dt and Dr are the translational and
+rotational diffusion constants, and F0 and T0 are the ac-
+tive force and torque, respectively. The random variables
+ξ(t) and ζ(t) are the Gaussian white noise terms with zero
+mean and unit variances given by ⟨ξ(t)⊗ξ(t′)⟩ = δ(t−t′)1
+and ⟨ζ(t)ζ(t′)⟩ = δ(t − t′), respectively, where 1 is the
+unit matrix.
+Such a set of coupled Langevin equa-
+tions can serve as models for a variety of active sys-
+tems, including systems subject to athermal noise [39–
+42], if {γt, γr, Dt, Dr} are regarded as independent pa-
+rameters, and Dt and Dr control the strength of noise
+terms. For systems with thermal noises that satisfy the
+fluctuation-dissipation relation, the Einstein relations ap-
+ply, Dt = kBT/γt and Dr = kBT/γr. The set of coupled
+Langevin equations (1) is a simplified version of a more
+general set of coupled Langevin equations that applies
+to asymmetric particles and accounts for translation-
+rotation coupling [43]. The set of general underdamped
+coupled Langevin equations for chemically-active colloids
+has been derived using fluctuating chemohydrodynam-
+ics [44] and molecular theory [45], where expressions for
+the active force and torque are given.
+The two-dimensional asymmetric and spatially peri-
+odic channel shown in Fig. 1 is specified as follows: for
+the upper wall, we have
+wu(x) =
+�
+wmin,
+x = 0
+wmax − (wmax − wmin) x
+L,
+0 < x ≤ L,
+(2)
+where L is the periodicity of the channel, and wmin and
+wmax refer to the minimum and maximum half-widths of
+the channel, respectively. The dimensionless aspect ratio
+of the channel is ϵ = wmin/wmax, where we set wmax = 1
+throughout the work. The local length of a cell of the
+channel is given by ∆(y) = (wmax −|y|)L/(wmax −wmin),
+where y is bounded between the lower and upper walls.
+Due to the symmetry about the principal axis of the
+channel, the lower wall is described by wl(x) = −wu(x).
+Consequently, 2 w(x) = wu(x)−wl(x) corresponds to the
+local width of the channel.
+When a particle encounters a channel wall, it is elas-
+tically reflected [31, 38, 46, 47], and its orientation θ is
+unchanged during the collision (sliding-reflecting bound-
+ary conditions [27, 38, 48]). So, the particle slides along
+the channel walls until a fluctuation in the orientation
+vector ˆn causes it to change so that it may move away
+from the wall.
+For translational motion described by Eqs. (1), inertial
+effects dominate frictional effects for times t ≪ M/γt ≡
+τv, while for orientational motion, they are important
+for times t ≪ I/γr ≡ τω, where τv and τω are the
+characteristic times for linear and angular velocity relax-
+ation, respectively. For active particles in a bulk medium,
+these times may be compared to the reorientation time
+τr = 1/Dr that determines the time scale on which orien-
+tation vector ˆn decays, and the time τa = 2a/v0 it takes
+a particle with speed v0 to move a distance equal to the
+particle diameter 2a.
+For the present study, where the dynamics takes place
+in a periodic channel, we are interested in the effects
+of inertia on time scales that reflect the motion on the
+length scale L of the periodic channel.
+We define the
+characteristic diffusion time τ = L2/Dt, which gauges the
+time the particle takes to diffuse one period of the channel
+length.
+This characteristic diffusion time is related to
+τv by τv/τ =
+�
+τv/τth
+�2, where τth = L/vth with vth =
+�
+kBT/M the thermal speed.
+Given these considerations, we use a dimensionless de-
+scription where lengths are scaled by the periodicity of
+the channel L, r′ = r/L, and time by τ, t′ = t/τ, analo-
+gous to that in Refs. [37, 49], so that Eqs. (1) read,
+M ∗¨r(t) = − ˙r(t) + f0ˆn(t) +
+√
+2 ξ(t),
+I∗¨θ(t) = − ˙θ(t) + t0 +
+√
+2α ζ(t),
+(3)
+and we dispensed with the primes in writing this coupled
+set of equations. The dimensionless mass and moment
+of inertia are given by M ∗ = τv/τ = MDt/(γtL2) and
+I∗ = τω/τ = IDt/(γrL2), respectively.
+In these vari-
+ables, the dimensionless mass M ∗ depends on physical
+mass M as well as Dt, γt, and L. Similarly, I∗ depends on
+the moment of inertia I as well as Dt, γr, and L. The di-
+mensionless active force and torque are f0 = F0L/(Dtγt)
+and t0 = T0L2/(Dtγr), respectively, and the parameter
+α = Drτ.
+In the following sections, we present results for the
+average velocity, effective diffusion coefficient, and other
+properties obtained from simulations of the coupled
+Langevin equations (3) in the channel
+[50]. At t = 0,
+the particles are uniformly distributed with random
+orientations in a periodic cell of the channel located
+between x = 0 and x = 1. The results are obtained from
+
+3
+averages over 104 stochastic trajectories.
+III.
+SPATIAL DISTRIBUTION
+−1
+0
+1
+0.01
+1
+100
+y
+Pst(y)
+M ∗ = 0.001
+M ∗ = 1
+M ∗ = 1000
+M ∗ = 0.001
+M ∗ = 1
+M ∗ = 1000
+f0 = 5, I∗ = 0.001
+t0 = 0
+α = 0.1, ǫ = 0.1
+(a)
+(b)
+(c)
+(d)
+FIG. 2. The steady state distribution of particles, mapped
+into a single cell of the channel, is depicted in (a)-(c) for
+different values of M ∗. The corresponding normalized prob-
+ability densities Pst(y) along the y direction are depicted in
+(d). The parameters are: I∗ = 0.001, f0 = 5, t0 = 0, α = 0.1,
+and ϵ = 0.1.
+In order to analyze the effects of inertia on the rec-
+tification and diffusion of self-propelled particles in an
+asymmetric channel, we first consider the spatial dis-
+tribution of particles mapped onto a single cell of the
+channel. Figure 2 shows the steady state distribution of
+particles and the corresponding normalized probability
+density Pst(y) along the y direction for different values
+of M ∗. For M ∗ → 0 in the overdamped limit, the dis-
+tribution of particles is inhomogeneous, and most of the
+particles tend to accumulate near the left corners of the
+cell (see Fig. 2(a) and the Pst(y) plots in (d)). The in-
+homogeneous distribution of particles can be ascribed to
+the active motion due to broken detailed balance and
+the presence of the spatial asymmetry imposed by the
+shape of the channel [27, 31]. It is interesting to see that
+on increasing M ∗ further, most of the particles quickly
+accumulate at the channel walls; while the rest of the
+particles adopt the shape of a funnel about the principal
+axis of the channel at the middle region due to narrow
+bottleneck openings (see Fig. 2(b)). Most of the parti-
+cles accumulate at the left corners of the cell, and the
+distribution of particles is symmetric about the principal
+axis of the channel (see the Pst(y) plots, especially the
+curve for M ∗ = 1). Such an observation provides evi-
+dence that the rectification of particles in an asymmetric
+channel can be enhanced by increasing M ∗. For strongly
+underdamped situations where M ∗ → ∞, the qualitative
+behavior of the distribution of particles and the corre-
+sponding Pst(y) are very similar; however, the rectifi-
+cation and diffusion of particles approach zero because
+inertia dominates the self-propulsion mechanism [51].
+−1
+0
+1
+0.01
+1
+100
+y
+P(y)
+t = 0
+t = 0.5
+Steady State
+t = 0
+t = 0.5
+t = 1
+(a)
+(b)
+(c)
+(d)
+FIG. 3.
+The time evolution of the distribution of parti-
+cles with M ∗ = 1 is shown in (a)-(c) at increasing times,
+t = 0, 0.5, 1, respectively.
+The normalized probability den-
+sity P(y) versus y is plotted in (d).
+The parameters are:
+I∗ = 0.001, f0 = 5, t0 = 0, α = 0.1, and ϵ = 0.1.
+The probability density evolution to its steady state
+distribution is shown in Fig. 3 for a system with M ∗ = 1
+and parameters as in Fig. 2.
+At t = 0, particles are
+distributed uniformly inside the periodic cell consistent
+with thermodynamic equilibrium [33–35, 46, 52]. As time
+increases to t = 0.5, the distribution becomes inhomoge-
+neous: particles near the channel walls quickly begin ac-
+cumulating there, and the funnel-shaped distribution of
+particles in the middle region of the cell starts to develop.
+At a yet later time t = 1, the funnel-shaped distribution
+sharpens, and particles have continued to accumulate at
+the walls. The distribution at this time closely resembles
+the steady state distribution shown in Fig. 2(b). Note
+that the probability of accumulation at the left corners
+of the cell is much higher than in other regions, which is
+reflected in the structure of the probability density P(y)
+in Fig. 3(d).
+IV.
+RECTIFICATION AND EFFECTIVE
+DIFFUSION
+Figure 4 shows the dependence of the average velocity
+v and effective diffusion coefficient Deff on M ∗ for differ-
+ent values of the self-propulsion force f0. We observe that
+the particles exhibit rectification (v ̸= 0) in the positive
+x direction; v is positive due to the chosen shape of the
+channel. If the shape of the channel were inverted with
+respect to the y axis, then the magnitude of rectification
+would remain the same, but v would lie in the negative
+x direction. In the small M ∗ limit, the channel asym-
+metry affects the particle dynamics more strongly since
+most of the particles accumulate at the channel walls (see
+Figs. 2 and 3), resulting in an increase in v with M ∗. In
+the strongly underdamped limit, when M ∗ → ∞, inertia
+dominates self-propulsion resulting in a rapid decay of v
+and Deff with M ∗. Therefore, as one might expect, v has
+a peak at an optimal mass M ∗
+op; thus, in a mixture, recti-
+
+4
+0.001
+0.01
+0.1
+1
+10
+100
+1000
+25
+50
+M ∗
+Deff
+0
+0.5
+1
+v
+f0 = 1
+f0 = 5
+f0 = 10
+0.001
+1
+1000
+0.5
+1
+1.5
+α = 0.1, I∗ = 0.001
+t0 = 0, ǫ = 0.1
+(a)
+(b)
+FIG. 4. Average velocity v as a function of M ∗ is shown in
+(a) for different values of the self-propulsion force f0. The
+corresponding effective diffusion coefficient Deff is shown in
+(b). The inset plots Deff versus M ∗ for f0 = 1 on an expanded
+scale to show its structure. Here and below, the solid lines
+are guides to the eye, and the statistical errors for v and Deff
+are smaller than the symbol sizes. The other parameters are:
+I∗ = 0.001, t0 = 0, α = 0.1, and ϵ = 0.1.
+0.001
+0.01
+0.1
+1
+10
+100
+1000
+5
+10
+15
+M ∗
+Deff
+0
+0.3
+0.6
+v
+α = 0.1
+α = 1
+α = 10
+0.001
+1
+1000
+1
+2
+f0 = 5, I∗ = 0.001
+t0 = 0, ǫ = 0.1
+(a)
+(b)
+FIG. 5. Average velocity v and effective diffusion coefficient
+Deff as a function of M ∗ are plotted in (a) and (b), respec-
+tively, for different values of the scaled rotational diffusion
+rate α.
+The inset in (b) shows Deff versus M ∗ for α = 1
+and α = 10 on an expanded scale. The other parameters are:
+I∗ = 0.001, t0 = 0, f0 = 5, and ϵ = 0.1.
+fied particles with M ∗
+op will have a higher speed compared
+to particles with other M ∗ values. The peak is more pro-
+nounced as f0 increases, and the value of the optimal
+mass can be controlled by changing f0. For the effective
+diffusion coefficient, we see that Deff initially decreases
+with increasing M ∗, but on increasing M ∗ further, Deff
+exhibits an enhanced diffusion peak, which is a signature
+of the accumulation of most of the particles at the chan-
+nel walls [38]. As expected, Deff increases monotonically
+with f0. As f0 → 0, the motion of self-propelled parti-
+cles approaches that of passive Brownian motion; thus,
+v will tend to zero, and the enhanced diffusion peak will
+no longer be present (see the inset of Fig. 4(b)).
+The variation of v and Deff with M ∗ for different val-
+ues of the rotational diffusion rate α is shown in Fig. 5.
+The qualitative trends for different α are the same as
+those described above for different f0; however, the peak
+is more pronounced as α decreases, and now M ∗
+op can be
+changed by tuning α. As expected, Deff decreases mono-
+tonically with increasing α.
+In particular, as α → ∞
+where reorientation is rapid, the self-propelled motion
+tends to passive Brownian motion, v tends to zero, and
+the enhanced diffusion peak vanishes (see the inset of
+Fig. 5(b)).
+0.001
+0.01
+0.1
+1
+10
+100
+1000
+0
+5
+10
+15
+M ∗
+Deff
+0
+0.3
+0.6
+v
+t0 = 0
+t0 = 1
+t0 = 10
+0.001
+1
+1000
+0
+0.5
+f0 = 5, α = 0.1
+I∗ = 0.001, ǫ = 0.1
+(a)
+(b)
+FIG. 6. Average velocity v and effective diffusion coefficient
+Deff as a function of M ∗ are plotted in (a) and (b), respec-
+tively, for different values of the active torque t0. The inset
+in (b) shows Deff versus M ∗ for t0 = 1 and t0 = 10 on an ex-
+panded scale. The parameters are: I∗ = 0.001, α = 0.1, f0 =
+5, and ϵ = 0.1.
+Figure 6 shows v and Deff versus M ∗ for different val-
+ues of the active torque t0. Again one observes that the
+qualitative trends are the same as those discussed above,
+and the values of M ∗
+op can also be changed by changes in
+the active torque t0.
+Finally, we point out that the qualitative behavior of
+v and Deff versus M ∗ remains the same for different val-
+ues of the moment of inertia I∗ and aspect ratio of the
+
+5
+channel ϵ. In addition, M ∗
+op is found to be independent
+of I∗, and it is only weakly dependent on ϵ.
+0.1
+1
+10
+0.01
+0.1
+1
+α
+M ∗
+op
+2
+5
+10
+0.1
+0.2
+0.5
+1
+f0
+M ∗
+op
+0.1
+0.2
+0.5
+0.4
+0.6
+0.8
+ǫ
+M ∗
+op
+0.1
+1
+100.001
+0.01
+0.1
+1
+t0
+M ∗
+op
+(a)
+(b)
+(c)
+(d)
+FIG. 7. Dependence of M ∗
+op on the rotational diffusion rate
+α (a), self-propulsion force f0 (b), aspect ratio of the channel
+ϵ (c), and active torque t0 (d). The moment of inertia I∗ =
+0.001.
+The dependence of M ∗
+op on the rotational diffusion rate
+α, self-propulsion force f0, aspect ratio of the channel ϵ,
+and active torque t0 is shown in Fig. 7. For fixed values of
+f0, ϵ, and t0, M ∗
+op decreases monotonically with α (panel
+(a)). From panel (b), for fixed values of α, ϵ, and t0, M ∗
+op
+varies nonmonotonically with f0 with the appearance of
+a peak. Note that M ∗
+op is weakly dependent on ϵ (panel
+(c), observe change in the ordinate scale); for fixed values
+of α, f0, and t0, M ∗
+op has a minimum. Lastly, from panel
+(d), M ∗
+op monotonically decreases with t0 for fixed values
+of α, f0, and ϵ.
+V.
+REMARKS AND CONCLUSION
+This study of the rectification and diffusion of self-
+propelled particles in a two-dimensional asymmetric
+channel showed that the inclusion of inertia leads to sev-
+eral distinctive features. In particular, most of the parti-
+cles accumulate at the channel walls with increasing par-
+ticle mass, while the remaining particles are distributed
+in a funnel-shaped region about the principal axis of the
+channel.
+This effect leads to enhanced rectification of
+heavier particles. The presence of a maximum in the ef-
+fective diffusion coefficient as a function of the mass is
+also a consequence of the accumulation of most of the
+particles at the channel walls.
+Furthermore, for vari-
+ous parameter values, the average particle velocity has a
+maximum as a function of the mass, indicating that par-
+ticles with an optimal mass M ∗
+op drift faster than other
+particles; hence, they can be sorted from a mixture with
+particles of different masses.
+While the Langevin model (1) can describe a wide va-
+riety of physical systems whose active agents are powered
+by various mechanisms and are subject to either thermal
+or athermal noise [16], it is instructive to discuss possible
+experimental realizations of the rectification effects de-
+scribed above. The asymmetric channels we considered
+can be constructed by microprinting on a substrate, and
+the effects of inertia can be determined from measure-
+ments of the average velocity and effective diffusion co-
+efficient [53–55]. From the results presented in the text,
+one can see that the effects of inertia described above will
+manifest themselves only for certain values of the system
+parameters, in particular, the particle mass, friction coef-
+ficients, diffusion constants, self-propulsion force, solvent
+viscosity, etc. The ability to control all of these parame-
+ters within desirable ranges places limits on the physical
+systems.
+A class of systems that may be of interest in this con-
+text are aerosols [26], where diffusiophoresis has been
+used to separate micrometer-scale particles [56, 57]. As
+an example, consider identically-sized active particles
+with radii a ∼ 200 nm, mass M ∼ 10−15 kg, mo-
+ment of inertia I ∼ 10−31 kg m2 in air at room tem-
+perature and pressure p = 104 − 105 Pa.
+The vis-
+cosity is given by η ∼ 10−5 kg/(m s), independent of
+pressure, with translational and rotational friction coeffi-
+cients, γt ∼ 10−11 kg/s and γr ∼ 10−24 kg m2/s, respec-
+tively. The active force lies in the range F0 ∼ 0.1−1 pN,
+and the active torque is taken to be zero. The transla-
+tional and rotational noise strengths are then determined
+by the values of Dt and Dr, respectively. The ratchet
+channel parameters are L = 10 µm, wmax = 10 µm, and
+ϵ = 0.1.
+For thermal noise, the Einstein relations hold, and
+Dt = kBT/γt ∼ 10−9 m2/s and Dr = kBT/γr ∼ 103 s−1.
+In the dimensionless units introduced earlier, we have
+τ ∼ 0.1 s, M ∗ ∼ 10−3, I∗ ∼ 10−6, f0 ∼ 102 − 103, and
+α ∼ 102. From these parameter values, we can see that
+the regime where inertial effects play a role cannot be
+accessed.
+For athermal noise, take Dt ∼ 10−8 − 10−6 m2/s and
+Dr ∼ 103 s−1. We then have τ ∼ 10−2 − 10−4 s, M ∗ ∼
+0.01 − 1, I∗ ∼ 10−5 − 10−3, f0 ∼ 100 − 0.1, and α ∼ 10 −
+0.1. Under these conditions, the interesting regime where
+inertial effects lead to an optimal mass can be reached,
+provided the system is driven by external noise sources.
+These results could stimulate the development of
+strategies for controlling the diffusion of active particles
+in entropic ratchet systems. Moreover, since the rectifi-
+cation of particles strongly depends on their mass, the
+model could be used to design lab-on-a-chip devices and
+artificial channels for the mass-based separation of par-
+ticles.
+VI.
+ACKNOWLEDGMENT
+This
+work
+was
+supported
+in
+part
+by
+the
+Nat-
+ural
+Sciences
+and
+Engineering
+Research
+Coun-
+cil
+(NSERC)
+of
+Canada
+and
+Compute
+Canada
+(www.computecanada.ca).
+
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+

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@@ -0,0 +1,2041 @@
+arXiv:2301.01987v1  [cs.IT]  5 Jan 2023
+1
+Energy Efficient Semantic Communication over
+Wireless Networks with Rate Splitting
+Zhaohui Yang, Mingzhe Chen, Zhaoyang Zhang, and Chongwen Huang
+Abstract—In this paper, the problem of wireless resource
+allocation and semantic information extraction for energy effi-
+cient semantic communications over wireless networks with rate
+splitting is investigated. In the considered model, a base station
+(BS) first extracts semantic information from its large-scale data,
+and then transmits the small-sized semantic information to each
+user which recovers the original data based on its local common
+knowledge. At the BS side, the probability graph is used to extract
+multi-level semantic information. In the downlink transmission,
+a rate splitting scheme is adopted, while the private small-sized
+semantic information is transmitted through private message
+and the common knowledge is transmitted through common
+message. Due to limited wireless resource, both computation
+energy and transmission energy are considered. This joint
+computation and communication problem is formulated as an
+optimization problem aiming to minimize the total communica-
+tion and computation energy consumption of the network under
+computation, latency, and transmit power constraints. To solve
+this problem, an alternating algorithm is proposed where the
+closed-form solutions for semantic information extraction ratio
+and computation frequency are obtained at each step. Numerical
+results verify the effectiveness of the proposed algorithm.
+Index Terms—Rate splitting multiple access, semantic commu-
+nication, energy efficient design.
+I. INTRODUCTION
+The rapid development of emerging applications such as
+digital twin, edge learning, and metaverse requires wireless
+networks to support high transmission data rate, ultra low
+latency, and seamless connectivity [1]–[4]. However, due to
+limited wireless resources such as frequency and time, con-
+ventional orthogonal multiple access schemes cannot support
+massive connectivity concern for next-generation wireless
+communication networks [5]. Through using the same time
+This work was supported in part by National Natural Science Foundation
+of China under Grant 61725104 and U20A20158, and National Key R&D
+Program of China under Grant 2018YFB1801104 and 2020YFB1807101. The
+work of Prof. Huang was supported by the China National Key R&D Program
+under Grant 2021YFA1000500, National Natural Science Foundation of
+China under Grant 62101492, Zhejiang Provincial Natural Science Foundation
+of China under Grant LR22F010002, National Natural Science Fund for
+Excellent Young Scientists Fund Program (Overseas), Zhejiang University
+Education Foundation Qizhen Scholar Foundation, and Fundamental Research
+Funds for the Central Universities under Grant 2021FZZX001-21.
+Z. Yang is with Zhejiang Lab, Hangzhou, Zhejiang, 311121, China. Z.
+Yang, Z. Zhang, and C. Huang are with the College of Information Sci-
+ence and Electronic Engineering, Zhejiang University, Hangzhou, Zhejiang
+310027, China, and Zhejiang Provincial Key Lab of Information Processing,
+Communication and Networking (IPCAN), Hangzhou, Zhejiang, 310007,
+China. (e-mails: yang [email protected], ning [email protected], chong-
+[email protected])
+M. Chen is with the Department of Electrical and Computer Engineering
+and Institute for Data Science and Computing, University of Miami, Coral
+Gables, FL, 33146 USA (e-mail:[email protected])
+or frequency resource, multiple users can be served in non-
+orthogonal multiple access (NOMA) [6]–[11], where users can
+be split in the power or code domain. Since additional users
+can be served with superposition coding at the transmitter
+and successive interference cancellation (SIC) at the receiver,
+the spectral efficiency of NOMA is generally higher than
+conventional orthogonal multiple access schemes.
+In downlink NOMA transmission, the receiver side decodes
+the interference for all received strong messages [11]. Thus,
+the computation capacity of NOMA decoding is generally
+high. To balance the decoding tradeoff of intended signal
+and interference signal, the concept of rate splitting multiple
+access (RSMA) was introduced in [12]–[15]. For downlink
+RSMA transmission, the transmission message intended for
+each user is divided into both common and private parts.
+All users intend to receive the common part of the message,
+i.e., common message, while only part of the users wish to
+receive and decode the specific private part of the message,
+i.e., private message. At the user side, the common message
+is decoded first with regarding all private messages as inter-
+ference, while the intended private message is decoded with
+only considering the private messages intended for other users
+as interference. Through dynamically controlling the split of
+private and common messages, the computation complexity
+of RSMA can be adjusted to achieve the specific spectral
+efficiency requirements. To implement RSMA for wireless
+communication systems, there are still many challenges, which
+include the resource allocation for private and common mes-
+sages, decoding order optimization, system design in imperfect
+channel and hardware mismatch cases.
+There are many contributions investigating the problems
+of RSMA in wireless communication systems. The general
+challenges of RSMA were pointed out in [14] for multiple
+input multiple output (MIMO) communication systems. To
+maximize the sum rate of all users, a distributed rate splitting
+technique was proposed in [16]. For a two-receiver multiple
+input single output (MISO) communication system with lim-
+ited rate feedback, the rate analysis was investigated in [17].
+Compared with NOMA and space-division multiple access
+(SDMA), it was shown in [18] that RSMA can achieve the
+best performance in terms of spectral and energy efficiency
+[19]. In particular, the energy efficiency optimization for
+RSMA and NOMA transmissions in a unmanned aerial vehicle
+assisted wireless communication system was investigated in
+[20] . Considering wireless energy transfer and information
+transmission, the linear precoding method for RSMA was
+investigated in [21]. For the case with imperfect channel state
+information, the sum rate maximization with partial channel
+
+2
+state information for RSMA was studied in [22], while a
+downlink MISO RSMA system with bounded channel errors
+was investigated in [23].
+The interplay between rate splitting with emerging tech-
+nologies has been investigated. With the help of reconfig-
+urable intelligent surface, the energy efficient resource allo-
+cation for reconfigurable intelligent surface assisted RSMA
+was investigated in [24], where the phase shift, rate alloca-
+tion, and trasnmit beamforming were jointly scheduled. The
+learning based traffic prediction method was studied in [25]
+for unmanned aerial vehicle enabled wireless communication
+system with rate splitting. The neural network was proposed
+in [26] to solve the user clustering problem in hierarchical
+rate splitting communication systems. Due to coupled rate
+and power allocation relationship, the resource allocation
+of RSMA usually leads to the nonconvex problem, which
+can be solved by utilizing the learning techniques such as
+deep reinforcement learning. Several deep learning algorithms
+were designed to solve various complex resource allocation
+problems for RSMA, which include total power minimization
+problem [27], joint power control, beamforming design, and
+splitting optimization problem [28], [29], power allocation
+problem with limited channel state information knowledge
+[30], [31], joint transmit power, user clustering, and resource
+block allocation problem [32], joint passive precoding at the
+reconfigurable intelligent surface and active precoding at the
+transmitter [33]. In the federated learning frameworks [34], the
+authors in [35] utilized RSMA for uplink model transmission
+to minimize the total delay of the whole system. A model-
+based deep learning algorithm was developed to solve the
+receiver design problem of RSMA in [36].
+Recently, semantic communication has attracted a lot of
+attention [37]–[46]. For the wireless communication system
+characterized by Shannon capacity, the receiver side needs
+to recover the information that is exactly the same as the
+transmitted information. However, in the emerging wireless
+applications such as virtual reality, personalized healthcare,
+autonomous driving, and the Internet-of-Everything (IoE), the
+wireless communication systems aim to meet the multimodal
+quality-of-experience (QoE) requirements with massive data,
+which makes the traditional Shannon capacity characterized
+transmission infeasible. Especially in human-computer inter-
+action scenarios, humans can control multiple IoE devices
+simultaneously through voice and augmented/virtual reality
+commands, making communication ubiquitous in small-range
+wireless networks, which poses severe challenges to tradi-
+tional bit-oriented communication challenge. Supporting real-
+time human-machine interaction and machine-to-machine in-
+teraction through the use of text, speech, images, and aug-
+mented/virtual reality is important for future wireless commu-
+nications. In order to support this interaction, the important
+information finally received depends mainly on the intent,
+rather than the bit information dependence of common sense.
+These applications use advanced signal processing to facilitate
+the development of task-oriented semantic communication [3],
+[47], [48]. In semantic communication, both transmitter and
+receiver share common knowledge, which can be used to
+extract small-size information at the transmitter and recover
+the original information at the receiver [49]. Similarly, in
+downlink RSMA, all users also need to receive both common
+information and private information. Due to the inherent sim-
+ilarity of common knowledge and common message, RSMA
+can be utilized to enhance the system performance of downlink
+semantic communication. To our best knowledge, there is
+no prior works that consider the integration of semantic
+communication and RSMA.
+The main contributions of this paper include:
+• The problem of wireless resource allocation and semantic
+information extraction for energy efficient semantic com-
+munications over wireless networks with rate splitting is
+investigated. In the considered model, the BS first extracts
+the semantic information from its large-scale data, and
+then transmits the small-sized semantic information to
+each user which recovers the original data based on the
+local common knowledge.
+• In the downlink transmission, the rate splitting scheme
+is adopted, while the private small-sized semantic in-
+formation is transmitted through private message and
+the common knowledge is transmitted through com-
+mon message. Due to limited wireless resource, both
+computational energy and transmission energy must be
+considered. This joint computation and communication
+problem is formulated as an optimization problem whose
+goal is to minimize the total energy consumption of the
+network under a latency constraint.
+• To solve this problem, an iterative algorithm is proposed
+where, at every step, closed-form solutions for semantic
+information extraction ratio and computation frequency
+are derived. Numerical results show the effectiveness of
+the proposed algorithm.
+The rest of this paper is organized as follows. The system
+model and problem formulation are described in Section II.
+The algorithm design is presented in Section III. Simulation
+results are analyzed in Section IV. Conclusions are drawn in
+Section V.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+Consider a downlink semantic wireless communication
+(SWC) network with one multiple-antenna BS and K single-
+antenna users, as shown in Fig. 1. The BS is equipped with
+N antennas and the set of users is denoted by K. Each user k
+has a large-sized data Dk to receive. Due to limited wireless
+resource, the BS needs to extract the small-sized semantic
+information from the original data Dk. In the considered
+model, the BS first extracts the semantic information based on
+directional probability graph and then transmits the semantic
+information via rate splitting technique.
+A. Semantic Communication Model
+In this part, we utilize the directional probability graph
+to characterize the inherent information of the transmitted
+information. In the directional probability graph, each vertex
+represents the semantic entity with different semantic levels.
+The higher level the semantic level is, the more complicated
+
+3
+Large-
+sized data
+Semantic 
+information
+User 
+��
+Fig. 1. Illustration of the considered SWC network with rate splitting.
+the semantic information is. The link between any two vertexes
+represents the probability.
+To construct the directional probability graph, we use the
+deep neural network to train the stored dataset, which includes
+three main steps. In the first step, the semantic entity is
+recognized from the dataset, where the semantic entity means
+the names in text, including person names, place names, etc.
+The name of semantic entity is highly open (various types,
+flexible lengths, unregistered words), contains rich knowledge
+and highlights individuality. Three common methods, i.e., rule
+method, taxonomy method, and sequence labeling [50] can be
+used to identify the semantic entity. The semantic entity is
+presented as a vertex in the directional probability graph. In
+the second step, the link between any two vertexes means the
+probability that one vertex can be linked with the other vertex.
+Through training the dataset, the probability between two
+vertexes can be calculated via convolutional neural networks.
+In the third step, the semantic information fusion is conducted.
+For two vertexes, if the link probabilities between these two
+vertexes are higher than a predefined threshold. As a result, the
+final directional probability graph becomes a multi-tier graph,
+as shown in Fig. 2.
+To obtain the small-size semantic communication, the ex-
+traction process includes two parts, as shown in Fig. 3. In the
+first part, the directional probability graph is used to extract
+semantic information and the output is denoted by G(Dk). To
+efficiently transmit information, in the second part, a subset
+Sk out of G(Dk) is selected at user k, which is used for data
+transmission.
+At the user side, each user utilizes the shared common di-
+rectional probability graph to recover the original data and the
+recovered data is denoted by R(Sk). The semantic accuracy
+of the recovered data
+uk(Dk, Sk) =
+�|R(Sk)|
+i=1
+min{σ(R(Sk), s′
+ki), σ(Dk, s′
+ki)}
+�|R(Sk)|
+i=1
+σ(R(Sk), s′
+ki)
+,
+(1)
+where |R(Sk)| is the number of bits in R(Sk), s′
+ki denotes
+the i-th word in text or frame in video in R(Sk), and
+ 
+�
+   
+�
+�
+ 
+�
+   
+�
+�
+ 
+�
+   
+�
+�
+ 
+�
+   
+�
+�
+�
+High semantic 
+layer
+Low semantic 
+layer
+Fig. 2. An example of the multi-level semantic information extraction.
+��
+�
+Large-sized data
+Semantic 
+information
+Recovered data
+Received 
+semantic 
+information
+�
+�
+�
+�
+�
+�
+Dk
+G(Dk)
+Sk
+�
+Selected 
+semantic 
+information
+R(Sk)
+Sk
+Transmitter
+Receiver
+Fig. 3. Illustration of the SWC model.
+σ(R(Sk), s′
+ki) is the number of occurrences of s′
+ki in R(Sk).
+B. RSMA Model
+In RSMA, the message intended for each user can be split
+into two parts, i.e., common part and private part [51]. The
+common parts from all users are collected and combined into
+a common message. Through sharing the same codebook for
+all users, the common message is encoded into the common
+message s0, which all users need to decode. The private part
+of each user k is encoded into the private stream sk, which is
+intended for the specific user k. As a result, the transmitted
+signal x of the BS can be written as:
+x = √p0w0s0 +
+K
+�
+k=1
+√pkwksk,
+(2)
+where w0 is the transmit beamforming of the common mes-
+sage s0 , wk is the transmit beamforming of the private
+message sk intended for user k, p0 is the transmit power of
+the common message s0, and pk is the transmit power of the
+private message sk.
+
+4
+For user k, the received message can be represented by:
+hH
+k x + nk = hH
+k
+√p0w0s0 +
+K
+�
+j=1
+√pkhH
+k wksj + nk,
+(3)
+where hk stands for the channel between user k and the BS.
+To decode the common message s0, the rate of user k can be
+given by:
+ck = B log2
+�
+1 +
+p0|hH
+k w0|2
+�K
+j=1 pj|hH
+k wj|2 + σ2
+�
+.
+(4)
+where B is the bandwidth of the BS. Note that all users need
+to decode the same common message. To ensure that all users
+can successfully decode the common message, the rate of the
+common message can be set as [18]
+c0 = min
+k∈K ck.
+(5)
+In our considered SWC with rate splitting, the common
+knowledge is shared by all users. Thus, the common knowl-
+edge required for semantic communication can be encoded
+in the common message. Besides, the common message also
+includes the parts that are allocated for different users, i.e., the
+rate in the common message allocated to user k is denoted by
+ak. As a result, the rate constraint for the common message
+can be given by
+a0 +
+K
+�
+k=1
+ak ≤ ck,
+∀k ∈ K,
+(6)
+where a0 is the rate allocated to updated common knowledge
+that all users need to receive. In SWC, a0 represents the rate of
+transmitting the information of updated directional probability
+graph.
+For each user, the common message is decoded first, and
+then the common message can be subtracted for decoding the
+private message. As a result, the rate for decoding the private
+message for user k can be calculated as
+rk = B log2
+�
+1 +
+pk|hH
+k wk|2
+�K
+j=1,j̸=k pj|hH
+k wj|2 + σ2
+�
+.
+(7)
+C. Transmission and Computation Model
+For each user k, the computation time for extracting seman-
+tic information from data Dk is
+t1k = y1k(Dk, Sk)
+fk
+,
+(8)
+where y1k(Dk, Sk) is the required amount of CPU cycles for
+calculating Sk out of Dk, and fk is the computing capacity of
+user k. The local computation energy can be given by:
+E1k = κy1k(Dk, Sk)f 2
+k,
+(9)
+where κ is a constant coefficient to measure the effective
+switched capacitance.
+With private rate (7) and allocated common rate ak, the
+downlink transmission time for transmitting Sk is given by
+t2k1 = Z(Sk)
+rk + ak
+,
+(10)
+��
+Computing and Transmitting at the BS
+Computing at Users
+Time
+Frequency
+b2
+bK
+b1
+B
+t12
+t22
+t11
+t21
+t2K
+t1K
+Computation Time
+Transmission Time
+t31
+t32
+t3K
+Fig. 4. Illustration of the computation and communication time.
+where Z(Sk) is the data size of set Sk. To transmit the
+renewed information about the knowledge base, i.e., updated
+information of directional probability graph, the transmission
+time of all users can be formulated as
+t0 = K0
+a0
+,
+(11)
+where K0 is the size of updated information of directional
+probability graph. Combining (10) and (11), the downlink
+transmission time for user k is
+t2k = max{t2k1, t0}.
+(12)
+The transmission energy for sending Sk is
+E2k = t2k1pk,
+(13)
+and the transmission energy for broadcasting updated infor-
+mation of directional probability graph is
+E20 = t0p0.
+(14)
+At user k, to recover the original data, the user needs to
+compute the semantic information Sk. The computation time
+of user k
+t3k = y2k(Dk, Sk)
+gk
+,
+(15)
+where y2k(Dk, Sk) is the number of computation cycles of
+recovering Dk from Sk and gk is the computation capac-
+ity at user k. The total complete time for user k includes
+computation time at the BS, downlink transmission time, and
+computation time at user k, as shown in Fig. 4. The overall
+complete time of user k including both computation and
+computation is
+tk = t1k + t2k + t3k
+= y1k(Dk, Sk)
+fk
++ max
+� Z(Sk)
+rk + ak
+, K0
+a0
+�
++ y2k(Dk, Sk)
+gk
+.
+(16)
+The energy consumption at user k is
+E3k = κy2k(Dk, Sk)g2
+k.
+(17)
+
+5
+With the above considered model, the total communication
+and computation energy consumption of the system is
+E =
+K
+�
+k=1
+(E1k + E2k + E3k) + E0
+=
+K
+�
+k=1
+�
+κy1k(Dk, Sk)f 2
+k + Z(Sk)pk
+rk + ak
++ κy2k(Dk, Sk)g2
+k
+�
++ K0p0
+a0
+.
+(18)
+We aim to minimize the total energy consumption of the
+whole system with considering the completion time, transmit
+information accuracy, computation capacity, rate allocation,
+and power allocation constraints. Mathematically, the formu-
+lated total energy minimization problem can be given by:
+min
+S,f,g,p,a,w E,
+(19)
+s.t. y1k(Dk, Sk)
+fk
++ max
+� Z(Sk)
+rk + ak
+, K0
+a0
+�
++ y2k(Dk, Sk)
+gk
+≤ T,
+∀k ∈ K,
+(19a)
+uk(Dk, Sk) ≥ Ak,
+∀k ∈ K,
+(19b)
+Sk ⊆ G(Dk)
+∀k ∈ K,
+(19c)
+a0 +
+K
+�
+k=1
+ak ≤ ck,
+∀k ∈ K,
+(19d)
+K
+�
+k=1
+fk ≤ F max
+(19e)
+K
+�
+k=0
+p0 ≤ P max
+(19f)
+ak, fk, pk ≥ 0,
+∀k,
+(19g)
+∥wk∥ = 1,
+∀k ∈ K ∪ {0},
+(19h)
+0 ≤ gk ≤ gmax
+k
+,
+∀k ∈ K,
+(19i)
+where S = {S1, · · · , SK}, f = [f0, f1, · · · , fK]T , g =
+[g1, · · · , gK]T , p = [p1, · · · , pK]T , a = [a0, · · · , pK]T ,
+w = [w0; w1; · · · ; wK], T is the maximum communication
+delay of the system, Ak is the minimum semantic accuracy
+for user k, F max is the maximum computation capacity at
+the BS, P max is the transmission power of the BS, and
+gmax
+k
+is the maximum local computation capacity of user
+k. Since both objective function and constraints (19a)-(19c)
+are nonconvex, it is generally hard to solve this problem. To
+solve this problem, we propose an iterative algorithm using
+the alternating method and successive convex approximation
+(SCA) approach.
+III. ALGORITHM DESIGN
+In this section, an alternating algorithm is proposed to
+iteratively solve problem (19) through optimizing three sub-
+problems, i.e., semantic information extraction subproblem,
+computation capacity subproblem, joint power control, rate
+allocation and beamforming design subproblem.
+Accuracy 
+k
+r
+Extraction rate
+Computation
+k
+r
+Extraction rate
+(
+,
+)
+k
+k
+k
+uk
+k
+k
+k
+k
+k
+,
+2 (
+,
+)
+k
+k
+y
+k
+1 (
+,
+)
+k
+k
+k
+y
+k
+k
+Fig. 5.
+Illustration of the accuracy and computation functions versus the
+extraction rate.
+A. Semantic Information Extraction
+With given computation capacity, power control, rate alloca-
+tion, and beamforming design, problem (19) can be simplified
+as
+min
+S
+K
+�
+k=1
+�
+κy1k(Dk, Sk)f 2
+k + Z(Sk)pk
+rk + ak
++ κy2k(Dk, Sk)g2
+k
+�
++ K0p0
+a0
+(20)
+s.t. y1k(Dk, Sk)
+fk
++ max
+� Z(Sk)
+rk + ak
+, K0
+a0
+�
++ y2k(Dk, Sk)
+gk
+≤ T,
+∀k ∈ K,
+(20a)
+uk(Dk, Sk) ≥ Ak,
+∀k ∈ K,
+(20b)
+Sk ⊆ G(Dk)
+∀k ∈ K.
+(20c)
+Problem (20) is hard to solve because of two general diffi-
+culties. The first difficulty lies in the discrete value space of
+variable Sk, which leads to the discrete optimization problem
+and the complexity to find the optimal solution is usually
+extremely too high. The second difficulty is the implicit
+expressions of accuracy function uk(Dk, Sk) and computation
+functions f1k(Dk, Sk) and f2k(Dk, Sk).
+To handle the first difficulty, we introduce the new variable,
+extraction rate ρk, which is defined as
+ρk =
+Z(Sk)
+Z(G(Dk)).
+(21)
+Absolutely, the value of ρk lies in (0,1]. In the following, we
+use variable to replace Sk for the purpose of obtaining the
+insights about extraction rate.
+To handle the second difficulty, we first analyze the trend
+of accuracy and computation functions. For the accuracy
+function, the accuracy always increases with the extraction
+rate since more information can be used to recover the original
+data, as shown in Fig. 5. As a result, the minimum accuracy
+constraint (20c) can be equivalent to
+ρk ≥ Γk,
+(22)
+where
+Γk
+is
+the
+minimum
+extraction
+rate
+satisfying
+uk(Dk, Γk) = Ak. For computation function y1k(Dk, ρk)
+is the number of required CPU cycles for computing the
+
+SS
+DS
+DSS
+Ds
+DSS
+DS
+D6
+information with extraction rate ρk out of Dk, y1k(Dk, Sk)
+includes two parts. The first part is computing the directional
+probability graph, which can be modeled as a function only
+related to the size of Dk, i.e., y3k(Dk). The second part is
+selecting the information with extraction rate ρk out from the
+directional probability graph. Since ρk = 0 or ρk = 1, the
+selection scheme is straightforward, which leads to the lowest
+computation cycles. Hence, the computation of the second
+part first increases and then decreases with the extraction rate
+ρk. As an example, computation function y1k(Dk, ρk) can be
+expressed as
+y1k(Dk, ρk) = y3k(Dk) + Ck1(ρk − Ck2)Ck3,
+(23)
+where Ck1 > 0, Ck2 ∈ (0, 1), and Ck3 > 0 are constant
+parameters and theses parameters can be obtained through sim-
+ulations. For computation function y2k(Dk, ρk), the number
+of computation cycles decreases with ρk since more semantic
+information can be helpful in recovering the original infor-
+mation. As an example, the computation function y2k(Dk, ρk)
+can be expressed as
+y2k(Dk, ρk) = Ck4ρ−Ck5
+k
+,
+(24)
+where Ck4 > 0 and Ck5 > 0 are constant parameters through
+simulations.
+With the above variable substitution (21) and expressions
+(22)-(24), problem (20) can be reformulated as:
+min
+ρρρ
+K
+�
+k=1
+�
+κf 2
+k(y3k(Dk) + Ck1(ρk − Ck2)Ck3)
++ Z(G(Dk))pkρk
+rk + ak
++ κCk4ρ−Ck5
+k
+g2
+k
+�
++ K0p0
+a0
+(25)
+s.t. y3k(Dk) + Ck1(ρk − Ck2)Ck3
+fk
++ max
+�Z(G(Dk))ρk
+rk + ak
+, K0
+a0
+�
++ Ck4ρ−Ck5
+k
+gk
+≤ T,
+∀k ∈ K,
+(25a)
+Γk ≤ ρk ≤ 1,
+∀k ∈ K,
+(25b)
+where ρ = [ρ1, · · · , ρK]T . Since both objective function and
+feasible set are convex, problem (25) is a convex problem.
+Thus, we can apply the dual method to obtain the Karush-
+Kuhn-Tucker (KKT) point. To calculate the solution of prob-
+lem (25), we can obtain the following theorem.
+Theorem 1. The optimal solution of problem (25) is
+ρ∗
+k =
+�
+ρ∗
+k1(λ1k1)
+if ρ∗
+k1(λ11) ≥ K0(ak+rk)
+a0Z(G(Dk))
+ρ∗
+k2(λ1k2)
+if ρ∗
+k2(λ12) < K0(ak+rk)
+a0Z(G(Dk))
+,
+(26)
+where ρ∗
+k1(λ1k) and ρ∗
+k2(λ1k) are respectively the solutions to
+∂L1(ρ,λ1k)
+∂ρk
+= 0 in (30) and (32), λ1k1 and λ1k2 respectively
+satisfy
+y3k(Dk) + Ck1(ρ∗
+k1(λ1k1)|1
+Γk − Ck2)Ck3
+fk
++ Z(G(Dk))
+rk + ak
++ Ck4(ρ∗
+k1(λ1k1)|1
+Γk)−Ck5
+gk
+= T,
+(27)
+y3k(Dk) + Ck1(ρ∗
+k2(λ1k2)|1
+Γk − Ck2)Ck3
+fk
++ K0
+a0
++ Ck4(ρ∗
+k2(λ1k2)|1
+Γk)−Ck5
+gk
+= T,
+(28)
+with a|c
+b = min{max{a, b}, c}.
+Proof. Denoting λ11, · · · , λ1K > 0 as the Lagrange multi-
+plier variables associated with constraint (25a), we obtain the
+Lagrange function of problem (25) as
+L1(ρ, λ1) =
+K
+�
+k=1
+�
+κf 2
+k(y3k(Dk) + Ck1(ρk − Ck2)Ck3)
++ Z(G(Dk))pkρk
+rk + ak
++ κCk4ρ−Ck5
+k
+g2
+k
+�
++ K0p0
+a0
++
+K
+�
+k=1
+λ1k
+�y3k(Dk) + Ck1(ρk − Ck2)Ck3
+fk
++ max
+�Z(G(Dk))ρk
+rk + ak
+, K0
+a0
+�
++ Ck4ρ−Ck5
+k
+gk
+− T
+�
+,
+(29)
+where λ1 = [λ11, · · · , λ1K]T . The first derivative of (29)
+becomes
+∂L1(ρ, λ1)
+∂ρk
+= κf 2
+kCk1Ck3(ρk − Ck2)Ck3−1
++ Z(G(Dk))pk
+rk + ak
+− κCk4Ck5ρ−Ck5−1
+k
+g2
+k
++ λ1k
+�Ck1Ck3(ρk − Ck2)Ck3−1
+fk
++ Z(G(Dk))
+rk + ak
+− Ck4Ck5ρ−Ck5−1
+k
+gk
+�
+(30)
+for
+ρk ≥ K0(ak + rk)
+a0Z(G(Dk)) ,
+(31)
+and
+∂L1(ρ, λ1)
+∂ρk
+= κf 2
+kCk1Ck3(ρk − Ck2)Ck3−1
++ Z(G(Dk))pk
+rk + ak
+− κCk4Ck5ρ−Ck5−1
+k
+g2
+k
++ λ1k
+�Ck1Ck3(ρk − Ck2)Ck3−1
+fk
+− Ck4Ck5ρ−Ck5−1
+k
+gk
+�
+(32)
+for
+ρk < K0(ak + rk)
+a0Z(G(Dk)) ,
+(33)
+Denote the solution of ∂L1(ρ,λ1)
+∂ρk
+= 0 to equations (30) and
+(32) by ρ∗
+k1(λ1) and ρ∗
+k2(λ1k), respectively. Note that the
+left hand sides of (30) and (32) are monotonically increasing
+with respect to ρk, solutions ρ∗
+k1(λ1k) and ρ∗
+k2(λ1k) can be
+obtained via the bisection method. Considering constraints
+(25b), (31), and (33), the Lagrange multiplier should meet
+the KKT condition, i.e., the optimal solution of problem can
+be presented in (26).
+
+7
+B. Optimal Computation Capacity
+With given semantic information extraction, power control,
+rate allocation, and beamforming design, problem (19) can be
+simplified as
+min
+f,g
+K
+�
+k=1
+�
+κy1k(Dk, Sk)f 2
+k + Z(Sk)pk
+rk + ak
++ κy2k(Dk, Sk)g2
+k
+�
++ K0p0
+a0
+(34)
+s.t. y1k(Dk, Sk)
+fk
++ max
+� Z(Sk)
+rk + ak
+, K0
+a0
+�
++ y2k(Dk, Sk)
+gk
+≤ T,
+∀k ∈ K,
+(34a)
+K
+�
+k=1
+fk ≤ F max
+(34b)
+fk ≥ 0,
+∀k,
+(34c)
+0 ≤ gk ≤ gmax
+k
+,
+∀k ∈ K.
+(34d)
+The Language function of problem (34) can be given by
+L2(f, g, λ2, λ3) =
+K
+�
+k=1
+�
+κy1k(Dk, Sk)f 2
+k + Z(Sk)pk
+rk + ak
++ κy2k(Dk, Sk)g2
+k
+�
++ K0p0
+a0
++
+K
+�
+k=1
+λ2k
+�y1k(Dk, Sk)
+fk
++ max
+� Z(Sk)
+rk + ak
+, K0
+a0
+�
++ y2k(Dk, Sk)
+gk
+− T
+�
++ λ3
+� K
+�
+k=1
+fk − F max
+�
+,
+(35)
+where λ2 = [λ21, · · · , λ2K]T is the Language multiplier
+associated with constraint (34a) and λ3 > 0 is the Language
+multiplier associated with constraint (34b). The first derivative
+of (35) becomes
+∂L2(f, g, λ2, λ3)
+∂fk
+=2κy1k(Dk, Sk)gk − λ2ky1k(Dk, Sk)
+g2
+k
++ λ3
+(36)
+∂L2(f, g, λ2, λ3)
+∂gk
+=2κy2k(Dk, Sk)fk − λ2ky2k(Dk, Sk)
+f 2
+k
+(37)
+Setting ∂L2(f,g,λ2,λ3)
+∂fk
+= 0 and ∂L2(f,g,λ2,λ3)
+∂gk
+= 0 yields
+2κy1k(Dk, Sk)f 3
+k + λ3f 2
+k − λ2ky1k(Dk, Sk) = 0,
+(38)
+gk =
+�λ2ky2k(Dk, Sk)
+2κy2k(Dk, Sk)
+� 1
+3
+.
+(39)
+The value of fk can be obtained via solving the cubic function
+in (38). Having obtained the value of computation capacity fk
+and gk, the value of Language multiplier can be updated via
+the gradient method. In the t-th iteration, the value of λ2k and
+λ3 are updated by
+λ2k(t) =
+�
+λ2k(t − 1) − υ(t)
+�y1k(Dk, Sk)
+fk
++
+max
+� Z(Sk)
+rk + ak
+, K0
+a0
+�
++ y2k(Dk, Sk)
+gk
+− T
+��+
+,
+(40)
+and
+λ3(t) =
+�
+λ3(t − 1) − υ(t)
+� K
+�
+k=1
+fk − F max
+��+
+,
+(41)
+where [a]+ = max a, 0 and υ(t) > 0 is the dynamic step size.
+Through iteratively updating (fk, gk) and (λ2k, λ3), the overall
+procedure yields the global optimal solution of problem (34).
+C. Joint Power Control, Rate Allocation, and Beamforming
+Design
+With given semantic information extraction and computa-
+tion capacity, problem (19) can be simplified as
+min
+p,a,w
+K
+�
+k=1
+�
+κy1k(Dk, Sk)f 2
+k + Z(Sk)pk
+rk + ak
++ κy2k(Dk, Sk)g2
+k
+�
++ K0p0
+a0
+,
+(42)
+s.t. y1k(Dk, Sk)
+fk
++ max
+� Z(Sk)
+rk + ak
+, K0
+a0
+�
++ y2k(Dk, Sk)
+gk
+≤ T,
+∀k ∈ K,
+(42a)
+a0 +
+K
+�
+k=1
+ak ≤ ck,
+∀k ∈ K,
+(42b)
+K
+�
+k=0
+p0 ≤ P max
+(42c)
+a0, ak, pk ≥ 0,
+∀k,
+(42d)
+∥wk∥ = 1,
+∀k ∈ K ∪ {0},
+(42e)
+Problem (42) is nonconvex owing to the nonconvex objec-
+tive function and constraints (42a), (42b) and (42e). To handle
+the nonconvexity of the objective function, we introduce new
+variable rk and use variable p2
+k to replace power pk. Thus,
+
+8
+problem (42) can be equivalently transformed to
+min
+p,a,r,w
+K
+�
+k=1
+�
+κy1k(Dk, Sk)f 2
+k + Z(Sk)p2
+k
+rk + ak
++ κy2k(Dk, Sk)g2
+k
+�
++ K0p2
+0
+a0
+,
+(43)
+s.t. y1k(Dk, Sk)
+fk
++ max
+� Z(Sk)
+rk + ak
+, K0
+a0
+�
++ y2k(Dk, Sk)
+gk
+≤ T,
+∀k ∈ K,
+(43a)
+a0 +
+K
+�
+k=1
+ak ≤ B log2
+�
+1 +
+p2
+0|hH
+k w0|2
+�K
+j=1 p2
+j|hH
+k wj|2 + σ2
+�
+,
+∀k ∈ K,
+(43b)
+rk ≤ B log2
+�
+1 +
+p2
+k|hH
+k wk|2
+�K
+j=1,j̸=k p2
+j|hH
+k wj|2 + σ2
+�
+,
+∀k ∈ K,
+(43c)
+a0, ak, pk ≥ 0,
+∀k,
+(43d)
+∥wk∥ ≤ 1,
+∀k ∈ K ∪ {0},
+(43e)
+where r = [r0, r1, · · · , rK]T , the objective function is convex,
+and constraint (43e) is replaced by the inequality without loss
+of generality. In problem (43), we only need to deal with
+the nonconvexity of constraints (43b) and (43c) . Through
+introducing slacking variables γk and ηk, problem (43) can
+be reformulated as:
+min
+p,a,r,w,γ,η
+K
+�
+k=1
+�
+κy1k(Dk, Sk)f 2
+k + Z(Sk)p2
+k
+rk + ak
++ κy2k(Dk, Sk)g2
+k
+�
++ K0p2
+0
+a0
+,
+(44)
+s.t. y1k(Dk, Sk)
+fk
++ max
+� Z(Sk)
+rk + ak
+, K0
+a0
+�
++ y2k(Dk, Sk)
+gk
+≤ T,
+∀k ∈ K,
+(44a)
+a0 +
+K
+�
+k=1
+ak ≤ B log2 (1 + ηk) ,
+∀k ∈ K,
+(44b)
+rk ≤ B log2 (1 + γk) ,
+∀k ∈ K,
+(44c)
+a0, ak, pk ≥ 0,
+∀k,
+(44d)
+∥wk∥ ≤ 1,
+∀k ∈ K ∪ {0},
+(44e)
+p2
+k|hH
+k wk|2
+�K
+j=1,j̸=k p2
+j|hH
+k wj|2 + σ2 ≥ γk,
+∀k ∈ K,
+(44f)
+p2
+0|hH
+k w0|2
+�K
+j=1 p2
+j|hH
+k wj|2 + σ2 ≥ ηk,
+∀k ∈ K, (44g)
+where γ = [γ1, · · · , γK]T and η = [η1, · · · , ηK]T . In problem
+(44), the objective function is transformed into convex. Be-
+cause of nonconex constraints (44f) and (44g), problem (44)
+is nonconvex. In the following, we utilize the SCA method to
+handle these two nonconvex constraints.
+For constraint (44f), it can be equivalent to
+p2
+k|hH
+k wk|2 ≥ γkαk,
+(45)
+K
+�
+j=1,j̸=k
+p2
+j|hH
+k wj|2 + σ2 ≤ αk,
+(46)
+where αk is a nonnegative slack variable. In (45), we can al-
+ways choose the term hH
+k wk as a real value through changing
+the phase of beamforming wk. Thus, constraint (45) can be
+rewritten as
+R(hH
+k wk) ≥
+√γkαk
+pk
+,
+(47)
+where the left hand side is convex now. Through using the
+first-order Taylor series to replace the right hand side of (47),
+constraint (47) can be approximated by
+R(hH
+k wk) ≥
+�
+γ(n)
+k α(n)
+k
+p(n)
+k
++
+�
+γ(n)
+k
+2p(n)
+k
+�
+α(n)
+k
+(αk − α(n)
+k )
++
+�
+α(n)
+k
+2p(n)
+k
+�
+γ(n)
+k
+(γk − γ(n)
+k ) −
+�
+γ(n)
+k
+α(n)
+k
+(p(n)
+k )2
+(pk − p(n)
+k ), (48)
+where the superscript (n) means the value of the variable in
+the n-th iteration. Moreover, (46) can be reformulated as
+K
+�
+j=1,j̸=k
+1
+4((p2
+j + |hH
+k wj|2)2 − (p2
+j − |hH
+k wj|2)2)
+=
+K
+�
+j=1,j̸=k
+p2
+j|hH
+k wj|2 + σ2 ≤ αk.
+(49)
+Through replacing the left hand side of (49) with its first-order
+Taylor approximation, we can obtain
+K
+�
+j=1,j̸=k
+1
+4
+�
+((p(n)
+j
+)2 + |hH
+k w(n)
+j
+|2)2 + 4((p(n)
+j
+)2
++ |hH
+k w(n)
+j
+|2)p(n)
+j
+(pj − p(n)
+j
+) − (p2
+j − |hH
+k wj|2)2
++ 4((p(n)
+j
+)2 + |hH
+k w(n)
+j
+|2)(R(hH
+k w(n)
+j
+hH
+k wj) − |hH
+k w(n)
+j
+|2)
+�
++ σ2 ≤ αk.
+(50)
+Similarly, we can introduce slack variable βk and constraint
+(44g) can be rewritten as:
+1
+4((p2
+0 + |hH
+k w0|2)2 − (p2
+0 − |hH
+k w0|2)2)
+=p2
+0|hH
+k w0|2 ≥ βkηk = 1
+4((βk + ηk)2 − (βk − ηk)2), (51)
+K
+�
+j=1
+p2
+j|hH
+k wj|2 + σ2 ≤ βk.
+(52)
+Note that we cannot make hH
+k w0 as real values for all k
+through changing the phase of w0. To handle the nonconvexity
+of (51), we use first-order Taylor approximation on both
+sides of (51), which is different from the method in [52].
+
+9
+Considering the first-order Taylor approximation on both sides,
+(51) can be transformed to
+((p(n)
+0 )2 + |hH
+k w(n)
+0 |2)2 + 4((p(n)
+0 )2
++|hH
+k w(n)
+0
+|2)p(n)
+0 (p0 − p(n)
+0 ) − (p2
+0 − |hH
+k w0|2)2
++4((p(n)
+0 )2 + |hH
+k w(n)
+0
+|2)(R(hH
+k w(n)
+0 hH
+k w0) − |hH
+k w(n)
+0 |2)
+≥(βk + ηk)2 − (β(n)
+k
+− η(n)
+k )(βk − ηk) + (β(n)
+k
+− η(n)
+k )2,
+(53)
+For constraint (52), we can use the similar method to handle
+the nonconvexity of (46). Thus, (52) can be approximated by
+K
+�
+j=1
+1
+4
+�
+((p(n)
+j
+)2 + |hH
+k w(n)
+j
+|2)2 + 4((p(n)
+j
+)2
++|hH
+k w(n)
+j
+|2)p(n)
+j
+(pj − p(n)
+j
+) − (p2
+j − |hH
+k wj|2)2
++4((p(n)
+j
+)2 + |hH
+k w(n)
+j
+|2)(R(hH
+k w(n)
+j
+hH
+k wj) − |hH
+k w(n)
+j
+|2)
+�
++ σ2 ≤ βk.
+(54)
+With the above approximations, we can approximate the
+nonconvex constraints (44f) and (44g) with the corresponding
+convex approximation terms. Thus, the original problem (44)
+can be approximated by the following convex one:
+min
+p,a,r,w,γ,α,η,β
+K
+�
+k=1
+�
+κy1k(Dk, Sk)f 2
+k + Z(Sk)p2
+k
+rk + ak
++ κy2k(Dk, Sk)g2
+k
+�
++ K0p2
+0
+a0
+,
+(55)
+s.t. (44a) − (44e), (48), (50), (53), (54),
+(55a)
+αk ≥ 0, βk ≥ 0,
+∀k ∈ K,
+(55b)
+where α
+= [α, · · · , α]T and β
+=
+[β1, · · · , βK]T . The
+convex problem (55) can be solved by the existing convex
+optimization toolbox.
+D. Algorithm Analysis
+The overall joint communication and computation resource
+allocation for SWC with RSMA is presented in Algorithm 1.
+According to Algorithm 1, the complexity of solving problem
+(19) lies in solving three subproblems at each iteration. For
+the semantic information extraction subproblem, the optimal
+solution is calculated by (26) in Theorem 1 with complexity
+O(K log2(1/ǫ1)), where O(log2(1/ǫ1)) is the complexity of
+solving (27) and (28) with the bisection method of accuracy
+ǫ1. For the computation capacity subproblem, the complexity
+is O(N1K), where N1 denotes the number of iterations of
+using the dual method for solving the computation capacity
+subproblem. For the joint power control, rate allocation, and
+beamforming design subproblem, the complexity lies in solv-
+ing the approximated convex problem (55). The complexity
+of obtaining the solution of problem (55) is O(M 2
+1 M2) [53],
+where M1 = (N+7)K+N+2 is the total number of variables
+and M2 = 13K+1 is the total number of constraints. The total
+complexity of solving the joint power control, rate allocation,
+and beamforming design subproblem is O(N2N 2K3), where
+N2 is the number of iterations for the SCA method. As a
+Algorithm 1 Joint Communication and Computation Resource
+Allocation for SWC with RSMA
+1: Initialize S(0), f (0), g(0), p(0), a(0), w(0). Set iteration
+number n = 1.
+2: repeat
+3:
+With
+given
+f (n−1), g(n−1), p(n−1), a(n−1), w(n−1),
+solve the semantic information extraction subproblem
+and obtain the solution S(n).
+4:
+With given S(n), p(n−1), a(n−1), w(n−1), solve the
+computation capacity subproblem and obtain the solu-
+tion f (n), g(n).
+5:
+With given S(n), f (n), g(n), solve the joint power con-
+trol, rate allocation, and beamforming design subprob-
+lem, of which the solution is p(n), a(n), w(n).
+6:
+Set n = n + 1.
+7: until the objective value (19) converges.
+TABLE I
+MAIN SYSTEM PARAMETERS
+Parameter
+Value
+Bandwidth of the BS B
+20 MHz
+Power spectral density of the noise power
+-174 dBm/Hz
+Maximum transmit power P max
+30 dBm
+Effective switched capacitance κ
+10−28
+Number of users K
+5
+result, the total complexity of the proposed Algorithm 1 is
+O(N3K log2(1/ǫ1) + N1N3K + N2N3N 2K3), where N3 is
+the number of outer iterations of Algorithm 1.
+IV. SIMULATION RESULTS
+In the simulations, there are K = 5 users in the considered
+area. For the pathloss model between each user and the BS,
+we set 128.1 + 37.6 log10 d (d is in km) [54] and the standard
+deviation of shadow fading is 4 dB [52]. Furthermore, the
+total bandwidth of the system is B = 20 MHz and the
+power spectral density of the noise power is −174 dBm/Hz.
+Unless specified otherwise, we set maximum transmit power
+P max = 30 dBm, the effective switched capacitance in local
+computation is κ = 10−28, maximum local computation
+capacity gmax
+1
+= · · · = gmax
+K
+= 2 GHz. For the considered
+semantic information task, we consider the same parameters
+as in [55]. The main system parameters are summarized in
+Table I.
+The proposed joint communication and computation re-
+source allocation for SWC with RSMA is labeled as ‘RSMA’.
+To compare the results of the proposed scheme, we con-
+sider the conventional orthogonal multiple access, frequency
+division multiple access (FDMA) [56], which is labeled as
+‘FDMA’, the total energy minimization problem for NOMA
+[57], which is labeled as ‘NOMA’. To better show the perfor-
+mance of multiple antenna scheme, we consider the SDMA
+system as in [18].
+Fig. 6 illustrates that the total communication and com-
+putation energy changes as the maximum transmit power of
+each user varies. According to this figure, the EXH-RSMA
+scheme stands for the exhaustive search method, which can
+
+10
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+Maximum transmit power (dBm)
+180
+200
+220
+240
+260
+280
+300
+Total communication and computation energy
+RSMA
+NOMA
+FDMA
+EXH-RSMA
+SDMA
+Fig. 6. Total communication and computation energy vs. maximum transmit
+power.
+yield a near globally optimal solution through running the
+proposed algorithm with 1000 initial solutions. It can be
+shown from this figure that the total energy decreases with
+the maximum transmit power of the BS. This is due to the
+fact that large transmit power can lead to low transmit time,
+which allows more time for computation and yields low total
+energy consumption. It is observed that the proposed RSMA
+outperforms FDMA, NOMA, since RSMA can achieve higher
+spectral efficiency than FDMA and NOMA. Compared to
+SDMA, RSMA can still achieve better energy consumption,
+in particular the maximum transmit power is high. The reason
+is that SDMA is more likely to serve the users with higher
+channel gains, while the users with poor channel gains tend
+to have long transmit time and high computation power is
+needed for task computation, thus leading to higher total
+energy consumption than RSMA. It can be also found that
+the proposed RSMA achieves near performance as the EXH-
+RSMA, which indicates the effectiveness of the proposed
+scheme.
+Fig. 7 shows the total energy versus bandwidth of the
+system. Based on this figure, the total communication and
+computation energy decreases as the bandwidth of the system
+increases for all schemes. This is because high bandwidth
+decreases the transmit time between users and the BS, which
+allows long computation time and consequently reduces the
+local computation energy consumption.
+Fig. 8 illustrates the trend of total communication and
+computation energy with the transmit data size of each user. It
+is observed that the total energy increases as the data size for
+all schemes. This is due to the fact that more information needs
+to be transmitted, thus increasing the transmit and computation
+power. It can be found that the growing speed of total energy
+od the proposed RSMA is slower than that of NOMA and
+FDMA, which shows the robustness of the RSMA.
+To show how the computation capacity affects the system
+performance, Fig. 9 presents the total communication and
+computation energy versus the maximum computation capac-
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+Bandwidth (MHz)
+95
+100
+105
+110
+115
+120
+125
+130
+Total communication and computation energy
+RSMA
+NOMA
+FDMA
+Fig. 7.
+Total communication and computation energy vs. bandwidth of the
+system.
+50
+100
+150
+200
+250
+300
+350
+400
+450
+500
+Data size (Kbits)
+110
+120
+130
+140
+150
+160
+170
+Total communication and computation energy
+RSMA
+NOMA
+FDMA
+Fig. 8. Total communication and computation energy vs. transmit data size
+of each user.
+ity of each user. According to this figure, the total energy first
+decreases rapidly and then the total energy tends to approach
+a fixed value. The reason lies in that for small computation
+capacity region, the increase of maximum computation ca-
+pacity can greatly decrease the computation time and more
+time can be used for transmission, thus reducing the transmit
+power and total energy. For high computation capacity region,
+each user has chosen its optimal computation capacity and the
+increase of maximum computation capacity does not affect the
+computation capacity allocation result, thus leading to stable
+energy consumption.
+V. CONCLUSIONS
+In this paper, the problem of wireless resource alloca-
+tion and semantic information extraction for energy efficient
+semantic communications over wireless networks with rate
+splitting is investigated. In the considered model, the BS
+first extracts the semantic information from its large-scale
+
+11
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Maximal computation capacity (GHz)
+100
+200
+300
+400
+500
+600
+700
+800
+Total communication and computation energy
+RSMA
+NOMA
+FDMA
+Fig. 9. Total communication and computation energy vs. maximum compu-
+tation capacity of each user.
+data, and then transmits the small-sized semantic information
+to each user which recovers the original data based on the
+local common knowledge. In the downlink transmission, the
+rate splitting scheme is adopted, while the private small-sized
+semantic information is transmitted through private message
+and the common knowledge is transmitted through common
+message. Due to limited wireless resource, both computational
+energy and transmission energy need to be considered. This
+joint computation and communication problem is considered
+as an optimization problem whose goal is to minimize the
+total energy consumption of the network under both task
+completion and semantic accuracy constraints. An iterative
+algorithm is presented to solve this problem, where at each
+step, the optimal solutions for semantic information extraction
+ratio and computation frequency are derived. Numerical results
+show the effectiveness of the proposed algorithm.
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+

L9FQT4oBgHgl3EQfUzaG/content/tmp_files/2301.13298v1.pdf.txt

@@ -0,0 +1,2125 @@
+LONGEVAL: Guidelines for Human Evaluation of
+Faithfulness in Long-form Summarization
+Kalpesh Krishna♠ ∗
+Erin Bransom♦
+Bailey Kuehl♦
+Mohit Iyyer♠
+Pradeep Dasigi♦
+Arman Cohan♦♥
+Kyle Lo♦
+♠University of Massachusetts Amherst, ♦Allen Institute for AI, ♥Yale University
+{kalpesh,miyyer}@cs.umass.edu
+{erinbransom,baileyk,pradeepd,armanc,kylel}@allenai.org
+Abstract
+While human evaluation remains best prac-
+tice for accurately judging the faithfulness of
+automatically-generated summaries, few solu-
+tions exist to address the increased difficulty
+and workload when evaluating long-form sum-
+maries.
+Through a survey of 162 papers
+on long-form summarization, we first shed
+light on current human evaluation practices
+surrounding long-form summaries.
+We find
+that 73% of these papers do not perform any
+human evaluation on model-generated sum-
+maries, while other works face new difficul-
+ties that manifest when dealing with long
+documents (e.g., low inter-annotator agree-
+ment). Motivated by our survey, we present
+LONGEVAL, a set of guidelines for human
+evaluation of faithfulness in long-form sum-
+maries that addresses the following challenges:
+(1) How can we achieve high inter-annotator
+agreement on faithfulness scores?
+(2) How
+can we minimize annotator workload while
+maintaining accurate faithfulness scores? and
+(3) Do humans benefit from automated align-
+ment between summary and source snippets?
+We deploy LONGEVAL in annotation studies
+on two long-form summarization datasets in
+different domains (SQuALITY and PubMed),
+and we find that switching to a finer granu-
+larity of judgment (e.g., clause-level) reduces
+inter-annotator variance in faithfulness scores
+(e.g., std-dev from 18.5 to 6.8).
+We also
+show that scores from a partial annotation
+of fine-grained units highly correlates with
+scores from a full annotation workload (0.89
+Kendall’s τ using 50% judgments). We release
+our human judgments, annotation templates,
+and our software for future research.1
+1
+Introduction
+Human judgments are considered the gold
+standard for evaluating model-generated sum-
+1https://github.com/martiansideofthemoon/
+longeval-summarization
+*Work done in an AI2 internship, author contributions here.
+maries (Kryscinski et al., 2019; Fabbri et al., 2021)
+and generated text more broadly (Celikyilmaz et al.,
+2020). Unfortunately, human evaluation tends to
+be labor-intensive, expensive to scale, and diffi-
+cult to design. This is problematic as a large num-
+ber of judged examples is needed to draw statisti-
+cally significant conclusions about system perfor-
+mances (Wei and Jia, 2021) or correlations between
+human judgments and automatic metrics (Deutsch
+et al., 2021). Human evaluation is especially chal-
+lenging when long sequences of generated text
+need to be evaluated, due to the inherent subjectiv-
+ity in the task (Karpinska et al., 2021; Clark et al.,
+2021; Krishna et al., 2021; Goyal et al., 2022).
+To better understand the challenges of human
+evaluation on long-form summaries (150 words
+or longer), we first conduct a comprehensive sur-
+vey of 162 publications and preprints on long-form
+summarization (Section 2). We find that 119 pa-
+pers (73%) do not perform human evaluation on
+long-form summaries, while the remaining papers
+deviate significantly from suggested best practices
+for reproducibility (Gehrmann et al., 2022). Cur-
+rent human evaluation setups lack standardization
+in their design decisions (such as annotation gran-
+ularity), some of which can significantly impact
+inter-annotator agreement (Section 3.1). Finally,
+20 papers explicitly mention human evaluation is
+expensive, difficult, and time-consuming due to the
+long length of summaries and source documents.
+To move towards a more consistent and efficient
+human evaluation, we present LONGEVAL, a set of
+guidelines for human evaluation of faithfulness in
+long-form summarization (Section 3). We empiri-
+cally evaluate LONGEVAL using human annotation
+studies on two long-form summarization datasets:
+SQuALITY (Wang et al., 2022) and PubMed (Co-
+han et al., 2018). We provide an overview of our
+main research questions and findings in Figure 1
+and enumerate them here:
+arXiv:2301.13298v1  [cs.CL]  30 Jan 2023
+
+…. He recognized her as old Hazeltyne 's daughter Harriet , no doubt 
+come to see justice done . She did n't have the hothouse - flower look 
+Asa would have expected in a girl whose father owned the most 
+valuable of the planetary franchises . She was not afraid to meet his 
+eye , the eye of a judicially certified criminal . There was , perhaps , a 
+crease of puzzlement in her brow , as if she had thought crimes were 
+committed by shriveled , rat - faced types , and not by young biological 
+engineers who still affected crewcuts . Tom Dorr , Hazeltyne 's general 
+manager , was her escort . Asa felt certain , without proof , that Dorr 
+was the man who had framed him for the charge of grand theft by 
+secreting a fresh Slider egg in his laboratory . The older man stared at 
+Asa coldly as he was led out of the courtroom and down the corridor 
+back to jail . Jumpy , Asa 's cellmate , took one look at his face as he 
+was put back behind bars . " Guilty , " Jumpy said ….Asa took four 
+steps to the far wall of the cell , stood there briefly with his head bent 
+and turned to face Jumpy . " Nope , " Asa said softly . " I 'm going into 
+a conversion tank . I 'm going to be a muck man , Jumpy . I 'm going 
+out to Jordan 's Planet and hunt Slider eggs . " " Smuggling ? It wo n't 
+work . " Asa did n't answer . The Hazeltyne company had gone after 
+him because he had …
+Asa Graybar is a biological engineer 
+who studies keeping Slider eggs 
+alive and he is accused of a crime at 
+the opening of the story . He thinks 
+he was framed by Tom Dorr , 
+Hazeltyne ’s general manager . He 
+was offered one year as a “ 
+changeling ” on another planet or 5 
+years in rehabilitation on Earth . He 
+elects to do the one year , and 
+thinks that he will get into smuggling 
+Slider eggs on Jordan ’s planet ….. 
+Source document (4.8K words)
+Summary (270 words)
+Alignment 
+Is this span fully supported 
+by the source?
+Yes
+No
+FINE-grained
+Q3: Is it helpful to automatically align summary 
+units with the long source document?
+Q2: Can annotator workload 
+be reduced by annotating just a 
+fraction of the long summary?
+Q1: Can inter annotator 
+agreement be improved with 
+fine-grained annotations?
+COARSE-grained
+How well is the summary 
+supported by the source? 
+Figure 1: Overview of research questions considered in LONGEVAL. Example summary taken from SQuALITY.
+RQ1: Can inter-annotator agreement be improved
+while evaluating faithfulness of long-form sum-
+maries via fine-grained annotations?
+Finding: Annotating faithfulness of individual
+summary clauses and aggregating them leads to sig-
+nificantly higher inter-annotator agreement, com-
+pared to the dominant paradigm of evaluating
+whole summaries at once via Likert ratings (std-dev
+18.5 to 6.8 on SQuALITY).
+RQ2:
+Can we reduce annotator workload by
+partially annotating a long summary while
+maintaining accurate faithfulness scores?
+Finding: Despite annotating a fraction of summary
+clauses, faithfulness scores under a reduced work-
+load maintain high correlation with those from a
+full workload (0.89 Kendall’s τ at 50% workload).
+RQ3:
+Do humans benefit from automatically
+aligning summary units to relevant sentences in
+the source document?
+Finding: Unlike suggestions in prior work on
+short-form summarization (Hardy et al., 2019;
+Kryscinski et al., 2020), aligning parts of the sum-
+mary to source document is only useful when the
+summary is highly extractive or mostly correct.
+Overall, our contributions are:
+(1) a 162-paper survey of current human evaluation
+practices in long-form summarization;
+(2) LONGEVAL, a set of three guidelines for evalu-
+ating faithfulness in long-form summarization;
+(3) an empirical validation of LONGEVAL guide-
+lines on two long-form summarization datasets in
+different domains (SQuALITY and PubMed);
+(4) A dataset with 3-way fine-grained human faith-
+fulness judgments for 120 SQuALITY & PubMed
+summaries annotated using LONGEVAL which can
+be used for benchmarking automatic metrics.
+We open-source our human evaluation data, an-
+notation interface, and code for future research.1
+2
+Survey of human evaluation practices
+Before discussing LONGEVAL, we first attempt to
+understand current human evaluation practices in
+long-form summarization through a comprehensive
+survey of 162 papers. Our survey reveals several
+concerning trends: absence of human evaluation,
+non-reproducible experimental setups, lack of stan-
+dardization, and complaints of long summaries be-
+ing challenging and expensive to evaluate. These
+results show an urgent need to develop more effi-
+cient and standardized human evaluation protocols.
+Selection of papers: We consider existing summa-
+rization datasets with an average summary length
+of at least 150 words, which includes several pop-
+ular datasets like arXiv (Cohan et al., 2018), Bill-
+Sum (Kornilova and Eidelman, 2019) and Multi-
+News (Fabbri et al., 2019); see Table 1 for a full list.
+For our survey, we select all papers that evaluated
+summarization models using at least one of these
+datasets.2 All of these papers were published be-
+tween June 2018 and September 2022, after the first
+long-form summarization datasets were released
+(PubMed / arXiv). Most of the 162 surveyed papers
+2We exclude five papers which used long-form summariza-
+tion data for pre-training only, like Wei et al. (2022).
+
+X0
+1
+2
+3
+4
+5were published in major NLP/ML venues, but we
+also include newer preprints from 2022.
+Long-form summaries are rarely evaluated by
+humans. We find that 101 out of 162 papers (62%)
+do not perform any human evaluation. 17 papers
+(11%) only perform human evaluation on short
+summaries (datasets like XSUM, Narayan et al.,
+2018), for which human evaluation is much easier.
+Human evaluation studies of long-form sum-
+maries are not reproducible. We further analyze
+the 44 papers performing human evaluation of long-
+form summaries to observe how often they follow
+reproducible practices from Gehrmann et al. (2022).
+Overall, we find that most studies do not follow
+these guidelines. Only 2 of the 44 papers release
+their raw human annotation data for further analy-
+sis. Only 9 papers provide details of their annotator
+instructions or interface, and just 12 papers perform
+any kind of statistical analysis, despite most papers
+annotating less than 50 summaries. While 33 pa-
+pers report using multiple annotators per summary,
+only 12 report inter-annotator agreement. Finally,
+just 14 papers conduct human evaluation on more
+than one dataset (more statistics in Appendix C).
+Existing human evaluation setups lack stan-
+dardization. In Table 2, we catalog the wide spec-
+trum of human evaluation setups in the surveyed pa-
+pers. 37 papers collect judgments of the full-length
+summary at once (“COARSE-grained”), while 6 pa-
+pers collect judgments at a finer granularity such as
+sentences or entities (“FINE-grained”). Even within
+a granularity, setups differ: Likert-scale (24 pa-
+pers), A/B testing (13 papers), binary per-sentence
+labels (4 papers) are the dominant protocols. In
+Section 3.1, we will see that this design decision
+is critical since COARSE annotations have much
+lower inter-annotator agreement than FINE.3
+Human evaluation of long-form summaries is
+challenging and expensive. Several of the sur-
+veyed papers discuss challenges in human evalua-
+tion of long-form summaries. 13 papers mention
+that expert annotators are necessary for human eval-
+uation of long-form summaries, especially in tech-
+nical domains like PubMed. 20 papers report that
+human evaluation of long-form summarization was
+3Besides granularity, we also observe a large spectrum of
+annotator qualifications in our survey, ranging from MTurkers
+to expert graduates (Appendix C). Since non-experts are
+known to be unsuitable for this task (Gillick and Liu, 2010;
+Fabbri et al., 2021), we use experts in our work (Appendix B).
+Dataset
+|source|
+|summary|
+papers
+(words)
+(words)
+PubMed (2018)
+3092
+205
+59
+arXiv (2018)
+5906
+163
+55
+BillSum (2019)
+1284
+174
+19
+MultiNews (2019)
+2103
+263
+54
+GovReport (2021)
+7551
+547
+16
+BookSum (2021)
+5102
+505
+4
+SummScreen (2022)
+6965
+227
+11
+SQuALITY (2022)
+5194
+227
+1
+Table 1: List of long-form summarization datasets con-
+sidered in our survey along with average source docu-
+ment and summary lengths. Each dataset considered
+has at least 150 word summaries on average.
+Type of human evaluation
+# papers
+% papers
+None
+101
+62%
+Short-form summaries only
+17
+11%
+Likert-scale COARSE-grained
+24
+15%
+A/B testing COARSE-grained
+13
+8%
+Extrinsic evaluation
+1
+1%
+Binary per sentence FINE-grained
+4
+2%
+QA-based FINE-grained
+2
+1%
+Table 2: Human evaluation setup in 162 summarization
+papers that evaluate long-form summaries. 73% of the
+papers do not evaluate long-form summaries with hu-
+mans, while others vary significantly in their setups.
+time-consuming, challenging, and expensive, pri-
+marily due to the long length of the summary and
+source document. To tackle the issue of high an-
+notator workload, we propose a partial annotation
+method in Section 3.2 and report high correlation
+to a full workload. Additionally, in Section 3.3
+we investigate the usefulness of highlighting sen-
+tences to help annotators navigate the long source
+document. While this has been advocated for in
+short-form summary evaluation (Hardy et al., 2019;
+Kryscinski et al., 2020) and used in 3 surveyed
+long-form papers, we find that it is only helpful
+when summaries are mostly correct and extractive.
+3
+The LONGEVAL guidelines for
+faithfulness human evaluation
+In Section 2, we report several concerning issues
+with current human evaluation practices in long-
+form summarization. To move towards more ef-
+ficient, reproducible and standardized protocols
+for human evaluation, we develop the LONGEVAL
+guidelines (Section 3.1-3.3, see Figure 1 for an
+overview). We focus on human evaluation of faith-
+fulness, which Wang et al. (2022) define as:
+
+“Checking the factual errors in the summary,
+where a factual error is a statement that con-
+tradicts the source document, or is not directly
+stated, heavily implied, or logically entailed by
+the source document”
+We conduct human annotation studies to empiri-
+cally motivate LONGEVAL. Our experiments are
+on two long-form summarization datasets span-
+ning diverse domains and levels of abstractiveness:
+(1) SQuALITY (Wang et al., 2022) is a summa-
+rization dataset in the literary domain (avg. sum-
+mary length of 227 words) where summaries de-
+scribe the plots of English science fiction sto-
+ries. SQuALITY is highly abstractive: on average
+just 16% of bigrams in the summary are present
+in the source document. We closely follow the
+human evaluation setup in Wang et al. (2022),
+and use BART (Lewis et al., 2020) and BART-
+DPR (Karpukhin et al., 2020) as our summarization
+models along with human-written summaries.
+(2) PubMed (Cohan et al., 2018) is a summariza-
+tion dataset in the scientific domain (avg. summary
+length of 205 words) that pairs English biomedical
+articles from PubMed4 with their abstracts as sum-
+maries. Compared to SQuALITY, PubMed is more
+extractive: 54% of summary bigrams are present in
+the source. We use BigBird-PEGASUS-large (Za-
+heer et al., 2020) and LongT5-large (Guo et al.,
+2022) as our summarization models,5 along with
+human written summaries. By default, LongT5 /
+BigBird were highly extractive compared to human-
+written PubMed summaries (87% / 74% vs 54%
+bigram overlap with source). Hence, for half the
+generations we block 6-grams from being copied
+from the source,6 reducing extractiveness to ∼54%.
+We call this setting “PubMed-ngram-block”.
+3.1
+RQ1: Does inter-annotator agreement
+improve using fine-grained annotations?
+In Section 2, we found that the dominant paradigm
+in literature (37 out of 44 papers) is to evaluate
+the whole summary at once (“COARSE”-grained,
+Figure 1 top left). 6 papers instead obtain fine-
+grained annotations for individual units (e.g., sen-
+tences) and average them (FINE, Figure 1 top right).
+4https://pubmed.ncbi.nlm.nih.gov/
+5LongT5 is the best publicly available PubMed summa-
+rizer. BigBird is a popular long-form summarization baseline.
+6Reducing extractiveness / copying is also a suggestion for
+fair-use of copyrighted work (Harvard, 2016; UMGC, 2020).
+Intuitively, FINE annotation has many advantages
+for longer summaries — it is less subjective than
+COARSE, since shorter spans needs to be judged
+rather than a long summary, and it helps localize
+model errors. However, the distinction between
+COARSE and FINE is never justified in literature,
+and inter-annotator agreement is rarely reported to
+understand the task subjectivity in each setup. To
+better understand the tradeoff, in this section we
+conduct human evaluations annotating the same set
+of summaries using these two different protocols.
+Task formulation: Let Fsumm denote the faithful-
+ness score of a summary. For COARSE, k-point
+Likert scale ratings are obtained for the summary
+(Fsumm ∈ {0, 1...k}), based on the faithfulness def-
+inition provided earlier. For FINE, we collect binary
+judgments of individual units in the summary and
+average them,
+Fsumm =
+1
+|Csumm|
+�
+c∈Csumm
+Fc, Fc ∈ {0, 1}
+where Csumm is a set of units in the summary and
+Fc is the faithfulness judgment for the unit c. In
+both protocols, the faithfulness score of a system is
+defined as
+1
+|S|
+�
+summ∈S Fsumm where S is the set
+of summaries generated by the system.7
+While sentences are a popular granularity for
+FINE (4 of the 6 surveyed papers), we found that
+summary sentences in both datasets were over-
+loaded with information. Hence, we segment sen-
+tences on conjunctions and punctuation to obtain
+more atomic units as Csumm. These units are often
+clauses,8 similar to summary content units (SCUs)
+in Pyramid (Nenkova and Passonneau, 2004).
+Collecting COARSE annotations: For SQuAL-
+ITY, we re-use the annotations provided by Wang
+et al. (2022) for faithfulness assessments. In their
+data, three annotators give each summary a 1-100
+direct assessment rating (Bojar et al., 2016). An-
+notators with experience in professional copyright-
+ing and editing were hired on Upwork,9 and these
+annotators were also involved in the creation of
+SQuALITY. Unfortunately, none of the surveyed
+papers that reported human evaluation results on
+7We assume all summary units get an equal weight. How-
+ever, some units may be more important than others, we dis-
+cuss this in the Limitations section.
+8An even finer granularity is entities / numbers. We avoid
+this due to prohibitive annotation cost on long summaries.
+9https://www.upwork.com/
+
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pearson correlation
+rouge-1_f1
+rouge-2_f1
+rouge-l_f1
+bart_score
+bart_sc_pb
+sent_bleu
+bert_score
+bleurt
+SQuALITY
+FINE
+COARSE
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pearson correlation
+rouge-1_f1
+rouge-2_f1
+rouge-l_f1
+bart_score
+bart_sc_pb
+sent_bleu
+bert_score
+bleurt
+PubMed
+FINE
+COARSE
+Figure 2: 95% confidence intervals of Pearson correlations between various automatic evaluation metrics and using
+human evaluation data collected with FINE (blue) and COARSE (orange) annotation methods. In both datasets, FINE
+annotations lead to much narrower CIs than COARSE annotations. See Appendix G for plot with Kendall’s Tau.
+0
+20
+40
+60
+80
+100
+Mean human score
+bart
+bart_dpr
+human
+SQuALITY
+FINE
+COARSE
+0
+20
+40
+60
+80
+100
+Mean human score
+longt5
+bigbird
+human
+PubMed
+FINE
+COARSE
+0
+20
+40
+60
+80
+100
+Mean human score
+longt5
+bigbird
+human
+PubMed-ngram-block
+FINE
+COARSE
+Figure 3: 95% confidence intervals of estimated model performances using FINE (blue) and COARSE (orange)
+annotation methods. Intervals calculated using bootstrap resampling across annotators (Appendix A). While both
+annotation granularities lead to similar relative ordering of systems, FINE annotations have narrower confidence
+intervals. The higher LongT5 score vs human in PubMed is due to highly extractive LongT5 summaries (Section 3).
+PubMed released their raw human annotations.10
+Hence, we collect our own COARSE evaluations on
+PubMed summaries on Upwork, using freelancers
+with professional experience reading and writing
+research papers (details in Appendix B.2). We col-
+lect 3 annotations per summary and use a 5-point
+Likert scale, the most common choice for COARSE
+assessment in our survey (18 out of 38 papers). In
+total, 120 summaries are evaluated.
+Collecting FINE annotations: For both SQuAL-
+ITY and PubMed, we collect FINE annotations on
+Upwork (3 annotators per FINE unit) for the same
+set of 120 summaries evaluated using COARSE an-
+notations. For SQuALITY, we hire freelancers with
+professional experience in English, creative writ-
+ing, or education. For PubMed, we hire freelancers
+with prior experience analyzing biomedical arti-
+cles. See Appendix B.1 for details of our annotator
+10In our email correspondence with authors of these works,
+they mentioned losing access or compliance issues as reasons
+for not sharing human evaluations. We received some exam-
+ples from Guo et al. (2021) and Ju et al. (2021) for reference.
+Dataset
+COARSE
+FINE
+SQuALITY
+18.5
+6.8
+PubMed
+11.8
+7.3
+PubMed + ngram block
+11.7
+9.3
+Average
+14.0
+7.8
+Table 3: Average standard deviation of faithfulness
+scores across annotators on a 100-point rating scale.
+Lower variation means higher agreement. Overall, we
+find that FINE-grained annotations have higher inter-
+annotator agreement than COARSE-grained annotations.
+Note that all FINE units of a summary were annotated
+to obtain these results (f = 1.0 in Section 3.2).
+screening process, compensation, instructions, and
+screenshots of our annotation interface.
+FINE annotations have higher inter-annotator
+agreement than COARSE annotations.
+This
+leads to more confident downstream estimates.
+We present our results in Table 3. Overall, we ob-
+serve that across all settings, FINE annotations have
+
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
+Fraction of units
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+KT correlation to full annotation
+PubMed
+SQuALITY
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Fraction of units
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+22.5
+25.0
+Inter-annotator standard deviation
+FINE - PubMed
+FINE - SQuALITY
+COARSE - PubMed
+COARSE - SQuALITY
+Figure 4: Accuracy and variance after annotating a fraction of units per summary (X-axis) with FINE. Despite
+annotating just a fraction of the summary, we observe a high segment-level Kendall tau correlation with a full
+annotation (left). However we observe higher inter-annotator variance as the fraction reduces (right). Confidence
+intervals shown are 95% and computed across 1000 random subsets (see Appendix F for left plot with Pearson).
+lower standard deviation (and thus higher agree-
+ment) in faithfulness scores than COARSE anno-
+tations (7.8 vs 14.0 average on 100-point scaled
+ratings). To illustrate the importance of higher
+agreement, we measure its effect on two down-
+stream statistics that human evaluation is primarily
+used for: (1) correlation with automatic metrics;
+and (2) mean system performance. We adapt the
+bootstrap resampling analysis11 of Deutsch et al.
+(2021) to estimate confidence intervals of these two
+downstream statistics for COARSE and FINE.
+In Figure 2, we plot the 95% confidence inter-
+vals of the Pearson correlation of various auto-
+matic evaluation metrics against FINE-grained and
+COARSE-grained human evaluation data. Across
+both datasets, FINE data leads to much narrower
+confidence intervals (0.15 vs 0.35 average uncer-
+tainty in Pearson correlation on PubMed) for the
+same number of summaries, implying higher sta-
+tistical power. In Figure 3, we observe a simi-
+lar trend with mean system performance. Inter-
+estingly, both annotation methods give the same
+relative ordering of systems (human > bart-dpr >
+bart for SQuALITY, human > longT5 > BigBird
+for PubMed-block), confirming the alignment of
+FINE and COARSE judgments on average.
+Recommendation: Unlike the dominant trend in
+prior work, FINE-grained evaluations should be pre-
+ferred over COARSE grained evaluation for long-
+11We slightly modify the algorithm in Deutsch et al. (2021)
+for inter-annotator variance, see Appendix A.
+form summaries. FINE annotations have lower inter-
+annotator variance than COARSE annotations and
+help localize model errors. In our setup we assume
+all FINE units are equally weighted while aggre-
+gating them to the final summary score. Despite
+this assumption, in our results we observe a consis-
+tent relative ordering of systems/metrics between
+COARSE and FINE annotations. Nevertheless, non-
+uniform weighing of units is an interesting future
+work direction; more in the Limitations section.
+3.2
+RQ2: Can we reduce annotator workload
+by partially annotating a long summary?
+In Section 3.1, we found that FINE annotations have
+lower variance than COARSE annotations. However,
+long summaries may be composed of several units
+(sentences or phrases) which each require FINE
+annotation. This could make FINE annotation very
+expensive for longer summaries (as also noted in
+our survey). What if we instead annotate a random
+subset of units from the summary? While this will
+lower annotation cost, how accurate would these
+partial annotations be? We explore this tradeoff by
+re-using the annotations collected in Section 3.1.
+For every summary, we randomly sample a fraction
+of units f ∈ {0.1, 0.2...0.9} and then measure its
+correlation to the full set of annotations collected.
+Each annotator gets a different random sample of
+units for the same summary. In initial experiments,
+we found that this yielded higher accuracy than
+when keeping the same set of units per annotator.
+
+Partial annotation has a high correlation to full
+annotation, but higher variance: In Figure 4
+(left) we plot the segment level Kendall’s τ correla-
+tion (relative ordering of summary scores) between
+a partial annotation and full annotation for different
+values of f. Overall, we observe a high correlation
+across different values of f. Despite annotating just
+half the summary (f = 0.5), in both datasets we
+observe a high correlation of 0.78-0.89 Kendall’s
+τ (95% interval) with a full annotation. Does a
+partial annotation preserve the variance benefits of
+FINE vs COARSE? In Figure 4 (right) we plot the
+inter-annotator variance for different values of f.
+In both datasets we find that a partial annotation
+has a higher variance than a full annotation. While
+for all values of f in SQuALITY we find that FINE
+annotations still have lower variance than COARSE,
+in PubMed COARSE has lower variance than FINE
+for f <= 0.3 with 95% confidence.
+Recommendation: Having annotators judge a ran-
+dom subset of units in a long-form summary is a
+simple way to reduce FINE annotation cost, and has
+high correlation with a full annotation. However,
+it increases inter-annotator variance. Annotating
+50% of the summary results in 0.78-0.89 Kendall’s
+τ correlation, with a 30-40% increase in standard
+deviation compared to full FINE annotation. Partial
+annotation may be limited in its ability to identify
+issues in summaries with very few errors. However,
+we find that this is not the case in current systems,
+which are abundant in faithfulness errors.
+3.3
+RQ3: Is it useful to align summary units
+to sentences in the source document?
+So far, we have focused on design decisions on the
+summary side of evaluation. However, evaluating
+faithfulness requires a comparison of facts between
+a summary and a source document. Long-form
+summaries tend to have long source documents
+(Table 1): 3.1K words for SQuALITY and 5.1K
+words for PubMed. In Section 2, we found sev-
+eral mentioned human evaluation is challenging
+since annotators need to read long source docu-
+ments. Some prior work has suggested highlight-
+ing spans in the source document that align with
+the summary (Hardy et al., 2019; Kryscinski et al.,
+2020; Vig et al., 2021) as shown in Figure 1. How-
+ever, these efforts have exclusively focused on news
+summarization with relatively short source docu-
+ments, like CNN/DM (804 words) (Nallapati et al.,
+2016) or XSUM (438 words) (Narayan et al., 2018).
+Algorithm
+R@3
+R@5
+R@10
+BM25 (1995)
+0.38
+0.46
+0.56
+ROUGE-1 (2004)
+0.31
+0.34
+0.46
+SIM (2019)
+0.37
+0.52
+0.60
+DPR (2020)
+0.29
+0.31
+0.41
+BERTScore-DB-XL (2020)
+0.30
+0.37
+0.46
+SummaC-NLI (2022)
+0.22
+0.26
+0.34
+MultiVers-FEVER (2022)
+0.47
+0.58
+0.71
+SuperPAL (2021)
+0.61
+0.68
+0.77
+Table 4:
+A comparison of algorithms finding the
+top source document sentences for summary units in
+SQuALITY. R@k (recall@k) denotes the fraction of
+times the gold sentence was in the top-k predictions.
+Hints
+Acc. (↑)
+Agree. (↑)
+Time (secs) (↓)
+(2-way)
+(Fleiss)
+All
+First 5
+None
+93%
+0.71
+41.4
+115.6
+SuperPAL
+92%
+0.64
+48.2
+84.6
+Gold
+92%
+0.63
+40.4
+60.4
+Table 5: Annotator performance (accuracy, agreement,
+median time) in detecting summary errors with differ-
+ent types of source document highlight hints. Overall,
+we see little difference across the three settings.
+How useful is highlighting based on alignment,
+or “hints”, when the spans are chosen from much
+longer documents?
+What is the best highlighting algorithm? We
+conduct a study to identify the alignment algorithm
+best suited for highlighting hints. We manually
+annotate 125 FINE units from human-written sum-
+maries of the SQuALITY validation split, marking
+the sentences best supporting them from the source
+document. We then test several candidate meth-
+ods for linking summary units to the source doc-
+ument. These include token overlap methods like
+ROUGE (Lin, 2004), retrievers (Karpukhin et al.,
+2020), and fact verifiers (Wadden et al., 2022). In
+Table 4, we find that SuperPAL (Ernst et al., 2021),
+a weakly supervised linking algorithm, performs
+best (0.61 recall@3 vs the next best 0.47). To im-
+prove precision, we filter matches scoring less than
+0.3 on SuperPAL, and show at most five highlights.
+Do highlighted hints improve summary error
+detection? To answer this question, we manu-
+ally perturb 50 FINE summary units in SQuALITY
+validation summaries, introducing entity errors or
+negations like Kryscinski et al. (2020). We mod-
+ify the summary context of the perturbed unit to
+ensure summaries are self-consistent. Annotators
+
+Question & TL;DR response
+Response Snippets
+Q: Did you find the highlighted
+hints useful while making your
+judgment?
+TL;DR: 4 out of 5 annota-
+tors said Sometimes, 1 said Yes.
+More useful for SQuALITY,
+summary units copied verbatim
+from source, correct summaries.
+“With summaries that had poor correctness, the hints were often a mess, and even correct spans had to be
+carefully checked. In summaries that were more correct, I could often just read the span and remember that
+it was correct, and then the hints helped me find the right source position, or refresh my memory about
+details.”
+“They were more useful when the summary was a near verbatim source reproduction.”
+“Yes, they were useful. Often they would highlight the exact passage needed to support the summary span.”
+“In PubMed, they were a little more chaotic, even for good summaries.”
+“SQuALITY summaries consisted of sentences or parts of sentences taken straight from the story (wording was
+exactly as in the text). So hints often lead to the exact place.”
+“For SQuALITY, they were mostly accurate and helpful. For PubMed, they were less accurate and relevant.”
+Q: Would the highlights have
+been sufficient to make judg-
+ments, or was reading the entire
+source document necessary?
+TL;DR: 3 out of 5 annota-
+tors said No, 2 said sometimes
+in SQuALITY. Reading the
+entire document was critical.
+“Reading the entire source document was very helpful to understand the basic story plot”
+“Even when the hints were relevant, sometimes they left out information (like character name)...”
+“Initially I tried skimming ... then concluded it’s easier to read the entire document first.”
+“With SQuALITY there were cases where almost all of the highlights did not make any sense and nothing of
+that was even mentioned in the story. With PubMed, it was even more difficult to find hints that support the
+text”
+“Reading the entire document was essential to understanding the whole process, the hints in isolation were
+not good enough. The hints and the summary often confused similar objects, especially when pronouns were
+involved, from different parts of the source. In PubMed a similar thing happened when the source discussed
+what other papers had done – punctuation, acronyms, and abbreviations played a big role in providing context.”
+Q: Did you use Ctrl+F searches
+in the source document while
+making judgments?
+TL;DR: 4 out of 5 annota-
+tors said Yes, 1 said yes only
+for PubMed.
+Ctrl+F helped
+locate synonyms, entities.
+“Yes, all the time. It was usually a safer bet than using the hints. The hints are given out of context of the
+whole SQuALITY story. There were a lot of problems with the PubMed hints involving numbers, which I
+often searched for. They were very rarely supported by the document, or contained wrong symbols (= instead
+of >).”
+“Yes, mostly in cases the highlight did not support the summary unit partially or entirely.”
+“I used Ctrl+F when looking for very specific words, like names. Searching was less helpful when it came to
+words that had synonyms or emotions.”
+“I did Ctrl+F on keywords taken directly from the summary unit as well as synonyms and any specific words
+that I remembered from the story that could help me get to that place in the source document quickly.”
+Table 6: Results and snippets from our questionnaire with FINE annotators. Overall, annotators find hints only
+sometimes useful, and mention reading the entire source document along with keyword searches.
+are shown 50 perturbed and 50 un-perturbed sum-
+maries, and asked to annotate whether the summary
+units are faithful to the source in three settings:12
+(1) no highlighted hints; (2) SuperPAL highlighted
+hints; (3) gold hints manually annotated by us. In
+Table 5, we show accuracy, inter-annotator agree-
+ment, and median time13 for each setting.
+Highlighted hints have almost no effect in eval-
+uating long-form summaries: Surprisingly, we
+observe that in all three metrics (accuracy, agree-
+ment, median time taken), scores are quite similar
+across the three settings. In fact, the “no-hint” set-
+ting scores slightly higher than the SuperPAL hint
+settings (93% vs 92% accuracy, 0.71 vs 0.64 Fleiss
+κ) and takes annotators less time (41.4 vs 48.2 sec-
+onds per unit). However, we find that hints helped
+annotate the first few units of a summary quicker
+(84.6 secs vs 115.6 secs per unit). We attribute our
+findings to a learning effect over time. FINE anno-
+tation of long-form summaries requires annotation
+12To prevent any bias, each annotator receives only one of
+these settings for a particular summary.
+13Calculated using the method in Akoury et al. (2020).
+of several units for the same document - summary
+pair. As annotation progresses, annotators get more
+familiar with the contents of the source document
+and summary, reducing the need for hints over time.
+See Appendix E for learning trajectory plots.
+Questionnaire with FINE annotators confirm
+limited utility of hints: Our evaluation so far is
+limited to perturbed human summaries. How effec-
+tive are hints on model-generated summaries? To
+answer this, we ask five of our FINE Upwork anno-
+tators (from Section 3.1) a set of three questions
+about their experiences using highlighted hints.14
+Detailed questionnaire results along with answer
+snippets are shown in Table 6. Overall, annota-
+tors find hints were useful only sometimes. Hints
+were less useful when (1) the summary unit was not
+supported in the source; (2) the summary unit was
+highly abstractive compared to the source; (3) pro-
+nouns, numbers, or abbreviations were involved;
+and (4) Pubmed summaries were annotated. Al-
+14The FINE annotations in Section 3.1 were shown hints in
+the source document. Since hints may not be helpful, annota-
+tors were told not to solely rely on hints for annotation.
+
+most all annotators said it was necessary to read
+the entire source document before annotation to get
+an overall idea of the plot and resolve coreferences.
+Nearly all annotators used “Ctrl+F” searches along
+with hints to search for specific keywords while
+making judgments. This was especially true when
+the summary unit was incorrect, since the source
+document had to be thoroughly searched (beyond
+the hints) before confidently marking “Incorrect”.
+Recommendation: In contrast to recommendations
+in prior work, automatically highlighted hints are
+useful only in some specific cases of long-form
+summarization: mostly correct summaries, almost
+verbatim copied sentences. Annotators should be
+instructed to read the entire source document and
+to not rely solely on highlighted hints, since that
+could bias their judgments. Based on a small-scale
+study, we found SuperPAL (Ernst et al., 2021) to
+be the most accurate method for finding hints, but
+its performance (61% recall@3) is far from ideal.
+3.4
+To what extent do our findings generalize
+to short-form summarization?
+In this work, we exclusively focus on summariza-
+tion datasets with an average summary length of at
+least 150 words. This constraint excludes two pop-
+ular benchmarks in summarization research over
+the last five years: CNN/DM (Nallapati et al., 2016)
+and XSUM (Narayan et al., 2018). How relevant
+are our research questions (RQs) and findings for
+these short-form summarization benchmarks?
+On average, XSUM (24 words) and CNNDM
+(60 words) contain much shorter summaries than
+SQuALITY (237 words). XSUM outputs typically
+contain only 1 sentence or roughly 2-3 FINE units
+per summary. This blurs the distinction between
+FINE and COARSE units, which makes it less use-
+ful to study RQ1 in these short-form settings. The
+shorter length of outputs also implies that evalu-
+ation is less expensive and consumes less time,
+which makes our RQ2 less relevant. Finally, on
+average, XSUM (440 words) and CNNDM (800
+words) also have much shorter source documents
+than datasets like SQuALITY (5200 words), reduc-
+ing the need for alignment (the main premise for
+RQ3). The main motivation behind our study is
+that human evaluation of long-form summarization
+datasets like SQuALITY and PubMed is challeng-
+ing and expensive due to the long length of the gen-
+erated text. Overall, our research questions and
+findings are more relevant for long-form sum-
+marization datasets than for short-form sum-
+marization datasets like XSUM and CNNDM.
+4
+Related Work
+A large body of recent work has focused on new
+automatic evaluation methods for summarization
+via NLI-based algorithms (Falke et al., 2019; La-
+ban et al., 2022) or QA-based algorithms (Wang
+et al., 2020; Fabbri et al., 2022). Our work focuses
+on the much less studied area of human evaluation,
+the gold standard for developing automatic met-
+rics. A notable effort in this space is the Pyramid
+method (Nenkova and Passonneau, 2004), along
+with work improving Pyramid efficiency (Shapira
+et al., 2019; Zhang and Bansal, 2021). Efficient
+Pyramid-like protocols have been used to collect
+large-scale datasets human judgments (Bhandari
+et al., 2020; Liu et al., 2022) in short-form news
+summarization tasks like CNN/DM. While these
+efforts focus on salience evaluation and assume ac-
+cess to multiple references, our work focuses on
+faithfulness and operates in a reference-free set-
+ting. Moreover, we focus on long-form summariza-
+tion tasks like SQuALITY and PubMed, which are
+much more challenging and expensive to evaluate.
+Evaluating summary faithfulness relates to fact
+verification (Vlachos and Riedel, 2014), where
+claim sentences are checked against a large knowl-
+edge source (Wikipedia). Prior work (Nakov et al.,
+2021) attempts to simplify the human fact checking
+process by methods like knowledge source snip-
+pets (Fan et al., 2020), similar to hint highlights
+(§3.3). Faithfulness in summarization differs from
+fact verification in three ways: (1) summaries are
+paragraph-long and contextual compared to sin-
+gle sentence stand-alone claims in fact verification;
+(2) summaries are grounded to a source document,
+compared to a large knowledge source in fact veri-
+fication; (3) summaries are model-generated com-
+pared to human-written claims in fact checking
+datasets (Thorne et al., 2018; Wadden et al., 2020).
+5
+Conclusion
+We present the LONGEVAL guidelines, a set of
+recommendations for moving towards standardized
+human evaluation of long-form summarization. We
+empirically analyze each recommendation on two
+datasets. Overall, we find that (1) FINE-grained an-
+notations have lower inter-annotator variance than
+COARSE-grained annotations; (2) partially annotat-
+
+ing a summary reduces annotator workload while
+maintaining accuracy; (3) highlighting hints in the
+source document has limited usefulness for evaluat-
+ing long-form summaries. As future work, we plan
+to conduct experiments on other aspects of summa-
+rization evaluation like salience and coherence.
+Limitations
+Human evaluation is a noisy process with many
+confounding variables. Some of these variables
+were kept constant among experiments on a dataset,
+but modifying them could change the trends in the
+results. These include: (1) number of annotations
+per summary; (2) the specific annotation interface
+used; (3) granularity for FINE evaluation (sentences
+vs phrases); (4) Number of points in the Likert scale
+for COARSE evaluation; (5) set of summarization
+systems evaluated; and finally (6) relative (eg: A/B
+tests) vs absolute evaluation (eg: Likert), which has
+been discussed in Tang et al. (2022) for short-form
+news summarization datasets like CNN/DM.
+Our paper is limited to faithfulness evaluation,
+but summaries are typically evaluated for salience,
+fluency, coherence as well (Fabbri et al., 2021).
+While fluency may be less of an issue due to
+large-scale language model pretraining (Dou et al.,
+2021), coherence and salience are important as-
+pects to evaluate especially in long-form summa-
+rization (Goyal et al., 2022). Our findings may not
+generalize to evaluation of coherence or salience.
+Our experiments in Section 3.1 assigned an
+equal weight to each FINE unit while calculating
+the overall score of the summary. However, the
+faithfulness of some FINE units may be more impor-
+tant than others. A non-uniform weighing of FINE
+units may be a good strategy if there is a notion
+of how critical a particular unit is for a summary’s
+correctness. For example: (1) PICO units are criti-
+cal in medical summaries (DeYoung et al., 2021);
+(2) the Pyramid scheme (Nenkova and Passonneau,
+2004) uses a reference frequency-based unit impor-
+tance, assuming access to multiple gold references.
+However, a consistent notion of importance is diffi-
+cult to establish across different domains, and also
+depends on an individual consumer’s preferences.
+Designing non-uniform weighing schemes is an
+interesting direction for future research.
+Ethical Considerations
+All experiments involving human evaluation in this
+paper were exempt under institutional IRB review.
+We fairly compensated each Upwork freelancer in-
+volved in this study, at a rate of 15-20$ per hour
+(respecting their suggested Upwork hourly wage).
+For each round of annotation, we estimated the
+average amount of time the task would take (by
+running pilots among ourselves), and provided an-
+notators with the estimated time requirement. Most
+freelancers finished the task within the time win-
+dow, but sometimes exceeded it by 0.5-1 hr. We
+compensated freelancers based on the actual time
+they took and their hourly wage, rather than a fixed
+amount per annotation.
+Acknowledgments
+First and foremost, we would like to thank all the
+nine Upwork freelancers who contributed human
+annotations to this project. We are very grateful
+to Yixiao Song, Alex Wang, John Giorgi, Dustin
+Wright, Yulia Otmakhova, Daniel Deutsch, Arie
+Cattan, Shiyue Zhang, Tanya Goyal, Greg Durrett,
+Marzena Karpinska, Ankita Gupta, Nader Akoury
+and the Semantic Scholar team for several useful
+discussions at various points during the project.
+This work was mostly done while Kalpesh Krishna
+(KK) was an intern at the Allen Institute for Arti-
+ficial Intelligence. KK was partly supported by a
+Google PhD Fellowship awarded in 2021.
+Author Contributions:
+Kalpesh Krishna led the
+project and performed all the technical contribu-
+tions including literature review, dataset collection
+and processing, model implementation, annotation
+interface development, running experiments, and
+data analysis. Kalpesh also contributed to project
+scoping and ideation and led the writing of the pa-
+per. Erin Bransom and Bailey Kuehl helped with
+obtaining human judgements, including piloting
+the task and giving feedback, performing the anno-
+tation themselves, and hiring and managing anno-
+tators on Upwork. Pradeep Dasigi, Arman Cohan,
+and Kyle Lo were mentors of the project during and
+after Kalpesh’s internship, contributing equally to
+project scoping, experimental design, ideation and
+direction throughout the course of the project and
+paper writing. Mohit Iyyer provided mentorship
+after the internship, in particular providing impor-
+tant feedback and direction on data analysis and
+contributing to paper writing.
+
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+Association for Computational Linguistics.
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+son Phang, and Samuel R Bowman. 2022. Squality:
+Building a long-document summarization dataset
+the hard way. arXiv preprint arXiv:2205.11465.
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+rater kappas. Advances in data analysis and classifi-
+cation, 4(4):271–286.
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+Shiyue Zhang and Mohit Bansal. 2021. Finding a bal-
+anced degree of automation for summary evaluation.
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+Conference on Learning Representations.
+
+Appendix
+A
+Bootstrap analysis of inter-annotator
+variance
+We utilize the bootstrap resampling (Tibshirani and
+Efron, 1993) technique described in Deutsch et al.
+(2021) to estimate confidence intervals for human
+evaluation data. At a high level, bootstrap resam-
+pling helps capture the uncertainty in a downstream
+test statistic by repeatedly sampling from the data
+with replacement. We consider two downstream
+test statistics in our work — (1) average system
+level performance; (2) correlation of human judge-
+ments to automatic metrics.
+While Deutsch et al. (2021) were primarily in-
+terested in uncertainty due to the specific instances
+and systems evaluated, our goal is to capture uncer-
+tainty due to the inter-annotator variance. Hence
+unlike Deutsch et al. (2021), we sample with re-
+placement from the set of annotators for every in-
+stance. Our precise formulation can be found in
+Algorithm 1, which operates on a X ∈ RN×M ma-
+trix of human annotations where N is the number
+of summaries, and M the number of annotators.
+Algorithm 1 Bootstrap Confidence Interval
+Input: X ∈ RN×M, k ∈ N, α ∈ [0, 1].
+N is summaries, M is annotators
+Output: (1 − α) × 100%-confidence interval
+1: samples ← an empty list
+2: for k iterations do
+3:
+Xs ← empty N × M matrix
+4:
+for i ∈ {1, . . . , N} do
+5:
+D ← samp. {1, . . . , M} w/ repl. M times
+6:
+for j ∈ {1, . . . M} do
+7:
+Xs[i, j] ← X[i, D[j]]
+8:
+end for
+9:
+end for
+10:
+Calculate test statistic on Xs and append to samples
+11: end for
+12: ℓ, u ← (α/2) × 100 and (1 − α/2) × 100 percentiles of
+samples
+13: return ℓ, u
+B
+Human evaluation details
+B.1
+FINE-grained evaluations of SQuALITY
+and PubMed summaries
+We interviewed a total of 9 Upwork freelancers
+for the position, offering a compensation of $15-
+16.5 / hr (depending on their Upwork hourly rate).
+The screening procedure involved a qualification
+task on synthetically perturbed summaries from
+the SQuALITY dataset validation split. Similar to
+the final annotation task, annotators were shown a
+F-κ
+R-κ
+all agree
+Random
+0.00
+0.00
+25%
+SQuALITY
+0.74
+0.76
+82%
+PubMed
+0.53
+0.65
+74%
+Table 7: Fleiss kappa (F-κ), Randolph kappa (R-κ), and
+agreement scores of our FINE annotation per summary
+unit. All κ scores are well above a random annotation
+baseline, indicating good agreement.
+highlighted clause from the summary, and asked
+to mark whether or not it is supported by the
+source document. 50% of the clauses were synthet-
+ically perturbed (via negation or entity swapping as
+in Kryscinski et al., 2020) and manually checked
+to ensure they were not supported by the source
+document. A total of 6 freelancers scored 85% or
+better, and were recruited for the main set of exper-
+iments. All 9 freelancers were compensated for the
+screening round at the rate of 15$ USD / hr.
+All six hired annotators are native or bilingual
+English speakers. All annotators have completed
+a degree at the undergraduate level and three also
+have Masters degrees, with the most common fo-
+cuses of the degrees being English/creative writing
+and education. The annotators’ common profes-
+sional experiences include copywriting, editing,
+proofreading, writing, and teaching. Finally, for
+PubMed annotations we re-hired three annotators
+from the pool of six SQuALITY annotators who
+mentioned they had experience reading and ana-
+lyzing biomedical articles. These three annotators
+were provided with an additional bonus of $30 after
+they completed all annotations.
+Annotators are provided with a detailed anno-
+tation guideline along with examples of faithful-
+ness (Table 10). Our guidelines are mostly con-
+sistent with a recently proposed set of guidelines
+for checking attribution in text generation (Rashkin
+et al., 2021). The final annotation interface is im-
+plemented in AMT Sandbox, as shown in Figure 8.
+Inter-annotator agreement (binary): Much of
+the analysis in Section 3 uses standard deviation
+across summaries scores to measure inter-annotator
+agreement. However, another way to calculate
+inter-annotator agreement for FINE annotations is
+measuring agreement on individual units which
+received a Yes / No judgment.
+In Table 7 we
+show these inter-annotator agreement statistics. We
+measure Fleiss Kappa (Fleiss, 1971), Randolph
+
+Kappa (Randolph, 2005; Warrens, 2010), and the
+fraction of sentence pairs with total agreement.15
+In the table we can see all agreement statistics are
+well away from a uniform random annotation base-
+line, indicating good agreement.
+B.2
+COARSE-grained evaluation of PubMed
+summaries
+None of the surveyed papers evaluating PubMed
+summaries with humans released their human eval-
+uation data. Hence, we decided to collect our own
+COARSE annotations. Since FINE annotations (Sec-
+tion B.1) may have biased our original set of an-
+notators, we hire three new annotators to perform
+overall assessments on a 5-point Likert scale. In
+other words, we use a “between-subject” experi-
+ment design to compare FINE against COARSE.
+We hired three freelancers on Upwork, all of
+whom have extensive professional experience read-
+ing research papers (two of them had PhDs in
+biomedical fields). All annotators were compen-
+sated at a rate of 20$ USD / hr, their hourly rate on
+Upwork. All three annotators had been previously
+screened and hired by us for different projects in
+the past. Two of them had assisted us in an an-
+notation task involved reading short summaries of
+biomedical academic papers and evaluating them
+for fluency, accuracy, correctness.
+Annotators are provided with a detailed anno-
+tation guideline along with examples of faithful-
+ness (Table 11). Our guidelines are mostly con-
+sistent with a recently proposed set of guidelines
+for checking attribution in text generation (Rashkin
+et al., 2021). The final annotation interface is im-
+plemented in LabelStudio, as shown in Figure 9.
+B.3
+Crowdworkers or expert annotators?
+Several prior works have raised the issue of low
+inter-annotator agreement and poor accuracy with
+non-expert annotators (eg: MTurk crowdworkers)
+in human evaluation of summarization (Gillick and
+Liu, 2010; Fabbri et al., 2021; Falke et al., 2019)
+and open-ended long-form generation (Karpinska
+et al., 2021; Clark et al., 2021). In our survey
+(Table 9), we found the type of annotators used in
+long-form summarization is often not specified (16
+/ 43 papers). Among other papers, 10 papers use
+non-experts while 17 papers use expert annotators
+(often graduate students).
+15The κ scores are measured using the library https://
+github.com/statsmodels/statsmodels.
+Overall, we echo the concerns with non-expert
+annotators and recommend hiring freelancers on
+Upwork (or experts) who are well-versed with
+the domain for annotation. In initial experiments,
+we attempted to recruit Amazon Mechanical Turk
+crowdworkers filtered by the “Master’s qualifica-
+tion” and having a 90%+ approval rating. In our
+qualification task of error detection in syntheti-
+cally perturbed SQuALITY summaries, MTurkers
+scored just 62% (binary classification) with a three-
+annotator Fleiss κ of 0.15. On the other hand, Up-
+work freelancers (with professional writing experi-
+ence) an accuracy 90% with a high inter-annotator
+agreement (Fleiss κ = 0.71).
+C
+Additional Survey Statistics
+In Table 8 and Table 9 we document some addi-
+tional statistics for the 44 papers conducting human
+evaluation of long-form summarization.
+Best practice
+# papers
+Raw human evaluation data released
+2 / 44
+Interface or instructions provided
+9 / 44
+Inter-annotator agreement reported
+12 / 44
+Statistical analysis conducted
+12 / 44
+Multiple datasets are human evaluated
+14 / 44
+Multiple annotators per summary
+33 / 44
+Annotator background reported
+33 / 44
+Specific summary aspects evaluated
+42 / 44
+Table 8: Fraction of surveyed papers following the best
+practices recommended by Gehrmann et al. (2022). We
+include only the 44 papers here which conducted a hu-
+man evaluation of long-form summarization.
+Type of annotator
+# papers
+No details specified
+11 / 44
+Native English speaker**
+5 / 44
+Mechnical Turk crowdworker
+9 / 44
+Non-expert volunteers
+1 / 44
+Extensive prior experience**
+3 / 44
+Graduate students / researchers
+13 / 44
+Upwork freelancers
+2 / 44
+Table 9: The types of annotators used across different
+long-form summarization papers. ** - No additional
+details were specified.
+D
+Automatic summarization metrics
+used for evaluation
+The following metrics are considered while measur-
+ing Pearson’s correlation with our human evalua-
+tion data (Figure 2) — ROUGE-1 / 2 / F (Lin, 2004),
+
+BARTScore / BARTScore-Parabank (Yuan et al.,
+2021), Sentence-BLEU (Papineni et al., 2002),
+BERTScore (Zhang et al., 2020) and BLEURT (Sel-
+lam et al., 2020). A number of metrics were calcu-
+lated using the SacreROUGE repository (Deutsch
+and Roth, 2020).
+E
+Learning effect while annotating
+long-form summaries
+In Section 3.3 we discussed a learning effect where
+annotators get more familiar with the contents of
+a source document as they annotate more FINE-
+grained units in a long-form summary. To better
+understand this effect, in Figure 5 we plot the av-
+erage time taken by annotators as they progress in
+their annotation of a summary. Overall, we find that
+annotators get significantly faster in annotating the
+summary after the first 20% units. We hypothesize
+that annotators get pretty familiar with the general
+topics in the source document after the first few
+annotations, speeding up subsequent annotations.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fraction of summary units annotated
+50
+100
+150
+200
+250
+Time taken (seconds)
+pred_hints
+gold_hints
+no_hints
+Figure 5: Learning effect over time while evaluating
+long-form summaries with FINE annotation. As the an-
+notators evaluate more summary units, they learn the
+document better and are much faster at annotation irre-
+spective of whether hints are shown to them.
+F
+Partial summary annotation with
+pearson correlation
+See Figure 6.
+G
+Metric correlations using Kendall’s
+Tau
+See Figure 7.
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
+Fraction of units
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pearson correlation to full annotation
+PubMed
+SQuALITY
+Figure 6: A version of Figure 4 using Pearson correla-
+tion instead of Kendall Tau correlation.
+
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+KT correlation
+rouge-1_f1
+rouge-2_f1
+rouge-l_f1
+bart_score
+bart_sc_pb
+sent_bleu
+bert_score
+bleurt
+SQuALITY
+FINE
+COARSE
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+KT correlation
+rouge-1_f1
+rouge-2_f1
+rouge-l_f1
+bart_score
+bart_sc_pb
+sent_bleu
+bert_score
+bleurt
+PubMed
+FINE
+COARSE
+Figure 7: A version of Figure 2 using Kendall’s Tau correlation instead of Pearson’s correlation.
+Figure 8: The AMT Sandbox annotation interface used for FINE evaluation of SQuALITY and PubMed summaries
+(Appendix B.1).
+
+Select an option
+Summary (focus on highlighted span): < pad> background : the aim of this study is to compare the learning and memorization rates of english using authentic and
+Yes
+1
+traditional assessment methods. methods: the research method is semi experimental and the sample included 60 students chosen by simple random sampling.
+No
+2
+results: the rate of students learning andmemorization is more in authentic assessment than traditional methods.the rate of memorizing is more in authentic methods
+than traditional ones. conclusion: the results of this research indicate that the rate of students learning and memorization is more in authentic methods than traditional
+methods.the students attitude toward test is more positive than traditional ones.
+Source Document:
+Prev HintNext Hint(4 hint(s) available)Store Source Position
+NOTE: hints may or may not be useful, please skim through document yourself (or search for keywords with Ctrl + F) if hints are unhelpful.
+is cancelled because it emphasized on loading learner s accumulator minds and interrogating in tests as jean piaget said the main goal of education must be training
+of innovators who could think not to repeat .
+it is training of investigators and researchers not those who adopt whatever is said ( 1 ) .
+obviously , it is not possible to train new generation for the living in a changeable society for present and next days without new planning to change functions and
+worksof educationsystem includinggoals,contents,teachingandlearningmethodsalso evaluation and assessmentmethods（2)contemporary,indomainof
+trainingscience,evaluationhasreceivedsuchan importancethat identifiedasan independentandspecialized scientificfield
+because of the importance of evaluation some specialists in domain of curriculum development like tiler presented it as a center of education process ( 3 )
+evaluation indicated as the last link for the teaching - learning process which is used in the end of educational period to separate students with different learning
+process
+if it could present suitable information permanently through feedbacks to students , it would be more effective to improve learning .
+foreignlanguageIn this task, you will be shown a long document ("Source Document") and its Summary. A span of text will be
+highlighted in the summary, and the goal is to check if this span is factually supported by the source document. You
+will need to choose one of two options:
+1. Yes: if all the facts in the highlighted summary span are supported by the source document
+2. No: if the highlighted summary span presents some information that is not supported by the source document (either
+a direct contradiction, or not present)
+In addition to the source document, you will be provided with some highlighted text ("hints") in the source document
+which may help you in making a decision. Press the "Next Hint" button to scroll through the highlighted hints. Source
+document hints may or may not be helpful. Do not make a judgment solely based on these hints. Skim through the
+source document yourself / search for keywords with Ctrl + F if the hints are not helpful.
+Below you can find some short representative examples.
+Example 1
+Summary (only highlighted span shown) = ... Retief is not Lemuel’s cousin. ...
+Source Document (snippets shown) = He eyed Retief ... "He ain’t no cousin of mine," Lemuel said slowly.
+Supports = Yes
+Example 2
+Summary (only highlighted span shown) = ... Lemeul knocks down Retief. ...
+Source Document (snippets shown) = Retief’s left fist shot out, smacked Lemuel’s face dead center. He stumbled back,
+blood starting from his nose; ... He caught himself, jumped for Retief ... and met a straight right that snapped him onto
+his back: out cold. "Wow!" said Potter. "The stranger took Lem ... in two punches!"
+Supports = No (Reason: Retief knocks down Lemeul, not the other way around.)
+Example 3
+Summary (only highlighted span shown) = ... Potter and his team do not trust the Embassy. ...
+Source Document (snippets shown) = Lemme up. My name’s Potter. Sorry ’bout that. I figured it was a Flap-jack boat;
+looks just like ’em . He waved a hand toward the north, where the desert lay.
+Supports = No (Reason: The claim is irrelevant to the evidence.)
+Table 10: Annotation guidelines provided to annotators for FINE-grained evaluation of SQuALITY and PubMed
+summaries. (Appendix B.1).
+
+Figure 9:
+The LabelStudio annotation interface used for COARSE evaluation of PubMed summaries (Ap-
+pendix B.2).
+
+Summary to annotate
+On a scale of 0-5, how factually correct is the summary with respect to
+thesourcedocument?
+introductionout - of - hospital cardiac arrest has a low survival rate to hospital discharge.recent studies
+0
+O 2
+04
+O5
+compareda simplifiedform of cpr,basedon chest compressionalone versus standard cpr including
+out - of hospital cpr for non traumatic cardiac arrest in different databases.resultswe identified only three
+randomized trials on this topic, including witnessed and not -witnessed cardiac arrests .when pooling
+evidence to support the superiority of the compression-only cpr interms of survival at hospital
+discharge,as 211/1842(11.5%)patients in the chest compression alone group versus 178/1895(9.4%
+)in thestandard cprgroup werealiveat hospital discharge:oddsratio from both petoand dersimonian
+laird methods = 0.80 (95% confidence interval 0.65 -0.99),p for effect = 0.04,p for heterogeneity = 0.69
+, inconsistency = 0%).conclusionsavailable evidence strongly support the superiority of bystander
+compression-only cpr.reasons forthe best efficacy of chest compression-only cpr include a better
+willingness to start cpr by bystanders,the low quality of mouth-to -mouth ventilation and a detrimental
+effect of too long interruptions of chest compressions during ventilation . based on our findings,
+compression - only cpr should be recommended as the preferred cpr technique performed by untrained
+bystander .
+SourceDocument
+Comments (optional)
+out -of -hospital cardiac arrest is still a major public health issue,claiming hundreds of thousands
+of livesworldwideyearly
+bystander - initiated cardiopulmonary resuscitation (cpr ) is essential to increase the chance of
+survival and neurologicalrecovery
+despite huge efforts to train laypeople to recognize and treat cardiac arrest , incidence of bystander
+reluctancetoperformmouth-to-mouthventilation is one of themajorreason
+pv
+whereas cpr including ventilation is still considered the gold standard approach before advanced life
+support can be instituted , a growing number of studies compared a simplified form of cpr , based
+on chest compression alone versus standard cpr including ventilation
+animal studies showed no difference in survival or even worse outcomes when ventilation was
+added to chest compressions ; nevertheless , in animal models of cardiac arrest due to respiratory
+inhumans
+observational studies of bystander - initiated cpr comparing standard and compressions - only cpr
+reported similar survival rates ; however , interpretation of the results is made difficult due to the
+highheterogeneityofthecausesof cardiac arrestandoftherescuecharacteristics
+chest compression - only cpr is simpler than standard cpr to teach ( during courses but even by
+dispatchersunderrealconditions),andlikelyahigherpercentageofbystanderswouldacceptto
+performitwhileavoidingmouth-to-mouthcontact:thedemonstrationthatitis(atleast)as
+effective as standard cpr can be crucial to improve survival rate in out - of - hospital cardiac arrest .
+with the underlying hypothesis that out - of - hospital cardiac arrest bystander - initiated
+compression
+- only cpr is equivalent to cpr including ventilation（standard cpr）,we performed a comprehensive
+systematicreview andmeta-analysis of randomizedcontrolledtrials,focusingonsurvival at
+hospital discharge
+pertinent studies were independently searched in biomedcentral , central , and pubmed ( updated
+september1,2010)by several trained investigatorInstructions for Likert-scale evaluation. Please read all instructions before starting the annotation.
+Setup
+1. Start by signing up on Label Studio, you will need to provide an email ID and password. It’s okay to use a
+non-existent throw-away email ID here. Also, do not use any personal / sensitive passwords (but make sure to remember
+your email / password for logging in next time!). Click on the box saying “<your name> — Summarization Evaluation”
+2. In this batch a total of 30 summaries need to be evaluated. Every three consecutive rows are different summaries of
+the same source document. You can evaluate a summary by clicking on a row, and annotating it. Optionally, you can
+click on “Label All Tasks” at the top of the screen.
+Annotation Task
+Each summary needs to be evaluated for its “correctness”. You need to provide a 0-5 judgment for the entire summary,
+where “correctness” can be defined as, “The absence of factual errors in the summary, where a factual error is a
+statement that contradicts the source document, or is not directly stated, heavily implied, or logically entailed by the
+source document”. For example,
+Source Document (snippet shown) = ..... Vitamin C was discovered in 1912, isolated in 1928, and, in 1933, was
+the first vitamin to be chemically produced. It is on the World Health Organization’s List of Essential Medicines.
+Vitamin C is available as an inexpensive generic and over-the-counter medication. Partly for its discovery, Albert
+Szent-Györgyi and Walter Norman Haworth were awarded the 1937 Nobel Prizes in Physiology and Medicine and
+Chemistry, respectively. Foods containing vitamin C include citrus fruits, kiwifruit, guava, broccoli, Brussels sprouts,
+bell peppers, potatoes, and strawberries. Prolonged storage or cooking may reduce vitamin C content in foods. . . . .
+Summary 1 (snippet shown) = ... Chicken contains vitamin C . . .
+Summary 2 (snippet shown) = ... Albert Szent-Györgyi won the 1955 Nobel Prize for discovering Vitamin C . . .
+Summary 3 (snippet shown) = ... Vitamin C was the first chemically produced Vitamin . . .
+Summary 4 (snippet shown) = ... Apple contains vitamin C . . .
+Errors marked in red. Here, the snippets for summary 1 are incorrect, summary 2 partially correct, and summary 3
+completely correct with respect to the source document. Summary 4 is incorrect with respect to the source document
+(since it’s never discussed), but a globally correct fact. You should treat such a summary as incorrect since it is not
+mentioned in the source document.
+(This is an illustrative example only, the actual annotation task has much longer summaries / source documents.)
+The rating scale is from 0 to 5, where 0 is the lowest possible rating (most or all of the summary is wrong / irrelevant to
+the source document), and 5 is the highest rating (most or all of the summary is correct).
+While it is compulsory to provide a judgment from 0 to 5 for each summary, you can optionally provide additional
+comments in your annotation. For instance, if the judgment needs to be more nuanced than a 5-point scale, you prefer
+to mark something like “3.5”, or you would like to add some other notes about your judgment.
+Press “Submit” after you have provided your annotation.
+Suggested workflow
+Every three consecutive rows contain different summaries for the same source document. We suggest the following
+workflow while annotating documents —
+1. Spend the first 15 minutes reading the source document and getting a general sense of the facts mentioned in the
+document.
+2. Spend 5 minutes to read and annotate the summaries in each of the three consecutive rows which correspond to the
+same document. Add optional comments / notes if necessary.
+3. In the last 5 minutes, re-calibrate your ratings across the three rows if needed (for instance, you significantly preferred
+the correctness of summary 1 vs summary 2, but you gave it the same rating in the initial pass). Add optional comments
+/ notes if necessary.
+Following this workflow, it should take 35 minutes to annotate each set of 3 rows. For 30 rows, this should take 6 hrs.
+Table 11: Annotation guidelines provided to annotators for COARSE evaluation of PubMed summaries (Ap-
+pendix B.2).
+

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+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+1
+InstructTTS: Modelling Expressive TTS in Discrete
+Latent Space with Natural Language Style Prompt
+Dongchao Yang*, Songxiang Liu*, Rongjie Huang, Guangzhi Lei, Chao Weng, Helen Meng, Fellow, IEEE and
+Dong Yu, Fellow, IEEE
+Abstract—Expressive text-to-speech (TTS) aims to synthesize
+different speaking style speech according to human’s demands.
+Nowadays, there are two common ways to control speaking styles:
+(1) Pre-defining a group of speaking style and using categorical
+index to denote different speaking style. However, there are
+limitations in the diversity of expressiveness, as these models can
+only generate the pre-defined styles. (2) Using reference speech
+as style input, which results in a problem that the extracted
+style information is not intuitive or interpretable. In this study,
+we attempt to use natural language as style prompt to control
+the styles in the synthetic speech, e.g., “Sigh tone in full of
+sad mood with some helpless feeling”. Considering that there
+is no existing TTS corpus which is proper to benchmark this
+novel task, we first construct a speech corpus, whose speech
+samples are annotated with not only content transcriptions but
+also style descriptions in natural language. Then we propose
+an expressive TTS model, named as InstructTTS, which is
+novel in the sense of following aspects: (1) We fully take the
+advantage of self-supervised learning and cross-modal metric
+learning, and propose a novel three-stage training procedure to
+obtain a robust sentence embedding model, which can effectively
+capture semantic information from the style prompts and control
+the speaking style in the generated speech. (2) We propose to
+model acoustic features in discrete latent space and train a
+novel discrete diffusion probabilistic model to generate vector-
+quantized (VQ) acoustic tokens rather than the commonly-used
+mel spectrogram. (3) We jointly apply mutual information (MI)
+estimation and minimization during acoustic model training to
+minimize style-speaker and style-content MI, avoiding possible
+content and speaker information leakage from the style prompt.
+Extensive objective and subjective evaluation has been conducted
+to verify the effectiveness and expressiveness of InstructTTS.
+Experimental results show that InstructTTS can synthesize high-
+fidelity and natural speech with style prompts controlling the
+speaking style. Synthesized samples are available 1.
+Index Terms—Text to speech, prompt-based learning, diffusion
+model, metric learning
+I. INTRODUCTION
+T
+EXT-to-speech (TTS) aims to generate human-like
+speech from input text, which attracts broad interest in the
+audio and speech processing community. Nowadays, the state-
+of-the-art TTS systems [1]–[3] are able to produce natural
+and high-quality speech. However, there still exists a big gap
+between TTS-synthetic speech and human speech in terms of
+Dongchao Yang and Helen Meng are with the Chinese University of Hong
+Kong. This work was done when Dongchao Yang was an intern at Tencent AI
+Lab. * denotes equal contribution with order determined by alphabetic order.
+Songxiang Liu, Guangzhi Lei, Chao Weng and Dong Yu are with Tencent
+AI Lab.
+Rongjie Huang is with the Zhejiang University, China.
+Songxiang Liu is the corresponding author.
+1http://dongchaoyang.top/InstructTTS/
+expressiveness, which limits the broad applications of current
+speech synthesis systems. Many researchers now focus on
+a more challenging task, i.e., expressive TTS, which aims
+to modeling and control the speaking style (e.g., emotion,
+speaking-rate and so on) in the generated speech according to
+human’s demands. We note that there are generally two types
+of methods in the literature to learn speaking style information:
+one uses auxiliary categorical style labels as the condition of
+the framework [4], [5], the other imitates the speaking style
+of a reference speech [6]–[9]. However, there are limitations
+in the diversity of expressiveness when categorical style labels
+are used, as these models can only generate a few pre-defined
+styles from the training set. Although TTS models using a
+reference utterance to model style generation can be trained
+in an unsupervised manner and generalizable to out-of-domain
+speaking styles, style information in the reference speech is
+not intuitive and interpretable. Moreover, it is hard to choose
+a reference speech sample in precise accordance of a user’s
+demand.
+For the first time, we study the modelling of expressive
+TTS with style prompt in natural language, where we meet
+with the following research problems: (1) how to train a
+language model that can capture semantic information from
+the natural language prompt and control the speaking style in
+the generated speech; (2) how to design an acoustic model
+to effectively model the challenging one-to-many learning
+problem of expressive TTS. In this paper, we will address
+these two challenges.
+The main contributions of this study are summarized as
+follows:
+(1) For the first time, we study the modelling of expressive
+TTS with natural language prompt, which brings us a step
+closer to achieve user-controllable expressive TTS.
+(2) We introduce a novel three stage training strategy to obtain
+a robust sentence embedding model, which can effectively
+capture semantic information from the style prompts.
+(3) Inspired by the success of large-scale language models,
+e.g., GPT3 and ChatGPT [10], we propose to model acoustic
+features in discrete latent space and cast speech synthesis
+as a language modeling task. Specifically, we train a novel
+discrete diffusion model to generate vector-quantized (VQ)
+acoustic feature rather than to predict the commonly-used mel-
+spectrogram.
+(4) We explore to model two types of VQ acoustic fea-
+ture: mel-spectrogram based VQ features and waveform-based
+VQ features. We prove that the two types of VQ features
+can be effectively modeled by our proposed novel discrete
+arXiv:2301.13662v1  [cs.SD]  31 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+2
+Discrete Diffusion
+Model
+Variance Adaptor
+Positional 
+Encoding
+Positional 
+Encoding
+Mel-spectrogram or waveform
+SALN adaptor
+Text Embedding
+Layer
+Content Prompt
+Speaker
+Embedding
+Layer
+(a) The overview of InstructTTS.
+Speaker
+ID
+Style
+Encoder
+Style
+Prompt
+Mel-spectrogram
+Phoneme Encoder
+Content 
+Encoder
+(b) The details of style encoder.
+Mel-VQ-Diffusion transformer decoder
+Token
+Embedding
+Encoder
+5
+1
+7 39
+M
+M
+0 M
+Diffusion 
+process
+5
+1
+7 39
+Decoder
+(c) The details of Mel-VQ-Diffusion.
+Text Embedding
+Layer
+Content Prompt
+ 
+Audio Encoder
+Mel-spectrogram 
+Style prompt
+Adaptor
+ Prompt Encoder
+Fig. 1. (a) shows the model architecture of our proposed InstructTTS. Where SALN denotes the style-adaptive layer normalization adaptor [14]. (b) shows
+the details of our proposed style encoder, which aims to extract style features from GT mel-spectrogram (training stage) or style prompt (inference stage). In
+Figure 1 (c), we give an example of discrete diffusion decoder to generate VQ mel-spectrogram acousic features (we name it as Mel-VQ-Diffusion).
+diffusion model. We must state that our waveform-based
+modelling method only needs one-stage training and it is
+a non-autoregressive model, which is far different from our
+concurrent work AudioLM [11], VALL-E [12] and MusicLM
+[13].
+(5) We jointly apply mutual information (MI) estimation
+and minimization during acoustic model training to minimize
+style-speaker and style-content MI, which avoiding possi-
+ble content and speaker information leakage from the style
+prompt.
+The rest of this paper is organized as follows: In Section II,
+we motivate our study by introducing the background and
+related work. In Section III, we present the details of datasets.
+In Section IV, we introduce the details of our proposed
+methods. The experimental setting, evaluation metrics and
+results are presented from Section V to Section VII. The study
+is concluded in Section VIII.
+II. RELATED WORK AND BACKGROUND
+This study is built on several previous works on cross-
+modal representation learning, vector quantization, diffusion
+probabilistic models and expressive TTS. We briefly introduce
+the related studies to set the stage for our research and
+rationalize the novelty of our contributions.
+A. Cross-modal Representation Learning
+Cross-modal representation learning aims to learn a com-
+mon latent space for different modal data (e.g. text and image,
+text and speech). In general, two different modal encoders are
+used to extract deep feature representation, then a variety of
+supervised or unsupervised strategies are devised to align the
+two modal representation spaces [15]–[17]. In our study, we
+expect to control the acoustic features (such as pitch, emotion
+and speed) by a natural language sentence. To realize this
+target, we turn to cross-modal representation learning. The
+details will be discussed later.
+B. Vector Quantization
+Vector quantization technique has been used in various
+fields, such as image [18]–[20] and speech processing [21]–
+[24]. VQ-VAE [18] was proposed to train an encoder to
+compress the image into a low-dimensional discrete latent
+space, then a decoder is used to recover the image from
+a group of discrete tokens. Inspired by VQ-VAE, a series
+of works adopt the idea to reconstruct mel-spectrogram or
+linear-spectrogram [25], [26]. Recently, a lot of works focus
+on reconstruct waveform by VQ-VAE. To supplement the
+information loss during the VQ process, a residual-VQ (R-
+VQ) [24] technique is proposed, which uses multiple different
+codebooks to encode the audio information. Nowadays, the
+majority of TTS systems focus on using an acoustic model
+(AM) to directly predict mel-spectrogram, then uses a pre-
+trained vocoder to recover waveform from the predicted mel-
+spectrogram [2], [27], [28]. However, the mel-spectrogram is
+highly correlated along both time and frequency axes in a
+complicated way, leading to a great difficulty for the AM
+to predict. Furthermore, the gap between the ground-truth
+(GT) mel-spectrogram and the predict one from AM degrades
+the performance due to the vocoder is trained on GT mel-
+spectrogram. In this study, instead of using AM to predict mel-
+spectrogram, we turn to predict learnable and vector-quantized
+acoustic representation, which is transformed to a discrete
+latent space.
+C. Expressive Text-to-speech
+Expressive TTS models have been studied for decades in the
+TTS community: Wang et al. [6] propose to use global style
+tokens to control and transfer the global style. Li et al. [29]
+adopt a multi-scale style encoder to assist expressive speech
+synthesis. Min et al. [14] propose Meta-StyleSpeech, which
+uses a meta-learning training strategy for multi-speaker TTS
+synthesis. SC-GlowTTS [30] proposed a speaker-conditional
+architecture that explores a flow-based decoder in a zero-
+shot scenario. Zhou et al. [31] propose a mixed emotion
+speech synthesis model, which can control multiple different
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+3
+emotions in one synthetic speech sample. Huang et al. [32]
+propose a multi-level style adaptor to transfer speaking style.
+Yang et al. [9] propose NoreSpeech, which can robust transfer
+style information from noisy reference speech. Liu et al. [33]
+attempt to use robust style descriptors to transfer style learned
+from low-quality but expressive speech data to a target voice.
+The most related to our work are Style-Tagging-TTS (ST-TTS)
+[34] and PromptTTS [35]. ST-TTS proposes to use style tag
+to guide the speaking style of synthsized speech, where style
+tag denotes short phrase or word representing the style of an
+utterance, such as emotion, intention, and tone of voice. In
+this study, we attempt to use longer natural language as style
+descriptions to control the styles in the synthetic speech, which
+is more complicated due to longer natural language prompts
+carry out more abundant semantic information and results in
+more complicated acoustic characteristic. Our concurrent work
+PromptTTS [35] proposed a similar idea with us, using a
+sentence as style prompt to control the style information in
+TTS systems. They define 5 different style factors (gender,
+pitch, speaking speed, volume, and emotion), and they assume
+the prompts have obvious style factor words, such as low-pitch,
+high-speaking speech and so on, which means that model can
+get style information from local-level description. Different
+from PromptTTS, our study does not apply constraint on the
+form of the style prompts and allows the user to use any free-
+form natural language to describe a speaking style, resulting
+in a much more challenging machine learning problem. Fur-
+thermore, we focus on Mandarin Chinese TTS and construct
+the first Mandarin Chinese speech corpus applicable for style-
+prompt-controllable expressive TTS.
+D. Diffusion Probabilistic Models
+Diffusion generative models are first proposed in [36] and
+achieve strong results in image generation [37]–[40] and
+speech synthesis [41]–[44]. Diffusion models with discrete
+state spaces are first introduced by Sohl-Dickstein et al.
+[36], who considered a diffusion process over binary random
+variables. Hoogeboom et al. [45] extend the model to categor-
+ical random variables with transition matrices characterized
+by uniform transition probabilities. Jacob et al. [46] further
+improve and extend discrete diffusion models by using a more
+structured categorical corruption process to corrupt the forward
+process. Many works have successfully applied discrete diffu-
+sion models in image or sound generation, e.g., D3PMs [46]
+VQ-Diffusion [38], DiffSound [26]. However, no one attempts
+to apply the discrete diffusion model to synthesize speech. In
+the following, we briefly review background knowledge of
+diffusion models.
+1) Vanilla Diffusion Model: A diffusion model consists of
+forward process and reverse process. The forward process
+attempts to corrupt the original data x0 into the noisy latent
+variable xT which follows a simple stationary distribution
+(e.g., Gaussian distribution), and the reverse process learns
+to recover the original data x0 from xT .
+Forward process Given the audio data x0, the forward
+process aims to corrupt the data x0 ∼ q(x0) into a sequence of
+increasingly noisy latent variables x1:T = x1, x2, ..., xT . Each
+of the noisy latent variables xt has the same dimensionality
+as x0. The forward process from data x0 to the variable xT
+can be formulated as a fixed Markov chain
+q(x1:T |x0) =
+T
+�
+t=1
+q(xt|xt−1).
+(1)
+Following [36], Gaussian noise is selected in each step, so
+that the conditional probability distribution is modeled as
+q(xt|xt−1) = N(xt; √1 − βtxt−1, βtI), where βt is a small
+positive constant. According to the pre-defined noise schedule
+β1, β2, ..., βT , the overall process gradually converts clean x0
+to a latent variable with an isotropic Gaussian distribution of
+p(xT ) = N(0, I). Due to the properties of the Markov chain,
+the probability distribution q(xt|x0) can be conveniently de-
+rived as
+q(xt|x0) = N(xt; √αtx0, (1 − αt)I),
+(2)
+where αt = 1 − βt and αt = �t
+s=1 αs.
+Reverse process The reverse process converts the latent
+variable xT ∼ N(0, I) into x0, whose jointly probability
+follows
+pθ(x0:T ) = p(xT )
+T
+�
+t=1
+pθ(xt−1|xt),
+(3)
+where pθ(·) is the distribution of the reverse process with
+learnable parameters θ. The posterior q(xt−1|xt, x0) can be
+derived according to the Bayes formula. In order to optimize
+the generative model pθ(x0) to fit the data distribution q(x0),
+one typically optimizes a variational upper bound on the
+negative log-likelihood [47].
+2) Discrete Diffusion model: In discrete diffusion model,
+a transition probability matrix is defined to indicate how x0
+transits to xt for each step of forward process. Assuming that
+x0 ∈ ZN and xk
+0 ∈ {1, 2, ..., P}, without introducing confu-
+sion, we omit the superscript k in the following presentation.
+The matrices [Qt]ij = q(xt = i|xt−1 = j) ∈ RP ×P defines
+the probabilities that xt−1 transits to xt. Then the forward
+process for the whole token sequence can be written as
+q(xt|xt−1) = c⊤(xt)Qtc(xt−1),
+(4)
+where c(x) denotes the one-hot column vector of x. The cate-
+gorical distribution over xt is given by the vector Qtc(xt−1).
+Due to the property of Markov chain, one can marginalize
+out the intermediate steps and derive the probability of xt at
+arbitrary timestep directly from x0 as
+q(xt|x0) = c⊤(xt)Qtc(x0), with Qt = Qt . . . Q1.
+(5)
+Besides, q(xt−1|xt, x0) can be derived according to the Bayes
+formula:
+q(xt−1|xt, x0) = q(xt|xt−1, x0)q(xt−1|x0)
+q(xt|x0)
+=
+�
+c⊤(xt)Qtc(xt−1)
+��
+c⊤(xt−1)Qt−1c(x0)
+�
+c⊤(xt)Qtc(x0)
+.
+(6)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+4
+TABLE I
+EXAMPLE STYLE PROMPTS FROM DIFFERENT CORPORA. SINCE NLSPEECH CORPUS IS IN MANDARIN CHINESE, WE ADDITIONALLY PROVIDE THE
+TRANSLATED VERSION. THE FSNR0 IS IN KOREAN, WE PROVIDE THE TRANSLATED ONES IN THE TABLE.
+FSNR0 [34]
+PromptSpeech [35]
+NLSpeech (translated)
+Seem sad
+A distressful male sound appeared in low volume
+The tone of the shock question revealed the sad feelings
+Bitter
+He sadly turns down his volume, pitch and speed
+It was a fiery expression of disapproval and condemnation,
+with a palpable sense of irony, a tinge of disgust and disdain
+Pleased
+The ladylike person made an increment of the volume and pitch
+There was a sense of joy in the words, an expression of
+joy in the heart, mixed with pride.
+In a hurry
+Men, low tone, said loudly and quickly
+His voice grew more agitated,
+and his tone revealed an urge and urgency.
+III. DATASET
+We use an internally collected Mandarin Chinese speech
+corpus named NLSpeech to conduct experimental evaluation
+since there is no openly available Mandarin Chinese speech
+corpus with rich style prompts. The corpus contains 44 hours
+of speech data (in total 32k utterances) from 7 speakers
+(5 female and 2 male). Audio waveform has a sampling
+rate of 24kHz. We randomly spare 0.1 hours of data as the
+validation set, another 0.1 hours of data as the test set and
+the remaining data as the training set. Each utterance has 5
+style prompts labeled by different annotators. To obtain high-
+quality annotations, we ask annotators to follow three steps of
+annotation strategy:
+• Step-1: The annotators first use one word to describe the
+overall perceived emotion of an utterance;
+• Step-2: The annotators then listen to the utterance care-
+fully and describe the emotion level of the utterance with
+one word;
+• Step-3: The annotators write a complete sentence in
+natural language to describe the style of the utterance.
+Note that we ask annotators to not care about the speech
+content, which may influence the perception of emotion and
+style. Table I shows example style prompts in our dataset, and
+we also compare NLSpeech with other existing related cor-
+pora, including the FSNR0 corpus [34] and the PromptSpeech
+corpus [35]). We note that the style prompts in NLSpeech
+are in free-form natural language sentences which are more
+consistent with those used in our daily life, while those in
+the FSNR0 and the PromptSpeech corpora are somewhat
+constraint to some degree. Meanwhile, this also brings us a
+challenging TTS problem since natural language sentences al-
+low for expressing virtually any concepts. Compact and infor-
+mative representation of style prompt is therefore paramount
+to achieve effective style controlling during speech synthesis.
+IV. PROPOSED METHOD
+The overall architecture of the proposed InstructTTS frame-
+work is demonstrated in Figure 1, which consists of five parts
+including a content encoder, a style encoder, a speaker encoder,
+a style-adaptive layer normalization (SALN) adaptor and a
+discrete diffusion decoder. The detailed design of each part
+will be introduced in this section.
+A. Content Encoder
+The content encoder aims to extract content representation
+from the content prompts. We follow the architecture of Fast-
+Prompt Encoder
+Audio Encoder
+Style Prompt 
+Audio embedding
+Style embedding
+Metric
+Learning
+Objective
+Audio
+Semantic Hyper-Sphere
+Fig. 2. The model architecture of cross-modal representation learning.
+Speech2, which consists of 4 feed-forward transformer. The
+hidden size, number of attention heads, kernel size and filter
+size of the one-dimensional convolution in the FFT block are
+set as 256, 2, 9 and 1024, respectively. After that, a variance
+adaptor is used to predict information such as duration and
+pitch that is closely related to the style of synthetic speech.
+B. Style encoder
+The style encoder module, as shown in Fig. 1 (b), includes
+three parts: A pre-trained robust style prompt embedding
+model, an adapt layer to map the style embedding into a
+new latent space, an audio encoder that used to encode style
+information from the target mel-spectrogram. Note that our
+pre-trained robust prompt embedding model is fixed when
+we train our TTS system. In the training stage, one of the
+training target is to minimize the distance between style
+prompt embedding and audio embedding. We note that the
+audio encoder may encode speaker and content information.
+To make sure the audio encoder only encodes the style-
+related information, during training, we jointly minimize the
+style-speaker mutual information (i.e., I(ze; zsid)) and style-
+content mutual information (i.e., I(ze; c)). Mutual information
+(MI) is a key measurement of correlation between random
+variables. However, MI of high-dimensional random variables
+with unknown distribution is intractable to compute. Previous
+works focus on either estimating the MI lower bound or the
+MI upper bound. MINE [48] and InfoNCE [18] compute a
+lower bound as the MI estimators while CLUB [49] computes
+an upper bound as the MI estimator. In this work, we use
+the CLUB method to minimize style-speaker and style-content
+MI to avoid content and speaker information leakage from the
+mel-spectrogram during training.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+5
+C. Style Prompt Embedding Model
+To extract style representation from the style prompts,
+we adopt a RoBERTa model [50] as prompt embedding
+model. Assuming we have a style prompt sequence S =
+[S1, S2, ..., SM], where M denotes the sequence length. We
+add a [CLS] token to the start of prompt sequence, and then
+feed into the prompt embedding model. After that, we take
+the representation of [CLS] token as the style representation
+of this sentence. In order to stably control the style in the
+output of TTS through natural language description, the quality
+of prompt embedding is of great importance, which should
+satisfy two conditions: (1) the learned style prompt space must
+be able to contain important semantic information; (2) the
+distribution of prompt embedding space should be relatively
+uniform and smooth, and the model can be generalized to the
+style description not seen in the training. To realize this target,
+we propose a novel three-stage training-fine-tuning strategy.
+The details are shown as follow.
+1) Training a base language model for Chinese: Given that
+most of open-source pre-trained language models are trained
+on English data, we first train a RoBERTa model on Chinese
+data.
+2) Fine-tuning the pre-trained language model on labeled
+data: We use a small amount of Chinese natural language
+inference (NLI) to fine-tune the model parameters in a super-
+vised way to achieve a better semantic representation of the
+model. Specifically, we follow the training strategy proposed
+in SimCSE [51], which using an InfoNCE loss [52] objective
+to fine-tune our pre-trained RoBERTa model.
+3) Cross-modal
+representation
+learning
+between
+style
+prompts and speech: We hope that the prompt embedding vec-
+tor from the style prompt sentence and the style representation
+vector from the speech can be mapped to the shared semantic
+space, so that we can control the style in the TTS output
+through the style description when testing. Thus, we propose
+a cross-modal representation learning process based on metric
+learning, as Figure III shows. Specifically, we build a audio-
+text retrieval task based on the style-prompt and audio pair
+in our NLSpeech dataset. For any style prompt, we randomly
+choose N − 1 negative audio samples, combined with one
+positive audio sample to build a training batch. Similarly, for
+one audio sample, we can also build a training batch that
+including one positive style prompt and N − 1 negative style
+prompts. Inspired by previous audio-text retrieval works [16],
+[53], we adopt contrastive ranking loss [54] and InfoNCE loss
+[52] as the training objective respectively. Experiments results
+show that InfoNCE loss brings better retrieval performance.
+The details will be introduced in Experiments part.
+D. Modelling Mel-spectrograms in Discrete Latent Space
+In this part, we introduce our hypothesis: modelling mel-
+spectrograms in discrete latent space is a suitable way for
+expressive TTS. Then we introduce how to utilize VQ-VAE as
+intermediate representations to help model mel-spectrogram.
+Lastly, we introduce our proposed non-autoregressive mel-
+spectrogram token generation model, which is based on dis-
+crete diffusion models.
+Mel-VQ-VAE
+Decoder
+Mel-spectrogram Codebook
+Z2 Z3
+Z1
+Z4
+ZK
+Discriminator
+mel-spectrogram
+tokens
+VQ(.)
+Encoder
+5
+1
+7 39
+5
+1
+7 39
+Neural audio codec
+Encoder
+VQ1(.)
+VQ2(.)
+5
+1
+7 39
+Residual 1
+1
+2
+3 33
+Residual 2
+VQ8(.)
+12
+0
+7 4
+5
+1
+7 39
+1
+2
+3 33
+12
+0
+7 4
+Decoder
+Residual 7
+Fig. 3.
+The overall architecture of the VQ-VAE and Neural audio codec
+models.
+Most of the text-to-speech (TTS) methods [2], [42], [55]
+directly learn the mapping from text to mel-spectrogram in
+continuous space. Then they use a pre-trained vocoder to de-
+code the predicted mel-spectrogram into waveform. However,
+frequency bins in a mel-spectrogram are highly correlated
+along both time and frequency axes in a complicated way,
+especially when the speech sample conveys highly expres-
+sive emotions and speaking styles, leading to a challenging
+modeling problem. Furthermore, the gap between the ground-
+truth mel-spectrogram and the predicted one also influence
+the synthesis performance [56]. In this study, we propose
+to model mel-spectrogram in discrete latent space, but still
+use a HiFi-GAN vocoder [56] to recover waveform from
+mel-spectrogram. Specifically, we first pre-train a VQ-VAE
+with a large-scale speech dataset, so that the pre-trained Mel-
+VQ-VAE encodes all of the linguistic, pitch, energy, emotion
+information into the latent codes. Then we regard the vector
+quantized latent codes as the predicting targets and hence
+model the mel-spectrogram in the discrete latent space. A
+similar idea modeling mel-spectrogram in discrete latent space
+is applied in VQ-TTS [57], which utilizes self-supervised VQ
+acoustic feature (vq-wav2vec [21]) rather than traditional mel-
+spectrogram as intermediate prediction target. VQ-TTS builds
+an autoregressive classification model for prosody label and
+VQ acoustic feature. Different from VQ-TTS, we still use
+mel-spectrogram as intermediate acoustic feature and use a
+VQ-VAE model to transform mel-sepctrogram into a latent
+discrete space for reducing the time-frequency correlations.
+As Figure 3 shows, a mel-spectrogram can be represented by
+a group of mel-spectrogram tokens. Thus, the mel-spectrogram
+synthesis problem transfers to predicting a group of discrete
+tokens, which can be seen as a language modeling problem.
+In the following, we will introduce the details of VQ-VAE,
+then we introduce our proposed Mel-VQ-Diffusion decoder.
+1) VQ-VAE: VQ-VAE is trained to approximate an input
+using a compressed intermediate representation, retrieved from
+a discrete codebook. VQ-VAE consists of three parts: an
+encoder Evq, a decoder G and a codebook Z = {zk}K
+k=1 ∈
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+6
+RK×nz containing a finite number of embedding vectors,
+where K denotes the size of the codebook and nz is the
+dimension of codes. Given a mel-spectrogram s ∈ RF ×L, the
+input s is firstly encoded into a lower-dimension representation
+ˆz = Evq(s) ∈ RF ′×L′×nz where F ′ × L′ represents the
+reduced frequency and time dimension . Then a spatial-wise
+quantizer Q(.) is used to map each spatial feature ˆzij into
+its closest codebook entry zk to obtain a spatial collection of
+spectrogram tokens zq
+zq = Q(ˆz) :=
+�
+arg min
+zk∈Z
+||ˆzij −zk||2
+2 for all (i, j) in (F ′, L′)
+�
+(7)
+Lastly, the mel-spectrogram can be faithfully reconstructed
+via the decoder, i.e., ˆs = G(zq). In this study, we follow
+VQGAN [20], which adds an adversarial loss [58] to improve
+the reconstruction performance.
+2) Mel-VQ-Diffusion decoder: With help of the pre-trained
+Mel-VQ-VAE, we transfer the problem of mel-spectrogram
+prediction into that of predicting a group of quantization
+tokens. To generate high-quality mel-spectrogram tokens while
+maintaining fast inference speed, we propose a Mel-VQ-
+Diffusion decoder. In the following, we first introduce the
+basic idea of Mel-VQ-Diffusion, then we summarize the
+training target. Lastly, we introduce a classifier-free guidance
+to enhance the connection between conditional information
+and training target.
+Given the paired training data (x0, y), where y denotes
+the combination of phone features, style features and speaker
+features, x0 denotes the ground truth mel-spectrogram tokens.
+We first build a diffusion process, which corrupts the distribu-
+tion of p(x0) into a controllable stationary distribution p(xT ).
+Then we build a Transformer-based neural network [59] to
+learn to recover the p(x0) conditioned on the y. Inspired by
+previous works [26], [60], we utilize a mask and uniform
+transition matrix to guide the diffusion process. The transition
+matrices Qt ∈ R(K+1)×(K+1) is defined as
+Qt =
+�
+����
+αt + βt
+βt
+βt
+· · ·
+0
+βt
+αt + βt
+βt
+· · ·
+0
+...
+...
+...
+...
+...
+γt
+γt
+γt
+· · ·
+1
+�
+���� .
+(8)
+The transition matrix denotes that each token has a probability
+of γt to transition to [MASK] token, a probability of Kβt be
+resampled uniformly over all the K categories and a probability
+of αt = 1 − Kβt − γt to stay the same token. Based on
+the transition matrix, we can derive the stationary distribution
+p(xT ) as
+p(xT ) = [βT , βT , · · · , γT ],
+(9)
+where αT = �T
+t=1 αt, γT = 1 − �T
+t=1(1 − γt) and βT =
+(1 − αT − γT )/K. We can calculate q(xt|x0) according to
+following formula:
+Qtc(x0) = αtc(x0) + (γt − βt)c(K + 1) + βt.
+(10)
+Decoder Training Target We train a network pθ(xt−1|xt, y)
+to estimate the posterior transition distribution q(xt−1|xt, x0).
+The network is trained to minimize the variational lower bound
+(VLB).
+Ldiff =
+T −1
+�
+t=1
+�
+DKL[q(xt−1|xt, x0)||pθ(xt−1|xt, y)]
+�
++ DKL(q(xT |x0)||p(xT )),
+(11)
+Enhancing the connection between x0 and y Based on
+previous discussion, we can see that the conditional informa-
+tion y inject into the network, to help optimize p(xt−1|xt, y).
+However, in the last few steps, when xt includes enough infor-
+mation, the network may ignore the conditional information y
+in the training stage. To solve this problem, we introduce the
+classifier free guidance [61], [62] to enhance the connection
+between x0 and y. Specifically, instead of only optimizing
+p(x|y), we expect to optimize the following target function:
+log(p(x|y)) + λ log(p(y|x)),
+(12)
+where λ is a hyper-parameter to control the degree of poste-
+rior constraint. Using Bayes’s theorem, Formula (12) can be
+derived as:
+arg max
+x
+[log p(x|y) + λ log p(y|x)]
+= arg max
+x
+[(λ + 1) log p(x|y) − λ log p(x)]
+= arg max
+x
+[log p(x) + (λ + 1)(log p(x|y) − log p(x))].
+(13)
+To predict the unconditional mel-spectrogram token, we follow
+[62] to use a learnable null vector n to represent unconditional
+information y. In the training stage, we set 10% probability
+to use null vector n. In the inference stage, we first generate
+the conditional mel-spectrogram token’s logits pθ(xt−1|xt, y),
+then predict the unconditional mel-spectrogram token’s logits
+pθ(xt−1|xt, n). Based on formula (13), the next step sample
+probability pθ(xt−1|xt, y) can be re-write as:
+pθ(xt−1|xt, n) + (λ + 1)(pθ(xt−1|xt, y) − pθ(xt−1|xt, n))
+(14)
+E. Modelling Waveform in Discrete Latent Space Via Residual
+Vector Quantizer
+Inspired by the success of neural audio codec models, such
+as Soundstream [24] and Encodec [23]. In this study, we
+additionally investigate directly predicting waveform in the
+discrete latent space with the help of large-scale pre-trained
+neural audio codec models.
+Recently, many methods have been proposed to generate
+speech using neural codec model, e.g. AudioLM [11] trains
+speech-to-speech language models on both k-means tokens
+from a self-supervised model and acoustic tokens from a
+neural codec model, leading to high-quality speech-to-speech
+generation. The concurrent work VALL-E [12] is the most
+related to ours, VALL-E proposes to train a two-stage models
+to synthesize speech based on text input and reference audio.
+However, VALL-E needs a two-stage training strategy, and
+the first stage is an autoregressive language model, which
+significant influence the synthesis speed. In this study, we pro-
+pose a non-autoregressive model based on discrete diffusion
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+7
+Token
+Embedding
+Encoder
+5
+1
+7 39
+1
+2
+3 33
+12
+0
+7 4
+1
+M
+M M
+M
+M
+3 M
+M
+M
+M 0
+Diffusion 
+process
+Wave-VQ-
+Diffusion
+Transformer
+Mutil-head
+output layer
+5
+1
+7 39
+1
+2
+3 33
+12
+0
+7 4
+Decoder
+Convolution layer
+Downsample
+Copy and Add
+Upsample
+L denotes the number of tokens generated by one
+codebook, Nq denotes the number of codebook.
+Fig. 4. The framework of our proposed U-transformer-based discrete diffusion decoder.
+model, which significantly improve the synthesis speed while
+maintaining high-quality synthesis performance.
+As Figure 3 shows, compared to Mel-VQ-VAE, the neural
+audio codec model includes more codebooks. Although using
+more codebooks can bring better reconstruction performance,
+it also raise a new research problem: How to model such long
+sequence by transformer? As we known, the computational
+complexity of transformer is related to the sequence length.
+For a 10s speech with 24k sampling rate, if we use 8
+codebooks and set 240 times downsampling in the encoder,
+we will get 8000 tokens. Using a transformer based model
+to handle such long sequence is challenging due to GPU
+memory limitation, thus it is necessary to seek novel strategy
+for long sequence modelling. In this study, we propose a
+U-Transformer architecture to simultaneously model multiple
+codebooks. As Figure 4 shows, we first use several convolution
+layers to downsample the input codebook matrix along the
+codebook number dimension, after the convolution layers,
+we use a transformer to model the relationship of tokens in
+latent space. After that, we use several convolution layers and
+upsampling layers to recover the codebook number dimension.
+Lastly, we use different output layers to output prediction
+results for each codebook simultaneously.
+1) Wave-VQ-Diffusion:
+There are three differences in
+Wave-VQ-Diffusion comparing to Mel-VQ-Diffusion: (1) We
+adopt a U-transformer architecture to model multiple code-
+books simultaneously. Note that we use the same transformer
+architecture as that in Mel-VQ-Diffusion. (2) We use different
+embedding table for different codebook, due to the fact that
+tokens from different codebooks follows different data distri-
+butions. (3) We design an improved mask and uniform strategy
+for the diffusion process, which is based on a principle that
+the information included in the codebook is gradually decrease
+from codebook 1 to Nq. The first codebook includes the
+most of text, style, speaker identity information, the following
+codebooks mainly include the fine-grained acoustic details,
+which is crucial for the speech’s quality. We conjecture that
+the first codebook’s tokens are easy to recover conditioned
+on y, instead the following codebook’s tokens are hard to
+recover due to the they have not obvious connection with y.
+Following the easy-first-generation principle, we should mask
+the last codebook (e.g. codebook Nq) at the start of the forward
+process and mask the foremost codebook (e.g. codebook 1) at
+the end of the forward process such that the learnable reverse
+process follows an easy-first generative behavior. However,
+previous commonly-used mask and uniform strategy assumes
+all of the token in the sequence are of the same importance,
+which violates the easy-first-generation principle. To solve this
+problem, we propose an improved mask and uniform strategy,
+whose details are presented in the following.
+Improved Mask and uniform strategy We dynamically al-
+locate different weights for different codebooks when we pre-
+define the transition matrix. Considering these aforementioned
+properties, we construct αi
+t , γi
+t and β
+i
+t as follows
+αi
+t = 1 − t
+T −
+exp( i%Nq
+2∗Nq )
+2 ∗ T
+,
+γi
+t = t
+T +
+exp( i%Nq
+2∗Nq )
+2 ∗ T
+,
+β
+i
+t = (1 − αi
+t − γi
+t)/K,
+(15)
+where Nq denotes the the number of codebooks in neural audio
+codec model, i denotes the token position in the sequence. In
+our study, we concatenate all of the tokens from codebook 1
+to codebook Nq.
+F. The Training and Inference Details
+In this section, we summarize the overall training objective
+and the inference process.
+1) Training objective: Our proposed InstructTTS can be
+trained in an end-to-end manner. The overall training objective
+is as follows:
+L = Ldiff + Lvar + λ1I(ze; c) + λ2I(ze; zsid)+
+λ3DEuc(zc, ze) − β1F1(θ1) − β2F2(θ2).
+(16)
+where Ldiff denotes the diffusion loss, Lvar denotes the dura-
+tion, pitch and energy reconstruction loss. I(.) denotes mutual
+information, DEuc denotes the L2 loss. F1(θ1) and F2(θ2)
+denote the likelihood approximation model of qθ1(zsid|ze)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+8
+Algorithm 1 Training of the InstructTTS.
+Require:
+Pre-trained prompt encoder, A transition matrix Qt,
+timestep T, network parameters θ, training epoch N,
+NLSpeech dataset D, the encoder of VQ-VAE Evq.
+1: for i = 1 to N do
+2:
+for (conetent prompt, style prompt, audio) in D do
+3:
+mel = get mel spectrogram(audio);
+4:
+x0 = Evq(mel);
+5:
+c =ContentEncoder(content prompt);
+6:
+ze =AudioEncoder(mel);
+7:
+zp =PromptEmb(style prompt);
+8:
+zs =SpeakerEmb(speaker id);
+9:
+y = c + ze + zs;
+10:
+sample t from Uniform(1, 2, 3, ..., T);
+11:
+sample xt from q(xt|x0) based on formula (10);
+12:
+estimate pθ(xt−1|xt, y);
+13:
+calculate loss according to formula (16);
+14:
+update network θ;
+15:
+end for
+16: end for
+17: return network θ.
+Algorithm 2 Inference of the InstructTTS.
+Require:
+Time stride ∆t, timestep T, Content Prompt, Style
+Prompt, the decoder of VQ-VAE G, network θ, stationary
+distribution p(xT );
+1: t = T, c =ContentEncoder(content prompt);
+2: zs =SpeakerEmb(speaker id);
+3: zp =PromptEmb(style prompt);
+4: y = c + zp + zs;
+5: sample xt from p(xT );
+6: while t>0 do
+7:
+sample xt based on formula (14)
+8:
+t ← (t − ∆t)
+9: end while
+10: return G(xt).
+and qθ2(ze|c) respectively. Details about the MI estimation
+and minimization can be found in [49]. The whole training
+process is summarized on Algorithm 1. Note that we assume
+a Mel-VQ-Diffusion decoder is used in Algorithm 1. When we
+use a Wave-VQ-diffusion decoder, a similar process is used.
+2) Inference: In the inference process, we directly use the
+feature extracted by style prompt embedding model as the style
+features. In our experiments, we set the T = 100 and ∆t = 1.
+The whole inference process is summarized on Algorithm 2.
+V. EXPERIMENTAL SETUP
+A. Dataset and Data Pre-processing
+1) Dataset for Vector Quantization Pre-training: To obtain
+a robust and acoustic-informative Vector Quantization model,
+we combine one internal dataset with three commonly-used
+public-available TTS datasets: (1) Our internal dataset, which
+is a Mandarin Chinese speech corpus, including 300h speech
+data. (2) The VCTK dataset 2. (3) The AISHELL3 dataset [?].
+(4) The LibriTTS clean dataset [63]. In total, the training set
+has 669 hours speech data.
+2) Dataset for InstructTTS: We use our internal dataset
+NLSpeech as our training and testing dataset. The details can
+refer to Section III.
+3) Data pre-processing: All audio clips have a sampling
+rate of 24kHz. For Mel-VQ-VAE pre-training, the log mel-
+spectrograms extracted using a 1024-points Hanning window
+with 240-points hop size and 80 mel bins. The PyWorld toolkit
+3 is used to compute F0 values from speech signals. Energy
+features are computed by taking the l2-norm of frequency bins
+in STFT magnitudes.
+B. Implementation Details
+We first pre-train the Mel-VQ-VAE and neural audio codec
+models. Then we fix the pre-trained model, and train the
+InstructTTS model in an end-to-end manner. In the following,
+we will introduce the details of network structure and training
+strategy.
+1) VQ-VAE: In this study, we follow VQ-GAN [20], [25],
+adopting similar network architecture for the VQ-VAE encoder
+Evq, decoder G, and discriminator D. To preserve more time-
+dimension information, we set a downsampling factor of 2
+along the time axis, and a downsampling factor of 20 along
+the frequency axis. For the codebook Z, the dimension of each
+code word vector nz is set as 256, and the codebook dictionary
+size K is set as 512. The learning rate is fixed and determined
+as a product of a base learning rate, the number of GPUs used
+and the batch size. In our experiments, the base learning rate
+is set as 1 × E−6. The Adam optimizer [64] (the betas are
+0.5 and 0.9) is adopted to optimize weights. We train VQ-
+VAE with batches of 24 mel-spectrograms on 8 Nvidia V100
+GPUs. The training takes about 3 days on the our dataset.
+2) Neural Audio Codec Model: Inspired by the success of
+Encodec [23] and SoundStream [24], we adopt similar network
+architecture with the Encodec model. Specifically, the encoder
+model E consists with a 1D convolution layer with 32 hidden
+channels and a kernel size of 7, followed by 4 convolution
+blocks. Each convolution block is composed of a single
+residual unit followed by a down-sampling layer consisting
+with a strided convolution layer, with a kernel size twice of
+the stride. The residual unit contains two convolution layers
+and a skip connection. The number of channels is doubled
+whenever down-sampling occurs. The convolution blocks are
+followed by a two-layer LSTM for sequence modeling and a
+final 1D convolution layer with a kernel size of 7 and 32 output
+channels. In our study, we set the strides as S = [6, 5, 4, 2]. We
+set the maximum codebook size as 12 in the training process.
+Similar to SoundStream [24], quantizer dropout is used. To
+maintain high-quality audio reconstruction performance, a
+multi-scale STFT-based (MS-STFT) discriminator is used. We
+train the neural audio codec model with batches of 32 audio
+segments (we randomly crop 1 second segment from a audio
+sample) on 32 Nvidia V100 GPUs. The training takes about
+3 days for 300 epochs in our dataset.
+2https://datashare.ed.ac.uk/handle/10283/2651
+3https://github.com/JeremyCCHsu/Python-Wrapper-for-World-Vocoder
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+9
+3) InstructTTS: Our proposed InstructTTS consists of three
+main parts: style encoder, content encoder and discrete diffu-
+sion decoder. For the content encoder, we follow the Fast-
+Speech2 [2], using the same architecture for the phoneme
+encoder and the variance adaptor. For the style encoder, which
+includes a pre-trained Prompt encoder model (the details refer
+to Section IV-C) and an audio encoder. The audio encoder
+consists of two convolution layers and one multi-head attention
+module. For the discrete diffusion model, we follow the similar
+architecture as [26], we built a 12-layer 8- head transformer
+with a dimension of 256 for the decoder. Each transformer
+block contains a full-context attention, a linear fusion layer
+to combine conditional features and a feed-forward network
+block. For the default setting, we set timesteps T = 100. We
+adopt the linear schedule strategy, which linearly increase γt
+and βt from 0 to 0.9 and 0.1, and decrease αt from 1 to 0. We
+optimize our network using the AdamW optimizer [65] with
+β1 = 0.9 and β2 = 0.94. The basic learning rate is 3 × E−6,
+and batch size is 16 for each GPU.
+C. Baseline Approach
+In the literature, there is no existing expressive TTS model
+using natural language style prompt to control stylish gener-
+ation. Following traditional neural TTS paradigm which pre-
+dicts intermediate acoustic features, such mel-spectrograms,
+from text input, we adapt the StyleSpeech model proposed
+in [14] as the baseline approach. We replace the Mel-Style-
+Encoder in the StyleSpeech model with the same style encoder
+module used in InstructTTS, making the comparison as fair
+as possible. The baseline model uses the same HiFi-GAN
+vocoder to generate waveform as the proposed model.
+VI. EVALUATION METRIC
+A. Objective Evaluation
+We evaluate the synthesized speech from two aspects:
+speech quality and prosody similarity. For speech quality, we
+adopt Mel-cepstral distorion (MCD) [66], structural similarity
+index measure (SSIM) [67] and Short-Time Objective Intelli-
+gibility (STOI) [68] to evaluate the speech quality. For prosody
+similarity, we use three pitch-related metrics: Gross Pitch Error
+(GPE), Voicing Decision Error (VDE) [69] and F0 Frame
+Error (FFE) [70]. GPE, VDE and FFE are widely applied to
+evaluate the performance of expressive TTS. The details of
+these metrics will be introduced as follows.
+1) Mel-cepstral distorion: Spectral features, based on the
+short-term power spectrum of sound, such as Mel-cepstral
+coefficients (MCEPs), contain rich information about expres-
+sivity and emotion [71]. Mel-cepstral Distortion (MCD) [66]
+is a widely adopted metric to measure the spectrum similarity,
+which is computed as
+MCD = 1
+T
+T −1
+�
+t=0
+�
+�
+�
+�
+M
+�
+m=1
+(cm,t − ˆcm,t)2,
+(17)
+where cm,t and ˆcm,t denote the m-th mel-frequency cepstral
+coefficient (MFCC) of the t-th frame from the reference and
+synthesized speech. We sum the squared differences over the
+first M MFCCs. In this study, we set M = 24.
+2) SSIM and STOI: Structural similarity index measure
+(SSIM) [67] and Short-Time Objective Intelligibility (STOI)
+[68] are effective metrics to evaluate the speech clarity and
+intelligibility. Following previous work [72], we also adopt
+them as one of the metrics for speech quality.
+3) Prosody-related metrics: Given that pitch is considered
+as a major prosodic factor contributing to speech emotion and
+closely correlated to the activity level [73], [74], in this study,
+we adopt three common pitch similarity metrics to evaluate the
+synthesis results, which are Gross Pitch Error (GPE), Voicing
+Decision Error (VDE) and F0 Frame Error (FFE) [70], with
+detailed descriptions as follows:
+1) Gross Pitch Error (GPE): measures the pitch similarity
+between a pair of compared utterances.
+2) Voice Decision Error (VDE): measures the difference of
+voiced/unvoiced decision between a pair of compared
+utterances.
+3) F0 Frame Error (FFE): reflects both pitch similarity and
+voiced/unvoiced decision differences between a pair of
+compared utterances.
+The metrics of GPE, VDE and FFE have been used as common
+objective evaluation metric for expressive TTS [7].
+B. Subjective Evaluation
+To further validate the effectiveness of our proposed method,
+we conduct subjective evaluation from two aspects: speech
+quality and style relevance.
+1) Speech quality: We first conduct the Mean Opinion
+Score (MOS) test to evaluate speech quality, which aims to
+evaluate the speech’s naturalness, fidelity and intelligibility.
+All participants are asked to listen to the reference speech
+(“Ground truth”) and the synthesized speech and score the
+“quality” of each speech sample on a 5-point scale (‘5’ for
+excellent, ‘4’ for good, ‘3’ for fair, ‘2’ for poor, and ‘1’ for
+bad). Each audio sample is rated by at least 20 testers.
+2) The style’s relevance between synthesized speech and the
+natural language prompt: We conduct RMOS (relevance mean
+opinion score) for speaking style relevance on the testing set
+to evaluate the relevance between synthesized speech and the
+prompt. All participants are asked to read the natural language
+prompt and then listen to the synthesized speech. After that,
+the participants are asked to score the “relevance” of each
+speech sample on a 5-point scale (‘5’ for excellent, ‘4’ for
+good, ‘3’ for fair, ‘2’ for poor, and ‘1’ for bad). Each audio
+sample is rated by at least 20 testers.
+3) AXY test: We propose to use AXY test [7] to assess
+the style relevance between the generated speech with its
+corresponding natural language style prompt. An AXY test
+aims to assess the style transfer performance, where raters
+are asked to rate a 7-point score (from -3 to 3) and choose
+the speech samples which sound closer to the target style in
+terms of style expression. For each reference (A), the listeners
+are asked to choose a preferred one among the samples
+synthesized by the baseline model (X) and proposed Method
+(Y), from which AXY preference rates are calculated. The
+scale ranges of 7-point are from “X is much closer” to “Both
+are about the same distance” to “Y is much closer”, and can
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+10
+TABLE II
+OBJECTIVE AND SUBJECTIVE EVALUATION AS WELL AS MODEL SIZE RESULTS. MCD, SSIM, STOI, GPE, VDE AND FFE ARE ADOPTED AS OBJECTIVE
+METRICS. GT DENOTES THE GROUND TRUTH SPEECH, GT (VOC) DENOTES THAT WE USE PRE-TRAINED VOCODER (HIFI-GAN) RECOVER SPEECH FROM
+MEL-SPECTROGRAM. MOS ANS RMOS AS THE SUBJECTIVE METRIC, IS PRESENTED WITH 95% CONFIDENCE INTERVALS.
+Model
+Decoder
+MCD(↓)
+SSIM(↑)
+STOI (↑)
+GPE(↓)
+VDE(↓)
+FFE(↓)
+MOS(↑)
+RMOS(↑)
+GT
+-
+4.62 ± 0.05
+4.65 ± 0.05
+GT (voc)
+-
+5.02
+0.695
+0.893
+0.006
+0.076
+0.08
+4.41 ± 0.07
+4.61 ± 0.07
+Baseline
+Mel-decoder
+5.75
+0.385
+0.613
+0.476
+0.347
+0.42
+4.04 ± 0.08
+3.85 ± 0.1
+InstructTTS
+Mel-VQ-Diff
+5.69
+0.387
+0.607
+0.479
+0.343
+0.40
+4.35 ± 0.07
+4.22 ± 0.09
+Wave-VQ-Diff
+5.77
+0.365
+0.587
+0.433
+0.343
+0.39
+3.59 ± 0.08
+4.27 ± 0.07
+TABLE III
+THE AXY PREFERENCE TEST RESULTS FOR SPEAKING STYLE
+RELEVANCE.
+X
+Y
+7-point score
+Baseline
+InstructTTS (Mel)
+0.72
+InstructTTS (Wave)
+0.84
+TABLE IV
+THE EMOTION CLASSIFICATION PROBABILITY (%) COMPARISON
+BETWEEN OUR PROPOSED METHODS AND THE BASELINE. FOR EACH TYPE
+OF EMOTION, WE CHOOSE 15 SAMPLES. THE TABLE REPORTS THE
+AVERAGED PROBABILITY VALUES OF 15 UTTERANCES.
+Model
+Sad
+Happy
+Angry
+Overall
+GT
+100
+88.80
+94.70
+95.20
+Baseline
+64.28
+66.60
+68.15
+66.70
+InstructTTS (Mel)
+71.42
+66.60
+68.40
+69.10
+InstructTTS (Wave)
+71.42
+55.50
+84.21
+71.42
+naturally be mapped on the integers from -3 to 3. Note that we
+do not use the ground truth speech as reference, instead we
+ask raters to read the natural language style prompt, and then
+evaluate which synthesized speech is closer to the prompt in
+terms of semantic meaning in emotion and style.
+C. Emotion Perception Test
+Given that speaking style is related with emotion. We choose
+three types of test samples (happy, sad and angry) from
+our test set based on the natural language prompt. Then we
+expect our proposed methods can generate similar emotional
+speech with the guidance of natural language prompt. We
+propose to use emotion classification probability to validate the
+emotion perception performance. Intuitively, the classification
+probabilities summarize the useful emotion information from
+the previous layers for final output layer. Thus, we believe
+that the classification probabilities can be an effective tool
+to justify the synthesized speech’s performance. To realize
+this, we first pre-train an emotion classification model in our
+internal emotion classification dataset. We adopt a pre-trained
+wav2vec2 [75] model as feature extractor, and then we add
+two linear layers and one softmax layer.
+VII. RESULTS AND ANALYSIS
+In this section, we conduct experiments to verify the ef-
+fectiveness of our proposed InstructTTS. We first compare
+the performance between our proposed InstructTTS and the
+baseline. Then we conduct ablation studies to validate the
+effectiveness of each part of our proposed methods.
+TABLE V
+THE ABLATION STUDY FOR CROSS-MODAL REPRESENTATION LEARNING.
+WE EVALUATED WITH THE TEST SET OF CHINESE STS-B CORPUS. SCC
+DENOTES SPEARMAN CORRELATION COEFFICIENT.
+Model
+SCC (%)
+w/o cross-modal learning
+80.4
+w cross-modal learning
+80.94
+A. The comparison between proposed InstructTTS and Base-
+line
+1) The analysis of objective metrics: Table II shows the
+objective metrics (MCD, SSIM, STOI, GPE, VDE, FFE) com-
+parison between our proposed InstructTTS and the baseline
+system. We have the following observations: (1) Our proposed
+InstructTTS achieves better performance than the baseline
+system in terms of speech quality and prosody. (2) Using Mel-
+VQ-Diffusion as decoder can realize better speech quality than
+Wave-VQ-Diffusion, but Wave-VQ-Diffusion is superior in
+maintaining prosody details. One of the reasons is that the pre-
+trained Mel-VQ-VAE downsamples 20 times in the frequency
+dimension, which may harm the pitch information. Instead,
+Wave-VQ-Diffusion directly models all of the information
+in time domain, prosody-related information can be well
+reserved, but some acoustic details may loss.
+2) Subjective Evaluation: We conduct crowd-sourced mean
+opinion score (MOS) tests to evaluate the quality of the
+synthesized speech perceptually. Furthermore, we also conduct
+crowd-sourced relevance mean opinion score (RMOS) tests
+to evaluate the relevance between the synthesized speech and
+the prompt. The results are shown on Table II. We can see
+that InstructTTS (mel) gets the best MOS performance, and
+InstructTTS (wave) gets the best RMOS performance. We can
+see that both of our proposed InstructTTS obtain better RMOS
+performance than the baseline. The subjective evaluation re-
+sults are consist with the objective evaluation results. We can
+also observe that the speech quality of InstructTTS (wave) still
+has room for improvement in quality, on which we will further
+study in our future work.
+We additionally conduct AXY preference test to compare
+InstructTTS and the baseline in terms of the naturalness of
+prosody in their generated speech. From Table III, we can see
+that the raters show much higher preference to the proposed
+InstructTTS (Mel) and InstructTTS (Wave) than to the baseline
+model.
+3) Emotion Perception Evaluation: To further evaluate the
+expressiveness in modeling speaking emotion and styles with
+InstructTTS, we conduct perception evaluation with a speech
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+11
+TABLE VI
+THE TEXT-TO-AUDIO RETRIEVAL PERFORMANCE IN THE TEST SET. WE
+USE RECALL AT RANK K (R@K) AS THE METRICS.
+Loss Type
+R@1
+R@5
+R@10
+Contrastive Loss
+11.62
+42.97
+61.72
+InfoNCE
+15.62
+42.97
+63.67
+TABLE VII
+THE ABLATION STUDY FOR MUTUAL INFORMATION MINIMIZATION
+(MIM) TRAINING STRATEGY.
+Model
+MIM
+MCD(↓)
+SSIM(↑)
+FFE(↓)
+InstructTTS (Mel)
+5.94
+0.368
+0.44
+✓
+5.69
+0.387
+0.40
+InstructTTS (Wave)
+5.91
+0.355
+0.43
+✓
+5.77
+0.365
+0.39
+emotion classification model. The details are introduced in
+Section VI-C. The results are reported in Table IV. We can
+see that our pre-trained speech emotion classification (SEC)
+model obtains a good classification performance in the ground
+truth set, which proves that our SEC model is effective.
+Furthermore, we can observe that our proposed InstructTTS
+got better classification performance, with the InstructTTS
+(Wave) model getting the best performance. We note that the
+evaluation results are consistent with the FFE results.
+B. Ablation studies for InstructTTS
+1) The impact of cross-modal representation learning for
+robust style embedding: In this section, we explore the effec-
+tiveness of our proposed cross-modal representation learning
+in Section IV-C. Table V presents the results. We can see
+that, after finetuning with our proposed cross-modal repre-
+sentation learning, the performance of the RoBERTa even
+achieves better performance in STS task than only finetuning
+with SimCSE. Furthermore, we also evaluate the text-to-audio
+retrieval performance in the test set. As Table VI shows, we
+can see that using InfoNCE loss as training objective can bring
+better retrieval performance than the contrastive ranking loss.
+2) The Impact of Mutual information minimization (MIM)
+training: In this study, we propose to using mutual information
+minimization strategy to constrain the encoded information
+by the audio encoder, we expect the audio encoder only
+encodes the style-related information. In this part, we conduct
+ablation studies to investigate whether our proposed MIM
+strategy can bring better performance. The experiments results
+report on Table VII. We can see that using MIM training
+strategy can significant improvement in both speech quality
+and pitch similarity, which proved that effectiveness of feature
+disentangled strategy.
+3) The effectiveness of classifier-free guidance: In this
+study, we propose to use classifier-free guidance (CFG) strat-
+egy to enhance the connection between conditional informa-
+tion and the predicted results. To validate the effectiveness of
+classifier-free guidance, we conduct ablation study, the exper-
+iments are shown on Table VIII. We can see that using CFG
+strategy can bring better performance due to it enhancing the
+connection between conditional information and the predicted
+TABLE VIII
+THE ABLATION STUDY FOR THE EFFECTIVENESS OF CLASSIFIER-FREE
+GUIDANCE (CFG).
+Model
+CFG
+MCD(↓)
+SSIM(↑)
+FFE(↓)
+InstructTTS (Mel)
+5.75
+0.36
+0.413
+✓
+5.69
+0.387
+0.40
+InstructTTS (Wav)
+5.75
+0.353
+0.41
+✓
+5.77
+0.365
+0.39
+                         Frame
+                         F0 (Hz)
+Fig. 5. Pitch tracks. We present the F0 contours of 10 different runs with the
+same text input, speaker id and style prompt conditioning.
+results, which forces the model to better utilize the conditional
+information.
+4) The effectiveness of improved diffusion strategy for
+Wave-VQ-Diffusion decoder: Table IX shows the experiments
+results when we use different diffusion strategies. We can see
+that our proposed improved mask and uniform strategy can
+bring better performance. The experimental results validate
+our proposed easy-first-generation principle.
+5) Discuss how many codebooks (Nq) we should use when
+we train InstructTTS (Wave): As we discuss in Section V-B,
+we train a neural audio codec model using 12 codebooks in
+total. In practice, we do not need to use all of 12 codebooks.
+Although using more codebooks can bring better speech
+quality, it also bring burden for the network. To choose a
+suitable Nq, we follow two principles: (1) make sure using Nq
+codebooks can get satisfactory reconstruction performance. (2)
+Nq should be as small as possible. We use an out-of-domain
+test set (including 1024 high-quality 24kHz audio samples) to
+evaluate the reconstruction performance of our neural audio
+codec model and the pre-trained Encodec model
+4. Table
+X shows the experimental results. We can see that when
+using 8 codebooks, we can get a comparable reconstruction
+performance with Encodec. Thus, we set Nq = 8 when we
+train InstructTTS (Wave) in this study. We also explore using
+Nq = 12 but receive not obvious performance boost. We
+conjecture that the amount of data in the NLSpeech dataset
+is still in-sufficient and using a larger-scale dataset can bring
+extra improvement.
+C. Synthesis Variation
+Unlike the baseline systhem, which output is uniquely
+determined by the input text and other conditional informa-
+tion (such as speaker identity, natural language prompt) at
+inference, InstructTTS takes sampling processes at denoising
+steps and can inject some variations into the generated speech.
+To demonstrate this, we run a InstructTTS (mel) model 10
+times for a particular input text, speaker and natural language
+4https://github.com/facebookresearch/encodec
+
+400
+350
+300
+250
+!
+200
+150
+·
+100
+50
+20
+40
+60
+80
+100
+120
+140
+160JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+12
+TABLE IX
+THE ABLATION STUDY FOR DIFFERENT DIFFUSION STRATEGY. MAR
+REPRESENTS THE MASK AND REPLACE STRATEGY. I-MAR DENOTES OUR
+IMPROVED MASK AND REPLACE STRATEGY. NOTE THAT WE CONDUCT
+EXPERIMENTS ON WAVE-VQ-DIFFUSION BASED INSTRUCTTTS IN THIS
+PART.
+Model
+MCD
+SSIM
+STOI
+FFE
+InstructTTS (MAR)
+5.85
+0.354
+0.565
+0.42
+InstructTTS (I-MAR)
+5.77
+0.365
+0.587
+0.39
+TABLE X
+THE NEURAL AUDIO CODEC’S RECONSTRUCTION PERFORMANCE
+COMPARISON. Nq DENOTES WE USE Nq CODEBOOKS TO
+RECONSTRUCTION THE AUDIO.
+Model
+Nq
+PESQ
+STOI
+Neural Audio Codec Model (ours)
+1
+1.974
+0.802
+2
+2.632
+0.869
+4
+3.240
+0.910
+8
+3.54
+0.932
+12
+3.62
+0.937
+Encodec
+24
+3.206
+0.965
+prompt, and then compute the F0 contours of the generated
+speech samples. We visualize in Figure 5 and observe that
+InstructTTS can synthesize speech with diverse pitches.
+VIII. CONCLUSION
+In this work, we present InstructTTS, which can synthesize
+expressive speech with the natural language prompt. To our
+best of knowledge, this is the first work to use long and
+complex natural language prompt to control the speaking style.
+In terms of acoustic model, we propose a novel perspective
+to model expressive TTS: we propose to model expressive
+TTS in the discrete latent space and cast speech synthesis as
+a language modeling task. We explore two kinds of modelling
+methods: (1) modelling mel-spectrogram with the help of
+a pre-trained Mel-VQ-VAE model; (2) modeling waveform
+with the help of a pre-trained neural audio codec model. In
+terms of model structure, we propose a novel U-transformer,
+which can effectively model long-sequence. Our experiments
+demonstrate the advantages of our proposed method.
+This work still has some limitations that need to be ad-
+dressed in our future work: (1) The inference speed is limited
+due to the diffusion step is large (we use 100 diffusion steps).
+(2) We will build large-scale dataset to train the InstructTTS
+models, similar to VALL-E and AudioLM. We believe that
+InstructTTS is expected to be more robust when the amount
+of training data increases.
+ACKNOWLEDGMENTS
+We thank the help of our colleagues Mingjie Jin and Dan
+Su for this paper. They help us build the NLSpeech dataset.
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+How, where and when do cosmic rays reach ultrahigh
+energies?
+James H. Matthews𝑎,∗ and Andrew M. Taylor𝑏
+𝑎Department of Physics, Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road,
+Oxford, OX1 3RH, UK
+𝑏Deutsches Elektronen-Synchrotron, Platanenallee 6, Zeuthen, Germany
+E-mail: [email protected]
+Understanding the origins of ultrahigh energy cosmic rays (UHECRs) – which reach energies
+in excess of 1020 eV – stretches particle acceleration physics to its very limits. In this review,
+we discuss how such energies can be reached, using general arguments that can often be derived
+on the back of an envelope. We explore possible particle acceleration mechanisms, with special
+attention paid to shock acceleration.
+Informed by the arguments derived, we discuss where
+UHECRs might come from and which classes of powerful astrophysical objects could be UHECR
+sources; generally, we favour radio galaxies, GRB afterglows and other sources which are not too
+compact and dissipate prodigious amounts of energy on large scales, allowing them to generate
+large products 𝛽𝐵𝑅 without the CRs undergoing restrictive losses. Finally, we discuss when
+UHECRs are accelerated by highlighting the importance of source variability, and explore the
+intriguing possibility that the UHECR arrival directions are partly a result of “echoes” from
+magnetic structures in the local Universe.
+27th European Cosmic Ray Symposium - ECRS
+25-29 July 2022
+Nijmegen, the Netherlands
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.02682v1  [astro-ph.HE]  6 Jan 2023
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+1.
+Introduction
+The origin of ultrahigh energy cosmic rays (UHECRs) has remained an open question ever
+since their discovery by Linsley [1]. Together with Scarsi, Linsley measured the energy spectrum
+above 1 EeV and even performed an analysis of arrival directions with a sample of 97 events.
+Since these pioneering results, decades of experimental and theoretical work have been dedicated
+to understanding the phenomenology and physics of UHECRs (see historical reviews by Watson
+[2, 3]) – the highest energy particles in nature. Despite Herculean efforts, the sources of UHECRs
+are not yet known, nor is the physics of their acceleration understood.
+The current state of the art UHECR observatories are the Pierre Auger Observatory (PAO), in
+Malargüe, Argentina (detection area ≈ 3000 km2), and the Telescope Array (TA) in Millard County,
+Utah, USA (detection area ≈ 700 km2). Both observatories have been critical for measuring the
+spectrum, composition and anisotropy of UHECRs over the past decade. The CR spectrum is
+characterised by a smooth power-law over 11 decades in energy, with a series of inflection points;
+in the UHE regime the most relevant features are the ankle, a hardening at ≈ 4 EeV, and a cutoff
+or flux suppression at ≈ 40 EeV [e.g. 4, see Fig. 1]. In terms of composition, combined fits of the
+spectrum and the distribution or moments of the depths of the air shower maxima, 𝑋max, suggest a
+composition that gets heavier with energy above the ankle [4–6, see also section 2.2]. Finally, the
+question of UHECR anisotropy has seen particularly exciting recent progress, with PAO reporting a
+dipole anisotropy at 5.2𝜎 significance [7], together with less significant indications of anisotropies
+on smaller scales from both PAO [8, 9] and TA [10, 11]. However, working backwards from arrival
+directions to uncover the sources of UHECRs remains challenging given the limited statistics at such
+high energies, uncertain detailed composition and, in particular, the obfuscating effect of (poorly
+constrained) intergalactic and Galactic magnetic fields.
+On the theoretical and modelling side, there have also been many recent advances (see, e.g.,
+chapters 5 & 6 of the EuCAPT white paper [12]). Building on the foundational theory of shock
+acceleration [13–16], particle-in-cell (PIC) simulations have provided unprecedented insights into
+the nonlinear plasma physics at work during shock acceleration [e.g. 17, 18] and magnetic recon-
+nection [e.g. 19, 20]. Cosmic-ray propagation codes, such as CR-Propa [21], now provide flexible
+frameworks for treating the propagation of UHECRs from source to detector. This mature and
+well-tested suite of computational tools are essential for understanding the theoretical cosmic-ray
+landscape, but one particular feature of UHECR acceleration is the vast range of scales at work;
+a mildly relativistic proton with 𝛾 ∼ 10 must increase it’s energy (and thus Larmor radius for a
+constant magnetic field strength) by a factor of 108 to reach the UHE (≳ 1018 eV) regime. Such a
+dynamic range is out of reach of even the most ambitious simulator.
+It is not yet possible to make unambiguous inferences about UHECR sources from data or
+theory alone. As a result, we must consider the whole picture, taking into account plasma physics,
+astrophysics and multimessenger astronomy, when interpreting the experimental data. The need for
+such a holistic view makes studying UHECRs challenging, but also particularly rich and rewarding
+(in our opinion!). We will try to convey some of that excitement in this review contribution to
+the proceedings of the European Cosmic Ray Symposium (ECRS) 2022. Our review is structured
+as follows, mirroring the corresponding ECRS talk.
+We start (section 2) by going over some
+UHECR fundamentals, to establish the basic assumptions we will make. We then discuss each
+2
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+of the interrogatives in our title: we explore how UHECRs might be able to reach such extreme
+energies (section 3), where they might be coming from (section 4), and when they might have been
+accelerated, with a biased focus on a specific ‘echoes’ model for their origin (section 5). Finally,
+in section 6, we conclude and comment on the future outlook. We generally adopt CGS units and
+Gaussian units for electromagnetism, but we often give energies in eV or EeV and rigidities in V or
+EV. We use the symbols E for electric field, 𝐸 for energy, 𝒗 for velocity, and define 𝛽 ≡ 𝑣/𝑐.
+2.
+UHECR Fundamentals
+We define ultrahigh energy cosmic rays (UHECRs) as charged particles (protons or nuclei)
+reaching an energy in excess of 1018 eV, although a successful UHECR source must be able to
+accelerate particles right up to ∼ 100 EeV in order to explain the full energy range of UHECR data.
+The spectrum of UHECRs arriving at Earth as measured by PAO is shown in the left hand panel of
+Fig. 1, with the main features labelled.
+The Larmor radius (or gyroradius) of such an ultra-relativistic particle with energy 𝐸 = 𝑝𝑐
+(where 𝑝 is momentum) is given by 𝑟𝑔 = 𝐸/(𝑍𝑒𝐵). This is the radius of gyration when undergoing
+circular rotation in a uniform magnetic field. Writing down the Larmor radius already gives us
+useful insights. Given in scaling relation form for characteristic UHECR energies, the Larmor
+radius is
+𝑟𝑔 = 10.8 kpc
+�
+𝐸
+10 EeV
+� � 𝐵
+𝜇G
+�−1
+𝑍−1.
+(1)
+This gives a (very minimal) condition for UHECR sources – a source must be able to confine a
+particle before it can accelerate it. However, for acceleration to take place, it is the electric field
+that really matters, and it is useful to think of the maximum rigidity, rather than maximum energy,
+associated with astrophysical acceleration sites. We define rigidity, which we quote in Volts (V), as
+R = 𝐸
+𝑍𝑒 .
+(2)
+The maximum rigidity, Rmax, is an important quantity because it relates directly to particle accel-
+eration in electric or magnetic fields. The energy gained by moving a particle a distance 𝑅 in an
+electric field of strength E is 𝑍𝑒E𝑅, so the maximum rigidity should be an intrinsic quantity of the
+accelerator; it depends only on the size of the region, and the electric field available.
+2.1 UHECR losses, propagation and ‘horizons’
+UHECRs are attenuated or degraded from interaction with the cosmic microwave background
+(CMB) and extragalactic background light (EBL). Protons can undergo resonant photopion con-
+version, which in this context is known as the Greisen-Zat’sepin-Kuzmin (GZK) effect [23, 24],
+while heavier nuclei undergo photodisintegration [25] and all nuclei are subject to Bethe-Heitler pair
+production. The photopion, pair production and photodisintegration processes impose composition-
+dependent energy loss lengths or mean free paths, making it difficult for UHECRs to reach us from
+very distant sources. These length-scales are often referred to as limits or horizons, but as with any
+opacity source there is a chance, however slim, that an UHECR can travel a considerable distance
+beyond this length scale. The energy loss lengths for protons and a few different ion species (He,
+3
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+1018
+1019
+1020
+E (eV)
+1027
+1028
+E3 J (eV2 cm−2s−1sr−1)
+‘Ankle’
+‘Flux Suppression’
+‘Instep?’
+PAO Data (ICRC 2019)
+1018
+1019
+1020
+E (eV)
+101
+102
+103
+104
+Energy loss length (Mpc)
+CoG/Local Sheet
+SG Structure
+Cyg A
+p
+He
+N
+Fe
+Figure 1: Left: The UHECR energy spectrum measured by PAO as presented at ICRC 2019 [38]. The
+spectrum is shown in 𝐸3 𝐽 units where 𝐽 is the differential flux spectrum in units of particles per unit area
+per unit energy per unit solid angle. As plotted, a 𝑑𝑁/𝑑𝐸 ∝ 𝐸−3 spectrum shows up as a horizontal line.
+Right: Energy loss lengths as a function of CR energy calculated for the same four species shown in Fig. 2.
+The energy loss length is defined in the text. The loss lengths are calculated by considering photopion,
+photodisintegration and pair production interactions with the CMB and EBL, using the EBL model of [22].
+N, Fe) are shown in Fig. 1 for the EBL model of [22]. The energy loss length here is defined as
+the average distance necessary for an UHECR to propagate in order for its energy to decrease to
+𝑒−1 of its original value. The separate bumps in the curves are attributed to different processes
+and radiation fields, with the CMB photopion and photodisintegration processes dominating at the
+highest energies. We have also marked on the figure the distance to Cygnus A of ∼ 240 Mpc
+[26], the typical distance to objects in the ‘Council of Giants’ (CoG) or ‘Local Sheet’ [27], and
+the characteristic scale length of ∼ 100 Mpc associated with the supergalactic plane [28]; these
+distances are all relevant to discussions here and in section 4. Fig. 1 highlights how difficult it is
+to accurately characterise the UHECR source population from the spectrum alone given that its
+form depends on the source spectrum, spatial distribution or redshift evolution, and composition.
+Nevertheless, broadly speaking, the various CMB and EBL interactions impose a characteristic
+length scale ∼ 100 Mpc within which the dominant UHECR sources are most likely to lie.
+Combined knowledge of source timescales, UHECR propagation and anisotropy can impose
+additional constraints on UHECR source distances. For example, Eichmann and collaborators have
+explored a model where Cygnus A was the dominant source in the sky up to tens of EeV [29].
+Cygnus A is compelling as an UHECR source because it is unusually powerful for a radio galaxy,
+and although its distance of ∼ 240 Mpc might appear restrictive, at 1019 eV energy loss lengths are
+quite large, and Cygnus A’s potentially vast UHECR luminosity could still produce the magnitude
+of the observed UHECR flux. However, in a follow-up paper [30], Eichmann showed that Cygnus
+A cannot account for an isotropic CR component at these energies, because the CRs would not have
+had time to isotropise in the extragalactic magnetic field in the time the source has been active; one
+should instead see an anisotropic signal pointing towards Cygnus A. This difficulty could in principle
+4
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+1018
+1019
+1020
+E (eV)
+600
+650
+700
+750
+800
+850
+900
+⟨Xmax⟩ (g cm−2)
+p
+He
+N
+Fe
+PAO 2019 (stat.)
+sys.
+1018
+1019
+1020
+E (eV)
+20
+30
+40
+50
+60
+70
+σXmax (g cm−2)
+Figure 2: Composition diagnostics from extensive air showers detected by the PAO, showing how the
+composition gets heavier at higher energies. Data taken from the PAO contribution to ICRC 2019 [38]. Left:
+The ⟨𝑋max⟩ distribution from PAO data, defined as the mean depth of the air shower maximum, as a function
+of energy. Right: 𝜎𝑋max, the standard deviation of 𝑋max as a function of energy. In both cases the coloured
+bands show the predictions for four pure composition scenarios, with the range shown corresponding to that
+spanned by three hadronic interaction models (QGSJet II-04, EPOS-LHC and Sibyll 2.1), as calculated using
+the parameterisation from Ref. [39], with adjusted parameters from S. Petrera (priv. comm.).
+be alleviated if the Galactic halo magnetic field can isotropise the signal on shorter timescales, but
+the general principle of ‘diffusive’ horizons for UHECR production is nevertheless important (see
+also Refs. [31, 32]).
+Following the cumulative composition and spectral measurements made by the PAO over the
+last 15 years, a growing body of evidence has amounted suggesting that UHECR at the highest
+energies must have a rather local origin [33, 34]. This finding is particularly interesting along with
+other suggestions that not many sources should be contributing to the UHECR spectrum in this
+high energy range [35]. Collectively this suggests that a local UHECR source may dominate the
+contribution to the UHECR spectrum at the highest energies. This finding may also be consistent
+with the UHECR dipole strength recently detected by the PAO [36], with the main contribution to
+the dipole being driven by the presence of this local source [37].
+2.2 Composition and Maximum Rigidity
+Since the maximum rigidity, rather than maximum energy, is an intrinsic property of a cosmic
+acceleration site, it follows that the charge on the nuclei (or the atomic composition of UHECRs) is
+important for establishing possible UHECR sources. At CR energies below the knee, experiments
+such as the Alpha Magnetic Spectrometer [AMS; 40] and Cosmic Ray Energetics and Mass exper-
+iment [CREAM; 41] can provide direct measurements of CR charge and therefore decompose the
+CR spectrum into different species. However, at ultrahigh energies, the CRs are detected through
+extensive air showers, and the main diagnostic of composition is a more indirect measure: the
+distribution of the depths of air shower maxima, 𝑋max. We show the first two moments of the 𝑋max
+5
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+distribution in Fig. 2 from the data released as part of the PAO contribution to ICRC 2019. The data
+are compared to theoretical distributions for 𝑋max for three different hadronic interaction models.
+The general trend is a gradual change from nearly pure protons around ∼ 3 EeV to a heavier com-
+position at higher energies. Such a trend might suggest maximum rigidities of Rmax ∼ 3 − 10 EV,
+which is broadly consistent with other studies: the combined fit of the spectrum and composition
+PAO data finds a cutoff rigidity of Rmax = 4.79 EV [4], and Ref. [42] find maximum energies for
+Fe nuclei at source of ≈ 300 EeV suggesting Rmax ≈ 11 EV. There is some wiggle room in this
+quantity but it cannot be too much lower than 10 EV in order to explain the observed > 100 EeV
+UHECRs, so we will adopt 10 EV(1019 V) as our ‘target’ rigidity when discussing UHECR sources.
+3.
+Acceleration of UHECRs (How?)
+3.1 The Hillas energy and power requirement
+The maximum characteristic energy associated with a particle acceleration process is the Hillas
+energy [43], given by
+𝐸𝐻 = 9.25 EeV
+�
+𝐵
+10 𝜇G
+� � 𝑅
+kpc
+�
+𝑍𝛽,
+(3)
+where 𝑍 is the dimensionless charge on the particle, 𝛽 = 𝑣/𝑐 is the velocity of the accelerator in
+units of 𝑐, 𝐵 is the magnetic field and 𝑅 is the characteristic size. The above equation can also be
+equivalently written as a Hillas rigidity in the form R𝐻 = 𝛽𝐵𝑅, which is the basic figure of merit for
+an UHECR accelerator. The Hillas condition is not the same as having a Larmor radius equal to the
+size of the acceleration region; there is an additional factor such that the acceleration region must be
+larger than the Larmor radius of the highest energy particles by a factor of 1/𝛽. The Hillas energy
+can be arrived at in various ways, but is perhaps best understood in terms of a particle travelling a
+distance 𝑅 in an optimally arranged −(𝒗/𝑐) × 𝑩 electric field. It is is only a characteristic maximum
+energy, and as we shall see in the next section is a necessary, but not sufficient criterion that is
+only reached under certain conditions. One can construct scenarios in which the Hillas energy is
+exceeded; for example, if a CR can be confined to a perpendicular shock for a very long time then
+in principle the CR can cross the shock on many occasions without escaping. However, in practice
+this is likely to require specialised shock or magnetic field geometries to avoid drifts, diffusion or
+advection removing the particle from the acceleration site. We will therefore proceed under the
+expectation that the Hillas energy really is a maximum or cutoff energy.
+The Hillas energy can be used to derive a minimum magnetic or kinetic power that a source
+must possess. A similar requirement was, to our knowledge, first discussed by Lovelace [44], but
+also forms the basis for Hillas’ figure 6 in the 1984 paper [43]. The power requirement is therefore
+sometimes referred to as a ‘Hillas-Lovelace limit’ (although see also Refs. [45–49]). The basic idea
+is that a source must be able to supply enough magnetic energy per unit time that a given product
+𝛽𝐵𝑅 can be maintained in the acceleration site. In the non-relativistic case, a limit on the kinetic
+power 𝑄𝑘 can be derived,
+𝑄𝑘 ≳ 1044 erg s−1 𝛽−1
+� 𝐸/𝑍
+1019eV
+�2
+𝜖𝑏 𝜂2,
+(4)
+6
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+where 𝜖𝑏 is the ratio of magnetic to kinetic power (which, in shocks, can be thought of as an
+efficiency of magnetic field amplification), and 𝜂 is an efficiency factor defined in section 3.2.1
+which describes how close the diffusion is to the Bohm regime (equation 7).
+In relativistic particle accelerators, the above expressions can be modified slightly to account
+for special relativistic effects. Ref. [50] gives the Hillas energy in the form 𝐸𝐻 = Γ𝑍𝑒𝐵𝛽𝑅, where
+Γ is the bulk Lorentz factor of the shock, and 𝑅 and 𝐵 are given in the co-moving frame. Whether
+the particle really gains this Lorentz boost likely depends on the nature of the turbulence generated
+at the shock, and the details of the particle transport in the source region [51]. However, such a
+boost may potentially be important particularly if GRB internal shock or afterglow models are to
+reach rigidities of 10 EV (see section 4.2.3). The power requirement can also include an additional
+Γ2 factor [49, 52], although this cancels with the outflow opening angle, Θ, if Θ ∝ Γ−1. In any case,
+it is hard for a relativistic accelerator to reach optimal conditions with 𝜂 ≈ 1 (see section 3.2.1), and
+so we take equations 3 and 4 as our basic energetic requirements.
+3.2 Particle acceleration mechanisms
+Astrophysical fluids are often hot and ionized plasmas in which electrons and ions are unbound
+and free to move. The motion of these free charges tends to rapidly damp or screen any local
+electrostatic field present in the plasma. However, bulk, differential motions of the plasma, with a
+velocity 𝒗, lead to a −(𝒗/𝑐) × 𝑩 electric field which can accelerate particles, where the velocity can
+be thought of as the characteristic velocity of ‘scattering centres’, in Hillas’ language [43]. Indeed,
+this electric field is the origin of the 𝛽𝐵 term in the Hillas energy above. Rather than acceleration
+in some spark gap or monolithic electrostatic field, particles are thought to acquire nonthermal
+energies through interactions with magnetised plasma that lead to a so-called ‘Fermi’ process: a
+gradual, stochastic acceleration in a −(𝒗/𝑐) × 𝑩 electric field.
+Fermi originally proposed that CRs gain energy from interactions with magnetised clouds [53],
+which, together with its derivatives, is now referred to as second-order Fermi acceleration because
+the fractional energy gain per scatter is proportional to 𝛽2. In the late 1970s, a series of authors
+proposed first-order Fermi acceleration at shocks [13–16], and since then Fermi processes have
+been extensively studied from various perspectives and are the subject of a number of review papers
+[e.g. 48, 54, 55]. We refer the reader to these reviews for a detailed discussion. Here, we discuss
+some of the basic reasoning behind first-order Fermi processes and the physical mechanisms at
+work, as well as the astrophysical sites in which they can operate.
+3.2.1 Shock Acceleration
+The most famous example of first-order Fermi acceleration is shock acceleration. A shock
+is a converging flow, with ∇ · 𝒗 < 0, and particles that cross the shock front gain a momentum
+boost proportional to the shock velocity 𝛽. The theory was laid out in the aforementioned series of
+papers, with Bell [13] providing a ‘microscopic’, test-particle description, and Blandford & Ostriker
+[14] a ‘macroscopic’ description using the Fokker-Planck equation. The basic result is that CRs
+crossing the shock front get an energy boost each time they do so, with a mean fractional energy
+gain ⟨Δ𝐸/𝐸⟩ = 𝛽. At the same time, CRs are being swept away from the front at a rate which is
+also proportional to 𝛽. It is straightforward to show [e.g. 13, 48] that the competition between these
+7
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+two effects – energy gain, and escape – leads to a power-law CR distribution of the form
+𝑑𝑁
+𝑑𝐸 ∝ 𝐸−𝑞,
+(5)
+with 𝑞 = 2 for the idealised example considered here. There are various effects that lead to a
+steepening of this spectral index, such as energy exchange with turbulent magnetic fields [56, 57],
+and in ultra-relativistic shocks an index of 𝑞 ≈ 2.2 − 2.3 is expected [50, 58].
+By treating the crossing of the shock and the scattering by magnetic irregularities as a diffusive
+process with coefficient 𝐷 = 𝜆𝑐, where 𝜆 is the mean free path, it is possible to derive an acceleration
+time. In detail, this acceleration time should allow for different upstream and downstream diffusion
+coefficients [59], but the basic form is
+𝜏acc ∼ 𝐷
+𝑣2𝑠
+≡
+𝜆
+𝛽2𝑠𝑐 .
+(6)
+The CR energy is maximised when the acceleration time is shortest, requiring small diffusion
+coefficients and large shock velocities (in the non-relativistic regime). The diffusion coefficient is
+often written in the form
+𝐷 ∼ 𝜂𝑟𝑔𝑐
+(7)
+where 𝜂 is the so-called gyrofactor; 𝜂 = 1 is the optimal Bohm regime where 𝜆 ≈ 𝑟𝑔, and 𝜂 > 1
+otherwise leading to slower acceleration. It is easy to show that the Hillas energy is necessary but
+not sufficient by combining equations 1, 6 and 7, and equating 𝜏acc with 𝑅/𝑣𝑠, giving the equation
+𝐸max ∼ 𝜂−1𝑍𝑒𝛽𝐵𝑅.
+(8)
+Thus, the Hillas energy is only reached when 𝜂 = 1 and Bohm diffusion applies, that is when
+𝜆 ≈ 𝑟𝑔. For this to happen, there must be strong turbulence with 𝛿𝐵/𝐵 ∼ 1 and structure in this
+turbulence on scales of the Larmor radius. These considerations show why the plasma physics of
+CR instabilites and the nonlinear, coupled acceleration process are important for understanding the
+maximum energy/rigidity attainable in a given accelerator.
+It was realised early on that CR-excited waves or MHD turbulence of some kind were needed
+to confine the CRs at the shock, allowing the CRs to cross many times and facilitate acceleration to
+high energies. Originally, Alfvén waves driven by the resonant CR instability [60] were invoked,
+but a new non-resonant or Bell instability was discovered [61, 62]. The non-resonant instability has
+a number of advantages; it grows faster than the resonant instability and creates turbulence on the
+scale of the Larmor radius of the particles driving the instability, providing a self-regulated process
+that allows acceleration to proceed close to the Bohm regime. The instability is thought to operate
+in supernova remnant (SNR) shocks where it is critical for providing the necessary magnetic field
+amplification, and growing the turbulent magnetic field to the Larmor radius of the highest energy
+particles. There is observational evidence that the Bohm regime is realised in SNR shocks [63, 64],
+suggesting they can get close to the special conditions needed for the Hillas energy to apply.
+In some sense it is natural to appeal to relativistic shocks as UHECR accelerators, given
+that some of the most powerful phenomena in the Universe involve ultrarelativistic outflows and
+invariably produce radiation from nonthermal electrons. The ultrarelativistic version of shock ac-
+celeration or first-order Fermi acceleration differs somewhat from its nonrelativistic counterpart.
+8
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+The expected spectral index is slightly steeper than the canonical shock acceleration value, with
+𝑞 ≈ 2.2 − 2.3, the compression ratio is higher, the shock is quasi-perpendicular and significant
+anisotropies develop in the particle distribution function [50, 58]. One might think the ultrarel-
+ativistic shocks are the most obvious sites for UHECR acceleration, since 𝛽 → 1 maximises the
+Hillas energy, but a number of authors have shown that relativistic shocks have difficulties reaching
+ultrahigh energies [51, 65–67]. In particular, Ref. [67] shows that the maximum energy is likely
+to be many orders of magnitude below EeV energies. This happens because the CR spectrum is
+steeper, so there is less energy to drive turbulence on UHECR Larmor radius scales, and the CRs
+also do not have time to drive large-scale turbulence, because they penetrate less far upstream and
+are quickly advected away downstream. There may be ways around these issues – for example, if
+there is pre-existing turbulence in the upstream medium (see also Refs. [68, 69] for relevant recent
+studies) – but we will refer to these collective difficulties as the ‘relativistic shock problem’ for
+UHECRs.
+3.2.2 Magnetic Reconnection
+Magnetic reconnection – the resistive dissipation of magnetic fields – is another mechanism
+that can accelerate particles. In this case the transfer of energy is from magnetic energy to thermal
+and kinetic, a fraction of which can be passed on to nonthermal particles.
+Reconnection has
+received a lot of attention as a particle acceleration mechanism recently, for various reasons. The
+last decade has seen dramatic progress in using PIC (as well as test particle and hybrid MHD-
+PIC) simulations to study first 2D, and subsequently 3D, reconnection sites. A variety of particle
+acceleration mechanisms can operate in these reconnection sites; unlike shock acceleration there is
+a current sheet involved, and the electric field close to the reconnection X-point can inject particles
+or accelerate them to modest energies. After injection, Fermi mechanisms can take over. Various
+Fermi mechanisms and models have been proposed, with acceleration taking place by traversing
+the converging flows either side of the X-point [70–72] or within contracting plasmoids [19].
+It is not yet clear how relevant magnetic reconnection is to the UHE, multi-EeV regime, though
+a number of authors have proposed it as a possible UHECR acceleration mechanism [71, 73, 74].
+One potential difficulty is arranging for structure (and energy density) in the magnetic field to be
+present on a wide range of scales from the resistive scale up to the Larmor radii of UHECRs.
+In shock acceleration, the magnetic field is amplified and stretched via, e.g., the CR-driven non-
+resonant instability, but we (the authors) do not know of a convincing mechanism to arrange for
+such a magnetic field structure in reconnection sites at this stage. However, that does not mean one
+does not exist or will not be forthcoming in the future.
+3.2.3 Other Mechanisms and General Comments
+As well as from shocks and reconnection, there are various other ways in which particles can
+gain large amounts of energy. Shear acceleration involves scattering across a shear layer [75],
+in a similar manner to shock acceleration, and has been proposed as a possible mechanism for
+UHECR acceleration at the edge of a relativistic jet. A detailed discussion of shear acceleration
+pertaining to UHECR acceleration in AGN jets is given by Rieger in a recent review [49], who
+highlights some recent studies proposing one-shot or ‘espresso’ acceleration in relativistic AGN
+jets [76–78]. Alternatively, shear acceleration can be rather gradual, and the details of the process
+9
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+depend on the thickness of the shear layer and the Kelvin-Helmholtz instabilities operating in the
+region [49, 79, 80]. Various authors have also discussed second-order Fermi acceleration (Fermi
+II) by MHD turbulence in, for example, giant radio lobes [81–83, see also section 4.2.2]. Finally,
+there is the possibility of acceleration by ‘unipolar induction’, whereby a rapidly rotating magnetic
+field in, e.g., a pulsar magnetosphere generates a large potential difference [45, 84–86]. We do
+not provide a detailed account of these three processes (shear, Fermi II, unipolar induction), and
+neither is this an exhaustive list, but we we do touch on some of them in more detail, with the
+relevant astrophysical context, in section 4. In our discussions hereafter, we will try to keep the
+arguments general, based on the basic energetics and physical conditions of the system, but a bias
+in emphasis towards shock acceleration is probably inevitable given the background of the authors,
+and the relative maturity of each mechanism at the time of writing.
+3.3 UHECR Losses and Escape
+The Hillas energy and power requirements above and the maximum energy derived from shock
+acceleration apply when the CR maximum energy is limited by either the escape time or dynamical
+time. In sources with strong magnetic fields or intense radiation fields, losses can instead limit
+the maximum energy. Synchrotron losses for CR nuclei with relative atomic mass 𝐴 occur on a
+timescale 𝜏sync = 142 yr (𝐴/𝑍)4 𝐸−1
+EeV𝐵−2, where 𝐸EeV is the energy in EeV. By equating this with
+the acceleration time (equation 6), we can write the maximum energy in a magnetic field of strength
+𝐵 as
+𝐸max,sync = 200 EeV
+� 1
+𝜂𝑍3
+�1/2 � 𝐵
+G
+�−1/2
+𝛽𝐴2,
+(9)
+which can be restrictive even for optimum acceleration conditions in strong magnetic field sources.
+Equivalently, we can invert this equation to write a maximum magnetic field strength for acceleration
+to a given energy,
+𝐵max = 400 G
+� 1
+𝜂𝑍3
+� �
+𝐸
+10 EeV
+�−2
+𝛽2𝐴4,
+(10)
+which illustrates the challenge of accelerating UHECRs in highly magnetised environments as
+discussed by various authors [87–92, see section 4 for source implications]. Interactions with
+ambient radiation fields during both the acceleration and escape of CRs can also limit the maximum
+energy through the same processes described in section 2.1, although the details depend on the
+radiation field shape and intensity considered. Considering photopion losses from protons, Ref.
+[87] derives an approximate limit on the source radiative luminosity at a distance 𝑅 from the source
+given by 𝐿𝛾 < 5 × 1044 erg s−1 ¯𝜖 (𝑅/1017cm) where ¯𝜖 is the energy of the maximum of the
+integral photon spectrum in eV. Such an upper limit might seem counter-intuitive given that we also
+found a lower limit on power from equation 4, but the latter is a magnetic power limit as opposed
+to a radiative one. The acceleration of UHECRs therefore favours sources which are neither too
+radiatively efficient nor too compact – ideally we need large amounts of energy to be dissipated so
+that a large product 𝛽𝐵𝑅 can be maintained, but without the energy densities in magnetic fields or
+radiation fields becoming too large.
+We close this section by noting that the losses within, and escape from, the acceleration
+site and immediate environment can have interesting implications for the emergent spectrum and
+10
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+composition of the UHECRs. For example, in the Unger-Farrar-Anchordoqui model [93], photo-
+disintegration in the source environment can naturally reproduce the location of the UHECR ankle,
+shape of the UHECR spectrum and composition trends (see also Ref. [94]). Similarly, diffusive
+escape modifies the spectral shape below a critical energy at which the escape time, 𝜏esc, is equal to
+the source age [e.g. 95, 96]. Above this energy, the at-source spectrum is gradually recovered. Fur-
+thermore, high rigidity CRs escape more quickly than low rigidity CRs, so the escaping UHECRs
+can be lighter than the internal CRs, although the details depend on a complex interplay between
+the source activity, and the cooling/escape timescales [96].
+3.4 UHECR Source Checklist
+With the above arguments in mind, and referring the reader to the references given for greater
+detail, we propose that the following criteria form a basic ‘checklist’ that a source or source
+population must satisfy to be a realistic UHECR candidate:
+• The source must have a large product 𝛽𝐵𝑅 to satisfy the Hillas condition (equation 3)
+• The source must dissipate a large amount of power in (for example) a shock or a site of
+magnetic reconnection (equation 4)
+• If the acceleration is diffusive, the diffusion coefficient must approach the Bohm regime or
+near-optimal conditions (e.g. equation 8) across a range of energies
+• If the acceleration is at a shock, the shock probably cannot be highly relativistic
+• The CRs must not undergo restrictive losses due to, e.g. curvature radiation, synchrotron
+radiation, adiabatic expansion, or interactions with photons
+• The source must be within a composition- and energy-dependent horizon from the Earth, or
+produce UHECRs with such efficacy that a substantial UHECR luminosity still reaches us.
+• The source must be common and powerful enough to produce the observed UHECR flux.
+Meeting all of these criteria turns out to be a challenge for any astrophysical source. However, it
+is also difficult to assess the relative merit of the sources given that (i) the underlying physics is
+complex and far from settled, and (ii) many of the estimates of velocities, magnetic field strengths,
+and jet/outflow powers are subject to large astrophysical uncertainties. Nevertheless, we will soldier
+on and discuss possible UHECR sources with this list of requirements as our guide.
+4.
+Astrophysical Sources of UHECRs (Where?)
+The detections of anisotropies in UHECR data from the Pierre Auger Observatory (PAO) and
+Telescope Array mean that we are entering an exciting era for UHECR astrophysics. In this section,
+we will first look at anisotropy data to see what the data alone tell us, before discussing the overall
+prospects of a host of astrophysical candidates.
+11
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+150◦
+120◦
+90◦
+60◦
+30◦
+0◦
+330◦
+300◦
+270◦
+240◦
+210◦
+-75°
+-60°
+-45°
+-30°
+-15°
+0°
+15°
+30°
+45°
+60°
+75°
+0
+2
+4
+6
+8
+10
+12
+14
+16
+Flux (10−3 km−2 sr−1 yr−1)
+Figure 3: UHECR flux map, in Galactic coordinates, from PAO above 41 EeV comprising 1274 events,
+produced from the data made publicly available by PAO [9]. A top-hat smoothing of radius 25◦ has been
+applied. The supergalactic plane and PAO exclusion are marked in green and grey, respectively.
+4.1 UHECR anisotropies
+Detecting statistically significant anisotropies in the arrival directions of UHECRs is a key
+goal of TA and PAO, and is an essential step for uncovering the origin of UHECRs. In 2017,
+PAO reported a large-scale anisotropy in the arrival directions of 30,000 CRs above 8 EeV at 5.2𝜎
+significance. The anisotropy is well-described by a dipole with 6.5% anisotropy. TA has also
+undertaken large-scale anisotropy searches [97], with results that are consistent with both isotropy
+and the PAO dipole. The detection of a significant UHECR dipole is a spectacular and important
+result. It more-or-less confirms some aspects of UHECR origins which had long been suspected:
+that UHECRs are extragalactic in origin, and are not isotropic. However, it is extremely difficult to
+pinpoint UHECR sources from a large-scale dipole on the sky. Moving to higher energies results
+in a trade-off. On the one hand, higher energy typically means higher rigidity, resulting in smaller
+magnetic deflections and anisotropies that emerge on smaller angular scales. These anisotropies
+can feasibly be correlated with astrophysical source catalogues. On the other, the statistics drop off
+markedly and so the number of events one is able to analyse can become prohibitively small.
+Both TA and PAO have reported indications of anisotropy on intermediate angular scales
+(≈ 5◦ − 25◦ search radii) at ≳ 40 EeV energies. PAO found (model-dependent) correlations with
+catalogues of star-forming galaxies (4𝜎) and AGN (3.2𝜎) [98], with anisotropic fraction of around
+5 − 10%. Here a ‘UHECR luminosity proxy’ must be adopted and PAO used gamma-ray and radio
+fluxes as weights for the relative luminosity of each source in UHECRs. In a more recent study [9],
+PAO presented a detailed investigation of 2635 events with reconstructed energy > 32 EeV. The
+12
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+arrival directions from this study are shown in Fig. 3. An excess in flux can be seen just above the
+Galactic plane, approximately in the direction of Centaurus, with a hint of additional excesses at
+southern Galactic latitudes and just below the PAO exclusion area. Using updated catalogues for
+gamma-ray luminosities as well as considering other catalogues with different luminosity proxies,
+Ref. [9] found post-trial 𝑝-values of < 10−3 for catalogues comprising jetted AGN, galaxies traced
+by near-infrared emission, X-ray AGN and star-forming galaxies. Each of these searches resulted
+in top-hat search radii in the range 22◦ − 25◦ and threshold energies of 38 − 40 EeV, and the most
+significant correlation was with the star-forming galaxies traced by their radio emission, giving
+𝑝 = 3.2 × 10−5 corresponding to > 4𝜎 significance. The same study also conducted a series of less
+model-dependent searches for correlations with structures such as the supergalactic and Galactic
+plane; although none of these were statistically significant, they did find a > 4𝜎 correlation with
+the Centaurus region on the sky, which is responsible for driving much of the correlation seen in
+the catalogue searches (since Cen A, NGC 4945 and M83 all lie in this area). More details on this
+study can be found in the contribution of C. Galleli, on behalf of PAO, to these proceedings.
+TA have also reported anisotropies on intermediate angular scales, with a particular excess
+that is often referred to as the ‘TA hotspot’ [10]. TA reported an excess of events above 57 EeV
+fairly close to the supergalactic plane, at RA = 146.7◦, Dec. = 43◦, which is roughly the direction
+of M82. Originally, the post-trial (local Li-Ma) significances was 3.4𝜎 (5.1𝜎), while more recent
+updates put the post-trial significance at 2.9−3.2𝜎 [99, 100]. TA have also reported another excess
+at slightly lower energies, in the approximate direction of the Perseus-Pisces supercluster [11].
+Finally, in addition to the TA and PAO searches, joint PAO and TA efforts have been undertaken
+to build full-sky maps of UHECR arrival directions by correcting for the different exposures and
+systematics between the two experiments [101, 102]. The results emerging from this show excesses
+roughly correlated with the supergalactic plane (or local sheet/CoG, which follows a similar path
+on the sky), with particular hotspots in the direction of NGC 253/Fornax, Cen A/NGC2945, and
+M82. Although the statistical significance of any correlations with these local planar structures
+is still fairly marginal (∼ 3𝜎) [102], the full-sky UHECR arrival directions seem to highlight the
+importance of nearby sources and/or structures, especially when considered in tandem with the
+arguments for local sources discussed in section 2.1.
+4.2 Possible Source Classes
+At this stage, unambiguous source identifications from anisotropy data alone are not possible.
+We must therefore consider carefully the underlying astrophysics when considering the best can-
+didate UHECR sources. Here, we will make a whistle-stop tour of possible sources based on the
+physics underpinning particle acceleration to ultrahigh energies discussed in the previous section.
+4.2.1 Star-forming Galaxies
+Particle acceleration in star-forming galaxies has been extensively studied, but the indication of
+anisotropy reported by PAO [98], with a 4𝜎 correlation with a ‘starburst’ catalogue, ignited interest
+in this class of objects as UHECR sources. We prefer to use the term ‘star-forming’ rather than
+‘starburst’ for this catalogue, as many of the galaxies have fairly typical star formation rates rather
+than the extreme values normally associated with starbursts. A number of local star-forming galaxies
+13
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+are known to be gamma-ray emitters and include bona fide starbursts such as M82 [103, 104]. It is
+therefore clear that high-energy particle acceleration does take place in these sources.
+A number of authors have either discussed the physics of particle acceleration in these starburst
+‘superwinds’ or proposed them as UHECR sources [105–108]. The superwinds are thought to be
+caused by dramatic bursts of star formation, with the combined effect of supernovae and stellar
+winds (and possibly also CR pressure from low energy CRs) driving a powerful kpc-scale outflow
+[109, 110]. The kinetic powers are estimated at ∼ 1042 erg s−1 for sources like NGC 253 and M82,
+with shock velocities of ∼ 1000 km s−1 [105, 110, 111]. Taking these numbers as characteristic we
+find a maximum rigidity estimate of
+Rmax ∼ 0.15 EV
+�� 𝜖𝑏
+0.1
+� �
+𝑄𝑘
+3 × 1042 erg s−1
+� �
+𝑣
+1000 km s−1
+��1/2
+𝜂−1
+(11)
+for particles accelerated by starburst superwinds, once again emphasizing the importance of the
+electric field and associated velocity term. This estimate is on the optimistic end of the more
+detailed calculations presented by Ref. [105], and would require quite efficient magnetic field
+amplification. UHECRs at our target maximum rigidity of 10 EV would therefore appear to be
+beyond the capabilities of starburst superwinds. Star-forming galaxies may still be the sources
+of UHECRs on the sky through their accumulated populations or historical record of magnetised
+neutron stars (magnetars and/or pulsars), gamma-ray bursts or tidal disruption events. We discuss
+these source classes in sections 4.2.4, 4.2.3 and 4.2.5, respectively. In addition, star-forming galaxies
+are polluters of the circumgalactic medium [CGM; e.g. 112, 113], a process which could produce
+magnetic fields on large scales and act as barrier to, or reflector of, UHECRs (see section 5).
+4.2.2 Radio Galaxies and AGN
+Radio galaxies (using our adopted definition) are active galaxies which produce giant, kpc-
+scale jets that emit synchrotron radiation in the radio band [114]. They have been commonly
+suggested as UHECR sources [29, 43, 83, 115–120], and the local radio galaxy Centaurus A (Cen
+A) is a particularly compelling candidate [81, 121, 122]. Radio galaxies are known to accelerate
+non-thermal particles as revealed by their radio, X-ray and gamma-ray emission, but the particle
+acceleration mechanism and sites of energy dissipation vary from source-to-source. Their radio
+morphology gives important clues as to their particle acceleration physics, and is broadly split
+into two Fanaroff-Riley (FR) classes [123]: FR-I sources are brighter in the centre and resemble
+a disrupted plume of jet material, whereas FR-II sources are brightest far from the nucleus and
+remain well-collimated until they reach the termination shock, producing a radio hotspot. The FR
+dichotomy is thought to be caused by a combination of jet power and environment [114], with
+FR-II sources generally being associated with powerful sources and/or poorer group or cluster
+environments (although Ref. [124] has shown FR-II morphologies can be produced down to rather
+low radio luminosities). From a particle acceleration perspective, it is generally thought that FR-
+II sources primarily accelerate the synchrotron-emitting electrons in the hotspot associated with
+the jet termination shock [e.g. 125–127] with additional particle acceleration along the jet itself
+[128–130] and in the lobes or backflows [83, 131, 132]. By contrast, many FR-I sources appear
+to continuously accelerate nonthermal electrons along their length in a distributed, in-situ process
+[133, 134], although they can also have features such as knots [135, 136] or bow shocks [137, 138].
+14
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+Given their higher average jet powers and strong termination shocks, FR-II radio galaxies
+are the natural class of UHECR accelerators from an energetic perspective [43, 115, 116]. The
+maximum energy in radio hotspots may be severely limited due to their relativistic nature [139],
+which motivated Matthews et al. to propose a model in which UHECRs are accelerated in multiple
+shocks of 𝛽 ≈ 2 in supersonic backflows. The maximum rigidity in these backflows is estimated,
+using hydrodynamic simulations, at 50 𝐸𝑉. However, powerful radio galaxies are also relatively
+rare, with only a handful of FR-II sources within a few hundred Mpc [140, 141]. This has lead
+various authors to consider local FR-I radio galaxies such as Cen A and Fornax A, which also have
+compelling associations with UHECR anisotropies [141]; however, in Cen A the current jet power
+is thought to be ∼ 1043 erg s−1 [83] and the bow-shock associated with the current activity is not fast
+or powerful to accelerate UHECRs [137]. Thus, although particle acceleration in mildly or non-
+relativistic shocks associated with radio galaxies seems quite attractive as an UHECR origins story,
+this requires flickering-type variability or jet powers which were significantly higher in the past
+[96, 141, 142]. A further challenge is to make sure that, if the jet is initially leptonic, then it entrains
+a significant hadronic component [83, 119]. Alternatively, Hardcastle, Wykes and collaborators
+[81, 83, 119] have suggested Fermi II acceleration in Cen A’s turbulent lobes, which could produce
+UHECRs for very fast Alfvén speeds. Furthermore, Eichmann and collaborators have shown that
+the UHECR spectrum can be reproduced from a population of radio galaxies including local sources
+like Cen and Fornax A, even when taking into account the detailed source physics and propagation
+[29, 120, 143].
+Finally, AGN winds, which are responsible for X-ray ‘ultra-fast outflow’ features [144] and
+for quasar blue-shifted broad absorption lines [145, 146], can drive shocks into their surroundings.
+These shocks may accelerate particles and produce synchrotron and inverse Compton emission
+[147].
+Although there is no definitive detection, there are observational hints of quasar wind
+contributions to radio and gamma-ray emission [148–151].
+Quasar winds might reach 𝑄𝑘 ∼
+5 × 1044 erg s−1 [147], if the outflow is 5% efficient compared to the bolometric power.
+The
+maximum shock velocity at the reverse shock is ≲ 0.1𝑐, suggesting a maximum rigidity of a few
+EeV from equation 4. For estimates based on Ref. [147] of 𝐵 ∼ 1 mG and 𝑅 ∼ 0.1 kpc, we
+obtain R𝐻 ∼ 10 EV. AGN winds were investigated as UHECR sources by Ref. [152], who found
+maximum proton energies of ∼ 100 EeV for optimistic parameters. Thus, AGN winds merit further
+investigation, but it is hard to draw any firm conclusions here, and the current lack of evidence for
+nonthermal particles makes them less compelling as UHECR accelerators.
+4.2.3 Gamma-ray bursts
+Gamma-ray bursts (GRBs) are violent, catastrophic explosions that release flashes of gamma-
+rays and prodigious amounts of energy. The general paradigm [e.g. 153] is that short GRBs are
+associated with merging neutron stars, and long GRBs with ‘collapsars’, in which a massive stellar
+core collapses to form a black hole; the latter are more common and so probably more interesting
+as UHECR sources.
+GRBs are known to accelerate particles, most likely in shocks, with the
+prompt emission thought to be produced by highly relativistic colliding internal shocks, giving
+way to a longer-lived afterglow phase powered by a forward shock. Additionally, emission from
+the reverse shock can be detected at early times [154]. MAGIC has detected early time afterglow
+emission at hundreds of GeV coincident with GRB 190114C [155], and HESS detected late time
+15
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+afterglow emission up to 3 TeV from GRB 190829A [156]. Furthermore, during the writing of
+these proceedings, LHAASO has reported an exciting detection of early time emission up to 18 TeV
+associated with the brightest (but not most intrinsically luminous) GRB to-date, GRB 221009A
+[157]. GRBs are clearly excellent particle accelerators up to at least the TeV regime – but can they
+reach ultrahigh energies?
+GRBs have been regularly discussed as UHECR sources, since a few years after workable
+theoretical models for the GRB engine and resulting fireball emerged [47, 158, 159]. They are
+attractive on account of their energetics; a GRB releases a total isotropic energy of up to 𝐸iso ∼
+1054 erg [e.g. 160] in a relatively short amount of time, and a large fraction of this power is dissipated
+through a strong shock. Waxman [158] and Vietri [159] argued that protons could reach energies in
+excess of 1020 eV and that the observed UHECR flux above 1020 eV could be explained as long as
+GRBs release similar amounts of energy in > 1020 eV CRs to the amount released in gamma-rays.
+More recent studies have confirmed the need for rather efficient UHECR production, as well as high
+baryon loading, to produce the UHECR flux at Earth [47, 91, 161]. However, if these efficiencies
+can be accommodated, then GRB models can in principle reproduce the observed UHECR spectrum
+rather well [91].
+As mentioned in section 3.2.1, an additional difficulty with UHECR acceleration in GRB blast
+waves arises from the ultra-relativistic nature of the shock. Even GRB afterglows have initial shock
+Lorentz factors of ∼ 100, approximately evolving with observer time, 𝑡obs as Γsh ∝ 𝑡−3/8
+obs
+in the case
+of a uniform density medium [162, 163]. This evolution leads to a relativistic-Newtonian transition
+around a year after the initial burst [163], and this non-relativistic phase dissipates significantly
+less power through the shock. For GRB afterglow shocks to accelerate UHECRs, one therefore
+either needs to circumvent the relativistic shock problem, or find another way to reach the UHE
+regime (e.g. by reconnection or shear acceleration). While in some sense this is the same problem
+discussed above for radio galaxies, in radio galaxies the jet bulk Lorentz factors are thought to be
+much lower and the presence of mildly or non-relativistic shocks in backflows and, perhaps in some
+cases, jets themselves, makes them more conducive to UHECR acceleration than GRBs. Further
+work is clearly needed to better understand the particle acceleration and shock physics in GRBs,
+but in principle the high energy leptonic emission is able to provide a useful probe of the turbulent
+plasma conditions. This was demonstrated recently by Ref. [68], with the recent TeV gamma-ray
+emission detected perhaps already challenging our understanding of the nature of turbulence in
+relativistic shocks.
+4.2.4 Magnetised Neutron Stars
+A highly magnetised, rapidly rotating neutron star can accelerate particles with a voltage drop
+of 𝜙 = Ω2𝜇/𝑐2, where Ω is the angular velocity and 𝜇 = 𝑅3
+NS𝐵 is the magnetic moment. This voltage
+is time-dependent since the NS can spin down (or up) and thus Ω can change. For sufficiently small
+periods and large magnetic dipole moments, the maximum energy is 𝐸max ∼ 1021 eV 𝑍 𝜇33 𝑃−2
+ms
+[e.g. 92], where 𝜇33 is the magnetic dipole moment in units of 1033 G cm3, typical for newly born
+magnetars with 𝐵 ∼ 1015 G [164], and 𝑃ms is the orbital period in milliseconds. For young pulsars
+with 𝐵 ∼ 1012 G the maximum energy is a factor of 1000 lower. For magnetars, this characteristic
+energy is easily within the UHECR regime for reasonable parameters; however, the problem with
+this mechanism is that particles accelerated in the NS magnetosphere (inside the light cylinder)
+16
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+undergo debilitating curvature losses, while CRs also undergo rapid synchrotron losses in such
+strong magnetic fields. Furthermore, the strong magnetic fields can be screened by prolific pair
+production [45, 84]. These limitations have instead led various authors to consider acceleration in
+the pulsar wind termination shock or nebula associated with a very young, rapidly spinning pulsar
+[85, 86, 165] or magnetar [92, 166], or from ∇𝐵 drifts in the wind nebula [167]. The termination
+shocks are likely to be highly relativistic and so the same issues discussed above for radio galaxies
+and GRBs apply. However, the Crab nebula does seem to be able to accelerate particles in close to
+optimal conditions; LHAASO have detected PeV emission from the Crab nebula [168] that requires
+𝜂−1 = 0.15 in our notation. It seems fairly likely that Crab-like PWNe are perhaps PeVatrons,
+but probably not UHECR sources, with a maximum proton energy of ∼ 30 PeV [45]. The higher
+spin-down luminosities of rapidly spinning magnetars make them more attractive, and the winds
+they power are feasible sites of particle acceleration to rigidities of ∼ 10 EV [92].
+4.2.5 Other Objects
+A veritable menagerie of other astrophysical objects have been proposed as UHECR sources;
+we only briefly discuss tidal disruption events (TDEs) and clusters of galaxies. TDEs are transient
+events that occur when a star passes close to a supermassive black hole and is ripped apart and
+accreted [169].
+Radio emission from jets in TDEs has been detected in a handful of sources
+[170, 171], and these jets could be interesting as sites of UHECR acceleration providing the CRs
+can survive in the intense radiation field and the relativistic shock problem can be overcome.
+Ref. [172] finds the former should be possible in the reverse shock and estimates a maximum energy
+of 64 EeV for protons (see also Ref. [173]). Cluster shocks – associated with gas accretion or
+galaxy mergers – are potential sites of UHECR acceleration, because they are extremely large (virial
+radii ∼ 1 Mpc). The intracluster medium (ICM) magnetic field is ∼ 1 𝜇G and the shock velocity is
+∼ 1000 km s−1, leading to a Hillas rigidity of 𝑅H ∼ 3 EV. This estimate is roughly in line with the
+maximum proton energy suggested by Kang for acceleration by multiple weak shocks in the ICM
+[174], and the calculation by Ref. [175], who find cluster accretion shocks can produce UHECRs
+from ∼ 2 − 10 EeV but struggle to produce the very highest energies. However, Ref. [176] show
+that UHECRs with energies up to ∼ 60 EeV could be accelerated in larger (𝑅 ∼ 5 Mpc) accretion
+shocks associated with ‘caustics’ if Bohm diffusion applies, finding that the Virgo cluster could
+make a substantial contribution to the observed UHECR flux at ∼ 30 EeV.
+4.3 A Modified Hillas Diagram
+In Hillas’ 1984 work, a number of useful diagnostic plots for identifiying UHECR sources
+were provided. Figure 1 of the same paper is often referred to as a ‘Hillas diagram’, in which
+characteristic size is plotted along the 𝑥-axis, and magnetic field along the 𝑦-axis. One can then
+obtain a rough estimate of the feasibility of UHECR sources by drawing diagonal lines of constant
+rigidity, an exercise that has been regularly used to inform discussions of UHECR origins [e.g.
+43, 177, 178]. However, technically the diagram only conveys the ability of a source to confine
+UHECRs of a given rigidity – it does not include the characteristic velocity of the accelerator and
+thus does not capture the impact of the electric field, E ∝ 𝛽𝐵. The velocity of the scatterers is
+plotted in Hillas’ figure 6, which is perhaps more informative in terms of which sources can actually
+accelerate to the UHE regime. Here we instead show a ‘modified Hillas diagram’ in Fig. 4, in which
+17
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+105
+109
+1013
+1017
+1021
+1025
+R (cm)
+10°10
+10°6
+10°2
+102
+106
+1010
+1014
+bB (G)
+1 EV
+10 EV
+100 EV
+Starburst 
+Winds#
+GRBs*†
+Magnetised 
+NS*†
+Radio 
+Galaxies†
+10 EeV p
+100 EeV Fe
+Magnetars
+Magnetar Nebulae/Winds 
+RG Hotspots
+& Backflows
+RG Lobes
+Clusters
+GRB Prompt
+TDEs
+GRB Afterglows
+AGN Winds
+Starbursts
+Figure 4: Modified Hillas plot showing the maximum electric field available, 𝛽𝐵, plotted against charac-
+teristic size. The three diagonal lines show parameters needed to achieve the labelled maximum rigidity.
+The coloured points show the values from table 1, the circles give a feel for the uncertainties, and points
+connected by lines have multiple estimates. The symbols (†, ∗, #) represent caveats; the meanings of † and
+∗ are defined in the text, while # means the source does not dissipate sufficient kinetic or magnetic power
+(equation 4). The lines with hashed regions mark the maximum magnetic field for acceleration of 10 EeV
+protons and 100 EeV Fe nuclei (equation 10). Broadly speaking, a vaiable UHECR source must sit in the
+region that is shaded in translucent green.
+the electric field available, 𝛽𝐵, is made explicit and plotted directly on the 𝑦-axis. The overall result
+is rather similar to the usual diagram, except that some sources move downwards in the parameter
+space (if they have 𝛽 < 1). The main source classes discussed in the text have been given extra
+prominence in their labelling and the values used in our estimates are given in table 1. The plot
+does not capture the detailed physics of UHECR acceleration, or the relativistic shock problem,
+and nor does it contain information about the number density or luminosity density of sources;
+nevertheless, this modified Hillas diagram acts as a useful summary of the maximum rigidities
+attainable in UHECR candidate sources, while emphasizing the need for more accurate estimates
+of 𝛽 and 𝐵 in cosmic accelerators.
+5.
+Source Variability and UHECR Echoes (When?)
+‘When’ is perhaps an unusual interrogative to apply to the origin of UHECRs. Our motive for
+using it is to briefly describe our ‘echoes’ model for UHECR production in dormant or declining
+sources, but time-dependence is an intrinsic part of any UHECR study, for a number of reasons.
+UHECR deflections in magnetic fields inevitably lead to a time delay with respect to any associated
+electromagnetic or neutrino signal [96, 181]. Variability on a wide range of timescales is ubiquitous
+in accreting systems such as AGN [182, 183] and the fuelling of the AGN might be expected to
+cause flickering or stochastic activity [184, 185]. Star formation also varies over time due to various
+18
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+Source Class
+𝛽
+𝑅 (cm)
+𝐵 (G)
+Ref.
+Section
+Magnetars†,∗
+1
+106
+1014
+[92]
+4.2.4
+Magnetar Winds†,∗
+1
+1014, 1016
+104, 10
+[92]
+4.2.4
+GRB Prompt/Internal†,∗
+1
+1013, 1016
+106, 100
+[91, 179]
+4.2.3
+GRB Afterglows†
+1
+1016
+1
+[179]
+4.2.3
+RG Hotspots†
+1
+1021
+3 × 10−4
+[139]
+4.2.2
+RG Backflows
+0.2
+1021
+10−4
+[131]
+4.2.2
+RG Lobes
+0.01
+1022
+10−5
+[180]
+4.2.2
+Starburst Winds
+0.004
+1021
+10−4
+[105, 110]
+4.2.1
+TDE Jets†
+1
+1014
+100
+[172]
+4.2.5
+Cluster Shocks
+0.004
+1024
+10−6
+[175, 176]
+4.2.5
+AGN Winds
+0.1
+1020
+10−3
+[147]
+4.2.2
+Table 1: Table of characteristic values of parameters adopted for the modified Hillas diagram (Fig. 4). In
+some cases multiple values are quoted designed to crudely represent the range spanned by evolving sources
+or literature values. Sources marked with a dagger (†) have relativistic characteristic speeds and so may have
+to contend with the relativistic shock problem discussed in section 3.2.1. Sources marked with an asterisk
+(∗) have strong magnetic fields and are subject to significant curvature or synchrotron losses which limit the
+maximum energy, so the Hillas energy is often a significant overestimate. A reference is given for each set of
+estimates, and the sub-section in which the source class is discussed is also labelled. The estimates presented
+here are inhomogenous and subject to large uncertainties and biases, and should not be taken as authoritative.
+triggers and/or regulators [186, 187], and some of the sources mentioned in the previous section
+are catastrophic events. With this in mind it seems reasonable to consider the time variability as an
+important factor for UHECR sources, as discussed before for transients like GRBs [158, 188, 189].
+Recently, Ref. [190] showed that a feasible model for intermediate UHECR anisotropies could
+be constructed, in which UHECRs are accelerated in a powerful outburst in a nearby source (Cen
+A), and the UHECRs then ‘echo’ or scatter off nearby magnetic structures, in this case associated
+with the circumgalactic medium of star-forming galaxies that lie within a few Mpc in the CoG/Local
+Sheet. A schematic depicting this scenario is shown in Fig. 5. The motivation here is that Cen A
+is unique in the CoG structure in having powerful AGN jet activity, and the model provides a way
+for arrival directions to be correlated with star-forming galaxies without requiring acceleration in
+the star-forming galaxies themselves. A workable model requires Cen A to have had a UHECR
+luminosity 20 Myr ago that is ∼ 200 times that currently reaching Earth from the source, and also
+necessitates magnetic field strengths of ∼ 10 − 20 nG out to a distance of 400 − 800 kpc in the
+CGM. M82 could well be able to maintain a large magnetic field on this scale through advection or
+amplification of magnetic fields [113, 190, 191], in which case the UHECR echo could explain the
+TA hotspot. When star-forming galaxies such as NGC 253 and IC 342 are included, an anisotropy
+pattern can be produced that is similar to the all-sky anisotropy reported by PAO and TA [101, 102].
+The echoes model should be testable through the use of UHECR ‘composition clocks’. Rigidity-
+dependent propagation and species-dependent loss lengths (Fig. 1) mean that, in principle, we should
+expect different compositions from the echo waves compared to the direct wave due to the different
+19
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+Star-forming galaxy with 
+magnetized CGM 
+(e.g M82)
+Centaurus A
+Earth
+“Direct” wave
+“Echo” wave
+CRs from powerful 
+past outburst
+L/c ~ 33 Myr
+L/c ~ 12 Myr
+Figure 5: Schematic depicting the basic principle of the echoes model proposed by Ref. [190] to explain
+intermediate scale anisotropies in PAO and TA data without requiring acceleration in starburst galaxies
+directly.
+path lengths travelled (see Fig. 5). In particular, species with short photodistintegration loss lengths
+could be under-represented in the echo wave, while high rigidity particles might be able to escape
+the source more quickly and be over-represented in the echo wave. We will present quantitative
+investigations of both of these effects in a future study (Taylor et al., in prep), which are exciting
+given the improved composition diagnostics of AugerPrime [192].
+We note that many authors have discussed the impact of extragalactic magnetic field structures
+on the arrival directions of UHECRs. For example, Kotera & Lemoine [193] suggest “the possibility
+that the last scattering center encountered by a CR be mistaken with the source of this CR”, while
+Kim et al. [194] explore a similar effect encountered if UHECRs travel along magnetic filaments
+before scattering towards Earth.
+The deflection of UHECRs by magnetic fields is a generic
+difficulty associated with UHECR searches, which is exacerbated by how difficult it is to glean
+accurate knowledge of astrophysical magnetic field strengths and structures.
+6.
+Conclusions and Future Outlook
+The origins of UHECRs remain elusive, despite extensive efforts, and the study of their
+acceleration and propagation is a rich topic that has synergies with a whole host of subfields
+of astrophysics and particle physics. In this review, we have described how particles might be
+accelerated to super-EeV energies and discussed the basic energetic requirements for this to happen,
+informed in a large part by Hillas’ 1984 work [43]. We have touched on more detailed aspects of the
+plasma physics of particle acceleration, particularly relating to shock acceleration, that are critical to
+the particle acceleration process. We used these physical arguments to write a ‘checklist’ for UHECR
+sources, which we applied to a range of astrophysical candidates such as AGN, GRBs, magnetised
+neutron stars and star-forming galaxies. Even in sources that dissipate energy at an astonishing rate,
+accelerating UHECRs still often requires particle acceleration physics to be stretched to its limits.
+20
+
+How, where and when do cosmic rays reach ultrahigh energies?
+James H. Matthews
+As we have alluded to on multiple occasions in this review, we are presently in an excit-
+ing era for UHECR experiment and theory, as well as particle acceleration more generally. The
+UHECR spectrum is well-characterised, 𝑋max distributions at ultrahigh energies are constraining
+the UHECR composition, and UHECR anisotropies are finally beginning to emerge above the noise
+at statistically significant levels. Nearly simultaneously, we have entered a ‘four-messenger’ era of
+high-energy astrophysics through the observation, sometimes with electromagnetic counterparts,
+of >TeV neutrinos by IceCube [195] and gravitational waves by LIGO and VIRGO [196]. In
+the future, the observational capabilities of the TAx4 [197] and AugerPrime [192] upgrades will
+dramatically improve our view of the UHECR sky, and the Cherenkov Telescope Array will shortly
+offer unprecendented sensitivity to TeV gamma-rays [198]. In combination with a rapidly ma-
+turing theoretical landscape [12], these transformative instruments offer great prospects for fully
+understanding the ‘origin story’ of UHECRs.
+Acknowledgements
+We would like to thank Tony Bell, Lauren Rhodes, Frank Rieger, Claudio Galelli, Sergio
+Petrera and Alan Watson for helpful discussions. We are extremely grateful to Jörg Horandel and
+the organising committee for ECRS 2022 for an excellent conference. JM acknowledges funding
+from the Royal Society and, previously, from the Herchel Smith fund at Cambridge.
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+25
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+X-ray Thomson scattering spectra from DFT-MD simulations based on a modified
+Chihara formula
+Maximilian Sch¨orner,1, ∗ Mandy Bethkenhagen,2, 3 Tilo D¨oppner,4
+Dominik Kraus,1, 5 Siegfried H. Glenzer,6 and Ronald Redmer1
+1University of Rostock, Institute of Physics, 18051 Rostock, Germany
+2 ´Ecole Normale Sup´erieure de Lyon, Laboratoire de G´eologie de Lyon LGLTPE UMR 5276,
+Centre Blaise Pascal, 46 all´ee d’Italie Lyon 69364, France
+3Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
+4Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
+5Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
+6SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
+(Dated: January 5, 2023)
+We study state-of-the-art approaches for calculating x-ray Thomson scattering spectra from den-
+sity functional theory molecular dynamics (DFT-MD) simulations based on a modified Chihara
+formula that expresses the inelastic contribution in terms of the dielectric function. We compare
+the electronic dynamic structure factor computed from the Mermin dielectric function using an
+ab initio electron-ion collision frequency to computations using a linear response time dependent
+density functional theory (LR-TDDFT) framework for hydrogen and beryllium and investigate the
+dispersion of free-free and bound-free contributions to the scattering signal. A separate treatment of
+these contributions in the Mermin dielectric function shows excellent agreement with LR-TDDFT
+results for ambient-density beryllium, but breaks down for highly compressed matter where the
+bound states become pressure ionized. LR-TDDFT is used to reanalyze x-ray Thomson scattering
+experiments on beryllium demonstrating strong deviations from the plasma conditions inferred with
+traditional analytic models at small scattering angles.
+I.
+INTRODUCTION
+X-ray Thomson scattering (XRTS) has been one of the
+premier diagnostic tools for warm dense matter (WDM)
+experiments, enabling measurements of the electron den-
+sity, temperature and ionization state [1–3]. The states
+reached in these experiments are characterized by tem-
+peratures of a few electronvolts (eV) and around solid
+densities, which constitutes strongly correlated plasmas
+with non-negligible degeneracy. This prevents the appli-
+cation of ideal plasma theory for the analysis of these
+experiments, and rather requires a quantum mechani-
+cal treatment in a many-body framework.
+Knowledge
+of equation of state data as well as thermal and electrical
+transport properties for warm dense hydrogen and beryl-
+lium is essential for modelling astrophysical objects [4, 5]
+and inertial confinement fusion [6], where hydrogen is
+used as fuel while beryllium often serves as ablator ma-
+terial [7, 8]. Furthermore, hydrogen and beryllium are
+excellent test cases for new theoretical approaches. The
+analytical behavior in many limiting cases for fully ion-
+ized hydrogen plasmas are known and beryllium can be
+used to test the treatment of bound states in a simple
+low-Z material. WDM is typically opaque in the optical
+regime, as the light frequency is smaller than the plasma
+frequency ωpl of these plasmas.
+Therefore, it is indis-
+pensable to have diagnostic tools at experiments that are
+well understood, both experimentally and theoretically.
+∗ [email protected]
+XRTS has proven to overcome many of the experimen-
+tal challenges of probing WDM. The high energy x-ray
+photons can penetrate dense plasmas and since the ad-
+vent of free electron lasers (FEL), rep-rated x-ray sources
+with sufficient brilliance for probing short-lived transient
+states are available in addition to laser-plasma sources
+which only allow a limited number of experiments and re-
+quire complex sample assemblies. New FEL techniques
+like self-seeding [9, 10] have also resulted in much nar-
+rower bandwidths of the x-ray source, enabling the mea-
+surement of phonons and ion acoustic modes [11, 12] and
+a better resolution of density and temperature-sensitive
+regions in the XRTS spectrum.
+Due to the steadily improving quality of collected spec-
+tra, it is vital to have accurate theoretical modeling of the
+scattering. While in the past, the resolution of XRTS
+spectra often did not allow for discrimination between
+different theoretical approaches, now, fitting experimen-
+tal spectra to theoretical models has allowed predictions
+of electron temperature and density to within a few per-
+cent uncertainties [13–15]. As a result, the fidelity of the
+theoretical model used is now the limiting factor in de-
+termining the correct plasma parameters in experiments
+that employ XRTS as a diagnostic tool. Most approaches
+rely on the semi-classical Chihara decomposition [16, 17]
+of the spectrum into three distinct contributions which
+originates from distinguishing between free and bound
+electrons in a chemical picture. An analogous fully quan-
+tum mechanical description has also been proposed [18].
+The standard approach for modeling XRTS spectra in
+the Chihara description is a combination of theories to
+describe each component individually [19]. The ion dy-
+arXiv:2301.01545v1  [physics.plasm-ph]  4 Jan 2023
+
+2
+namics are usually described by the hypernetted-chain
+approximation with different expressions for the inter-
+action potential while the form factors are described by
+a screened hydrogenic approximation to the wave func-
+tions [20] and the Debye-H¨uckel approximation for the
+screening cloud. The plasmon can be described by the
+random phase approximation (RPA) or the Mermin di-
+electric function in order to also include electron-ion col-
+lisions which can also be approximated to different de-
+grees [21]. Further electron correlations can be accounted
+for by local field corrections [22]. Contributions that are
+related to bound-free transitions are treated within the
+impulse approximation [23] which is sometimes modified
+by the ionization potential depression and normalized
+according to different sum rules.
+Each of these theo-
+ries entails various approximations and achieves different
+degrees of fidelity in a wide range of temperatures and
+densities which severely complicates gauging sources of
+errors.
+Furthermore, the ionization degree is often left as an
+unconstrained parameter in the fitting procedure which
+risks overfitting and neglects its dependence on tempera-
+ture and density. In recent years, this approach has been
+partially replaced by ab initio descriptions like density
+functional theory molecular dynamics (DFT-MD) sim-
+ulations and real time or linear response time depen-
+dent DFT (RT/LR-TDDFT) computations. Witte et al.
+successfully used electron-ion collision frequencies deter-
+mined by DFT to accurately model the plasmon of an
+aluminum plasma [24]. This approach was subsequently
+compared to LR-TDDFT by Ramakrishna et al. for am-
+bient and extreme conditions in aluminum [25] and car-
+bon [26], which was then used to discern miscibility in
+an XRTS experiment [13].
+Baczewski et al.
+went be-
+yond the Chihara decomposition by simulating the real
+time propagation of the electronic density using RT-
+TDDFT [27].
+Path integral Monte Carlo simulations
+have delivered approximation-free results for the uniform
+electron gas [28] and hydrogen plasmas [29], but are cur-
+rently unable to describe heavier elements.
+In this work, we describe the calculation of XRTS spec-
+tra in an ab initio framework where each contribution
+to the Chihara decomposition is extracted from a DFT-
+MD simulation that is based on well defined approxima-
+tions and limits the free parameters to the mass density
+and temperature by constraining the number of free elec-
+trons per atom according to a new technique recently
+proposed by Bethkenhagen et al. [30]. The capability of
+DFT-MD to compute ion dynamics and the form factors
+was already demonstrated and tested in previous pub-
+lications [31–33].
+Therefore, we focus on the inelastic
+contribution that can be computed from a self-consistent
+DFT cycle using the Kubo-Greenwood formula or LR-
+TDDFT. We give an overview of the theoretical founda-
+tion for computing the electronic dynamic structure fac-
+tor from the Mermin dielectric function with a dynamic
+complex collision frequency and apply this framework
+to extract a DFT-based collision frequency in Secs. II A
+and II B. In Sec. II D we give the details of the simula-
+tion method. We compute DFT-based collision frequen-
+cies for a hydrogen plasma and compare them to sev-
+eral analytic approaches in Sec. III and we study the im-
+pact of these collision frequencies on DSFs for hydrogen
+and beryllium plasmas in Secs. IV A, IV B, and IV C. In
+Sec. V, we apply LR-TDDFT to XRTS experiments on
+beryllium to evaluate their impact on the inferred plasma
+parameters.
+II.
+THEORETICAL BACKGROUND
+A.
+Dynamic structure factor
+The electronic dynamic structure factor (DSF) [1]
+Stot
+ee (⃗k, ω) =
+1
+2πNe
+� ∞
+−∞
+dt⟨ne
+⃗k(τ)ne
+−⃗k(τ + t)⟩τ eiωt
+(1)
+is the central quantity representing the spatially resolved
+power spectrum of an electronic system, describing its dy-
+namics at given temporal and spatial periodicities given
+by the frequency ω and the wave vector ⃗k, respectively.
+The number of considered electrons is Ne and the spatial
+Fourier components of the electron density are given by
+ne
+⃗k. The time is described by t and τ, where ⟨...⟩τ de-
+scribes a time average over τ. Experimentally, Stot
+ee (⃗k, ω)
+can be used to identify how strong a photon will cou-
+ple to density fluctuations at a given energy transfer and
+scattering angle [1].
+In this work we will use a slight
+modification of the common decomposition of Eq. (1) in-
+troduced by Chihara [16, 17]:
+Stot
+ee (⃗k, ω) = |fi(⃗k) + q(⃗k)|2Sii(⃗k, ω)+
++ ZfS0
+ee(⃗k, ω) + ZbSbf(⃗k, ω)
+�
+��
+�
+Z Set(⃗k,ω)
+.
+(2)
+The first term refers to the elastic response of the elec-
+trons which follow the ion motion described by the ion-
+ion structure factor Sii(⃗k, ω). Here, fi(⃗k) describes the
+contribution of tightly bound electrons and q(⃗k) repre-
+sents the loosely bound screening cloud around the ions.
+The second term, called the electron feature, arises from
+the collective behavior of the free electrons in the system
+undergoing transitions to different free-electron states.
+The number of free electrons per atom is labeled Zf and
+their DSF is denoted by S0
+ee(⃗k, ω).
+The last term in
+Eq. (2) is the bound-free contribution. In the original
+work, Chihara clearly separates free and bound electrons
+and describes this term as a convolution of the DSF of
+the core electrons with the self-part of the ionic DSF [17].
+We treat the bound-free contribution on the same footing
+as the free electron contibution and introduce the bound-
+free DSF Sbf(⃗k, ω) and the number of bound electrons per
+atom Zb. Both the free electron and bound-free contri-
+butions arise due to inelastic transitions of the electrons
+
+3
+and can, therefore, be combined into one DSF Set(⃗k, ω)
+that accounts for all electronic transitions. This avoids
+the artificial separation into bound and free electrons for
+both the charge state Z and the DSF. According to the
+fluctuation-dissipation theorem [34], this combined DSF
+can be related the dielectric response described by the
+dielectric function ϵ(⃗k, ω) via
+Set(⃗k, ω) = − ϵ0ℏ⃗k2
+πe2ne
+Im
+�
+ϵ−1(⃗k, ω)
+�
+1 − exp
+�
+−ℏω
+kBTe
+�.
+(3)
+The vacuum permittivity is denoted by ϵ0, the reduced
+Planck constant is ℏ and e is the elementary charge. The
+electron density is given by ne, the electron tempera-
+ture is Te and the Boltzmann constant is kB. At which
+conditions the separation into free and bound-free part
+in Eq. (2) is justified and yields the same results as the
+combined approach is discussed in Secs. IV B and IV C.
+B.
+Dielectric Function with electron-ion collisions
+The dielectric function ϵ(⃗k, ω) is a central material
+property that is connected to other material properties,
+like the electrical conductivity σ(ω) in the long wave-
+length limit or the DSF via the fluctuation dissipation
+theorem from Eq. (3). One of the first approaches that
+produced collective features of the electron system, like
+plasmons, is the Lindhard dielectric function [35]
+ϵRPA(⃗k, ω) = lim
+η→0
+�
+1−
+− 2e2
+ϵ0k2
+�
+d3q
+(2π)3
+f⃗q−
+⃗k
+2 − f⃗q+
+⃗k
+2
+ℏ (ω + iη) + E⃗q−
+⃗k
+2 − E⃗q+
+⃗k
+2
+�
+.
+(4)
+which accounts for electric field screening in the Random
+Phase Approximation (RPA). The arguments ⃗k and ω
+are the wave vector and the angular frequency, respec-
+tively while the electron charge is denoted by e. E⃗q and
+f⃗q are the kinetic energy and the Fermi occupation of
+an electron with wave vector ⃗q in the unperturbed free
+electron gas. The small imaginary contribution to the
+frequency η is introduced to avoid the pole in the in-
+tegration and approaches zero thereafter. However, for
+degenerate, strongly correlated systems electron-ion in-
+teractions, which are neglected in Eq. (4), have to be ac-
+counted for in order to accurately describe the dielectric
+function.
+It was shown that electron-ion collisions can be in-
+cluded via a dynamic collision frequency ν(ω) in the
+framework of the Mermin dielectric function [36–39]
+ϵMermin(⃗k, ω; ν(ω)) = 1+
++
+�
+1 + i ν(ω)
+ω
+� �
+ϵRPA(⃗k, ω + iν(ω)) − 1
+�
+1 + i ν(ω)
+ω
+ϵRPA(⃗k,ω+iν(ω))−1
+ϵRPA(⃗k,0)−1
+.
+(5)
+Extensive work has been performed on the evaluation
+of different analytic collision frequencies and local field
+corrections [21, 22, 40], as well as first attempts to incor-
+porate ab initio results to determine collision frequen-
+cies [41].
+We present the derivation of the RPA dielectric func-
+tion in the presence of a dynamic complex collision fre-
+quency in Appendix A. Equations (5), (A7) and (A8)
+are the basis for calculating the Mermin dielectric func-
+tion for a given dynamic collision frequency ν(ω). In the
+following, because we are dealing with isotropic systems,
+we will only consider the magnitude of wave vector ⃗k and
+drop the vector notation.
+One of the most prominent approximations for the col-
+lision frequency is the Born collision frequency [21], the
+combination of which with the Mermin dielectric func-
+tion in Eq. (5) is called the Born-Mermin approxima-
+tion (BMA). It is widely used in the analysis of XRTS
+spectra in the WDM field.
+We give the exact equa-
+tions used in this work in Appendix B. However, complex
+many-particle effects, as they are considered in ab initio
+simulations, cannot be accounted for by this approach.
+FIG. 1. Schematic work flow for determining the dynamic col-
+lision frequency and k-dependent dielectric function via DFT.
+In Fig. 1, we show the schematic procedure to compute
+a DFT-based collision frequency from an electrical con-
+ductivity in the optical limit. In essence, we construct a
+complex collision frequency for which the Mermin dielec-
+tric function coincides with the ab initio dielectric func-
+tion in the optical limit. As input, the temperature and
+electron density of the plasma are needed for the Mer-
+min dielectric function and the real part of the electrical
+conductivity is needed from the simulation. According to
+
+DFT-MD
+input
+Consistency check
+KK-Transform
+Re[α(k = O, w)]
+Im[o(k = 0,w)]
+Imle
+Re[o]
+Im[o] = Eo w (1 - Re[e])
+mO
+KK-Transform
+Im[e(k = 0,w)]
+Re[e(k = 0,w)]
+DFT(k = 0,w)
+Mermin(k, w; v(w))
+eMermin(k, w; vDFT(w))
+T,n
+DFT()
+Input4
+the Kubo-Greenwood formula [42, 43] the conductivity is
+Re [σ(k = 0, ω)] = 2πe2
+3ωΩ
+�
+⃗g
+w⃗g
+N
+�
+j=1
+N
+�
+i=1
+3
+�
+α=1
+×
+×
+�
+f(ϵj,⃗g) − f(ϵi,⃗g)
+�
+| ⟨ψj,⃗g|ˆvα|ψi,⃗g⟩ |2δ(ϵi,⃗g − ϵj,⃗g − ℏω).
+(6)
+The indices i and j run over the eigenstates, α runs over
+the spatial orientations and ⃗g denotes the reciprocal vec-
+tors in the Brillouin zone where the wave functions ψi,⃗g
+are evaluated. The Fermi-Dirac occupation at a given
+eigenenergy ϵj,⃗g is described by f(ϵj,⃗g) and ˆvα is the ve-
+locity operator in the direction α.
+The normalization
+volume is denoted by Ω and w⃗g is the weigthing of each
+k-point. We translate the electrical conductivity to the
+imaginary dielectric function via
+Im [ϵ(k = 0, ω)] =
+1
+ϵ0ω Re [σ(k = 0, ω)]
+(7)
+and use the Kramers-Kronig transformation to compute
+the corresponding real part, leading to a complex dielec-
+tric function ϵDFT(k = 0, ω). If we require an equivalence
+between the DFT result and the Mermin dielectric func-
+tion in the optical limit
+ϵDFT (k = 0, ω)
+!= lim
+k→0 ϵMermin (k, ω; ν (ω)) ,
+(8)
+the real and imaginary parts must be equal simultane-
+ously. This can be achieved by adjusting the real and
+imaginary part of the dynamic collision frequency which
+feeds into the Mermin dielectric function, leading to a
+two dimensional optimization problem. The result of this
+optimization is a collision frequency νDFT for which the
+analytic Mermin dielectric function yields the same re-
+sults as DFT in the macroscopic limit. Because there is
+no notion of bound states in the theoretical framework
+of the Mermin dielectric function, the electrical conduc-
+tivity must only originate from free or quasi-free states.
+For this purpose, the conductivity in Eq. (6) can be split
+into different contibutions, see Ref. 30 for details.
+Figure 2 shows the convergence of the Mermin dielec-
+tric function and DSF to the DFT result in the opti-
+cal limit for a beryllium plasma at ρ = 5 g/cm3 and
+T = 100 eV. Due to the presence of bound states in
+beryllium at these conditions, only the electrical conduc-
+tivity due to free electrons can be used as an input to the
+workflow depicted in Fig. 1 and all quantities in Fig. 2 are
+free-electron contibutions. The DFT result for the DSF
+SDFT and the dielectric function ϵDFT are only available
+at k = 0 and are shown as a constant reference for the
+various k depicted in Fig. 2. In both panels, it is appar-
+ent that, with the correct collision frequency νDFT, the
+Mermin result converges to the optical limit described
+by DFT. In practice, the limit k → 0 is reached at wave
+numbers that correspond to length scales that are sig-
+nificantly larger than any characteristic length scales of
+the studied system. For beryllium at these conditions,
+the convergence is reached for wave numbers smaller or
+equal to 10−4 ˚A−1 as depicted in Fig. 2. The dielectric
+functions in the upper panel are connected to the DSF
+in the lower panel by Eq. (3). However, it is apparent
+that the dynamic dielectric function in the upper panel
+of Fig. 2 is more sensitive to changes in the wave num-
+ber than the DSF shown in the bottom panel, which is
+dominated by the pole in ϵ−1(k, ω).
+0
+5
+10
+15
+20
+25
+30
+35
+−50
+0
+50
+100
+150
+200
+ϵ(k, ω)
+k = 10−4 ˚A−1
+k = 10−1 ˚A−1
+k = 0.5 ˚A−1
+Re[ϵDFT(k = 0)]
+Im[ϵDFT(k = 0)]
+Re[ϵMermin(k)]
+Im[ϵMermin(k)]
+0
+20
+40
+60
+80
+100
+¯hω [eV]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+S0
+ee(k, ω) [a.u.]
+k = 10−4 ˚A−1
+k = 10−1 ˚A−1
+k = 0.5 ˚A−1
+k = 1 ˚A−1
+k = 3 ˚A−1
+SDFT(k = 0, ω)
+SMermin (k, ω; νDFT)
+FIG. 2. The top panel shows the free-electron part of the di-
+electric function ϵ(k, ω) in a beryllium plasma at ρ = 5 g/cm3
+and T = 100 eV. The DFT results are given at k = 0,
+where the solid lines are the real part and the dash-dotted
+lines are the imaginary part. The Mermin dielectric function
+from Eq. (5) is calculated with the DFT collision frequency
+νDFT. The colors represent different values for k, while the
+real and imaginary parts are given by the circles and crosses,
+respectively. The bottom panel shows the free-electron DSF
+S0
+ee(k, ω) computed from DFT (solid lines) at k = 0 and from
+the Mermin dielectric function (circles) at various k.
+The
+DSFs are scaled to the same magnitude and the dielectric
+function and DSFs are shifted by 75 and 0.5 a.u., respectively,
+with respect to the next lowest wave number for readability.
+
+5
+C.
+Linear response time dependent density
+functional theory
+In the framework of LR-TDDFT the density response
+of the non-interacting Kohn-Sham system can be evalu-
+ated at a finite momentum transfer as [44, 45]:
+χKS(⃗k, ω) = 1
+Ω
+�
+⃗g,i,j
+f(ϵi,⃗g) − f(ϵj,⃗g+⃗k)
+ω + ϵi,⃗g − ϵj,⃗g+⃗k + iη ×
+× ⟨ψi,⃗g|e−i⃗k⃗r|ψj,⃗g+⃗k⟩ ⟨ψi,⃗g|ei⃗k⃗r|ψj,⃗g+⃗k⟩ .
+(9)
+The quantities in this equation are defined analogously to
+the Kubo-Greenwood formula in Eq. (6). This response
+function can be related to the full density response χ via a
+Dyson equation [44], with different levels of approxima-
+tion for the exchange-correlation kernel fXC. A closed
+expression can be written as
+χ(⃗k, ω) =
+χKS(⃗k, ω)
+1 −
+�
+v(⃗k) + fXC(⃗k, ω)
+�
+χKS(⃗k, ω)
+,
+(10)
+where v(⃗k) is the Fourier transform of the Coulomb po-
+tential. The level of the RPA is achieved for fXC = 0, for
+which the dielectric function can be computed as
+ϵRPA
+KS (⃗k, ω) = 1 − 4π
+|⃗k|2 χKS(⃗k, ω).
+(11)
+Because the Mermin dielectric function accounts for elec-
+tron interactions on the level of the RPA, we set fXC = 0
+and use Eq. (11) in Secs. IV A, IV B and IV C to facili-
+tate comparisons. In Sec. V, we use the adiabatic local
+density approximation [44, 46].
+D.
+Computational details
+All DFT-MD simulations for this work were per-
+formed with the Vienna ab initio simulation package
+(VASP) [47–49]. The electronic and ionic parts are de-
+coupled by the Born-Oppenheimer approximation and,
+for fixed ion positions, the electronic problem is solved in
+the finite temperature DFT approach [50]. In VASP, the
+electronic wave functions are expanded in a plane wave
+basis set up to a energy cutoff Ecut. After the electronic
+ground state density is determined self-consistently at
+every time step, the forces on the ions via Coulomb in-
+teractions with other ions and the electron cloud are com-
+puted and the ions are moved according to Newton’s sec-
+ond law.
+The temperature control in the MD simula-
+tion is performed via the Nos´e-Hoover algorithm [51, 52]
+with a mass parameter corresponding to a temperature
+oscillation period of 40 time steps. All simulations are
+performed using the exchange-correlation functional of
+Perdew, Burke, and Ernzerhof (PBE) [53].
+For beryl-
+lium, we use the PAW_PBE Be_sv_GW 31Mar2010 poten-
+tial with an energy cutoff of 800 eV for all simulations
+apart from the compressed case in Sec. IV C for which we
+use a Coulomb potential with a cutoff of 10 000 eV. For
+further details on the hydrogen simulation parameters,
+see Ref. 54.
+The dynamic electrical conductivity, that is the input
+for the scheme presented in Fig. 1, was computed from
+the eigenfunctions and eigenenergies of separate DFT cy-
+cles with a more precise energy convergence criterion
+via the Kubo-Greenwood formula (6).
+These simula-
+tions were performed on at least five snapshots taken
+at equidistant time steps from the DFT-MD simulation.
+The scheme described in Sec. II B was implemented us-
+ing the NumPy software package [55] for arrays to store
+the dynamic properties and for the evaluation of sim-
+ple numerical integration. More elaborate integrals, such
+as in Eqs. (A7) and (A8), were evaluated using Gaus-
+sian quadrature from the SciPy software package [56].
+The Kramers-Kronig transformation between the real
+and imaginary part of the dynamic dielectric function
+and the electrical conductivity was performed according
+to Maclaurin’s formula from Ref. 57.
+The
+linear
+response
+time
+dependent
+DFT
+(LR-
+TDDFT) calculations were performed in the GPAW
+code [45, 58–60]. The same snapshots as for the Kubo-
+Greenwood calculations were used and a 2x2x2 or 4x4x4
+Monkhorst-Pack grid [61] was employed for calculations
+of k-dependent dielectric functions. For the considered
+conditions, already the Baldereschi mean value point [62]
+yields converged optical conductivities for the Kubo-
+Greenwood calculations.
+For hydrogen, the dielectric
+function was computed with a plane-wave energy cut-
+off of at least 50 eV, while for beryllium at least 250 eV
+were used.
+III.
+DYNAMIC COLLISION FREQUENCY
+The work flow presented in Fig. 1 results in a complex
+dynamic collision frequency νDFT(ω). To study how this
+collision frequency compares to different levels of analytic
+approximations, we determine the real part of νDFT for a
+hydrogen isochore at ρ = 2 g/cm3 from 5 to 100 eV (see
+Ref. 54 for numerical details). This temperature range
+was chosen to illustrate the transition from the WDM
+regime to the ideal plasma regime. In Fig. 3 we compare
+these collision frequencies to the Lenard-Balescu (LB)
+collision frequency, the T-Matrix (TM) approach and
+the Gould-DeWitt (GDW) approach. The LB approach
+goes beyond the Born collision frequency by including
+dynamic screening, while the TM approach accounts for
+strong binary collisions by summing up ladder diagrams
+in the perturbation expansion [63]. The GDW scheme
+combines the dynamic screening of the LB approach with
+the strong collisions of the TM treatment and should,
+in principle, give the most accurate results. For further
+details on the analytic approaches see Refs. 21, 63–66.
+The aforementioned approaches solely describe electron-
+ion collisions, but electron-electron (e-e) collisions can
+
+6
+0
+1
+2
+3
+4
+Re[¯hν] [eV]
+T = 100 eV
+DFT
+LB
+TM
+GDW
+GDW with e-e
+LR-TDDFT
+0
+2
+4
+6
+8
+Re[¯hν] [eV]
+T = 50 eV
+0
+2
+4
+6
+8
+Re[¯hν] [eV]
+T = 25 eV
+0
+100
+200
+300
+400
+500
+600
+700
+¯hω [eV]
+0
+2
+4
+6
+8
+Re[¯hν] [eV]
+T = 5 eV
+FIG. 3. The real part of the dynamic collision frequency of hy-
+drogen plasmas at ρ = 2 g/cm3 for temperatures ranging from
+5 to 100 eV. The DFT and LR-TDDFT collision frequencies
+determined via Eq. (8) from their respective electrical con-
+ductivities are shown in black and pink, respectively. The LB
+collision frequency is shown in blue with crosses and the T-
+Matrix approach is shown in yellow with plus symbols. The
+GDW collision frequencies with and without electron-electron
+collisions are depicted in red as a dotted line and as a solid
+line with filled circles, respectively.
+be included by modulating the collision frequency with
+a renormalization factor [21].
+The GDW collision fre-
+quency including e-e collisions is also indicated in Fig. 3
+by the red dotted lines. It is apparent that although the
+DFT predictions agree well with the TM and GDW ap-
+proach at high temperatures, it deviates significantly at
+lower temperatures where complex many-body and quan-
+tum effects contribute strongly. At T = 100 eV, the colli-
+sion frequency is dominated by strong collisions between
+ions and electrons. However, the inclusion of e-e collisions
+via the renormalization factor leads to worse agreement
+with the DFT results, which is in agreement with recent
+observations that the Kubo-Greenwood formula applied
+to DFT lacks e-e collisions [54, 67].
+Furthermore, we
+apply the work flow presented in Fig. 1 to the electri-
+cal conductivity in the optical limit computed by LR-
+TDDFT to extract a collision frequency which we show
+as the pink dashed lines in Fig. 3. At all temperatures,
+its behavior is very similar to the Kubo-Greenwood re-
+sults which indicates that electron-electron collisions are
+also not included in this description of transport prop-
+erties. It is remarkable that at high frequencies the LR-
+TDDFT collision frequency lies significantly below the
+Kubo-Greenwood results for all considered temperatures.
+In our tests, this could not be attributed to a lack of con-
+vergence in number of bands or cutoff energy.
+IV.
+DYNAMIC STRUCTURE FACTOR
+A.
+Hydrogen
+Given a dynamic collision frequency ν(ω), Eqs. (3) and
+(5) can be used to compute the electronic DSF See(k, ω)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Set(k, ω) [a.u.]
+0.67 ˚A−1
+1.15 ˚A−1
+1.89 ˚A−1
+2.40 ˚A−1
+T = 5 eV
+LR-TDDFT
+Mermin + DFT
+Mermin + GDW (e-e)
+Mermin + Born
+0
+20
+40
+60
+80
+100
+120
+¯hω [eV]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Set(k, ω) [a.u.]
+0.67 ˚A−1
+1.15 ˚A−1
+1.89 ˚A−1
+2.40 ˚A−1
+T = 50 eV
+FIG. 4. The inelastic electronic DSF Set(k, ω) of a hydrogen
+plasma at ρ = 2 g/cm3 and T = 5 eV (upper panel) and T =
+50 eV (lower panel) from k = 0.67 ˚A−1 to k = 2.40 ˚A−1. The
+solid line denotes the direct computation from LR-TDDFT
+at the respective wave numbers, while the other lines denote
+DSFs computed from the Mermin dielectric function with the
+DFT collision frequency (dashed lines), the GDW collision
+frequency including electron-electron collisions (dash-dotted
+lines) and the Born collision frequency (dotted lines). The
+DSFs are shifted by 0.5 with respect to the next lowest wave
+number for readability.
+
+7
+where the k dependence only enters through the Mermin
+dielectric function. The LR-TDDFT approach allows di-
+rect access to the dielectric function at finite k by com-
+puting transitions matrix elements between Kohn-Sham
+states at different k points [45]. In Fig. 4, we show the
+electronic DSF of a hydrogen plasma at ρ = 2 g/cm3 and
+T = 50 eV (lower panel) and T = 5 eV (upper panel).
+The direct computations through LR-TDDFT are shown
+as solid lines, while we also present DSFs computed via
+the Mermin dielectric function in conjunction with the
+DFT and GDW collision frequencies shown in Fig. 3 as
+dashed and dash-dotted lines, respectively. Additionally,
+we show the results from the Mermin dielectric function
+with the Born collision frequency (see Eq. (B1)), which
+constitutes the often used Born-Mermin approach, as
+dotted lines. At the lowest wave number shown in Fig. 4,
+k = 0.67 ˚A−1, we are considering the collective behavior
+where collision are important, as can be seen from the
+dimensionless scattering parameter α (see Ref. 1 for def-
+inition) which is 4.17 and 2.84 for T = 5 and T = 50 eV,
+respectively.
+As expected for a fully ionized hydrogen plasma, the
+k dependence encoded by the Mermin dielectric function
+agrees well with the direct computation via LR-TDDFT
+for all considered collision frequencies at both conditions.
+However, at T = 5 eV, the damping of the plasmon pre-
+dicted by LR-TDDFT can only be captured with the
+DFT collision frequency, especially at small k. The Born
+collision frequency leads to a vast overestimation of the
+plasmon magnitude for k below 2.4 ˚A−1 and also the
+GDW approach with renormalization overestimates the
+magnitude by a factor of 2 for k below 1.15 ˚A−1. With
+increasing wave numbers, the collisions become less sig-
+nificant, and the DSFs for all collision frequencies start
+to converge to the same result. At T = 50 eV, the col-
+lisions play a smaller role, which is demonstrated by the
+largely identical predictions from all collision frequen-
+cies for k above 1.15 ˚A−1. It is notable that although
+the inclusion of electron-electron collisions leads to sig-
+nificant discrepancies between the dynamic collision fre-
+quencies in Fig. 3, these differences cannot be observed in
+the DSF, given the numerical noise. In the LR-TDDFT
+data, a small additional contribution at ℏω = 0 eV ap-
+pears, which has also recently been seen in path inte-
+gral Monte Carlo simulations [68]. This bump is not in-
+cluded in the Mermin formalism and appears more pro-
+nounced at higher temperatures and lower densities (also
+see Sec. IV B and IV C), leading us to propose that it
+is connected to bound-bound transitions without energy
+transfer.
+B.
+Isochorically heated beryllium
+To investigate the impact of tightly bound states on
+the presented procedure, we study a beryllium plasma
+at ρ = 1.8 g/cm3 and T = 12 eV, for which the ap-
+proach of Ref. 30 predicts a charge state Z = 2.1. The
+bound 1s states are energetically clearly separated from
+the free electrons. The collision frequency can either be
+determined from the full dynamic electrical conductivity
+that includes the transitions from the bound 1s states to
+the conduction band, or from the free-free electrical con-
+ductivity by restricting the transition matrix elements
+in Eq. (6) to transitions orginating and ending in the
+conduction band (for details on this decomposition, see
+Ref. 30). In the latter case, only the free-free contribution
+to the DSF is considered within the Mermin dielectric
+function, while the bound-free contribution must be ap-
+proximated by its behavior at k → 0. In Fig. 5, we show
+the comparison of these two approaches to the direct
+computation of the electronic DSF using LR-TDDFT.
+At the lowest wave number k = 0.49 ˚A−1, shown in the
+upper panel, all approaches agree well, as expected due
+to the construction of the collision frequency which re-
+quires equivalence in the limit of small k (see Eq. (8)).
+The separation of the conductivity into a free-free and a
+bound-free contribution allows us to clearly identify the
+different terms of the Chihara formula (2) in the DSF.
+The dotted line represents the bound-free contribution,
+which agress exactly with the LR-TDDFT data above
+∼ 90 eV, and the dashed line represents the free-free
+contribution (plasmon), which matches the LR-TDDFT
+results below ∼ 90 eV. Remarkably, the prefactors Zf
+and Zb in Eq. (2) which give the respective weighting
+of these two features come out of the definition of the
+charge state described in Ref. 30 and agree virtually ex-
+actly with the direct computation including all transi-
+tions in LR-TDDFT.
+At k = 1.47 ˚A−1 in the middle panel of Fig. 5, the devi-
+ation of the approach using the full collision frequency to
+the other approaches becomes apparent. The bound-free
+dominated DSF above ∼ 90 eV is still well approximated
+by both the full collision frequency and the bound-free
+feature at k → 0.
+Below ∼ 90 eV, however, the ap-
+proach using the full collision frequency, denoted by the
+dash-dotted line, deviates strongly (note the logarithmic
+scale) from the LR-TDDFT result.
+The free-free fea-
+ture computed solely from the collision frequency based
+on free-free transitions, denoted by the dashed line, still
+agrees very well with the LR-TDDFT calculation in this
+energy regime.
+The bottom panel of Fig. 5, showing the DSF at
+k = 3.42 ˚A−1, highlights the complete breakdown of
+the approach using the full collision frequency.
+While
+the DSF is still described adequately above ∼ 90 eV, its
+shape is very different from the LR-TDDFT result below
+that energy. On the other hand, the separate description
+of free-free and bound-free contributions again describes
+the DSF accurately compared to the LR-TDDFT data.
+However, the approximation of the bound-free feature by
+its k → 0 limit starts to deteriorate at this wave number.
+At the highest energy shift shown in Fig. 5, this approxi-
+mation underestimates the LR-TDDFT value by a factor
+of almost 2. Additionally, at the onset of the bound-free
+feature around 100 eV, it overestimates the DSF com-
+
+8
+pared to the LR-TDDFT as can be seen in Fig. 6 which
+shows the DSF on a linear scale. The fast deterioration
+beyond the k → 0 limit of the approach using the full
+collision frequency is expected because the framework of
+the Mermin dielectric function, which encodes the k de-
+pendence, does not include the existence of bound states.
+Therefore, any such states that are artificially introduced
+via the collision frequency cannot be handled correctly in
+the k dependence.
+Furthermore, in Fig. 6, we show the DSFs computed
+from the Mermin dielectric function with Born collision
+frequencies for a plasma with a charge state Z = 2 and
+Z = 4.
+The position of the plasmon peak for Z = 2
+agrees well with the DFT spectra, while the position of
+the Z = 4 plasma is consistently at too high energies, as
+expected due to the higher free-electron density. How-
+ever, at low k, the dampening of the plasmon peak due
+to the Born collision frequency is too low compared to the
+DFT data, similar as observed for hydrogen in Fig. 4. At
+the higher wave numbers, the plasmon-peak position of
+the DFT results agrees well with Mermin function using
+the Born collision frequency at Z = 2, clearly indicating
+that the bound 1s states do not contribute to this feature.
+10−3
+10−1
+Set(k, ω) [a.u.]
+k = 0.49 ˚A−1
+LR-TDDFT
+DFT(b-f) at k = 0
+Mermin + DFT(f-f)
+Mermin + DFT(full)
+10−4
+10−2
+100
+Set(k, ω) [a.u.]
+k = 1.47 ˚A−1
+−50
+0
+50
+100
+150
+200
+250
+300
+¯hω [eV]
+10−2
+10−1
+100
+Set(k, ω) [a.u.]
+k = 3.42 ˚A−1
+FIG. 5. The inelastic electronic DSF Set(k, ω) of a beryllium
+plasma at ρ = 1.8 g/cm3 and T = 12 eV for various k values
+on a logarithmic scale. The solid lines are direct computations
+at the given k using LR-TDDFT. The dash-dotted and the
+dashed lines denote DSFs computed from the Mermin dielec-
+tric function with the full DFT collision frequency, determined
+from the electrical conductivity including bound-free transi-
+tions, and the free-free collision frequency, determined from
+the electrical conductivity including only free-free transitions,
+respectively. The dotted lines denote the DSF computed di-
+rectly from the bound-free conductivity at k = 0 ˚A−1.
+−50
+0
+50
+100
+150
+200
+250
+300
+¯hω [eV]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Set(k, ω) [a.u.]
+k = 0.49 ˚A−1
+k = 1.47 ˚A−1
+k = 2.44 ˚A−1
+k = 3.42 ˚A−1
+LR-TDDFT
+(Mermin + f-f) + b-f(k = 0)
+Mermin + Born (Z = 2)
+Mermin + Born (Z = 4)
+−100
+0
+E − µ [eV]
+0.0
+0.5
+1.0
+D(E) [a. u.]
+FIG. 6. The inelastic electronic DSF Set(k, ω) of a beryllium
+plasma at ρ = 1.8 g/cm3 and T = 12 eV for various k values.
+The solid lines are direct computations at the given k using
+LR-TDDFT, while the dotted and dash-dotted lines denote
+DSFs computed from the Mermin dielectric function with the
+Born collision frequency for a plasma with a charge state of
+Z = 2 and Z = 4, respectively. The dashed lines represent
+the sum of the DSF computed through the Mermin dielectric
+function using the free-free collision frequency and the bound-
+free DSF at k = 0 ˚A−1. The DSFs are shifted by 0.5 with
+respect to the next lowest wave number for readability. In
+the inset, the solid line shows the density of states, while the
+shaded area denotes the occupied density of states.
+The inset in Fig. 6 shows the density of states (DOS) of
+the beryllium plasma which shows a clear separation be-
+tween the narrow 1s band, which is fully occupied, and
+the conduction band. This clear distinction is the rea-
+son why the separate treatment of free-free and bound-
+free contibutions is successful.
+The bound-free feature
+does not exhibit a strong k dependence up to high k
+values [23, 69], and the plasmon occurs energetically sep-
+arated in the DSF.
+C.
+Compressed beryllium
+With increasing density and temperature the notion
+of bound states becomes ill-defined in WDM. The in-
+set in Fig. 7 shows the DOS of a beryllium plasma at
+T = 50 eV and ρ = 40 g/cm3 which demonstrates the
+closing of the band gap compared to the inset in Fig. 6.
+Furthermore, the former 1s states broaden significantly
+
+9
+0
+100
+200
+300
+400
+500
+600
+700
+800
+¯hω [eV]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Set(k, ω) [a.u.]
+k = 1.37 ˚A−1
+k = 4.12 ˚A−1
+k = 6.87 ˚A−1
+(Mermin + f-f) + b-f(k = 0)
+Mermin + DFT(full)
+LR-TDDFT
+−200
+0
+200
+E − µ [eV]
+0.0
+0.5
+1.0
+D(E) [a. u.]
+FIG. 7. The inelastic electronic DSF Set(k, ω) of a beryllium
+plasma at ρ = 40
+g/cm3 and T = 50 eV for various k val-
+ues. The solid lines are direct computations at the given k
+using LR-TDDFT, while the dashed lines represent the sum
+of the DSF computed through the Mermin dielectric function
+using the free-free collision frequency and the bound-free DSF
+at k = 0 ˚A−1. The dotted lines denote the DSF computed
+through the Mermin dielectric function with the full collision
+frequency. The DSFs are shifted by 0.5 with respect to the
+next lowest wave number for readability.
+In the inset, the
+solid line shows the density of states, while the shaded area
+denotes the occupied density of states.
+into a band and the DOS converges towards the
+√
+E be-
+havior of a free electron gas. Because the band gap is still
+clearly identifiable the separate treatment of bound-free
+and free-free contributions to the DSF presented in the
+previous section can also be applied to these conditions.
+Figure 7 shows the results of this separate treatment,
+as well as the direct computation using LR-TDDFT and
+the DSF from the Mermin dielectric function using the
+full collision frequency. While the separate treatment of
+bound-free and free-free contributions yields excellent re-
+sults for the near-ambient density case in Fig. 5, it poorly
+approximates the LR-TDDFT results in strongly com-
+pressed beryllium shown in Fig. 7. The plasmon peak at
+k = 1.37 ˚A−1 is severly underdamped due to the missing
+bound-free transitions in the collision frequency, which
+occur in the same energy range as the free-free transi-
+tions at these conditions. The use of the Born collision
+frequency in lieu of the free-free DFT collision frequency
+leads to an increase of the plasmon peak magnitude by a
+factor of 2 (not shown in Fig. 7). The broader peak aris-
+ing around ∼ 130 eV for k = 4.12 and k = 6.87 ˚A−1 is due
+to the insufficient approximation of the bound-free fea-
+ture by its value at k = 0 ˚A−1. As can be seen from the
+LR-TDDFT data, the bound-free features merges with
+the free-free feature to form one homogeneous feature.
+At these conditions, using the full collision frequency in
+the Mermin dielectric function gives better results, which
+is expected as the former 1s states lose their bound char-
+acter due to the higher compression and higher temper-
+ature. For all considered wave numbers, this approach
+yields good agreement with the LR-TDDFT data above
+∼ 200 eV, and approximates the trends below that energy
+fairly well. Solely at ∼ 100 eV this approach predicts
+a feature that is not visible in the LR-TDDFT results
+across the considered k range.
+V.
+APPLICATION TO EXPERIMENTS
+We
+reanalyze
+previous
+XRTS
+experiments
+by
+D¨oppner et al. [70] and Kritcher et al. [71] using LR-
+TDDFT to evaluate the influence of advanced methods
+on the initially inferred plasma parameters. In general,
+temperature and density of the target must be consid-
+ered simultaneously. However, since D¨oppner et al. used
+detailed balance in their forward scattering experiment
+to determine the temperature as T = 18 eV, we use this
+value and vary the density to find the best agreement
+with the experimental data.
+To justify this approach
+we show the results of a recently suggested model-free
+temperature diagnostic [72] in Appendix C. For the
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+ρ [g/cm3]
+1.0
+1.2
+χ2
+2920
+2940
+2960
+2980
+3000
+¯hω [eV]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Intensity [a.u.]
+ρ = 1.0 g/cm3
+ρ = 1.4 g/cm3
+ρ = 1.8 g/cm3
+ρ = 2.2 g/cm3
+Inelastic
+2920
+2930
+2940
+2950
+0.0
+0.2
+FIG. 8.
+The lower panel shows the scattering intensity of
+an isochorically heated beryllium target at T = 18 eV from
+Ref. 70. The colors of the solid lines encode different densities
+used in the LR-TDDFT simulations. The dotted lines denote
+the inelastic contributions.
+The upper panel shows the χ2
+deviation depending on the density used in the simulation
+where the colored dots correspond to the spectra shown in the
+lower panel and the black curve is achieved by interpolating
+to 40 evenly spaced densities between these spectra.
+
+10
+other experiment, we include the temperature in the
+analysis.
+Firstly, in Fig. 8, we show simulated XRTS spectra
+with densities ranging from 1.0 to 2.2 g/cm3 at T = 18 eV
+together with the forward XRTS spectrum recorded by
+D¨oppner et al. [70], which was collected at the Omega
+laser facility at the Laboratory for Laser Energetics at
+the University of Rochester. The experiment probed a
+scattering vector of approximately k = 1 ˚A−1, enabling
+access to collective behavior of the plasma. In the original
+analysis of the experiment a density of 1.17 g/cm3 was
+determined by D¨oppner et al. [70]. The electron feature
+was treated on the level of the RPA wihtout including
+electron-ion collisions and the ionization was assumed
+to be Zf = 2.3.
+We compute the electron feature for
+various densities from LR-TDDFT while including local
+field corrections via the adiabatic local density approxi-
+mation [44, 46]. The magnitude of the ion feature is left
+as a free parameter in the χ2 minimization. Although
+none of the computed spectra capture the plasmon at
+2930 eV perfectly, the spectrum at ρ = 1.8 g/cm3 yields
+a 5% lower χ2 deviation than any of the other consid-
+ered densities.
+The ionization state at this density is
+Z = 2.14, determined via the Thomas-Reiche-Kuhn sum
+rule [30], which is approximately 7% lower than the value
+used by D¨oppner et al. [70]. Furthermore, the here com-
+puted density of the sample is more than 50% higher
+than the originally determined value, indicating the need
+to use sophisticated methods to achieve reliable results
+in forward scattering experiments.
+For the experiment by Kritcher et al. [71], the tempera-
+ture cannot reliably be inferred from the detailed balance
+relation and the temperature must, therefore, be included
+in the analysis. Furthermore, the instrument and source
+functions were not available and must be modeled ex-
+plicitly in the analysis. To analyze the experiments, we
+simulate spectra on a sufficiently large temperature and
+density grid and interpolate between them [73] to model
+arbitrary ρ − T combinations in this range. Due to the
+high number of parameters involved in this sort of anal-
+ysis, we employ Bayesian inference [74] implemented in
+the PyMC3 software package [75] and use the sequential
+Monte Carlo algorithm [76, 77] for sampling the parame-
+ter space. In Fig. 9, we consider the backward XRTS ex-
+periment at k = 8.42 ˚A−1 on imploding beryllium shells
+by Kritcher et al. [71], which was also performed at the
+Omega laser facility. To analyze the experiment, we sim-
+ulate spectra on a grid ranging from 2 to 32 g/cm3 and
+from 0.1 to 25 eV. No instrument or source function was
+supplied in Ref. 71. We, therefore, use the parametriza-
+tion of a zinc source given in Ref. 14 and include all
+the parameters of the instrument response function in
+the Bayesian analysis. We also replace the Gaussian de-
+scribing the source broadening by a skewed Gaussian to
+account for the asymmetry observed in the ion feature.
+Thus, ten parameters determine the shape of the spec-
+trum, including the physical parameters describing the
+density and temperature of the sample and the magni-
+−600
+−400
+−200
+0
+200
+¯hω [eV]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Intensity [a.u.]
+Exp. (t = 3.1 ± 0.1 ns, Kritcher et al.)
+Total spectrum
+Inelastic contribution
+Elastic contribution
+5
+6
+7
+8
+9
+10
+ρ [g/cm3]
+0.0
+0.2
+0.4
+p(ρ|Exp.)
+ρ = 7.9+1.0
+−0.8 g/cm3
+80 %
+FIG. 9.
+Scattering intensity of imploding beryllium shells
+from Ref. 71. The upper panel shows the experimental data
+at a delay t = 3.1 ± 0.1 ns and the posterior prediction for
+the elastic and inelastic contributions based on LR-TDDFT
+simulations. The thin lines are 100 spectra computed from
+parameters randomly sampled from the posterior probility
+distribution.
+The shaded areas show the region below the
+average posterior predictions. The lower panel shows the re-
+duced posterior probability distribution in the density param-
+eter ρ where the dark shaded area under the curve indicates
+the 80% highest posterior density interval.
+tude of the ion feature, and 7 parameters describing the
+experimental setup. The upper panel of Fig. 9 shows an
+XRTS spectrum collected from an imploding beryllium
+shell at a delay of t = 3.1 ± 0.1 ns and the posterior
+prediction for the elastic and inelastic contribution to
+the simulated scattering spectrum.
+The posterior pre-
+dictions are obtained by sampling parameters according
+to the posterior probability distribution and using these
+parameters to simulate the spectrum.
+The agreement
+between the simulated spectrum and the experimental
+data is excellent. The bottom panel of Fig. 9 shows the
+reduced posterior probability distribution in the density
+parameter ρ, which is the full probability distribution
+integrated over all other parameters. The inferred den-
+sity ρ = 7.9+1.0
+−0.8 g/cm3 corresponds to the maximum
+aposteriori probability and the uncertainties are deter-
+mined from the 80% highest posterior density interval.
+With an assumed ionization state Z = 2, the original
+analysis by Kritcher et al. [71] resulted in estimates of
+ρ = 8.23 ± 2.24 g/cm3 and T = 14 ± 3 eV. The density,
+which is the most sensitive plasma parameter with re-
+spect to the Compton feature at these conditions, agrees
+very well with our current study. However, Kritcher et al.
+also used a temperature-dependent model for the ion fea-
+ture, while we keep the ion feature as a free parameter.
+Therefore, the inferred temperature is mainly determined
+from the relative magnitude of the ion feature and Comp-
+
+11
+ton feature. Because the shape of the Compton feature is
+not very sensitive to the temperature at these conditions,
+we cannot reliably determine the electron temperature.
+VI.
+CONCLUSION
+In this work, we presented the theoretical basis for
+computation of DSFs using the Mermin dielectric func-
+tion with a dynamic complex collision frequency and
+showed how this framework can be used to extract col-
+lision frequencies from DFT simulations. We compared
+these collision frequencies to several analytic approaches
+for hydrogen plasmas at ρ = 2 g/cm3 and, for temper-
+atures approaching the ideal plasma limit, found good
+agreement with models that incorporate strong collisions.
+Furthermore, we studied how different collision frequen-
+cies impact the DSF calculated from the Mermin dielec-
+tric function and compared these results to the direct
+computation of the DSF at the given wave numbers us-
+ing LR-TDDFT. For hydrogen, we find good agreement
+for all collision frequencies at high k, while at small k, es-
+pecially the frequently used Born approximation leads to
+underdamped plasmon peaks. For beryllium, we showed
+that a separate treatment of free-free and bound-free con-
+tributions to the DSF yields excellent agreement with the
+LR-TDDFT for near-ambient densities up to moderate
+wave numbers (k = 3.42 ˚A−1), while it disagrees signif-
+icantly for highly compressed beryllium because bound-
+free transition interact with the free-free transitions to
+dampen the plasmon. Therefore, in order to get accurate
+DSFs over a wide range of wave numbers in extreme con-
+ditions, it is imperative to employ ab initio approaches
+like LR-TDDFT or path integral Monte Carlo simula-
+tions. We applied LR-TDDFT to XRTS experiments on
+beryllium and found significant deviations of more than
+50% in inferred density for small k for D¨oppner et al. [70]
+and found good agreement with analytical approaches for
+backscattering with large k for Kritcher et al. [71].
+ACKNOWLEDGMENTS
+We want to thank P. Sperling, B. Witte, M. French,
+G. R¨opke, H. J. Lee and A. Cangi for many helpful
+discussions.
+M. S. and R. R. acknowledge support by
+the Deutsche Forschungsgemeinschaft (DFG) within the
+Research Unit FOR 2440.
+All simulations and analy-
+ses were performed at the North-German Supercomput-
+ing Alliance (HLRN) and the ITMZ of the University
+of Rostock. M. B. gratefully acknowledges support by
+the European Horizon 2020 programme within the Marie
+Sk�lodowska-Curie actions (xICE grant 894725) and the
+NOMIS foundation. The work of T. D. was performed
+under the auspices of the U.S. Department of Energy
+by Lawrence Livermore National Laboratory under Con-
+tract No. DE-AC52-07NA27344.
+Appendix A: Derivation of real and imaginary part
+of RPA dielectric function
+The collision frequency is generally a complex number
+ν(ω) = ν1(ω) + i ν2(ω),
+(A1)
+meaning that its imaginary part acts as a shift of the
+frequency that enters into ϵRPA in Eq. (5) and its real
+part takes on the role of the artificial damping η that
+was introduced in Eq. (4).
+However, in this case, the
+damping is not set to zero after the integration.
+Now, we will split Eq. (4) into its real and imaginary
+part and consider the modulation of the input frequency
+ω by the complex frequency from Eq. (A1) where the
+argument of ν is dropped for readability:
+Re
+�
+ϵRPA(⃗k, ω + i ν)
+�
+= 1 − 2e2
+ϵ0k2
+�
+d3q
+(2π)3 ×
+×
+�
+f⃗q−
+⃗k
+2 − f⃗q+
+⃗k
+2
+� �
+ℏ˜ω −
+�
+E⃗q+
+⃗k
+2 − E⃗q−
+⃗k
+2
+��
+�
+ℏ˜ω −
+�
+E⃗q+
+⃗k
+2 − E⃗q−
+⃗k
+2
+��2
++ ℏ2ν2
+1
+,
+(A2)
+Im
+�
+ϵRPA(⃗k, ω + i ν)
+�
+= 2e2
+ϵ0k2
+�
+d3q
+(2π)3 ×
+× ℏν1
+f⃗q−
+⃗k
+2 − f⃗q+
+⃗k
+2
+�
+ℏ˜ω −
+�
+E⃗q+
+⃗k
+2 − E⃗q−
+⃗k
+2
+��2
++ ℏ2ν2
+1
+.
+(A3)
+The shifted frequency ˜ω = ω − ν2 is introduced here.
+These integrals are performed across the entire momen-
+tum space and can therefore be shifted by an arbitrary
+vector ⃗y because for an integral of a function G(⃗x), which
+goes to 0 as |⃗x| → ∞, it holds that
+�
+R3 d3x G(⃗x) =
+�
+R3 d3x G(⃗x − ⃗y),
+with |⃗y| < ∞. (A4)
+Therefore, we can separate the integrand in Eqs. (A2)
+and (A3) into two summands with the Fermi occupation
+of the up- and down-shifted momentum, respectively. We
+further use Eq. (A4) to shift the momenta in the argu-
+ment of the Fermi occupation to ⃗q in order to get f⃗q as
+a common prefactor for both summands. The momenta
+in the subscripts of the energy have to be shifted accord-
+ingly. This gives
+Re
+�
+ϵRPA(⃗k, ω + i ν)
+�
+= 1−
+− 2e2
+ϵ0k2 2π
+� ∞
+0
+dq
+(2π)3 q2fq
+me
+ℏ2k ×
+×
+� 1
+−1
+dz
+�
+κ − 1
+2 (k + 2qz)
+�
+κ − 1
+2 (k + 2qz)
+�2 + ∆2 −
+−
+κ − 1
+2 (−k + 2qz)
+�
+κ − 1
+2 (−k + 2qz)
+�2 + ∆2
+�
+(A5)
+
+12
+for the real part and
+Im
+�
+ϵRPA(⃗k, ω + i ν)
+�
+=
+4π e2
+ϵ0k2
+� ∞
+0
+dq
+(2π)3
+ν1m2
+e
+ℏ3k2 q2fq×
+×
+� 1
+−1
+dz
+�
+1
+�
+κ − 1
+2 (k + 2qz)
+�2 + ∆2 −
+−
+1
+�
+κ − 1
+2 (−k + 2qz)
+�2 + ∆2
+�
+(A6)
+for the imaginary part.
+Here, ⃗k was fixed in the q3-
+direction and z = cos θ where θ is the angle between
+⃗q and ⃗k. The shorthands κ = ˜ωme
+ℏk
+and ∆ = meν1
+ℏk
+with
+the electron mass me are introduced. The Fermi occu-
+pation can be pulled out of the angle integration as it
+only depends on the magnitude of the momentum. The
+integral over the angle can be performed analytically in
+Eqs. (A5) and (A6), giving
+Re
+�
+ϵRPA(⃗k, ω + i ν)
+�
+= 1 + 2π mee2
+ϵ0ℏ2k3
+� ∞
+0
+dq
+(2π)3 qfq×
+× ln
+�
+∆2 +
+�
+κ − k
+2 − q
+�2� �
+∆2 +
+�
+κ + k
+2 + q
+�2�
+�
+∆2 +
+�
+κ − k
+2 + q
+�2� �
+∆2 +
+�
+κ + k
+2 − q
+�2�
+(A7)
+for the real part, and
+Im
+�
+ϵRPA(⃗k, ω + i ν)
+�
+= −4π mee2
+ϵ0ℏ2k3
+� ∞
+0
+dq
+(2π)3 qfq×
+×
+�
+arctan
+�
+κ − k
+2 − q
+∆
+�
++ arctan
+�
+κ + k
+2 + q
+∆
+�
+−
+− arctan
+�
+κ − k
+2 + q
+∆
+�
+− arctan
+�
+κ + k
+2 − q
+∆
+� �
+(A8)
+for the imaginary part of the RPA dielectric function
+modulated by a complex frequency. The remaining in-
+tegration over q has to be performed numerically.
+Appendix B: Expressions for the Born collision
+frequency
+One of the most prominent approximations for the col-
+lision frequency is the Born collision frequency [21]
+νBorn(ω) = −i
+ϵ0niΩ2
+6π2e2neme
+� ∞
+0
+dq q6×
+× V 2
+ei(q)Sii(q) 1
+ω
+�
+ϵRPA(q, ω) − ϵRPA(q, 0)
+�
+,
+(B1)
+with the ion density ni, the electron density ne and
+the normalization volume Ω.
+There are different ap-
+proximations for the electron-ion potential Vei and the
+−20
+0
+20
+40
+¯hω [eV]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Intensity [a.u.]
+Exp.
+Inst.
+20
+30
+40
+¯hω [eV]
+5
+10
+15
+20
+25
+30
+35
+40
+45
+Te [eV]
+Te = 19 ± 1.5 eV
+Without
+instrument
+With
+instrument
+FIG. 10. The left panel shows the scattering intensity and
+the instrument function from Ref. 70. The right panel shows
+the inferred electron temperature accoring to Ref. 72.
+static structure factor Sii.
+The potential can be ap-
+proximated by the screened Coulomb potential with the
+Debye-H¨uckel or Thomas-Fermi screening parameter de-
+pending on the density and temperature regime consid-
+ered. Approaches to the structure factor range from the
+assumption of a homogeneous electron gas (Sii(q) = 1)
+or analytic models like the Debye-H¨uckel theory to more
+sophisticated methods like the hypernetted-chain (HNC)
+equation or MD simulations. Here, we use the potential
+Vei(q) = V Coulomb
+ei
+(q)
+ϵRPA(q, 0)
+= −eeei
+ϵ0Ω
+1
+q2 ϵRPA(q, 0)
+(B2)
+and the static structure factor we calculate from our
+DFT-MD simulations.
+Equation (B1) is computed by
+directly calculating its real part and subsequently per-
+forming the Kramers-Kronig [78, 79] transformation to
+arrive at the imaginary part.
+Appendix C: Temperature determination via
+Laplace transform
+We employ the recently proposed temperature diag-
+nostic based on a two-sided Laplace transform [72] to
+inferr the temperature from experiment performed by
+D¨oppner et al.
+The left panel of Fig. 10 shows the
+scattering data and the instrument function, while the
+right panel shows the inferred temperature according to
+the procedure described in Ref. 72. The x axis denotes
+the energy up to which the two-sided Laplace trans-
+form is performed.
+A convergence is observed beyond
+40 eV and the electron temperature is determined to be
+19 ± 1.5 eV. This values agrees within errorbars with the
+electron temperature of 18 eV, originally determined by
+D¨oppner et al. We, therefore, exclude the electron tem-
+perature from the fitting procedure for this experiment.
+
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+arXiv:2301.13675v1  [math.AC]  31 Jan 2023
+ON TIGHT SUBMODULES
+OF MODULES OVER VALUATION DOMAINS
+PETER DANCHEV AND L´ASZL ´O FUCHS
+Abstract. This note offers an unusual approach of studying a class of mod-
+ules inasmuch as it is investigating a subclass of the category of modules over a
+valuation domain. This class is far from being a full subcategory, it is not even
+a category.
+Our concern is the subclass consisting of modules of projective
+dimension not exceeding one, admitting only morphisms whose kernels and
+cokernels are also objects in this subclass. This class is still tractable, several
+features are in general simpler than in module categories, but lots of familiar
+properties are lost. A number of results on modules in this class are similar to
+those on modules over rank one discrete valuation domains (where the global
+dimension is 1). The study is considerably simplified by taking advantage of
+the general theory of modules over valuation domains available in the litera-
+ture, e.g. in [14]-[15]. Our main goal is to establish the basic features and have
+a closer look at injectivity, pure-injectivity, and cotorsionness, but we do not
+wish to enter into an in-depth study of these properties.
+1. Introduction
+Everybody familiar with module theory over integral domains knows well that
+the theory simplifies tremendously if the ring is a Dedekind domain; i.e. the modules
+have projective dimension (p.d.)
+at most one.
+That the condition ‘p.d.1’ is so
+powerful was recognized by E. Matlis who developed interesting properties under
+the hypothesis that the field of quotients as a module has p.d.1; see [17] (in [15] these
+rings were named Matlis domains in his honor). In order to understand p.d.1 better
+and to learn more about it, it is natural to try to find out how the theory would
+look like over a general integral domain if one deals only with its modules of p.d.
+not exceeding 1, and ignores modules of higher projective dimensions. This means
+to investigate a class where the objects are modules of p.d.≤ 1 and morphisms
+are required to have both kernels and cokernels of p.d.≤ 1. This is what we are
+planning to do in this article over an arbitrary valuation domain (i.e. an integral
+domain where the ideals form a chain with respect to inclusion). Valuation domains
+are the first and obvious choice for such a study, as they are sufficiently general,
+but still manageable, and luckily, there is an extensive literature available on them
+that reveals a lot of information one can take advantage of.
+The selected subclass of the module category is not a category; a primary reason
+for it is that the usual composition rule of mappings works only under an additional
+condition. When dealing with modules of such a class, it is soon realized that one
+has to reassess familiar facts and obvious concepts to fit into the new situation. The
+usual sum of two morphisms is rarely another one, submodules that belong to this
+Date: February 1, 2023.
+2020 Mathematics Subject Classification. Primary 13C05, 13F99. Secondary 13G05.
+1
+
+subclass only exceptionally form a lattice, the tensor product of two objects may
+not belong to this class, etc. But on the other hand, some nice features of discrete
+rank one valuation domains carry over, like the equality of injectivity and divisi-
+bility or the pure-injectivity of cyclically presented torsion modules. We will also
+pay attention to pure-injectivity and to the cotorsion property (Sections 7 and 8).
+Though these concepts are defined to have close resemblance to the familiar module
+concepts, one should always keep in mind that they are not exactly the same. The
+discussion of the submodules in direct sums of cyclically presented modules (Section
+5) already shows the huge difference from the traditional treatment of modules.
+Throughout the symbol V will denote a valuation domain (commutative), and Q
+its quotient field. We will use the notation K = Q/V . The symbol V × denotes the
+multiplicative monoid of non-zero elements of V . Torsion and torsion-free modules
+have the usual meaning.
+The abbreviation p.d.
+denotes projective dimension.
+The global weak dimension of a valuation domain is 1. Likewise, |X| denotes the
+cardinality of the set X, and ω is the smallest infinite ordinal. We also abbreviate
+gen M to stand for the minimal cardinality of generating sets of M.
+We use
+“countably generated” to mean gen M = ℵ0 (not finite).
+Torsion-freeness and
+flatness are equivalent over valuation domains; consequently, relative divisibility
+and purity have the same meaning. For the construction of valuation domains with
+prescribed value groups, see e.g. [15, Chap. II, Section 3].
+For a valuation domain V , we consider the category CV as the category V -Mod
+of V -modules with the usual morphisms. Our main goal is to study the subclass
+C∗
+V whose objects are the V -modules of p.d.≤ 1 and whose morphisms are required
+to have kernels and cokernels that also have p.d.≤ 1. The subobjects are the tight
+submodules (see Section 2). While our results are restricted to this subclass, in
+proofs we will often use arguments and concepts from the covering category CV .
+Modules of projective dimension one have been discussed briefly in an earlier pa-
+per [9] with emphasis on Pr¨ufer domains, and some of our present results appeared
+there in a different context.
+The idea of investigating the subclass C∗
+V comes from a recent paper [12] where
+tight submodules played a dominating role. To deal with classes of modules where
+both the kernels and the cokernels of the maps were restricted looked strange, but
+at the same time interesting and challenging. We admit, we were first hesitating
+to get involved in an uncharted territory with no immediate applications in the
+horizon, but in spite of this we decided to start working on this class, since we
+believed that the results could be helpful in a better understanding of the impact
+of projective dimension one as well as of the role of tightness in submodules. In
+addition, they might provide counterexamples in unusual situations. In this paper
+we begin with exploring this idea, and accordingly, we have been trying to find out
+not only what can be verified, but also what is no longer valid in comparison to
+conventional module theory.
+2. Tight Submodules
+The fundamental concept we are using throughout is the tightness of submodules
+in modules of p.d.≤ 1. The following definition applies to all rings. Let B < A
+be modules such that p.d.A ≤ 1. B is called tight in A if p.d.A/B ≤ 1. Then
+also p.d.B ≤ 1. It is evident that direct summands are tight submodules, but tight
+2
+
+submodules need not be summands. The tightness for p.d.1 that was introduced
+in [8] and immediately generalized to higher p.d.s in [1], was slightly different: its
+definition required that p.d.A/B ≤ p.d.A. The only difference between the two
+variants is that in a free V -module cyclically presented a free submodule is not
+tight in the sense of [8], but is in this paper.
+(The present version was called
+t-submodule in [9].)
+The following easily verifiable basic rules will be applied, often without explicit
+reference:
+Lemma 2.1. Let C < B < A be V -modules and p.d.A ≤ 1.
+(a) If C is tight in B and B is tight in A, then C is tight in A.
+(b) If C and B are tight in A, then C is tight in B.
+(c) If C and B are tight in A, then B/C is tight in A/C.
+(d) If C is tight in A and B/C is tight in A/C, then B is tight in A.
+□
+As already mentioned before, the symbol CV will denote the category of V -
+modules with the usual morphisms, while C∗
+V is the subclass whose objects are
+the V -modules of p.d.≤ 1. E.g. the frequently used localizations VS at countable
+multiplicative submonoids S of V × are objects of our class C∗
+V . The subobjects of
+an object M ∈ C∗
+V are the tight submodules of M. The morphisms in C∗
+V are those
+module homomorphisms φ : M → N (M, N ∈ C∗
+V ) for which Im φ is tight in N.
+This also means that Ker φ should be a tight submodule of M. Thus morphisms in
+C∗
+V have tight kernels and images. Clearly, a submodule A of M is tight if and only
+if the inclusion morphism A → M belongs to C∗
+V . In order to check whether or not
+C∗
+V is a category, one ought to examine the axioms of category theory; in particular,
+the critical one that says that if α : A → B and β : B → C are morphisms, then so
+is the composite map βα : A → C. Unfortunately, this property is seldom satisfied
+in C∗
+V . As we will see in a moment, this property is related to the one that the sum
+of two tight submodules is again tight – which holds very rarely. In the next lemma
+this property is compared to similar properties, in particular to the tightness of the
+intersection of two tight submodules.
+Lemma 2.2. Let M be a V -module of p.d.≤ 1, and A, B two tight submodules of
+M. Consider the following conditions:
+(i) A ∩ B is tight in A;
+(ii) A ∩ B is tight in B;
+(iii) A is tight in A + B;
+(iv) B is tight in A + B;
+(v) A ∩ B is tight in M;
+(vi) A ∩ B is tight in A + B;
+(vii) A + B is tight in M.
+Then conditions (i)-(v) are equivalent, (vi) follows from each of them, and (vii)
+implies each of them.
+Proof. (i) ⇔ (iv) as well as (ii) ⇔ (iii) follows from Noether’s isomorphism theorem.
+(i) ⇔ (v). The third non-zero module in the exact sequence
+0 → A/(A ∩ B) → M/(A ∩ B) → M/A → 0
+has p.d.≤ 1. We deduce, by the well-known Kaplansky’s lemma on p.d.’s in short
+exact sequences (see e.g.
+[15, Lemma 2.4, p.
+202]) that the first two non-zero
+modules in the last exact sequence are simultaneously of p.d. ≤ 1.
+3
+
+(vii) ⇒ (ii). From the exact sequence
+0 → (A + B)/A → M/A → M/(A + B) → 0
+where M/A and M/(A+B) have p.d.≤ 1 we argue, by the same Kaplansky’s lemma
+that p.d.(A + B)/A = 1. Hence p.d.B/(A ∩ B) = 1, and A ∩ B is tight in B.
+(ii) ⇒ (v). By hypothesis A ∩ B is tight in B, and B is tight in M. Therefore,
+by Lemma 2.1(a), A ∩ B is tight in M. Similar argument yields (i) ⇒ (v).
+(vii) ⇒ (iii). Because of Lemma 2.1(b), A and A + B tight in M implies A is
+tight in A + B.
+(i)+(ii) ⇒ (vi). If A/(A ∩ B) and (A + B)/A ∼= B/(A ∩ B) have p.d.≤ 1, then
+the middle term in the exact sequence
+0 → A/(A ∩ B) → (A + B)/(A ∩ B) → (A + B)/A → 0
+is likewise of p.d.≤ 1.
+□
+We are indebted to L. Salce for furnishing us with an example showing that the
+sum of two tight submodules of a module of p.d.1 need not be tight even if their
+intersection is tight (the converse is ruled out by the implication (vii) ⇒ (v) in
+Lemma 2.2). Since the proof requires several results from the theory of valuation
+domains that are not needed in this paper, we skip the details.
+The relation between the module and its tight submodules is a fundamental
+issue. The following simple fact might provide useful information.
+Lemma 2.3. Let M be a V -module of p.d.≤ 1. A tight submodule C of M satisfies
+gen C ≤ gen M.
+Proof. If gen M = n is an integer, then by Warfield [21] M is the direct sum of
+n cyclic submodules, so its Goldie-dimension is n (the Goldie-dimension — to be
+abbreviated as Gd — of a module M is the supremum of the cardinalities of the
+sets of non-zero summands in direct sums ⊕i∈IMi contained in M). A submodule
+C cannot have larger Goldie-dimension, thus gen C ≤ n.
+0
+0
+�
+�
+H
+H
+�
+�
+0 −−−−→ C −−−−→ N −−−−→
+F
+−−−−→ 0
+||
+�
+�ψ
+0 −−−−→ C −−−−→ M
+φ
+−−−−→ M/C −−−−→ 0
+�
+�
+0
+0
+If gen M = κ is an infinite cardinal, then consider a free resolution 0 → H →
+F → M/C → 0 where gen F = κ.
+The pullback N of φ : M → M/C and
+ψ : F → M/C fits in the vertical exact sequence 0 → H → N → M → 0. (See the
+commutative diagram with exact rows and columns.) Obviously, the free submodule
+H of the free F satisfies gen H ≤ gen F = κ, whence gen N ≤ gen H + gen M = κ
+4
+
+and gen N ≥ gen M = κ follow. Thus gen N = κ, and as N ∼= F ⊕C, the inequality
+gen C ≤ gen N = κ becomes evident.
+□
+Next we verify a result on the tightness of modules in a chain. Keep in mind
+that being tight is not an inductive property. (Continuity in the next lemma means
+that Mρ = ∪σ<ρMσ whenever ρ is a limit ordinal.)
+Lemma 2.4. Suppose
+(1)
+0 = M0 < M1 < · · · < Mσ < · · · < Mτ = M
+is a continuous well-ordered ascending chain of submodules of the module M with
+union M such that the modules Mσ (σ < τ) have p.d.1. If each module is tight in
+its immediate successor, then each module is tight in all of the successor modules
+of the chain, and also in M. Furthermore, p.d.M = 1.
+Proof. Apply Auslander’s familiar criterion on the p.d. of the union of a chain (see
+e.g. [15, Chap. VI, Lemma 2.6]) to the subchain starting at Mσ, to argue that
+p.d.Mσ+ρ+1/Mσ+ρ ≤ 1 for all ordinals ρ ≥ 0 implies that Mσ is tight in every
+successor in the chain. That p.d.M = 1 follows from the same lemma.
+□
+Let us agree that when we say “object”, we will mean an object of the subclass
+C∗
+V of CV (remember: all objects have p.d.≤ 1, and all subobjects are tight!).
+Manifestly, C∗
+V is closed under direct summands and direct sums; the projections
+onto direct summands and the embeddings of direct summands are morphisms in
+C∗
+V . In the opposite direction, we observe that neither the sum nor the intersection
+of two subobjects is necessarily a subobject, and the union of an ascending chain
+of objects need not be an object. The class C∗
+V is not additive in general; it is of
+course if V is a discrete valuation domain (DVD).
+3. Fundamental Objects
+Because of the restrictive composition rule of morphisms in our class, we have
+to learn from scratch if and how some of the familiar basic concepts have to be
+modified to fit in. One should not be surprised if in some cases hard-to-believe
+situations have to be accepted.
+Cyclically presented objects. The simplest objects in our class C∗
+V are the
+cyclically presented modules: V r/V s (r, s ∈ V ) where V s ≤ V r are principal ideals.
+All cyclic submodules of V r/V s are subobjects. Later on we will see that these
+are the only subobjects of V r/V s ∼= V (rs−1) ≤ V . Accordingly, only the principal
+ideals are subobjects in object V .
+Multiplications by ring elements induce endomorphisms of V r/V s whose kernels
+and images are subobjects. Morphisms between cyclically presented modules are
+the same in C∗
+V as in CV , therefore the V - as well as the C∗
+V -endomorphism ring of
+a cyclically presented object is local.
+Finitely generated objects. The finitely generated objects behave very nicely
+in C∗
+V . It is an elementary result that a finitely generated module over a valuation
+domain is of p.d.≤ 1 if and only if it is finitely presented (see [14, Proposition 4.1, p.
+83]), and then it is the direct sum of a finite number of cyclically presented modules
+(see Warfield [21], or [15, p.159]). Thus in our class C∗
+V , the finitely generated objects
+are the finitely presented V -modules. Note that the annihilators of elements in such
+a module are principal ideals (the ideal 0 is included), thus subobjects.
+5
+
+Example 3.1. Let {x, y, z, w} be a generating set of the module M with defining
+relations rx = 0, sy = 0, ax + by + cz = 0, ex + dw = 0 where r, s, a, b, c, d, e are
+non-units in V ×. The structure of M depends to a great extent on the divisibility
+relations between these ring elements. Assume e.g. the following proper divisibility
+relations: c | d | b | e | a | s | r.
+We can choose a different generating set:
+{x, y, z′, w′} with z′ = z + ac−1x + bc−1y, w′ = w + ed−1x; then the new relations
+will be rx = 0, sy = 0, cz′ = 0, dw′ = 0. This shows that the finitely presented
+module M is the direct sum of four cyclically presented submodules generated by
+x, y, z′, w′ with annihilators V r, V s, V c, V d, respectively.
+□
+If B is a finitely generated submodule of a finitely presented module A, then A/B
+is finitely presented, so it has p.d.≤ 1. Thus B is a subobject in A. If C is tight
+in a finitely presented module A, then p.d.A/C ≤ 1 implies that A/C is finitely
+presented. Hence C is finitely generated. Therefore, the subobjects in a finitely
+generated object are precisely the finitely generated submodules. An immediate
+consequence of this fact is the following corollary. It is telling us that the subclass
+of finitely presented objects behaves the same manner in C∗
+V as in CV .
+Corollary 3.2. 1) Let α : A → B and β : B → C be morphisms between finitely
+presented objects in C∗
+V . Then the composite map βα is also a morphism in C∗
+V .
+2) The endomorphisms of a finitely presented object in C∗
+V form a ring.
+Proof. It suffices to show that the intersection of two finitely generated subobjects
+is also finitely generated (the sum of two finitely generated submodules is evidently
+again finitely generated). The implication (vii)⇒(vi) in Lemma 2.2 ensures that
+such an intersection is tight also in the sum of the objects. Tight in finitely gener-
+ated is finitely generated.
+□
+Important observations on finitely generated submodules in objects are recorded
+in the following proposition.
+Proposition 3.3. (i) Finitely generated submodules in any object are (finitely
+presented) subobjects. In a finitely generated object they are the only subobjects.
+(ii) The finitely presented subobjects of an object M in the class C∗
+V form a
+sublattice in the lattice of submodules of M.
+(iii) The sum and intersection of a finitely presented subobject with any subobject
+are also subobjects.
+Proof. (i) This is an immediate consequence of [15, Lemma 6.4, p. 217], which
+ensures that finitely generated submodules of modules of p.d.≤ 1 are tight. The
+second part of claim (i) was already stated above.
+(ii) This follows from Corollary 3.2.
+(iii) Let A be a finitely presented and B an arbitrary subobject of M.
+The
+module (A + B)/B is a finitely generated submodule of M/B, therefore, it is tight
+in M/B. Hence A + B is tight in M. The rest follows from Lemma 2.2.
+□
+Uniserial objects.
+Most important objects are the uniserial modules (also
+called serial modules in the literature): these are defined as modules whose sub-
+modules form a chain with respect to inclusion. The obvious examples in CV are
+the so-called standard uniserial modules: the field of quotients, Q, as well as its
+submodules and submodules of their epic images, i.e. modules of the form J/I
+where 0 ≤ I < J ≤ Q are submodules. By Osofsky (see [18] or also [15, Chap.
+6
+
+VI, Sect. 3]), the p.d. of a submodule J of Q over a valuation domain is an in-
+teger n ≥ 0 if and only if gen J = ℵn−1 (ℵ−1 means “finite”). Hence uniserial
+objects in C∗
+V are at most countably generated. However, — as noted before — a
+countably generated ideal J is not a subobject of V in C∗
+V ; indeed, as p.d.V = 0
+and p.d.J = 1 imply p.d.V/J = 2. A torsion uniserial ought to have p.d.≤ 1 to
+belong to C∗
+V , therefore only those standard uniserials J/I are objects for which J
+is at most countably generated and I is cyclic, since p.d.J/I ≤ 1 implies that J/I
+is coherent; see e.g. [14, Chap. IV, Theorem 4.3]. Thus all the annihilator ideals
+of elements in a uniserial object are principal ideals. Hence it follows that the only
+proper subobjects of uniserial objects are cyclically presented. The non-standard
+uniserials (that play an important role in the theory of valuation domains, cf. [2]
+or [15]) will now be ignored, since their p.d. always exceeds 1.
+A convenient way to deal with uniserial objects is to view them in the form
+J (torsion-free case) or J/V (torsion case), where J is either Q (provided it is
+countably generated) or an at most countably generated proper submodule of Q
+containing V . Then it is trivial to answer the question of isomorphism of uniserial
+modules: J ∼= J′ if and only if J = rJ′ or J′ = rJ for some r ∈ V × (i.e. J′ = qJ
+for some 0 ̸= q ∈ Q), while J/V ∼= J′/V if and only if J = J′. Also, it makes sense
+to talk about total order even in the set of torsion uniserial objects in C∗
+V : the order
+relation being induced by the natural inclusion relation of the numerators J.
+Let U be a uniserial object and r ∈ V ×. It is obvious what is meant by rU. But
+it also makes sense to write r−1U. Indeed, this denotes the uniserial object U ′ that
+satisfies rU ′ = U; it is unique up to isomorphism.
+From the description of the proper subobjects of a countably generated uniserial
+object U ∈ C∗
+V it follows that their only endomorphisms are the automorphisms
+and the map to the zero submodule, that is, EndC∗
+V (U) = AutV (U)∪{0}. It is well
+known that the endomorphism ring of a uniserial module in CV is a local ring (see
+Shores–Lewis [19]), and modules with local endomorphism rings enjoy the Exchange
+Property.
+The Exchange Property is one of the properties most frequently investigated
+about the behavior of summands. Recall that a module A has the (finite) Exchange
+Property if direct decompositions M = A ⊕ B = C ⊕ D of any module M imply
+that there is another decomposition of the form M = A ⊕ C1 ⊕ D1 such that
+C1 ≤ C, D1 ≤ D. Furthermore, A has the Cancellation Property if A ⊕ B ∼= A ⊕ C
+for arbitrary modules B, C implies B ∼= C.
+Finally, A enjoys the Substitution
+Property if M = A1 ⊕ B = A2 ⊕ C with A1 ∼= A2 ∼= A implies the existence of a
+submodule A′ ≤ M such that A′ ∼= A and M = A′ ⊕ B = A′ ⊕ C. (See [6].) We
+note that, if an object A ∈ C∗
+V has either the Exchange, or the Cancellation, or the
+Substitution Property as a module in CV , then it also displays the same property
+in the class C∗
+V , since this class is closed under direct summands and direct sums.
+In view of the preceding remarks, from [15, Corollary 2.3, p. 342] and [15, p.181]
+we derive the following theorem.
+Theorem 3.4. Finite direct sums of uniserial objects in the class C∗
+V enjoy all of
+the cancellation, exchange and substitution properties.
+□
+Maximal uniserial submodules. By a maximal uniserial submodule in M
+we mean a uniserial submodule that is not properly contained in another uniserial
+submodule of M.
+7
+
+Theorem 3.5. Suppose M is a V -module of p.d.≤ 1, and U is a uniserial sub-
+module in M.
+(i) If U is maximal uniserial in M, then it is at most countably generated and
+has p.d.≤ 1.
+(ii) If U is countably generated and tight in M, then it is a maximal uniserial
+submodule in M.
+(iii) If U is a countably generated maximal uniserial submodule in a tight sub-
+module of M, then it is also maximal in M.
+Proof. (i) See [15, Chap. VI, Lemma 6.7].
+(ii) By way of contradiction assume U is not maximal in M, i.e. there exists a
+uniserial U ′ ≤ M that contains U properly. We may assume that U ′ is cyclic, say,
+U ′ = V a for a ∈ M. Then V a/U is a non-zero cyclic submodule in the module
+M/U which has p.d.1 by hypothesis. Hence it follows that V a/U is tight in M/U,
+so U must be cyclic. This contradiction completes the proof of (ii).
+(iii) The proof is similar to that of (ii). If U is a maximal uniserial in a tight
+submodule N of M and contained in a larger uniserial U ′ ≤ M that is cyclic, then
+U ′/U ∼= (U ′ + N)/N is a cyclic submodule in M/N, so cyclically presented. Again
+we can conclude that U must be cyclic.
+□
+An immediate consequence of this theorem is that in a direct sum of cyclically
+presented objects all uniserial subobjects are also cyclically presented.
+Mixed modules as objects. An object in C∗
+V that is mixed in the usual sense
+(i.e. neither torsion nor torsion-free) need not have a 1-dimensional torsion part.
+Actually, the torsion submodule of a mixed module of p.d.1 can have any p.d.
+not exceeding the maximal p.d. of torsion-free V -modules minus 1 whenever this
+number is ≥ 1. Indeed, select a torsion-free V -module N of p.d. n ≥ 2 and set
+N = F/G with a free V -module F. Let H be an essential free submodule of G, and
+define M = F/H. Then M is an object with torsion submodule G/H of p.d. n − 1.
+Even if the torsion submodule of a mixed object is an object, it need not be a
+subobject. Therefore, it seems reasonable to consider an object mixed in C∗
+V if its
+torsion submodule is a non-zero proper subobject. For an injective object in C∗
+V
+that is a mixed module in CV , but not in C∗
+V , see Example 6.9 infra.
+Countably generated objects. Suppose M is the union of a countable as-
+cending chain Mn (n < ω) of finitely presented modules. Then each Mn is tight
+in its immediate successor and hence also in M which will have p.d.1 (cf. Lemma
+2.4). Moreover, since every finitely generated submodule of M is contained in some
+Mn, all finitely generated submodules of M are tight in M and finitely presented,
+so subobjects. Cyclic subobjects are cyclically presented, thus the annihilators of
+elements in M are principal ideals (i.e. also objects). Submodules that are count-
+ably generated may or may not be subobjects in countably generated objects. E.g.
+the countably generated ideals of V are objects, but not subobjects of V .
+Claim (i) in our next theorem shows that the module M in the preceding para-
+graph is a typical countably generated object.
+Theorem 3.6. (i) A countably generated V -module is of p.d.≤ 1 if and only if it
+is the union of a countable ascending chain of finitely presented (tight) submodules.
+(ii) Let A, B be tight submodules in a V -module M of p.d.≤ 1 such that B is at
+most countably generated. Then
+8
+
+(a) C = A ∩ B is also at most countably generated, and tight in B and also
+in A and in M;
+(b) A + B is of p.d.≤ 1, and A is tight in it.
+Proof. (i) One way the claim follows from Proposition 3.3, while the converse is
+taken care of by the last but one paragraph before this theorem.
+(ii) (a) B is the union of a countable chain {Bn | n < ω} of finitely presented
+submodules. Hence B/C = B/(A ∩ B) ∼= (A + B)/A = ∪n(A + Bn)/A, where the
+last quotient modules are finitely generated (epic images of the Bn in M/A), and
+therefore tight in M/A. The p.d. of their union (A + B)/A equals 1, which means
+p.d.B/C = 1. Thus C is tight in B, and hence in M, and then also in A.
+(b) The sum A + B is the union of the chain of the extensions A + Bn of A by
+finitely presented modules, so its p.d. is ≤ 1.
+□
+Next we give more examples of countably generated objects.
+By choosing a
+suitable value group, it is easy to construct valuation domains that have the required
+properties.
+Example 3.7. Consider a uniserial object U generated by {ui (i < ω)}. Attach to
+each ui ∈ U a cyclically presented module, say, generated by bi such that ribi = ui
+where the non-units ri ∈ V × are required to satisfy the condition that V/V ri is
+properly embeddable in U/V ui. Then U is a pure subobject of B = ⟨U, bi (i < ω)⟩
+such that B/U is a direct sum of cyclically presented modules ∼= V bi/V ui. Since
+pure extensions by a cyclically presented module are obviously splitting, we obtain:
+B ∼= U ⊕ (⊕i<ωV bi/V ui).
+□
+Example 3.8. Let J, L denote submodules of Q containing V that are at most
+countably generated, and let r ∈ V × be a non-unit. Then one of J/V r and L/V r,
+say, the former, admits an isomorphic embedding in the latter, φ : J/V r → L/V r,
+that is the identity on V/V r. Define N as the push-out of the embeddings φ :
+V/V r → J/V r and ψ : V/V r → L/V r.
+Then N is an extension of its pure
+submodule L/V r by J/V . A fast calculation shows that N = L/V r ⊕ J′/V where
+J′/V = {(x, φ(x)) | x ∈ J/V } ≤ J/V ⊕ L/V .
+□
+The abundance of subobjects. It is a remarkable fact that even if V is not a
+discrete rank one valuation domain, objects in C∗
+V contain a large number of tight
+submodules of all possible sizes. From our discussion above this is clear for finitely
+and countably generated objects, while for uncountably generated objects it is a
+consequence of the existence of tight systems. Every module M of p.d.≤ 1 admits
+a tight system T (over any integral domain). This is defined as a G(ℵ0)-family of
+tight submodules; see [15, Chap. VI, Sect. 5]. Recall that, for an infinite cardinal
+κ, by a G(κ)-family G of submodules of a module M is meant a set of submodules
+such that the following conditions are satisfied: 1) 0, M ∈ G; 2) G is closed under
+unions; 3) if X is a subset of M of cardinality ≤ κ and A ∈ G, then there exists
+a B ∈ G such that X ∪ A ⊆ B and gen(B/A) ≤ κ. It follows that in case T is
+a tight system of M, then A < C (A, C ∈ T ) implies A is tight in C. Moreover,
+under the canonical map the complete preimages of the tight submodules in C/A
+are subobjects of C; they are also subobjects in M.
+An immediate corollary of the existence of tight systems is the next result.
+Corollary 3.9. Let M be an object, and N a submodule of M. Then N is contained
+in a tight submodule N ∗ of M such that gen N ∗ ≤ max{gen N, ℵ0}.
+9
+
+Proof. Every generator of N is contained in some countably generated tight sub-
+module that belongs to a fixed tight system T of M. The union of all these members
+of T is a member N ∗ of T containing N. By construction, N ∗ can be generated by
+gen N · ℵ0 elements.
+□
+A tight system T of object M allows us to build a continuous well-ordered
+ascending chain (1) of tight submodules Mσ ∈ T for some ordinal τ, such that
+gen(Mσ+1/Mσ) ≤ ℵ0 for all σ < τ. Moreover, since a countably generated object
+is the union of a chain of finitely presented subobjects (Theorem 3.6), the chain (1)
+can be refined so as to have all of the quotients Mσ+1/Mσ finitely, or even cyclically
+presented.
+Another important consequence of the existence of tight systems is that we have
+already formulated in Proposition 3.3: every finitely generated submodule of a
+module M (of any size) of p.d.1 is finitely presented and tight in M.
+Projective objects. A finitely generated torsion-free module over a valuation
+domain is free. A useful fact: a finite rank pure submodule in a free V -module is
+a summand; see e.g. [14, Chap. XIV, Theorem 6.1]. By Kaplansky [16], projective
+V -modules are free. Evidently, they are projective objects in our class C∗
+V as well.
+Dimension calculation shows that a tight submodule of a free module must have
+zero p.d. Therefore, we can state:
+Theorem 3.10. The subobjects of free V -modules are the free submodules.
+□
+Hence we conclude that every object M in C∗
+V admits a free resolution in the
+form of a short exact sequence: 0 → H → F → M → 0 where H, F are free
+V -modules, i.e. free objects.
+4. More Fundamental Concepts
+Continuing the review of the basics, we would like to establish more results
+concerning the objects in the class C∗
+V , but in order to deal with the objects more
+efficiently, we need several tools available in [14] and in [15]. In this section we
+review some concepts and facts we shall need.
+Annihilators of elements. The study of objects in C∗
+V is greatly simplified by
+the fact that the annihilator ideals of elements in objects are not just objects, but
+they are even principal ideals. This has been pointed out before, but let us give a
+formal proof of this property.
+Lemma 4.1. Let M be an object in C∗
+V . Then for any element a ∈ M, the anni-
+hilator annM(a) = {r ∈ V | ra = 0} is a principal ideal of V .
+Proof. M has a tight system, so a ∈ M is included in a countably generated tight
+submodule N of M, and hence also in a finitely presented submodule (see Theorem
+3.6 above). For finitely presented objects the claim has been established before.
+□
+Heights of elements.
+The principal information in describing the way an
+element is located in the module is stored in its height. Heights of elements are
+defined by using uniserial modules, see [14]. The uniserials that occur as possible
+heights for valuation domains have been studied in [1] and [2].
+Fortunately, in
+modules of p.d.≤ 1 only most tractable heights can occur.
+10
+
+Suppose M is an object and 0 ̸= a ∈ M. Consider maps φJ : J → M of the
+submodules J of Q containing V such that φJ(1) = a. For a fixed a, the union in
+Q of those J’s for which such a φJ exists is a submodule HM(a) of Q, called the
+height-ideal of a ∈ M. The module
+hM(a) = HM(a)/V
+is defined as the height of a ∈ M. We call hM(a) non-limit height or limit-height
+according as HM(a) is one of the J’s or is not. In the limit case we write hM(a) =
+U −. Note that hM(a) is always a uniserial torsion module; it is of the form U = J/V
+with J ⊆ Q (equality only in case Q ∈ C∗
+V ). In the non-limit case, the element
+a is contained in a uniserial module W that is a maximal uniserial in M such
+that hM(a) = W/V a.
+The heights of elements in a non-standard uniserial are
+uncountable limit heights — these are out of question in C∗
+V . The set of heights
+occurring in C∗
+V is totally ordered in the obvious way once we declare the non-
+limit height J/V to be larger than the corresponding limit height (J/V )−. The
+minimum height is 0 (this is the height of the generator in a cyclic module), and
+we set h(0) = ∞ as the maximum height.
+Example 4.2. To give an example of a limit height, consider a countably generated
+submodule J of Q containing V , and choose a properly ascending chain of fractional
+ideals {V t−1
+i
+| i < ω} with union J (where ti ∈ V ×). Define a countably generated
+object X as follows: the generators are xi (i < ω) with the defining relations:
+(2)
+rt0x0 = 0,
+t0x0 = tixi
+(i < ω)
+where r ∈ V × is arbitrary. The element t0x0 has limit height, namely (J/r−1V )−.
+To get an idea of what kind of module X is, observe that the cyclic submodules
+generated by the elements xi − (t−1
+i ti+1)xi+1 are summands of X for all i < ω
+(a complement is the submodule generated by all the given generators with xi
+removed). Actually, these cyclic modules generate their direct sum X′ in X. This
+X′ is tight and pure in X, and the quotient X/X′ is a countably generateduniserial
+module containing the coset x0 + X′.
+□
+Next we prove the following:
+Theorem 4.3. A non-zero element in an object of the class C∗
+V has one of the
+following heights:
+(i) cyclic height;
+(ii) countably generated non-limit height;
+(iii) arbitrary limit height of standard type.
+Elements in a finitely generated module cannot have limit heights.
+Proof. To begin with, observe that for each of (i)-(iii) we already had examples
+above, so it remains only to show that (i)-(iii) is a complete list. The only other
+heights in CV are uncountably generated non-limit heights. Working toward con-
+tradiction, suppose that for some a ∈ M ∈ C∗
+V , we have hM(a) = J/V with an
+uncountably generated submodule J of Q. There is a homomorphism φJ : J → M
+such that φJ(1) = a. The maximal property of J as height implies that φJ(J) must
+be a maximal uniserial in M. Therefore, by Theorem 3.5 it is at most countably
+generated — a contradiction, completing the proof of the first claim.
+11
+
+That (iii) cannot occur in a finitely generated module is an immediate conse-
+quence of the simple fact that limit heights require infinite Goldie-dimension, as is
+clear from Example 4.2.
+□
+Height-gaps. Suppose U is a uniserial object and V r ̸= 0 is the annihilator of
+a ∈ U. Then hU(a) = rU and hU(sa) = s−1rU provided that sa ̸= 0 for s ∈ V .
+If U is contained in a V -module M, then the heights of these elements may be
+larger in M. In general, in every module M, for an element a ∈ M and its multiple
+ra (r ∈ V ×) the inequality
+hM(a) ≤ r−1hM(ra)
+holds. We say that M has a height-gap at 0 ̸= a ∈ M if, hM(a) > shM(x) holds
+whenever sx = a for some x ∈ M and for a non-unit s ∈ V .
+Example 4.4. To illustrate height-gaps, let U be a uniserial module, and x1, x2, x3
+symbols. Suppose that the non-units si, ti ∈ V × satisfy the following proper divis-
+ibility relations: s1 | s2 | s3 and t1 | t2 | t3. Pick some u ∈ U such that s3u ̸= 0,
+and define a module N to be generated by U and by the given symbols subject to
+the relations siu = sitixi (i = 1, 2, 3). The height-gaps in the submodule U are at
+s1u, s2u, s3u, and at 0.
+□
+Purity. The main point about this widely used concept that we are emphasiz-
+ing repeatedly is that in valuation domains it is equivalent to the simpler relative
+divisibility (see [20]). Thus a submodule N is pure in a V -module M if and only
+if rN = N ∩ rM holds for every r ∈ V ×. Equivalently, for all r ∈ V , the map
+V/V r ⊗V N → V/V r ⊗V M induced by the inclusion N → M is monic. This is
+tantamount to the injectivity of the map HomV (V/V r, M) → HomV (V/V r, M/N)
+for all r ∈ V × induced by the natural homomorphism M → M/N. A pure-exact
+sequence 0 → A → B → C → 0 is an exact sequence in which the image of the map
+A → B is pure in B.
+Lemma 4.5. (i) Let U be a uniserial submodule of an object M, and a ∈ U. If
+hM(a) = U/V a, then U is a maximal uniserial in M, and there is no height-gap in
+U at a and above.
+(ii) If U is a maximal uniserial in M and is torsion with no height-gaps other
+than the one at 0, then U is pure in M.
+Proof. (i) This is rather obvious.
+(ii) U is not pure in M means that there is u ∈ U such that hU(u) < hM(u).
+Hence there must be a height-gap in U at u or above, because by maximality, some
+of the generators of U have the same height in U as in M.
+□
+We recall the definition of Pext1
+V (X, M): it is a sub-bifunctor of Ext1
+V (X, M),
+consisting of those non-equivalant extensions of M by X in which M is a pure
+submodule (see e.g. [15, p. 45]). In the commutative case, Pext is a V -module.
+5. Theorems on Torsion and Torsion-free Modules
+In this section, we discuss briefly a few fundamental results on torsion and
+torsion-free objects. An in-depth study that would require more research and in-
+teresting applications is planned in the future.
+We start with the following simple observation.
+12
+
+Theorem 5.1. A pure and tight finitely generated submodule in an object is a direct
+summand.
+Proof. Suppose a finitely generated module N is pure and tight in a module M
+of p.d.≤ 1. By the tightness of N, M/N has p.d.≤ 1, and by Corollary 7.7 N is
+pure-injective. All this combined implies that N is a summand of M.
+□
+We continue with typical examples of direct sums of cyclically presented mod-
+ules:
+the pure-projective objects.
+These are defined as objects P that satisfy
+Pext1
+V (P, M) = 0 for all objects M ∈ C∗
+V .
+Theorem 5.2. A V -module is pure-projective if and only if it is a direct sum of
+cyclically presented modules.
+Proof. This is a special case of a well-known theorem. E.g. it follows from [15,
+Chap. VI, Theorem 12.2].
+□
+Concerning direct sums of uniserials, a most important result is the following
+theorem (this is not related to tightness).
+Theorem 5.3. (i) The uniserial summands in a direct sum of uniserial modules
+are unique up to isomorphism.
+(ii) Summands of a direct sum of uniserial modules are themselves direct sums
+of uniserials.
+Proof. These are well-known immediate consequences of the fact that the endomor-
+phism rings of uniserial modules are local (Theorem 3.4).
+□
+Let r ∈ V × be a non-unit. By a V/V r-homogeneous module we mean a V -module
+H such that each element is contained in a submodule of H that is ∼= V/V r. Then
+H satisfies rH = 0, and any cyclic submodule of H that is ∼= V/V r must be pure
+in H. Moreover, by Theorem 5.1 it is then a summand.
+Proposition 5.4. Suppose M is a V -module of p.d.1.
+(i) If rM = 0, then a V/V r-homogeneous tight submodule is a summand of M.
+(ii) If M is V/V r-homogeneous, then it is the direct sum of cyclically presented
+submodules, all isomorphic to V/V r.
+(iii) If D is a divisible object, then for every r ∈ V ×, D[r] is V/V r-homogeneous,
+so a direct sum of cyclically presented submodules isomorphic to V/V r.
+Proof. For (i)-(ii) we refer to [15, Chap. XII, Theorems 2.2 and 2.3], and for (iii)
+to [15, Chap. XIV, Corollary 2.4].
+��
+We note that the number of summands ∼= V/V r in (iii) is the same for every r
+provided that D[r] ̸= 0: it is the Goldie-dimension of D.
+An important theorem in abelian group theory, due to L. Ya. Kulikov, states
+that a subgroup of a direct sum of cyclic groups is likewise a direct sum of cyclic
+groups (see, e.g., [11]). An analogue in C∗
+V would state that a tight submodule of a
+direct sum of cyclically presented modules is also such a direct sum. This is indeed
+true for torsion-free modules: a tight submodule of a free module in C∗
+V is again
+free. It was conjectured that this holds in C∗
+V also in the torsion case. (For torsion
+abelian groups, see e.g. [11, Chap. 3, Theorem 5.7].) However, we claim that the
+module X in Example 4.2 refutes this conjecture. In order to prove this, consider
+the module Y in the following example.
+13
+
+Example 5.5. The countably generated torsion object Y is defined just as the
+module X in Example 4.2: it is generated by the same set {xi | i < ω} with the
+same defining relations, but there is a single modification: we replace t0 ∈ V × by
+s ∈ V × that is picked such that J < V s−1. In this case, V x0 is a pure and tight
+submodule in Y (a summand), and the elements xi −(t−1
+i s)x0 for all i > 0 generate
+cyclic direct summands of Y such that Y is the direct sum of V x0 and these cyclic
+submodules. (Another, but less explicit argument to obtain the structure of Y is as
+follows. After observing that the cyclic submodule V x0 is pure in Y , it only remains
+to point out that moreover, it is a summand of Y , since Y/V x0 is pure-projective
+as the direct sum of cyclically presented modules ∼= V ti (i > 0).)
+□
+To argue that the object X of Example 4.2 cannot be a direct sum of cyclically
+presented objects, appeal to Theorem 4.3. The element t0x0 ∈ X is of countable
+limit height, and as such it cannot belong to a direct sum of the stated kind: it would
+be contained already in a finitely generated summand with the same limit height.
+However, this is impossible as is demonstrated by the cited theorem. Thus the
+object X that is (isomorphic to) a tight submodule (observe that Y/X ∼= V s/V t0
+is cyclically presented) in a direct sum Y of cyclically presented modules fails to be
+a direct sum of such modules.
+Hence it is obvious that this theorem of Kulikov cannot have the suspected
+analogue in C∗
+V without additional hypotheses. Looking for simple conditions that
+would lead us to a Kulikov-type theorem for subobjects in direct sums of cyclically
+presented objects, we selected (b), in addition to the obvious (a), that seems natural
+to assume:
+(a) The non-zero elements have cyclic heights.
+(b) The uniserial submodules admit but a finite number of height-gaps.
+Under the hypothesis of (a)-(b), we will prove a desired analogue for the count-
+ably generated torsion objects (see Theorem 5.8 below). But first we deal with
+preliminary lemmas.
+We need a definition. Similarly to [13], we will call a V -module M cyclically
+separable if every finite set of its elements can be embedded in a finitely generated
+summand of M, i.e. in a summand that is the direct sum of a finite number of cycli-
+cally presented modules. Observe that in order to verify the cyclic separability of
+a torsion object, it suffices to check the defining property only for one element sub-
+sets, as every finitely generated object is a finite direct sum of cyclically presented
+objects. Hence it is evident that summands of modules inherit cyclical separability.
+We now prove a crucial lemma.
+Lemma 5.6. Let M denote a torsion object in C∗
+V . If M satisfies conditions (a)-
+(b), then it is a cyclically separable V -module.
+Proof. Assume M has properties (a)-(b), and let 0 ̸= a ∈ M. By (a), a is contained
+in a cyclically presented submodule C = V c that is maximal uniserial in M. If C
+contains no height-gap strictly between a and 0, then C is pure in M, and hence a
+summand of M (Theorem 5.1). Thus in this case a embeds in a cyclically presented
+summand of M, and we are done. If there are height-gaps in C between a and 0,
+then by (b) there is one, say at rc (r ∈ V ), such that no height-gap exists strictly
+between rc and 0. Then by the previous argument there is a cyclically presented
+summand B = V b of M that contains rc, M = V b ⊕ M ′. If b′ ∈ V b is such that
+V rb′/V c ∼= V r, then V (c) + V (b) = V (c − b′) ⊕ V (b). In this case, the projection of
+14
+
+V (c − b′) in M ′ contains the coordinate of a with a smaller number of height-gaps
+below it. Repeating this process for the coordinates of a a finite number of times,
+we get a finitely generated summand of M that contains the selected element a.
+□
+Lemma 5.7. Let M be a torsion object satisfying conditions (a)-(b).
+If M is
+countably generated, then it is a direct sum of cyclically presented objects.
+Proof. By Lemma 5.6, M is cyclically separable. It is a simple exercise to prove that
+a countably generated cyclically separable module is a direct sum of cyclics.
+□
+The following analogue of Kulikov’s theorem is now easily established.
+Theorem 5.8. Assume M is a direct sum of cyclically presented torsion objects,
+and N is a countably generated subobject satisfying condition (b) above. Then N
+is likewise a direct sum of cyclically presented subobjects.
+Proof. Owing to Theorem 2.3 and Lemma 5.6 it suffices to show that N satisfies
+condition (a). But this is immediate by virtue of Theorem 3.5 (iii).
+□
+We are asking the obvious question: do the preceding lemma and theorem hold
+for larger cardinalities? The answer is: for Lemma 5.7 counterexamples are torsion-
+complete abelian p-groups with countable unbounded basic subgroups; cf.
+[11,
+Chap. 10, sect. 3]. For Theorem 5.8 we do not know the answer.
+We record the following two parallel questions.
+Problem 5.9. Are pure subobjects in direct sums of torsion uniserial (resp. count-
+ably generated) objects also direct sums of the same kind?
+□
+Next we want to get an idea of the torsion-free modules in C∗
+V . It is a pleasant
+surprise that all countably generated torsion-free modules in CV are objects in C∗
+V .
+This is evident from the following theorem.
+Theorem 5.10. (i) A torsion-free V -module A has p.d.≤ 1 if and only if every
+rank one pure submodule is at most countably generated.
+(ii) A torsion-free V -module A is of p.d.≤ 1 if and only if it admits a well-ordered
+ascending chain of tight pure submodules Aα such that for each α, Aα+1/Aα is of
+rank one and of p.d.1 (thus cyclic or countably generated torsion-free).
+Proof. It suffices to refer to [7, Corollary 4.5] and to [15, Chap. VI, Lemma 6.6],
+respectively.
+□
+We continue with a theorem that resembles Pontryagin’s theorem on countable
+free abelian groups. A similar result for the projective dimension one case is also
+included.
+Theorem 5.11. A torsion-free module of countable rank in C∗
+V is free (is an object
+in C∗
+V ) if and only if its finite rank pure submodules are free (have p.d.≤ 1).
+Proof. See [15, Chap. VI, Corollary 3.12].
+□
+15
+
+6. Divisible and Injective Objects
+The theory of divisibility and injectivity clearly illustrates a fundamental differ-
+ence between the classes C∗
+V and CV .
+Divisible objects.
+Divisibility of modules is defined as usual: D ∈ C∗
+V is
+divisible if rD = D for all r ∈ V ×. Equivalently, the equality Ext1
+V (V/V r, D) = 0
+holds for all r ∈ V ×. The prototype of divisible modules, the quotient field Q of
+V as a V -module, is in general not an object. It is exactly when Q is a countably
+generated V -module (then p.d.Q = 1, i.e. V is a Matlis domain). But the module
+∂V (see [8]), the generator of the subcategory of the divisible modules in V -Mod
+has p.d.1, so it is an object in C∗
+V . Recall that ∂V is generated by the k-tuples
+(r1, . . . , rk) for all k ≥ 0 of non-unit elements ri ∈ V ×, subject to the defining
+relations
+(3)
+rk(r1, . . . , rk−1, rk) = (r1, . . . , rk−1)
+(k > 0)
+for all choices of the ri.
+The generator w = (∅) generates a submodule of ∂V
+isomorphic to V such that ∂V0 = ∂V /V w is a divisible torsion module of p.d.1
+(which is a generator of the subcategory of divisible torsion modules in CV ). See
+[8] or [15] for more details.
+As far as the structure of divisible objects is concerned, the following information
+is crucial. (Pay attention to the enormous simplification over Matlis domains.)
+Theorem 6.1. (i) An object in C∗
+V is divisible if and only if it is a summand of a
+direct sum of copies of the module ∂V .
+(ii) If V is a Matlis domain, then an object is divisible if and only if it is the
+direct sum of copies of Q and/or K.
+Proof. (i) In [8, Theorem 18] it is shown that over a Pr¨ufer domain (and hence over
+a valuation domain) a divisible module has p.d.1 if and only if it is a summand of
+a direct sum of copies of ∂. (By the way, this holds for all integral domains.)
+(ii) See [15, Chap. VII, Theorem 3.5].
+□
+In order to obtain a full set of invariants for a divisible object D, we introduce
+two cardinal invariants measuring the size of its torsion and torsion-free parts. One
+is κ = rk D, the torsion-free rank of D, the number of generators of a maximal
+size free submodule contained in D. The other invariant is λ = gen D[r] for any
+non-unit r ∈ V ×. Thus λ is the cardinality of the set of summands ∼= V/V r in a
+direct decomposition of D[r] into indecomposable summands. These two cardinals
+form a complete set of invariants characterizing divisible objects in C∗
+V . In fact,
+Theorem 6.2. Assume D and D′ are divisible objects in C∗
+V . Then D ∼= D′ if and
+only if
+(i) their ranks are equal: rk D = rk D′; and
+(iI) for some, and hence for each r ∈ V ×, gen D[r] = gen D′[r].
+Proof. See [10, Theorem C] or [15, Chap. VII, Theorem 3.4].
+□
+We also state the existence theorem accompanying this structure theorem.
+Theorem 6.3. Given the cardinals κ, λ, there exists a divisible object D in class
+C∗
+V such that rk D = κ and gen D[r] = λ if and only if
+(i) in case p.d.Q = 1: both κ and λ are arbitrary;
+(ii) in case p.d.Q > 1: κ is arbitrary and λ ≥ max{κ, gen Q}.
+16
+
+Proof. We refer to [10, Theorem 3] or to [15, Chap. VII, Theorem 3.8].
+□
+From the foregoing results we can draw the conclusion that in case p.d.Q > 1
+every divisible D ̸= 0 in C∗
+V satisfies Gd(D) ≥ gen Q. Furthermore, no indecom-
+posable divisible object exists in C∗
+V .
+We also have the embedding result as expected:
+Theorem 6.4. Every object in C∗
+V is a subobject of a divisible object.
+Proof. Write M ∈ C∗
+V as M = F/H with free V -modules F, H. If F = ⊕i∈IV xi
+with V xi ∼= V , then define G = ⊕i∈I∂i with ∂i ∼= ∂V , and embed F in G by
+identifying the generator xi with the generator wi ∈ ∂i, for all i. Then F becomes
+a subobject of G, and G/H will be a divisible module of p.d.1 that contains a copy
+of M as a subobject.
+□
+h-divisibility of a V -module H is defined by the extendibility of the homomor-
+phisms V → H to Q → H (see [17] or [15, p. 38]). With the exception of the next
+proposition, this concept will not be discussed, considering that h-divisible modules
+rarely exist in C∗
+V , even injective objects are not h-divisible whenever p.d.Q > 1.
+Proposition 6.5. (i) h-divisible objects exist in C∗
+V if and only if V is a Matlis
+domain, in which case all divisible modules are h-divisible.
+(ii) If V is a Matlis domain, then an h-divisible object of p.d.1 is the direct sum
+of copies of Q and K.
+Proof. (i) It is well known that over a domain, divisibility and h-divisibility are
+equivalent if and only if p.d.Q ≤ 1 (see e.g. [15, Chap. VII, Theorem 2.8]).
+(ii) See [15, Chap. VII, Sect. 2].
+□
+Injective objects. The role of injective modules is played by objects E ∈ C∗
+V
+that satisfy Ext1
+V (A, E) = 0 for all A ∈ C∗
+V ; i.e. whenever E is a subobject, it
+must be a summand.
+Luckily, this property is equivalent to the more familiar
+extensibility of morphisms into E from subobjects to objects. But this equivalence
+comes with a caveat: the extended map need not be a C∗
+V -morphism, since its image
+might not be tight. Perhaps unexpectedly, injectivity and divisibility turn out to
+be equivalent.
+Theorem 6.6. The following conditions are equivalent for an object E ∈ C∗
+V .
+(i) Ext1
+V (C, E) = 0 for all C ∈ C∗
+V ;
+(ii) every morphism φ : A → E in C∗
+V extends to a homomorphism ψ : B → E
+whenever A, B are in C∗
+V and A is tight in B;
+(iii) E is a divisible object.
+Proof. In the category CV we have an exact sequence
+(4)
+HomV (B, E) → HomV (A, E) → Ext1
+V (B/A, E) → . . .
+(i) ⇒ (ii). Hypothesis implies that Ext in (4) vanishes, so the map between the
+two Homs is surjective.
+(ii) ⇒ (iii). Condition (ii) ensures that for every r ∈ V ×, every map φ : V r → E
+extends to V → E (note that φ ∈ C∗
+V ). This is equivalent to the divisibility of E
+by r.
+17
+
+(iii) ⇒ (i). By Bazzoni–Herbera [3], over an integral domain R, a module E
+is divisible if (and only if) Ext1
+R(C, E) = 0 holds for all R-modules C of p.d.≤ 1.
+(This implication was proved in [9, Theorem 6] for Pr¨ufer domains.)
+□
+As an immediate corollary to the preceding theorem we obtain:
+Corollary 6.7. Direct sums of injective objects are likewise injective. In particular,
+injective objects are Σ-injective, i.e. any direct sum of copies of an injective object
+is injective.
+□
+A well-known test for the injectivity of a module is that its extensions by cyclic
+modules are splitting. In C∗
+V this criterion simplifies to cyclically presented modules:
+Theorem 6.8. An object E ∈ C∗
+V is injective if and only if Ext1
+V (C, E) = 0 holds
+for all cyclically presented objects C.
+□
+It is well known in commutative module theory that every module contains a
+unique maximal divisible submodule. This is not true in C∗
+V in general, because the
+relevant property that the sum of two divisible objects is again one fails; indeed,
+the property of being of p.d. at most 1 is frequently lost when forming the sum.
+Example 6.9. We exhibit an injective object which is a mixed V -module, but
+neither torsion, nor torsion-free, nor mixed as an object of C∗
+V . Let V be a valuation
+domain such that p.d.Q = 3, and consider the divisible V -module ∂V defined above.
+As p.d.∂V = 1, we have ∂V ∈ C∗
+V . The torsion submodule T of ∂V has p.d.2, since
+∂V /T ∼= Q. Therefore ∂V , as an object of C∗
+V , is neither torsion, nor torsion-free,
+nor mixed.
+□
+The following corollary is obvious in view of our discussion of the divisible ob-
+jects. (For the following (i), cf. Theorem 6.4.)
+Corollary 6.10. (i) Every object embeds as a subobject in an injective object.
+(ii) Objects that are epic images of injective objects (modulo subobjects) are them-
+selves injective.
+(iii) Every object M admits an injective resolution, that is an exact sequence
+0 → M → A → B → 0 of modules of p.d.≤ 1 where A, B are injective objects.
+(iv) The injective dimension of any object in C∗
+V is 0 or 1.
+□
+Let us pause for a moment to answer a question concerning the existence of
+injective envelopes in the class C∗
+V . By the injective envelope of an object M we
+mean an injective object E(M) containing M as a subobject such that for every
+injective object E containing M as a subobject, the identity map of M extends to
+a tight embedding E(M) → E. Of course, if an envelope exists, it is then unique
+up to isomorphism.
+Theorem 6.11. All modules in the class C∗
+V admit injective (divisible) envelopes
+if and only if V is a rank one discrete valuation domain.
+Proof. If V is a DVD, then CV and C∗
+V are identical, and the claim in CV is well-
+known. If V is a Matlis domain, then all divisible modules are h-divisible, they are
+direct sums of copies of Q and/or K. If V is not a DVD, then it contains a countably
+generated ideal, and such an ideal cannot be tight in a direct sum of Qs. Finally,
+if p.d.Q > 1, then the C∗
+V -injective envelope of V should be an indecomposable
+summand of ∂V , but — as observed above — such a summand does not exist.
+□
+18
+
+7.
+Pure-Injectivity
+Pure-injectivity in C∗
+V can be developed like in traditional module theory (see e.g.
+[14, Chap. XI, Section 2] and [15, Chap. XIII, Sections 2-3]). As expected, there
+are several changes, so we provide the whole proof, but skip routine arguments.
+By a system of equations (with unkowns xj (j ∈ J)) over a module M we mean
+a system of linear equations
+(5)
+�
+j∈Ji
+rijxj = ai ∈ M
+(rij ∈ V, i ∈ I)
+for i ∈ I, j ∈ J where I, J are arbitrary index sets, and Ji is a finite subset
+of J for each i ∈ I. Let F denote the free module generated by the unknowns
+xj (j ∈ J) and H its submodule generated by the left sides of the equations for all
+i ∈ I. We consider only consistent systems; i.e., systems that do not contain hidden
+contradiction. This means that we get a genuine homomorphism φ : H → M by
+mapping the generators of H onto the elements of M as shown by the equations, i.e.
+φ : �
+j∈Ji rijxj �→ ai. It is easy to check that the system has a solution in M if and
+only if φ extends to a homomorphism φ∗ : F → M, in which case xi = φ∗(xi) ∈ M
+are the solutions. It is pretty obvious that a consistent system defines an extension
+M ∗ of M by F/H by adjoining to M the unknowns xi as generators subject to the
+defining relations (5). Furthermore, it is straightforward to check that M will be
+pure in M ∗ exactly if (5) is finitely solvable in M, i.e. the finite subsystems of (5)
+admit solutions in M.
+We define p.d.(F/H) as the projective dimension of the system (5). It will be
+convenient to call (5) an adequate equation system if 1) it is consistent; 2) its p.d.
+is ≤ 1; and 3) it is finitely solvable.
+An object M ∈ C∗
+V is said to be pure-injective if it has either one of the equivalent
+properties listed in the following theorem.
+Theorem 7.1. For an object M, (α)-(γ) are equivalent properties:
+(α) Pext1
+V (C, M) = 0 for all objects C.
+(β) If A is a pure subobject of object B, then every C∗
+V -map A → M extends to
+a homomorphism B → M (that need not be a C∗
+V -map).
+(γ) Every adequate equation system over M has a global solution in M.
+Proof. All modules in this proof are objects in C∗
+V .
+(α) ⇒ (β) Assuming (α), consider the following push-out diagram where the top
+sequence is pure-exact and ζ is a C∗
+V -morphism.
+0 −−−−→ A −−−−→ B −−−−→ C −−−−→ 0
+�ζ
+�
+���
+0 −−−−→ M −−−−→ N −−−−→ C −−−−→ 0
+Then the bottom row is also pure-exact, so it splits by hypothesis. Hence there is
+a homomorphism B → M making the upper triangle ABM commute. This is an
+extension of ζ, proving (β).
+(β) ⇒ (γ) Given an adequate equation system (5) over M, consider the corre-
+sponding free module F and its submodule H. If (5) is viewed as a system over M ∗,
+then by construction, there is an extension ψ : F → M ∗ of φ : H → M ≤ M ∗. This
+means that (5) is solvable in M ∗. Hypothesis (β) implies that M is a summand
+19
+
+of M ∗. Hence ψ followed by the projection M ∗ → M yields a desired extension
+F → M of φ.
+(γ) ⇒ (α) In the next diagram, let the bottom row represent a pure extension
+of M by C, and the top row a free resolution of C. The map φ∗ exists because F
+is free (making the right square commute). It is evident that its restriction φ to
+H makes the left square commute. As H is tight in F, the pair {H, F} along with
+φ defines a system (5) of equations that is finitely solvable in M, due the purity
+of the bottom sequence. Thus (5) is an adequate system, and hence condition (γ)
+implies that there exists a map F → M that makes the maps in the upper triangle
+HFM commute. Then the bottom sequence splits, establishing (α).
+0 −−−−→ H −−−−→ F −−−−→ C −−−−→ 0
+φ
+�
+�φ∗
+���
+0 −−−−��� M −−−−→ N −−−−→ C −−−−→ 0
+□
+Evidently, divisible (i.e. injective) objects are pure-injective. Moreover, they
+contain a lot of pure-injective subobjects — as is shown by the following theorem.
+Theorem 7.2. Let D be a divisible object in C∗
+V . For every r ∈ V ×, the submodule
+D[r] = {d ∈ D | rd = 0} is a pure-injective object.
+Proof. From the isomorphism D/D[r] ∼= D and from p.d.D = 1 we infer that D[r]
+is tight in D. Let A be a pure subobject of object B, and ξ : A → D[r] a C∗
+V -map.
+ξ induces a map ξ′ : A/rA → D[r] that extends (by purity) to ξ′′ : B/rB → D.
+Evidently, Im ξ′′ ≤ D[r] as well, so the canonical map B → B/rB followed by ξ′′
+yields a desired extension B → D[r] of ξ.
+□
+Imitating the proof of Eklof–Mekler [5, Chap. V, Corollary 1.3], we verify:
+Theorem 7.3. Every object in C∗
+V is a pure subobject in a pure-injective object.
+Proof. Select any cardinal κ > max{|V |, ℵ0}. Given a module M ∈ C∗
+V , we define a
+continuous well-ordered ascending chain {Mσ | σ < κ} of length κ as follows. Start
+with M0 = M. If for some σ the modules Mρ of p.d.≤ 1 have been constructed
+for all ρ ≤ σ, then define Mσ+1 by adjoining to Mσ the unknowns (as additional
+generators) of every adequate equation system with defining relations given by
+the systems. It is readily checked that then p.d.Mσ+1 ≤ 1 as well, and Mσ will
+be tight and pure in Mσ+1 such that all the adequate equation systems over Mσ
+are solvable in Mσ+1. At limit ordinals, we take the union which will again have
+p.d.≤ 1 and will contain all previously constructed Mρs as tight pure submodules.
+It is straightforward to check that the union of the constructed chain will satisfy
+condition (γ), and thus it will be a pure-injective object containing M as a pure
+subobject. (The process can stop at systems with λ unknowns, where λ is any
+uncountable cardinal > |V |.)
+□
+The next proposition implies that the first Ulm-submodule of a pure-injective
+object is an injective object whenever its p.d. is ≤ 1.
+Proposition 7.4. The first Ulm-submodule M 1 = ∩r∈V ×rM of a pure-injective
+object M satisfies Ext1
+V (C, M 1) = 0 for all objects C.
+20
+
+Proof. Let {ri | i ∈ I} be a list of the non-unit elements of V ×. We have to show
+that for any given a ∈ M 1, for each rj the equation rjx = a is solvable for x ∈ M 1.
+For each j ∈ I, consider the following system of linear equations
+rjx = a,
+rixi = x
+(i ∈ I).
+An easy calculation confirms that the p.d. of this system is 1. Furthermore, since
+the chosen element a is divisible by rjri for all indices, each system is finitely
+solvable in M. By hypothesis, it has a solution in M.
+Clearly, each solution x
+belongs to M 1, so M 1 is a divisible submodule.
+□
+For every valuation domain V of global dimension ≥ 2, we exhibit an example
+of a pure-injective object in C∗
+V whose injective submodule is an object, but neither
+a subobject nor a summand in C∗
+V .
+Example 7.5. Let V be as stated. We form the following diagram with pure-exact
+rows and commutative squares.
+0 −−−−→ A −−−−→ B
+−−−−→ C −−−−→ 0
+�
+�
+���
+0 −−−−→ D −−−−→ B′ −−−−→ C −−−−→ 0
+���
+�
+�
+0 −−−−→ D −−−−→ B′′ −−−−→ C′ −−−−→ 0
+For the top row select a pure-exact sequence of torsion V -modules such that A, B
+are of p.d.1, and p.d.C = 2. We can make the selection such that the first Ulm-
+submodule of C is 0. Embed A in a divisible object D as a tight submodule, and
+get the middle row as a pure-exact sequence via pushout. The next step is the
+application of the embedding process of Theorem 7.3 to C (it works even if C is
+not an object) to obtain a pure extension C′ by a module H of p.d.1 such that C′
+has the property that its pure extensions by V -modules of p.d.≤ 1 are splitting.
+Since p.d.H = 1, there is a module B′′ making (though not uniquely) the bottom
+sequence pure-exact and the diagram commutative. The middle vertical arrows are
+injections and the projective dimensions of B, B′/B, B′′/B′ are all 1. Hence we
+have p.d.B′′ = 1 as well. Furthermore, as a pure extension of D by C′, the module
+B′′ has the property that its pure extensions by V -modules of p.d.≤ 1 are splitting.
+This means that B′′ is a pure-injective object in C∗
+V . Its injective submodule D has
+p.d.1, but, since p.d.C′ = 2, it is not tight in B′′, so it is not a summand in C∗
+V .
+□
+We close this section with an example and its corollary.
+Example 7.6. An explicit example of a pure-injective object is a direct sum
+C = ⊕i∈IV/V r for any non-unit r ∈ V × and any index set I. This follows from
+Proposition 5.4 (iii).
+□
+Consequently, we can state the following corollary (the case for finitely presented
+objects has been stated above in Theorem 5.1):
+Corollary 7.7. Every V/V r-homogeneous (r ∈ V ×) torsion module is Σ-pure-
+injective.
+21
+
+Proof. If D is any direct sum of copies of ∂V , then the submodule D[r] is V/V r-
+homogeneous, so a direct sum of copies of V/V r. Moreover, it is pure-injective by
+Example 7.6. Summands of D[r] as well as finite direct sums of pure-injectives are
+also pure-injective.
+□
+8.
+Cotorsion Modules
+We shall call an object C cotorsion if Ext1
+R(U, C) = 0 holds for all uniserial
+(i.e. rank one) torsion-free objects U. Evidently, it suffices to demand this only for
+countably generated uniserials. Readers familiar with cotorsion theory immediately
+recognize that this cotorsion concept corresponds to Warfield-cotorsion (where split-
+ting is required for extensions by all torsion-free modules). This claim will become
+even more transparent in light of the following general statement.
+Lemma 8.1. An object C ∈ C∗
+V is cotorsion if and only if it satisfies the equation
+Ext1
+V (A, C) = 0 for all torsion-free objects A.
+Proof. Definition settles the claim in one direction. For the ’only if’ part, assume
+that C is cotorsion and A is torsion-free of p.d.≤ 1. We know from Theorem 5.10
+that then A is the union of a continuous well-ordered ascending chain of torsion-free
+submodules Aα (α < κ) that are pure and tight in A such that all the quotients
+Aα+1/Aα are torsion-free of rank 1. We now refer to Eklof’s theorem (see [4]) on the
+extension by the union of a chain to conclude that A satisfies the quoted equation,
+as all the mentioned quotients in the chain satisfy it.
+□
+In order to characterize cotorsion objects in terms of solvability of systems of
+equations, consider a torsion-free uniserial object U.
+If it is not cyclic, then it
+is generated by a countable set {un | n < ω} such that rnun+1 = un for some
+rn ∈ V × for all n < ω. Therefore, an extension B of module C by U looks like
+B = ⟨C, bn (n < ω)⟩ with defining relations given by the equations rnbn+1−bn = cn
+for certain cn ∈ C. Clearly, C is cotorsion if and only if C is a summand of B for
+all permissible choices of the Us and the cns if and only if each consistent countable
+system of equations of the form
+(6)
+rnxn+1 − xn = cn
+with cn ∈ C (n < ω)
+is solvable in C. In the last case, a solution xn yields a complement to C in B: the
+submodule generated by the elements an = xn − bn (n < ω). This leads us to the
+following theorem.
+Theorem 8.2. An object C is cotorsion if and only if all consistent systems of
+linear equations of the form (6) constructed with torsion-free uniserial objects U
+are solvable in C.
+□
+More useful information about cotorsion objects is provided by the next result.
+Theorem 8.3. (i) All pure-injective objects in C∗
+V are also cotorsion objects.
+(ii) Every object is a subobject of a cotorsion object with torsion-free cokernel.
+Proof. (i) is an immediate consequence of the definitions, since all extensions by
+torsion-free V -modules are pure-extensions.
+(ii) This can be verified easily, just copy the proof of Theorem 7.3, using (6)
+in place of linear systems of p.d.≤ 1, mutatis mutandis. (By the way, the mere
+embedding property follows already from (i) and Theorem 7.3.)
+□
+22
+
+The next lemma is a convincing evidence that in some respect the cotorsion
+objects behave like ordinary cotorsion modules, though several relevant features
+are missing.
+Lemma 8.4. (i) Extension of cotorsion by cotorsion is again cotorsion.
+(ii) Modules of p.d.1 that are epimorphic images of cotorsion objects (modulo
+tight submodules) are likewise cotorsion objects.
+Proof. (i) is obvious.
+(ii) This is evident considering that if C → C′ is a surjective map, then for
+every module A of p.d.≤ 1, the induced map Ext1
+V (A, C) → Ext1
+V (A, C′) is also
+surjective.
+□
+In order to demonstrate that not all cotorsion objects are pure-injective, take e.g.
+a torsion object T whose first Ulm-submodule T 1 is a subobject, but not divisible.
+Then the embedding process mentioned in the proof of Theorem 8.3(ii) yields a
+cotorsion object T containing T as a tight submodule such that T /T is torsion-
+free. This T cannot be not pure-injective, because its Ulm-submodule contains its
+torsion submodule T 1 that is a subobject, but is not injective (cf. Theorem 7.4).
+(Another proof can be given by displaying an extension of a pure-injective by a
+pure-injective (that is necessarily cotorsion) which fails to be pure-injective.)
+We raise the following problem on cotorsion modules in C∗
+V .
+Problem 8.5. Are the Ulm-submodules of cotorsion modules cotorsion and the
+Ulm-factors pure-injective in C∗
+V as in the case of DVD?
+□
+Acknowledgment. We would like to thank Luigi Salce for his numerous helpful
+comments.
+Correction. (by L. Fuchs) Non-standard uniserial modules appear frequently in the study of
+modules over valuation domains, we could not avoid mentioning them in our study either.
+I
+would like to correct erroneous statements on them in the literature.
+In her very interesting
+papers on non-standard uniserials (Bull. Amer. Math. Soc. 25 (1991) and Contemporary Math.
+124 (1992)) B. Osofsky stated that non-standard uniserials were investigated because of their
+connection to Kaplansky’s problem on the existence of valuation rings that are not homomorphic
+images of valuation domains. This incorrect claim (with another mistaken statement) was restated
+in the review of Osofsky’s first article by R. G¨obel in Math. Reviews. The fact is that the problem
+of existence of non-standard uniserials was raised in 1980 by L. Salce during our joint investigation
+of modules over valuation domains, and it was him who named them ”non-standard”. S. Shelah
+was told about non-standard uniserials only at the Udine Conference in April 1984 just before
+the night he succeeded in establishing their existence. Then neither Shelah nor anybody else at
+the well-attended conference could claim that Kaplansky’s problem had been solved, since at this
+point nobody suspected that it was related to non-standard uniserials. The connection became
+known only three months later when we solved the Kaplansky problem, and the solution relied
+on non-standard uniserials (see the original solution published in [14]).
+23
+
+References
+[1] S. Bazzoni and L. Fuchs, On modules of finite projective dimension over valuation domains,
+Abelian Groups and Modules, CISM Courses and Lectures 287 (1984), 361–371.
+[2] S. Bazzoni, L. Fuchs, and L. Salce, The hierarchy of uniserial modules over a valuation
+domain, Forum Math. 7 (1995), 247–277.
+[3] S. Bazzoni and D. Herbera, Cotorsion pairs generated by modules of bounded projective
+dimension, Israel J. Math. 174 (2009), 119–160.
+[4] P.C. Eklof, Homological algebra and set theory, Trans. Amer. Math. Soc. 227 (1977), 207–225.
+[5] P.C. Eklof and A. Mekler, Almost Free Modules. Set-theoretic Methods (North Holland, 1990).
+[6] L. Fuchs, On a substitution property of modules, Monatsh. Math. 75 (1971), 198–204.
+[7] L. Fuchs, On projective dimensions of modules over valuation domains, in: Abelian Group
+Theory, Lecture Notes Math., 1006 (Springer, 1983), 589–598.
+[8] L. Fuchs, On divisible modules over domains, Abelian Groups and Modules, CISM Courses
+and Lectures 287 (1984), 341–356.
+[9] L. Fuchs, Note on modules of projective dimension one, in: Abelian Group Theory, Gordon
+and Breach Sci. Publ., (New York, 1987), 425–432.
+[10] L. Fuchs, On divisible modules over valuation domains, J. Algebra 110 (1987), 498–506.
+[11] L. Fuchs, Abelian Groups, Springer (Cham, Switzerland, 2015).
+[12] L. Fuchs, Torsion-free extensions of projective modules by torsion modules, J. Commut.
+Algebra, to appear (2023).
+[13] L. Fuchs and L. Salce, Separable torsion modules over valuation domains, Archiv Math. 41
+(1983), 17–24.
+[14] L. Fuchs and L. Salce, Modules over Valuation Domains, Lecture Notes in Pure and Applied
+Math. 97, Marcel Dekker Inc. (New York, Basel, 1985).
+[15] L. Fuchs and L. Salce, Modules over non-Noetherian Domains, Mathematical Surveys and
+Monographs 84, Amer. Math. Soc. (Providence, RI, 2001).
+[16] I. Kaplansky, Projective modules, Ann. Math. 68 (1958), 372–377.
+[17] E. Matlis, Divisible modules, Proc. Amer. Math. Soc. 11 (1960), 385–391.
+[18] B. Osofsky, Global dimension of valuation rings, Trans. Amer. Math. Soc. 127 (1967), 136–
+149.
+[19] T.S. Shores and W.J. Lewis, Serial modules and endomorphism rings, Duke Math. J. 41
+(1974), 889–909.
+[20] R.B. Warfield, Jr., Purity and algebraic compactness for modules, Pac. J. Math. 28 (1969),
+263–276.
+[21] R.B. Warfield, Jr., Decomposability of finitely presented modules, Proc. Amer. Math. Soc. 25
+(1970), 167–172.
+Institute of Mathematics & Informatics, Bulgarian Academy of Sciences, 1113 Sofia,
+Bulgaria
+Email address: [email protected]; [email protected]
+Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, USA
+and 1724 Pinetree Cir NE, Atlanta, Georgia 30329, USA
+Email address: [email protected]
+24
+

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+1
+Coffee stain effect on a fibre from axisymmetric
+droplets
+Marie Corpart1, Frédéric Restagno1 and François Boulogne1†
+1Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France.
+(Received xx; revised xx; accepted xx)
+The so-called coffee stain effect has been intensively studied over the past decades, but
+most of the studies are focused on sessile droplets. In this paper, we analyse the origin of
+the difference between the deposition of suspended particles in a sessile drop and in an
+axisymmetric drop deposited on a fibre. First, we model the shape of a drop on a fibre
+and its evaporative flux with some approximations to derive analytical calculations. Then,
+for pinned contact lines, we solve the hydrodynamics equations in the liquid phase under
+the lubrication approximation to determine the flow velocity toward the contact lines. We
+comment these results by comparison to a sessile drop of similar evaporating conditions,
+and we show that the substrate curvature plays a role on the contact line depinning, the local
+evaporative flux, and the liquid flow field. The competition between the advection and the
+Brownian motion indicates that the transport of the particles toward the contact line occurs
+in a volume localised in the close vicinity of the contact lines for a drop on a fibre. Thus, the
+fibre geometry induces a weaker accumulation of particles at the contact line compared to a
+sessile drop, leading to the more homogeneous deposit observed experimentally.
+Key words: Authors should not enter keywords on the manuscript, as these must be chosen by
+the author during the online submission process and will then be added during the typesetting
+process (see Keyword PDF for the full list). Other classifications will be added at the same
+time.
+1. Introduction
+The deposition of a material on surfaces in a controlled manner is a key aspect in a broad
+range of applications. Among the coating techniques, the deposition of suspended particles,
+which can also be solute particles such as salts and polymers, through the evaporation of
+the carrying liquid here called the solvent is a common approach (Routh 2013; Brutin &
+Starov 2018). When a volatile drop is deposited on a surface, particles are carried toward
+the contact line. This so-called coffee stain effect introduced by Deegan et al. (1997) and
+recently reviewed by Gelderblom et al. (2022) and Wilson & D’Ambrosio (2023), is the
+consequence of a radial flow induced by evaporation. To rationalise this phenomenon, the
+† Email address for correspondence: [email protected]
+arXiv:2301.04053v1  [physics.flu-dyn]  10 Jan 2023
+
+2
+fluid flow in the drop must be determined to explain the dynamics of the particle motion. The
+resolution of the liquid flow is subordinated to the knowledge of the drop shape resulting from
+capillary phenomena, which is to a good approximation a spherical cap, for a drop on a flat
+substrate, when its size is smaller than the capillary length. In addition, the derivation of the
+evaporative flux is necessary. The evaporative flux of a circular disk, i.e. a sessile drop with a
+vanishing contact angle, has been derived analytically by Cooke (1967), and generalisations
+to non-zero contact angle are also available in the literature (Sreznevsky 1882; Picknett &
+Bexon 1977). Thus, the fluid flow in the sessile drop has been derived theoretically through
+different contributions, e.g. Deegan et al. (2000); Hu & Larson (2005); Popov (2005); Zheng
+(2009); Larson (2014).
+In addition, Hamamoto et al. (2011) and then Marin et al. (2011a,b) revealed the rush-hour
+effect, which consists in an increase of the average particle velocity toward the contact line
+as the contact angle decreases in time. This particle velocity increase has an impact on the
+ordering of the particles in the final deposit (Marin et al. 2011a,b). The time evolution of
+the particle accumulation at the contact line has been satisfactorily predicted and measured
+by various authors (Popov 2005; Deegan 2000; Monteux & Lequeux 2011; Berteloot et al.
+2012; Larson 2014; Boulogne et al. 2017). The resulting deposition pattern forms a ring shape
+(Deegan 2000; Routh 2013; Brutin & Starov 2018) that triggered various investigations to find
+strategies for tuning, limiting, or suppressing this effect. For instance, studies focused on the
+liquid properties especially with solutal Marangoni flows (Kajiya et al. 2009; Sempels et al.
+2013; Kim et al. 2016; Pahlavan et al. 2021), on the substrate hydrophobicity (Gelderblom
+et al. 2011), on the substrate permeability (Boulogne et al. 2015), and on the multiple drops
+interaction (Pradhan & Panigrahi 2015; Wray et al. 2020, 2021). See Mampallil & Eral
+(2018) for a recent review.
+Most of the literature is focused on drops on flat surfaces. However, drops on fibres also
+represent a relevant situation for applications such as the drying of filters, clothing (Duprat
+2022), and insulating materials (Sauret et al. 2015). A liquid deposited on fibres can adopt
+a rich variety of morphologies. We distinguish the clamshell shape, which corresponds to
+a small drop wetting a portion of the fibre perimeter from the barrel shape where the drop
+wets the fibre as a pearl on a necklace (Chou et al. 2011). More complex liquid shapes can be
+obtained on fibrous networks. Fibres can be either crossed or parallel, which leads to a rich
+variety of equilibrium morphologies including liquid columns, distorted drops, and drops
+coexisting with columns (Protiere et al. 2012; Sauret et al. 2015).
+The deposition of particles dissolved in an evaporating clamshell drop on a fibre has
+already been investigated by Pham et al. (2002), where a similar behaviour to sessile drops
+has been observed. In the barrel case, the liquid morphology is remarkable due to the
+substrate geometry. The fibre curvature induces an inflection point of the interface and a drop
+aspect ratio of order of the unity even for a perfectly wetting fluid in contrast to sessile drops
+(Carroll 1976; Brochard-Wyart et al. 1991; Lorenceau et al. 2006). By minimising the surface
+energy, Carroll (1976) has obtained an analytical expression of the drop profile. In addition,
+Corpart et al. (2022) recently obtained by numerical calculations the evaporative flux of an
+axisymmetric drop on a fibre, demonstrating that the divergence of the evaporative flux is
+localised near the contact line. Thus, due to the differences in shape and evaporative flux
+induced by the substrate, we propose to investigate theoretically the particle deposition from
+a drop deposited on a fibre. In this paper, our approach will favour analytical calculations
+when possible and we will limit ourselves to the regime where the contact line is pinned on
+the substrate. Thus, we will rationalise the particle transport and the deposit left at the initial
+position of the contact lines, which will be compared to the well-established sessile drop.
+To do so, we investigate theoretically the transport of particles due to evaporation in a drop
+wetting a fibre in an axisymmetric barrel configuration. The obtained model is compared to
+
+3
+Figure 1: (a) Sketch presenting the notations of an axisymmetric drop of length 2𝐿 on a
+fibre of radius 𝑎. 𝑐sat is the saturation vapour mass concentration, the vapour mass
+concentration in air at the liquid air/interface, and 𝑐∞ is the mass concentration of vapour
+far from the droplet. (b) Comparison of the analytical solution obtained by Carroll (1976)
+of the dimensionless profile ℎ/𝑎 at 𝐿/𝑎 = 6.5 (solid lines), to the proposed ansatz (2.1)
+with 𝐿/𝑎 ≈ 6.7 (dotted lines) for 𝜃 = 0, 1, 5, 10, 15◦. The apex height is obtained by
+fitting Carroll’s profile by equation (2.1) and the length 𝐿/𝑎 = 6.7 is chosen to be the
+fitted length obtained for 𝜃 = 0. The dimensions chosen here corresponds approximately
+to a microlitre drop deposited on a glass fibre of a hundred micrometers radius. The curves
+are arbitrarily shifted for clarity.
+the case of a sessile drop in terms of volume loss dynamics, characteristic velocities, and
+efficiency of the particles transport toward the contact line. In Section 2, we introduce the
+phenomenological equations for the drop shape and the evaporative flux in order to obtain
+analytical predictions. Then, we derive the hydrodynamics equation in the liquid phase
+under the lubrication approximation to provide the velocity field toward the pinned contact
+line. In Section 3, we compare the drop on a fibre with the sessile drop, and we comment
+about the main differences between these systems. In Section 4, we present some qualitative
+experimental observations on the particle dynamics in both geometries that we compare to
+the theoretical investigations.
+2. Fluid flow of an evaporating drop on a fibre
+We consider a volatile drop of density 𝜌, viscosity 𝜂, surface tension 𝛾, and volume Ω, on a
+fibre of radius 𝑎. The contact angle of the liquid on the material, defined by Young’s law, is
+denoted 𝜃. On a horizontal single fibre, two configurations exist: a drop pierced by a fibre,
+the barrel shape, or a drop wetting a portion of the fibre circumference, the clamshell shape.
+As shown by Chou et al. (2011), this barrel shape is stable for a certain range of contact
+angles whose upper limit depends on the drop volume. For sufficiently small contact angles,
+this configuration is therefore realistic. Also, studies indicate that gravity, when sufficient,
+can off-centre the drop leading to an asymmetric shape (Chou et al. 2011; Gupta et al.
+2021). The role of gravity over capillarity can be quantified by the Worthington number
+Wo = 𝜌𝑔Ω/(𝜋𝛾𝑎) (Worthington 1885; Ferguson 1912). We assume a negligible effect of the
+gravity, i.e. Wo ≪ 1. The drop is therefore considered perfectly axisymmetric in this study.
+Thus, we adopt the cylindrical coordinate system shown in figure 1(a). The profile of
+the liquid-vapour interface is described by a function ℎ(𝑧, 𝑡). The evaporation of the liquid
+generates a volume variation, and therefore an internal flow. When the liquid is seeded with
+particles of radius 𝑏, the liquid flow can carry these particles. Our aim is to predict analytically
+the particle transport in the liquid phase. In the present work, we focus our interest on the
+regime for which the contact line is pinned, i.e. the length 𝐿 is constant. In practice, this
+
+4
+pinning occurs due to the contact angle hysteresis and the additional pinning force due to
+the particles accumulated at the contact line creating defects (Joanny & de Gennes 1984;
+di Meglio 1992; Boulogne et al. 2016). As evaporation proceeds, the drop height and the
+contact angle decrease.
+The profile of the liquid interface (Carroll 1976) and evaporative flux (Corpart et al. 2022)
+are particularly complex in this geometry. Consequently, to perform an analytical analysis, we
+will model the drop shape and the evaporative flux with phenomenological expressions, for
+which the validity will be discussed. In addition, we will write the hydrodynamics equations
+to derive the main component of the liquid flow, along the fibre, while the contact lines
+remain pinned. Then, we will be in position to discuss the particle transport in the drop.
+2.1. Profile of the liquid-vapour interface
+An analytical solution of the drop shape has been derived by Carroll (1976), but the elliptical
+integral involved in this solution would limit the analytical derivation of the present problem.
+Thus, we choose to describe the drop profile with an ansatz fitting Carroll’s solution
+ℎ(𝑧, 𝑡) = ℎ0(𝑡)
+�
+1 − 𝑧2
+𝐿2
+�2
+,
+(2.1)
+where ℎ0(𝑡) is the drop height at the apex 𝑧 = 0 and 𝐿 is the distance from the apex to the
+contact line (Fig. 1(a).). We consider the regime for which contact lines are pinned, so the
+length 𝐿 remains constant. The contact angle is defined by Young’s law and depends only on
+the liquid/solid properties. For an axisymetric drop on a fibre, Carroll (1976) demonstrated
+that the relation between the drop shape and the contact angle is not straightforward and in
+particular, the contact angle cannot be read directly from the slope between the interface
+and the surface of the fibre as it is for sessile drops. Typically, for contact angles between
+0 and 30◦, the slope of the interface vanishes in the vicinity of the contact line as shown
+in figure 1(b) where we compare drop profiles obtained from Carroll (1976) at 𝐿/𝑎 = 6.5
+to equation 2.1 for 𝜃 = 0, 5, 10, and 15◦. To describe the drop profile, we fit Carroll’s
+profile with equation 2.1 where 𝐿 and ℎ0 are the two adjustable parameters. To keep the
+wetted-length constant over the dynamics, we choose to set 𝐿 as the fitted length obtained
+for 𝜃 = 0, 𝐿/𝑎 ≈ 6.7. Then, we use the values of ℎ0 obtained by the fit to calculate drop
+profile with equation (2.1) (see dashed lines in Fig. 1(b)). As shown in figure 1(b), the
+proposed description of the drop profile is reasonably close to the analytical solution and
+describes its main features. We tested the validity of the ansatz for various drop profiles and
+we found a good agreement for drop with small contact angles, typically lower than 30◦ and
+dimensionless volume Ω/𝑎3 < 1000.
+The liquid volume is defined as
+Ω(𝑡) =
+∫
+𝑎+ℎ(𝑧,𝑡)
+𝑎
+∫ 2𝜋
+0
+∫
+𝐿
+−𝐿
+𝑟 d𝑟 d𝜃 d𝑧 = 32
+315𝜋𝐿ℎ0(𝑡) (21𝑎 + 8ℎ0(𝑡)) .
+(2.2)
+With equation (2.1) and equation (2.2), we can estimate the volume of the drop and compare
+it to the volume calculated by Carroll (1976) from his analytical description of the drop
+profile. The ansatz gives a good approximation of the drop volume with a relative error of
+about 10 % for the largest contact angle tested, 𝜃 = 15◦, for which the difference between the
+fit and the analytical profile is the largest.
+2.2. Evaporative flux
+The evaporation process is assumed to be limited by the diffusion of vapour in air (Cazabat
+& Guéna 2010), described by the vapour diffusion coefficient Dv. The steady-state regime is
+
+5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1 − z2
+L2
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+2.4
+2.6
+ve(z)
+v0e
+Figure 2: Local dimensionless evaporation velocity 𝑣e(𝑧)/𝑣0e as a function of 1 − 𝑧2/𝐿2
+where 𝑣0e is the evaporation velocity at the drop apex. The points are obtained from
+numerical simulations using finite element methods presented in Corpart et al. (2022) for
+𝑎 = 125 µm, 𝜃 = 10◦, Ω = 1 µL, which corresponds to 𝐿/𝑎 ≈ 6.7. The dashed line is
+equation (2.3) with the fitted parameters 𝛼 = 0.7 and 𝛽 = 0.1.
+reached on a timescale 𝐿2/Dv, which is, in most situations, short compared to the observation
+timescale. Thus, the vapour mass concentration field 𝑐(𝑟, 𝑧) is the solution of the Laplace
+equation △𝑐 = 0. The boundary conditions are 𝑐 = 𝑐sat at the liquid-vapour interface,
+𝑐 = 𝑐∞ far from the interface, and a no flux condition 𝒏 · ∇𝑐 = 0 on the solid-gas interface
+characterised by its normal 𝒏 (see Fig. 1(a)). Once the mass concentration field is obtained,
+the evaporation velocity at the liquid-vapour interface is defined as 𝑣e(𝑧) = Dv
+𝜌 𝒏 · ∇𝑐|ℎ(𝑧,𝑡),
+where the derivative is calculated at the liquid-vapour interface and 𝜌 is the density of the
+liquid.
+Corpart et al. (2022) have recently performed numerical simulations using finite elements
+method to determine the local evaporative flux of a drop on a fibre. In particular, this
+study revealed that the evaporation velocity differs from the well-known sessile drop in two
+aspects. First, the divergence is localised in the close vicinity of the contact line, and second,
+the contact angle has a weak effect on the evaporation velocity. A typical result of these
+computations is presented in figure 2. The localisation of the divergence near the contact line
+implies that the evaporation velocity cannot be written as a power law (Corpart et al. 2022).
+Instead, we propose to fit the results of the numerical simulations by
+𝑣e(𝑧) = 𝑣0
+e
+�
+𝛽
+�
+1 − 𝑧2
+𝐿2
+�−𝛼
++ 1 − 𝛽
+�
+,
+(2.3)
+where 𝛼 and 𝛽 are constants, independent of the contact angle, for 𝜃 < 20◦ (Corpart
+et al. 2022). The prefactor 𝑣0
+e = 𝑣e(𝑧 = 0) is the single parameter that depends on the
+environmental conditions and is proportional to Dv(𝑐sat − 𝑐∞)/𝜌. The total evaporative flux
+writes 𝑄e(𝑡) =
+∫
+𝑣ed𝑆 = 2𝜋
+∫ 𝐿
+−𝐿 𝑣e(𝑧) (𝑎 + ℎ(𝑧, 𝑡))d𝑧, which leads after integration to
+𝑄e(𝑡) = 2𝜋𝑣0
+e𝐿 (𝐴1𝑎 + 𝐴2ℎ0(𝑡)) ,
+(2.4)
+
+6
+with
+𝐴1 = √𝜋𝛽 Γ(1 − 𝛼)
+Γ( 3
+2 − 𝛼)
++ 2(1 − 𝛽),
+(2.5a)
+𝐴2 = √𝜋𝛽 Γ(3 − 𝛼)
+Γ( 7
+2 − 𝛼)
++ 16
+15 (1 − 𝛽),
+(2.5b)
+where Γ is the Gamma-function (Abramowitz & Stegun 1972).
+2.3. Hydrodynamics
+2.3.1. Lubrication approximation
+Now that the geometry and the evaporation dynamics are described, we analyse the flow in
+the liquid phase. By considerations of symmetries, the velocity field can be written in the
+cylindrical coordinates (𝑣𝑟 (𝑟, 𝑧), 𝑣𝑧(𝑟, 𝑧)). The continuity equation is
+1
+𝑟
+𝜕(𝑟𝑣𝑟)
+𝜕𝑟
++ 𝜕𝑣𝑧
+𝜕𝑧 = 0,
+(2.6)
+and the Navier-Stokes equations in the stationary regime are
+−𝜕𝑝
+𝜕𝑟 + 𝜂
+�1
+𝑟
+𝜕
+𝜕𝑟
+�
+𝑟 𝜕𝑣𝑟
+𝜕𝑟
+�
++ 𝜕2𝑣𝑟
+𝜕𝑧2 − 𝑣𝑟
+𝑟2
+�
+= 0,
+(2.7a)
+−𝜕𝑝
+𝜕𝑧 + 𝜂
+�1
+𝑟
+𝜕
+𝜕𝑟
+�
+𝑟 𝜕𝑣𝑧
+𝜕𝑟
+�
++ 𝜕2𝑣𝑧
+𝜕𝑧2
+�
+= 0,
+(2.7b)
+where 𝑝(𝑟, 𝑧) is the liquid pressure. The boundary conditions are no fluid slippage on the
+fibre and no stress at the liquid-vapour interface, i.e.
+𝑣𝑧(𝑟, 𝑧) = 0,
+on
+𝑟 = 𝑎
+and
+|𝑧| ⩽ 𝐿,
+(2.8a)
+𝜕𝑣𝑧
+𝜕𝑟 = 0,
+on
+𝑟 = 𝑎 + ℎ(𝑧, 𝑡)
+and
+|𝑧| ⩽ 𝐿.
+(2.8b)
+Equation 2.8b means that tracers are assumed to have no surfactant effect, due to their
+size. We apply the lubrication approximation to equations (2.7a, 2.7b), which is valid for
+ℎ0
+𝐿
+𝜌ℎ(𝑧,𝑡)𝑣𝑧
+𝜂
+≪ 1 (Batchelor 2000) and we get
+d𝑝
+d𝑧 = 𝜂
+�1
+𝑟
+𝜕
+𝜕𝑟
+�
+𝑟 𝜕𝑣𝑧
+𝜕𝑟
+��
+.
+(2.9)
+Based on the differential equation (2.9) and the boundary conditions (2.8a) and (2.8b), we
+can calculate the velocity field 𝑣𝑧
+𝑣𝑧(𝑟, 𝑧, 𝑡) = −1
+𝜂
+d𝑝
+d𝑧
+� (𝑎 + ℎ(𝑧, 𝑡))2
+2
+ln
+� 𝑟
+𝑟0
+�
+− 𝑟2
+4
+�
+,
+(2.10)
+with 𝑟0 = 𝑎 exp �−𝑎2/(2(𝑎 + ℎ(𝑧, 𝑡))2)�. We remark that the radial dependence of 𝑣𝑧 is not a
+quadratic function in contrast to the sessile drop that will be recalled thereafter in Section A.
+2.3.2. Mass conservation
+To fully obtain the fluid velocity 𝑣𝑧, the pressure gradient d𝑝/d𝑧 must be determined. To do
+so, we write the mass conservation over a slice between 𝑧 and 𝑧 + d𝑧,
+𝜕ℎ
+𝜕𝑡 +
+1
+2(𝑎 + ℎ)
+𝜕
+𝜕𝑧
+�
+(2𝑎ℎ + ℎ2) ¯𝑣𝑧
+�
++ 𝑣e(𝑧) = 0,
+(2.11)
+
+7
+where ¯𝑣𝑧(𝑧, 𝑡) is the velocity 𝑣𝑧(𝑟, 𝑧, 𝑡) averaged over a cross-section perpendicular to the
+fibre,
+¯𝑣𝑧(𝑧, 𝑡) =
+2
+2𝑎ℎ + ℎ2
+∫
+𝑎+ℎ(𝑧,𝑡)
+𝑎
+𝑣𝑧(𝑟, 𝑧, 𝑡) 𝑟 d𝑟.
+(2.12)
+Now, we can establish the relation between ¯𝑣𝑧 and 𝑣𝑧. First, we integrate equation (2.12)
+with equation (2.10) to express ¯𝑣𝑧 as a function of d𝑝/d𝑧. Equation (2.12) is written
+¯𝑣𝑧(𝑧, 𝑡) = −1
+𝜂
+d𝑝
+d𝑧 G(𝑎, ℎ(𝑧, 𝑡)),
+(2.13)
+where the geometrical function G(𝑎, ℎ(𝑧, 𝑡)) is given by
+G(𝑎, ℎ(𝑧, 𝑡)) =
+1
+4(2𝑎ℎ + ℎ2)
+�𝑎4
+2 + (𝑎 + ℎ)4
+�
+𝑎2
+(𝑎 + ℎ)2
+�1
+2 − ln 𝑎
+𝑟0
+�
++ ln 𝑎 + ℎ
+𝑟0
+− 1
+��
+.
+(2.14)
+Substituting the pressure derivative from (2.10) in (2.13) yields
+𝑣𝑧(𝑟, 𝑧, 𝑡) =
+ℎ¯𝑣𝑧
+G(𝑎, ℎ(𝑧, 𝑡))
+� (𝑎 + ℎ(𝑧, 𝑡))2
+2
+ln
+� 𝑟
+𝑟0
+�
+− 𝑟2
+4
+�
+.
+(2.15)
+In the next subsections, we derive the time evolution of the drop profile 𝜕ℎ/𝜕𝑡, which will
+be used along the evaporation velocity 𝑣e(𝑧) in the mass conservation (2.11) to obtain the
+average velocity ¯𝑣𝑧 as a function of the evaporation dynamics.
+2.3.3. Liquid evaporation
+First, we calculate the time derivative of the liquid profile, 𝜕ℎ/𝜕𝑡, which appears in equation
+(2.11). The time derivative of the drop profile defined in equation (2.1) considering a constant
+drop length 𝐿 gives
+𝜕ℎ(𝑧, 𝑡)
+𝜕𝑡
+= dℎ0
+d𝑡
+�
+1 − 𝑧2
+𝐿2
+�2
+.
+(2.16)
+In addition, the loss of liquid by evaporation compensates the total evaporative flux as
+dΩ/d𝑡 = −𝑄e(𝑡). Substituting the volume defined in equation (2.2), we have
+dℎ0
+d𝑡 = − 315
+32𝜋
+𝑄e(𝑡)
+𝐿(21𝑎 + 16ℎ0(𝑡)) ,
+(2.17)
+which fully defines the time derivative of the liquid profile.
+Integrating equation 2.17 from 0 to 𝑡 and ℎi to ℎ0 leads to
+16
+𝐴2
+(ℎi − ℎ0(𝑡)) + 𝑎(21𝐴2 − 16𝐴1)
+𝐴2
+2
+ln
+�
+𝐴1𝑎 + 𝐴2ℎi
+𝐴1𝑎 + 𝐴2ℎ0(𝑡)
+�
+= 315
+16 𝑣0
+e𝑡.
+(2.18)
+At the first leading order in (ℎi − ℎ0)/ℎi, we obtain the time variation of the height of the
+apex
+ℎ0(𝑡) ≈ ℎi − 315
+16
+𝐴1𝑎 + 𝐴2ℎi
+21𝑎 + 16ℎi
+𝑣0
+e 𝑡.
+(2.19)
+2.3.4. Mean liquid velocity
+Now, we derive the mean liquid velocity ¯𝑣𝑧. By substituting equations (2.16) and (2.17) in
+the mass conservation (2.11), we have
+
+8
+1
+2(𝑎 + ℎ)
+𝜕
+𝜕𝑧
+�
+(2𝑎ℎ + ℎ2) ¯𝑣𝑧
+�
+= 315
+32𝜋
+𝑄e(𝑡)
+𝐿(21𝑎 + 16ℎ0)
+�
+1 − 𝑧2
+𝐿2
+�2
+− 𝑣e(𝑧).
+(2.20)
+The integration of this differential equation from 0 to 𝑧 yields
+(2𝑎ℎ + ℎ2) ¯𝑣𝑧 = 315
+16𝜋
+𝑄e(𝑡)
+(21𝑎 + 16ℎ0)
+𝑧
+𝐿
+�
+𝑎 P1
+� 𝑧
+𝐿
+�
++ ℎ0 P2
+� 𝑧
+𝐿
+��
+− 2
+∫
+𝑧
+0
+(𝑎 + ℎ)𝑣ed𝑧, (2.21)
+where P1(𝑥) = 1 − 2
+3𝑥2 + 1
+5𝑥4 and P2(𝑥) = 1 − 4
+3𝑥2 + 6
+5𝑥4 − 4
+7𝑥6 + 1
+9𝑥8.
+The plane perpendicular to the fibre at 𝑧 = 0 is a plane of symmetry, thus ¯𝑣𝑧(𝑧 = 0) = 0.
+From equation (2.3), we can calculate the remaining integral of equation (2.21)
+2
+∫
+𝑧
+0
+(𝑎 + ℎ)𝑣e d𝑧 = 2𝑣0
+e𝑧
+�
+𝑎
+�
+𝛽 2𝐹1
+�1
+2, 𝛼; 3
+2; 𝑧2
+𝐿2
+�
++ (1 − 𝛽)
+�
++ ℎ0
+�
+𝛽 2𝐹1
+�1
+2, 𝛼 − 2; 3
+2; 𝑧2
+𝐿2
+�
++ (1 − 𝛽) P1
+� 𝑧
+𝐿
+���
+,
+(2.22)
+where the hypergeometric function 2𝐹1(𝑎, 𝑏; 𝑐; 𝑧) writes (Abramowitz & Stegun 1972)
+2𝐹1(𝑎, 𝑏; 𝑐; 𝑧) =
+∞
+∑︁
+𝑛=0
+(𝑎)𝑛(𝑏)𝑛
+(𝑐)𝑛
+𝑧𝑛
+𝑛! = 1 + 𝑎𝑏
+𝑐
+𝑧
+1! + 𝑎(𝑎 + 1)𝑏(𝑏 + 1)
+𝑐(𝑐 + 1)
+𝑧2
+2! + · · · .
+The combination of equations (2.21) and (2.22) provides the mean velocity ¯𝑣𝑧(𝑧, 𝑡). The
+velocity profile 𝑣𝑧(𝑟, 𝑧, 𝑡) is therefore obtained by a straightforward substitution in equation
+(2.15). This description of the fluid flow permits to analyse the induced particle transport,
+which will be discussed in Section 3.
+3. Comparison between geometries
+In this Section, we discuss the results obtained in Section 2 on a fibre that we compare to
+the well-established results for a drop on a flat surface (See Appendix A) (Deegan 2000;
+Popov 2005; Stauber et al. 2014; Larson 2014; Boulogne et al. 2017). To adopt versatile
+notations for the two geometries, let 𝑥 be the 𝑧 or 𝑟 coordinate for the fibre or the sessile case,
+respectively. Hence, the velocity toward the contact line is denoted ¯𝑣𝑥. Similarly, the length
+L denotes the length 𝐿 or 𝑅, respectively.
+First, we compare the time evolution of the drop profiles for pinned contact lines to reveal
+the effect of the substrate curvature on the duration of this regime of evaporation with respect
+to the total time of evaporation. Next, we analyse the fluid velocity toward the contact line
+and its efficiency to transport the particles through a Péclet number. Finally, we compute to
+total number of particles accumulated at the contact line during the pinned regime.
+3.1. Methodology
+To be able to compare the two geometries, a choice on the initial parameters has to be made.
+The possible parameters are 𝜃i the initial contact angle, L the wetted length, Ω(𝑡 = 0) the
+initial volume, and 𝑄e(𝑡 = 0), the initial evaporation rate. A common practice for comparison
+is to consider the same liquid-solid system such that the initial contact angle 𝜃i is fixed. In
+addition, a similar evaporation rate 𝑄e for both systems brings the advantage of a comparable
+driving force for the particle transport. With equation (2.4), we can calculate the evaporation
+
+9
+rate of a drop on a fibre whose geometry is given by equation (2.1) at 𝐿/𝑎 ≈ 6.7 and
+𝜃 = 𝜃i = 15◦ (see Fig. 1(b)). Numerical simulation in a previous study by Corpart et al.
+(2022) provides 𝑣0
+e for a water droplet in the initial geometry described here and evaporating
+in dry air (𝑐∞ = 0) at 20 ◦C (𝑐sat = 1.72×10−2 kg/s and Dv = 2.36×10−5 m2/s) (Lide 2008).
+We obtained 𝑣0
+e ≈ 4 × 10−7 m/s. From equation (A 4), we can calculate the wetted length
+of the sessile droplet having the same evaporation rate for the same ambient conditions. We
+find a similar lengthscale 𝑅 ≈ 𝐿. For instance, the barrel-shaped drop on a fibre of radius
+𝑎 = 125 µm has, initially, a wetted-length 𝐿 ≈ 8.4 × 10−4 m and a sessile drop evaporating
+at the same initial rate has a wetted length 𝑅 ≈ 8.9 × 10−4 m. We thus choose to make
+comparisons at the same wetted length 𝐿/𝑎 ≈ 6.7 which is given by fitting Carroll’s profile
+with equation (2.1) as explained in Section 2 (see Fig. 1(b)). For the same wetted length
+and the same initial contact angle, the initial volume of the droplet is different in the two
+geometries. For example, for 𝐿 = 𝑅 ≈ 8.4 × 10−4 m and 𝜃i = 15◦, the initial volume is
+Ω(𝑡 = 0) ≈ 0.1 µL for the sessile case and Ω(𝑡 = 0) ≈ 0.6 µL for the drop on fibre. The
+sessile drop has a smaller initial volume and thus a shorter lifetime than the drop on a fibre
+even if the initial evaporation rate is the same in the two geometries. Additionally, since the
+particle transport depends on the particle size, we consider a particle diameter 2𝑏 = 1 µm
+motivated by the large number of studies on the coffee-stain using micrometer-sized particles.
+To summarise, we choose to compare drops of same wetting length and same initial
+contact angle. This corresponds to drops having different initial volumes but the same initial
+evaporation rate for the two different geometries when evaporating in the same ambient
+conditions. In this framework, we can compare the mean velocities toward the contact line
+for the two systems. The competition between the particle transport and the Brownian motion
+will be rationalised by a Péclet number.
+3.2. Time evolution of the drop shapes
+We analyse the evolution of the drop shape in both configurations. The temporal evolution of
+drop heights, plotted in figure 3(c) are given by equation 2.18 (blue solid line) approximated
+by equation 2.19 (dashed line) for the drop on a fibre and equation A 2 (black line) for the
+sessile drop.
+The evolution of the drop shape for different contact angles for a sessile drop is illustrated
+in figure 3(a). Due to the particle accumulation at the contact line that increases the pinning
+force (Joanny & de Gennes 1984; di Meglio 1992; Boulogne et al. 2016), we consider that
+the depinning nearly occurs at a zero contact angle, which corresponds to a vanishing drop
+volume. Now, considering the drop on a fibre, we plot in figure 3(b) the drop shape on a
+fibre having the same wetted length and the same contact angles as the sessile drops in
+figure 3(a). By fitting the profiles of figure 3(b) with equation 2.1 we get the apex heights
+and the corresponding times by inserting them in equation 2.18. The results are represented
+by the circles in figure 3(c). This figure shows that the temporal evolution of the apex height
+of a drop on a fibre is well described by the approximated equation 2.19 (in dashed-line in
+Fig 3(c)) during the pinned-regime.
+From figures 3(a) and (b), we also get the height of the apex at the depinning ℎdep
+0
+=
+ℎ0(𝜃 = 0). In the example studied here, we get ℎdep
+0
+= 0 for the sessile drop and ℎdep
+0
+= 2.4𝑎
+(corresponding to ℎdep
+0
+= 0.8ℎi) for the drop on a fibre. From that we get the duration of
+the pinned regime 𝜏dep of the drop on a fibre by inserting ℎdep
+0
+into equation 2.18. In the
+example studied here, we find 𝜏dep ≈ 110 s. In the case of the sessile drop the volume tends
+to zero at the end of the pinned regime, meaning that 𝜏dep = 𝜏e (Eq. A 5) the lifetime of
+a sessile drop evaporating entirely at constant contact radius. In the example studied here
+
+10
+−5
+0
+5
+r/a
+−4
+−2
+0
+2
+4
+6
+h/a
+(a)
+θ = 0◦
+θ = 5◦
+θ = 10◦
+θ = 15◦
+−5
+0
+5
+z/a
+−4
+−2
+0
+2
+4
+h/a
+(b)
+0
+25
+50
+75
+100
+t (s)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+h0/hi
+(c)
+Fibre
+Sessile
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t/τdep
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+θ(◦)
+(d)
+θ = 0◦
+θ = 0.1◦
+θ = 1◦
+θ = 5◦
+θ = 10◦
+θ = 15◦
+θ = θi
+�
+1 −
+t
+τdep
+�
+Figure 3: (a) – (b) Evolution of the drop shape at different contact angles in (a) the sessile
+case (Eq. A 1) and (b) the fibre configuration for which the drop profile is described by
+elliptic integrals given by Carroll (1976). (c) Time evolution of the drop height at the apex
+ℎ0 normalised by the initial height ℎi for the two configurations. The black line
+corresponds to the sessile drop (Eq. A 2). The blue lines are for an axisymmetric drop on a
+fibre, the solid line is equation 2.18 and the dashed line is the approximation given by
+equation 2.19. (d) Dynamics of the contact angle 𝜃 for a sessile drop. (c) – (d) Circles
+represent the apex heights (c) and contact angles (d) of the profiles of drops on fibres
+obtained with our ansatz (Eq. 2.1) and plotted in figure 1(b). Here, the comparisons are
+performed for the same wetted length L/𝑎 ≈ 6.7 and an initial contact angle 𝜃i = 15◦.
+This corresponds to water on glass substrate of initial volume Ω(𝑡 = 0) ≈ 0.6 µL for the
+fibre configuration (𝑎 = 125 µm) and Ω(𝑡 = 0) ≈ 0.1 µL for the sessile case. The initial
+evaporation rate 𝑄e(𝑡 = 0) is nearly the same for the two configurations. For both
+geometries the ambient conditions are taken to be those of water evaporating in dry air at
+20◦C, 𝑐sat = 1.72 × 10−2 kg/m3, 𝑐∞ = 0 and Dv = 2.36 × 10−5 m2/s (Lide 2008). In this
+conditions, for a drop on a fibre (Ω = 1 µL, 𝜃 = 10◦ and 𝑎 = 125 µm) we obtained by
+numerical simulations 𝑣0e ≈ 4 × 10−7 m/s, 𝛼 = 0.7 and 𝛽 = 0.1 (Fig. 2).
+we find 𝜏e ≈ 90 s. To establish the relationship between time and contact angle, we plot on
+figure 3(d) (circles) the temporal evolution of contact angle obtained from the drop profiles
+represented in figure 3(b) for which we know ℎ0 and 𝜃. In solid black line, we represent
+𝜃(𝑡) = 𝜃i
+�1 − 𝑡/𝜏dep
+� valid for a sessile drop (Eq. A 2) which also describes very well the
+temporal evolution of the contact angle of a drop on a fibre. We thus find for both geometries
+that ℎ0 ∝ 𝜃 ∝ 𝑡.
+Moreover, as observed in figure 3(b, c), the relative variation of the drop height ℎ0 (and
+
+11
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x/L
+10−8
+10−6
+10−4
+10−2
+¯vx(m/s)
+(a)
+θ = 0.1◦
+θ = 1◦
+θ = 5◦
+θ = 10◦
+θ = 15◦
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x/L
+10−4
+10−1
+102
+105
+Pe
+(b)
+Figure 4: (a) Mean velocity of the flow toward the contact line ¯𝑣𝑥 as a function of the
+dimensionless coordinate 𝑥/L along the solid surface. The comparison is performed for
+water drops in both configurations with the same dimensionless wetted length L/𝑎 ≈ 6.7
+and various contact angles (see caption). Dashed lines correspond to ¯𝑣𝑟 (Eq. (A 6)), the
+average fluid velocity toward the contact line in sessile drops. Solid lines are ¯𝑣𝑧, the
+average fluid velocity in drop on fibre, given by equation (2.21). The dependence of ¯𝑣𝑧 in
+𝜃 is implicit and contained in the dimensions of the drop which are obtained by fitting
+Carroll’s drop profile with equation (2.1) for each contact angle as done on Fig. 1(b). (b)
+Péclet number Pe deduced from the mean velocity and drop profile (Eqs. (2.1) and (A 1))
+as a function of 𝑥/L. Solid lines correspond to drop on fibre and dashed lines to sessile
+drops having the same wetted length. The solid red line corresponds to Pe = 1. For both
+geometries the ambient conditions are taken to be those of water evaporating in dry air at
+20◦C, described in the paragraph 3.1 and in the figure 3.
+drop volume) during the pinned regime, is small for the drop on the fibre. During the pinned
+regime, a small amount of the initial volume has evaporated, which means that the duration
+of the pinned regime is short compared to the drop lifetime 𝜏dep ≪ 𝜏e. The remarkable
+difference with the sessile drop is the significant liquid volume remaining at a zero contact
+angle, the contact angle at which the depinning is supposed to occur. We can estimate the
+remaining volume at the end of the pinned regime from equation (2.2). For the geometry
+represented in figure 3(b), we find that only 35 % of the initial volume has evaporated before
+the contact line depinning.
+From this comparison, we can state that the geometry imposes constraints on the dynamics
+of the drop profile. In particular, in contrast to the sessile drop, only a small fraction of the
+liquid volume can evaporate in the constant wetted length regime. Also, on a fibre, a drop
+has two contact lines such that we expect that only one of the two lines depins. At this
+time, the lower quantity of evaporated liquid implies that only a fraction of the particles
+are accumulated at the contact lines and that the complementary fraction of particles is still
+dispersed in the liquid phase. However to compute the number of accumulated particles at
+the contact line the velocity field inside the drop needs to be described which is the object of
+the next section.
+3.3. Transport of particles toward the contact line
+3.3.1. Mean flow velocity toward the contact line
+In figure 4(a), we plot ¯𝑣𝑥 for both geometries, 𝑥 being either 𝑧 or 𝑟 according to the geometry.
+In the centre, 𝑥 = 0, the fluid velocity is equal to zero by symmetry. We observe that, at the
+beginning of evaporation, i.e. for 𝜃 = 15◦ the flow towards the contact line is one order of
+magnitude higher in the sessile drop in most of the radial positions. Near the contact line, the
+
+12
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t/τdep
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x/L
+Fibre
+Sessile
+x = x0
+x = x⋆
+Figure 5: Temporal evolution of the dimensionless positions of the advected fluid layer
+𝑥0/L (solid lines) and of 𝑥★/L (dashed lines) the positions beyond which the Péclet
+number is greater than unity. Blue lines are used for axisymmetric drops on fibres, and
+black lines for sessile drops. Here, the comparison between the two geometries is
+performed in the conditions described in the paragraph 3.1 and in figure 3.
+mean liquid velocity diverges due to the vanishing liquid thickness for both geometries. As
+the liquid evaporates i.e. 𝜃 decreases, the so-called rush-hour effect (Hamamoto et al. 2011;
+Marin et al. 2011a) occurs in the sessile drop, i.e. ¯𝑣𝑟 increases in time. However, ¯𝑣𝑧 remains
+nearly constant for a drop on a fibre. This difference is due to the curvature of the substrate
+that enables the existence of the peculiar axisymmetric barrel morphology, for which the
+variation of the drop profile remains limited when 𝜃 decreases (see Fig. 3(b)) unlike the case
+of the spherical cap on a flat substrate (Fig. 3(a)).
+3.3.2. Péclet number
+To describe the effective transport of particles toward the contact line, the action of the liquid
+flow on the particles must be compared to the Brownian motion. Particles are transported by
+the shear flow in the drop characterised by the mean shear rate ¯�𝛾 = ¯𝑣(𝑥, 𝑡)/ℎ(𝑥, 𝑡) and also
+by the Brownian motion D = 𝑘B𝑇/(6𝜋𝜂𝑏) where 𝑘B is the Boltzmann constant and 𝑇 the
+temperature. To compare these two competing forces, we introduce the mean Péclet number
+Pe defined as Pe = ¯�𝛾𝑏2/D (Bossis & Brady 1989).
+In figure 4(b), we plot the mean Péclet number Pe as a function of the dimensionless
+position along the solid surface 𝑥/L. As previously we observe that, initially, the mean
+Péclet number is comparable in the two geometries and diverges in the vicinity of the triple
+line where the liquid height vanishes. As the liquid evaporates the mean Péclet number
+strongly increases in the sessile case while it remains constant for the barrel-shaped drop on
+a fibre. Again, we attribute these differences to the difference of morphology between the
+droplets due to the curvature of the substrate.
+3.4. Number of particles accumulating at the contact line
+3.4.1. Advection of a slice of fluid
+We consider a fluid layer at a position 𝑥0 of an infinitesimal width d𝑥, which is advected
+toward the contact line at a velocity ¯𝑣𝑥(𝑥0, 𝑡). The advection of a fluid layer is described
+
+13
+by its position 𝑥0(𝑡) that satisfies (Deegan 2000; Popov 2005; Monteux & Lequeux 2011;
+Berteloot et al. 2012)
+d𝑥0
+d𝑡 = ¯𝑣𝑥(𝑥0, 𝑡).
+(3.1)
+For the sessile drop, the solution is recalled in Appendix A (Eq. A 9) and is represented in
+figure 5 in solid black line. For the fibre, the complexity of equation 2.21 makes us unable
+to find an analytical solution of the differential equation (3.1). Instead, we proceed to a
+numerical integration with odeint from scipy (Jones et al. 2001–). The solution is plotted in
+solid blue line in figure 5.
+We define 𝑥★ as the position along the solid surface for which Pe(𝑥★) = 1. For 𝑥 > 𝑥★, we
+have Pe > 1 such that the particles advection by the flow overcome their Brownian diffusion.
+The particles in the volume of liquid enclosed between 𝑥 = 𝑥★ and 𝑥 = L are advected by
+the flow and transported to the triple line.
+In figure 5, we plot in dashed lines 𝑥★/L as a function of the dimensionless time for the
+two geometries. At the beginning of the drying, 𝑥★/L is closed to unity for both geometries.
+This means that the region over which the particles are transported by the liquid flow,
+corresponding to the region between 𝑥 = 𝑥★ and 𝑥 = L, is localised in the close vicinity
+of the contact line at the beginning of evaporation. We even note that initially this region is
+smaller for a sessile drop than for a drop on a fibre 𝑟★/𝑅 < 𝑧★/𝐿. Thus, at the beginning
+of evaporation, the transport of the particles toward the contact line is more efficient for the
+drop on a fibre. However, as the sessile drop evaporates, the region boundary position 𝑟★/𝑅
+continuously decreases to reach zero at the end of the pinned regime (𝑡 = 𝜏dep). On the fibre,
+𝑧★/𝐿 remains nearly constant, close to unity. In other words, the width of the attraction zone
+in a sessile drop grows as the liquid evaporates. Progressively, this zone occupies the entire
+drop, which leads to transport of the majority of the suspended particles toward the contact
+line. This is not the case for the drop on a fibre for which the zones in which the particles are
+effectively transported by the flow to the contact lines remain small and located in the close
+vicinity of the triple line.
+The curves presented in figure 5 provide a comparison of the relative positions of 𝑥★
+(dashed lines) and 𝑥0 (solid lines). During most of the pinned contact line regime, the area
+bounded by the position of the advected fluid layer 𝑥0 includes small and large Péclet numbers
+domains (𝑥0 < 𝑥★ < L). Exceptions are noticed at short timescales after evaporation starts
+and at the end of the pinned regime for the sessile drop.
+3.4.2. Particle accumulation dynamics in the large Péclet domain – 𝑥★ < 𝑥0
+If 𝑥★ < 𝑥0, i.e. Pe > 1 between 𝑥0 and L, the particles in this layer are transported toward the
+contact line such that their number is conserved. On the fibre, from the initial concentration
+(number of particles per unit volume) 𝑐i and the initial liquid profile ℎ(𝑧, 𝑡 = 0), the number
+of particles 𝑁CL accumulated at each contact line can be written
+𝑁fibre
+CL (𝑡) = 𝑐i
+∫
+𝑎+ℎ(𝑧, 𝑡=0)
+𝑎
+∫ 2𝜋
+0
+∫
+𝐿
+𝑧0(𝑡)
+𝑟d𝑟 d𝜃 d𝑧,
+(3.2)
+which gives,
+𝑁fibre
+CL (𝑡) = 𝜋𝑐i ℎi 𝐿
+�
+ℎi P3
+� 𝑧0
+𝐿
+�
++ 2𝑎 P4
+� 𝑧0
+𝐿
+��
+.
+(3.3)
+with P3(𝑥) = − 𝑥9
+9 + 4𝑥7
+7 − 6𝑥5
+5 + 4𝑥3
+3 − 𝑥 + 128
+315 and P4(𝑥) = − 𝑥5
+5 + 2𝑥3
+3 − 𝑥 + 8
+15. The above
+equation is shown as dashed blue line in figure 6. The equivalent calculation for the sessile
+drop is recalled in Appendix A.1 (Eq. A 11) and is represented in dashed grey line in figure 6.
+
+14
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t/τdep
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+NCL/Ntot
+0
+25
+50
+75
+100
+t (s)
+0
+5000
+10000
+NCL
+Fibre
+z⋆ ≤ z0
+Sessile
+r⋆ ≤ r0
+Figure 6: Time evolution of the dimensionless number of particles at the contact line
+𝑁CL/𝑁tot. Blue lines represent axisymmetric drops on fibres and black lines sessile drops.
+Solid lines are obtained from equation 3.3 (fibre) or equation A 11 (sessile) when 𝑥0 ⩾ 𝑥★
+and from equation 3.5 (fibre) or equation A 13 (sessile) when 𝑥0 ⩽ 𝑥★. The dashed lines
+represent the results obtained under the assumption that all particles contained between 𝑥0
+and L are transported to the contact line and are plotted from Eq. (A 11) for a sessile drop
+(grey dashed line) and Eq.(3.3) for a drop on a fibre (blue dashed line). Here the
+comparison between the two geometries is performed in the conditions described in the
+paragraph 3.1 and in figure 3. The inset shows the temporal evolution of the number of
+particles 𝑁CL plotted in solid lines in the main figure for an initial particle concentration
+𝑐i = 1 × 108 particles/mL.
+3.4.3. Particle accumulation dynamics over small and large Péclet domains – 𝑥★ ⩾ 𝑥0
+Now, if positions 𝑥★ and 𝑥0 are swapped, the number of accumulated particles is decomposed
+from two contributions. The first contribution, from 𝑥★ to L is equivalent to equation
+3.2. In this large Péclet domain, all the particles are advected with an increasing particle
+concentration as evaporation proceeds. Between 𝑥0 and 𝑥★, the small Péclet number indicates
+that Brownian motion maintains the particle concentration uniform. Once the fluid layer
+reaches 𝑥★, particles are advected and concentrated as it is between 𝑥★ and L. Therefore,
+we evaluate the number of particles in the fluid layer at the position 𝑥★ with a particle
+concentration 𝑐i. The sum of these two contributions gives
+𝑁fibre
+CL (𝑡) = 𝑐i
+∫
+𝑎+ℎ(𝑧,𝑡=0)
+𝑎
+∫ 2𝜋
+0
+∫
+𝐿
+𝑧★ 𝑟d𝑟 d𝜃 d𝑧 + 𝑐i
+∫
+𝑎+ℎ(𝑧★,𝑡=0)
+𝑎
+∫ 2𝜋
+0
+∫
+𝑧★
+𝑧0(𝑡)
+𝑟d𝑟 d𝜃 d𝑧.
+(3.4)
+After integration, we obtain
+𝑁fibre
+CL (𝑡) = 𝜋𝑐i ℎi 𝐿
+������
+ℎi ��
+�
+P3
+� 𝑧★
+𝐿
+�
++
+�
+1 −
+� 𝑧★
+𝐿
+�2�4 � 𝑧★
+𝐿 − 𝑧0
+𝐿
+�
+��
+�
++2𝑎 ��
+�
+P4
+� 𝑧★
+𝐿
+�
++
+�
+1 −
+� 𝑧★
+𝐿
+�2�2 � 𝑧★
+𝐿 − 𝑧0
+𝐿
+�
+��
+�
+������
+.
+(3.5)
+The same calculation is performed in Appendix A.1 for the sessile drop.
+
+15
+Using the appropriate conditions according to the relative positions of 𝑥0 and 𝑥★ shown in
+figure 5, we plot in solid lines in figure 6 the dimensionless number of particles accumulated
+at the contact line. In the inset of figure 6, we plot the number of particles accumulated at
+the triple line over time for an initial concentration of 𝑐i = 1 × 108 particles/mL.
+Figure 6 shows that the classical calculation (Deegan et al. 2000; Berteloot et al. 2012;
+Popov 2005; Monteux & Lequeux 2011; Boulogne et al. 2017), made in the literature for a
+sessile drop, valid if all the particles contained between 𝑥0 and L are transported by the flow,
+leads to overestimate the number of particles accumulated at the triple line during the first
+part of the pinned regime. However, at the end of the drying the liquid height tends towards
+0 which leads to Pe > 1 in almost the entire drop (cf. Fig. 5) i.e. 𝑟★ ⩽ 𝑟0. The result is that
+almost all the particles are transported and deposited at the initial position of the contact line
+during drying, which corresponds to a typical density of 𝑁tot/2𝜋𝑅 ≈ 2 particles/µm in the
+final deposit. In practice, we must note also that the threshold value Pe = 1 is arbitrary and
+must be adjusted for a fine quantitative description.
+For a drop on a fibre, on the other hand, 𝑧★ is almost constant and therefore during most
+of the pinned regime, 𝑧★ > 𝑧0 which means that the classical calculation of equation 3.3,
+represented in blue dashed line in figure 6, overestimates the number of particles accumulated
+at the contact line. Indeed, taking into account the fact that Brownian diffusion dominates
+in the zone between 𝑧0 and 𝑧★, we obtain a number of particles accumulated at the edge of
+the drop that is ten times lower than the one obtained by considering that all the particles
+between 𝑧0 and 𝑧★ are advected by the flow.
+Figure 6 also shows that the particles contained in the drop on a fibre are transported toward
+the contact lines. At the end of the pinned regime, the number of particles accumulated at the
+initial positions of the triple lines of a drop on a fibre corresponds to 𝑁CL(𝜏dep)/2𝜋𝑎 = 0.2
+particles/µm, which is about 10 times lower than the sessile drop. As shown in the inset, the
+duration of the pinned regime is approximately the same in both geometries 𝜏fibre
+dep ≈ 110 s
+and 𝜏sessile
+dep
+≈ 90 s, but the rate of accumulation of particles at the contact line is lower in the
+drop on fibre than in the sessile drop. This lower rate can be attributed to the overall lower
+fluid velocity and the narrow size of the advection-dominated domain in the fibre geometry.
+Next we want to compare the results of the calculations with what is observed experimen-
+tally.
+4. Experimental observations
+4.1. Materials and method
+We used fluorescent particles of polystyrene (Lifetechnologies) of diameter 2𝑏 = 1 𝜇m,
+diluted in pure water to a concentration of 𝑐i = 1 × 108 particles/mL. The experiments are
+performed at 20 ◦C and at a relative humidity between RH = 30 % and RH = 47 %. The
+drops are deposited on the substrates using a micropipette (Eppendorf 0.1 - 2.5 𝜇L). Images
+are recorded by using a camera (ORCA-Flash4.0, Hamamatsu).
+For the sessile drop experiment, the drop is placed on a glass microscope slide washed with
+distilled water and soap and rinsed with acetone (Fisher, purity ⩾ 99,8 %) and with anhydrous
+ethanol (Carlo Erba). The observations are made from below using an inverted fluorescence
+microscope (IX83, Olympus) equipped with a 4× magnification objective (Olympus). For
+the drop-on-fibre experiments, fibres of radius 𝑎 = 125 𝜇m are supplied by Saint-Gobain and
+activated by a plasma generator (Electro-Technics Products) prior to the experiments. The
+drop is observed from the side with a custom horizontal fluorescence microscope equipped
+with a 5× magnification objective (Mitutoyo). The initial volumes are chosen to have the
+
+16
+Figure 7: Experimental observations of particle transport induced by evaporation. (a,b)
+Temporal evolution of a drop on a fibre (side view) and on a flat surface (bottom view),
+respectively. The initial wetted length is 𝑅 = 𝐿 ≈ 0.75 mm. The times framed in red
+correspond to the moment when the depinning of one of the contact lines occurs. (c,d)
+Photographs of the corresponding final deposit. The direction of gravity is shown in the
+pictures. The scale bars represent 0.2 mm and the yellow arrows indicate the initial
+position of the triple lines. Movies are provided in supplementary materials.
+same initial wetted length in both geometries, such that 0.7 𝜇L is deposited on the fibre and
+0.1 𝜇L on the microscope slide.
+4.2. Observations
+An example of the temporal evolution of the system is shown in figure 7(a) for a drop on
+a fibre and 7(b) for a sessile drop. The Worthington number associated to the drop on the
+fibre is Wo ≈ 0.2, validating the small effect of gravity on the drop shape as observed in
+figure 7(a).
+One of the main differences between the two geometries is that there is only one contact
+line in a sessile drop, whereas two independent contact lines exist in a drop on a fibre. In
+both cases, the contact line depinning from its initial position is indicated by red frames in
+figures 7(a) and 7(b). The exact depinning time 𝜏dep of one of the two triple lines of a drop on
+a fibre is difficult to determine experimentally due to the curvature of the substrate and the
+interface. However, it is observed that, on a fibre, the time during which the wetted length is
+constant is small compared to the total lifetime of the drop and represents about 1–10 % of
+the lifetime. This is not observed for a sessile drop where the triple line remains pinned for
+the majority of the drying time, i.e. 80–90 % of the lifetime. The lifetime of the sessile drop
+is shorter than the lifetime of the drop on a fibre because its initial volume is lower. Corpart
+et al. (2022) have shown that the evaporative flux of a barrel-shaped droplet on a fibre is
+
+17
+correctly approximated by the one of a spherical droplet in the same condition, such that
+the lifetime of a drop on a fibre can be estimated as 𝜌𝐿2/(2Dv[𝑐sat − 𝑐∞]). Comparing this
+result to the lifetime of a sessile drop defined in equation A 5 for the experimental condition
+tested here, we obtain 𝜏fibre
+e
+/𝜏sessile
+e
+≈ 8/(𝜋𝜃i) ≈ 10 for 𝜃i = 15◦, which is in good agreement
+with what is measured experimentally as we get 𝜏fibre
+e
+/𝜏sessile
+e
+≈ 11.
+The final deposition is shown in figure 7(c) and figure 7(d) for these two geometries. During
+the pinned regime of a drop on a fibre, it is observed that the areas over which particles are
+transported to the triple lines are small and remain localised in the vicinity of the contact
+line, so that few particles are deposited at the contact lines. Conversely, in a sessile drop
+evaporating at constant wetted length, we observe that the zone of attraction of the triple
+line progressively increases over time to finally invade the entire liquid. The particles are
+therefore mainly transported and deposited at the initial position of the contact line.
+When one of the two contact lines unpins of the fibre, which is beyond the proposed model,
+the particles continue to accumulate at the stationary contact line, while the movement of
+the other triple line generates a complex flow in the drop. This contact line can then anchor
+again and a complex alternation of the movement of the left and right lines can be noticed,
+leading to the typical final morphology of the deposit observed in figure 7(c).
+4.3. Comparison with the proposed model
+There is a qualitative agreement between the observations and the predictions of the
+calculation. Indeed, for the drop on a fibre we observe experimentally that the zone over
+which the particles are transported towards the triple line is small and remains localised at
+the edge of the drops during the pinned regime and that there are few particles accumulated
+at the contact line during the pinned regime. The duration of the pinned regime is also short
+compared to the lifetime of the drop and when the depinning occurs, there is a significant
+volume of liquid remaining on the fibre.
+However, we were unable to obtain a quantitative agreement between the theory and the
+experiments because of the difficulty of tracking the particles due to the curvature of the
+substrate and the liquid/air interface. In addition, the area where we can measure the velocity
+is located at the edge of the drop where the velocity diverges, so we cannot see any difference
+between the two geometries.
+5. Conclusions
+We conducted a theoretical investigation of the particle accumulation at the contact lines
+during the pinned contact line regime of an evaporating axisymmetric barrel-shaped drop
+on a fibre. First, to obtain analytical expressions, we defined a phenomenological equation
+for the drop shape that we compared to the exact solution. We used a phenomenological
+model for the evaporation velocity along the liquid-vapour interface, which is supported by a
+previous study (Corpart et al. 2022). Within the lubrication approximation, we calculated the
+velocity field toward the contact line. As the advection of particles competes with Brownian
+motion, we quantify the ability for the liquid flow to effectively transport the particles with
+a Péclet number. We compared our results to the well-known sessile drop geometry.
+In our analysis, we highlighted that the liquid morphology is strongly different for both
+systems due to the fibre curvature. A first consequence is that a large liquid volume remains
+on the fibre when one of the two contact lines unpins, whereas nearly all the sessile drop
+is evaporated. A second consequence is that the divergence of the evaporation velocity is
+localised in the close vicinity of the triple line contrarily to the sessile drop. From these two
+observations and our calculations, we have shown that the liquid flow velocity far from the
+contact line is order of magnitudes lower on the fibre, although the initial total evaporation
+
+18
+rates are similar. Nevertheless, the velocity field is not sufficient to obtain a description on
+the particle transport. As the advection of particles competes with Brownian motion, a Péclet
+number indicates how effective the particle transport is. In a sessile drop, the domain where
+advection dominates diffusion grows in time, until a full invasion of the drop at the final stage,
+which is in perfect agreement with the common observation of an outward radial motion of
+particles. On the fibre, the situation is strikingly different: advection remains located in a
+small region near the contact line for the same variation of the contact angle. Therefore the
+calculated rate of particle accumulation at the contact line is lower in a drop on a fibre than
+in a sessile drop.
+As a result, the number of particles accumulated at the contact line at the end of the pinned
+regime on a fibre is weaker. A unique and rich feature of the fibre geometry is the existence
+of two contact lines, topologically disconnected. One of them remains pinned and particles
+keep accumulated in time, while the other one recedes, which brings an interesting dynamics,
+more complex than the sessile drop. The evaporation dynamics continues with a succession
+of contact line pinning-depinning, leading to the succession of ring-like deposits shown in
+figure 7(c). The fine description of the final pattern requires the modelling of the receding
+contact line (Freed-Brown 2014). Elucidating the receding dynamics of the contact lines
+induced by evaporation coupled with the particle deposition is an important consideration
+for future studies.
+Supplementary data. Supplementary material and movies are available at
+Acknowledgements. We thank Saint-Gobain and ANRT for funding this study and J. Delavoipière and M.
+Lamblet for useful discussions.
+Declaration of interests. The authors report no conflict of interest.
+Appendix A. Fluid flow of an evaporating sessile drop
+In this Appendix, we recall some results formerly obtained on the evaporation of a sessile
+drop with a pinned contact line. These results will be used for quantitatively comparing the
+evaporation of sessile drop and a drop on a fibre in the next Section.
+We consider a drop of volatile liquid of the same properties as in the main text and we recall
+some results in the former works of Deegan (2000); Popov (2005); Stauber et al. (2014);
+Larson (2014); Boulogne et al. (2017) and summarised recently by Gelderblom et al. (2022).
+The drop is sitting on a flat surface with a contact angle 𝜃 and a constant contact radius 𝑅.
+We consider small volume such that the gravitational effects are negligible and the geometry
+of the liquid is well described by a spherical cap. We assume a small contact angle, which
+simplifies the description of the drop shape and the evaporative flux. Thus, the drop profile
+is
+ℎ(𝑟, 𝑡) = ℎ0(𝑡)
+�
+1 − 𝑟2
+𝑅2
+�
+,
+(A 1)
+where ℎ0(𝑡) ≈ 𝑅𝜃(𝑡)/2 is the height at the apex. In the pinned regime,
+ℎ0(𝑡) = ℎi
+�
+1 −
+𝑡
+𝜏dep
+�
+,
+(A 2)
+with ℎi the initial height of the apex and 𝜏dep the duration of the evaporation process in the
+pinned regime. For a sessile drop, the receding contact angle tends to zero, in particular due
+to the presence of particles (Joanny & de Gennes 1984; di Meglio 1992; Boulogne et al.
+2016). Therefore, we can estimate with a good approximation that 𝜏dep ≈ 𝜏e, the lifetime of
+the drop.
+
+19
+In these conditions, the evaporation velocity is
+𝑣e(𝑟) = 𝑣0
+e
+�
+1 − 𝑟2
+𝑅2
+�−1/2
+,
+(A 3)
+with the characteristic evaporation velocity 𝑣0
+e = 2Dv(𝑐sat − 𝑐∞)/(𝜋𝜌𝑅), where Dv is the
+diffusion coefficient of the vapour in the air. Then, the total evaporative flux is
+𝑄e = 4Dv(𝑐sat − 𝑐∞)𝑅.
+(A 4)
+From equation (A 4), we can write the evaporative time assuming that all the liquid evaporates
+during the pinned-regime
+𝜏e =
+𝜋𝜌ℎi𝑅
+8Dv(𝑐sat − 𝑐∞) .
+(A 5)
+Under the lubrication approximation, the mean radial velocity is
+¯𝑣𝑟 (𝑟, 𝑡) = 𝑣0
+e
+𝑅2
+𝑟ℎ(𝑟, 𝑡)
+��
+1 − 𝑟2
+𝑅2
+�1/2
+−
+�
+1 − 𝑟2
+𝑅2
+�2�
+,
+(A 6)
+and the radial velocity field corresponds to half of a Poiseuille flow, given by
+𝑣𝑟 (𝑟, 𝑧, 𝑡) = 3
+2
+𝑅2𝑣0
+e
+𝑟ℎ(𝑟, 𝑡)3
+��
+1 − 𝑟2
+𝑅2
+�1/2
+−
+�
+1 − 𝑟2
+𝑅2
+�2� �
+𝑧2 − 2ℎ(𝑟, 𝑡)𝑧
+�
+.
+(A 7)
+A.1. Particle accumulation dynamics
+The number of particles 𝑁CL(𝑡) accumulating at the contact line is the sum of the particles
+contained in the volume between 𝑟0(𝑡) and 𝑅 where 𝑟0(𝑡) is defined as
+d𝑟0
+d𝑡 = ¯𝑣𝑟 (𝑟0(𝑡), 𝑡),
+(A 8)
+which leads after integration to
+𝑟0(𝑡)
+𝑅
+=
+�
+�
+�
+1 −
+�
+1 −
+�
+1 − 𝑡
+𝜏𝑒
+�3/4�2/3
+.
+(A 9)
+As presented in Section 3.4.2, the number of accumulated particles in the case 𝑟★ < 𝑟0
+writes
+𝑁sessile
+CL
+(𝑡) = 2𝜋𝑐i
+∫
+𝑅
+𝑟0(𝑡)
+ℎ(𝑟′, 𝑡 = 0) 𝑟′d𝑟′,
+(A 10)
+where 𝑐i is the initial particle concentration.
+𝑁sessile
+CL
+(𝑡) = 2𝜋𝑐iℎi𝑅2
+�1
+4 − 1
+2
+𝑟0(𝑡)2
+𝑅2
++ 1
+4
+𝑟0(𝑡)4
+𝑅4
+�
+,
+(A 11)
+For the other case where 𝑟★ > 𝑟0, we have
+𝑁sessile
+CL
+(𝑡) = 𝑐i
+∫
+𝑅
+𝑟★
+∫ 2𝜋
+0
+∫ ℎ(𝑟,𝑡=0)
+0
+𝑟d𝑟 d𝜃 d𝑧 + 𝑐i
+∫ 𝑟★
+𝑟0
+∫ 2𝜋
+0
+∫ ℎ(𝑟★,𝑡=0)
+0
+𝑟d𝑟 d�� d𝑧, (A 12)
+
+20
+which writes after integration:
+𝑁sessile
+CL
+(𝑡) = 2𝜋𝑐iℎi𝑅2
+�
+1
+4 − 1
+4
+𝑟★(𝑡)4
+𝑅4
+− 1
+2
+𝑟2
+0(𝑡)
+𝑅2
+�
+1 − 𝑟★(𝑡)2
+𝑅2
+��
+.
+(A 13)
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+

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+arXiv:2301.08705v1  [nlin.SI]  20 Jan 2023
+Multi-soliton solutions of the sine-Gordon equation with
+elliptic-function background
+Daisuke A. Takahashi1,2∗
+1Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1,
+Yokohama, Kanagawa 223-8521, Japan
+2Department of Physics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551,
+Japan
+E-mail: [email protected]
+Abstract.
+The multi-soliton solution of the sine-Gordon equation in the presence of elliptic-
+function background is derived by the inverse scattering method.
+The key tool in our
+formulation is the Lax pair written by 4×4 matrix differential operators given by Takhtadzhyan
+and Faddeev in 1974, which enables us to use the conventional form of the integral
+representation of the Jost solutions and Krichever’s theory of commuting differential operators.
+As a by-product we also provide generalized orthogonality and completeness relations for
+eigenfunctions associated with indefinite inner product. The multi-soliton solution is expressed
+by a determinant of theta functions and the shift of the background lattice due to solitons is
+also determined using addition formula. One kink and one breather solutions are presented by
+animated gifs.
+1. Introduction
+The sine-Gordon (SG) equation has a wide variety of applications in physics and other natural
+sciences. For example, it appears as a continuum limit of the Frenkel-Kontorowa model
+describing the commensurate-incommensurate transition (Ref. [1], chapter 6 and Ref. [2],
+chapter 10), the effective model for the chiral magnet and soliton propagation on it [3, 4],
+and kinks in the long Josephson junction of superconductors [5, 6, 7]. The model where the
+partial derivative is changed from hyperbolic to elliptic type has also been solved and applied
+to vortices and dislocations in spatially two-dimensional systems [8, 9, 10]. The rogue-wave
+solutions with unstable background has been recently investigated [11]. From the view of
+the theory of classical integrable systems, the SG equation is one of the earliest equations
+formulated via the zero-curvature expression, known as the Ablowitz-Kaup-Newell-Segur
+(AKNS) formalism [12, 13]. The analogical integrable models with higher group symmetries
+have been also investigated (Ref. [14] and references found in Ref. [15], part II, chapter I, §8).
+In some of the above-mentioned works, the multi-soliton excited states not only for
+uniform background but also for oscillating non-uniform background attract considerable
+physical attention. While the multi-soliton solutions of an integrable equation in the presence
+of the elliptic-function, or more general quasiperiodic Riemann theta function background,
+can be in principle obtained by taking a special limit of the finite-zone quasiperiodic
+solutions [16], the procedure is often complicated. Furthermore, in physical applications, the
+eigenfunctions of the Lax pair, which is necessary in construction of soliton solutions, often
+∗ The current primary affiliation is 2.
+
+2
+play additional roles in, e.g., calculation of dispersion relations of linearized waves and linear
+stability analysis. Therefore, constructing the soliton solutions with non-uniform background
+not from a degenerate case of quasiperiodic solutions but by adding solitons to the stationary
+background using eigenfunctions of the Lax pair via various conventional methods provides
+helpful by-products in investigation of physical phenomena.
+Motivated by these circumstances, in this paper, we derive the multi-soliton solution
+of the SG equation in the presence of elliptic-function background. Though this problem
+itself has been solved in many references including the above-mentioned ones, we believe
+that the following features (i)-(iii) will bring some new light. (i) We use the Lax formalism
+based on 4 × 4 matrix differential operator by Takhtadzhyan and Faddeev [17], which is a
+variant of the Lax pair written by an integral operator [18]. While the famous treatment
+of the SG equation in soliton theory is the zero-curvature expression using 2 × 2 matrices
+which depend on the spectral parameter [12, 13], the use of this method has the following
+advantages: (a) the theory of commuting differential operators [19] can be applied and (b)
+the common integral representation of Jost solutions can be used without modification so we
+do not need trial and error to find suitable form. (ii) The final expression of the multi-soliton
+solution, which is obtained by formulating the inverse scattering method (ISM) and solving
+the Gelfand-Levitan-Marchenko (GLM) equation, is compactly summarized as a determinant
+of theta functions, and its asymptotic form is also derived using the addition formulas in
+Ref. [20]. (iii) The orthogonal and completeness relations of eigenfunctions arising from the
+indefinite inner product is discussed in detail, which is necessary in formulation of the ISM
+and determination of the possible emerging patterns of discrete eigenvalues corresponding
+to solitons.
+This treatment is regarded as a generalization of the corresponding finite-
+dimensional linear algebra [21] to continuous space and differential operators.
+The organization of this paper is as follows. In Sec. 2, we write down the Lax pair of the
+SG equation. In Sec. 3, we point out that the Lax operators are “σ-self-adjoint”, and introduce
+the associated indefinite inner product which defines the generalized orthogonal relations to
+eigenfunctions. In Sec. 4, we derive simultaneous eigenfunctions of the Lax pair for the
+stationary soliton lattice solution, and identify the corresponding algebraic curve, based on
+the theory of commuting differential operators [19]. The eigenvalues and eigenfunctions are
+parametrized by uniformization variable on torus, and their symmetries are also presented.
+In Sec. 5, we define left and right Jost solutions and the scattering matrix for the system
+where the background potential asymptotically tends to the stationary soliton lattice at spatial
+infinities. In Sec. 6, we introduce the integral representation of the Jost solution, and present
+the GLM equation which determines the kernel function of the integral representation from the
+scattering data. In Sec. 7, we solve the GLM equation for reflectionless case and determine the
+multi-soliton solution, which is expressed by determinant of theta functions. The phase shift
+of the background lattice is also found. In Sec. 8, by solving the time-dependent Lax equation,
+we determine the time-evolution of the scattering matrix, and using this, we derive the time-
+dependent multi-soliton solution for the SG equation. In Sec. 9, we provide the constraint
+that the discrete eigenvalues must satisfy in order for the resultant solution to become real
+and bounded. In Sec. 10, we address the gif animations of the soliton solutions generated
+by Mathematica.
+The method of visualization and used numerical values are presented.
+Sec. 11 is devoted to summary and discussion. The appendices discuss several technical
+details.
+In Appendix A, the integration formula necessary to obtain the eigenfunction is
+presented. In Appendix B, we show the detailed derivation of the eigenfunctions given in
+Sec. 4. In Appendix C, we derive the completeness relation for eigenfunctions of the soliton
+lattice potential. There, we also discuss several technical important points on, e.g., the zero-
+norm eigenfunction defined by indefinite inner product and the necessity of the expression
+
+3
+by meromorphic integrand. In Appendix D, we provide the detailed derivation of the GLM
+equation. In Appendix E, we calculate the determinant of theta functions appearing in the
+asymptotic form of the soliton solution using addition formulas.
+2. Lax pair
+We write an n × n identity and zero matrix as In and On. Let σ1, σ2, and σ3 be the Pauli
+matrices. The Lax pair and the Lax equation for the SG equation is given by [17]
+4i∂ ˆL
+∂t = [ ˆL, ˆB],
+(2.1)
+ˆL = −4i∂x
+�σ3
+O2
+�
++
+� iσ1w
+e−iφσ2/2
+e−iφσ2/2
+O2
+�
+,
+w = φx + φt,
+(2.2)
+ˆB = 4i∂x
+�I2
+−I2
+�
+− 2
+�
+σ3e−iφσ2/2
+e−iφσ2/2σ3
+�
+.
+(2.3)
+Equation (2.1) then reduces to the SG equation
+φtt − φxx + sin φ = 0.
+(2.4)
+Here we slightly changed prefactors of operators and choice of the SU(2) basis in the way
+(σ1, σ2, σ3) → (σ1, σ3, −σ2) from the original work Ref. [17]. This makes no essential
+difference in formulation.
+3. Symmetry of the Lax operator and orthogonality of eigenfunctions
+Henceforth we simply write σ := σ3 ⊕ σ3 = I2 ⊗ σ3. Then, ˆL and ˆB are both “σ-self-
+adjoint”[21], i.e.,
+σ ˆL†σ = ˆL,
+σ ˆB†σ = ˆB.
+(3.1)
+Therefore the orthogonal and completeness relations of eigenfunctions of these operators are
+defined through the indefinite inner product, called “σ-inner product” in [21] : (f1, f2)σ :=
+�
+dxf †
+1 σf2. The correct identification of the completeness relation is important in derivation
+of the GLM equation (Appendix C and Appendix D).
+Following the established procedure of the ISM, we first consider the eigenvalue problem
+of ˆL. Since the highest-order coefficient matrix of ˆL is given by σ3 ⊕ O2, which obviously
+has rank 2, there exist two linearly independent eigenfunctions for a given eigenvalue λ. They
+can be written in the form f =
+�
+g
+λ−1e−iφσ2/2g
+�
+with g a two-component column vector. If we
+rewrite the equation with respect to g using the light-cone coordinate x′ = x+t
+2 , t′ = x−t
+2 , the
+famous 2 × 2 AKNS form [12] is reproduced. Let f1 =
+�
+g1
+1
+λ e−iφσ2/2g1
+�
+and f2 =
+�
+g2
+1
+λ e−iφσ2/2g2
+�
+be
+eigenfunctions of ˆL with eigenvalues λ1 and λ2. Then, we can show
+4i
+�
+f †
+1 (I2 ⊕ O2)f2
+�
+x = (λ∗
+1 − λ2)f †
+1 σf2.
+(3.2)
+Therefore, if λ∗
+1 � λ2, then we have
+�
+dxf †
+1 σf2 = 0, which is the above-mentioned
+orthogonality.
+On the other hand, when λ∗
+1 = λ2, we obtain f †
+1 (I2 ⊕ O2)f2 = g†
+1g2 =
+(x-independent). We can also show that if λ1 = λ2, det(g1, g2) does not depend on x, which
+
+4
+can be more easily shown using the AKNS form.
+In addition to Eq. (3.1), ˆL further has the following symmetry:
+(I2 ⊗ σ2) ˆL(I2 ⊗ σ2) = ˆL∗,
+(σ3 ⊗ σ2) ˆL(σ3 ⊗ σ2) = − ˆL,
+(3.3)
+which immediately means
+ˆLf = λf
+↔
+ˆL[(I2 ⊗ σ2)f ∗] = λ∗(I2 ⊗ σ2)f ∗
+↔
+ˆL[(σ3 ⊗ σ2)f] = −λ(σ3 ⊗ σ2)f
+↔
+ˆL[(σ3 ⊗ I2)f ∗] = −λ∗(σ3 ⊗ I2)f ∗.
+(3.4)
+Thus the eigenvalues λ, λ∗, −λ, −λ∗ always appear simultaneously.
+ˆL also has a little unfamiliar symmetry:
+ˆL f = λf
+↔
+˜L[(σ2 ⊗ σ3)f] = −λ−1(σ2 ⊗ σ3)f,
+(3.5)
+˜L := an operator such that w in ˆL is replaced by ˜w = 2φx − w.
+(3.6)
+We thus find the identity
+ˆL−1 = −(σ2 ⊗ σ3) ˜L(σ2 ⊗ σ3).
+(3.7)
+It is unusual that the inverse of a matrix differential operator can be written down explicitly,
+and furthermore, its expression does not include an integral operator. Such situation seems to
+occur when the highest-order coefficient matrix is not full-rank, which is −4iσ3 ⊕ O2 for the
+present ˆL.
+In particular, for the stationary solution of the SG equation φt = 0, we find w = ˜w = φx
+and hence ˆL = ˜L. In this case ˆL−1 = −(σ2 ⊗ σ3) ˆL(σ2 ⊗ σ3) holds and the eigenvalues λ
+and −λ−1 appear in pairs. We, however, again emphasize that this relation does not hold for
+general w � φx, and therefore when we consider the scattering matrix S in the ISM in Sec. 5,
+we must not impose this symmetry to ˆL.
+4. Soliton lattice solution, Riemann surface, and eigenfunctions
+Let us consider the eigenfunction for the stationary solution, which we henceforth write
+φ0(x). Since the Lax pair commutes [ ˆL, ˆB] = 0 for the stationary solution, we consider the
+simultaneous eigenfunction:
+ˆLf0 = λf0,
+ˆBf0 = ωf0.
+(4.1)
+We first determine φ0(x). The first integral for the stationary SG equation is
+φ2
+0x + 2 cos φ0 = 4
+m − 2
+↔
+( 1
+2φ0x)2
+1 − m sin2 φ0−π
+2
+= 1
+m.
+(4.2)
+Integrating this equation once again yields the soliton lattice solution as
+φ0(x) = π + 2 am
+�
+x
+√m
+���m
+�
+,
+(4.3)
+φ′
+0(x) =
+2
+√m dn
+�
+x
+√m
+���m
+�
+,
+cos φ0(x)
+2
+= − sn
+�
+x
+√m
+���m
+�
+,
+sin φ0(x)
+2
+= cn
+�
+x
+√m
+���m
+�
+.
+(4.4)
+Here and henceforth, we use the notations of elliptic functions in the Abramowitz-Stegun
+book [22] unless otherwise noted. If 0 < m < 1, it is the rotating solution. If m = 1, it
+
+5
+represents the stationary one-kink solution, and if m > 1, it is an oscillating solution. In
+this paper we only consider the rotating background 0 < m < 1. Generally, the pair of
+commuting differential operators satisfies an algebraic relation P( ˆL, ˆB) = 0 [19]. Strictly
+speaking, Krichever’s original paper [19] only consider the case where the highest-order
+coefficient matrices of the matrix differential operators are invertible, but similar nature can
+be seen even if this assumption is not satisfied and an algebraic curve can be defined in many
+cases. In the present case, P(λ, ω) = 0 gives a genus-one curve, i.e., an elliptic curve:
+λ2ω2 = λ4 + 2
+� 2
+m − 1
+�
+λ2 + 1.
+(4.5)
+It can be parametrized by elliptic functions:
+λ(z) = −i √m sn(iz|m) cn(iz|m)
+dn(iz|m)
+= i √m sn(iz|m) sn(iz − K|m),
+(4.6)
+ω(z) = sn2(iz|m) − sn2(iz − K|m)
+λ(z)
+,
+(4.7)
+where K = K(m) and K′ = K(1 − m). The eigenfunction f0(x, z) for the above parametrized
+eigenvalue λ = λ(z) and the corresponding crystal momentum k(z) is
+k(z) = Z(2iz + iK′|m) + Z(2iz − iK′|m)
+4i √m
+,
+(4.8)
+f0(x, z) = eik(z)xΘ4(0)
+2Θ4( x
+√m)
+
+Θ1( x
+√m − iz)/Θ4(iz)
+Θ2( x
+√m − iz)/Θ3(iz)
+−iΘ4( x
+√m − iz)/Θ1(iz)
+−iΘ3( x
+√m − iz)/Θ2(iz)
+
+.
+(4.9)
+Here, we define the scaled theta functions by Θi(u|m) := [ϑi( πu
+2K , q)]Abramowitz-Stegun with the
+nome q = e−πK′/K, and the second argument m is omitted. Using this scaled theta function,
+the Jacobi zeta function is expressed as Z(u|m) =
+d
+du ln Θ4(u|m). The derivation of Eqs. (4.8)
+and (4.9) are given in Appendix B.
+λ(z), ω(z), k(z) satisfy the following (quasi-)periodicity, parity, and the complex
+conjugation relations:
+λ(z) = (−1)l+nλ(z + nK′ + ilK)(−1)n = −λ(−z) = λ(z∗)∗,
+(4.10)
+ω(z) = (−1)nω(z + nK′ + ilK) = −ω(−z) = ω(z∗)∗,
+(4.11)
+k(z) = k(z + nK′ + ilK) +
+nπ
+2K √m = −k(−z) = k(z∗),
+(4.12)
+for n, l ∈ Z. The eigenfunction f0(x, z) also has the symmetries
+f0(x, z + nK′ + ilK) = il �
+(σl
+3σn
+2) ⊗ (σl
+2σn
+3)
+�
+f0(x, z),
+(4.13)
+f0(x, −z∗) = (σ3 ⊗ I2)f0(x, z)∗.
+(4.14)
+The two linearly independent solution for a given eigenvalue λ = λ(z) are f0(x, z) and
+f0(x, −z − iK), because λ(z) = λ(−z − iK).
+To cover all eigenfunctions, the region we must consider is the rectangle with vertices
+z = ±K′ ± iK, which is the fundamental period parallelogram of λ(z).
+The bounded
+
+6
+eigenfunctions with k(z) ∈ R, i.e., the scattering eigenstates, lie on the lines z ∈ R + iK
+2 Z,
+and there are four independent lines in one fundamental period parallelogram. Among them,
+R, R−iK gives monotonically increasing k(z) and real λ(z), while R± iK
+2 gives monotonically
+decreasing k(z) and pure imaginary λ(z).
+The region satisfying Im k(z) > 0 is given by
+−K < Im z < − K
+2 , 0 < Im z < K
+2 , and the positions of zeros of a(z) corresponding to bound
+eigenstates are chosen from this region in Sec. 6.
+The completeness relation satisfied for scattering eigenstates is
+δ(x − y)I4 =
+�
+C1+C2
+dz
+2π f0(x, z)f0(y, z∗)†σ.
+(4.15)
+Here, C1 is the rectangular contour passing through the vertices −K′ → K′ → K′ + iK
+2 →
+−K′ + iK
+2 → −K′ and C2 is the one −K′ − iK → K′ − iK → K′ − iK
+2 → −K′ − iK
+2 → −K′ − iK.
+Note that the integrations along vertical lines cancel out due to the periodicity. Therefore,
+the actual path with non-zero contribution is
+�
+C1+C2 =
+� K′
+−K′ −
+� K′+ iK
+2
+−K′+ iK
+2 +
+� K′−iK
+−K′−iK −
+� K′− iK
+2
+−K′− iK
+2 . The
+derivation of Eq. (4.15) is given in Appendix C.
+5. Scattering matrix
+Henceforth, we consider the scattering problem under the condition that the background
+potential tends to the stationary soliton lattice at spatial infinities x → ±∞. Let the potential
+φ(x) satisfy the boundary condition
+φ(x) →
+
+φ0(x)
+(x → −∞),
+φ0(x − x0)
+(x → +∞),
+(5.1)
+where x0 represents the phase shift of the background lattice caused by solitons and ripple
+waves. We define the left and right Jost solutions by the asymptotic form
+f−(x, z) → f0(x, z)
+(x → −∞),
+(5.2)
+f+(x, z) → f0(x − x0, z)
+(x → +∞).
+(5.3)
+The scattering matrix S (z) is then defined by the linear transformation between these two:
+�
+f+(x, z)
+f+(x, −z − iK)
+�
+=
+�
+f−(x, z)
+f−(x, −z − iK)
+�
+S (z).
+(5.4)
+The uniqueness of the definition of the Jost solution by its asymptotic form and the properties
+of eigenfunctions (4.13) and (4.14) imply
+f±(x, z) = (−iσ3 ⊗ σ2)l f±(x, z + 2nK′ + ilK) = (σ3 ⊗ I2)f±(x, −z∗)∗,
+n, l ∈ Z.
+(5.5)
+Note that while f0(x, z) has an additional symmetry with respect to translation of z by
+(2n + 1)K′ (see Eq. (4.13)), the corresponding symmetry does not exist for Jost solutions
+f±(x, z), because this additional symmetry is coming from special nature of φ0(x) satisfying
+ˆL = ˜L in Eqs. (3.5)-(3.7). Combining Eqs. (5.4) and (5.5), we have
+S (z) = σl
+3S (z + 2nK′ + ilK)σl
+3 = S (−z∗)∗ = σ1S (−z − iK)σ1,
+n, l ∈ Z.
+(5.6)
+Equating the integration constant obtained by integrating Eq. (3.2) at x → ±∞, we find
+S (z)−1 = S (z∗)†.
+(5.7)
+
+7
+We can also show det S = 1 from the constancy of the Wronskian det(g1, g2) (see the sentence
+after Eq. (3.2)). Therefore, S and its inverse matrix has the forms
+S (z) =
+�a(z)
+−b(z∗)∗
+b(z)
+a(z∗)∗
+�
+,
+(5.8)
+S (z)−1 =
+�a(z∗)∗
+b(z∗)∗
+−b(z)
+a(z)
+�
+,
+(5.9)
+with matrix elements satisfying
+a(z)a(z∗)∗ + b(z)b(z∗)∗ = 1,
+(5.10)
+a(z) = a(−z∗)∗ = a(z − iK),
+b(z) = b(−z∗)∗ = −b(z − iK).
+(5.11)
+From this, the zeros of a(z) always appear simultaneously at four points z, −z∗, z−iK, −z∗−iK,
+and the orders of these zeros are the same.
+6. Integral representation of Jost solution and Gelfand-Levitan-Marchenko equation
+Let us introduce the integral representation for the left Jost solution
+f−(x, z) = f0(x, z) +
+� x
+−∞
+dyΓ(x, y)f0(y, z).
+(6.1)
+The kernel function Γ(x, y) is a 4 × 4 matrix and assumed to decrease exponentially in the
+limits x, y → −∞.
+In order for this integral to converge, the integrand must decrease
+y → −∞, and this condition is satisfied at least for eigenfunctions with Im k ≤ 0, i.e.,
+−K/2 ≤ Im z ≤ 0, K/2 ≤ Im z ≤ K. From Eq. (3.4), we immediately conclude that
+Γ(x, y) = (σ3 ⊗ σ2)Γ(x, y)(σ3 ⊗ σ2) = (σ3 ⊗ I2)Γ(x, y)∗(σ3 ⊗ I2) = (I2 ⊗ σ2)Γ(x, y)∗(I2 ⊗ σ2).
+(6.2)
+Let us now write
+ˆL0(x) = −4iσ3 ⊕ O2 + U0(x),
+U0(x) =
+� iσ1w0
+e−iφ0σ2/2
+e−iφ0σ2/2
+O2
+�
+,
+(6.3)
+ˆL(x) = −4iσ3 ⊕ O2 + U(x),
+U0(x) =
+� iσ1w
+e−iφσ2/2
+e−iφσ2/2
+O2
+�
+.
+(6.4)
+Following the same derivation as Ref. [23], section 2.8, (In the present case, the first-order
+coefficient matrix of ˆL is not full-rank, but it makes no difference to the proof.)
+U(x) − U0(x) = 4i�σ3 ⊕ O2, Γ(x, x)�,
+(6.5)
+ˆL(x)Γ(x, y) = Γ(x, y) ˆL0(y).
+(6.6)
+From Eq. (6.5),
+w = w0 + 2Γ12,
+(6.7)
+eiφ/2 = eiφ0/2 + 4(iΓ13 + Γ14).
+�
+↔ cos φ
+2 = cos φ0
+2 + 4iΓ13, sin φ
+2 = sin φ0
+2 − 4iΓ14.
+�
+,
+(6.8)
+where we briefly write the matrix element of Γ(x, x) as Γi j = [Γ(x, x)]i j. Note that Γ12, iΓ13,
+and iΓ14 are real-valued functions due to Eq. (6.2). Using these relations, we can re-construct
+
+8
+the potential from the kernel Γ, which is determined from scattering data by solving the GLM
+equation shown below.
+The integral kernel Γ(x, y) is uniquely determined from the following scattering data: (i)
+the values of the reflection coefficient for scattering states r(z) = b(z)/a(z), where scattering
+states mean the bounded eigenfunction s.t. k(z) ∈ R, appearing on lines z ∈ R + iK
+2 Z, and (ii)
+the list of the zeros of a(z), which we write z1, . . . , zn, and the normalization factor c2
+j := b(z j)
+i˙a(z j),
+where zj’s are to be chosen from the region Im k(zj) > 0, and due to the property (5.11),
+the zeros simultaneously appear at z, −z∗, z − iK, −z∗ − iK. The values of c2
+j also have some
+constraint, whose detail will be described in Sec. 9. Here, we only treat the case where all
+zeros of a(z) are first order. For uniform background, the solitons corresponding to higher-
+order zeros are discussed in Ref. [24]. The case of unstable uniform background is considered
+in Ref. [25].
+The GLM equation that determines the kernel Γ(x, y) from the above-mentioned
+scattering data is given by
+Γ(x, y) + Ω(x, y) +
+� x
+−∞
+dwΓ(x, w)Ω(w, y) = 0,
+y ≤ x.
+(6.9)
+Here, Ω = Ωbd + Ωsc with
+Ωbd(x, y) := W(x)W(y)T(I2 ⊗ σ1),
+(6.10)
+W(x) := −iσ3 ⊗ σ2 (f0(x, −z1)c1, . . ., f0(x, −zn)cn) ,
+(6.11)
+Ωsc(x, y) :=
+
+� K′
+−K′ −
+� K′+ iK
+2
+−K′+ iK
+2
++
+� K′+iK
+−K′−iK
+−
+� K′− iK
+2
+−K′− iK
+2
+
+dz
+2π f0(x, −z − iK)r(z)f0(y, z∗)†σ.
+(6.12)
+Ωbd represents the contribution from bound states, and Ωsc is that from scattering states. The
+derivation of the GLM equation (6.9) with (6.10)-(6.12) is given in Appendix D.
+7. Reflectionless potentials
+Let us solve the GLM equation (6.9) for the reflectionless case, i.e., Ωsc = 0. The ansatz for
+the kernel Γ(x, y) is as follows. Let us introduce the notation H(x) = (h1(x), . . ., hn(x)), where
+each hi(x) is four-component column vector. We then assume
+Γ(x, y) = H(x)W(y)T(I2 ⊗ σ1).
+(7.1)
+Substituting it to the GLM equation, we obtain the equation for H(x):
+H(x) + W(x) + H(x)G(x) = 0,
+(7.2)
+where we define the n × n Gram matrix by
+G(x) =
+� x
+−∞
+dyW(y)T(I2 ⊗ σ1)W(y).
+(7.3)
+From Eq. (7.2), H = −W(In+G)−1. Let Gl j(x) be the (l, j)-component of G(x). Using Eq. (3.2)
+and the symmetries of λ(z) and f0(x, z), Eqs. (4.10), (4.13), and (4.14), we find
+Gl j(x) = clc j
+4 f0(x, −zl)T(σ2 ⊕ O2)f0(x, −zj)
+λ(zl) − λ(zj)
+.
+(7.4)
+
+9
+It can be simplified using the Weierstrass three-term formula in Ref. [20], Eq. (3.8). The result
+is
+Gl j(x) = clc je−i[k(zl)+k(z j)]x Θ2Θ4Θ3( x
+√m + i(zl + zj))
+Θ3Θ2(i(zl + zj))Θ4( x
+√m) .
+(7.5)
+Thus,
+Γ(x, y) = −W(x)[In + G(x)]−1W(y)T(I2 ⊗ σ1).
+(7.6)
+Let us write W as an array of row vectors: W =
+
+W1
+W2
+W3
+W4
+, where each Wi is a 1 × n row vector.
+Then we get the expressions Γ13 = −W1(In + G)−1WT
+4 , Γ14 = −W1(In + G)−1WT
+3 . Using the
+relation (6.8) and the Woodbury-type identity 1 + ⃗a†A−1⃗b = det(A+⃗b⃗a†)
+det A
+,
+cos φ
+2 = cos φ0
+2 + 4iΓ13 =
+�
+cos φ0
+2
+� det(I + P)
+det(I + G),
+Pi j = Gi j −
+4i
+cos φ0
+2
+f03(x, −zi)f02(x, −zj),
+(7.7)
+sin φ
+2 = sin φ0
+2 − 4iΓ14 =
+�
+sin φ0
+2
+� det(I + Q)
+det(I + G),
+Qi j = Gi j −
+4i
+sin φ0
+2
+f04(x, −zi)f02(x, −zj),
+(7.8)
+where f0 j(x, z) represents the j-th component of f0(x, z) defined by (4.9).
+These can
+be simplified by using Ref. [20], Eq. (3.5b).
+Rewriting C j =
+1
+c2
+j
+√m, and introducing
+new matrices G, P, Q through the relations Gi j
+=
+m1/4cic je−i[k(zi)+k(z j)]xGi j,
+Pi j
+=
+m1/4cic je−i[k(zi)+k(z j)]xPi j, Qi j = m1/4cic je−i[k(zi)+k(z j)]xQi j, we obtain the final expressions
+cos φ
+2 =
+�
+cos φ0
+2
+� det(E + P)
+det(E + G),
+(7.9)
+sin φ
+2 =
+�
+sin φ0
+2
+� det(E + Q)
+det(E + G),
+(7.10)
+where the components of n × n matrices E, G, P, and Q are given by
+Ei j = δi jC je2ik(z j)x,
+(7.11)
+Gi j =
+Θ4Θ3( x
+√m + i(zi + zj))
+Θ2(i(zi + zj))Θ4( x
+√m) ,
+(7.12)
+Pi j =
+�Θ4(izi)
+Θ1(izi)
+� 
+−Θ4Θ2( x
+√m + i(zi + zj))
+Θ2(i(zi + zj))Θ1( x
+√m)
+
+�Θ2(izj)
+Θ3(izj)
+�
+,
+(7.13)
+Qi j =
+�Θ4(izi)
+Θ2(izi)
+� 
+Θ4Θ1( x
+√m + i(zi + zj))
+Θ2(i(zi + zj))Θ2( x
+√m)
+
+�Θ1(izj)
+Θ3(izj)
+�
+.
+(7.14)
+Since the wavenumbers used to make bound states all satisfy Im k(zj) > 0,
+E →
+
+∞
+(x → −∞),
+0
+(x → +∞).
+(7.15)
+
+10
+Therefore, the asymptotic form of φ is given by
+cos φ
+2 →
+
+cos φ0
+2
+(x → −∞),
+cos φ0
+2 det PG−1
+(x → +∞),
+(7.16)
+sin φ
+2 →
+
+sin φ0
+2
+(x → −∞),
+sin φ0
+2 det QG−1
+(x → +∞).
+(7.17)
+The determinant appearing in x → +∞ is calculated in Appendix E. The resultant asymptotic
+form is
+φ(x) →
+
+φ(x)
+(x → −∞),
+φ(x − x0)
+(x → +∞),
+(7.18)
+x0 := −2i √m
+n
+�
+j=1
+zn.
+(7.19)
+x0 has the meaning of the phase shift of the background lattice caused by solitons. The
+realness of x0 follows from the fact that the zeros of a(z) always simultaneously emerge at
+four points z, z − iK, −z∗, and −z∗ − iK.
+8. Time evolution
+Finally, let us determine the time dependence of scattering matrix if the system obeys the Lax
+equation (2.1). Let f be a time-dependent eigenfunction of ˆL. Then, the Lax equation implies
+ˆLf = λf,
+(8.1)
+4ift = − ˆBf.
+(8.2)
+We now define the time-dependent eigenfunction by the initial condition ˜f+(t = 0, x, z) =
+f+(x, z). Though ˆB is now a time-dependent operator for finite x, it is time-independent at
+x → ±∞. Therefore, the asymptotic forms of this ˜f+(t, x, z) is given by
+( ˜f+(t, x, z), ˜f+(t, x, −z − iK)) →
+
+(f0(x, z), f0(x, −z − iK))eiω(z)tσ3/4S (z)
+(x → −∞),
+(f0(x − x0, z), f0(x − x0, −z − iK))eiω(z)tσ3/4
+(x → +∞).
+(8.3)
+Here, we have used the relation ω(−z − iK) = −ω(z). Therefore, if we define the time-
+dependent right Jost solution by the asymptotic form f+(t, x, z) → f0(x − x0, z) for x →
++∞, which is different from ˜f+(t, x, z), then the relation ( ˜f+(t, x, z), ˜f+(t, x, −z − iK)) =
+(f+(t, x, z), f+(t, x, −z − iK))eiω(z)t/4 holds. Then the time evolution of the scattering matrix
+is given by
+S (t, z) = eiω(z)tσ3/4S (z)e−iω(z)tσ3/4,
+(8.4)
+or for each component,
+a(t, z) = a(z),
+b(t, z) = e−iω(z)t/2b(z).
+(8.5)
+Since C j is defined by C j =
+1
+c2
+j
+√m =
+i˙a(z j)
+√mb(z j), its time evolution is written as
+C j(t) = C jeiω(z j)t/2.
+(8.6)
+
+11
+Therefore, we can define the time-dependent Ei j(t) as follows. In Eq. (7.11), we replace C j
+by C j(t) = C jeiω(z j)t/2, i.e.,
+Ei j(t) = δi jC je2ik(z j)x+iω(z j)t/2.
+(8.7)
+Then the time-dependent solution of the SG equation is given by replacing E with E(t) in Eqs.
+(7.9) and (7.10):
+cos φ
+2 =
+�
+cos φ0
+2
+� det(E(t) + P)
+det(E(t) + G),
+(8.8)
+sin φ
+2 =
+�
+sin φ0
+2
+� det(E(t) + Q)
+det(E(t) + G),
+(8.9)
+where the definition of G, P, and Q remains unchanged (Eqs. (7.12)-(7.14)).
+9. Limitation to eigenvalues zj’s and normalization coefficients C j’s
+In order for the function φ(x, t) to be real and bounded, zj’s, which are the zeros of a(z)
+corresponding to the discrete eigenvalues for bound states, and the normalization coefficients
+C j’s must satisfy several conditions.1 Here we describe it.
+First, depending on its value, zj’s must satisfy the following (i) or (ii):
+(i) zj, zj − iK, −z∗
+j, −z∗
+j − iK appear simultaneously. Due to this symmetry, one of these
+four zj can be chosen from 0 ≤ Re zj ≤ K′, 0 < Im zj < K
+2 without loss of generality.
+(ii) If, as a special case, zj = irj or zj = irj + K′ with rj ∈ (0, K
+2 ), then just two zeros zj and
+zj − iK appear simultaneously.
+The case (i) corresponds to the breather solution and the case (ii) the traveling one kink
+solution. The total number of zj’s are always even. If there are breathers and even number of
+kinks, the number is a multiple of four, while if there exists odd number of kinks, the number
+is of the form 4n + 2. The velocity of the soliton is given by v j = − Im ω(z j)
+4 Im k(z j). In particular,
+for one kink solution, it reduces to v j = − ω(z j)
+4k(z j), which is negative if zj = irj and positive if
+zj = irj + K′.
+Next let us consider the condition for C j’s. By Eq. (5.11), the derivative of a(z) satisfies
+˙a(z) = −˙a(−z∗)∗ = ˙a(z − iK),
+(9.1)
+and thus if we define
+C(z) :=
+i˙a(z)
+√mb(z),
+(9.2)
+then it has the symmetry
+C(z) = C(−z∗)∗ = −C(z − iK).
+(9.3)
+This symmetry is the same as that of b(z) given in Eq. (5.11). Therefore, for the breather
+solution where four zeros appear, if we temporarily write these four as zj, zj+1 = zj−iK, zj+2 =
+−z∗
+j, zj+3 = −z∗
+j − iK, then we should choose the corresponding coefficients as C j+1 =
+−C j, C j+2 = C∗
+j, C j+3 = −C∗
+j. For the kink solution, where two zeros zj and zj+1 = zj − iK
+appear, C j+1 = −C j are both real and must have the opposite sign. Whether the solution
+becomes kink or anti-kink, i.e., the phase rotation becomes counterclockwise or clockwise, is
+determined by which one is chosen positive.
+1 If we consider application to scientific problems where the complex-valued and/or divergent solutions have some
+appropriate physical interpretations, the restriction stated here can be loosened.
+
+12
+Figure 1. A snapshot of one kink solution. The number of zeros is n = 2 and the parameters
+are m = 0.95, z1 = 0.02iK + K′, z2 = z1 − iK, C1 = −1, C2 = −C1. The snapshot at t = 1.
+10. Animation of soliton solutions
+Here, we present a few examples of the soliton solutions, Eqs. (8.8) and (8.9), by gif
+animations. We visualize the solution through the 3D plot (x, cos φ(x, t), sin φ(x, t)). An
+example of snapshot is shown in Fig. 1. See attached files, testx.gif with x=1,2, and 3. Below,
+we provide the parameters for each solution. We write K = K(m) and K′ = K(1 − m).
+(i) test1.gif: one kink solution.
+The number of zeros is n = 2. The parameters are m = 0.99, z1 = 0.02iK + K′, z2 =
+z1 − iK, C1 = −1, C2 = −C1. The time range is −10 ≤ t ≤ 10.
+(ii) test2.gif: one anti-kink solution.
+C1 = 1, C2 = −C1, and other parameters are the same as (i).
+(iii) test3.gif: one breather solution.
+The number of zeros is n = 4. The parameters are m = 0.99, z1 = 0.1iK + 0.5K′, z2 =
+z1 − iK, z3 = −z∗
+1, z4 = −z∗
+2, C1 = −1, C2 = −C1, C3 = C∗
+1, C4 = C∗
+2. The time range
+is −10 ≤ t ≤ 10.
+Solutions with more number of solitons will be generated using the Mathematica file in google
+drive.2
+11. Summary and discussion
+In this paper, we have derived the multi-soliton solutions of the SG equation with elliptic-
+function background by the ISM. The result is expressed by a determinant of theta functions,
+i.e., Eqs. (8.8) and (8.9). The shift of the background lattice caused by solitons (7.18) with
+(7.19) has also been found using the addition formula of theta functions. One key tool in our
+work is the Lax pair represented by 4×4 matrix differential operators (2.2) and (2.3), originally
+introduced in Ref. [17]. This most conventional but seemingly outdated approach makes
+it possible to use the common form of the integral representation of the Jost solution (6.1)
+2 In this file, since the default build-in Jacobi elliptic functions are a little slow, the functions defined by the ratio of
+theta functions are used alternatively.
+
+-10
+.5
+10 -213
+without modification and the formulation of the ISM is simplified. Also, the completeness
+relations in an indefinite inner product space, which is necessary in the formulation of the
+ISM, has been discussed in detail (Appendix C). Application to various physical phenomena
+including the dynamics of defects with periodic background, extension to unstable oscillating
+background (m > 1) and studying multi rogue waves, and the solitons associated with higher-
+order zeros of a(z), are left as possible future works.
+Before concluding, we provide several technical remarks and perspectives.
+In this
+work, we can introduce the concept of orthogonality between eigenfunctions based on the
+indefinite inner product, since the Lax operators have the symmetry (3.1). If the Lax pair
+has no such symmetry and any inner product cannot be defined, the completeness relation
+will be constructed from the set of left and right eigenfunctions, an analog of left and right
+eigenvectors for finite dimensional matrix. Such basis is sometimes called a bi-orthogonal
+basis.
+Whether we can always reduce the classical integrable systems written by zero-curvature
+condition (or a compatibility condition) and/or the Lax pair including an integral operator to
+the Lax formalism with differential operators seems to be unclear. (The inverse operation
+is easy; if one has an integrable equation written by the Lax form using matrix differential
+operators, one can immediately rewrite it in a zero-curvature expression.) If we can do it,
+the integral representation of the Jost solution will be widely applicable and multi-soliton
+solutions will be easily obtained by the dressing method [26], where we can even omit the
+full formulation of the ISM if we do not have an interest in general initial-value problem. We
+also note that, as emphasized in Secs. 3 and 4, the highest-order coefficient matrix of the Lax
+operator ˆL is not full-rank and ˆL−1 can be written down explicitly without using an integral
+operator. Though we do not show detail here, a similar property also emerges in the derivative
+nonlinear Schr¨odinger equation, which is usually treated by the Kaup-Newell form. That is,
+if we rewrite it to the Lax form, the highest-order coefficient matrix is not full-rank. Even in
+these systems, the algebraic curve from the pair of commuting operators can be appropriately
+defined (see Eq. (4.5)), though, strictly speaking, these cases are not included in the seminal
+work [19]. Exploring these examples more comprehensively and exhaustively might bring a
+slight extension to the theory of commuting differential operators.
+Acknowledgments
+This work was supported by MEXT-Supported Program Grant No. S1511006 and JSPS
+KAKENHI Grant No. JP19H05821.
+Appendix A. Integral often emerging in calculation of eigenfunctions for elliptic
+potentials
+Let us define scd(z|m) := sn(z|m) cn(z|m) dn(z|m). When an elliptic-function potential is given
+in some hierarchy of integrable systems, the calculation of its eigenfunctions often reduces to
+the following integral:
+�
+dxα[scd(αx) + scd β]
+sn2(αx) − sn2 β
+= αZ(β)x + ln Θ1(αx − β)
+Θ4(β)Θ4(αx),
+(A.1)
+even if the corresponding Riemann surface has higher genus g > 1. This formula can be
+derived by using [27], appendix B. If we interpret this formula using Weierstrass’s functions,
+it reduces to the formula given in Ref. [28], II, Kap. 6, §3, and is called the standard form
+
+14
+of the elliptic integral of the third kind. Using it, the solution to the first-order differential
+equation
+fx
+f = γ + α[scd αx + scd β]
+sn2 αx − sn2 β
+(A.2)
+is given by
+f(x) = Ceikx Θ1(αx − β)
+Θ4(β)Θ4(αx), k = −i(γ + αZ(β)).
+(A.3)
+If m ∈ [0, 1], α > 0, and k is real, then f becomes a twisted periodic function
+f
+�
+x + 4Kl
+α
+�
+= eik 4Kl
+α f(x),
+l ∈ Z.
+(A.4)
+Therefore, if k can be physically interpreted to be crystal momentum of the Bloch function, it
+is defined up to mod απ
+2K and the corresponding Brillouin zone is given by [− απ
+4K , απ
+4K ].
+Appendix B. Eigenfunctions for soliton lattice
+In this appendix, we derive the simultaneous eigenfunction f0(x, z) with its crystal momentum
+k(z) in Eqs. (4.8) and (4.9) for the time-independent Lax pair ˆL and ˆB with the soliton lattice
+potential (4.3), when the eigenvalues are parametrized by Eqs. (4.6) and (4.7). The method is
+in fact the special case of the one given by Krichever [19].
+Let us write f0 =
+�
+g
+1
+λ e−iφσ2g
+�
+, where g is a two-component vector g = � g1
+g2
+�. Eliminating g′
+1
+and g′
+2 using the eigenequation of ˆL in Eq. (4.1), and substituting them to that of ˆB, one obtains
+two linear relations with respect to g1 and g2. The determinant of this coefficient matrix must
+vanish, which gives an elliptic curve (4.5). Using it, one can eliminate either of g1 and g2 and
+obtain the first-order differential equation only containing one function:
+g′
+1
+g1
+= −i(2λ − ω)
+4
++
+i(−1 + λ4 − 2λ3ω + λ2ω2)
+2λ(−1 + λ2 − λω + 2 cos2 φ0
+2 )
++
+−φ0x cos φ0
+2 sin φ0
+2
+−1 + λ2 − λω + 2 cos2 φ0
+2
+,
+(B.1)
+g′
+2
+g2
+= i(2λ + ω)
+4
++ −i(−1 + λ4 + 2λ3ω + λ2ω2)
+2λ(−1 + λ2 + λω + 2 cos2 φ0
+2 )
++
+−φ0x cos φ0
+2 sin φ0
+2
+−1 + λ2 + λω + 2 cos2 φ0
+2
+.
+(B.2)
+Using Eq. (4.4) and the parametrizations (4.6) and (4.7), the above equations reduce to
+g′
+1
+g1
+= −i(2λ − ω)
+4
++
+1
+√m
+scd( x
+√m) + scd(iz)
+sn2( x
+√m) − sn2(iz) ,
+(B.3)
+g′
+2
+g2
+= i(2λ + ω)
+4
++
+1
+√m
+scd( x
+√m) + scd(iz − K)
+sn2( x
+√m) − sn2(iz − K) .
+(B.4)
+Therefore, using the formula in Appendix A, we find
+g1 = C1eik1x Θ1( x
+√m − iz)
+Θ4(iz)Θ4( x
+√m), k1 = ω − 2λ
+4
+−
+i√mZ(iz),
+(B.5)
+g2 = C2eik2x Θ2( x
+√m − iz)
+Θ3(iz)Θ4( x
+√m), k2 = ω + 2λ
+4
+−
+i√mZ(iz − K).
+(B.6)
+
+15
+Due to the Bloch (Floquet) theorem, g1 and g2 must share the same crystal momentum, i.e.,
+k1 and k2 must be related by k1 ≡ k2 mod
+π
+2 √mK . In fact, using the periodicity of the Jacobi
+elliptic and zeta functions, we can check k1 = k2. Therefore, henceforth we simply write
+k1 = k2 = k(z). The ratio C1/C2 must also be fixed. Rewriting g1/g2 using the Jacobi elliptic
+function, and checking its consistency with the linear relation between g1 and g2 obtained
+from the eigenequation of ˆB, we find C1/C2 = 1.
+The crystal momentum k(z) can be further simplified as follows. The derivative k′(z) can
+be simplified using the double-angle formula of the cs function:
+k′(z) =
+1
+√m
+�
+− cs2(2iz) − E
+K
+�
+=
+1
+2 √m
+�
+dn2(2iz + iK′) + dn2(2iz − iK′) − 2E
+K
+�
+.
+(B.7)
+Since Z′ = dn2 − E
+K , integrating this expression soon gives the Jacobi zeta function. Further-
+more, the constant of integration can be fixed using some specific value at any point, e.g.,
+k1( K′
+2 ) = −
+π
+4K √m. Thus we obtain Eq. (4.8).
+The third and fourth components of the eigenfunction, g3 = λ−1
+�
+− sn( x
+√m)g1 − cn( x
+√m)g2
+�
+and g4 = λ−1
+�
+cn( x
+√m)g1 − sn( x
+√m)g2
+�
+, can be simplified using another expression of Eq. (4.6),
+λ(z) = −i Θ1(iz)Θ2(iz)
+Θ3(iz)Θ4(iz), and the addition formula of theta functions, in particular, Ref. [20],
+Eqs. (3.5b) and (3.8). The final expression is then given by Eq. (4.9).
+Appendix C. Completeness relation
+In this appendix, we derive the completeness relation (4.15) for eigenfunctions of the soliton
+lattice potential φ0(x) of Eq. (4.3).3
+The first important point is that ˆL is not self-adjoint, i.e., ˆL† � ˆL and instead satisfies
+ˆL† = σ ˆLσ with σ = I2 ⊗ σ3 as stated in Sec. 3. For such operator, the eigenfunction satisfies
+the orthogonal relation with respect to the indefinite inner product (f, g)σ =
+�
+dxf †σg.
+The normalization of eigenfunction is also made based on (f, f)σ and all eigenfunctions
+are classified into the ones possessing positive, negative, and zero norm. The completeness
+relation for the case of finite-dimensional linear algebra is shown in section 3 of Ref. [21], and
+3 In order to eliminate confusion, we should explain the difference of the usage of the terminology “eigenvalues an
+eigenfunctions” between classical integrable systems and other fields. In theoretical physics, particularly in quantum
+mechanics, when one finds a solution of an eigenequation of a given differential operator ˆL f = λf , the operand
+function f is called an eigenfunction only when it is a bounded function and only in this case λ is included to the set
+of eigenvalues. The eigenfunctions possessing everywhere-bounded plane-wave type behavior are called a scattering
+state and constitute the set of continuous eigenvalues, i.e., a band. The eigenfunction with localized profile and finite
+norm is called a bound state and make a discrete eigenvalue. The completeness relation, which is in bra-ket notation
+of quantum mechanics often expressed as 1 = �
+n |n⟩ ⟨n|, is constructed by gathering all these scattering and bound
+eigenstates.
+In classical integrable systems, the exponentially divergent solution of an eigenequation, which is usually not
+counted as an eigenfunction, plays an important role in generating multi-soliton solutions by various methods. An
+elementary example is the divergent solution of the Schr¨odinger operator −fxx = −κ2 f with eigenvalue λ = −κ2 < 0
+and f = e±κx, which is indeed used to construct the multi-soliton solution of the KdV equation. Thus, identification
+of all solutions of the eigenequation for any complex λ is important, and therefore, in the context where no
+confusion occurs, these divergent solutions f and corresponding values λ are also sometimes called eigenfunctions
+and eigenvalues, without distinction. This manuscript also adopts this loose use of terminology in several sections,
+e.g., in Sec. 4. To preserve the logical unambiguity, the “genuine” eigenfunctions with no divergent behaviors are
+explicitly referred to as scattering and bound eigenstates. The completeness relation includes only these “genuine”
+eigenfunctions.
+The solution of eigenequation for arbitrary complex eigenvalue is also often called the Baker-
+Akhiezer function, whose original meaning is the single-valued function defined on a Riemann surface and possessing
+finitely many essential singular points but meromorphic excepting these points.
+
+16
+we now need to consider its analog in continuous space for a differential operator. (A specific
+example for a differential operator in continuous space is found in Refs. [29, 30], though
+not fully general.) In mathematical physics, a space equipped with indefinite inner product
+is called the Krein space. Eigenfunctions with nonzero norm can be almost analogously
+treated to the case of self-adjoint operators whose eigenvalues are real and eigenfunctions
+are normalized by positive-definite norm. On the other hand, we must carefully consider
+eigenfunctions with zero norm. In particular, the eigenfunction with complex eigenvalue
+always has zero norm due to Eq. (3.2). In this case, in order to prepare a σ-orthogonal basis,
+we should construct positive- and negative-norm functions by linear combination of a pair of
+zero-norm eigenfunctions with mutually complex conjugate eigenvalues.
+The second important point in writing down the completeness relation supposed to be
+applied in the ISM is that the integrand must be expressed by meromorphic function with
+respect to spectral variable λ or its parametrization variable z, because, in the derivation
+of the GLM equation in Appendix D, we want to use the residue theorem. Therefore, we
+must eliminate z∗ from the integrand using the complex conjugation relations described in
+Eqs. (4.10)-(4.14). The key relation is as follows. A little calculation using the addition
+formula of theta functions shows
+f0(x, z∗)†σf0(x, z) =
+4
+�
+i=1
+(−1)i−1gi(x, z∗)∗gi(x, z) = −1
+√m
+�
+dn2( x
+√m) + cs2(2iz)
+�
+.
+(C.1)
+This quantity is meromorphic, i.e., only including z due to Eq. (4.14). The spatial average of
+this quantity is, using dn2 = E
+K and recalling Eq. (B.7),
+f0(x, z∗)†σf0(x, z) =
+1
+√m
+�
+− E
+K − cs2(2iz)
+�
+= k′(z),
+(C.2)
+which is used to normalize scattering eigenstates and rewrite the k-integral to z-integral. As
+shown below, the spatial average of the norm density f0(x, z)†σf0(x, z) for scattering states all
+reduces to Eq. (C.2).
+Thirdly, if we have a σ-orthogonal basis of the Bloch-type functions parametrized
+by crystal momentum k and band index α, and all of them have non-vanishing norm and
+are normalized so that the spatial average becomes φ±(k, α)†σφ±(k, α) = ±1, where ±
+represents the sign of norm, the completeness relation is given by �
+α
+�
+dk
+2π(φ+(k, α)φ+(k, α)† −
+φ−(k, α)φ−(k, α)†)σ = Id. This fact can be proved by considering the finite-length system
+where the scattering states can be explicitly normalizable and their spectrum is discretized,
+and then taking the infinite-length limit. Here we omit the detail of this argument, since it
+is tedious but straightforward. For readers’ reference, we note that a similar discussion in
+another physical problem for discretization of scattering states of a self-adjoint operator and
+its infinite limit can be found in Ref. [31], section 3.
+Keeping in mind the above-mentioned things, let us now write down the completeness
+relation. Since the potentials (4.4) are periodic, there are only scattering states by the Bloch
+theorem and no bound state exist.
+Hence, the completeness relation is written only by
+scattering states. As stated in Sec. 4, up to periodicity, there are four distinct lines in z-plane
+where crystal momentum becomes real k(z) ∈ R, representing the scattering states. Here we
+discuss the scattering states on each line in detail.
+(i) z ∈ R.
+We can check λ(z) ∈ R and f0(x, z) has positive norm, so f0(x, z∗) = f0(x, z) and hence
+
+17
+f0(x, z∗)†σf0(x, z) = f0(x, z)†σf0(x, z) = k′(z) > 0. Therefore, the contribution of these
+states to completeness relation is
+�
+dk
+2π
+f0(x,z)f0(y,z)†σ
+k′(z)
+=
+� K′
+−K′
+dz
+2π f0(x, z)f0(y, z∗)†σ.
+(ii) z ∈ R − iK.
+We can check λ(z) ∈ R and f0(x, z) has negative norm, and f0(x, z∗) = −f0(x, z)
+using periodicity (4.13). Thus, f0(x, z)†σf0(x, z) = −f0(x, z∗)†σf0(x, z) = −k′(z) < 0.
+Therefore, the contribution to these states to completeness relation is
+�
+dk f0(x,z)f0(y,z)†σ
+−k′(z)
+=
+� K′+iK
+−K′−iK
+dz
+2π f0(x, z)f0(y, z∗)†σ.
+(iii) z ∈ R ± iK
+2 .
+We can check λ(z) ∈ iR and hence f0(x, z) has zero norm. Therefore we must make
+nonzero-norm functions by linear combination of a pair of eigenfunctions with mutually
+complex conjugate eigenvalues. Since λ(z∗) = λ(z)∗ and k(z∗) = k(z)∗, this pair share the
+same wavenumber. Now let us assume that f1 and f2 are zero-norm eigenfunctions of
+the Bloch type with complex eigenvalues λ and λ∗ and share the same and real-valued
+wavenumber k. By definition the spatial average of norm density is zero: f †
+i σfi =
+0, i = 1, 2, but f †
+1 σf2 � 0 unless they belong to nontrivial Jordan blocks. Then, if an
+overall prefactors of f1 and f2 are adjusted to satisfy the relation f †
+1 σf2 = f †
+2 σf1 > 0,
+then f± =
+f1± f2
+√
+2
+are positive- and negative-norm functions and σ-orthogonal to each
+other. Thus these two can be used as a σ-orthogonal basis, and the contribution of
+these states to completeness relation is written as
+�
+dk
+2π
+f+ f †
++σ− f− f †
+−σ
+f †
+2 σf1
+=
+�
+dk
+2π
+(f1 f †
+2 + f2 f †
+1 )σ
+f †
+2 σf1
+.
+Now let us take f1 = f0(x, z), z ∈ R + iK
+2 and f2 = −f0(x, z∗), z∗ ∈ R − iK
+2 . By
+Eq. (C.2), f †
+2 σf1 = −k′(z) > 0 is real and positive.
+(Note: k(z) is a decreasing
+function on z ∈ R ± iK
+2 .) Therefore, the contribution of these states to completeness
+relation is given by
+�
+dk
+2π
+(− f0(x,z)f0(y,z∗)†− f0(x,z∗)f0(y,z)†)σ
+−k′(z)
+= −
+� K′+ iK
+2
+−K′+ iK
+2
+dz
+2π(f0(x, z)f0(y, z∗)† +
+f0(x, z∗)f0(y, z)†)σ.
+Changing the dummy variable z → z∗, the latter term can be
+rewritten as −
+� K′− iK
+2
+−K′− iK
+2
+dz
+2π f0(x, z)f0(y, z∗)†σ.
+Summarizing all contributions from (i)-(iii), the completeness relation is given by
+δ(x − y)I4 =
+� K′
+−K′
+dz
+2π f0(x, z)f0(y, z∗)†σ +
+� K′+iK
+−K′−iK
+dz
+2π f0(x, z)f0(y, z∗)†σ
+−
+� K′+ iK
+2
+−K′+ iK
+2
+dz
+2π f0(x, z)f0(y, z∗)†σ −
+� K′− iK
+2
+−K′− iK
+2
+dz
+2π f0(x, z)f0(y, z∗)†σ.
+(C.3)
+Adding vertical paths which cancel out by periodicity, we obtain the desired form
+δ(x − y)I4 =
+�
+C1+C2
+dz
+2π f(x, z)f(y, z∗)†σ,
+(C.4)
+where the definitions of the contours C1 and C2 are already described in the sentences after
+Eq. (4.15).
+
+18
+Appendix D. Derivation of the GLM equation
+In this appendix we drive the GLM equation (6.9). By the relation between right and left Jost
+solutions (5.4) and the integral representation (6.1),
+f+(x, z) 1
+a(z) − f0(x, z) =
+� x
+−∞
+dyΓ(x, y)f0(y, z) +
+�
+f0(x, −z − iK) +
+� x
+−∞
+dyΓ(x, y)f0(y, −z − iK)
+� b(z)
+a(z).
+(D.1)
+Multiplying f(w,z∗)†σ
+2π
+, w ≤ x to both sides of the above equation from right, we integrate the
+expression by z along the contour C1 + C2. First, let us evaluate the right-hand side. Defining
+Ωsc(x, w) :=
+�
+C1+C2
+dz
+2π f0(x, −z − iK)b(z)
+a(z) f0(w, z∗)†σ,
+(D.2)
+and using the completeness relation (4.15), we find
+R.H.S. = Γ(x, w) + Ωsc(x, w) +
+� x
+−∞
+dyΓ(x, y)Ωsc(y, w).
+(D.3)
+In the contour integral of Ωsc, actually the integrals for scattering states only contribute, and
+hence it can be rewritten as Eq. (6.12).
+Next, let us evaluate the left-hand side. In the SG equation, generally a(z) can have
+higher-order zeros [24], but here we only consider the case where all zeros are of the first
+order. Let zeros of a(z) be z1, . . ., zn. As already mentioned before, by Eq. (5.11), the zeros
+of a(z) simultaneously emerge at z, −z∗, z − iK, −z∗ − iK. Therefore, not all zj’s can be freely
+chosen. Here, however, we formally assign independent labels to all zeros. Then, the left-hand
+side can be calculated using the residue theorem:
+L.H.S. =
+n
+�
+j=1
+if+(x, zj)f0(w, z∗
+j)†
+˙a(zj)
+=
+n
+�
+j=1
+if−(x, −zj − iK)b(zj)f0(w, z∗
+j)†
+˙a(zj)
+,
+(D.4)
+where the relation f+(x, zj) = b(zj)f−(x, −zj − iK) arising from the condition a(zj) = 0 has
+been used. If we define
+Ωbd(x, w) :=
+n
+�
+j=1
+f0(x, −zj − iK)c2
+j f0(w, z∗
+j)†σ,
+c2
+j := b(zj)
+i˙a(zj),
+(D.5)
+then
+L.H.S. = −Ωbd(x, w) −
+� x
+−∞
+dyΓ(x, y)Ωbd(y, w).
+(D.6)
+To summarize, writing Ω = Ωsc + Ωbd, we obtain the final result
+Γ(x, w) + Ω(x, w) +
+� x
+−∞
+dyΓ(x, y)Ω(y, w) = 0,
+w ≤ x.
+(D.7)
+If we simplify Eq. (D.5) using Eqs. (4.13) and (4.14), it is rewritten as Eq. (6.10) with (6.11).
+
+19
+Appendix E. Calculation of determinant
+Here we calculate the asymptotic form of the multi-soliton solution for x → +∞. In particular,
+we provide the shift of the lattice x0 in Eq. (7.19). First, we write P = S 1 ˜PS 2 and Q = S 3 ˜QS 4,
+where
+S 1 = diag
+�Θ4(iz1)
+Θ1(iz1), . . ., Θ4(izn)
+Θ1(izn)
+�
+,
+S 2 = diag
+�Θ2(iz1)
+Θ3(iz1), . . ., Θ2(izn)
+Θ3(izn)
+�
+,
+(E.1)
+S 3 = diag
+�Θ4(iz1)
+Θ2(iz1), . . ., Θ4(izn)
+Θ2(izn)
+�
+,
+S 4 = diag
+�Θ1(iz1)
+Θ3(iz1), . . ., Θ1(izn)
+Θ3(izn)
+�
+,
+(E.2)
+and
+˜Pi j =
+−Θ4Θ2( x
+√m + i(zi + zj))
+Θ2(i(zi + zj))Θ1( x
+√m) ,
+(E.3)
+˜Qi j =
+Θ4Θ1( x
+√m + i(zi + zj))
+Θ2(i(zi + zj))Θ2( x
+√m) .
+(E.4)
+Then, using the formula in Ref. [27], appendix D, we obtain
+det ˜P
+det G = (−1)n Θ4( x
+√m)Θ1( x
+√m + nK + 2i �n
+j=1 zj)
+Θ1( x
+√m)Θ4( x
+√m + nK + 2i �n
+j=1 zj),
+(E.5)
+det ˜Q
+det G = (−1)n Θ4( x
+√m)Θ2( x
+√m + nK + 2i �n
+j=1 zj)
+Θ2( x
+√m)Θ4( x
+√m + nK + 2i �n
+j=1 zj).
+(E.6)
+This relation itself is generally valid even when all zj’s are independent and n is not even.
+As discussed in Sec. 9, for the real and bounded solution of the SG equation, n is even and zj
+and zj − iK simultaneously appear. In this case, we label the zeros, e.g., as zn/2+ j = zj − iK,
+j =
+n
+2 + 1, . . ., n. Then, det S 1 = �n/2
+j=1
+Θ4(iz j)Θ3(iz j)
+Θ1(iz j)Θ2(iz j),
+det S 2 = (−1)n/2 det S −1
+1 ,
+det S 4 =
+�n/2
+j=1
+Θ1(iz j)Θ2(iz j)
+Θ3(iz j)Θ4(iz j),
+det S 3 = (−1)n/2 det S −1
+4 , and hence
+det P
+det G =
+Θ4( x
+√m)Θ1( x
+√m + 2i �n
+j=1 zj)
+Θ1( x
+√m)Θ4( x
+√m + 2i �n
+j=1 zj) = cos φ0(x−x0)
+2
+cos φ0(x)
+2
+,
+(E.7)
+det Q
+det G =
+Θ4( x
+√m)Θ2( x
+√m + 2i �n
+j=1 zj)
+Θ2( x
+√m)Θ4( x
+√m + 2i �n
+j=1 zj) = sin φ0(x−x0)
+2
+sin φ0(x)
+2
+.
+(E.8)
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+

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+Multi-task Highly Adaptive Lasso
+Ivana Malenica 1 Rachael V. Phillips 2 Daniel Lazzareschi 3 Jeremy R. Coyle 4 Romain Pirracchio 3
+Mark J. van der Laan 2
+Abstract
+We propose a novel, fully nonparametric approach
+for the multi-task learning, the Multi-task Highly
+Adaptive Lasso (MT-HAL). MT-HAL simultane-
+ously learns features, samples and task associ-
+ations important for the common model, while
+imposing a shared sparse structure among sim-
+ilar tasks. Given multiple tasks, our approach
+automatically finds a sparse sharing structure.
+The proposed MTL algorithm attains a power-
+ful dimension-free convergence rate of op(n−1/4)
+or better. We show that MT-HAL outperforms
+sparsity-based MTL competitors across a wide
+range of simulation studies, including settings
+with nonlinear and linear relationships, varying
+levels of sparsity and task correlations, and differ-
+ent numbers of covariates and sample size.
+1. Introduction
+Multi-task Learning (MTL) is a machine learning paradigm
+in which multiple tasks are simultaneously learned by a
+shared model. Originally motivated by insufficient data is-
+sues, MTL proved broadly effective over the years even as
+n got large —- attracting a lot of attention in the artificial in-
+telligence and machine learning (ML) communities (Ruder,
+2017; Crawshaw, 2020; Zhang & Yang, 2022). Applications
+of MTL are extensive and cross-disciplinary, ranging from
+reinforcement learning and speech recognition, to bioinfor-
+matics and systems biology (Cipolla et al., 2018; Tang et al.,
+2020; Dizaji et al., 2021; Sodhani et al., 2021; Zhang &
+Yang, 2022; Wang & Sun, 2022). Deep multi-task learning
+models have become particularly prevalent in computer vi-
+sion and natural language processing (Misra et al., 2016;
+Liu et al., 2019; Vandenhende, 2022).
+1Department of Statistics, Harvard University, Cambridge, MA,
+USA 2Division of Biostatistics, University of California at Berke-
+ley, Berkeley, CA, USA 3Department of Anesthesia and Periopera-
+tive Care, University of California at San Francisco, San Francisco,
+CA, USA 4Preva Group, Seattle, WA, USA. Correspondence to:
+Ivana Malenica <[email protected]>.
+The wide application of MTL can be attributed to its con-
+nection to how humans learn: a single complicated process
+can often be divided into several related (sub)tasks. Due to
+their common latent structure, learning from all (sub)tasks
+simultaneously proves to be more advantageous than simply
+learning each one independently. In particular, MTL utilizes
+shared knowledge to improve generalization performance,
+as it allows for common representations to be effectively
+leveraged by all considered tasks. This is in contrast to an-
+other ML paradigm with a similar setup, known as transfer
+learning — which seeks to improve performance of a sin-
+gle task via knowledge transfer from source tasks (Zhuang
+et al., 2019). Having access to kindred processes, MTL
+learns more robust representations — resulting in better
+knowledge sharing, and lower risk of catastrophic covariate
+shifts and overfitting. In addition, multi-task learning (1)
+increases the effective sample size for fitting a model, (2)
+ignores the task data-dependent noise, (3) biases the model
+to prefer representations that other tasks also prefer and (4)
+allows for “eavesdroping” across related tasks, as some rela-
+tionships might be easier to learn in particular tasks (Ruder,
+2017). Shared representations also increase data efficiency,
+and can potentially yield faster learning speed and better
+generalizations for future tasks (Crawshaw, 2020).
+The key challenge in MTL is what, how and when to share
+the common structure among all (or some subset of) tasks
+(Zhang & Yang, 2022). In the following, we pay particular
+attention on what to share, which implies how. More specif-
+ically, the MTL literature on common task representation
+typically focuses on instance, parameter or feature sharing.
+Instance-based MTL aims to detect important occurrences
+which can be useful for other tasks. Parameter-based MTL
+shares coefficients, weights or layers to help learn model
+parameters; the most common approaches include low-rank,
+task clustering, task relation learning and decomposition
+approaches (Zhang & Yang, 2022). In particular, the task
+relation approach learns model parameters and pairwise
+task relations simultaneously (Zhang et al., 2010; Zhang
+& Yeung, 2014). The feature-based MTL learns common
+predictors across tasks as a way to share knowledge. As
+tasks are related, it’s intuitive to assume that different tasks
+share some common feature representation. One particularly
+interesting way of thought is to learn a subset of original
+arXiv:2301.12029v1  [stat.ML]  27 Jan 2023
+
+Multi-task Highly Adaptive Lasso
+2
+features as the joint task representation; this has prompted
+a rich literature on sparse structure learning (Lounici et al.,
+2009; Obozinski et al., 2011; Sun et al., 2019; Wang & Sun,
+2022).
+In this work, we present the Multi-task Highly Adaptive
+Lasso (MT-HAL) algorithm, which represents a fully non-
+parameteric method for MTL. Most modern ML algorithms
+assume similar behavior for all points sufficiently close ac-
+cording to a given metric, referred to as local smoothness
+assumptions (e.g., adaptive selection of the size of a neigh-
+borhood). Reliance on local smoothness is often necessary
+in order to ensure favorable statistical properties of the esti-
+mator, especially in high dimensions. However, too much
+reliance on it can render an estimator inefficient. Instead
+of relying on parametric or local smoothness assumptions,
+the proposed MTL algorithm builds on the Highly Adaptive
+Lasso (HAL): a nonparametric estimator with remarkable
+convergence rates that does not rely on local smoothness
+assumptions and is not constructed using them (Benkeser
+& van der Laan, 2016; van der Laan, 2017). Instead, HAL
+restricts the global measure of smoothness by assuming that
+the truth is cadlag (right-hand continuous with left-hand
+limits) with a bounded sectional variation norm (“HAL” or
+H class). To the best of our knowledge, HAL is the only
+proven method which converges quickly enough for a large
+class of functions, independent of the dimension of the co-
+variate space (Tsiatis, 2006; van der Laan, 2017; Schuler &
+van der Laan, 2022).
+The proposed algorithm, MT-HAL, is a minimum loss-based
+estimator in the HAL class of functions with a bounded vari-
+ation norm. It depends on data-dependent basis functions
+constructed over all samples, predictors and tasks. The con-
+structed basis functions incorporate task interactions across
+available samples and predictors. In contrast to the usual
+feature-based MTL algorithms, MT-HAL does not require
+a parametric specification of the relationship between pre-
+dictors and outcome. MT-HAL also does not require any
+functional form, model or prior knowledge on the relation-
+ship between the different tasks. If prior knowledge on task
+relationships is available, functional forms can be supplied
+to MT-HAL and the generated basis functions will respect
+the provided form. Roughly speaking, MT-HAL restricts
+the amount of true function variation over its domain in a
+global sense, instead of locally. Practically speaking, it re-
+lies on the mixed-norm l2,1 to bound the variation norm and
+produce an unique solution with a similar sparsity pattern
+across tasks.
+At a high-level, MT-HAL operates as a fully nonparametric
+mixture of feature learning and task relation approach in
+MTL. It learns common data-dependent basis functions (aka,
+features based on original predictors), which are robust and
+invariant to all available tasks. To avoid including unrelated
+processes, MT-HAL also includes task interactions as a task
+membership indicator and across the domain space of each
+task. As such, it operates as a task relation parameter-based
+MTL as well — it learns pairwise task relations simultane-
+ously as model parameters. The final basis function coeffi-
+cients therefore give us an interpretable insight into a subset
+of features (and even feature domain), samples, and task
+relationships relevant for the shared model. The learned
+common representation reflects a joint sparsity structure as
+well, due to the final cross-validation based bound on the
+true variation norm.
+Despite being a powerful fully nonparametric approach to
+MTL, MT-HAL still preserves its dimension-free op(n−1/4)
+rate even in a multi-task setting. As such, MT-HAL is
+not only suited for finding a common model among tasks,
+but has convergence rates necessary for semi-parametric
+efficient estimation as well. We demonstrate superior per-
+formance of MT-HAL across many different simulations,
+testing how (non)linearity, level of sparsity, task relatedness,
+dimension of covariate space and sample size affect its per-
+formance. Finally, we apply MT-HAL to the benchmark
+Parkinson’s disease dataset, which predicts disease clinical
+symptom scores for each patient based on biomedical voice
+measurements (Tsanas et al., 2009; Jawanpuria & Nath,
+2012).
+2. Formulation of the Statistical Problem
+2.1. Data and Likelihood
+Let Ok denote data on the kth task, such that Ok =
+(T k, Xk, Y k) = (W k, Y k) for k = 1, . . . , K. Each Ok
+is of a fixed dimension, and an element of a Euclidean set
+O. We assume that different tasks for each k = 1, . . . , K,
+and the corresponding data {Ok}K
+k=1, are possibly related
+to each other.
+Let Xk denote a vector of baseline co-
+variates (“predictors”, “features”) for the kth task, such
+that Xk ∈ RPk and Xk = (Xk
+1 , . . . , Xk
+Pk). We denote
+nk as the number of samples for task k, which could po-
+tentially be different for each task. Then, we have that
+Xk
+p = {Xk
+1,p, . . . , Xk
+nk,p} where Xk
+i,p represents data on
+covariate p for sample i in task k. Without loss of generality,
+we assume all predictors in Xk have mean zero and standard
+deviation one, but otherwise make no assumptions on the
+dimension, form or correlation structure for the covariate
+process. As such, for all predictors p, we let
+nk
+�
+i=1
+Xk
+i,p = 0 and
+nk
+�
+i=1
+(Xk
+i,p)2 = 1.
+Further, we define Y k as a vector of outcomes, where
+Y k = (Y k
+1 , . . . , Y k
+nk), and Y k is a nk × 1 vector. We also
+specifically define T k as categorical variable indicating kth
+task membership: if sample i is part of task k, then T k
+i = k.
+
+Multi-task Highly Adaptive Lasso
+3
+If we want to emphasize the set of all covariates in task k,
+we write W k = (T k, Xk).
+In order to define the combined set of all tasks, we specify a
+more compact notation where n = �K
+k=1 nk. In particular,
+let Y = (Y 1, . . . , Y K) and T = (T 1, . . . , T K) where Y
+and T are n × 1 vectors. The matrix of covariates, X =
+(X1, . . . , XK) is a block-diagonal matrix with Xk being
+the kth block. Let d = �K
+k=1 Pk, so that X is a n × d
+matrix; if all tasks have the same covariates, then we define
+d as d = P. Consequently, we define n independent and
+identically distributed (i.i.d.) observations of Oi for i ∈ [n]
+where Oi = (Ti, Xi, Yi) and On = (O1, . . . , On) ∼ P0.
+With that, we assume that all K tasks are sampled from
+the same data-generating distribution P0, which is unknown
+and unspecified. As such, Oi reflects observed data for unit
+i sampled from P0, corresponding to a task specified by Ti.
+We denote by M the statistical model, which represents
+the set of laws from which On can be drawn. Intuitively,
+the more we know (or are willing to assume) about the
+experiment that produces the data, the smaller the statistical
+model M. In this work, we define M as a nonparametric
+model, with P0 ∈ M. We impose no assumptions on the
+true data-generating distribution P0, except that all tasks
+possibly share some unspecified structure. We further define
+P as any distribution which also lies in M, such that P ∈
+M. Let p0 denote the density of P0 with respect to (w.r.t) a
+measure µ that dominates all elements of M. The likelihood
+of On, p0(On), can be factorized according to the time-
+ordering as follows:
+n
+�
+i=1
+p0,(t,x)(Ti, Xi)p0,y(Yi | Ti, Xi),
+(1)
+where p0,(t,x) marks the probability density for all the base-
+line covariates, and p0,y is the conditional density of the
+outcome given all the predictors. At times, it proves useful
+to use notation from empirical process theory. Specifically,
+we define Pf to be the empirical average of the function
+f w.r.t. the distribution P, that is, Pf =
+�
+f(o)dP(o). We
+use Pn to denote the empirical distribution of the sample,
+which gives each observation weight 1/n, irrespective of
+the task membership.
+2.2. Target Parameter
+We define the relevant feature of the true data distribution
+we are interested in as the statistical target parameter (short,
+target parameter or estimand). We define a parameter map-
+ping Ψ : M → R, and a parameter value ψ := Ψ(P) for
+any given P ∈ M. The estimate, evaluated at (T, X), is
+written as ψ(T, X) := Ψ(P)(T, X). In some cases, we
+might be interested in learning the entire conditional distri-
+bution P0,y. However, frequently the actual goal is to learn
+a particular feature of the true distribution that satisfies a
+scientific question of interest. In a multi-task problem, we
+are interested in multivariate prediction of the entire set of
+outcomes collected, given the task and observed covariates.
+The target parameter then corresponds to
+Ψ(P0)(T, X) = EP0(Y | T, X) = EP0(Y | W),
+(2)
+where the expectation on the right hand side is taken
+w.r.t the truth, and the true parameter value is denoted
+as ψ0 = Ψ(P0). In words, we are interested in learning
+(T, X) �→ Ψ(P0)(T, X) using all the data and available
+tasks. The prediction function for unit i is then obtained
+with Ψ(P)(Ti, Xi), for any P ∈ M and i ∈ [n].
+2.3. Loss-based Parameter Definition
+We define L as a valid loss function, chosen in accordance
+with the target parameter. Specifically, we refer to a valid
+loss for a given target as a function whose expectation w.r.t.
+P0 is minimized by the true value of the parameter. Our
+accent on appropriate loss functions strives from their multi-
+ple use within our framework — as a theoretical criterion
+for comparing the estimator and the truth, as well as a way
+to compare multiple estimators of the target parameter (van
+der Laan & Dudoit, 2003; Dudoit & van der Laan, 2005;
+van der Vaart et al., 2006; van der Laan et al., 2006).
+Let L be a loss adapted to the problem in question,
+i.e.
+a function that maps every Ψ(P) to L(Ψ(P)) :
+(k, Xi, Yi) �→ L(Ψ(P))(k, Xi, Yi); note that we can equiv-
+alently write L(Ψ(P))(k, Xi, Yi) as L(Ψ(P))(Ti, Xi, Yi)
+or just L(Ψ(P))(Wi, Yi) for short notation. We define the
+true risk as the expected value of L(Ψ(P))(Ti, Xi, Yi) w.r.t
+the true conditional distribution P0 across all individuals
+and tasks:
+R(P0, ψ) = EP0[L(Ψ(P))(T, X, Y )].
+The notation for the true risk, R(P0, ψ), emphasizes that ψ
+is evaluated w.r.t. the true data-generating distribution (P0).
+As specified by the definition of a valid loss, we define ψ0
+as the minimizer over the true risk of all evaluated ψ in the
+parameter space ΨΨΨ,
+ψ0 = argmin
+ψ∈Ψ
+Ψ
+Ψ
+R(P0, ψ).
+The estimator mapping, ˆΨ, is a function from the em-
+pirical distribution to the parameter space. In particular,
+let Pn,K denote the empirical distribution of n samples
+collected across K tasks. Then, Pn,K �→ ˆΨ(Pn,K) rep-
+resents a mapping from Pn,K into a predictive function
+ˆΨ(Pn,K). Further, the predictive function ˆΨ(Pn,K) maps
+(Ti, Xi) into a subject-specific outcome, Yi.
+We write
+ψn(Ti, Xi) := ˆΨ(Pn,K)(Ti, Xi) as the predicted outcome
+for unit i of the estimator ˆΨ(Pn,K) based on (Ti, Xi). Fi-
+nally, we note that the true risk establishes a true measure
+
+Multi-task Highly Adaptive Lasso
+4
+of performance of the estimator. In order to obtain an un-
+biased estimate of the true risk, we resort to appropriate
+cross-validation (CV).
+2.4. Cross-validation
+Let C(i, k) denote, at a minimum, the task k- and unit i-
+specific record C(i, k, ·) = (Oi, k, ·). To derive a general
+representation for cross-validation, we define a split vector
+Bn, where Bn(i, k) ∈ {0, 1}n. A realization of Bn de-
+fines a particular split of the learning set into corresponding
+disjoint subsets,
+Bv
+n(i, k) =
+�
+0, C(i, k) in the training set
+1, C(i, k) in the validation set,
+where Bv
+n(i, k) denotes a v-fold assignment of unit i and
+task k for split Bv
+n. For example, we can partition the full
+data into V splits of approximately equal size, such that
+each fold v for v = 1, . . . , V has a balanced number of tasks
+members. We define P 0
+n,K,v and P 1
+n,K,v as the empirical
+distribution of the training and validation sets corresponding
+to sample split v, respectively. To alleviate notation, we let
+all sample indexes and tasks used for training be an element
+of B0
+v, and B1
+v if in a validation set for fold v. The total
+number of samples in the training and validation set are then
+written as n0 and n1.
+3. Highly Adaptive Lasso
+We define the multi-task setup and its objective as a statis-
+tical parameter estimated via a loss-based paradigm in a
+nonparametric model, which involves minimizing the cross-
+validated risk. In the following, we elaborate on a specific
+algorithm with favorable statistical properties we build on
+in order to accommodate the multi-task objective.
+Most modern nonparametric machine learning algorithms,
+including neural networks, tree-based models and histogram
+regression, assume similar behavior for all points sufficiently
+close according to a given metric (Benkeser & van der Laan,
+2016). We refer to such assumption as local smoothness,
+which typically translates to implicit or adaptive selection
+of the size of a neighborhood and continuous differentia-
+bility. Reliance on local smoothness is often necessary in
+order to ensure favorable statistical properties of the esti-
+mator, especially in high dimensions. However, too much
+reliance on local smoothness can render an estimator inef-
+ficient. While some methods provide a way to calculate
+neighborhood size that optimizes the bias-variance trade-off
+(e.g., kernel regression), it is generally unknown how to
+adjust local smoothness assumptions when constructing a
+nonparameteric estimator.
+In contrast to most commonly used machine learning meth-
+ods, the Highly Adaptive Lasso (HAL) is a nonparametric
+estimator that does not rely, and is not constructed, using
+local smoothness assumptions (Benkeser & van der Laan,
+2016; van der Laan, 2017). Instead, HAL restricts a global
+measure of smoothness by assuming that the true target
+parameter is cadlag (right-hand continuous with left-hand
+limits) with a bounded sectional variation norm (“HAL” or
+H class). In practice, such assumptions prove to be very
+mild as (1) cadlag functions are very general, even allowing
+for discontinuities; (2) the variation norm can be made ar-
+bitrarily large and adapted to the problem (Schuler & van
+der Laan, 2022). To the best of our knowledge, HAL is the
+only algorithm with fast-enough convergence rates to allow
+for efficient inference in nonparametric statistical models
+regardless of the dimension of the problem. In particular,
+its minimum rate does not depend on the underlying lo-
+cal smoothness of the true regression function, ψ0. In the
+following, we give a brief theoretical and practical descrip-
+tion of HAL, necessary in order to understand the proposed
+algorithm. The Highly Adaptive Lasso for the multi-task
+objective is presented in Section 4.
+3.1. Cadlag functions with finite variation norm
+We assume that ψ0 ∈ H, where H is a set of d-variate
+real valued cadlag functions with τ as an upper bound on
+all support. A function in H is right-continuous with left-
+hand limits, as well as left-continuous at any point on the
+right-edge of [0, τ]. In addition, we also assume that ψ0 has
+a finite sectional variation norm bounded by some univer-
+sal constant M < ∞. Note that any cadlag function with
+bounded variation norm generates a finite measure, result-
+ing in well defined integrals w.r.t the function. With that,
+we define a sectional variation norm of a multivariate real
+valued cadlag function ψ as
+∥ψ∥var = ψ(0) +
+�
+s⊂{1,...,d}
+� τs
+0s
+|ψs(dus)|,
+where the sum is taken over all subsets of {1, . . . , d}. In
+particular, for a given subset of s, we define us = (uj :
+j ∈ s) and u−s = (uj : j /∈ s). To illustrate for X =
+(X1, X2) and s = 1, we have that Xs = {X1} and X−s =
+{X2}. We define the section ψs(us) ≡ ψ(us, 0−s), where
+ψs varies along the variables in us according to ψ, but sets
+the variables in u−s to zero.
+3.2. Minimum Loss-based Estimator in the HAL class
+We denote a class of functions HM as HM
+= {ψ :
+∥ψ∥var < M}, where M is an arbitrary large and unknown
+constant. As introduced in subsection 2.3, we define the
+true minimizer of the average loss in class HM as
+ψ0,M = argmin
+ψ∈HM
+R(P0, ψ),
+
+Multi-task Highly Adaptive Lasso
+5
+with the MLE in HM defined as
+ψn = ψn,M = argmin
+ψ∈HM
+1
+n
+n
+�
+i=1
+L(ψ)(Ti, Xi, Yi).
+We emphasize that if ∥ψ0∥var < M, then ψ0,M = ψ0
+as ψ0 ∈ HM. As most functions with infinite variation
+norm tend to be pathological (e.g., sin(1/x) or x sin(1/x)),
+having ∥ψ0∥var < M is a very mild assumption.
+The true bound on the sectional variation norm is, however,
+unlikely to be known in practice. One way of choosing M
+is based on the cross-validated choice of the bound. To start,
+we consider a grid of potential values, where M1 is the small-
+est and MB the largest bound such that ∥ψ0∥var < MB. For
+each b = 1, . . . , B corresponding to bounds M1, . . . , MB,
+we have the parameter value ψn,Mb = ˆΨMb(Pn,K), where
+ψn,Mb is the MLE in the H class with variation norm smaller
+than Mb (HMb). Referring to the cross-validation notation
+introduced in subsection 2.4, the cross-validation selector
+Mn of M is the grid value with the lowest estimated cross-
+validated risk:
+Mn = argmin
+b
+1
+V
+V
+�
+v=1
+EP 1
+n,K,vL(ˆΨMb(P 0
+n,K,v)).
+Finally, in order to study the difference between an estima-
+tor and the truth (relevant for theoretical results presented in
+subsection 4.2), we construct loss-based dissimilarity mea-
+sures. In particular, for some ψ ∈ H, we define d0(ψ, ψ0)
+as the loss-based dissimilarity measure corresponding to the
+squared error loss where
+d0(ψ, ψ0) = EP0[L(ψ) − L(ψ0)] = ∥ψ − ψ0∥2
+P0.
+(3)
+4. Multi-task Highly Adaptive Lasso
+Keeping in mind the underpinnings presented in Section
+3, how can we construct a HAL estimator for the multi-
+task objective? Practically, the original HAL is a minimum
+loss-based estimator in the H class. Instead of specifying
+a relationship between outcome and covariates, HAL uses
+a special set of data-dependent basis functions. As such,
+all the estimation is done completely nonparametrically,
+only restricting the global amount of fluctuation of the true
+function over its domain (instead of locally).
+For the multi-task problem however, we want all tasks to be
+learned simultaneously, instead of independently. We also
+want it to be agnostic to the number of predictors and sam-
+ples per task, as well as to learn task-specific associations
+at the same time as the shared model. In case parameters
+for different tasks share the same sparsity pattern, we want
+the proposed algorithm to be able to learn that structure too.
+Hence, when estimating the target parameter in Equation
+(2), we opt for a joint sparsity regularization, instead of pe-
+nalizing each task separately. Similarly to the multivariate
+group lasso (Yuan & Lin, 2006; Obozinski et al., 2011; Si-
+mon & Tibshirani, 2012), we consider a l1/lq mixed-norm
+penalty with q > 1, which for l1/l2 corresponds to
+∥α∥2,1 :=
+�
+�
+d
+�
+p=1
+� K
+�
+k=1
+|αk
+p|2
+�1/2�
+� =
+d
+�
+p=1
+∥αp∥2,
+for some vector of coefficients α, K tasks, and d predictors.
+The αk
+p then represents coefficients for predictor p and task
+k. Intuitively, norm ∥αp∥2 has the effect of enforcing the el-
+ements of αp to achieve zeros simultaneously. The objective
+of the l2,1 norm is to minimize the l2 norm of the columns,
+followed by the l1 norm over all predictors — resulting in
+a matrix with a sparse number of columns and a small l2
+norm. The mixed-norm penalty allows us to impose a soft
+constraint on the structure of the sparse solution that we are
+looking for, thus promoting a group-wise sparsity pattern
+across multiple tasks.
+4.1. Algorithm
+In order to present the full algorithm, we first note that a
+function with finite variation norm can be represented as
+the sum over subsets of an integral w.r.t. a subset-specific
+measure. Also, each subset-specific measure can be ap-
+proximated by a discrete measure over support points; we
+elaborate on this derivation in the Appendix. As a conse-
+quence, we can define support by the actual n observations
+across K tasks, which reduces the optimization problem
+to a finite-dimensional one. Let Φ : W → {0, 1}N de-
+fine a mapping where neither Φ or N are pre-specified, but
+instead completely determined by the data (and thus data-
+adaptive). With that, let φs,t denote a particular basis func-
+tion, where s is a section in {1, . . . , d}, and t a knot point
+corresponding to observed Wi. To enumerate all bases, we
+have that Φ(W) = [φs1,t1(W), . . . , φsN,tN (W)], resulting
+in a total of N basis functions, where N = n(2d − 1). We
+emphasize that the bases depend only on observed values,
+across all samples and tasks. In particular, let ˜ws,i denote
+an observed value of W, where ˜ws,i = { ˜wc,i : c ∈ s} for
+subset s with i = 1, . . . , nk in task k. Then, we define
+φs,i = 1( ˜ws,i ≤ ws), where Φ constructs all the bases that
+can be constructed using each observed Wi as a knot point,
+across all possible sections. As such, applying Φ row-wise
+to W results in an output Φ(W) of zeros and ones (as per
+indicator function dependent on observed values across all
+samples and tasks) , which is of �k
+k=1 nk × N dimension:
+Φ(
+�
+��
+W 1
+...
+W k
+�
+��
+�
+��
+�
+n×d
+) →
+�
+��
+1
+. . .
+0
+0
+. . .
+0
+...
+0
+. . .
+0
+1
+. . .
+1
+�
+��
+�
+��
+�
+n×N
+
+Multi-task Highly Adaptive Lasso
+6
+Note that passing all the tasks to Φ at once allows us to share
+basis functions across tasks. What’s more, it allows us to
+create task-interaction bases with separate coefficients. We
+consider a minimization problem over all linear combina-
+tions of the basis functions w → φs,i(w) = 1( ˜ws,i ≤ ws)
+and corresponding coefficients βs,i, summed over all sub-
+sets s and for each i (as such, over all tasks). Here we
+emphasize that the sum of the absolute values of the coeffi-
+cients is the variation norm, and must therefore be bounded.
+A discrete approximation of ψ with support defined by the
+observed n points may then be constructed as
+ψβ = β0 +
+�
+s⊂{1,...,d}
+K
+�
+k=1
+nk
+�
+i=1
+βs,i1( ˜ws,i ≤ ws)
+= β0 +
+�
+s⊂{1,...,d}
+K
+�
+k=1
+nk
+�
+i=1
+βs,iφs,i,
+where β0 is the intercept corresponding to ψ(0). The empir-
+ical minimization problem over all discrete measures with
+support defined by the observed samples then corresponds
+to
+βn =
+argmin
+β,∥β∥2,1<M
+PnL(ψβ),
+where we remind that Pn denotes the empirical distribution
+of the sample and β is a vector of coefficients for each basis
+with a subspace
+Hn,M =
+�
+Φ(W)β
+s.t. ∥β∥2,1 ≤ M.
+As our parameter of interest is a conditional mean, we could
+use the weighted square error to define the loss. Accordingly,
+for the ith sample we have that
+L(ψβ)(Wi, Yi) = gk
+i (Y k
+i − ψβ(W k
+i ))2,
+with gk
+i as the sample- and task- specific weight. The multi-
+task HAL estimator is then formulated by minimizing the
+squared loss with the l2,1 penalty:
+ψβ = argmin
+ψβ∈HM
+K
+�
+k=1
+nk
+�
+i=1
+L(ψβ)(Wi, Yi) + λ
+N
+�
+p=1
+∥βp∥2,
+where
+∥βp∥2 =
+�
+�
+�
+�
+K
+�
+k=1
+|βkp|2 =
+�
+�
+�
+�
+K
+�
+k=1
+nk
+�
+i=1
+|βi,p|2.
+4.2. Theoretical Properties
+For the conditional mean as a target parameter, the MLE
+(ψn) converges to its M-specific truth in loss-based dissimi-
+larity at a rate faster than n1/2 (equivalently, no slower than
+Algorithm 1 Multi-task Highly Adaptive Lasso
+1: Input: data Ok = (W k, Y k) for tasks k = 1, . . . , K,
+loss function L, number of CV folds V
+Algorithm:
+2: Create a combined set of tasks O = (W, Y ).
+3: Define a data-adaptive mapping Φ : W → {0, 1}N.
+4: Generate basis functions across all tasks, predictors and
+samples: Φ(W) = [φs1,1, . . . , φsN,n].
+5: Define a grid M1, . . . , MB where ∥ψ0∥var < MB.
+6: for v = 1 to V do
+7:
+for b = 1 to B do
+8:
+Empirical minimization for the training set:
+βn.v,b =
+argmin
+β,∥β∥2,1<Mb
+P 0
+n,K,vL(ψβ)
+9:
+end for
+10: end for
+11: Pick the CV selector in the validation set:
+Mn = argmin
+b∈{1,...,B}
+1
+V
+V
+�
+v=1
+P 1
+n,K,vL(ψβn.v,b).
+n1/4 w.r.t to a norm l2 under P0). This result holds even
+if the parameter space is completely nonparametric, and,
+remarkably, regardless of the dimension of W. As such,
+the minimum rate of convergence does not depend on the
+underlying smoothness of ψ0, as is typically the case for
+machine learning algorithms. The result still applies even if
+ψ0 is non-differentiable (Benkeser & van der Laan, 2016;
+van der Laan, 2017). Under weak continuity conditions, the
+HAL based estimator is also uniformly consistent (van der
+Laan & Bibaut, 2017). Theorem 4.4 formalizes the theoret-
+ical properties of HAL with a multi-task objective; in the
+Appendix, we show that the proposed formulation adapted
+to the multi-task problem is another valid way of bounding
+the sectional variation norm. As such, the multi-task HAL
+is a fast converging algorithm regardless of the dimension
+of the covariate space or the considered number of tasks K.
+Assumption 4.1.
+sup
+ψ∈HM
+∥L(ψ)∥var
+∥ψ∥var
+< ∞.
+Assumption 4.2.
+sup
+ψ∈HM
+∥L(ψ) − L(ψ0,M)∥2
+P0
+∥ψ0 − ψ0,M∥2
+P0
+< ∞.
+Assumption 4.3.
+sup
+ψ∈H,o∈O
+|L(ψ)(o)| < ∞.
+
+Multi-task Highly Adaptive Lasso
+7
+Theorem 4.4 (Rate of convergence). For a square error
+loss and under Assumptions (4.1) and (4.2), we have that
+∥ψn,M − ψ0,M∥P0 = OP (n−1/4−1/(8(d+1))).
+If Mn is picked based on minimizing the CV risk, under
+additional Assumption (4.3), we then have that
+∥ψn,Mn − ψ0,M∥P0 =OP (n−1/4−1/(8(d+1)))
++ OP (n−1/2 log B),
+for b = 1, . . . , B where MB corresponds to the largest
+bound on the variation norm in a grid with ∥ψ0∥var < MB.
+Proof. Let l(ψn, ψ0)
+=
+L(ψn,M) − L(ψ0,M), where
+l(ψn, ψ0) falls in the P0-Donsker class with a variation
+norm smaller than M. Then,
+0 ≤ d0(ψn,M, ψ0,M) = EP0[l(ψn, ψ0)]
+= −EPn[l(ψn, ψ0)] + EP0[l(ψn, ψ0)] + EPn[l(ψn, ψ0)]
+= −(Pn − P0)[l(ψn, ψ0)] + Pn[l(ψn, ψ0)]
+≤ −(Pn − P0)[l(ψn, ψ0)],
+where first inequality follows from the definition of a
+loss-based dissimilarity defined in Equation 3.
+By As-
+sumptions (4.1) and (4.2), we have that P0[l(ψn, ψ0)] and
+∥l(ψn, ψ0)∥2
+P0 are Op(n−1/2). By results in empirical pro-
+cess theory, we know that √n(Pn − P0)l(ψn, ψ0) →p 0
+and P0l(ψn, ψ0)2 →p 0 as n → ∞ since l(ψn, ψ0) is
+Donsker (van der Vaart & Wellner, 2013). Therefore, we
+have that (Pn − P0)[l(ψn, ψ0)] = oP (n−1/2). As 0 ≤
+d0(ψn,M, ψ0,M) ≤ −(Pn − P0)[l(ψn, ψ0)], it follows that
+∥ψn,M −ψ0,M∥2
+P0 = oP (n−1/2) and ∥ψn,M −ψ0,M∥P0 =
+oP (n−1/4). If M is not known, it remains to compare
+the performance of ψn,Mn (where Mn is the choice of M
+which minimizes the CV risk) to the oracle M ∗
+n (where M ∗
+n
+is the choice of M which minimizes the true CV risk). The
+OP (n−1/2 log B) comes from oracle results for the CV se-
+lector w.r.t. a loss-based dissimilarity (Dudoit & van der
+Laan, 2005; van der Laan & Dudoit, 2003; van der Vaart
+et al., 2006; van der Laan et al., 2006). For the precise rate,
+we refer to van der Laan (2017).
+The
+result
+of
+Theorem
+4.4
+shows
+that,
+roughly,
+do(ψn,M, ψ0,M) = oP (n−1/2) or ∥ψn,M − ψ0,M∥P0 =
+oP (n−1/4).
+What’s more, Theorem 4.4 states that by
+choosing B such that n−1/2 log B → 0 as n → ∞, we
+preserve the rate even when the true variation norm is
+unknown.
+Note that if ∥ψ0∥var < M, it follows that
+∥ψn,M − ψ0∥P0 = OP (n−1/4−1/(8(d+1))).
+5. Simulations
+In this section, we report performance of the proposed multi-
+task HAL in various simulation settings. In total, we con-
+sider four different data-generating distributions and test
+popular multi-task algorithms based on sparsity assumptions
+in addition to MT-HAL (MT-lasso and MT-L21). Simula-
+tion setups are indicated by the first letter of each setting: (1)
+“N” for nonlinear data-generating process; (2) “H” for high
+level of sparsity (60%) vs. “L” for low sparsity (20%); (3)
+“S” for the same level of sparsity across tasks vs. “D” for
+different sparsity levels. As such, simulation setup “NHS”
+stands for nonlinear DGP with high level of sparsity that
+is shared across all tasks. In all simulations we consider
+K = 5 tasks and different number of samples for each task
+(n = {100, 100, 150, 150, 100} corresponding task 1 to 5).
+Although the number of covariates was the same across
+all tasks (d = 6), we emphasize that our setup allows for
+different (and non-overlapping) set of covariates. For each
+multi-task method and simulation setup, we report the mean
+squared error (MSE) and variable selection performance
+(precision and accuracy), calculated on a separate test data
+over 100 Monte Carlo simulations. Note that, in this setting,
+true positives denote the true estimated nonzero coefficients.
+Therefore, precision is calculated as true positives/(true pos-
+itives + false positives), whereas accuracy is reported as the
+number of true positives/(true positives + false negatives).
+The data-generating processes corresponding to simulation
+“NHS”, “NLS”, “NHD” and “NLD” are nonlinear models
+with d = 6 covariates and normally distributed coefficients.
+In particular, true coefficients for simulation “NHS” are
+β0,NHS = (β0,1, . . . , β0,|NHS|, 0, . . . , 0), where each β0,j
+for 1 ≤ j ≤ |NHS| is sampled from a standard normal
+distribution. Similarly, the error term is normally distributed
+(sampled from N(0, 0.1)), with coefficient 0.3. Covariates
+with nonzero coefficients, constituting of binary, categorical
+and continuous variables, are transformed via logarithmic,
+cosine and squared operations of predictor interactions. For
+simulations with the same level of sparsity across tasks,
+highest level of sparsity was 60% (corresponding to around
+3 nonzero coefficients per task), whereas the lowest was
+20% (5 nonzero coefficients per task). For different sparsity
+profiles across tasks, we considered random deviations from
+true level of sparsity across different tasks: 40% − 80% and
+0.05%−40% for high and low level of sparsity, respectively.
+Compared to other sparsity-based multi-task algorithms,
+MT-HAL results in the lowest MSE across all considered
+data-generating processes, often reducing the MSE by more
+than a half. On average, it also tends to produce less false
+negatives than considered competitors, resulting in high co-
+efficient accuracy and comparable precision (slightly more
+false positives on average in our simulations: this is to be
+expected due to a rich space considered, and could be me-
+diated with a finer grid). All algorithms perform best in
+low sparsity settings, particularly if the sparsity structure
+is shared across tasks. We report results for nonlinear sim-
+ulations at n = 600 in Table (1). Additional simulation
+results are available in the Appendix, and include perfor-
+
+Multi-task Highly Adaptive Lasso
+8
+mance of the proposed method in the following settings: (1)
+different sample sizes, including very small n; (2) linear
+data-generating processes with interactions and (3) high-
+dimensional nonlinear setup.
+Table 1. Mean squared error (MSE), precision (“Prec”) and accu-
+rarcy (“Accu”) for each of the nonlinear simulation setups with
+d = 6 covariates, K = 5 tasks, and total of n = 600 samples split
+between the K tasks as {100, 100, 150, 150, 100}.
+SETUP
+METHOD
+MSE
+PREC %
+ACCU %
+NHS
+MT-HAL
+0.68
+58.7
+75.0
+NHS
+MT-LASSO
+1.49
+64.4
+31.0
+NHS
+MT-L21
+1.48
+55.3
+40.9
+NLS
+MT-HAL
+0.68
+91.4
+80.5
+NLS
+MT-LASSO
+1.59
+98.5
+44.6
+NLS
+MT-L21
+1.54
+97.6
+60.5
+NHD
+MT-HAL
+0.45
+61.9
+69.0
+NHD
+MT-LASSO
+0.74
+65.5
+28.0
+NHD
+MT-L21
+0.71
+53.5
+36.7
+NLD
+MT-HAL
+0.67
+87.9
+80.5
+NLD
+MT-LASSO
+1.55
+98.0
+45.2
+NLD
+MT-L21
+1.52
+89.5
+59.9
+6. Data Analysis
+Using a range of biomedical voice measurements obtained
+from 42 adults with early-stage Parkinson’s disease (PD),
+we predicted two clinical symptom scores, motor Unified
+Parkinson’s Disease Rating Scale (UPDRS) and total UP-
+DRS (Tsanas et al., 2009). These data come from a six-
+month trial that aimed to assess the suitability of measure-
+ments of dysphonia (impairment of voice production) for
+telemonitoring of PD. The repeated measures data contains
+5,875 voice recordings from the 42 individuals. The follow-
+ing variables were considered as covariates for predicting
+the two outcomes: subject identifier, age, sex, and sixteen
+biomedical voice measures.
+For each MTL estimator considered in the simulations,
+we examined predictive performance in the data analysis
+in terms of the CV MSE. We considered a clustered CV
+scheme with respect to the subject identifier (each subject’s
+observations are always all together as training or test set ob-
+servations across all CV folds) of k-fold/V-fold CV with ten
+folds, and so the MSE reported represents an honest, inde-
+pendent evaluation on a test dataset that was not seen during
+training. The results from the data analysis are presented in
+Table (2).
+The MT-HAL resulted in the lowest cross-validated MSE
+out of all the MTL algorithms considered. However, we note
+that performance of all MTL algorithms in this setup is poor.
+This could be due to a low signal in collected covariates
+w.r.t. to the target outcomes. In a real-world application,
+we also typically advocate for the use of a CV-based “super
+Table 2. Predictive performance, measured in terms of the cross-
+validated mean squared error for motor UPDRS, total UPDRS and
+overall outcome, for each multi-task estimator considered in the
+data application for predicting of Parkinson’s disease symptom
+scores from biomedical voice measurements.
+METHOD
+MUPDRS
+TUPDRS
+OVERALL
+MT-HAL
+58.9
+101.0
+79.9
+MT-LASSO
+60.5
+103.9
+82.2
+MT-L21
+60.9
+104.2
+82.6
+learner” selector that, among a library of candidate MTL
+algorithms (including several variations of MT-HAL with
+different hyperparameter specifications), chooses the MTL
+algorithm with the lowest CV risk. A MTL “super learner”
+with a rich library of different MTL algorithms might be
+able to improve on the predictive performance, instead of
+considering each algorithm separately.
+7. Discussion
+In this work, we propose a fully nonparametric approach for
+the multi-task learning problem. Our proposed framework
+simultaneously learns features, samples and task associa-
+tions important for the common model, while imposing a
+shared sparse structure among similar tasks. The problem
+formulation imposes no assumptions on the relationship
+between predictors and outcomes, or task interconnections,
+other than a global bound on the variation norm and func-
+tion space; these assumptions prove to be gather general
+in nature, often resulting in pathological examples if not
+respected. The proposed MTL algorithm attains a powerful
+dimension-free op(n−1/4) (or better) convergence rate, and
+outperforms all considered sparsity-based MTL competitors
+across a wide range of DGPs: including nonlinear and lin-
+ear setups, different levels of sparsity and task correlations,
+dimension of covariate space and sample sizes.
+As part of future work, there are several directions to be
+explored. First, we can make the procedure more computa-
+tionally scalable by relying on empirical loss minimization
+within nested Donsker classes (Schuler & van der Laan,
+2022). For online or sequential data, one can consider for-
+mulating the multi-task problem as only part of the flexible
+ensemble model (Malenica et al., 2023). Along similar lines,
+the issue of “when to share” can be alleviated by formulating
+the question as a loss-based problem as well. In particular,
+one could define a problem setup where cross-validation is
+used to pick among single- and multi-task models, even con-
+sidering many different algorithms and task combinations.
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+Multi-task Highly Adaptive Lasso
+11
+A. Mixed-norm penalty controls the sectional variation norm
+In the following we give a sketch argument that the mixed-norm penalty l2,1 still controls the sectional variation norm of a
+cadlag function. First, let ψ ∈ H and ∥ψ∥var < ∞. Then by (Gill et al., 1995) we can write ψ(w) as
+ψ(w) = ψ(0) +
+�
+s⊂{1,...,d}
+� τs
+Os
+1(u ≤ ws)ψs(du).
+We can approximate ψ using a discrete measure ψm and m support points. In particular, let s denote a section in {1, . . . , d}
+with t a knot point, such that (us,t : t) denotes all the support. Correspondingly, let dψm,s,t denote the pointmass assigned
+to us,t by ψm, resulting in an approximation of ψm(w) written as
+ψ(0) +
+�
+s⊂{1,...,d}
+�
+t
+1(us,t ≤ ws)dψm,s,t.
+(4)
+Note that 1(us,t ≤ ws) is a basis function, with the corresponding coefficient dψm,s,t. The discrete approximation of ψ(w)
+is a linear combination of basis functions summed over all sections s and knots t. The sum of absolute values of dψm,s,t
+then corresponds to the variation norm of ψ,
+∥ψm∥var = ψ(0) +
+�
+s⊂{1,...,d}
+�
+t
+|dψm,s,t|.
+As defined in Section 4.1, let ˜ws,i denote an observed value ˜ws,i = { ˜wc,i : c ∈ s} for subset s with i = 1, . . . , nk in task k.
+Applying result in Equation (4) for the support defined by the nk samples for each task k, we derive to the following new
+approximation for ψ(w):
+ψ(0) +
+�
+s⊂{1,...,d}
+K
+�
+k=1
+nk
+�
+i=1
+1( ˜ws,i ≤ ws)dψm,s,t.
+where we can define φs,i = 1( ˜ws,i ≤ ws) and dψm,s,t = βs,i to ease notation. The variation norm of ψ is then
+∥ψm∥var = ψ(0) +
+�
+s⊂{1,...,d}
+K
+�
+k=1
+nk
+�
+i=1
+|βs,i|,
+which corresponds to the l1 norm for the following optimization problem
+ψn = argmin
+ψ∈HM
+PnL(ψ)
+with
+HM =
+�
+ψ ∈ H
+s.t. ∥ψ∥var ≤ M.
+If ψ ∈ H, we need the true variation norm of ψ to be smaller than some universal constant M, which is equivalent to the
+amount of ”allowance” given by the l1 penalty (allowed upper bound on the sum of absolute values of the coefficients). It’s
+straight forward to show that
+∥β∥1 =
+N
+�
+p=1
+∥βp∥1 =
+N
+�
+p=1
+� K
+�
+k=1
+�
+|βkp|2
+�
+≥
+N
+�
+p=1
+�
+�
+�
+�
+�
+�
+K
+�
+k=1
+|βkp|2
+�
+� =
+N
+�
+p=1
+∥βp∥2
+= ∥β∥2,1,
+proving that l2,1 norm is less than or equal to the variation norm, thus controlling the amount of allowed variation.
+
+Multi-task Highly Adaptive Lasso
+12
+B. Additional Simulations
+B.1. Different sample sizes
+In the following, we provide results of additional simulations at various sample sizes. In particular, we report performance
+of MT-HAL, MT-lasso and MT-L21 for n = (50, 100, 200, . . . , 800, 900, 1000). All additional simulations correspond to
+nonlinear data-generating processes (DGPs) as described in Section 5. We consider nonlinear DGPs across various n with
+(1) high level of sparsity (60%, “H”) vs. low sparsity (20%, “L” ); (2) tasks with the same level of sparsity (“S”) vs. distinct
+sparsity levels across tasks (“D”). For all simulations, we consider K = 5 tasks and different number of samples for each k,
+in order to encourage different level of importance for each task in the optimization procedure. The number of covariates
+remains the same across all tasks (d = 6). We report the mean squared error (MSE) at each sample size and DGP in Figure
+1, as well as coefficient precision and accuracy in Figures 2 and 3. All reported results are calculated on a separate test data
+over 100 Monte Carlo simulations.
+As reported for n = 600 in the main simulation results, MT-HAL achieves the lowest MSE across all DGPs and sample
+sizes considered. The improvement seen at n = 600 persists even for small sample sizes (n = 50), and consistently shows
+reduction in the MSE by more than a half. In terms of coefficient precision, the hardest settings across all algorithms and
+sample sizes are, as expected, DGPs with high sparsity, where MT-lasso seems to perform the best. However, accuracy
+results also demonstrate that MT-lasso tends to produce a lot of false negatives. Except for the very small samples sizes,
+MT-HAL results in best accuracy and comparable precision across all considered DGP settings.
+B.2. Linear Setup with Interactions
+The data-generating processes corresponding to simulation “LHS”, “LLS”, “LHD” and “LLD” are linear models with
+d = 6 covariates and normally distributed coefficients. The setup for linear simulations is the same as for nonlinear, we
+just omit the nonlinear transformations of the covariate space. For example, true coefficients for simulation “LHS” are
+β0,LHS = (β0,1, . . . , β0,|LHS|, 0, . . . , 0). Each β0,j for 1 ≤ j ≤ |LHS| is sampled from a standard normal distribution,
+and the error term is normal with final coefficient of 0.3. For simulations with the same level of sparsity across tasks, highest
+level of sparsity was 60%, whereas the lowest was 20%, as for the nonlinear DGPs. For different sparsity profiles across
+tasks, we considered random deviations from true level of sparsity across different tasks (40% − 80% for high level of
+sparsity and 0.05% − 40% for low). Covariates with nonzero coefficients consist of binary, categorical and continuous
+variables. Final outcome regression consist of only linear terms, but we add few interactions as well (up to second order).
+High-sparsity settings put most nonzero coefficients on terms that are not interactions, while low-sparsity setup includes
+interactions. We report results for linear simulations at n = 600 in Table (3).
+As expected, there is a vast improvement in overall performance for all methods for an easier (linear) setup. For high-sparsity
+simulations (LHS and LHD), the true model is linear with no interactions, hence both MT-lasso and MT-L21 operate within
+the true model. As MT-HAL starts from a nonparametric space, it is not surprising to see MT-lasso and MT-L21 perform
+better in this setup; it is, however, usually unrealistic to know the true DGP in advance. It is worth nothing that, even
+in a completely linear setting, as soon as the true DGP contains few interactions, performance of MT-lasso and MT-L21
+significantly drops in terms of the MSE. As observed in nonlinear simulations, MT-HAL preserves its advantage in terms of
+MSE performance for high-sparsity simulations; when MT-lasso and MT-l21 operate within their true model, MT-HAL
+remains very competitive.
+B.3. High-dimensional Setup
+Finally, we demonstrate performance of the multi-task HAL in high-dimensions in Table 4. In particular, the data-
+generating processes corresponding to simulation “HNHS”, “HNLS”, “HNHD” and “HNLD” are high-dimensional
+nonlinear models with d = 20 covariates per each task and normally distributed coefficients.
+The DGPs in the
+high-dimensional setting are as previously described in Section 5. The true coefficients for simulation “HNHS” are
+β0,HNHS = (β0,1, . . . , β0,|NHS|, 0, . . . , 0), where each β0,j for 1 ≤ j ≤ |HNHS| is sampled from a standard normal
+distribution. The corresponding error term is sampled from N(0, 0.1), with final coefficient of 0.3. Highest and lowest level
+of sparsity was 60% and 20% for simulations where a lot of sharing across tasks is expected. For different sparsity profiles
+across tasks, we considered random deviations from true level of sparsity across different tasks (40% − 80% for high level
+of sparsity and 0.05% − 40% for low). Covariates with nonzero coefficients are transformed via exponential, logarithmic,
+cosine and squared operations of predictors and predictor interactions. Final MSE, coefficient precision and accuracy results
+
+Multi-task Highly Adaptive Lasso
+13
+for high-dimensional nonlinear simulations at n = 600 are shown in Table (4).
+Similarly to other nonlinear simulations with smaller number of covariates (and less continuous predictors), MT-HAL
+remains the MTL algorithm with the smallest mean squared error. Across all DGPs considered, it consistently shows
+reduction in MSE by more than a half (as seen in previous simulations as well). Precision and accuracy show a significant
+decrease for all considered MTL algorithms, as expected for high-dimensional highly nonlinear settings. Despite not getting
+all non-zero coefficients right, MSE remains surprisingly low for all considered methods. As observed previously, MT-HAL
+universally produces better accuracy and comparable precision in terms of the non-zero coefficients, compared to considered
+competitors.
+Table 3. Mean squared error (MSE), precision (“Prec”) and accurarcy (“Accu”) for each of the linear simulation setups with d = 6
+covariates, K = 5 tasks, and total of n = 600 samples split between the K tasks as {100, 100, 150, 150, 100}. Reported results were
+generated across 100 Monte Carlo simulations.
+SETUP
+METHOD
+MSE
+PREC %
+ACCU %
+LHS
+MT-HAL
+0.062
+95.2
+85.7
+LHS
+MT-LASSO
+0.054
+100
+76.4
+LHS
+MT-L21
+0.014
+100
+90.3
+LLS
+MT-HAL
+0.144
+94.1
+74.1
+LLS
+MT-LASSO
+0.395
+99.8
+65.1
+LLS
+MT-L21
+0.324
+99.7
+74.9
+LHD
+MT-HAL
+0.060
+94.9
+85.2
+LHD
+MT-LASSO
+0.035
+100
+78.7
+LHD
+MT-L21
+0.018
+100
+85.7
+LLD
+MT-HAL
+0.284
+95.9
+78.0
+LLD
+MT-LASSO
+0.717
+100
+64.5
+LLD
+MT-L21
+0.644
+99.8
+77.2
+Table 4. Mean squared error (MSE), precision (“Prec”) and accurarcy (“Accu”) for each of the high-dimensional nonlinear simulation
+setups with d = 20 covariates, K = 5 tasks, and total of n = 600 samples split between the K tasks as {100, 100, 150, 150, 100}.
+Reported results were generated across 100 Monte Carlo simulations.
+SETUP
+METHOD
+MSE
+PREC %
+ACCU %
+HNHS
+MT-HAL
+0.446
+20.8
+37.8
+HNHS
+MT-LASSO
+0.863
+44.2
+27.5
+HNHS
+MT-L21
+0.808
+39.0
+35.7
+HNLS
+MT-HAL
+0.431
+38.2
+50.1
+HNLS
+MT-LASSO
+1.01
+46.1
+31.3
+HNLS
+MT-L21
+0.812
+45.4
+40.0
+HNHD
+MT-HAL
+0.425
+37.3
+36.7
+HNHD
+MT-LASSO
+0.841
+34.3
+26.4
+HNHD
+MT-L21
+0.809
+42.6
+33.7
+HNLD
+MT-HAL
+0.451
+36.5
+31.7
+HNLD
+MT-LASSO
+1.01
+46.1
+30.5
+HNLD
+MT-L21
+0.842
+42.9
+39.1
+
+Multi-task Highly Adaptive Lasso
+14
+Figure 1. Mean Squared Error (MSE) at sample sizes n = (50, 100, 200, . . . , 800, 900, 1000) for each of the nonlinear simulation setups:
+nonlinear, high sparsity, same sparsity profile across tasks (“NHS”); nonlinear, low sparsity, same sparsity profile across tasks (“NLS”);
+nonlinear, high sparsity, different sparsity profile across tasks (“NHD”); nonlinear, low sparsity, different sparsity profile across tasks
+(‘NLD”). All simulation setups contain d = 6 covariates and K = 5 tasks. Reported results were generated across 100 Monte Carlo
+simulations.
+
+(a) NHS
+(b) NLS
+3.0-
+2.5
+2.5
+2.0
+MSE
+2.0 
+1.5
+1.5
+1.0
+1.0
+0.5
+250
+500
+750
+1000
+250
+500
+750
+1000
+Method
+RMTL 121
+(c) NHD
+(d) NLD
+RMTLlasso
+MT HAL
+2.5
+1.2
+1.0
+2.0 
+ISE
+M
+0.8
+1.5
+0.6
+1.0
+0.4 -
+250
+500
+750
+1000
+250
+500
+750
+1000
+n
+nMulti-task Highly Adaptive Lasso
+15
+Figure 2. Coefficient precision at sample sizes n = (50, 100, 200, . . . , 800, 900, 1000) for each of the nonlinear simulation setups:
+nonlinear, high sparsity, same sparsity profile across tasks (“NHS”); nonlinear, low sparsity, same sparsity profile across tasks (“NLS”);
+nonlinear, high sparsity, different sparsity profile across tasks (“NHD”); nonlinear, low sparsity, different sparsity profile across tasks
+(‘NLD”). All simulation setups contain d = 6 covariates and K = 5 tasks. Reported results were generated across 100 Monte Carlo
+simulations.
+
+(a) NHS
+(b) NLS
+70
+100.0
+97.5
+65
+Precision
+95.0
+60
+92.5
+55
+90.0
+87.5
+250
+500
+750
+1000
+250
+500
+750
+1000
+Method
+RMTL 121
+(c) NHD
+(d) NLD
+RMTLlasso
+100
+MT_HAL
+65
+95
+Precision
+60
+90
+55
+85
+50
+250
+500
+750
+1000
+250
+500
+750
+1000
+n
+nMulti-task Highly Adaptive Lasso
+16
+Figure 3. Coefficient accuracy at sample sizes n = (50, 100, 200, . . . , 800, 900, 1000) for each of the nonlinear simulation setups:
+nonlinear, high sparsity, same sparsity profile across tasks (“NHS”); nonlinear, low sparsity, same sparsity profile across tasks (“NLS”);
+nonlinear, high sparsity, different sparsity profile across tasks (“NHD”); nonlinear, low sparsity, different sparsity profile across tasks
+(‘NLD”). All simulation setups contain d = 6 covariates and K = 5 tasks. Reported results were generated across 100 Monte Carlo
+simulations.
+
+(a) NHS
+(b) NLS
+0.8
+0.7
+0.6
+0.7
+Recall
+0.5
+0.6 -
+0.4
+0.5
+0.3
+250
+500
+750
+1000
+250
+500
+750
+1000
+Method
+RMTL 121
+(c) NHD
+(d) NLD
+RMTLlasso
+MT HAL
+0.7
+0.8
+0.6
+0.7-
+Recall
+0.5
+0.6 
+0.4
+0.5
+0.3
+0.4 -
+250
+500
+750
+1000
+250
+500
+750
+1000
+n
+n

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+arXiv:2301.01786v1  [hep-th]  4 Jan 2023
+Is Yang-Mills Theory Unitary in Fractional Spacetime Dimensions?
+Qingjun Jin,1, ∗ Ke Ren,2, 3, † Gang Yang,3, 4, 5, ‡ and Rui Yu6, 3, §
+1Graduate School of China Academy of Engineering Physics,
+No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
+2School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China
+3CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
+Chinese Academy of Sciences, Beijing, 100190, China
+4School of Fundamental Physics and Mathematical Sciences,
+Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
+5International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China
+6Beijing Computational Science Research Center, Beijing 100193, China
+We show that Yang-Mills theory is non-unitairy in non-integer spacetime dimensions. The viola-
+tion of unitarity is due to the existence of evanescent operators that vanish in four dimensions but
+are non-zero in general d dimensions. We show that the gluonic evanescent operators give rise to
+both negative-norm states and complex anomalous dimensions.
+INTRODUCTION
+While the real world lives in four spacetime dimensions,
+we often need to consider quantum field theories (QFTs)
+in non-integer dimensions. One example is the dimen-
+sional regularization method in which the spacetime is
+analytically continued to d = 4 − 2ǫ dimensions [1]. An-
+other example is the ǫ-expansion method used to com-
+pute critical exponents [2], where theories at Wilson-
+Fisher (WF) fixed points are defined in fractional dimen-
+sions depending on ǫ.
+Some other studies of QFTs in
+non-integer dimensions can be found in e.g. [3–9].
+Unitarity, as one fundamental physical assumption,
+plays a key role in many studies of QFTs. Surprisingly,
+it has been shown that the φ4 scalar field theory in non-
+integer dimensions is not unitary [10, 11]. The source of
+unitarity violation is the so-called evanescent operators,
+which are non-zero in general d dimensions but vanish
+in the limit d → 4. It was shown that these evanescent
+operators give rise to negative norm states and also lead
+to complex anomalous dimensions [10, 11]. The negative
+norm states were also found in the Gross-Neveu-Yukawa
+theory [12]. An important question is: does the similar
+mechanism for the unitarity violation happen for more
+general QFTs, in particular, the Yang-Mills (YM) theo-
+ries?
+In this paper, we address this question and provide
+concrete evidence that the pure YM theory is non-
+unitarity in 4 + ǫ dimensions.
+Because of the extra
+spin structure as well as the gauge symmetry, YM the-
+ories are much more complicated than the scalar theo-
+ries, making this generalization highly non-trivial. Our
+study is closely based on the recent construction of glu-
+onic evanescent operators in [13, 14]. We compute the
+Gram matrix of gluonic operators and find that there are
+negative norm states associated with the YM evanescent
+operators.
+Moreover, these operators lead to complex
+anomalous dimensions which we show in the one-loop or-
+der. Our new results for the YM theory thus suggest that
+the unitarity violation should be ubiquitous in QFT at
+non-integer spacetime dimensions.
+We point out that compared to the scalar theory case,
+there are important new features for the YM theory.
+First, both the negative norm states and the complex
+anomalous dimensions appear in the basis of dimension-
+12 operators (here we consider operators that are Lorentz
+scalars). In contrast, in the scalar theory, the negative
+norm states start at dimension-15 operators and complex
+anomalous dimensions appear at much higher dimension-
+23 [11]. Second, unlike the scalar theory having an in-
+frared (IR) fixed point, the pure YM theory is asymp-
+totically free and is expected to have an ultraviolet (UV)
+fixed point at spacetime dimension 4 + ǫ, see e.g. [15–
+17]. Our computation shows that there are negative norm
+evanescent states appearing for 4 < d < 5. Interestingly,
+the numbers of negative norm states match precisely with
+the numbers of pairs of complex anomalous dimensions
+for the length-4 operators (operators contain four Fµν’s)
+up to mass dimension 16.
+In the following we first give a brief review of the glu-
+onic evanescent operators, then we discuss the negative
+norm states and the complex anomalous dimensions, and
+finally, we comment on the implications of the violation
+of unitary for pure YM theory as well as some general-
+izations.
+GLUONIC EVANESCENT OPERATORS
+In the pure YM theory, the local gauge invariant oper-
+ators are built by taking products of the field strength
+Fµν and covariant derivatives Dµ. The simplest opera-
+tor is the term tr(FµνF µν) in the Lagrangian which has
+mass dimension four. A special class of operators is the
+so-called evanescent operators, which are defined in gen-
+eral d dimensions but vanish in the limit d → 4. The
+evanescent operators can be constructed by multiplying
+
+2
+∆0
+8
+10 12 14
+16
+N p
+4
+20 82 232 550
+N e
+0
+4
+25 92 259
+TABLE I. Counting of single-trace length-4 basis up to di-
+mension 16.
+N p and N e are the numbers of physical and
+evanescent operators, respectively.
+a tensor operator by a Kronecker symbol δµ1···µn
+ν1···νn with
+n ≥ 5 and then taking Lorentz contraction, for example,
+Oa = δµ1µ2µ3µ4µ5
+ν1ν2ν3ν4ν5 tr(Dν5Fµ1µ2Dµ5Fν1ν2Fµ3µ4Fν3ν4). (1)
+For convenience, we will refer to the non-evanescent op-
+erators as physical operators.
+In this paper we will focus on operators that are
+Lorentz scalars with all Lorentz indices contracted. Be-
+cause we study operators that are covariant in d di-
+mensional spacetime, the Lorentz contractions involve no
+Levi-Civita ǫ-tensor, thus all operators are parity even.
+Gluonic evanescent operators start to appear at canon-
+ical dimension 10 and involve at least four Fµν’s, such as
+the one in (1). For the purpose of later computations,
+it is important to have a classification of operators ac-
+cording to their mass dimension ∆0. Two operators are
+said to be equivalent if their difference is proportional to
+the equation of motion (DµF µν = 0) or Bianchi identity
+(DµFνρ + DνFρµ + DρFµν = 0). One can construct the
+basis of operators at a given mass dimension by elimi-
+nating such equivalence. We keep operators with total
+derivatives in the operator basis. A systematic classifica-
+tion of gluonic evanescent operators has been studied in
+[13].
+For the computation of the Gram matrix and one-loop
+anomalous dimensions, we note that there is no mixing
+between operators containing different numbers of Fµν.
+In this work, we will mainly focus on the single-trace
+length-4 operators (those containing four Fµν but an ar-
+bitrary number of Dµ), which will be sufficient for the
+study of unitarity violation. In Table I, we summarize
+the counting of length-4 basis (including both evanescent
+and physical operators) up to dimension 16.
+NEGATIVE NORM
+In this section, we show that the inner product of local
+operators is not positive definite in the YM theory in
+non-integer dimensions.
+This problem can be studied
+by considering the Gram matrix, which is a symmetric
+matrix defined as Gij in the two-point Green function
+⟨Oi(x)Oj(0)⟩ =
+Gij
+|x2|∆i .
+(2)
+We will show that the Gram matrix is not positive defi-
+nite, and so there are negative norm states which implies
+that the theory is non-unitary.
+For simplicity, we will compute the Gram matrix in
+the free YM theory by setting the YM coupling g = 0.
+This provides the leading order result of the Gram ma-
+trix. We mention that in principle one can compute the
+Gram matrix elements in (2) by Wick contraction, which
+is however cumbersome for high dimensional operators.
+Instead, we develop an efficient method for computing
+the Gram matrix based on form factors; details will be
+given in [18].
+For Lorentz invariant operators, Gram matrices only
+depend on the rank of gauge group Nc, the spacetime di-
+mension d, and the canonical dimensions of operators. As
+a simple example, the norm of the dimension-4 operator
+tr(FµνF µν) is
+GtrF 2,trF 2 = 8 N 2
+c d(d − 1)(d − 2)3 .
+(3)
+The interesting feature of Gram matrices is the ap-
+pearance of negative norms at d = 4 + ǫ for ǫ > 0. The
+lowest canonical dimension allowing negative norm states
+is ∆0 = 12. Take dimension-12 length-4 single-trace op-
+erators as an example. This set contains 107 independent
+operators (see Table I), and the corresponding Gram ma-
+trix has a negative eigenvalue and therefore is not positive
+definite. To explain the emergence of the negative-norm
+states, consider the following operator
+Ob = DρDσ
+�
+δµ1µ2µ4µ5ν1ρ
+µ3ν2ν3ν4ν5σ tr(Dµ1Fµ2µ3Fµ4µ5Dν1Fν2ν3Fν4ν5)
+�
+.
+(4)
+This operator contains a rank-6 Kronecker symbol, so it
+vanishes at d = 4 and d = 5. Explicitly, the norm of Ob
+is
+Gbb = 1152 N 2
+c (N 2
+c − 1)(3d + 8)(d + 2)(d + 1)d5
+× (d − 1)2(d − 2)5(d − 3)(d − 4)(d − 5) ,
+(5)
+which is negative when 4 < d < 5.
+Thus the Gram
+matrix for the ∆0 = 12 operators is non-positive definite
+at d = 4 + ǫ. As a comparison, for operators containing
+a rank-5 Kronecker symbol, their norms have no factor
+(d − 5) and still be positive for d = 4 + ǫ.
+The above operator example Ob also implies that there
+exist negative-norm states for operators with arbitrarily
+higher canonical dimensions. Such operators can be con-
+structed by inserting arbitrary pairs of Dµ (with µ con-
+tracted) in the dimension-12 operator (4). The norm of
+such an operator also has a factor linear in (d − 4) and
+(d − 5) and thus is negative at d = 4 + ǫ.
+We have computed Gram matrices for higher dimen-
+sion bases and find the number of negative-norm states
+grows as ∆0 increases. In Table II we show the num-
+bers of positive and negative norm states of single-trace
+length-4 operators up to to dimension-16, counted from
+positive and negative eigenvalues of Gram matrices. The
+number of negative eigenvalues can be also understood
+from the counting of evanescent operators: following the
+method of primitive operators in our previous work [13],
+
+3
+∆0
+8
+10
+12
+14
+16
+N p
++ = N p
+4
+20
+82
+232
+550
+N e
++
+0
+4
+24
+88
+246
+N e
+−
+0
+0
+1
+4
+13
+Nγ-complex
+0
+0
+1 × 2 4 × 2 13 × 2
+TABLE II. Counting of states with positive and negative
+norms for the single-trace length-4 basis up to mass dimen-
+sion 16. N+ and N− are the numbers of positive and negative
+norm states, respectively. Nγ-complex is the number of com-
+plext anomalous dimensions.
+all the basis operators with length-4 are constructed from
+Kronecker symbols with rank five or six, and we find
+that the numbers of negative eigenvalues always equal
+the numbers of linearly independent evanescent opera-
+tors containing rank-6 Kronecker symbols.
+To give a better understanding of the feature of the YM
+evanescent operators, let us make a comparison with the
+scalar theory. In the YM theory, both the field strengths
+Fµν and covariant derivatives Dµ carry Lorentz indices,
+while in the scalar theory, all Lorentz indices are only
+from the covariant derivatives. As a result, the evanes-
+cent operators in the scalar theory must have length
+L ≥ 5 (namely, containing at least five scalar fields).
+This should explain why the negative norm states ap-
+pear at higher-dimensional operators in the scalar the-
+ory, for example, in [11] it was found that the first Gram
+matrix block that contains both negative- and positive-
+norm evanescent states appears at canonical dimension
+∆0 = 18 and length L = 6 and in the scalar theory.
+In contrast, in the YM theory, the Lorentz structure of
+the operators is much richer, and as we discussed above,
+the similar Gram matrix with negative-norm states first
+appears at dimension 12 and with length L = 4.
+So far we have considered the Gram matrix in the
+“free” YM theory with g = 0. This leading order result
+is expected to capture the main feature. In the next sec-
+tion, we will consider the interaction theory and compute
+anomalous dimensions, and we will find the unitarity-
+violating effect that is consistent with the above free the-
+ory results.
+COMPLEX ANOMALOUS DIMENSIONS
+The scaling dimension ∆ of a local operator is defined as
+the sum of its canonical (classical) dimension ∆0 and a
+quantum correction part γ called anomalous dimension
+(AD). In a unitary CFT, the spectrum of scaling dimen-
+sions should be always real and bounded from below. In
+this section, we will show that gluonic evanescent oper-
+ators have complex ADs at the WF fixed point of the
+YM theory, thus showing manifestly that the unitarity
+property is violated.
+We first briefly review how to compute the ADs of
+local operators at the WF fixed point. For a set of bare
+operators basis {Oi} with the same canonical dimension
+∆0, one can define the renormalized operators {Oi,R} as
+OR
+i = Z j
+i Oj ,
+(6)
+where the renormalization matrix element Z j
+i
+represents
+the mixing from Oi to Oj and is determined by UV poles
+in ǫ in the minimal subtraction scheme. The dilatation
+matrix is defined to be
+D ≡ −µdZ
+dµ Z−1 .
+(7)
+The eigenvalues of the dilatation matrix give the ADs,
+which can be expanded as
+γ =
+�
+l=1
+�αs
+4π
+�l
+γ(l) ,
+(8)
+where αs is the renormalized YM coupling constant.
+At the WF fixed point [2], the beta function vanishes
+for the special value of the coupling α∗(ǫ) (we work in
+d = 4 + ǫ dimensions), which at the lowest order is given
+as
+α∗(ǫ) = 2πǫ
+β0
++ O(ǫ2) .
+(9)
+In terms of α∗, the dilatation matrix and the ADs are
+expanded in ǫ, and the leading order expansion is
+γ∗ = ǫγ(1)
+∗
++ O(ǫ2) ,
+γ(1)
+∗
+= γ(1)/(2β0) .
+(10)
+For example, for the operator tr(FµνF µν),
+γ(1)
+∗,tr(F 2) = −1 .
+(11)
+We stress that since YM theory is asymptotically free
+with β0 = 11Nc
+3
+> 0, the WF fixed point is at d > 4,
+i.e. ǫ > 0.
+Let us give a short description of the strategy we use
+to compute ADs, and one can refer to [13, 14] for de-
+tailed discussion. We first calculate the bare loop form
+factor [19], which is defined as
+FO(1, .., n; q) =
+�
+ddxe−iq·x⟨g1, . . . , gn|O(x)|0⟩ .
+(12)
+Next, we subtract the IR divergences according to
+Catani’s formulas [20] to get the UV divergence. The
+UV divergence has the structure as a linear combination
+of the tree-level form factors of the basis operators and
+the coefficients are matrix elements Z j
+i . Finally, one can
+calculate the dilatation matrix according to the definition
+(7) whose eigenvalues give ADs.
+Now we present the results of ADs.
+Since there is
+no evanescent-to-physical operator mixing at one loop
+
+4
+(namely, (Z(1)) p
+e
+= 0) [13], one can safely divide oper-
+ators into evanescent and physical sectors and compute
+their one-loop ADs separately.
+Our calculation shows
+that the one-loop complex ADs, which only happen in
+the evanescent sectors, begin to appear at canonical di-
+mension 12. This is consistent with the fact that negative
+norm states start at this dimension as discussed in the
+last section. The operator basis can be further classified
+into small sectors according to their parity under charge
+conjugation as well as Lorentz structures. The Z-matrix
+will take a blockwise structure, and the ADs for opera-
+tors in different sectors are just eigenvalues of different
+sub-blocks.
+As a concrete example, at length four, the lowest di-
+mensional sector including complex ADs is a dimension-
+12 sector containing eight evanescent operators:
+DνDρ
+�
+δ12456ν
+3789µρ
+�
+tr(D1F23F45D6F78F9µ) + Rev.)
+��
+, (13)
+DνDρ
+�
+δ1
+4δ2356ν
+789µρ
+�
+tr(D1F23F45D6F78F9µ) + Rev.)
+��
+,
+DνDρ
+�
+δ1
+4δ2356ν
+789µρ
+�
+tr(D1F23D4F56F78F9µ) + Rev.)
+��
+,
+DνDρ
+�
+δ1
+4δ2357ν
+689µρ
+�
+tr(D1F23D4F56F78F9µ) + Rev.)
+��
+,
+DνDρ
+�
+δ1
+4δ2367ν
+589µρ
+�
+tr(D1F23F45D6F78F9µ) + Rev.)
+��
+,
+DνDρ
+�
+δ1
+4δ2378ν
+569µρ
+�
+tr(D1F23D4F56F78F9µ) + Rev.)
+��
+,
+DνDρ
+�
+δ1
+5δ2347ν
+689µρ
+�
+tr(D1F23D4F56F78F9µ) + Rev.)
+��
+,
+DνDρ
+�
+δ2
+4δ1567ν
+389µρ
+�
+tr(D1F23F45D6F78F9µ) + Rev.)
+��
+.
+Note that the first operator in (13) is the only dimension-
+12 operator containing a tensor degree-6 Kronecker sym-
+bol and is responsible for the existence of a negative
+norm state. We mention that we have focused on op-
+erators that are Lorentz scalars. An alert reader may
+find that the above operators are actually total deriva-
+tives of dimension-10 tensor-2 operators. We will see in
+(15) below that the eigenvalue equation for these eight
+operators is not factorizable (with rational coefficients),
+which implies that their Z matrix cannot be decom-
+posed into smaller blocks. Thus there should exist eight
+dimension-10 tensor-2 primary operators which give the
+same anomalous dimensions.
+The one-loop Z-matrix of this sector reads
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+− 38
+3ǫ
+2
+ǫ
+− 13
+12ǫ
+0
+14
+3ǫ
+0
+14
+3ǫ
+28
+3ǫ
+− 1
+2ǫ
+− 85
+6ǫ
+2
+ǫ
+5
+6ǫ
+− 2
+3ǫ − 5
+12ǫ − 7
+3ǫ − 16
+3ǫ
+0
+− 4
+ǫ
+− 22
+3ǫ
+16
+3ǫ
+0
+− 4
+3ǫ
+0
+16
+3ǫ
+0
+− 4
+3ǫ
+7
+3ǫ
+− 34
+3ǫ
+0
+− 4
+3ǫ
+0
+0
+1
+12ǫ
+− 1
+12ǫ
+− 3
+8ǫ
+1
+12ǫ
+− 44
+3ǫ
+5
+8ǫ
+1
+2ǫ
+2
+ǫ
+0
+4
+3ǫ
+2
+3ǫ
+0
+0
+− 18
+ǫ
+0
+− 16
+3ǫ
+1
+6ǫ
+3
+2ǫ
+9
+16ǫ
+− 1
+2ǫ
+29
+6ǫ
+− 5
+12ǫ − 49
+6ǫ
+13
+3ǫ
+− 5
+6ǫ
+− 1
+3ǫ
+13
+32ǫ
+− 5
+6ǫ
+3
+4ǫ
+1
+4ǫ
+5
+12ǫ
+− 91
+6ǫ
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+.
+(14)
+The one-loop ADs γ(1)
+∗
+are roots of the following eigen-
+value equation:
+x8 − 609x7
+44
++ 160645x6
+1936
+− 2994289x5
+10648
++ 137886093x4
+234256
+− 1002685855x3
+1288408
++ 17961071517x2
+28344976
+− 22596287199x
+77948684
++ 195122885985
+3429742096
+= 0 ,
+(15)
+which is a non-trivial degree-8 polynomial equation. Re-
+markably, two of the roots are complex with numerical
+values:
+1.90386 ± 0.181142 i.
+(16)
+This provides further concrete evidence that pure YM
+theory is non-unitary in non-integer dimensions [21].
+We also compute the ADs for higher dimensional oper-
+ators and find more complex ones. For the length-4 op-
+erators up to ∆0 = 16, we observe an interesting pattern:
+the number of complex ADs is exactly twice the number
+of negative-norm states, which is summarized in Table II.
+This kind of match is not a general feature though; for
+example, the relation breaks down for the dimension-12
+length-5 operators where there are 8 negative-norm states
+but only 14 complex ADs. More details will be given in
+[18].
+Finally, we mention that the one-loop results already
+give important implications for the property at high
+loops. In particular, for a sector of operators that has
+no complex AD and also has no degeneracy of ADs at
+one loop, this sector will not have any complex AD at
+higher loop orders. This may be understood by following
+a standard perturbative calculation in quantum mechan-
+ics, see e.g. [22].
+DISCUSSION
+In this paper, we provide concrete evidence showing that
+the pure YM theory is non-unitarity in fractional space-
+time dimensions d = 4 + ǫ. In particular, we find that
+YM evanescent operators provide negative-norm states
+and also generate complex anomalous dimensions, gener-
+alizing the previous study for the scalar theory in [10, 11].
+As mentioned in the introduction, the pure YM the-
+ory is expected to have a UV conformal fixed point. If
+we couple YM with a sufficiently large number of matter
+fields (see e.g. [23]), the asymptotic freedom can disap-
+pear and the theory has an IR conformal fixed point at
+4 − ǫ.
+We expect that the unitarity violation still ex-
+ists in such cases.
+First, there are also negative-norm
+states corresponding to evanescent operators containing
+a rank-5 Kronecker symbol with the norm proportional to
+(d−4). Second, for the ADs, the operator mixing matrix
+will enlarge in general because of the appearance of new
+
+5
+operators containing matter fields. A preliminary study
+of the YM coupled to scalars shows that the complex
+anomalous dimensions persist; details will be presented
+in [18]. It would be interesting to check this for general
+gauge theory models.
+Besides the evidence of negative-norm states and com-
+plex ADs, it would be worthwhile to explore if the
+unitarity-violating effects could be manifest in other ob-
+servables, such as correlation functions or S-matrix [24].
+The unitarity violation certainly makes it questionable to
+apply the conformal bootstrap method [25] for general
+CFTs in non-integer dimensions.
+Note that the boot-
+strap method may provide good approximations in some
+cases [26] (see also [27–29]), and this may be explained
+by the fact that the unitarity violation effects occur at
+relatively high dimensional states such that the unitarity-
+violating effect is suppressed [10, 11]. In the YM the-
+ory, since the operators with negative norms or complex
+anomalous dimensions appear at lower dimensions than
+in the scalar theory, we expect that the unitarity violat-
+ing effects of the YM theory are stronger than the scalar
+theory. We mention that other bootstrap methods that
+do not rely on unitarity (see e.g. [30]) should still work
+in this case.
+Finally, it should be interesting to generalize our study
+to gravitational theories in which similar evanescent op-
+erators can be defined, for example, by replacing Fµν
+with Rµν.
+Note that local operators are not physical
+observables in gravity, and one may need to consider
+other observables like amplitudes (see e.g. [31, 32]) or
+non-local operators.
+Another important connection to
+gravity is through the holographic duality [33–35] which
+implies that a gauge theory in d dimensions would have
+a holographic dual of gravity (string) theory in d + 1 di-
+mensions. The non-unitary gauge theory in fractional d
+dimensions thus implies that the dual gravity theory also
+violates unitarity. It would be highly interesting to see
+the violating effect explicitly on the gravity side.
+Acknowledgments. We would like to thank Bo Feng,
+Yunfeng Jiang, Jianxin Lu, and Tao Shi for discussion.
+This work is supported in part by the National Natu-
+ral Science Foundation of China (Grants No. 11935013,
+12175291, 12047503, 12047502, 11947301).
+We also
+thank the support of the HPC Cluster of ITP-CAS.
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
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+Generated using the official AMS LATEX template v6.1 two-column layout. This work has been submitted for
+publication. Copyright in this work may be transferred without further notice, and this version may no longer be
+accessible.
+Surface heating steers planetary-scale ocean circulation
+Dhruv Bhagtani,a Andrew McC. Hogg,a Ryan M. Holmes,b Navid C. Constantinou,a
+a Research School of Earth Sciences & ARC Center of Excellence for Climate Extremes, Australian National University, Canberra,
+Australia
+b School of Geosciences, University of Sydney, Sydney, Australia
+ABSTRACT: Gyres are central features of large-scale ocean circulation and are involved in transporting tracers such as heat, nutrients, and
+carbon-dioxide within and across ocean basins. Traditionally, the gyre circulation is thought to be driven by surface winds and quantified
+via Sverdrup balance, but it has been proposed that surface buoyancy fluxes may also contribute to gyre forcing. Through a series of
+eddy-permitting global ocean model simulations with perturbed surface forcing, the relative contribution of wind stress and surface heat
+flux forcing to the large-scale ocean circulation is investigated, focusing on the subtropical gyres. In addition to gyre strength being linearly
+proportional to wind stress, it is shown that the gyre circulation is strongly impacted by variations in the surface heat flux (specifically, its
+meridional gradient) through a rearrangement of the ocean’s buoyancy structure. On shorter timescales (∼ decade), the gyre circulation
+anomalies are proportional to the magnitude of the surface heat flux gradient perturbation, with up to ∼ 0.15Sv anomaly induced per Wm−2
+change in the surface heat flux. On timescales longer than a decade, the gyre response to surface buoyancy flux gradient perturbations
+becomes non-linear as ocean circulation anomalies feed back onto the buoyancy structure induced by the surface buoyancy fluxes. These
+interactions complicate the development of a buoyancy-driven theory for the gyres to complement the Sverdrup relation. The present study
+challenges the conventional understanding of the interplay between fundamental drivers of the large-scale ocean circulation.
+SIGNIFICANCE STATEMENT:
+Ocean gyres are
+large swirling circulation features that redistribute heat
+across ocean basins.
+It is commonly believed that sur-
+face winds are the sole driver of ocean gyres, but recent
+literature suggests that other mechanisms could also be
+influential. Here, we perform a series of numerical sim-
+ulations in which we artificially change the winds or the
+heating at the ocean’s surface and investigate how each
+factor independently affects the ocean gyres. We find that
+gyres are steered by both winds and surface heating, and
+that the ocean circulation behaves differently to heating
+on short and long timescales. In addition, the circulation
+depends on where the heating is applied at the ocean’s sur-
+face. Through these simulations, we argue that a complete
+theory explaining the formation of gyres should consider
+heating at the ocean’s surface as a possible driver, in addi-
+tion to the winds.
+1. Introduction
+The large-scale ocean circulation derives its energy from
+a variety of sources including wind stress (Wunsch and
+Ferrari 2004; Hughes and Wilson 2008; Jamet et al. 2021),
+tidal forces (Oka and Niwa 2013), and surface and geother-
+mal buoyancy fluxes (Hughes et al. 2009; Hogg and Gayen
+2020). Gyres are fundamental elements of the large-scale
+circulation and play a crucial role in the biogeochemical
+and hydrological cycles in the ocean by transporting mo-
+mentum, heat, nutrients and chemicals within and across
+ocean basins (Webb 2017). In particular, gyres contribute
+Corresponding author: D. Bhagtani, [email protected]
+to global heat transport by transferring heat poleward (Pal-
+ter 2015; Zhang et al. 2021; Li et al. 2022). Despite their
+importance in regulating large-scale weather and climate
+patterns, the interplay between the processes leading to
+the formation and evolution of ocean gyres are not fully
+understood.
+Traditional oceanographic literature on the drivers of
+ocean circulation suggest that near-surface horizontal flows
+are primarily caused by mechanical forcing due to wind
+stress (Sverdrup 1947) and tidal forces (Wunsch and Ferrari
+2004; Oka and Niwa 2013), and the meridional overturning
+circulation (MOC) is chiefly driven by surface buoyancy
+fluxes (Stommel and Arons 1959). However, this simpli-
+fied viewpoint has been amended with time. Recent litera-
+ture points to a definitive role played by wind stress on the
+MOC through isopycnal upwelling in the Southern Ocean
+(Abernathey and Ferreira 2015; Hogg et al. 2017) and
+bottom-enhanced diapycnal mixing (Stanley and Saenko
+2014; Drake et al. 2020). Similarly, buoyancy forcing is
+thought to exert a significant control on horizontal cir-
+culation through the conversion from potential to kinetic
+energy (Tailleux 2009; Hughes et al. 2009) and control of
+the stratification in the ocean (Shi et al. 2020), consistent
+with the fundamental dynamics of rotating horizontal con-
+vection (Gayen and Griffiths 2021). In this paper, we aim
+to evaluate the interplay between wind stress and surface
+buoyancy forcing in driving various features of the large-
+scale ocean circulation, with an emphasis on basin-scale
+ocean gyres.
+Considerable progress has been made to elucidate the
+impact of surface wind stress on ocean gyres (Sverdrup
+1
+arXiv:2301.11474v1  [physics.ao-ph]  27 Jan 2023
+
+2
+1947; Stommel 1948; Munk 1950; Rhines and Young 1982;
+Luyten et al. 1983; Pedlosky 1986). Munk (1950) pro-
+posed the Sverdrup relation by constructing a relationship
+between the curl of the wind stress and the depth-integrated
+meridional geostrophic transport, valid in the ocean inte-
+rior away from coastlines,
+𝑉 = ˆz · (∇×τ)
+𝛽
+,
+(1)
+where 𝑉 = 𝜌0
+∫
+𝜐d𝑧 is the time-mean depth-integrated
+meridional mass transport with 𝜌0 the ocean’s reference
+density, ˆz is the unit vector in the vertical, τ = 𝜏𝑥 ˆx+ 𝜏𝑦 ˆy
+the time-mean horizontal wind stress at the ocean’s surface,
+and 𝛽 = 𝜕 𝑓 /𝜕𝑦 is the meridional gradient of the Coriolis
+frequency 𝑓 . The return flow occurs as an intense iner-
+tial western boundary flow (see, e.g., Hughes and Cuevas
+(2001)). The Sverdrup relation (1) describes the depen-
+dence of the horizontal structure of barotropic gyres on
+the wind stress curl, and to date, remains the corner-stone
+theory of wind-driven gyres.
+The Sverdrup relation focuses on understanding the hor-
+izontal structure of the vertically-integrated gyre transport.
+The ventilated thermocline theory, developed by Luyten
+et al. (1983), made further strides in interpreting the ver-
+tical structure of ocean gyres forced by wind stress. They
+obtained a layer-wise meridional transport for gyres but
+restricted gyre transport to only ventilated isopycnals; that
+is, isopycnals outcropping to the surface of the ocean, with
+non-ventilated isopycnals at rest. However, several studies
+point to a net re-circulatory transport in the non-ventilated
+isopycnal layers (Rhines and Young 1982; McDowell et al.
+1982) so long as they do not interact with the ocean’s sur-
+face or topography. Therefore, ocean gyres can be viewed
+as a combination of ventilation and recirculation regimes,
+and are controlled by surface wind stress curl.
+Wind-driven theories encapsulate gyre circulation to ze-
+roth order, however, observational studies show deviations
+from the Sverdrup relation.
+Gray and Riser (2014) ex-
+amined the validity of Sverdrup dynamics in a point-wise
+manner using observations from Argo floats (Roemmich
+et al. 2004) and found that it agrees well with observations
+in the interior subtropical gyres, with significant devia-
+tions in subpolar regions. Colin De Verdière and Ollitrault
+(2016) obtained a depth-integrated geostrophic transport
+using Argo data (Ollitrault and Rannou 2013) and World
+Ocean Atlas 2009 (Locarnini et al. 2010) to estimate a
+global Sverdrup streamfunction, which was found to un-
+derrepresent the subtropical and subpolar gyre strength by
+a factor of 2. These discrepancies indicate the presence of
+other processes playing a role in the vorticity balance, such
+as bottom pressure torques (Hughes and Cuevas 2001), di-
+apycnal mixing (Lavergne et al. 2021), buoyancy forcing
+(Hogg and Gayen 2020; Liu et al. 2022) and coupling
+with the meridional overturning (Klockmann et al. 2020;
+Berglund et al. 2022). At present, a unified theory en-
+capsulating all the previously stated mechanisms is not
+available, which limits our understanding of the coupling
+between these processes, as well as their combined effects
+on ocean circulation. The present paper makes an attempt
+to isolate and understand the roles of wind stress and sur-
+face buoyancy forcing in shaping the planetary-scale ocean
+gyres.
+Surface buoyancy forcing alters the ocean’s density
+structure through heat and freshwater fluxes (Large and
+Yeager 2009; Talley et al. 2011). Together with the ther-
+mal wind relation,
+𝑓 𝜕u
+𝜕𝑧 = ˆz ×∇𝑏,
+(2)
+where 𝑏(x,𝑡) = 𝑔(1 − 𝜌(x,𝑡)/𝜌0) is the buoyancy, with
+𝑔 the gravitational acceleration, these density structure
+changes can be used to understand how changes in surface
+buoyancy forcing might impact ocean circulation. How-
+ever, the relationship between surface buoyancy forcing
+and the horizontal circulation is complicated. For exam-
+ple, variations in mixed layer depth and non-linear feed-
+backs with the ocean circulation both influence how the
+surface buoyancy forcing is “felt” within the ocean. The
+mixed layer ingests a fraction of the surface buoyancy flux,
+which is reflected in the anomalous ocean’s density within
+the layer. The amount of buoyancy forcing reaching the
+layers below is thus inversely related to the mixed layer
+depth, with a deeper mixed layer taking a longer time to re-
+lay the excess buoyancy forcing into the subsurface layers
+(Xie et al. 2010) due to its higher effective heat capacity.
+Furthermore, the circulation modifies the influence of sur-
+face buoyancy forcing on the ocean’s buoyancy structure
+through heat advection (Bryden et al. 1991). Advection
+acts to alter the buoyancy structure remote from the forc-
+ing, which, in view of (2), would also cause anomalies
+in the ocean circulation in that remote location. In this
+paper, we evaluate the variability in ocean’s stratification
+over time to better understand the non-linear and non-local
+connection between the surface buoyancy forcing and the
+gyres.
+Past studies examined the role of surface buoyancy forc-
+ing in restructuring ocean gyres. Goldsbrough (1933) ob-
+served that freshwater fluxes can drive horizontal circu-
+lation via induced sea surface height anomalies. Luyten
+et al. (1985) and Pedlosky (1986) extended the ventilated
+thermocline theory (Luyten et al. 1983) to include an in-
+terfacial mass flux between various isopycnal layers (to
+represent surface buoyancy forcing), and using a simpli-
+fied ocean model, demonstrated a geostrophic baroclinic
+flow induced by the buoyancy flux and steered by wind
+stress. Colin de Verdière (1989) used a model similar to
+Cox and Bryan (1984) but coupled the surface buoyancy
+forcing to wind stress via a bulk formula and concluded
+
+3
+that the former drives a baroclinic mode to significantly
+recast the horizontal and vertical structure of subtropical
+gyres. Gjermundsen et al. (2018) instead considered simu-
+lations conducted in the absence of any surface momentum
+inputs and applied buoyancy flux at the ocean’s surface
+through a meridionally varying surface temperature restor-
+ing profile. They observed a broad eastward zonal flow as
+a consequence of the surface buoyancy flux structure and
+the thermal wind balance as well as a western boundary
+current. Hogg and Gayen (2020) conducted a series of
+numerical simulations with a restoring temperature profile
+and no wind stress in two configurations: a direct numer-
+ical simulation in a 3D box domain, and a layered gen-
+eral circulation model in a sector configuration, and found
+a double (resembling a subtropical and subpolar) gyre in
+both scenarios. With a multitude of contrasting viewpoints
+on the processes leading to the formation of gyres, we are
+motivated to examine further the role of surface buoyancy
+forcing in driving ocean gyres.
+It is worth emphasizing here that ocean gyres do not
+exist in isolation – they interact with other fundamental
+aspects of ocean circulation, such as the MOC. The MOC
+complements gyre circulation in transporting tracers such
+as heat, chemicals, and nutrients globally between ocean
+basins. It has been argued that thermohaline forcing is re-
+sponsible for driving the MOC (Stommel and Arons 1959;
+Stommel 1961), with gyres being driven by wind stress
+(Munk 1950). Luyten et al. (1985) contradicted this sim-
+plified view by establishing a link between wind stress and
+heat gain for the North Atlantic subpolar gyre. Yeager and
+Danabasoglu (2014) and Yeager (2015) used a coupled
+ocean-sea-ice configuration of the Community Earth Sys-
+tem Model with perturbed forcing, and found that most of
+the decadal variability in both the Atlantic MOC (AMOC)
+and the North Atlantic subpolar gyre was due to variability
+in surface buoyancy forcing, while changes in interannual
+variability were attributed to wind stress anomalies. Mod-
+eling studies (Yeager 2015; Larson et al. 2020) identify a
+direct relationship between the mid-depth overturning cell
+and the North Atlantic subtropical gyre transport on the
+basis that the two circulatory features are linked through
+the northward flowing Gulf Stream. Thus, ocean gyres
+and the MOC are coupled dynamical features (Klockmann
+et al. 2020; Berglund et al. 2022), and a thorough analysis
+of the drivers behind the formation of ocean gyres requires
+a quantitative understanding of other processes of ocean
+circulation.
+The objectives of the present paper are: (i) to quantify
+how the surface buoyancy forcing affects the structure of
+ocean gyres, and (ii) to understand how wind stress and
+surface buoyancy fluxes act in concert to drive large-scale
+ocean circulation. Section 2 outlines the simulation setup
+and a gyre metric used to analyze the ocean’s circulation.
+Section 3 examines the sensitivity of gyre circulation to
+changes in surface wind stress, followed by a brief discus-
+sion of the coupling between gyres and other large-scale
+circulation features in the ocean. Section 4 further inves-
+tigates the role of surface buoyancy forcing gradients in
+steering the ocean circulation. In section 5 we look at a
+uniform warming experiment to illustrate that changes in
+ocean circulation can also occur in the absence of a merid-
+ional gradient in buoyancy forcing anomaly. In section 6,
+we conclude by emphasizing the connected roles of wind
+stress and surface buoyancy fluxes in driving ocean cir-
+culation, the importance of surface buoyancy forcing in
+driving ocean gyres, and future directions to advance our
+understanding of ocean circulation.
+2. Models and Methods
+a. Flux-forced simulations
+Surface buoyancy fluxes in ocean-sea ice general cir-
+culation models are usually parameterized using bulk for-
+mulae (Large et al. 1994), and are therefore dependent on
+the model’s dynamic sea surface temperature, as well as
+the externally prescribed atmospheric winds, humidity, air
+temperature and radiative fluxes. Therefore, any changes
+in circulation (for example due to changes in wind stress)
+have the ability to alter the surface buoyancy forcing. To
+isolate the impacts of wind and surface buoyancy forc-
+ing from each other, in this study we construct a series of
+global simulations where we force the ocean using fluxes
+at the surface. We call them “flux-forced” simulations.
+The flux-forced simulations permit us to modify the sur-
+face boundary fluxes independently, in that they decouple
+the wind and surface buoyancy forcing from each other.
+Forcing for the flux-forced control experiment is
+constructed from a 200-year control simulation using
+ACCESS-OM2-025 (Kiss et al. 2020), a global ocean-
+sea ice model at 0.25◦ resolution and 50 vertical layers.
+ACCESS-OM2-025 is an amalgamation of the Modular
+Ocean Model v5.1 ocean model (Griffies 2012) and the
+CICE v5.1.2 (Hunke et al. 2015) sea ice model. We ap-
+ply a repeat-year atmospheric forcing using the JRA55-
+do v1.3 reanalysis product (Tsujino et al. 2018) to drive the
+ACCESS-OM2-025 control simulation. We use the period
+1st May 1990 to 30th April 1991 as the repeat year for
+atmospheric forcing following Stewart et al. (2020). The
+ACCESS-OM2-025 control experiment is initialized using
+temperature and salinity data from the World Ocean Atlas
+2013 (Locarnini et al. 2013; Zweng et al. 2013), and in-
+corporates the Gent-McWilliams parameterization (Gent
+and McWilliams 1990) (with a maximum diffusivity of
+200 m2 s−1) to complement partially resolved mesoscale
+eddy fluxes.
+Vertical mixing is parameterized using a
+slightly altered K-profile parameterization (Large et al.
+1994) (see Appendix). The 200-year control ACCESS-
+OM2-025 simulation is used to create a climatology of
+surface boundary fluxes at 3-hourly temporal frequency,
+
+4
+obtained by combining the last 20 years of surface forcing
+data.
+The flux-forced simulations apply the climatology of
+surface boundary fluxes to a stand-alone implementation
+of the Modular Ocean Model v5.1 (Griffies 2012). The
+flux-forced control experiment is initialized from the end
+of the ACCESS-OM2 control experiment and run for 100
+years, after which we branch off a series of flux-forced per-
+turbation simulations with modified surface fluxes. These
+perturbation experiments are run for another 100 years.
+Although 100 years is not sufficient for the deep ocean
+to reach equilibrium, it is enough for the upper and mid-
+ocean circulation to respond to changes in surface forcing
+(Saenko 2009).
+We conduct three types of sensitivity experiments:
+(i) Perturbations in surface wind stress,
+(ii) Perturbations in surface meridional heat flux gradi-
+ents, and
+(iii) A ‘uniform warming’ perturbation.
+A list of all flux-forced experiments is given in Table 1.
+Each set of experiments is described below.
+We perform wind perturbation experiments by increas-
+ing or decreasing the global wind stresses by a multiplica-
+tive factor of 0.5 or 1.5 respectively (Table 1).
+The construction of surface buoyancy flux gradient per-
+turbation experiments is based on the thermal wind rela-
+tion (2), which suggests a dependence of the ocean circu-
+lation on horizontal density gradients, which in turn could
+be modified by prescribing a spatially varying buoyancy
+flux perturbation at the ocean’s surface. Herein, we apply
+buoyancy flux perturbations by varying the prescribed sur-
+face heat fluxes. The largest heat losses in the ocean occur
+in focused regions over the subtropical western boundary
+currents (Fig. 1a). The fine-scale spatial structure in the
+surface heat fluxes differs from the broader patterns of the
+wind stress forcing, and for that reason, choosing a multi-
+plicative approach for surface buoyancy flux perturbations
+would enhance this fine-scale structure, which could po-
+tentially instigate spurious behavior in ocean circulation.
+Furthermore, a multiplicative approach would potentially
+lead to strong convection over the western boundary cur-
+rents where heat loss in the control is strongest (Fig. 1b).
+Therefore, for our buoyancy flux gradient perturbations we
+choose instead to add or subtract a broad surface heat flux
+pattern (Fig. 1) to the flux-forced control simulation, mul-
+tiplied by a constant flux amplitude for each experiment
+(Table 1) to enhance or reduce the meridional buoyancy
+gradients. We ensure that the global integral of the pat-
+tern is zero by adjusting the magnitude of the heat flux
+in subpolar and polar regions to be double the magnitude
+of heat flux in subtropical regions (which has twice the
+area), with zero perturbation in the tropics. To minimize
+any spurious behavior in ocean circulation due to the ap-
+plied buoyancy perturbation, we employ a hyperbolic tan-
+gent function (tanh [(𝑦 − 𝑦0)/Δ𝑦] over the latitude band of
+Δ𝑦 = 12.5◦, with 𝑦0 the transition latitude; see Fig. 1b) at
+the junction between subtropical and subpolar buoyancy
+anomalies. Buoyancy perturbation experiments for which
+the mask is multiplied by a positive value have surface
+heat fluxes that enhance the near-surface meridional buoy-
+ancy gradients, and are labeled as “increased buoyancy flux
+contrast” experiments. Conversely, all experiments where
+the buoyancy perturbation mask is multiplied by a nega-
+tive value are labeled as “reduced buoyancy flux contrast”
+experiments. We were unable to run a −30 Wm−2 simula-
+tion as it became unstable due to unrealistically warm sea
+surface temperatures at high latitudes.
+Finally, we conduct a globally uniform warming exper-
+iment, which differs from the surface meridional heat flux
+gradient perturbation experiments in that we do not ex-
+ternally induce a surface buoyancy gradient in the ocean.
+However, we still anticipate anomalies in the circulation
+owing to its non-local and non-linear feedback with the sur-
+face buoyancy forcing, and lateral variations in the mixed
+layer depth.
+Our flux-forced control simulation is not fully equili-
+brated as shown in Fig. 1c, where the mean sea surface tem-
+peratures gradually increase with time (≈ 0.1◦ Cdecade−1).
+The systematic increase is due to frazil formation in polar
+latitudes, which is modeled via an additional heat input.
+This heat gain is a proxy for heat transferred by a fictional
+ice model coupled to the flux-forced ocean model, as cold
+water is converted to ice. Frazil formation in our experi-
+ments is not prescribed like the other surface heat fluxes;
+instead, it depends on ocean temperature and acts to allevi-
+ate excessive cooling in polar regions. The dependence of
+frazil formation on the surface buoyancy forcing limits the
+magnitude of buoyancy flux perturbation we can apply at
+the ocean’s surface without the simulation becoming un-
+stable. Moreover, the heat gain due to frazil formation is
+amplified in increased buoyancy contrast experiments with
+a stronger heat loss in subpolar and polar regions. This net
+heat gain is partially mitigated by applying a globally uni-
+form heat loss of 1.28Wm−2 to all increased buoyancy flux
+contrast experiments. The resulting heat flux anomalies
+due to frazil formation are much smaller than the buoy-
+ancy perturbations applied in our sensitivity experiments,
+and therefore, we do not expect any significant departures
+in ocean circulation due to this frazil formation induced
+buoyancy gain.
+b. Gyre metrics
+Gyre strength is generally defined as the vertically inte-
+grated mass transport from surface to bottom . However,
+this procedure of estimating gyre strength disregards the
+baroclinic component of gyre strength, which integrates
+
+5
+Fig. 1: Model setup for sensitivity experiments. (a) Climatological net surface heating for flux-forced control simulation.
+(b) Surface buoyancy flux perturbation pattern. This pattern is multiplied by a scale factor and then applied to panel (a)
+to construct the surface buoyancy flux perturbation experiments. The perturbation pattern is −1.0 in subpolar and
+polar regions, and +0.5 in subtropical regions, with a 1.5 meridional contrast between the two extrema. A hyperbolic
+tangent function is used to smoothly connect: (i) subtropical and subpolar regions and (ii) subtropics and tropics.
+(c) Global average surface temperature from each simulation performed in this study, illustrating the model spinup
+method. Simulation time is referenced with respect to the beginning of the flux-forced perturbation experiments.
+Table 1: List of experiments. G denotes perturbation applied globally and G−T perturbations that exclude the tropics,
+defined as the equatorward extent of subtropical gyres in the equilibrated flux-forced control simulation (see Fig. 1b).
+Experiment
+Wind Factor
+Surface Buoyancy Flux Contrast Δ𝐵 (Wm−2)
+Region
+Control
+1
+0
+G
+0.5×W
+0.5
+0
+G
+1.5×W
+1.5
+0
+G
+−15 Wm−2
+1
+−15
+G−T
+−7.5 Wm−2
+1
+−7.5
+G−T
++7.5 Wm−2
+1
++7.5
+G−T
++15 Wm−2
+1
++15
+G−T
++30 Wm−2
+1
++30
+G−T
+Uniform warming
+1
+0, instead spatially uniform +5
+G
+to zero in depth. To circumvent this issue, we estimate
+the subtropical gyre strength using an “isopycnal outcrop-
+ping method". We integrate the horizontal mass transport
+from the surface only to the depth of the densest isopycnal
+(measured using potential density referenced to 2000 dbar
+and denoted as 𝜎max) that outcrops to the ocean’s surface
+
+a. Control surface heat flux
+b. Normalised buoyancy perturbation
+80
+0
+0
+W
+-0.5
+-80
+-1
+c. Spinup configuration
+40
+Surface temperature (°C)
+35
+30
+25
+20
+15
+-150
+-300
+-250
+-200
+-100
+-50
+0
+50
+100
+Time (years)
+ACCESS-OM2 control
+-15 Wm-2
++7.5 W m-2
++30 W m-2
+1.5xW
+Flux-forced control
+-7.5 W m-2
++15 W m-2
+0.5xW
+Uniform warming6
+in a given basin (marked by the red boxes in Fig. 2). For
+simplicity, we choose 𝜎max = 1035.8 kgm−3 for all four
+subtropical gyres in each flux-forced simulation. Then,
+to arrive at a single scalar estimate of the gyre’s strength,
+we select the 95th percentile (to filter out vigorous inertial
+re-circulating eddies near the western boundary region) of
+a 5-year running mean (to filter out transient eddies and
+seasonal isopycnal outcropping) of the resulting density-
+integrated horizontal transport streamfunction.
+The isopycnal outcropping method captures the baro-
+clinic component of gyres, and is used in all flux-forced
+simulations to compare the gyre strength.
+However, it
+suffers from two limitations:
+1. Surface buoyancy perturbations could restructure the
+ocean’s stratification, which may alter the isopyc-
+nal regime occupied by the gyres.
+These changes
+are not well-represented in the isopycnal outcropping
+method, since the method integrates the entire circu-
+lation from the surface to 𝜎max = 1035.8 kgm−3. We
+use a deep isopycnal for 𝜎max to ensure we fully cap-
+ture the subtropical gyres. In doing so, we may record
+a portion of abyssal circulation, which is usually
+much weaker than near-surface circulation. Thus, this
+method characterizes the subtropical gyre strength.
+2. Computing the streamfunction requires that the flow
+is divergence-free, which is not guaranteed due to the
+possibility of a net transport across the 𝜎max isopy-
+cnal. However, in the ocean’s interior, flow across
+isopycnals is much more restricted compared to flow
+along isopycnals (see, e.g., Abernathey et al. 2022)
+and, therefore, our streamfunction calculations are
+correct to leading order.
+3. Wind stress perturbation experiments
+In this section, we investigate two perturbation exper-
+iments wherein we change the magnitude of wind stress
+by 0.5 and 1.5 times the control experiment, which alters
+the time-mean vorticity input due to the wind stress curl
+by the same factor. We analyze short-term and long-term
+variations in the subtropical gyre transport for Atlantic and
+Pacific oceans for both hemispheres, along with a brief
+discussion of their coupling with the Meridional Overturn-
+ing Circulation (MOC) and Antarctic Circumpolar Current
+(ACC). Since our flux-forced experiments do not incorpo-
+rate sea-ice dynamics, changes in Weddell and Ross gyres
+are not reported in the paper, as sea-ice can significantly
+alter gyre dynamics in polar regions.
+The dashed lines in the time series in Fig. 2 show the
+expected transport as predicted by the Sverdrup linear scal-
+ing of the average gyre transport in the control experiment
+for the last 100 years of the simulation. Subtropical gyres
+follow Sverdrup scaling to a large extent, consistent with
+the ventilated thermocline theory (Luyten et al. 1983); de-
+viations are observed in the North Atlantic subtropical gyre
+in both wind perturbation simulations, and in the 0.5×W
+simulation in the North Pacific subtropical gyre.
+The gyre strengths adjust quickly to changes in wind
+forcing (solid curves in the time series in Fig. 2), which is
+likely due to a quick adjustment of the western boundary
+current by barotropic Rossby waves (Veronis and Stom-
+mel 1956; Anderson and Gill 1975). Subsequent smaller
+changes in gyre transports may be attributed to baroclinic
+Rossby waves, which propagate slowly and have a smaller
+effect on the gyre circulation (Anderson and Gill 1975).
+The time series also show that after the initial response,
+gyres in the Pacific Ocean are more stable than in the At-
+lantic Ocean.
+Next, we analyze the impact of the perturbations in
+wind stress forcing on other large scale flow metrics, start-
+ing with the AMOC. For the first 10 years, the AMOC
+strength is inversely related to the wind stress magnitude
+(Fig. 3a), which is likely linked to a change in northward
+Gulf Stream barotropic transport, part of which lies within
+the AMOC (evaluated between 𝜎2 ∈ [1035.5,1038] kgm−3
+density classes, which captures the bulk of the overturn-
+ing circulation).
+Similar results have been reported by
+Hazeleger and Drijfhout (2006) and Yang et al. (2016).
+After the initial transient response, we observe a slight de-
+cline in AMOC strength for the 0.5×W experiment com-
+pared with the control, consistent with previous studies
+(e.g. Lohmann et al. 2021). However, we also observe a
+reduction in the AMOC in the 1.5×W experiment in the
+first 60 years, while a gradual increase to about 10% com-
+pared with the control experiment is observed in the next
+40 years. These positive anomalies in AMOC in the last
+40 years with increased wind stress could be associated
+with eddy compensation in the Southern Ocean (Morrison
+and Hogg 2013), or other non-linear feedback in the ocean
+circulation.
+We observe only slight (≈ 4%) changes in the ACC trans-
+port in both 0.5×W and 1.5×W simulations compared to
+the control (Fig. 3b), which is consistent with several nu-
+merical studies that the ACC is weakly sensitive to the
+strength of the wind stress due to a process known as
+eddy saturation (Munday et al. 2013; Marshall et al. 2017;
+Constantinou and Hogg 2019). However, the validity of
+ACC transport (especially the inconsistent reduction in the
+1.5×W simulation) may be questioned on the basis that
+the control flux-forced experiment is not equilibrated, and
+declines steadily (≈ 9%) during the same time period.
+We observe a near-linear relationship between the glob-
+ally integrated kinetic energy (Fig. 3c) and wind stress
+magnitude. Winds supply energy to the ocean through gen-
+eration of eddies and enhanced circulation, which leads to
+an increase in the global kinetic energy (Wunsch and Fer-
+rari 2004). We discern that the change in kinetic energy
+due to wind stress is partially due to a change in the gyre
+
+7
+Fig. 2: The barotropic streamfunction averaged over the last 15 years of the flux-forced control experiment. The
+streamfunction is multiplied with sign( 𝑓 ). The subpanels show time series of gyre strength for the control (black),
+0.5×W (pink), and 1.5×W (purple) experiments in the (a) North Pacific subtropical gyre, (b) North Atlantic subtropical
+gyre, (c) South Pacific subtropical gyre, and (d) South Atlantic subtropical gyre. An equal y-tick spacing is used on the
+y-axis to facilitate comparison between the different gyres. Red boxes show the extent of each basin used to estimate the
+95th percentile subtropical gyre strength using the isopycnal outcropping method. Dashed lines show the gyre transport
+predictions based on Sverdrup linear scaling, i.e., the time-mean gyre strength in the control multiplied by the wind
+perturbation factor.
+circulation (Fig. 2) as well as mesoscale eddies, and only
+weakly related to the MOC and ACC (Fig. 3). A complete
+energy budget calculation is not presented here, as it is
+beyond the scope of the study.
+4. Surface buoyancy flux contrast perturbation experi-
+ments
+a. Reduced surface buoyancy flux contrast experiments
+In the previous section, we analyzed the effects of wind
+stress on the large-scale circulation. In this section, we
+discuss two sensitivity experiments wherein we reduce the
+
+8
+0
+20
+40
+60
+80
+100
+12
+14
+16
+18
+Transport (Sv)
+a. Atlantic Meridional Overturning Circulation
+0
+20
+40
+60
+80
+100
+85
+90
+95
+Transport (Sv)
+b. Antarctic Circumpolar Current
+0
+20
+40
+60
+80
+100
+Time (years)
+1500
+2000
+2500
+3000
+Energy (Joules)
+c. Globally Integrated Kinetic Energy
+0.5xW
+Control
+1.5xW
+Fig. 3: Monthly-mean time-series circulation metrics for
+the wind perturbation flux-forced simulations:
+0.5×W
+(pink), control (black) and 1.5×W (purple) after a 5-year
+rolling mean was applied. (a) Atlantic meridional over-
+turning circulation: integrated meridional transport for
+𝜎2 ∈ [1035.5,1038.0] kgm−3 at 26◦N for longitudes be-
+tween 103◦W and 5◦W. (b) Antarctic Circumpolar Current
+transport through the Drake Passage. (c) Globally inte-
+grated kinetic energy.
+intergyre surface meridional buoyancy difference at the
+poleward zonal peripheries of subtropical gyres by 7.5 and
+15Wm−2. The buoyancy contrast perturbation is expected
+to cause variations in horizontal density gradients: from
+the thermal wind relation (2), these variations could lead
+to anomalies in the ocean circulation.
+We analyze short (< 1 decade) and long (> 1 decade)
+time responses of the four subtropical gyres to delineate
+the linear and non-linear behavior of the ocean circulation
+due to the surface buoyancy flux gradient anomalies. The
+subtropical gyres, with the exception of the South Pacific
+gyre, initially reduce compared with the control simulation
+(Fig. 4). This first-decade reduction is approximately linear
+with respect to the magnitude of the surface buoyancy flux
+gradient anomaly, as shown by the red bars in Figs. 4a-
+c for the −7.5 Wm−2 and −15 Wm−2 experiments. The
+relaxation in gyre strength is consistent with the thermal
+wind relation (2): reduction in buoyancy gradients acts
+to reduce horizontal flow. The bar graphs alongside the
+gyre strength time series in Fig. 4 reveal that the Atlantic
+subtropical gyres initially react 2 to 4 times more strongly
+(measured by the percentage change in the gyre transport)
+to changes in surface heat fluxes than the Pacific subtropical
+gyres.
+However, with time, the Pacific gyres display a
+greater change than the Atlantic gyres. The time series
+reveal that the Atlantic gyres are more susceptible to a
+reduction in surface meridional buoyancy forcing contrast
+than the Pacific Ocean on short timescales.
+Figures 5 and 6 highlight spatial and temporal varia-
+tions in the ocean’s density structure due to the applied
+heat flux anomaly. Focusing on the −7.5 Wm−2 simula-
+tion, Figs. 5a and 6a highlight minor stratification anoma-
+lies in the ���rst 7 years of the simulation period, with the
+subtropical Atlantic Ocean demonstrating slightly larger
+meridional buoyancy gradient anomalies than the subtrop-
+ical Pacific Ocean. These gradients are consistent with
+stronger circulation anomalies (through (2)) in the two At-
+lantic subtropical gyres in the initial stages of the simula-
+tion.
+In addition to the gyre circulation being linear with re-
+spect to the magnitude of the surface buoyancy flux gra-
+dient perturbation in the first decade, anomalies in the
+ocean’s buoyancy structure develop linearly with time for
+the −7.5 Wm−2 simulation. As an example, the potential
+density latitude-depth transect for the −7.5 Wm−2 simula-
+tion at the end of 95 years (Figs. 5c) and 6c) is quite similar
+to the potential density transect for −15 Wm−2 simulation
+at the end of 50 years (Figs. 5e and 6e).
+The manifestation of surface buoyancy fluxes on the
+density structure of the ocean, especially on longer time
+scales, is not always linear, which may lead to a com-
+plex circulatory response. Unlike the −7.5 Wm−2 simu-
+lation, Figs. 5d-f and 6d-f suggest that the anomalies in
+the Atlantic and Pacific Ocean’s density structure in the
+−15 Wm−2 simulation evolve non-linearly with time in the
+latter stages of the simulation period due to heat advec-
+tion by the circulation. Comparing Fig. 5e and Fig. 5f, we
+notice an increase in potential density in the upper ocean
+subtropical region in year 95, which is overshadowed by a
+relatively stronger (to year 50) potential density increase in
+the subpolar region. The overall effect is a relative (to year
+50) increase in meridional density gradients, and therefore,
+spin up of the northern region of the subtropical gyre in
+the −15 Wm−2 simulation by ≈ 18% in the last 20 years
+of the simulation (Fig. 4a). In summary, surface buoyancy
+forcing anomalies alter the density structure of the mixed
+layer and gradually infiltrate to deeper layers. However,
+this downward infiltration is continuously modified by the
+ocean circulation through heat redistribution, leading to a
+
+9
+0
+20
+40
+60
+80
+100
+Time (years)
+20
+22
+24
+26
+28
+30
+Transport (Sv)
+a. North Atlantic
+-15
+-7.5
+B (W m
+2)
+28
+21
+14
+7
+0
+Fractional change (%)
+-1.8
+-1.0
+-3.4
+-2.1
+0
+20
+40
+60
+80
+100
+Time (years)
+20
+22
+24
+26
+28
+30
+Transport (Sv)
+b. North Pacific
+-15
+-7.5
+B (W m
+2)
+28
+21
+14
+7
+0
+Fractional change (%)
+-0.5
+-0.3
+-5.7
+-2.2
+0
+20
+40
+60
+80
+100
+Time (years)
+16
+18
+20
+Transport (Sv)
+c. South Atlantic
+-15
+-7.5
+B (W m
+2)
+28
+21
+14
+7
+0
+Fractional change (%)
+-1.8
+-0.5
+-1.9
+-1.7
+0
+20
+40
+60
+80
+100
+Time (years)
+10
+12
+14
+Transport (Sv)
+d. South Pacific
+-15
+-7.5
+B (W m
+2)
+28
+21
+14
+7
+0
+Fractional change (%)
+-0.1
+-0.2
+-3.1
+-2.3
+-15 W m
+2
+-7.5 W m
+2
+Control
+Fig. 4: Comparison of subtropical gyre strength for the reduced surface buoyancy flux contrast experiments. For each
+gyre, the left panel shows the time series for −15 Wm−2 (dark green), −7.5 Wm−2 (light green), and control (black)
+simulations, and the right panel shows the fractional change in gyre strength with respect to control for the first 10 (red)
+and last 10 (blue) years of the simulation. Values on each bar depict the absolute change in gyre strength (in Sv) relative
+to the control.
+non-linear evolution of the density structure, and hence the
+gyre circulation on longer timescales.
+In these experiments we observe a weakening of the
+AMOC with time (Fig. 7a), as a reduction in the sur-
+face buoyancy flux gradients causes the subpolar and polar
+regions to experience a stronger stratification.
+Stronger
+stratification suppresses deep water formation in the North
+Atlantic, which is a major source of the AMOC (Togg-
+weiler and Samuels 1995; Marshall and Speer 2012). The
+AMOC steadily reduces in the first 60 years of the re-
+duced buoyancy contrast simulations, after which it begins
+to recover in the −15 Wm−2 simulation, as opposed to the
+−7.5 Wm−2 where it continues to slow down. Moreover,
+we observe a temporal correlation between the North At-
+lantic subtropical gyre strength and the AMOC (Figs. 4a
+and 7a).
+There is a nominal (≈2.5%) reduction in the circumpolar
+transport for both −7.5 Wm−2 and −15 Wm−2 experiments
+at the end of 100 years (Fig. 7b). Several competing factors
+such as reduced meridional buoyancy gradients over the
+ACC latitude band (Hogg 2010), deepening and shoaling of
+the thermocline respectively over the northern and southern
+regions of the ACC, variations in lateral mixing (Ragen
+et al. 2020), and alterations to the Antarctic bottom water
+production (Morrison and Hogg 2013) could influence the
+ACC strength – a thorough analysis is beyond the scope of
+this study.
+Finally, we briefly discuss the changes in ocean circu-
+lation due to surface buoyancy forcing anomalies from an
+energetics perspective. In addition to anomalies observed
+in the large-scale circulatory features (Figs. 4 and 7a-b),
+the variations in globally integrated kinetic energy (Fig. 7d)
+supplement our understanding that surface buoyancy forc-
+ing is an important mechanism in steering the ocean cir-
+culation. A reduction in surface buoyancy flux gradients
+inhibits the production of available potential energy, which
+reduces the conversion from available potential energy to
+kinetic energy.
+b. Increased surface buoyancy flux contrast experiments
+In the previous subsection, we considered the short-term
+and long-term ramifications of reducing surface buoyancy
+flux gradients on the ocean circulation. A natural follow-
+up question is: How would the circulation respond to an
+increase in meridional surface buoyancy flux gradients?
+Here, we analyze two surface buoyancy perturbation ex-
+periments where we increase the meridional surface heat
+contrast by +7.5 Wm−2 and +15 Wm−2 at the latitude of
+western boundary separation for subtropical gyres using
+the heat flux perturbation map in Fig. 1b.
+Similar to the reduced surface buoyancy flux contrast
+experiments, the anomalies in the Atlantic Ocean in the in-
+creased surface buoyancy flux contrast experiments are in-
+duced more quickly than in the Pacific (compare the red bar
+
+10
+50
+0
+50
+200
+400
+600
+800
+1000
+a. -7.5 W m
+2: Year 7
+50
+0
+50
+b. -7.5 W m
+2: Year 50
+50
+0
+50
+c. -7.5 W m
+2: Year 95
+50
+0
+50
+200
+400
+600
+800
+1000
+d. -15 W m
+2: Year 7
+50
+0
+50
+e. -15 W m
+2: Year 50
+50
+0
+50
+f. -15 W m
+2: Year 95
+1.8
+1.2
+0.6
+0.0
+0.6
+1.2
+1.8
+2 (kg m
+3)
+Latitude (degrees)
+Depth (m)
+Fig. 5: Potential density (𝜎2) anomalies for a longitudinal slice of the upper Atlantic Ocean in the −7.5 Wm−2 (top row)
+and −15 Wm−2 (bottom row) experiments for year 7 (left column), year 50 (middle column) and year 95 (right column),
+obtained by averaging between 60◦W and 30◦W for all latitudes. Blue indicates increase in potential density, associated
+with cooling and/or salinification, whereas red indicates decrease in potential density, associated with heating and/or
+freshening.
+50
+0
+50
+200
+400
+600
+800
+1000
+a. -7.5 W m
+2: Year 7
+50
+0
+50
+b. -7.5 W m
+2: Year 50
+50
+0
+50
+c. -7.5 W m
+2: Year 95
+50
+0
+50
+200
+400
+600
+800
+1000
+d. -15 W m
+2: Year 7
+50
+0
+50
+e. -15 W m
+2: Year 50
+50
+0
+50
+f. -15 W m
+2: Year 95
+1.8
+1.2
+0.6
+0.0
+0.6
+1.2
+1.8
+2 (kg m
+3)
+Latitude (degrees)
+Depth (m)
+Fig. 6: Potential density (𝜎2) anomalies for a longitudinal slice of the upper Pacific Ocean in the −7.5 Wm−2 (top
+row) and −15 Wm−2 (bottom row) experiments for year 7 (left column), year 50 (middle column) and year 95 (right
+column), obtained by averaging between 220◦W and 140◦W for all latitudes. Blue indicates increase in potential density,
+associated with cooling and/or salinification, whereas red indicates decrease in potential density, associated with heating
+and/or freshening.
+
+11
+0
+20
+40
+60
+80
+100
+10
+12
+14
+16
+Transport (Sv)
+a. Atlantic Meridional Overturning Circulation
+0
+20
+40
+60
+80
+100
+87.5
+90.0
+92.5
+95.0
+Transport (Sv)
+b. Antarctic Circumpolar Current
+0
+20
+40
+60
+80
+100
+Time (years)
+1800
+1900
+2000
+Energy (Joules)
+c. Globally Integrated Kinetic Energy
+-15 W m
+2
+-7.5 W m
+2
+Control
+Fig. 7: Circulation metrics for the reduced surface buoy-
+ancy flux contrast simulations: −15 Wm−2 (light green),
+−7.5 Wm−2 (dark green), and control (black) after a 5-
+year rolling mean was applied.
+(a) Atlantic meridional
+overturning circulation: integrated meridional transport
+for 𝜎2 ∈ [1035.5,1038.0] kgm−3 at 26◦N for longitudes
+between 103◦W and 5◦W. (b) Antarctic Circumpolar Cur-
+rent transport through the Drake Passage.
+(c) Globally
+integrated kinetic energy.
+graphs in Fig. 8), with the Atlantic gyres in the +15 Wm−2
+simulation intensifying by ≈ 20% after 15 years. A linear
+regression model of the form:
+Δ𝜓 = 𝑚Δ𝐵,
+(3)
+is applied, using outputs from the first decade of both
+the reduced and increased surface buoyancy flux contrast
+experiments. In this equation, Δ𝜓 is the change in gyre
+circulation, and 𝑚 = Δ𝜓/Δ𝐵 represents the variation in
+circulation due to the surface buoyancy contrast Δ𝐵. The
+regression model predicts a strong linear behavior with
+reference to the applied surface buoyancy flux contrast for
+the Atlantic and North Pacific subtropical gyres (inferred
+from the high R2 scores in Table 2). Variations in the South
+Pacific subtropical gyre due to an applied surface buoyancy
+contrast in the first 10 years is negligible (Table 2).
+On longer timescales, the regression model (3) performs
+poorly, as the relationship between gyre circulation and
+anomalous surface buoyancy flux contrast becomes non-
+linear with time. This is accompanied by an oscillatory
+behavior in the gyre strength, as can be observed in the
+time series in Fig. 8. Although the North Atlantic gyre
+strength time series shows high variability, the circulation
+estimates for both +7.5 Wm−2 and +15 Wm−2 simula-
+tions are generally greater than the control.
+The North
+Pacific subtropical gyre strength anomalies monotonically
+increase with time for the +15 Wm−2 simulation (Fig. 8b),
+but show no temporal relation for the +7.5 Wm−2 sim-
+ulation. The South Atlantic subtropical gyre strength is
+enhanced in the first decade of the simulations (Fig. 8c),
+followed by a plateauing for about 20-25 years. The re-
+duction in the South Atlantic subtropical gyre strength in
+the last 50 years can be attributed to an inaccurate esti-
+mate of the circulation: integrating meridional transport
+for all isopycnals having 𝜎2 ≤ 1035.8 kgm−3 captures a
+part of the mid-depth overturning circulation. The South
+Pacific subtropical gyre strength anomalies are minimal for
+both +7.5 Wm−2 and +15 Wm−2 simulations in the first
+decade, but grow with time, with the gyre strength in the
++15 Wm−2 simulation fluctuating around the control. In
+conclusion, an oscillatory response is present in all four
+subtropical gyres, suggesting that there is a complicated
+feedback between ocean circulation and surface buoyancy
+forcing.
+Table 2: Linear regression model (8) for the subtropical
+gyre perturbations over the first 10 years. R2 score indi-
+cates the extent of gyre variability due to buoyancy forcing
+that is captured by the linear regression model.
+Gyre basin
+𝑚 (Sv m2W−1)
+R2 score
+North Atlantic
+0.15
+0.99
+South Atlantic
+0.10
+0.98
+North Pacific
+0.02
+0.91
+South Pacific
+0.00
+-0.65
+The North Atlantic subtropical gyre shows an oscilla-
+tory response in terms of strength (Fig. 8a) as well as the
+western boundary separation latitude (Fig. 9a). The con-
+traction of the northern extent of the subtropical gyre is
+accompanied by a shoaling and southward expansion of
+the cyclonic North Atlantic subpolar gyre (compare green
+contours in Figs. 9c and 9d). The shoaled subpolar gyre
+carries colder water to the subtropical regions in the near-
+surface layers, which sits above the less dense subtropical
+gyre. The unstable vertical structure leads to convection in
+the northern half of the western boundary region, resulting
+
+12
+0
+20
+40
+60
+80
+100
+Time (years)
+23
+26
+29
+32
+35
+Transport (Sv)
+a. North Atlantic
++7.5 +15
+B (W m
+2)
+20
+0
+20
+Fractional change (%)
+0.8
+2.2
+1.9
+-1.9
+0
+20
+40
+60
+80
+100
+Time (years)
+27
+30
+Transport (Sv)
+b. North Pacific
++7.5 +15
+B (W m
+2)
+20
+0
+20
+Fractional change (%)
+0.3
+0.3
+-0.1
+2.4
+0
+20
+40
+60
+80
+100
+Time (years)
+11
+14
+17
+20
+Transport (Sv)
+c. South Atlantic
++7.5 +15
+B (W m
+2)
+60
+40
+20
+0
+20
+Fractional change (%)
+0.5
+1.3
+-4.3
+-6.6
+0
+20
+40
+60
+80
+100
+Time (years)
+13
+16
+Transport (Sv)
+d. South Pacific
++7.5 +15
+B (W m
+2)
+20
+0
+20
+Fractional change (%)
+-0.0
+-0.0
+0.8
+-0.7
+Control
++7.5W m
+2
++15 W m
+2
+Fig. 8: Comparison of subtropical gyre strength for the increased surface buoyancy flux contrast experiments. For each
+gyre, the left panel shows the time series for control (black), +7.5Wm−2 (orange), and +15Wm−2 (red) simulations, and
+the right panel shows the fractional change in gyre strength with respect to control for the first 10 (red) and last 10 (blue)
+years of the simulation. Values on each bar depict the absolute change in gyre strength (in Sv) relative to the control.
+in excessively deep mixing layers during that time period
+(Fig. 9b).
+The onset of convection in the North Atlantic basin is
+associated with the development of an oscillating abyssal
+anticyclonic gyre below 4000m (Fig. 9e and Fig. 9f). The
+strength of the abyssal gyre correlates well with the mixing
+layer depth (Fig. 9b), which in turn is induced by convec-
+tion in the western boundary of the subtropical gyre. The
+formation of the abyssal gyre could be explained by two
+hypotheses: (i) Excessively deep mixing layers due to con-
+vection could relay vorticity input due to the surface wind
+stress directly to the abyssal circulation, or (ii) the south-
+ward extension of cyclonic subpolar gyre at the surface
+could baroclinically stimulate an anti-cyclonic circulation
+in the deeper layers. A complete analysis of the processes
+leading to the formation of the abyssal gyre is outside the
+scope of this study.
+In the reduced buoyancy contrast experiments, the den-
+sity anomalies for the −7.5 Wm−2 experiment at the end of
+100 years closely match with the −15 Wm−2 simulation at
+the end of 50 years, suggesting that small reductions in sur-
+face buoyancy flux gradients are manifested linearly in the
+ocean’s buoyancy structure. However, this is not true for
+the positive buoyancy flux anomaly experiments (compare
+the Atlantic basin in Figs. 10c and 10e, and the Pacific
+basin in Figs. 11c and 11e). Likewise, we observe a non-
+linear evolution of the ocean’s buoyancy structure in the
+Atlantic and Pacific basins for the +30 Wm−2 simulation
+(not presented here).
+The AMOC in the increased surface buoyancy flux con-
+trast simulations initially spins up with time, followed by an
+oscillatory behavior (Fig. 12a). The time taken to display
+this oscillatory behavior is inversely related to the magni-
+tude of the surface buoyancy flux gradient anomaly. The
+oscillation of the North Atlantic subtropical gyre strength
+(Fig. 8a) is similar to that of the AMOC, suggesting a direct
+relationship between the two. Increased cooling in subpo-
+lar regions stimulates the production of NADW, which may
+cause an acceleration of the mid-depth circulation (Morri-
+son et al. 2011).
+The reduced buoyancy flux contrast experiments re-
+vealed a weak dependence of circumpolar transport on
+meridional surface buoyancy gradients (Fig. 7c), with tem-
+poral variability within experiments being stronger than
+the anomalies. However, increasing the latitudinal surface
+buoyancy contrast produces a robust increase in the ACC
+(Fig. 12b), in agreement with Hogg (2010) and Morrison
+and Hogg (2013). An increase in the surface buoyancy
+flux gradients also promotes the conversion of available
+potential energy to kinetic energy (Fig. 12c), consistent
+with Tailleux (2009).
+5. Uniform warming perturbation experiment
+In the previous section, we showed that variations in
+surface heat fluxes, with opposite signed anomalies ap-
+plied over subpolar and subtropical regions, can produce
+anomalies in the ocean circulation. In this section we ask;
+
+13
+Fig. 9: (a) Western boundary separation latitude of the North Atlantic subtropical gyre for the control (black) and
++15 Wm−2 (red) simulations. (b) Temporal correlation between mixing layer depth (light red) and strength of the
+abyssal gyre (dark red) for +15 Wm−2 simulation. The mixing layer depth is estimated by spatially averaging over
+the red box in panel (f). The abyssal circulation is measured by taking the 95th percentile of all density classes lying
+below 3000m in red box in panel (f). (c)-(f) Streamfunction of the North Atlantic basin for the +15 Wm−2 simulation.
+Compare the emergence of abyssal gyre (red box in panel (f)) with the beginning of the simulation (panels e), when the
+entire anticyclonic circulation is confined in the upper 1500 m of the basin. The western boundary separation latitude
+time series in panel (a) denotes the times (see red dots in panel (a)) that correspond to these streamfunctions.
+is the same true if the surface heat flux anomalies are glob-
+ally uniform? We expect changes in the ocean circulation
+due to a spatially uniform surface heat flux due to sev-
+eral processes. Firstly, lateral variations in mixed layer
+depth imply that buoyancy anomalies induced by the uni-
+form heating will be non-uniform. In addition, changes
+in circulation continuously alter the buoyancy structure
+of the ocean through advection. To understand the com-
+bined effects of mixed layer depth variations and advective
+feedbacks on the ocean circulation, we analyze a uniform
+warming experiment, where a globally constant heat flux
+of +5 Wm−2 is applied at the ocean’s surface.
+Changes in the strength of each gyre do occur under the
+uniform warming perturbation (Fig. 13a-c). Focusing first
+on the North Pacific basin (Fig. 13b), we observe ∼22% in-
+tensification of the subtropical gyre, consistent with the re-
+sults of Sakamoto et al. (2005) and Chen et al. (2019). This
+increase could be attributed to spatial variations in mixed
+layer depth near the western boundary region. The mixed
+layer traps the excess heat received from the ocean’s sur-
+face and distributes only a fraction of this heat to the layer
+below. Furthermore, a deeper mixed layer has a higher
+heat capacity due to its ability to store more heat. Deep
+mixed layers at the western boundary of the North Pacific
+subtropical gyre moderately shield the region from devel-
+oping stratification. Conversely, shallower mixed layers to
+the east of the western boundary lead to more stratification
+in that region. The spatially uneven growth of stratification
+in the subtropical gyre strengthens zonal buoyancy gradi-
+ents near the western boundary region, which in view of
+the thermal wind relation, intensifies the meridional gyre
+flow.
+We record a strong (∼50%) intensification of the South
+Atlantic subtropical gyre (Fig. 13c) due to a similar later-
+ally varying mixed layer depth as observed in the North
+Pacific basin, which augments the zonal buoyancy gra-
+dients near the western boundary in response to surface
+heating. Finally, the South Pacific gyre anomalies show
+minor oscillations (with an amplitude of ∼ 1.5Sv) due to
+uniform surface heating (not shown).
+We observe positive anomalies in the North Atlantic
+subtropical gyre strength for the first 20 years of the uni-
+
+c. Top 1500 m, Year 10
+d. Top 1500 m, Year 82
+e. 1500-5000 m, Year 10
+f. 1500-5000 m, Year 82
+36
+73°
+Latitude (degrees)
+24
+Transport (Sv)
+65°N
+12
+0
+55°N
+40°N
+24
+-36
+20°N
+M.0E M.09 M.06
+0°
+M.0E M.09 M.06
+0°
+90°W 60°W
+30°W
+0°
+M.0E M.09 M.06
+0°
+Longitude (degrees)14
+50
+0
+50
+200
+400
+600
+800
+1000
+a. +7.5 W m
+2: Year 7
+50
+0
+50
+b. +7.5 W m
+2: Year 50
+50
+0
+50
+c. +7.5 W m
+2: Year 95
+50
+0
+50
+200
+400
+600
+800
+1000
+d. +15 W m
+2: Year 7
+50
+0
+50
+e. +15 W m
+2: Year 50
+50
+0
+50
+f. +15 W m
+2: Year 95
+1.8
+1.2
+0.6
+0.0
+0.6
+1.2
+1.8
+2 (kg m
+3)
+Latitude (degrees)
+Depth (m)
+Fig. 10: Potential density (𝜎2) anomalies for a longitudinal slice of the upper Atlantic Ocean for +7.5 Wm−2 (top row)
+and +15 Wm−2 (bottom row) simulation for year 7 (left column), year 50 (middle column) and year 95 (right column),
+obtained by averaging between 60◦W and 30◦W for all latitudes. Blue indicates increase in potential density, associated
+with cooling, whereas red indicates decrease in potential density, associated with heating.
+50
+0
+50
+200
+400
+600
+800
+1000
+a. +7.5 W m
+2: Year 7
+50
+0
+50
+b. +7.5 W m
+2: Year 50
+50
+0
+50
+c. +7.5 W m
+2: Year 95
+50
+0
+50
+200
+400
+600
+800
+1000
+d. +15 W m
+2: Year 7
+50
+0
+50
+e. +15 W m
+2: Year 50
+50
+0
+50
+f. +15 W m
+2: Year 95
+1.8
+1.2
+0.6
+0.0
+0.6
+1.2
+1.8
+2 (kg m
+3)
+Latitude (degrees)
+Depth (m)
+Fig. 11: Potential density (𝜎2) anomalies for a longitudinal slice of the upper Pacific Ocean for +7.5 Wm−2 (top row)
+and +15 Wm−2 (bottom row) simulation for year 7 (left column), year 50 (middle column) and year 95 (right column),
+obtained by averaging between 220◦W and 140◦W for all latitudes. Blue indicates increase in potential density, associated
+with cooling, whereas red indicates decrease in potential density, associated with heating.
+
+15
+0
+20
+40
+60
+80
+100
+15
+20
+25
+Transport (Sv)
+a. Atlantic Meridional Overturning Circulation
+0
+20
+40
+60
+80
+100
+90
+100
+110
+120
+Transport (Sv)
+b. Antarctic Circumpolar Current
+0
+20
+40
+60
+80
+100
+Time (years)
+2000
+2200
+2400
+Energy (Joules)
+c. Globally Integrated Kinetic Energy
+Control
++7.5 W m
+2
++15 W m
+2
+Fig. 12:
+Monthly-mean time-series circulation metrics
+for the wind perturbation flux-forced simulations: control
+(black), +7.5 Wm−2 (orange), and +15 Wm−2 (red) after
+a 5-year rolling mean was applied. (a) Atlantic meridional
+overturning circulation: integrated meridional transport
+for 𝜎2 ∈ [1035.5,1038.0] kgm−3 at 26◦N for longitudes
+between 103◦W and 5◦W. (b) Antarctic Circumpolar Cur-
+rent transport through the Drake Passage.
+(c) Globally
+integrated kinetic energy.
+form warming simulation due to differing heat capacities
+of mixed layer near the Gulf Stream (Fig. 13a). However,
+the initial spin-up is followed by a systematic slowdown
+over the next 75 years. Several modeling studies have re-
+ported a correlation between the North Atlantic subtropical
+gyre strength and the AMOC (Yeager 2015; Larson et al.
+2020), and we observe a reduction in the AMOC over the
+same time period (Fig. 13d). This reduction is explained
+by two processes: (i) the first 20 years of increased gyre
+strength transports a larger volume of warm water through
+the Gulf Stream from tropical and subtropical latitudes to
+the subpolar regions, and (ii) the uniform warming applied
+at the ocean’s surface promotes the generation of lighter
+waters in the subpolar regions at the ocean’s surface. These
+two processes limit North Atlantic deep water formation,
+causing an AMOC slowdown (Lohmann et al. 2008; Cheng
+et al. 2013).
+There is an intensification of the circumpolar transport
+(Fig. 13e), which could be linked to increased meridional
+buoyancy gradients due to uneven ingestion of surface
+buoyancy flux into deeper layers. These irregularities are
+caused by spatial variations in the mixed layer depth across
+the latitudinal band of the ACC.
+Finally, the globally integrated kinetic energy increases
+by almost 50% over the full experiment (Fig. 13f). We
+can ascribe the resulting kinetic energy increase to mean
+(for example, gyres, ACC) flows as well as mesoscale ed-
+dies, and is consistent with the energy conversion argument
+put forth by Tailleux (2009) that surface buoyancy forcing
+could induce kinetic energy in the system through a con-
+version from available potential energy.
+6. Summary and Discussion
+In this study, we conducted a series of perturbed forcing
+simulations using a partially eddy-resolving ocean model
+(at a 0.25◦ lateral resolution) to understand the importance
+of wind stress and surface buoyancy forcing in steering
+planetary-scale ocean circulation. These perturbation ex-
+periments (listed in Table 1) are forced for 100 years each
+using surface boundary fluxes (and are thus called “flux-
+forced simulations”) to separate the contribution of winds
+and surface buoyancy in driving the circulation, and are
+classified into three categories: (i) wind perturbation ex-
+periments, (ii) surface buoyancy flux contrast perturba-
+tions, and (iii) a spatially uniform warming perturbation.
+The flux-forced simulations illustrate that both wind
+stress (Fig. 2) and surface buoyancy forcing (Figs. 4, 8,
+and 13a-c) are crucial in shaping the planetary-scale sub-
+tropical gyres. We find that perturbations in surface buoy-
+ancy flux gradients modify the ocean’s buoyancy structure
+(Fig. 5, 6, 10, and 11), and thus the circulation through
+the thermal wind relation (2). In addition, anomalies in
+horizontal buoyancy gradients (and hence, the circulation
+(Fig. 13)) could also be induced using a spatially uniform
+surface heat flux due to lateral differences in mixed layer
+depth and heat advection by the circulation.
+The anomalous horizontal density gradients are propor-
+tional to the surface buoyancy flux gradient perturbations
+on short (< 1 decade) timescales (compare Figs. 5a and d;
+Figs. 6a and d; Figs. 10a and d; Figs. 11a and d). Through
+the thermal wind relation (2), we diagnose a linear rela-
+tionship (R2 > 0.9 in Table 2) between the anomalous gyre
+circulation and the magnitude of the surface buoyancy flux
+gradient perturbation on short timescales. Over this pe-
+riod, the Atlantic gyres are observed to be 2-4 times more
+susceptible to changes in surface buoyancy flux gradients
+than the Pacific gyres, with as much as a 0.15Sv anomaly
+per Wm−2 change in the subtropical/subpolar surface heat
+flux in the North Atlantic subtropical gyre.
+
+16
+0
+20
+40
+60
+80
+100
+23
+26
+29
+32
+Transport (Sv)
+a. North Atlantic
+0
+20
+40
+60
+80
+100
+25
+28
+31
+34
+37
+Transport (Sv)
+b. North Pacific
+0
+20
+40
+60
+80
+100
+18
+21
+24
+27
+30
+33
+Transport (Sv)
+c. South Atlantic
+0
+20
+40
+60
+80
+100
+6
+9
+12
+15
+18
+Transport (Sv)
+d. Atlantic Meridional Overturning Circulation
+0
+20
+40
+60
+80
+100
+Time (years)
+87
+90
+93
+96
+99
+102
+Transport (Sv)
+e. Antarctic Circumpolar Current
+0
+20
+40
+60
+80
+100
+Time (years)
+2000
+2250
+2500
+2750
+3000
+3250
+Energy (Joules)
+f. Globally Integrated Kinetic Energy
+Control
++5W m
+2 uniform
+Fig. 13:
+(a)-(c) Comparison of subtropical gyre strength time series for control (black) and uniform warm-
+ing (dark red) simulations.
+(d) Atlantic meridional overturning circulation:
+integrated meridional transport for
+𝜎2 ∈ [1035.5,1038.0] kgm−3 at 26◦N for longitudes between 103◦W and 5◦W. (e) Antarctic Circumpolar Current
+transport through the Drake Passage. (f) Globally integrated kinetic energy.
+Over time, the lateral buoyancy gradients become less
+proportional to the surface buoyancy flux perturbations
+(compare Figs. 5c and f; Figs. 6c and f; Figs. 10c and
+f; Figs. 11c and f). A divergence from this linear rela-
+tionship with time could be attributed to several factors
+such as lateral variations in mixed layer depth (Xie et al.
+2010) and advective feedbacks between the surface buoy-
+ancy forcing anomalies and the ocean circulation (Bryden
+et al. 1991). For example, in the −15Wm−2 simulation,
+this non-linear connection can be most clearly observed in
+the spin-up of the North Atlantic subtropical gyre in the
+last 20 years (Fig. 4a) and a surge in the AMOC in the
+last 40 years (Fig. 7a). The ocean circulation in the in-
+creased surface buoyancy flux contrast simulations is more
+non-linearly related to surface buoyancy flux gradient per-
+turbation than the reduced surface buoyancy flux contrast
+simulations. For example, we observe a slowdown in North
+Atlantic, South Atlantic, and South Pacific gyre strength in
+the last 40 years of the +15Wm−2 simulation (Fig. 8) and
+an oscillatory AMOC in both +7.5Wm−2 and +15Wm−2
+simulations (Fig. 12a).
+The flux-forced simulations allowed us to conduct per-
+turbation simulations in which each surface forcing could
+be altered independently. However, the present study has
+numerous caveats. Firstly, in reality the wind and buoyancy
+forcing are strongly coupled. In earlier experiments that
+used bulk formula for the surface heat fluxes (not shown),
+we found that decreasing the wind forcing strongly reduced
+the surface buoyancy fluxes as well due to the reduction in
+poleward heat transport. Next, the flux-forced simulations
+are conducted at 0.25◦ resolution and can only partially
+capture the mesoscale eddies. Moreover, the flux-forced
+control simulation is not fully equilibrated (see Fig. 1c) due
+to the dynamic frazil formation at high latitudes, which
+
+17
+continuously adds heat in the polar regions.
+The frazil
+formation limits the magnitude of surface buoyancy flux
+perturbation we can apply in the polar regions. We have
+partially muted the frazil heat gain in the increased sur-
+face buoyancy flux contrast experiments through adding a
+globally uniform heat loss. In regions of extreme buoyancy
+anomalies, the isopycnal outcropping method (section 2b)
+is prone to capturing other elements of ocean circulation
+especially in regions of heat gain, such as the deep cell of
+the AMOC, which may produce erroneous results. Finally,
+the surface buoyancy flux perturbation experiments have
+not equilibrated even after 100 years, and hence, should
+not be misunderstood as the final response.
+The present study reinforces recent evidence supporting
+the existence of a buoyancy-driven component in ocean
+gyres (Gjermundsen et al. 2018; Hogg and Gayen 2020; Liu
+et al. 2022). We envisage that a complete theory describing
+the formation of ocean gyres should incorporate the effects
+of surface buoyancy forcing, in addition to surface wind
+stress (Munk 1950).
+However, the influence of surface
+buoyancy forcing on gyres depends on the ocean state,
+allowing non-linear and non-local feedbacks with the ocean
+circulation that obscure the formulation of a simple theory.
+These feedbacks may include the role of the mixed layer in
+capturing excess heat and the horizontal transport of heat
+by the circulation, both of which influence the background
+stratification.
+Acknowledgments.
+We would like to thank the
+community
+of
+the
+Consortium
+for
+Ocean–Sea
+Ice
+Modeling in Australia (http://cosima.org.au) for
+helpful
+discussions
+and
+for
+the
+development
+and
+maintenance of cosima-cookbook package (https:
+//github.com/COSIMA/cosima-cookbook)
+and
+the
+cosima-recipes repository (https://github.com/
+COSIMA/cosima-recipes), both of which are indis-
+pensable for our workflow.
+D.B. would like to thank
+Bishakhdatta Gayen, Aviv Solodoch, Andy Thompson,
+and Christopher Wolfe for stimulating discussions during
+early stages of this work. D.B. expresses gratitude to the
+Computational Modeling Systems group at the Australian
+Research Council Center for Climate Extremes for their
+assistance in conducting the flux-forced simulations. Our
+analyses were facilitated with the Python packages dask
+(Rocklin 2015) and xarray (Hoyer and Hamman 2017).
+Computational resources were provided by the Australian
+National Computational Infrastructure at the ANU, which
+is supported by the Commonwealth Government of Aus-
+tralia.
+R.M.H. is supported by the Australian Research
+Council DECRA Fellowship DE210100004.
+N.C.C. is
+supported by the Australian Research Council DECRA
+Fellowship DE210100749.
+Data availability statement.
+Python code used for gen-
+erating figures are available at https://github.com/
+dhruvbhagtani/varying-surface-forcing. Output
+to reproduce figures will be available in a Zenodo repos-
+itory upon acceptance of manuscript.
+MOM5 source
+code for flux-forced simulations is available at https:
+//github.com/dhruvbhagtani/MOM5.
+APPENDIX
+K-profile ocean surface boundary layer
+parameterization
+The ACCESS-OM2 control and flux-forced simulations
+estimate the mixing layer depth using the K-profile param-
+eterization (Large et al. 1994). The mixing layer depth
+depends on many factors, such as Langmuir turbulence
+(Belcher et al. 2012), surface buoyancy forcing (Yoshikawa
+2015), wind stress (Grant and Belcher 2011), and convec-
+tion (Sohail et al. 2020). The effect of these processes on
+the mixing layer depth can be encapsulated in the Richard-
+son number, which is the ratio of stratification, 𝜕𝑧𝑏, and
+vertical flow shear squared, |𝜕𝑧u|2. The K-profile param-
+eterization uses the bulk Richardson number Ri𝑏 (Stull
+1988) defined over a depth ℎ:
+Ri𝑏(ℎ) =
+[𝑏(0) − 𝑏(−ℎ)]/ℎ
+|u(0) −u(−ℎ)|2/ℎ2 +𝑢2
+turb/ℎ2 ,
+(A1)
+where, e.g., 𝑏(−ℎ) ≡ 𝑏(𝑥, 𝑦,𝑧 = −ℎ,𝑡). In (A1), the numer-
+ator is the mean stratification averaged over depth ℎ and in
+the denominator, |u(0) −u(−ℎ)|/ℎ is the magnitude of the
+resolved velocity shear averaged over depth ℎ while term
+𝑢turb/ℎ quantifies the unresolved/turbulent velocity shear.
+The parameterization determines the mixing layer depth ℎ
+such that Ri𝑏(ℎ) is equal to a critical Richardson number,
+typically taken to be 0.25-0.3.
+Prior to creating the flux-forced simulations, we con-
+ducted wind sensitivity experiments using ACCESS-OM2-
+025 (not presented here) along with the traditional K-profile
+parameterization reported in Large et al. (1994). However,
+the surface buoyancy forcing was inadvertently modified
+in these sensitivity experiments through anomalies in the
+mixing layer depth and ocean circulation. In an attempt to
+minimize surface buoyancy flux variations in the ACCESS-
+OM2-025 wind stress sensitivity experiments, we recon-
+structed the resolved velocity shear term |u(0) −u(−ℎ)|/ℎ
+in (A1) in the K-profile parameterization, which was found
+to primarily cause mixing layer depth anomalies in the sen-
+sitivity experiments. We parameterized |u(0) −u(−ℎ)| as
+a function of the friction velocity, 𝑢∗ = (|τ|/𝜌0)1/2 and
+depth ℎ:
+|u(0) −u(−ℎ)|2 = (𝑐𝑎𝑢2
+∗ +𝑐𝑏𝑢∗)(1−e−𝑐𝑒ℎ/√𝑢∗),
+(A2)
+where 𝑐𝑎, 𝑐𝑏, and 𝑐𝑒 were coefficients found using multi-
+variate linear regression. Table A1 lists typical ranges of
+the three parameters for an optimal solution. The parame-
+terization (A2) performs well in the tropical and subtropical
+
+18
+regions. Errors in the polar regions are expected because
+our parameterization (A2) does not account for sea ice and
+marginal ice zone so as to stay consistent with the flux-
+forced experiments, which could alter resolved velocity
+shear in these regions.
+Table A1: Ranges of parameters for resolved velocity
+shear obtained using multi-variate linear-regression.
+Coefficient
+Range
+𝑐𝑎
+50−70
+𝑐𝑏
+0.8−1.2 m s−1
+𝑐𝑒
+0.009−0.011 (m s)−1/2
+The parameterization prevented discrepancies in the
+mixing layer depth due to alterations in the wind stress,
+and was implemented in the ACCESS-OM2-025 control
+experiment (Fig. 1c). To maintain consistency with the
+ACCESS-OM2-025 control simulation, we retained the
+resolved shear parameterization in the flux-forced simula-
+tions.
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