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+OPTIMISED MORSE TRANSFORM OF A GAUSSIAN PROCESS
+FEATURE SPACE
+Fabio E. A. Albertani∗† , Alex J. W. Thom∗
+January 5, 2023
+ABSTRACT
+Morse projections are well-known in chemistry and allow one, within a Morse potential approximation,
+to redefine the potential in a simple quadratic form. The latter, being a non-linear transform, is also
+very helpful for machine learning methods as they improve the performance of models by projecting
+the feature space onto more well-suited coordinates. Usually, the Morse projection parameters are
+taken from numerical benchmarks. We investigate the effect of changing these parameters latter
+on the model learning, as well as using the machine learning method itself to make the parameters
+decision. We find that learning is not necessarily improved by the latter and that general Morse
+projections are extremely susceptible to changes in the training data.
+1
+Introduction
+Machine learning, as in many fields of science, has rev-
+olutionised the way theoretical chemists approach the
+interpolation of molecular properties. The many methods
+encompassed by the machine learning framework pro-
+vide tools to construct models of the former with great
+accuracy1–3. A particular method that has seen success is
+the Gaussian process (GP) framework which has seen ex-
+tensive publications in machine learning potential energy
+surface applications4–12.
+The representation of the molecular geometry is an essen-
+tial part of the ML building process and has seen many
+“solutions” spring up through the years13. When using
+global or local descriptors of the atomic configuration
+of the system to build a “feature space”, one often uses
+the internuclear distances as an underlying coordinates.
+The latter are often transformed to improve the accu-
+racy of models since ML models do not perform equally
+when the training data is projected onto different feature
+spaces. One known projection of the feature space is
+the Morse transform of the internuclear distances which
+often improves one’s ability to learn the surface14.
+Given the ability of a GPs to learn the underlying pattern
+of the target function15,16, it is interesting to consider a
+GP which can change the underlying function in its opti-
+misation. This is done by making the distance fed to the
+kernel (see next section) transform with GP hyperparam-
+eters within the kernel itself.
+Many more feature space transformations could be con-
+sidered (these are also not restricted to transformations
+based on internuclear distances) but we will here discuss
+the effect of the added “transformation hyperparameters”
+on the GP optimisation process.
+2
+Gaussian Processes
+A Gaussian process is a machine learning regression
+method and is defined as a collection of random vari-
+ables, any finite number of which have a joint Gaussian
+distribution16. An essential part of a GP model is its
+kernel function which defines, over a feature space (the
+input space of the GP), a measure of similarity.
+There are many possible kernel functions one can defined,
+as they only need to adhere to a few simple rules16. We
+use here the Matérn class kernel multiplied by a constant
+kernel (CK) and summed with a White Kernel (WK) to
+model noise. The covariance between two vectors over
+the feature space, X and X′ here, is given by
+K(X, X′) = σ2 21−ν
+Γ(ν)
+�
+√
+2ν d
+ρ
+�ν
+Kν
+�
+√
+2ν d
+ρ
+�
++ λ2
+(1)
+∗ Yusuf Hamied Department of Chemistry, University of Cambridge, Cambridge, Lensfield Road, CB2 1EW
+† [email protected]
+arXiv:2301.02172v1  [physics.chem-ph]  5 Jan 2023
+
+Optimised Morse transform of a Gaussian process feature space
+where Γ is the gamma function, Kν is the modified
+Bessel function of the second kind of degree ν, ρ are
+length scales and d is the Euclidean distance in feature
+space |X − X′|. The ν parameter is not optimised and
+defines the smoothness of the kernel: a GP with a Matérn
+kernel of parameter ν = n + 0.5 is n-times differen-
+tiable↓. We also explore an infinitely smooth version of
+the Matérn kernel with ν → ∞, commonly known as the
+radial basis function (RBF) kernel.
+At a set of query points, forming a matrix Xp of size
+Np × Nfeatures, a GP model predicts a Gaussian distri-
+bution with a mean (sometimes called the latent func-
+tion), here denoted y(Xp), and a variance, here denoted
+∆(Xp), which is associated to the model confidence. For
+a set of prediction points, Xp, the predicted distribution
+are given by16:
+y(Xp) = Kpt K−1
+tt y
+∆(Xp) = Kpp − Kpt K−1
+tt Ktp
+(2)
+where the kernel matrices are subscripted with the ma-
+trices they evaluate (p for query points and t for train-
+ing) and the ijth element of the matrix Knm is given
+by K(Xn,i, Xm,j). A common metric, used by the ML
+community, to define the confidence in predictions is the
+∆95% confidence interval which is given as y ± 2∆ for
+GPs.
+GPs are optimised by finding the most suited hyperpa-
+rameters for its kernel. Using a Bayesian approach, one
+finds the latter by maximising the log-marginal likelihood
+(LML) defined as16
+LML = −1
+2yTK−1
+tt y − 1
+2log|Ktt| − n
+2 log(2π)
+(3)
+where Ktt, as before, is the covariance matrix of the
+training set to itself. The terms on the LHS of equa-
+tion 3 can be understood as a fit, a regularisation and a
+normalisation term respectively.
+Practically, the maximisation is done by minimising
+−LML but we will use the term LML as the surface
+we minimise and the term “minimum” as a set of hy-
+perparameters corresponding to a model selected by the
+GP.
+The LML exploration is done with the GMIN suite17–19
+which allows to give a full description of the minima,
+both global and local, of the surface as well as their
+connectivity. In order to visualise the surfaces, we use
+disconnectivity graphs20–22 which represent, on a −LML
+vertical scale, minima as vertical lines connected by tran-
+sition states shown by connecting those lines.
+3
+Methodology
+If one takes the coordinates to be specified as a vector, X,
+of N(N − 1)/2 internuclear distances, then the Morse
+transformed coordinates form a vector defined as
+T (X; M = {α, X0}) :
+RN(N−1)/2
+→ RN(N−1)/2
+Xi
+�→ exp
+�
+− (Xi − X0)/α
+�
+(4)
+where α is the Morse parameter and X0 is the Morse
+shift parameter. In order to simplify the notation, we will
+write the Morse transformed vector XJ as T XJ.
+The reasoning behind this transform is that an analytical
+Morse potential becomes quadratic when projected onto
+the coordinates T X. If one considers non-analytical po-
+tentials, one expects the potentials to closer to quadratic
+in T X than X. The simpler PES is better described by a
+GP since the length scale of the problem becomes more
+“unique”. Despite some specific bonds having chemically
+derived optimal Morse parameters, there is not always a
+straight-forward way to select those parameters. There
+are two ways of optimising those parameters: a numeri-
+cal optimisation to reduce the error on a testing set (the
+traditional “best-fit” approach) and a Bayesian approach
+with a Morse hyperparameter. As one does not want the
+number of hyperparameters to be too large and optimise
+in a very large space, we will set Morse hyparparameters
+to be equal for all feature dimensions.
+Taking a basic RBF kernel16, on a Morse transformed
+feature space, the kernel is evaluated as
+K( ˜XA = T XA, ˜XB = T XB; ρ) =
+exp
+�
+− 1
+2( ˜XA − ˜XB)TP( ˜XA − ˜XB)
+�
+where P = In
+�
+�����
+ρ−2
+1
+ρ−2
+2
+...
+ρ−2
+n
+�
+�����
+(5)
+where we now used the matrix notation of the RBF ker-
+nel and where ρi are the length scales along each feature
+dimension. We use here the ˜XJ notation to differentiate
+the fixed Morse projection of the kernel input (the Morse
+parameters do not appear in the evaluation of the ker-
+nel in the LHS of equation 5) from the projection taken
+within the kernel itself, like in equation 6.
+Instead of equation 5, one can use the internuclear dis-
+tances, X as an input and optimise the Morse parameters
+↓ For example, with ν = 2.5, one ensures that the GP latent function is physical since both the atomic forces (first derivative) and
+atomic Hessians (second derivative) are smooth w.r.t. geometrical changes
+2
+
+Optimised Morse transform of a Gaussian process feature space
+inside a “MorseRBF” kernel which is evaluated as
+K(XA, XB; M, ρ) =
+exp
+�
+− 1
+2(T XA − T XB)T P (T XA − T XB)
+�
+̸≡ exp
+�
+− 1
+2( ˜XA − ˜XB)T P ( ˜XA − ˜XB)
+�
+(6)
+where P is the same matrix as the one in equation 5. In
+the MorseRBF approach, the Morse parameters are hy-
+perparameters of the kernel alongside the length scales.
+The interesting approach of the kernel is that it does not,
+as is common practice, optimise to the Morse parameters
+that minimises the error on the testing set, the “best-fit”
+approach, but instead uses a Bayesian approach and a
+“statistically relevant” set of Morse parameters.
+Regarding the X0 parameter, one can see in figure 1-2
+that the covariance function drops very quickly for data
+points in the region X < X0 which could potentially lead
+to a loss of information. With very compressed kernel
+length scales around X < X0, data will not affect the
+latent function. Learning was also performed for those
+surfaces but were not considered further in this study.
+The derivatives of the kernel with respect to each hyper-
+parameter can be obtained analytically. The derivatives
+with respect to the length scales are not affected by the
+Morse transform and are equivalent to simply changing
+the feature space in the derivatives of the standard RBF
+kernel. The derivative with respect to the Morse parame-
+ter can also be analytical obtained and is given by
+∂αK(XA, XB) =
+�
+− 1
+2∂α
+��T XA − T XB
+�T�
+P
+�T XA − T XB
+��
+K(XA, XB)+
+�
+− 1
+2
+�T XA − T XB
+�T P
+∂α
+��T XA − T XB
+���
+K(XA, XB)
+=
+�� XA
+2α2
+T XA − XB
+2α2
+T XB
+�T P
+�T XA − T XB
+��
+K(XA, XB)+
+��T XA − T XB
+�T P
+� XA
+2α2
+T XA − XB
+2α2
+T XB
+��
+K(XA, XB)
+(7)
+where T XI are Morse transformed vectors of the original
+XI data points (to simplify the notation we did not write
+the dependency on α and ρ).
+In a very similar manner to the MorseRBF kernel, one
+can define a MorseMatérn kernel starting from equation
+1 and, despite analytical definitions of the gradient of the
+kernel with respect to the Morse hyperparameter being
+quite complicated, one can use numerical gradients and
+optimise the Morse transformed kernel.
+To understand the effect of the Morse kernels, we
+compare the shape of the kernel functions in the non-
+transformed space. Figure 1-2 shows a Matérn and a
+MorseMatérn, projected back to the non-transformed di-
+mension, to give a better insight.
+−1
+0
+1
+2
+3
+X
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+K(XA, XB)
+X0 = 0
+−1
+0
+1
+2
+3
+X
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+K(XA, XB)
+X0 = 1
+−1
+0
+1
+2
+3
+X
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+K(XA, XB)
+X0 = 2
+Figure 1: Covariance of the Matérn (ν = 2.5) kernel
+(black lines) compared to the MorseMatérn kernel (red
+lines), projected back onto X, for different X0 and with
+α = 2.0. The covariance is quite unsymmetrical an the
+forward influence is greater than the backward influence
+since the transform expands the dataset at large X values.
+The X0 parameter dampens the strong “elongation” of the
+covariance at small X values and also strongly contracts
+the covariance extent at X< X0 where the exponent in
+equation 4 becomes positive.
+−1
+0
+1
+2
+3
+X
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+K(XA, XB)
+X0 = 0
+−1
+0
+1
+2
+3
+X
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+K(XA, XB)
+X0 = 1
+−1
+0
+1
+2
+3
+X
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+K(XA, XB)
+X0 = 2
+Figure 2: Covariance of the Matérn (ν = 2.5) kernel
+(black lines) compared to the MorseMatérn kernel (red
+lines), projected back onto X, for a larger α = 5.0. As
+opposed to figure 1, the effect of the X0 does not affect
+the covariance as much. The widening seen from the
+previous figure is just a consequence of the length scale
+hyperparameter being equal since one cannot associate
+the Morse transformed length scale to the linear space
+one.
+Morse kernels allow the covariance to be unsymmetrical
+in the internuclear space, which affects the correlation of
+data in a very particular way over the feature space. As
+shown on figure 1, the forward and backward correlation
+differ and the extent of that effect depends greatly on the
+α parameter. This increased flexibility of the kernel in
+the model optimisation, allows to control the long range
+effect of the training data for PES modelling.
+3
+
+Optimised Morse transform of a Gaussian process feature space
+4
+Results
+We use a training set of 48 water geometries calculated
+UHF/aug-cc-pVDZ energies, sampled from a Boltzmann
+distribution using the Metropolis–Hastings algorithm
+with data up to 0.3 Ha above the equilibrium energy,
+using the Q-Chem software23. Firstly, the training data is
+projected on the 3 internuclear distances (ID) and Morse
+transformed according to equation 4 to create the feature
+space of the GPs. Secondly, the optimisable transforma-
+tion using GPs with both the MorseRBF kernel and the
+MorseMatérn class of kernel, we use the twice differen-
+tiable kernel with ν = 2.5. Finally, in order to assess
+the performance of each latent function, we define the
+MAE of predictions on a testing set also sampled from a
+Boltzmann distribution with data up to 0.2 Ha above the
+equilibrium energy.
+The Bayesian approach optimises the LML(θ) which
+only includes the training data. This is very different
+from optimising the MAE(θ) which only includes the
+testing data (we do not explore this surface here). Since
+we use GMIN and explore the whole LML landscape, one
+can combine those approaches and rank local minima of
+the LML surface with their respective MAEs. One then
+selects the minimum which has the lowest error. This
+gives an hybrid approach which optimises the MAE(θ |
+∂ LML(θ) = 0).
+The “best-fit” approach, given we use a single Morse
+parameter, is a 1D minimisation of the MAE(α). One
+also selects, for each GP trained with a different Morse
+parameter, the LML minimum with the lowest MAE. We
+first look at the results of the Matérn (ν = 2.5) kernel.
+As mentioned before, the better performing minimum
+of the LML is not always the global minimum. For the
+Matérn (ν = 2.5) kernel, this is seen in figure 3: from
+α = 2.2 it is a worse performing model that is lower on
+the LML surface while the best performing minimum can
+be followed. Even though there is no guarantee that fol-
+lowing the better performing minimum on the LML as α
+changes ensures selection of the best model, it also seem
+unlikely that an eventually better performing minimum
+at a different and larger α could not be followed back to
+smaller Morse parameters where it disappears (with the
+exception of α → 0).
+2
+5
+10
+α
+2.5
+5.0
+7.5
+MAE [mHa]
+5
+10
+α
+0
+1
+log(ρ0) = log(ρ1)
+0
+1
+log(ρ2)
+1
+5
+10
+α
+Figure 3: Optimised hyperparameters of the Matérn
+(ν = 2.5) kernel along the lengths scales representing
+the Morse transformed O-H distances (ρ0 = ρ1) and
+the Morse transformed H-H distance (ρ2) for different
+minima as well as the MAE (of each respective minima)
+on a test set. The blue-green dots represent the lower of
+the minima on the LML while the red-yellow dots are
+the second lowest minimum. The trajectory highlighted
+in black represents the models with the lowest MAE in
+both panels. Around α = 2.0, the two modes of selec-
+tion yield different models (the grey area is plotted to aid
+clarity of the switch between the two regimes).
+The overall behaviour of the trajectories in hyperparam-
+eters space of both LML minima represented in panel
+(b) of figure 3 is expected. As α increases, the length
+scales shorten. A larger Morse parameter compresses the
+training data, shortening the distance between data. As
+a consequence, a constant length scale would flatten the
+GP latent function. This is prevented by the data term
+of the LML, which causes the minima to move towards
+shorter length scales. Moreover, the minima trajectory
+can be observed to be almost linear towards larger Morse
+parameters as the transform of equation 4 becomes itself
+more linear since, in the limit of infinite α, the transform
+is linear:
+Xi �→ lim
+α→∞ exp(−Xi/α) =
+lim
+α→∞
+�
+1 − Xi
+α + X2
+i
+2α2 + . . .
+�
+≃ 1 − Xi
+α
+(8)
+This opens the question of redefining length scales to re-
+flect the change in α (for example as ρi → ρi/α) hyper-
+parameter. This does not seem to affect the optimisation
+process and will not be considered further.
+Despite figure 3 showing only two minima, the GP has
+multiple minima on the LML surface. However, only two
+minima provided PES models with low MAEs. Figure
+4 shows the disconnectivity graphs of the LML to show
+the complexity of the surface for the Matérn (ν = 2.5)
+kernel. It is surprising that the variations are so large
+despite the training data being unchanged.
+4
+
+Optimised Morse transform of a Gaussian process feature space
+Disconnectivity graphs show some surprising variations
+that are sometimes a simple consequence of TSs being
+very flat and hard to capture. This leads to the latter disap-
+pearing after small changes to the LML space which has
+strong consequences on the network of minima that can
+be shown. This is the case of the graphs for α = 0.2 and
+α = 0.4 in figure 4 where the latter finds TSs between
+LML minima more easily.
+epsilon
+11
+12
+19
+0.2
+epsilon
+2
+3
+4
+6
+7
+8
+9
+0.4
+epsilon
+4
+5
+6
+7
+1.2
+epsilon
+5
+7
+1.4
+epsilon
+4
+6
+2.2
+epsilon
+5
+7
+2.6
+epsilon
+10
+11
+2.8
+epsilon
+6
+7
+15
+3.0
+epsilon
+7
+8
+9
+3.2
+epsilon
+6
+7
+3.6
+epsilon
+1
+8
+9
+4.0
+epsilon
+1
+6
+4.4
+epsilon
+1
+6
+7
+11
+4.6
+epsilon
+2
+10
+4.8
+epsilon
+7
+8
+5.0
+epsilon
+9
+10
+11
+5.8
+epsilon
+3
+4
+7.4
+epsilon
+2
+3
+7.8
+epsilon
+10
+11
+12
+14
+8.2
+epsilon
+1
+7
+9.0
+epsilon
+5
+6
+9
+9.8
+epsilon
+1
+4
+10.2
+epsilon
+2
+3
+10.6
+Figure 4: LMLs disconnectivity graphs for the Matérn
+(ν = 2.5) kernels. Labels underneath each graph denote
+the α parameter for the corresponding GP LML.
+The minimum on the LML with the lowest MAE is al-
+ways shown on the graphs and, despite its connectivity to
+other LML minima changing, is easy to follow. GPs con-
+verge towards a “good” model when the Morse parameter
+is large enough and all latent function for GPs with α > 1
+perform similarly, as shown by the plateau in figure 3.
+The latent functions, given in figure 5, are also similar.
+The PES models are shown for the GP with a Morse
+parameter of α = 2.0 (as it has been used extensively in
+the previous chapter) and for comparison purposes for
+the GP with a Morse parameter of α = 5.0 where the
+MAE of the best model is reaching a “plateau”.
+MAE: 1.58 mHa
+0.25
+0.50
+0.75
+exp(−rO-H1/2.0)
+0.25
+0.50
+0.75
+exp(−rO-H2/2.0)
+-76.2
+-76.0
+-75.8
+-75.6
+-75.4
+-75.2
+MAE: 1.47 mHa
+0.75
+exp(−rO-H1/5.0)
+0.75
+exp(−rO-H2/5.0)
+-76.2
+-76.0
+-75.8
+-75.6
+-75.4
+-75.2
+Figure 5: Resulting PES, projected on the Morse trans-
+formed O-H nuclear distances, for Matérn kernels trained
+on Morse transformed spaces with parameters α =2.0
+(higher graph) and α = 5.0 (lower graph). The magenta
+lines are isovalue contours of the kernel function where
+the covariance to the highlighted point is equal nσ2/4
+for n = 3, 2, 1 where σ is the amplitude hyperparameter
+of equation 1.
+The RBF kernel seem to have a much less stable LML
+landscapes (see figure 6). However, the lowest MAE(θ)
+models is always found to be the lowest minimum on the
+LML(θ). As opposed to the Matérn kernel, the optimal
+Morse parameter value to minimise the MAE is more
+distinct (see figure 8).
+The Gaussian process for the optimal value seem to cor-
+respond to a value where the training data is not too
+compressed and does not allow the RBF kernel to over fit.
+Despite the MAE being similar to the Matérn kernel best
+performing models, the length scales are shorter and the
+latent function, displayed on figure 7, shows that the RBF
+model predictions are only reliable close to the training
+data and do not “carry” any of the information to longer
+bond lengths, like the Matérn kernel does.
+5
+
+Optimised Morse transform of a Gaussian process feature space
+epsilon
+15
+16
+17
+0.4
+epsilon
+9
+11
+0.6
+epsilon
+5
+12
+0.8
+epsilon
+4
+11
+1.2
+epsilon
+6
+13
+17
+1.4
+epsilon
+17
+19
+1.6
+epsilon
+5
+12
+16
+1.8
+epsilon
+19
+24
+25
+26
+118
+2.2
+epsilon
+22
+23
+2.4
+epsilon
+1
+4
+2.6
+epsilon
+28
+29
+41
+48
+2.8
+epsilon
+13
+16
+3.0
+epsilon
+13
+18
+3.2
+epsilon
+19
+20
+22
+3.6
+epsilon
+5
+17
+3.8
+epsilon
+13
+16
+4.0
+epsilon
+13
+18
+4.4
+epsilon
+13
+16
+4.6
+epsilon
+5
+13
+5.0
+epsilon
+7
+11
+12
+6.2
+epsilon
+1
+3
+6
+7
+8
+9
+7.0
+epsilon
+1
+4
+8
+10
+11
+12
+15
+16
+17
+18
+7.4
+epsilon
+4
+6
+7.8
+epsilon
+5
+12
+13
+8.2
+epsilon
+1
+4
+7
+9
+10
+11
+8.6
+epsilon
+7
+9
+9.4
+epsilon
+1
+8
+10
+11
+12
+9.8
+epsilon
+7
+9
+10
+10.2
+epsilon
+8
+9
+10.6
+Figure 6: LMLs disconnectivity graphs for the RBF ker-
+nels. Again, the labels underneath each graph denote the
+α parameter for the corresponding GP LML.
+Compared to the graphs of the Matérn kernel in figure
+4, the graphs for the RBF kernel in figure 6 show more
+minima and show stronger changes. This is due to the
+tendency for the RBF kernel to over fit data giving a more
+complex LML landscape in the region with short length
+scales. The TSs are also harder to optimise, in these short
+length scale regions, making the graphs rapidly changing.
+The most performant GP is found for α = 0.8 and one
+can see, in figure 8, that its MAE is similar to the best
+Matérn GPs. The resulting latent function is shown in
+figure 7 alongside the latent function for the GP trained
+with the Morse parameter set to α = 2.0 to compare with
+the model of the Matérn kernel in figure 5. One can see
+that, for α = 2.0, despite similar MAEs, the RBF kernel
+is more “local” and does not predict a meaningful PES at
+longer bond lengths.
+MAE: 1.60 mHa
+0.25
+0.50
+exp(−rO-H1/0.8)
+0.25
+0.50
+exp(−rO-H2/0.8)
+-76.2
+-76.0
+-75.8
+-75.6
+-75.4
+-75.2
+MAE: 3.52 mHa
+0.25
+0.50
+0.75
+exp(−rO-H1/2.0)
+0.25
+0.50
+0.75
+exp(−rO-H2/2.0)
+-76.2
+-76.0
+-75.8
+-75.6
+-75.4
+-75.2
+Figure 7: Resulting PES, projected on the Morse trans-
+formed O-H nuclear distances, for RBF kernels trained
+on Morse transformed spaces with parameters α = 0.8
+(higher graph) and α = 2.0 (lower graph) respectively.
+These correspond to the minimum along the MAE plot
+in figure 3 for the RBF and the Matérn kernels. The ma-
+genta lines are isovalue contours of the kernel function.
+In figure 7, as before, the contours represent an isocon-
+tour of the kernel from a given sample↓. One can see
+that despite the surfaces covering the same geometry
+stretches, for the larger α, the optimised length scales
+is much shorter and only allows a sample to span “ in-
+fluence” over a small part of the considered space. This
+leads to partial over fitting of the training data and a more
+complicated PES model.
+To summarise both the MAE optimisation of the GPs
+trained with RBF and Matérn kernels, we plot the MAE
+curves against the Morse parameter. This is the curve
+that one minimises in the “best-fit” approach and leads
+to selecting α = 0.8 for the RBF kernel and a larger
+↓ The first contour is where the covariance function evaluates to 0.75σ2, where σ is the amplitude hyperparameter, while the second
+one correspond to 0.5σ2.
+6
+
+Optimised Morse transform of a Gaussian process feature space
+α > 2.0 for the Matérn kernel. The latter produces a
+monotonically decreasing line which indicates that the
+optimal Morse transform is a linear transform (since the
+limit of α → ∞ reduces the Morse transform to the latter,
+as explained in equation 8). This is an indication that it
+is not optimal , for the Matérn kernel, to do the transfor-
+mation and that the initial internuclear distances produce
+a better feature space to learn on.
+1
+5
+10
+α
+2.5
+5.0
+7.5
+MAE [mHa]
+Figure 8: MAE curves for the RBF (blue) and Matérn
+(red) kernels against the Morse parameter. One can see
+that the Matérn curve does not show a minimum and thus
+indicates the best transform is the linear transform, i.e.
+the limit of the Morse transform when α → ∞.
+5
+Optimisable Morse Kernels
+We now explore the optimisation of GP with the same
+training data projected on the internuclear distances that
+are Morse transformed in the kernel, for example as given
+by equation 6 for the MorseRBF kernel. As usual the
+kernels are scaled by an optimisable CK and have an
+added optimisable noise given by a WK. The additional
+hyperparameter, α, means we approach the Morse pa-
+rameter optimisation in a fully Bayesian manner through
+the LML minimisation. As mentioned before this does
+remove the testing set in the optimisation and only the
+training data affects its optimisation.
+For the MorseRBF, multiple minima on the LML are
+obtained and, when ranked with their respective MAEs,
+the best GP models are found to be in the region of
+0.5 < α < 1.0. This is in accordance with the MAE
+curve, as seen in figure 8, of the GPs trained with standard
+RBF kernels and fixed Morse transforms. As expected,
+the MorseRBF GP best models latent function are very
+similar to the RBF GP latent function with small α param-
+eters↓: the lowest minima on the LML for the MorseRBF
+is shown in figure 9.
+0.5
+1.5
+2.5
+rO-H1 [˚A]
+0.5
+1.5
+2.5
+rO-H2 [˚A]
+-76.2
+-76.0
+-75.8
+-75.6
+-75.4
+-75.2
+0.5
+1.5
+2.5
+rO-H1 [˚A]
+0.5
+1.5
+2.5
+rO-H2 [˚A]
+-76.2
+-76.0
+-75.8
+-75.6
+-75.4
+-75.2
+Figure 9: Latent functions of GPs trained with a standard
+RBF kernel (higher graph) and a fixed Morse parameter
+close to the one exhibited by the lowest LML minimum
+of the other GP, trained with a MorseRBF kernel (lower
+graph). The change in length scale (shown by the kernel
+isovalue contour extending further out) is simply a conse-
+quence of the small difference in α value and the models
+are essentially the same.
+Obtaining MorseRBF models that resembles the RBF
+ones is important as it tells us that the added hyperpa-
+rameter dimension creates a convex LML hypersurface↓
+that can be optimised. The other hyperparameters of the
+kernel are quite close to the ones of the kernel that does
+not include the transformation when one fixes the latter
+with the parameters found by the Morse kernel.
+To summarise, the Morse kernels do optimise to lower
+values which agree better with the optimal MAE(α) for
+the RBF kernel but not for the Matérn kernel. Figure 10
+shows the disparity between the two Morse kernel ability
+to replicate the “best-fit” approach..
+1
+5
+10
+α
+2.5
+5.0
+7.5
+MAE [mHa]
+Figure 10: MAE curves for the RBF (blue) and Matérn
+(red) kernels against the Morse parameter. The dots rep-
+resent the optimised Morse hyperparameter of the Morse
+kernels (blue for MorseRBF and red for MorseMatérn)
+with grey line to aid clarity.
+6
+Changing the Training Data
+Since the feature space is optimised differently with re-
+spect to the selected training data, we will consider the
+effect of adding data to the previously discussed models.
+We still use the MAE of the final GP model but we use
+two different testing sets. The new set is also taken from a
+↓ The MorseRBF kernel is equal to a RBF kernel and a fixed Morse projection with the optimal Morse hyperparameter.
+↓ It should be made clear again that we are technically talking about the −LML surface which we are optimising. On the true LML
+surface this would be concave.
+7
+
+Optimised Morse transform of a Gaussian process feature space
+Boltzmann distibution but at a higher temperature which
+allows data to be sampled 0.4 Ha above the equilibrium
+energy.
+The training data is changed by adding data sampled
+from NM clusters↓. Two things are interesting to fol-
+low: the effect of increasing the size of the dataset on
+the MAE(α) curves as well as the progression of LML-
+optimised Morse hyperparameters.
+2
+5
+10
+α
+0
+2
+4
+MAElow [mHa]
+8
+16
+MAEhigh [mHa]
+(a) N = 29
+2
+5
+10
+α
+0
+2
+4
+MAElow [mHa]
+8
+16
+MAEhigh [mHa]
+(b) N = 39
+2
+5
+10
+α
+0
+2
+4
+MAElow [mHa]
+8
+16
+MAEhigh [mHa]
+(c) N = 49
+2
+5
+10
+α
+0
+2
+4
+MAElow [mHa]
+8
+16
+MAEhigh [mHa]
+(d) N = 59
+2
+5
+10
+α
+0
+2
+4
+MAElow [mHa]
+8
+16
+MAEhigh [mHa]
+(e) N = 69
+2
+5
+10
+α
+0
+2
+4
+MAElow [mHa]
+8
+16
+MAEhigh [mHa]
+(f) N = 79
+Figure 11:
+Different MAE(α) curves for different
+datasets. The two colours represent the MAE on different
+testing sets↓ to see the dependency of the minimum of
+the MAE(α) with respect to the chosen testing set. There
+is no clear choice for an α, although there seem to be
+a preference for small α values in the RBF kernel, un-
+til training data becomes rather large and most Morse
+parameters perform equally.
+A first observation that can be made from the curves,
+in figure 11, is that a different testing set can lead to
+a different optimal Morse parameter. A second impor-
+tant aspect is that small changes to the training set (in
+this case adding training data) can importantly alter the
+curves. For the latter, it is a surprising result since the
+new training data does not differ from the original train-
+ing data in terms of what it describes. The new sampled
+training data does not allow the GP to understand new
+patterns in the target function, which were not seen in
+the original set. One could expect that consequently the
+changes in the training data would not affect the relative
+performance of GPs with different Morse parameters.
+As data is added to the original training set, the MAE
+curves tend to flatten and stop exhibiting a clear mini-
+mum. The optimal Morse parameter is not well defined
+and GPs, despite having different projections on their
+feature spaces, perform similarly in terms of MAE. The
+sparsity of training data is reduced, which makes the fea-
+ture space less relevant: dense training data is likely to
+perform well in any way it is projected.
+Consequently, instead of additional data making the min-
+imum of the MAE(α) more and more distinct, one sees
+the minimum disappearing.
+0.2
+0.5
+ρ0 = ρ1
+0.2
+0.5
+ρ2
+5
+15
+25
+35
+45
+RBF
+0.5
+1.5
+ρ0 = ρ1
+0.5
+1.5
+ρ2
+5
+15
+25
+35
+45
+Matérn (ν = 2.5)
+Figure 12: Trajectories of optimal models in hyperpa-
+rameter space for different datasets (each dataset is rep-
+resented by a different colour and the dots follow their
+respective best minima on the LML landscapes). The
+end of the trajectory (where the label is given) is going
+towards larger Morse parameters and each trajectory has
+points, since the progression is smooth, ordered for in-
+creasing α along itself. The colourbar is given as greys
+since it has shared by all training sets.
+In figure 12, each progression in hyperparameter space
+seem to have an initial curved trajectory followed by a
+linear trajectory. This linear regime was already observed
+in figure 3 as a consequence of the Morse transform limit
+(see equation 8). If one considers the end of the trajectory
+of the Matérn GPs, one sees that the optimised length
+scales of GPs with the same Morse projection increases
+with the training set size. The trend is less clear for the
+RBF GPs where trajectories are not as well-behaved.
+Some latent functions of the Matérn GPs are plotted in
+figure 13. Despite the length scale growing larger, the
+PESs seem to strongly “oscillate”, as if they had a short
+length scale. This is particularly seen away from the data
+as seen in panel (e) and (f).
+MAE: 3.06 mHa
+MAE: 0.73 mHa
+MAE: 0.39 mHa
+0.25
+0.75
+exp(−rO−H1/2)
+0.25
+0.75
+exp(−rO−H2/2)
+-76.2
+-76.0
+-75.8
+-75.6
+-75.4
+-75.2
+(a) N = 29
+0.25
+0.75
+exp(−rO−H1/2)
+0.25
+0.75
+exp(−rO−H2/2)
+-76.2
+-76.0
+-75.8
+-75.6
+-75.4
+-75.2
+(b) N = 49
+0.25
+0.75
+exp(−rO−H1/2)
+0.25
+0.75
+exp(−rO−H2/2)
+-76.2
+-76.0
+-75.8
+-75.6
+-75.4
+-75.2
+(c) N = 69
+Figure 13: Resulting PES, projected on the Morse trans-
+formed O-H internuclear distances, for Matérn kernels
+trained on Morse transformed spaces with parameters
+α =2.0 for different training dataset sizes.
+If one thinks of the optimisation in a Bayesian sense, one
+would expect the opposite where minima on the LML
+become more well-defined as new data, if it still agrees
+↓ There is no overlap of the two training set. The additional data is added incrementally to the original training data with batches of
+5 random samples drawn from the NM clusters.
+8
+
+9.0
+6.0
+Q
+3.0Optimised Morse transform of a Gaussian process feature space
+with the hyperparameters of the model of that particular
+minimum, is added.
+6.1
+Optimisable Morse kernels for changing data
+Gaussian processes trained with a MorseMatérn (ν =
+2.5) kernel and a MorseRBF kernel are trained on the
+same datasets and compared to the MAE(α) trends of
+figure 11. These results do not use the full GMIN imple-
+mentation and use a basic sklearn24 L-BFGS approach.
+All reported minima have projected gradients converged
+to 10−2, which is not as tight a convergence criterion as
+LML minima that were found using the GMIN imple-
+mentation. Table 1 summaries the results for both the
+Morse-transformed kernels with the training data ranging
+from the initial set to the fully “merged” one in incre-
+ments of 5 data points.
+N
+MorseRBF: α
+MorseRBF: ρ0 = ρ1
+MorseMatérn: α
+MorseMatérn: ρ0 = ρ1
+29
+0.534
+0.41
+0.580
+2.08
+34
+0.774
+0.62
+0.960
+0.17
+39
+0.804
+0.39
+0.804
+0.86
+44
+0.937
+0.22
+0.514
+3.05
+49
+0.802
+0.18
+0.539
+0.88
+54
+0.806
+0.12
+0.569
+0.55
+59
+0.440
+0.42
+0.993
+2.40
+64
+0.391
+0.32
+0.622
+2.27
+69
+0.949
+0.23
+0.448
+0.15
+74
+0.750
+0.50
+0.525
+0.72
+79
+0.251
+0.12
+0.721
+0.30
+Table 1: Summary of Morse parameters and length scales for optimised Morse kernels with an increasing size of training
+set (number of samples given by N). When the Morse kernel performs better than its “standard” kernel counterpart, the
+values are written in green. One can see that the improvement is very rarely seen for the RBF kernel side and only seen
+about half the time for the kernel Matérn.
+There does not seem to be a smooth transition as data is
+added in terms of an optimal α and there does not seem
+to a consistently better method for choosing the Morse
+parameters in the Bayesian GP framework.
+Since N, the size of the training set, is not a continu-
+ous variable that changes the LML smoothly, it is not
+necessary that the change in α is smooth. One expects
+the N parameter to produce smooth changes of the LML
+minima. It is clear that MorseMatérn kernels, like the
+MorseRBF kernels, tend to favour small α values when
+optimised through the LML but it is hard to understand
+the reason for the progression seen in the tables above.
+These optimal values are not similar to the usual values
+used for Morse projections in the literature but it does
+not in terms of MAE discredit those choices.
+7
+Conclusion
+Optimising, with a Bayesian approach, the feature space
+of the GP to produce more performant latent functions,
+in terms of MAE, is not straight forward. The LML is
+made more complex by the additional DOF and there is
+a strong correlation between the hyperparameters. Differ-
+ent transforms might not necessarily suffer from this but,
+for the Morse transform, the relation between the Morse
+parameter and the length scales is evident.
+For the Morse transform, since the limit of its parameters
+tending to a certain value (α → ∞ here) give a linear
+transform, optimising the transform parameters can also
+inform us on the “usefulness” of the transform. The
+curves of figure 8, for example, show that the transform
+9
+
+Optimised Morse transform of a Gaussian process feature space
+is actually producing less performant GP models for this
+system.
+The “best-fit” approach also produced some interesting
+results regarding the choice of testing set to produce the
+MAE curves one minimises. A target function is as-
+sumed to have an optimal Morse parameter to project
+it to a “simpler” surface↓. However, a different testing
+set can significantly affect the result of the minimisation
+(this cannot be interpreted in the Bayesian approach since
+there is no testing set in the LML minimisation). This
+should not be the case if the testing set is “complete”,
+in the sense that new samples are drawn from the same
+distribution. It will be the case if the distribution change↓
+which suggests that one should use testing data that is
+suitable for the intended use of the GP model.
+Acknowledgments
+I would like to thank the Royal Society for funding as
+well as the Wales group of the University of Cambridge
+for providing access to the GMIN suite17. Moreover, I
+would like to thank Angelos Michaelides and Albert Par-
+tay Bartòk for fruitful discussion during my PhD viva
+that improved this work.
+References
+(1)
+J. Behler, The Journal of Chemical Physics, 2016,
+145, 170901.
+(2)
+F. Noé, A. Tkatchenko, K.-R. Müller and C.
+Clementi, Annual Review of Physical Chemistry,
+2020, 71, 361–390.
+(3)
+V. L. Deringer, A. P. Bartók, N. Bernstein, D. M.
+Wilkins, M. Ceriotti and G. Csányi, Chemical
+Reviews, 2021, 121, PMID: 34398616, 10073–
+10141.
+(4)
+A. P. Bartók and G. Csányi, International Journal
+of Quantum Chemistry, 2015, 115, 1051–1057.
+(5)
+S. R. Sourish Das and R. Sambasivan, Computing
+Research Repositry, 2015, abs/1509.05142, 1–17.
+(6)
+A. J. Cresswell, R. J. Wheatley, R. D. Wilkinson
+and R. S. Graham, Faraday Discussions, 2016,
+192, 415–436.
+(7)
+J. Cui and R. V. Krems, Journal of Physics B:
+Atomic, Molecular and Optical Physics, 2016, 49,
+224001.
+(8)
+E. Uteva, R. S. Graham, R. D. Wilkinson and R. J.
+Wheatley, The Journal of Chemical Physics, 2017,
+147, 161706.
+(9)
+B. Kolb, P. Marshall, B. Zhao, B. Jiang and H.
+Guo, The Journal of Physical Chemistry A, 2017,
+121, PMID: 28287725, 2552–2557.
+(10)
+E. Uteva, R. S. Graham, R. D. Wilkinson and R. J.
+Wheatley, The Journal of Chemical Physics, 2018,
+149, 174114.
+(11)
+D. Dragoni, T. D. Daff, G. Csányi and N. Marzari,
+Physical Review Materials, 2018, 2, 1–16.
+(12)
+J. Dai and R. V. Krems, Journal of Chemical The-
+ory and Computation, 2020, 16, 1386–1395.
+(13)
+M. F. Langer, A. Goeßmann and M. Rupp, arXiv,
+2020.
+(14)
+C. Qu, Q. Yu and J. M. Bowman, Annual Review
+of Physical Chemistry, 2018, 69, 151–175.
+(15)
+J. Sacks, S. B. Schiller and W. J. Welch, Techno-
+metrics, 1989, 31, 41–47.
+(16)
+C. E. Rasmussen and C. K. I. Williams, Gaus-
+sian Processes for Machine Learning (Adaptive
+Computation and Machine Learning), 2005.
+(17)
+D. J. Wales, GMIN: A program for finding global
+minima and calculating thermodynamic properties
+from basin-sampling.
+(18)
+D. J. Wales, OPTIM: A program for optimising
+geometries and calculating pathways.
+(19)
+D. J. Wales, PATHSAMPLE: A program for gen-
+erating connected stationary point databases and
+extracting global kinetics.
+(20)
+O. M. Becker and M. Karplus, The Journal of
+Chemical Physics, 1997, 106, 1495–1517.
+(21)
+D. J. Wales, M. A. Miller and T. R. Walsh, Nature,
+1998, 394, 758–760.
+(22)
+M. Miller, D. J. Wales and V. de Souza, disconnec-
+tionDPS: A program for creating disconnectivity
+graphs.
+(23)
+Y. Shao, Z. Gan, E. Epifanovsky, A. T. Gilbert,
+M. Wormit, J. Kussmann, A. W. Lange, A. Behn,
+J. Deng, X. Feng, D. Ghosh, M. Goldey, P. R.
+Horn, L. D. Jacobson, I. Kaliman, R. Z. Khali-
+ullin, T. Ku?, A. Landau, J. Liu, E. I. Proynov,
+Y. M. Rhee, R. M. Richard, M. A. Rohrdanz, R. P.
+Steele, E. J. Sundstrom, H. L. W. III, P. M. Zim-
+merman, D. Zuev, B. Albrecht, E. Alguire, B.
+Austin, G. J. O. Beran, Y. A. Bernard, E. Berquist,
+K. Brandhorst, K. B. Bravaya, S. T. Brown, D.
+Casanova, C.-M. Chang, Y. Chen, S. H. Chien,
+K. D. Closser, D. L. Crittenden, M. Diedenhofen,
+R. A. D. Jr., H. Do, A. D. Dutoi, R. G. Edgar, S.
+Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-
+Zadorozhnaya, J. Gomes, M. W. Hanson-Heine,
+P. H. Harbach, A. W. Hauser, E. G. Hohenstein,
+Z. C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K.
+Khistyaev, J. Kim, J. Kim, R. A. King, P. Klun-
+zinger, D. Kosenkov, T. Kowalczyk, C. M. Krauter,
+K. U. Lao, A. D. Laurent, K. V. Lawler, S. V.
+Levchenko, C. Y. Lin, F. Liu, E. Livshits, R. C.
+Lochan, A. Luenser, P. Manohar, S. F. Manzer,
+S.-P. Mao, N. Mardirossian, A. V. Marenich, S. A.
+↓ This is not guaranteed to a GP easier to train on that surface.
+↓ In the results presented in figure 11, it was the temperature of the Boltzmann distribution that changed between the distributions.
+10
+
+Optimised Morse transform of a Gaussian process feature space
+Maurer, N. J. Mayhall, E. Neuscamman, C. M.
+Oana, R. Olivares-Amaya, D. P. O?Neill, J. A.
+Parkhill, T. M. Perrine, R. Peverati, A. Prociuk,
+D. R. Rehn, E. Rosta, N. J. Russ, S. M. Sharada,
+S. Sharma, D. W. Small, A. Sodt, T. Stein, D.
+Stück, Y.-C. Su, A. J. Thom, T. Tsuchimochi, V.
+Vanovschi, L. Vogt, O. Vydrov, T. Wang, M. A.
+Watson, J. Wenzel, A. White, C. F. Williams, J.
+Yang, S. Yeganeh, S. R. Yost, Z.-Q. You, I. Y.
+Zhang, X. Zhang, Y. Zhao, B. R. Brooks, G. K.
+Chan, D. M. Chipman, C. J. Cramer, W. A. G. III,
+M. S. Gordon, W. J. Hehre, A. Klamt, H. F. S.
+III, M. W. Schmidt, C. D. Sherrill, D. G. Truhlar,
+A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer,
+A. T. Bell, N. A. Besley, J.-D. Chai, A. Dreuw,
+B. D. Dunietz, T. R. Furlani, S. R. Gwaltney, C.-P.
+Hsu, Y. Jung, J. Kong, D. S. Lambrecht, W. Liang,
+C. Ochsenfeld, V. A. Rassolov, L. V. Slipchenko,
+J. E. Subotnik, T. V. Voorhis, J. M. Herbert, A. I.
+Krylov, P. M. Gill and M. Head-Gordon, Molecu-
+lar Physics, 2015, 113, 184–215.
+(24)
+F. Pedregosa, G. Varoquaux, A. Gramfort, V.
+Michel, B. Thirion, O. Grisel, M. Blondel, P. Pret-
+tenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A.
+Passos, D. Cournapeau, M. Brucher, M. Perrot
+and E. Duchesnay, Journal of Machine Learning
+Research, 2011, 12, 2825–2830.
+11
+

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+arXiv:2301.02173v1  [nlin.AO]  22 Dec 2022
+Reconstruction of Phase Dynamics from Macroscopic Observations Based on Linear
+and Nonlinear Response Theories
+Yoshiyuki Y. Yamaguchi1∗ and Yu Terada2,3,4†
+1Department of Applied Mathematics and Physics,
+Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
+2Neurobiology Section, Division of Biological Sciences,
+University of California San Diego, La Jolla, CA 92093, United States of America
+3Institute for Physics of Intelligence, Department of Physics Graduate School of Science,
+The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
+4Laboratory for Neural Computation and Adaptation,
+RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
+We propose a novel method to reconstruct phase dynamics equations from responses in macro-
+scopic variables to weak inputs. Developing linear and nonlinear response theories in coupled phase-
+oscillators, we derive formulae which connect the responses with the system parameters including
+the time delay in interactions. We examine our method by applying it to two phase models, one
+of which describes a mean-field network of the Hodgkin–Huxley type neurons with a nonzero time
+delay.
+The method does not require much invasiveness nor microscopic observations, and these
+advantages highlight its broad applicability in various fields.
+Rhythmical phenomena have been ubiquitously ob-
+served in nature as well as in engineering systems and
+attracted a wide spectrum of interests [1–3].
+Specific
+rhythmical dynamics are believed to play crucial func-
+tional roles in information processing of the brain [4, 5].
+Theoretical analysis have contributed to understanding
+the nature of interacting rhythmical systems. One signif-
+icant success in theoretical researches is the phase reduc-
+tion, which reduces a high-dimensional rhythmic dynam-
+ical system to a one-dimensional phase-oscillator system
+by eliminating the other nonessential degrees of freedom
+[6–8].
+In this framework, a collective system of inter-
+acting units is described by a coupled phase-oscillator
+system, which consists of the natural frequency distribu-
+tion, coupling function, and time delay in interactions.
+A dynamical system behind an observed rhythmic phe-
+nomenon in the real world is mostly, however, unknown,
+while the knowledge helps to profoundly understand, pre-
+dict, and control it. This means high demand to specify
+the underlying coupled phase-oscillator system.
+As the reconstruction is a central issue in coupled
+phase-oscillator systems, many works have proposed re-
+construction methods [9–18]. However, there are mainly
+two rooms that should be addressed. The first is the as-
+sumption of accessibility to individual elements. The pre-
+vious works assume that time series of almost all elements
+are available, which implausible in some situations. For
+example, with electroencephalogram or functional mag-
+netic response imaging signals, we can obtain only meso-
+scopic or macroscopic activity of the nervous systems.
+The second is the inference of the time delay. The exis-
+tence of the time delay is in principle inevitable in real
+systems, and can drastically change dynamics [19, 20]. It
+∗ [email protected]
+† [email protected]
+is therefore a next step to develop a method that can be
+implemented with unknown interaction delay.
+Here, we utilize the linear response theory for coupled
+phase-oscillator systems [21–23] with the aid of a nonlin-
+ear response theory. We apply weak external forces into
+a system, and observe asymptotic responses of order pa-
+rameters, which are macroscopic variables. We note that
+it does not require time series of individual elements and
+that the time delay is tractable. Further, applied external
+forces are assumed substantially weak, since we focus on
+a regime where the linear response theory is valid. This
+assumption brings another advantage that our approach
+possesses, because strong inputs into a system may cause
+an undesirable change in states of a system. The essen-
+tial assumptions on models are that the system has the
+mean-field, all-to-all homogeneous interactions and that
+the system lies in the nonsynchronized state.
+For the
+first assumption, it is worth remarking that the all-to-
+all interaction may not be extremely special, because the
+criticality in the small-world network [24] belongs to the
+universality class of the all-to-all interaction [25, 26]. The
+mean-field analysis employed here could be extended by
+assuming statistics in couplings [27, 28]. The second as-
+sumption comes from the effectiveness of linear response
+theory developed in [23] and here.
+Based on the phase reduction [29] and following the
+first assumption, we describe the underlying coupled
+phase-oscillator system by
+dθj
+dt = ωj + 1
+N
+N
+�
+k=1
+Γ (θj(t) − θk(t − τ)) + H(θj(t), t; ωex).
+(1)
+The variable θj(t) represents the phase of the jth oscilla-
+tor at time t, the constant ωj is the natural frequency
+following the natural frequency distribution g(ω), the
+function Γ represents the coupling function, the constant
+τ is the time delay for the coupling.
+The function H
+
+2
+represents the external force and the constant ωex is its
+frequency. The system parameters g(ω), Γ, and τ are
+intrinsically determined but unknown, and we will infer
+them from observation of responses to the external force
+H by varying the controllable frequency ωex. The cou-
+pling function Γ(θ) is 2π-periodic and is expanded into
+the Fourier series as
+Γ (θ) = −
+∞
+�
+m=1
+Km sin (mθ + αm) ,
+(2)
+where Km is the coupling strength and αm is the phase-
+lag parameter for the mth Fourier component of Γ(θ).
+We here apply the external force as
+H (θ, t; ωex) = −Θ(t)
+∞
+�
+m=1
+hm sin [m (θ − ωext)] ,
+(3)
+where hm is the amplitude of the mth mode. The func-
+tion Θ(t) is the unit step function: The external force is
+off for t < 0 and kicks in at t = 0.
+The dynamics (A1) are described in the limit N → ∞
+by the equation of continuity [30] governing F(θ, ω, t),
+which is the probability density function at the time t
+and normalized as
+� ∞
+−∞ dω
+� 2π
+0
+dθ F(θ, ω, t) = 1.
+The
+nonsynchronized state specified as F0(ω) = g(ω)/(2π),
+which corresponds to the uniform distribution over θ, is
+a stationary solution to the equation of continuity. The
+order parameters, whose responses we observe, are de-
+fined by [31]
+zn(t) =
+� ∞
+−∞
+dω
+� 2π
+0
+dθ einθF(θ, ω, t).
+(4)
+Assuming that the external force h = (h1, h2, · · · ) is
+sufficiently small, we perturbatively analyze the equation
+of continuity by using the Fourier transform in θ and
+the Laplace transform in t. Supposing that F0 is stable,
+we obtain the asymptotic evolution of zn(t) in the lin-
+ear regime as e−inωextzn(t)
+t→∞
+−−−→ χn(ωex)hn + O(∥h∥2),
+where we suppose n > 0 hereafter [23]. Smallness of h
+ensures that observation of e−inωextzn provides a good
+approximation of χn(ωex)hn.
+Moreover, if we apply
+hm (m > 0) and observe e−inωextzn (n ̸= m), then
+we have a nonlinear response of order O(∥h∥2).
+Our
+goal is to obtain formulae that allow to reconstruct τ,
+Km’s, αm’s, and g(ω) from observation date of {χn(ωex)}
+and nonlinear responses for a set of external frequency,
+ωex ∈ {ω1
+ex, · · · , ωS
+ex}, where ω1
+ex < · · · < ωS
+ex. We call
+a sampling reliable, if the range ωS
+ex − ω1
+ex is sufficiently
+large and the gaps ωi+1
+ex
+− ωi
+ex are sufficiently small.
+The susceptibility χn(ωex) of the linear response reads
+[32]
+χn(ωex) =
+G(ωex)
+2 − Ln(ωex)G(ωex)
+(n > 0),
+(5)
+where Ln(ωex)
+=
+Kne−i(αn+nωexτ)
+and G(ωex)
+=
+πg(ωex) + i PV
+� ∞
+−∞ dω g(ω)/(ω − ωex). The symbol PV
+indicates the Cauchy principal value. We remark that
+G(ωex) does not depend on the mode number n. Thanks
+to this independence, once we obtain one of Lm’s, say Ln,
+the other coefficients are obtained thought the relation
+Lm(ωex) − Ln(ωex) =
+1
+χn(ωex) −
+1
+χm(ωex).
+(6)
+This is the key relation in our method.
+An obtained
+Lm infers the natural frequency distribution g(ω) from
+observation of the susceptibility χm(ωex) as
+g(ω) = 1
+π Re G(ω) = 1
+π Re
+�
+2χm(ω)
+1 + Lm(ω)χm(ω)
+�
+.
+(7)
+Our method is twofold: inference of τ (Procedure-1)
+and the others (Procedure-2).
+The latter is further
+decomposed into the two cases of τ > 0 (Procedure-
+2A) and τ = 0 (Procedure-2B).
+Procedure-1 performs a finite Fourier transform
+Lmn(t) =
+1
+ωSex − ω1ex
+� ωS
+ex
+ω1
+ex
+[Lm(ωex) − Ln(ωex)]eiωextdωex.
+(8)
+If the sampling of ωex is perfectly reliable so as to repro-
+duce the integral of (8) in the limit ωS
+ex − ω1
+ex → ∞, we
+have Lmn(t)
+ωS
+ex−ω1
+ex→∞
+−−−−−−−−→ Kme−iαmδt,mτ −Kne−iαnδt,nτ,
+where δt,t′ is the Kronecker delta. The absolute value
+|Lmn(t)| has one (τ = 0) or two (τ ̸= 0) peaks at t = mτ
+and t = nτ, and the peak positions infer the time delay τ.
+An actual sampling induces two types of errors from the
+above limit: One comes from boundedness of ωS
+ex − ω1
+ex,
+and the other from finiteness of the sample number. The
+latter type concerns errors of the numerical integration.
+Nevertheless, large peaks appear at t = mτ and t = nτ if
+the sampling is sufficiently reliable, and Km and Kn are
+sufficiently large comparing with the errors.
+Procedure-2A
+uses
+the
+relation
+Lmn(mτ)
+=
+Kme−iαm under a reliable sampling of ωex to infer Km
+and αnm. They with τ give the factor Lm(ω), and the
+natural frequency distribution g(ω) is inferred by (7). We
+remark that we solely used linear responses up to this
+procedure.
+Procedure-2B is for τ = 0, since the peak at t = 0
+mixes the modes m and n, Lmn(0) = Kme−iαm −
+Kne−iαn.
+The linear equations for Kme−iαm (m =
+1, 2, 3) obtained from L12(0), L13(0), and L23(0), for in-
+stance, are degenerate. We thus use a nonlinear response
+to infer, for example, L1:
+z2 in O(∥h∥2) can be ob-
+served by applying the external force in the first mode
+h = (h1, 0, 0, · · · ) as e−i2ωextz2(t)
+t→∞
+−−−→ χ11
+2 (ωex)h2
+1. The
+nonlinear response coefficient is theoretically obtained as
+[32]
+χ11
+2 (ωex) =
+2iG′(ωex)
+[2 − L2G(ωex)][2 − L1G(ωex)]2 ,
+(9)
+where G′(ωex) is the derivative of G(ωex) with respect
+to ωex. Solving (9) we have one expression of G′(ωex).
+
+3
+TABLE I. True and inferred parameter values of Model-1
+and Model-2. The inferred values are given for each sample
+set. NI means noninferred values, because there is no clear
+peak around t = 3τ in neither |L34| nor |L35|. Procedure-1
+implies that K4 should be sufficiently small from absence of
+clear peak of |L45(t)| [see Fig. 1(d)].
+Model-1 τ
+K1
+α1
+K2
+α2
+K3
+α3
+Truth
+2
+1.379 0.7884 0.568 -3.0316 0.154 -0.7546
+Ω50
+1
+1.987
+1.383 0.820
+0.596 -3.016
+0.153 -0.864
+Ω25
+1
+1.995
+1.381 0.793
+0.582 -3.111
+NI
+NI
+Model-2 τ
+K1
+α1
+K2
+α2
+Truth
+0
+1
+1
+0
+0
+Ω81
+2
+0.001
+0.958 1.001
+0.044 -2.119
+Ω41
+2
+-0.001 1.063 0.497
+0.521 -0.706
+We independently have another expression of G′(ωex)
+through solving (A31) by G and derivating it. The com-
+bination of the above two expressions of G′(ωex) gives
+L1 = K1e−iα1 =
+2χ11
+2 (ωex)
+iχ2(ωex)χ′
+1(ωex) −
+1
+χ1(ωex)
+(10)
+for τ = 0 [32]. We take the average over S estimated
+values of L1 from ω1
+ex, · · · , ωS
+ex.
+The other coefficients
+Lm (m > 1) are estimated from (6) by taking the av-
+erage. We remark that Procedure-2B is also applica-
+ble for τ > 0, where L1 is obtained as a solution to a
+quadratic equation. However, Procedure-2A provides
+higher performance in inference for a nonzero time-delay
+case as compared in an application [32].
+By employing the theory developed above, we tackle
+a reconstruction problem in two models: Model-1 has
+a delay, that is, τ > 0 and Procedure-2A is applied,
+while Model-2 does not and Procedure-2B is in use.
+Their system parameters are arranged in Table I. Nu-
+merical simulations of (A1) are performed in the use of
+the second-order Runge-Kutta algorithm with the time
+step ∆t = 0.01. Responses of order parameters are ob-
+tained as the average in the time interval (50, 150]. The
+number of oscillators is N = 105. All the numerical sim-
+ulations are performed by activating only one mode in
+h with strength 0.1: hm = 0.1 and hn = 0 (n ̸= m)
+for the mth mode. This strength is sufficiently small for
+the linear response but sufficiently large for overcoming
+finite-size fluctuation of order O(1/
+√
+N) by the second-
+order response of order O(∥h∥2).
+Model-1 is motivated by neurobiological systems and
+is connected directly to a network of the Hodgkin–Huxley
+neurons. As in [33, 34], the Fourier components of the
+modes m (m ≥ 4) are zero.
+The time delay is set as
+τ = 2, which is compatible with experimental observa-
+tions [35]. Taking another experimental observation [36]
+into account, we assume the log-normal natural frequency
+distribution
+g1(ω) =
+1
+ω
+�
+2πσ2
+1
+exp
+�
+−(ln ω − µ1)2
+2σ2
+1
+�
+(11)
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+0
+2
+4
+6
+8
+10
+(a)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+2
+4
+6
+8
+10
+(b)
+0
+0.05
+0.1
+0.15
+0.2
+0
+2
+4
+6
+8
+10
+(c)
+0
+0.05
+0.1
+0.15
+0.2
+0
+2
+4
+6
+8
+10
+(d)
+|L1n(t)|
+t
+|L2n(t)|
+t
+|L3n(t)|
+t
+|L4n(t)|
+t
+FIG. 1. Procedure-1 in Model-1. |Lmn(t)| (8) computed
+from the sample set Ω50
+1 . (a) m = 1 and n ∈ {2, 3, 4, 5}. (b)
+m = 2 and n ∈ {3, 4, 5}. (c) m = 3 and n ∈ {4, 5}. (c) m = 4
+and n ∈ {5}. The lines are n = 2 (purple chain), n = 3 (green
+broken), n = 4 (blue dotted), and n = 5 (orange solid). The
+vertical dashed black lines mark the inferred time-delay mτ,
+and the horizontal solid black lines the inferred Km.
+−1
+−0.5
+0
+0.5
+1
+1.5
+2
+−3
+−2
+−1
+0
+1
+2
+3
+(a)
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0
+2
+4
+6
+8
+10
+(b)
+Γ1(θ)
+θ
+Truth
+Ω50
+1
+Ω25
+1
+g1(ω)
+ω
+Truth
+From L1
+From L2
+From L3
+FIG. 2.
+Comparison between the truth (purple solid line)
+and the inference in Model-1 having τ > 0. (a) The coupling
+function Γ1(θ). The sample sets are Ω50
+1
+(green broken line)
+and Ω25
+1
+(blue chain line). (b) The natural frequency distri-
+bution g1(ω) (11) obtained from the inferred L1 (green filled
+circles), L2 (blue open circles), and L3 (orange triangles) by
+(7). The sample set is Ω50
+1 .
+with µ1 = ln 5 and σ1 = 1. The external frequency is
+sampled from the interval [0.2, 10] with the step ∆ωex =
+0.2 for the sample set Ω50
+1
+(S = 50), and ∆ωex = 0.4
+for the set Ω25
+1
+(S = 25). We start from Procedure-
+1. We approximately compute Lmn(t) (8) by using the
+midpoint algorithm, where a sampling point ωi
+ex is the
+midpoint. Absolute values |Lmn(t)| for the set Ω50
+1
+are
+reported in Fig. 1. We obtain the estimate τ = 1.987
+by taking the average over the largest peak positions for
+the pairs (m, n) = (3, 4) and (m′, n′) (m′ = 1, 2; n′ =
+m′ + 1, · · · , 5). A graph should have two large peaks at
+t = mτ and t = nτ, but some peaks are not visible in
+Fig. 1. No clear peak at t = nτ implies that Kn is smaller
+than the error level. Indeed, no clear peak of |L45(t)| in
+Fig. 1(d) is consistent with K4 = K5 = 0. Procedure-
+2A infers the coefficients Lm’s from the value of Lmn(t)
+at the peak position, where the above mentioned pairs
+
+4
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+−10 −8 −6 −4 −2
+0
+2
+4
+6
+8
+10
+(a)
+−8
+−6
+−4
+−2
+0
+2
+4
+6
+8
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+(b)
+|L12(t)|
+t
+ReL1, ImL1
+ωex
+ReL1
+ImL1
+0.5165
+-0.8068
+FIG. 3.
+Model-2.
+(a) Procedure-1.
+The peak position
+is τ = 0.001 and the peak height is 1.014. (b) Procedure-
+2B to infer L1 by (10) for each external frequency ωex. The
+real part ReLm (purple filled circles) and the imaginary part
+ImLm (green open circles). The purple and green horizontal
+solid lines mark the averaged values. The sample set is Ω81
+2 .
+are in use to take the average. Performing the same pro-
+cedure but using the set Ω25
+1 , we obtain another set of
+inferences. The inferences are compared with the true
+values in Table I. The coupling function Γ1(θ) is directly
+obtained from Lm’s, and the natural frequency distribu-
+tion g1(ω) is inferred through the relation (7). They are
+in good agreement with the true ones for the set Ω50
+1
+as
+exhibited in Fig. 2. Increasing the number of samples im-
+proves the inference, because the sampling set becomes
+more reliable.
+Model-2 is the Sakaguchi–Kuramoto model [37] which
+is specified by the parameter set (K1, α1) = (1, 1) and the
+other Fourier modes are zero. To demonstrate the ability
+of the proposed method for general natural frequency dis-
+tributions, a nonunimodal and asymmetric natural fre-
+quency distribution is assumed as
+g2(ω) = ae−(x−µ2)2/(2σ2
+2) + (1 − a)e−(x+µ2)2/(2σ2
+2)
+√
+2π
+,
+(12)
+where a = 0.8, µ2 = 2, and σ2 = 1. The external fre-
+quency is sampled from [−4, 4] with the step ∆ωex = 0.1
+for the sample set Ω81
+2
+(S = 81) and ∆ωex = 0.2 for the
+set Ω41
+2 (S = 41). To compute the derivative χ′
+1(ωex), we
+use the central difference except for the head and the end
+points, namely ω1
+ex and ωS
+ex, for which the forward and
+backward differences are in use, respectively.
+From now on, we concentrate on inferences of L1 and
+L2. Procedure-1 confirms that |L12(t)| has a large peak
+at t = 0.001 [see Fig. 3(a)], and hence we conclude no
+time-delay, τ = 0. The peak height 1.014 corresponds to
+|K1e−iα1 − K2e−iα2|, and the fact K2 = 0 implies that
+the peak height approximately infers the value of K1 = 1.
+However, we do not know the value of K2 a priori, and we
+cannot determine K1 yet. We thus use Procedure-2B,
+(10), for inferring L1, and (6) for L2. They are obtained
+as functions of ωex, and L1(ωex) is reported in Fig. 3(b).
+We determine the inferred values of the constants L1 and
+L2 by taking the average over ωex, and the constants
+Km and αm (m = 1, 2) from the averaged Lm.
+The
+inferred values are arranged in Table I. The set Ω81
+2 infers
+good values, while the set Ω41
+2
+does not provide good
+−1.5
+−1
+−0.5
+0
+0.5
+1
+−3
+−2
+−1
+0
+1
+2
+3
+(a)
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+(b)
+Γ2(θ)
+θ
+Truth
+Ω81
+2
+Ω41
+2
+g2(ω)
+ω
+Truth
+From L1
+From L2
+FIG. 4.
+Comparison between the truth (purple solid line)
+and the inference in Model-2 having τ = 0. (a) The cou-
+pling function Γ2(θ). The sample sets are Ω81
+2
+(green broken
+line) and Ω41
+2
+(blue chain line). (b) The natural frequency
+distribution g2(ω) (12) obtained from the inferred L1 (green
+filled circles) and L2 (blue open circles) through (7).
+The
+sample set is Ω81
+2 .
+inferences, due to the lack of precision in computation of
+the derivative χ′
+1(ωex). The inferred coupling function Γ2
+and the natural frequency distribution g2(ω) agree with
+the true ones as reported in Fig. 4.
+In summary, we proposed a method to reconstruct the
+underlying coupled phase-oscillator model of a collective
+rhythmic system by observing responses in order param-
+eters to a weak external force with varying its frequency.
+Non-invasivity is respected due to weakness of the exter-
+nal force, and we do not need to know activity of indi-
+vidual elements of the system. The proposed method is
+examined through numerical simulations in two models.
+The unknown system parameters including the time de-
+lay in interactions have been successfully inferred, when
+the sampling of the external frequency lies on a suffi-
+ciently large range with sufficiently small gaps. Finally,
+we remark on potential directions of development: ex-
+tensions to synchronized states, to noisy systems, and to
+network systems.
+Y.Y.Y. acknowledges the support of JSPS KAKENHI
+Grants No. 16K05472 and No. 21K03402. Y.T. is sup-
+ported by the Special Postdoctoral Research Program at
+RIKEN and JSPS KAKENHI Grant No. 19K20365.
+
+5
+Appendix A: Linear and nonlinear response theories
+1.
+Equations to analyze
+We consider the equation of motion
+dθj
+dt = ωj + 1
+N
+N
+�
+k=1
+Γ (θj(t) − θk(t − τ)) + H(θj, t; ωex),
+(j = 1, · · · , N).
+(A1)
+The variable θj is the phase of the jth phase-oscillator. The natural frequency ωj follows the natural frequency
+distribution g(ω). The function Γ is the coupling function and the constant τ is the time delay. We assume that the
+external force H is sufficiently small, i.e. ∥H∥ ≪ 1, where ∥H∥ is a certain norm of the function H. Dynamics of
+(A1) are described in the limit N → ∞ by the equation of continuity
+∂F
+∂t + ∂
+∂θ {[ω + v[F] + H(θ, t; ωex)] F} = 0,
+(A2)
+where
+v[F](θ, t; τ) =
+� ∞
+−∞
+dω
+� 2π
+0
+dθ Γ(θ − θ′)F(θ′, ω, t − τ).
+(A3)
+Suppose that the nonsynchronized state F0(ω) = g(ω)/(2π) is stable stationary under H ≡ 0. We expand F around
+F0 as
+F(θ, ω, t) = F0(ω) + f (1)(θ, ω, t) + f (2)(θ, ω, t) + · · · ,
+(A4)
+where f (k) = O(∥H∥k). Substituting the expansion (A4) into the equation of continuity (A2), we have
+∂f (1)
+∂t
++ ∂
+∂θ
+�
+ωf (1) +
+�
+v[f (1)] + H
+�
+F0
+�
+= 0
+(A5)
+in the order of O(∥H∥), and
+∂f (2)
+∂t
++ ∂
+∂θ
+�
+ωf (2) + v[f (2)]F0 +
+�
+v[f (1)] + H
+�
+f (1)�
+= 0
+(A6)
+in the order of O(∥H∥2). We analyze (A5) and (A6) through the Fourier series expansion in θ and the Laplace
+transform in t.
+2.
+Fourier series expansion
+The coupling function Γ, the external force H, and the perturbations f (k) are 2π-periodic functions with respect
+to θ, and they are expanded into the Fourier series as
+Γ (θ) = −
+∞
+�
+m=1
+Km sin (mθ + αm) = −
+�
+n̸=0
+Γneinθ,
+(A7)
+H (θ, t; ωex) = −Θ(t)
+∞
+�
+m=1
+hm sin [m (θ − ωext)] = −
+�
+n̸=0
+einθHn(t; ωex),
+(A8)
+and
+f (k)(θ, ω, t) =
+�
+n̸=0
+einθf (k)
+n (ω, t).
+(A9)
+
+6
+Here, we have the relations
+Γn = iKn
+2 eiαn,
+Γ−n = Γ∗
+n
+(n > 0)
+(A10)
+and
+Hn(t; ωex) = ihn
+2 Θ(t)e−inωext,
+H−n = H∗
+n
+(n > 0)
+(A11)
+where the superscript ∗ represents the complex conjugate. We assume that Γ0 = 0, since it is renormalized into ω, in
+other words, into a shift of the natural frequency distribution g(ω). Note that there is no external force of the zeroth
+mode: H0 ≡ 0. The order parameter functionals zn[f]’s are defined by
+zn[f](t) =
+� ∞
+−∞
+dω
+� 2π
+0
+dθ einθf(θ, ω, t) = 2π
+� ∞
+−∞
+f−n(ω, t).
+(A12)
+The Fourier series expansions give
+∂f (1)
+n
+∂t
++ in
+�
+ωf (1)
+n
++
+�
+Γnz(1)
+−n(t − τ) + Hn
+�
+F0
+�
+= 0
+(A13)
+in O(∥H∥) and
+∂f (2)
+n
+∂t
++ in
+�
+ωf (2)
+n
++ Γnz(2)
+−n(t − τ)F0 + N (2)
+n
+�
+= 0
+(A14)
+in O(∥H∥2). The symbol z(k)
+−n(t) = z−n[f (k)](t) was introduced to simplify the notation. The second-order nonlinear
+term N (2)
+n
+is defined by
+N (2)
+n (ω, t) =
+�
+m
+�
+Γmz(1)
+−m(t − τ) + Hm(t)
+�
+f (1)
+n−m(ω, t).
+(A15)
+3.
+Laplace transform
+From now on, the Laplace transform of a function is indicated by the upper hat symbol. For an arbitrary analytic
+function ϕ(t), the Laplace transform is defined by
+�ϕ(s) =
+� ∞
+0
+e−stϕ(t)dt,
+Re(s) > 0,
+(A16)
+where the domain Re(s) > 0 is introduced to ensure the convergence of integral. The perturbation f is zero at
+t = 0, since F0 is stable stationary and no external force is applied in t < 0. We hence have the Laplace transformed
+equations as
+(s + inω) �f (1)
+n
++ in
+�
+Γne−sτ �z(1)
+−n + �Hn
+�
+F0 = 0
+(A17)
+in O(∥H∥) and
+(s + inω) �f (2)
+n
++ in
+�
+Γne−sτ �z(2)
+−nF0 + �
+N (2)
+n
+�
+= 0
+(A18)
+in O(∥H∥2).
+4.
+Linear response : O(∥H∥)
+The equation (A17) is solved algebraically. Dividing s + inω, multiplying by 2π, and integrating over ω, we have
+�z(1)
+−n(s) = −
+�Hn(s)
+Λn(s) In(s),
+Re(s) > 0.
+(A19)
+
+7
+where the spectrum function Λn(s) (n ̸= 0) is
+Λn(s) = 1 + Γne−sτIn(s),
+Re(s) > 0.
+(A20)
+and the integral In(s) is
+In(s) =
+� ∞
+−∞
+g(ω)
+ω − is/n,
+Re(s) > 0.
+(A21)
+The domain Re(s) > 0 comes from the domain of the Laplace transform (A16).
+In(s), and Λn(s) and z(1)
+−n(s) accordingly, are analytically continued to the whole complex s plane as follows. The
+integrand of In(s) has the singularity at ω = is/n, which is located on the upper (lower) half of the complex ω plane
+for Re(s) > 0 and n > 0 (n < 0). Moving the singularity to the other half, we smoothly modify the integral contour,
+the real axis, so as to avoid the singularity. As a result, the residue is added, because the modified contour, denoted
+by L, encloses the singularity entirely for Re(s) < 0 and half for Re(s) = 0. The continued integral In(s) is therefore
+In(s) =
+�
+L
+g(ω)
+ω − is/ndω =
+
+
+
+
+
+
+
+
+���
+
+
+
+
+
+
+� ∞
+−∞
+g(ω)
+ω − is/ndω
+(Re(s) > 0)
+PV
+� ∞
+−∞
+g(ω)
+ω − is/ndω + sgn(n)iπg(is/n) (Re(s) = 0)
+� ∞
+−∞
+g(ω)
+ω − is/ndω + sgn(n)i2πg(is/n)
+(Re(s) < 0)
+(A22)
+where PV represents the Cauchy principal value, and sgn(n) is the sign of n representing the direction of the integral
+counter enclosing the singularity.
+Temporal evolution of z(1)
+−n(t) is obtained by performing the inverse Laplace transform as
+z(1)
+−n(t) =
+1
+2πi
+� σ+i∞
+σ−i∞
+est�z(1)
+−n(s)ds,
+(A23)
+where σ ∈ R is larger than the real parts of any singularities of �z(1)
+−n(s). The continuation of �z(1)
+−n(s) permits us to use
+the residue theorem by adding the half-circle lying in left-half of the complex s plane; The inverse Laplace transform
+picks up the singularity of �z(1)
+−n(s). The asymptotic behavior is determined by the pole of �z−n(s) which has the largest
+real part. Since we assumed that the reference state F0 is stable, all the roots of Λn(s) are in the region Re(s) < 0,
+which induce the Landau damping. The asymptotic behavior is hence determined by the poles of �Hn(s) and �H−n(s),
+which are
+�Hn(s) = ihn
+2
+1
+s + inωex
+,
+�H−n(s) = −ihn
+2
+1
+s − inωex
+,
+(n > 0).
+(A24)
+The continued integrals In(s) at the poles are
+In(−inωex) = iG∗(ωex),
+I−n(inωex) = −iG(ωex),
+(n > 0)
+(A25)
+where
+G(ωex) = πg(ωex) + iPV
+� ∞
+−∞
+g(ω)
+ω − ωex
+dω.
+(A26)
+The spectrum functions at the poles are
+Λn(−inωex) = 1
+2 [2 − L∗
+nG∗(ωex)] ,
+Λ−n(inωex) = 1
+2 [2 − LnG(ωex)] ,
+(n > 0)
+(A27)
+where Ln = Kne−i(αn+nωexτ).
+Putting all together, the asymptotic temporal evolution is for n > 0 is
+z(1)
+−n(t)
+t→∞
+−−−→ e−inωext
+G∗(ωex)
+2 − L∗nG∗(ωex)hn,
+z(1)
+n (t)
+t→∞
+−−−→ einωext
+G(ωex)
+2 − LnG(ωex)hn.
+(A28)
+
+8
+The susceptibility χm
+n (ωex) defined by
+e−inωextz(1)
+n (t)
+t→∞
+−−−→
+�
+m
+χm
+n (ωex)hm + O(∥H∥2),
+einωextz(1)
+−n(t)
+t→∞
+−−−→
+�
+m
+χ−m
+−n (ωex)h−m + O(∥H∥2),
+(n > 0)
+(A29)
+is hence
+χm
+n (ωex) = χn(ωex)δnm,
+χ−m
+−n (ωex) = χ−n(ωex)δnm,
+(n > 0),
+(A30)
+where
+χn(ωex) =
+G(ωex)
+2 − LnG(ωex),
+χ−n(ωex) =
+G∗(ωex)
+2 − L∗nG∗(ωex),
+(n > 0).
+(A31)
+5.
+Nonlinear response : O(∥H∥2)
+The same way as O(∥H∥) gives the Laplace transform �z(2)
+−n(s) as
+�z(2)
+−n(s) = −2π
+Λn(s)
+� ∞
+−∞
+�
+N (2)
+n (ω, s)
+ω − is/n dω.
+(A32)
+We need the Laplace transform of products, which appear in �
+N (2)
+n .
+a.
+Laplace transform of a product function
+For analytic functions f(t) and g(t), we have the relation
+�
+fg(s) =
+1
+2πi
+� σg+i∞
+σg−i∞
+�f(s − s′)�g(s′)ds′,
+(A33)
+where σg ∈ R is larger than the real parts of any singularities of �g(s). A proof of (A33) is straightforward. We denote
+the inverse Laplace transforms of �f(s) and �g(s) as
+f(t) =
+1
+2πi
+� σf +i∞
+σf −i∞
+es1t �f(s1)ds1,
+(A34)
+where σf ∈ R is larger than the real parts of any singularities of �f(s), and
+g(t) =
+1
+2πi
+� σg+i∞
+σg−i∞
+es2t�g(s2)ds2.
+(A35)
+Changing the variables as (s, s′) = (s1 + s2, s2), the product function (fg)(t) is expressed as
+(fg)(t) =
+1
+2πi
+� σf +σg+i∞
+σf +σg−i∞
+ds est
+�
+1
+2πi
+� σg+i∞
+σg−i∞
+ds′ �f(s − s′)�g(s′)
+�
+.
+(A36)
+The integral over s is the inverse Laplace transform of the inside of the square brackets, and hence we have the relation
+(A33).
+We note that we pick up the singularities of �g only in the integral with respect to s′. Let a be a pole of �f(s), and
+b of �g(s). By the definitions, we have Re(a) < σ1 and Re(b) < σ2. The convolution yields a pole of �f which lies on
+the right-side of the line Re(s′) = σg, since s′ = s − a = σf + σg − a > σg. Therefore, this singularity is not enclosed
+by the integral counter, which consists of the line Re(s′) = σg and the left half-circle passing through the point at
+infinity on the left-half complex s′ plane.
+
+9
+b.
+Convolution in �
+N (2)
+n
+Let us denote
+Vm(t) = Γmz(1)
+−m(t − τ) + Hm(t),
+(A37)
+which rewrite the nonlinear term N (2)
+n
+into
+N (2)
+n (ω, t) =
+�
+m
+Vm(t)f (1)
+n−m(ω, t).
+(A38)
+The Laplace transform �z(2)
+−n(s) is expressed as
+�z(2)
+−n(s) = −2π
+Λn(s)
+�
+m
+� ∞
+−∞
+L[Vmf (1)
+n−m](s)
+ω − is/n
+dω,
+(A39)
+where L represents the Laplace transform operator.
+The Laplace transform of Vm is
+�Vm(s) = Γme−sτ �z(1)
+−m(s) + �Hm(s) =
+�Hm(s)
+Λm(s) ,
+(A40)
+where we used (A19) and (A20). The Laplace transform �f (1)
+m (ω, s) is then from (A17)
+�f (1)
+m (ω, s) = −
+F0(ω)
+ω − is/m
+�Hm(s)
+Λm(s) .
+(A41)
+The Laplace transform of Vmf (1)
+n−m is
+L[Vmf (1)
+n−m](s) =
+1
+2πi
+� σ2+i∞
+σ2−i∞
+�Hm(s′)
+Λm(s′)
+F0(ω)
+ω − i s−s′
+n���m
+�Hn−m(s − s′)
+Λn−m(s − s′) ds′.
+(A42)
+Remembering the note at the end of Sec. A 5 a and keeping in mind that we are interested in the asymptotic temporal
+evolution, we pick up the pole of �Hm(s′) which is at s′ = −imωex. The principal part of the Laplace transform is
+then
+PPL[Vmf (1)
+n−m](s) =
+Res( �Hm)
+Λm(−imωex)
+�Hn−m(s + imωex)
+Λn−m(s + imωex)
+F0(ω)
+ω − i s+imωex
+n−m
+,
+(A43)
+where PP represents the principal part surviving in the limit t → ∞, and Res( �Hm) = sgn(m)ihm/2 is the residue of
+�Hm. Substituting the above expression into (A44), we have
+PP�z(2)
+−n(s) =
+−1
+Λn(s)
+�
+m
+Res( �Hm)
+Λm(−imωex)
+�Hn−m(s + imωex)
+Λn−m(s + imωex) Tn,m(s),
+(A44)
+where
+Tn,m(s) =
+�
+L
+g(ω)
+�
+ω − i s+imωex
+n−m
+� �
+ω − i s
+n
+�dω.
+(A45)
+We pick up the pole of �Hn−m(s + imωex), which is at s = −inωex, for the asymptotic temporal evolution. Then,
+einωextz(2)
+−n(t)
+t→∞
+−−−→
+−1
+Λn(−inωex)
+�
+m
+Res( �Hm)Res( �Hn−m)Tn,m(−inωex)
+Λm(−imωex)Λn−m(−i(n − m)ωex).
+(A46)
+We have to be careful for the value Tn,m(−inωex), because the integrand of Tn,m(−inωex) has the pole of order two
+at ω = ωex.
+
+10
+c.
+Nonlinear response coefficient
+From now on, we focus on the linear response of the mode 2 induced by the external force of the mode 1, i.e. h1 > 0
+and hl = 0 (l > 1). Setting n = 2 and m = 1 in (A46), we have
+e2iωextz(2)
+−2(t)
+t→∞
+−−−→
+T2,1(−2iωex)
+4Λ2(−2iωex)[Λ1(−iωex)]2 h2
+1.
+(A47)
+To obtain the value T2,1(−2iωex), we first perform the partial fraction decomposition as
+T2,1(s) =
+2
+i(s + 2iωex) [I1(s + iωex) − I2(s)] .
+(A48)
+In the limit s → −2iω′
+ex (ω′
+ex ̸= ωex) from the upper-half s plane, we have
+T2,1(−2iω′
+ex) =
+i
+ω′ex − ωex
+[G∗(2ω′
+ex − ωex) − G∗(ω′
+ex)] .
+(A49)
+Further taking the limit ω′
+ex → ωex, we have
+T2,1(−2iωex) = i (G∗)′ (ωex).
+(A50)
+The asymptotic temporal evolution of z(2)
+2 (t) is hence
+e−2iωextz(2)
+2 (t)
+t→∞
+−−−→ χ11
+2 (ωex)h2
+1 + O(∥H∥3),
+(A51)
+where
+χ11
+2 (ωex) =
+iG′(ωex)
+4Λ∗
+2(−2iωex)[Λ∗
+1(−iωex)]2 .
+(A52)
+Substituting (A27) into the above expression, we have
+χ11
+2 (ωex) =
+2iG′(ωex)
+[2 − L2(ωex)G(ωex)][2 − L1(ωex)G(ωex)]2 = 2iG′(ωex)
+[G(ωex)]3 χ2(ωex)[χ1(ωex)]2,
+(A53)
+where we used (A31).
+Appendix B: Inference of L1
+The nonlinear response coefficient (A53) gives
+G′(ωex) =
+χ11
+2 (ωex)[G(ωex)]3
+2iχ2(ωex)[χ1(ωex)]2 .
+(B1)
+Another expression of G′(ωex) is obtained by solving (A31) by G(ωex) as
+G(ωex) =
+2χn(ωex)
+1 + Ln(ωex)χn(ωex)
+(B2)
+and derivating it with respect to ωex as
+G′(ωex) = 2χ′
+n[1 + Lnχn] − χn[Lnχn]′
+[1 + Lnχn]2
+= χ′
+n(ωex) + inτLn[χn(ωex)]2
+2[χn(ωex)]2
+[G(ωex)]2.
+(B3)
+where we used the definition Ln = Kne−i(αn+nωexτ). The combination between (B1) and (B3) provides for n = 1
+G(ωex) = iχ2(ωex)[χ′
+1(ωex) + iτL1[χ1(ωex)]2]
+χ11
+2 (ωex)
+.
+(B4)
+This expression and (B2) for n = 1 give the equality
+1 + L1(ωex)χ1(ωex)
+2χ1(ωex)
+=
+χ11
+2 (ωex)
+iχ2(ωex){χ′
+1(ωex) + iτL1[χ1(ωex)]2}.
+(B5)
+This is the equation for determining L1.
+
+11
+1.
+For τ = 0
+In particular, L1 is uniquely determined for τ = 0 as
+L1 = K1e−iα1 =
+2χ11
+2 (ωex)
+iχ2(ωex)χ′
+1(ωex) −
+1
+χ1(ωex).
+(B6)
+2.
+For τ > 0
+We can infer L1 from the quadratic equation (B5) for τ > 0 as well as for τ = 0. The quadratic equation is rewritten
+into
+AL2
+1 + BL1 + C = 0,
+(B7)
+where
+A(ωex) = iτ [χ1(ωex)]2
+χ′
+1(ωex) ,
+B(ωex) = 1 + iτ χ1(ωex)
+χ′
+1(ωex),
+C(ωex) =
+1
+χ1(ωex) −
+2χ11
+2 (ωex)
+iχ2(ωex)χ′
+1(ωex).
+(B8)
+We have the two solutions to (B7), and we select the solution
+L1(ωex) = − B(ωex)
+2A(ωex)
+�
+1 −
+�
+1 − 4A(ωex)C(ωex)
+[B(ωex)]2
+�
+(B9)
+to have (B6) in the limit τ → 0, namely A → 0. The inferred L1 induces the other inferences of Lm’s through the
+relation
+Lm(ωex) − L1(ωex) =
+1
+χ1(ωex) −
+1
+χm(ωex)
+(m ≥ 2).
+(B10)
+The inferred parameter values are summarized in Table II for Model-1. The inferred coupling function Γ1(θ) and
+the natural frequency distribution g1(ω) are compared with the true ones in Fig. 5. We observe rather large errors in
+higher order modes in Γ1(θ), and precision is improved by truncating the Fourier series up to the mode-3. Moreover,
+the errors tend to decrease as the number of samples increases, and g1(ω) is well inferred irrespective of used modes.
+TABLE II. True and inferred parameter values of Model-1 from (B9) and (B10), by taking the average over ωex. The time
+delay τ is inferred by Procedure-1.
+Model-1 τ
+K1
+α1
+K2
+α2
+K3
+α3
+K4
+α4
+K5
+α5
+Truth
+2
+1.379 0.7884 0.568 -3.0316 0.154 -0.7546 0
+–
+0
+–
+Ω50
+1
+1.987 1.215 0.925
+0.683 -2.663
+0.257 0.694
+0.119 2.108 0.289 0.991
+���25
+1
+1.995 0.857 0.806
+0.956 -2.584
+0.414 1.004
+0.253 1.190 0.389 0.407
+[1] A.
+T.
+Winfree,
+The
+Geometry
+of
+Biological
+Time
+(Springer, New York, 2001).
+[2] S. H. Strogatz, Sync: How order emerges from chaos in
+the universe, nature, and daily life (Hyperion, New York,
+2003).
+[3] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza-
+tion: a universal concept in nonlinear sciences (Cam-
+bridge University Press, Cambridge, 2001).
+[4] A. Palmigiano, T. Geisel, F. Wolf, and D. Battaglia,
+Flexible information routing by transient synchrony, Nat.
+Neurosci. 20, 1014-1022 (2017).
+[5] G. Buzs´aki and E.I. Moser, Memory, navigation and
+theta rhythm in the hippocampal-entorhinal system,
+Nat. Neurosci. 16, 130-138 (2013).
+[6] Y. Kuramoto, Chemical oscillations, waves, and turbu-
+lence (Dover, New York, 2003).
+[7] H. Nakao, Phase reduction approach to synchronisation
+of nonlinear oscillators, Contemp. Phys. 57, 188 (2016).
+[8] Y. Kuramoto and H. Nakao, On the concept of dynamical
+reduction: the case of coupled oscillators, Phil. Trans. R.
+
+12
+−1.5
+−1
+−0.5
+0
+0.5
+1
+1.5
+2
+−3
+−2
+−1
+0
+1
+2
+3
+(a)
+−1.5
+−1
+−0.5
+0
+0.5
+1
+1.5
+2
+−3
+−2
+−1
+0
+1
+2
+3
+(b)
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0
+2
+4
+6
+8
+10
+(c)
+Γ1(θ)
+θ
+Truth
+Sample set Ω50
+1
+Sample set Ω25
+1
+Γ1(θ)
+θ
+Truth
+Sample set Ω50
+1 (up to mode-3)
+Sample set Ω25
+1 (up to mode-3)
+g1(ω)
+ω
+Truth
+From L1
+From L2
+From L3
+From L4
+From L5
+FIG. 5. Comparison between the truth and the inference in Model-1 having τ > 0. (a) The coupling function Γ1(θ) produced
+from the sample set Ω50
+1
+(green broken line), Ω25
+1
+(blue chain line). (b) Same as (a) but the inferred Γ1(θ) are truncated up to
+the Fourier mode-3. (c) The natural frequency distribution g1(ω) obtained from the inferred L1 (green filled circles), L2 (blue
+open circles), L3 (orange triangles), L4 (yellow inverse triangles), and L5 (dark-blue diamonds). The sample set is Ω50
+1 .
+Soc. A 377, 20190041 (2019).
+[9] R. F. Gal´an, G. B. Ermentrout, and N. N. Urban, Effi-
+cient estimation of phase-resetting curves in real neurons
+and its significance for neural-network modeling, Phys.
+Rev. Lett. 94, 158101 (2005).
+[10] J. Miyazaki and S. Kinoshita, Determination of a cou-
+pling function in multicoupled oscillators, Phys. Rev.
+Lett. 96, 194101 (2005).
+[11] I. T. Tokuda, S. Jain, I. Z. Kiss, and J. L. Hudson, Infer-
+ring phase equations from multivariate time series, Phys.
+Rev. Lett. 99, 064101 (2007).
+[12] B. Kralemann,
+L. Cimponeriu,
+M. Rosenblum,
+A.
+Pikovsky, and R. Mrowka, Uncovering interaction of cou-
+pled oscillators from data, Phys. Rev. E 76, 055201(R)
+(2007).
+[13] B. Kralemann,
+L. Cimponeriu,
+M. Rosenblum,
+A.
+Pikovsky, and R. Mrowka, Phase dynamics of coupled
+oscillators reconstructed from data, Phys. Rev. E 77,
+066205 (2008).
+[14] W. D. Penny, V. Litvak, L. Fuentemilla, E. Duzel, and
+K. Friston, Dynamic Causal Models for phase coupling,
+J. Neurosci. Methods 183, 19 (2009).
+[15] T. Stankovski, A. Duggento. P. V. E. McClintock, and
+A Stefanovska, Inference of Time-Evolving Coupled Dy-
+namical Systems in the Presence of Noise, Phys. Rev.
+Lett. 109, 024101 (2012).
+[16] K. Ota and T. Aoyagi, Direct extraction of phase dynam-
+ics from fluctuating rhythmic data based on a Bayesian
+approach, arXiv: 1405.4126 (2014).
+[17] A. Pikovsky, Reconstruction of a random phase dynamics
+network from observations, Phys. Lett. A 382, 147 (2018).
+[18] F. Mori and H. Kori, Noninvasive inference methods for
+interaction and noise intensities of coupled oscillators us-
+ing only spike time data, Proc. Natl. Acad. Sci. 119,
+e2113620119 (2022).
+[19] M. K. S. Yeung and S. H. Strogatz, Time delay in the
+Kuramoto model of coupled oscillators, Phys. Rev. Lett.
+82, 648 (1999).
+[20] E. Montbri´o, D. Paz´o, and J. Schmidt, Time delay in the
+Kuramoto model with bimodal frequency distribution,
+Phys. Rev. E 74, 056201 (2006).
+[21] H. Sakaguchi, Cooperative Phenomena in Coupled Oscil-
+lator Systems under External Fields, Prog. Theor. Phys.
+79, 39 (1988).
+[22] H. Daido, Susceptibility of large populations of coupled
+oscillators, Phys. Rev. E 91 012925 (2015).
+[23] Y. Terada and Y. Y. Yamaguchi, Linear response the-
+ory for coupled phase oscillators with general coupling
+functions, J. Phys. A: Math. Theor. 53, 044001 (2020).
+[24] D. J. Watts and S. H. Strogatz, Collective dynamics of
+‘small-world’ networks, Nature 393, 440 (1998).
+[25] H. Hong, M. Y. Choi, and B. J. Kim, Synchronization on
+small-world networks, Phys. Rev. E 65, 026139 (2002).
+[26] R. Yoneda, K. Harada, and Y. Y. Yamaguchi, Critical
+exponents in coupled phase-oscillator models on small-
+world networks, Phys. Rev. E 102, 062212 (2020).
+[27] H. Daido, Population Dynamics of Randomly Interacting
+Self-Oscillators. I: Tractable Models without Frustration,
+Prog. Theor. Phys. 77, 622 (1987).
+[28] T. Ichinomiya, Frequency synchronization in a random
+oscillator network, Phys. Rev. E 70, 026116 (2004).
+[29] F. C. Hoppensteadt and E.M. Izhikevich, Weakly con-
+nected neural networks (Springer, New York, 1997).
+[30] C. Lancellotti, On the Vlasov limit for systems of nonlin-
+early coupled oscillators without noise, Transport Theory
+and Statistical Physics 34, 523 (2005).
+[31] H. Daido, Order function and macroscopic mutual en-
+trainment in uniformly coupled limit-cycle oscillators,
+Prog. Theor. Phys. 88, 1213 (1992).
+[32] See the Supplementary Material at [URL].
+[33] D. Hansel, G. Mato, and C. Meunier, Phase Dynamics
+for Weakly Coupled Hodgkin-Huxley Neurons, Europhys.
+Lett. 23, 367 (1993).
+[34] D. Hansel, G. Mato, and C. Meunier, Synchrony in ex-
+citatory neural networks, Neural Comput. 7, 307-337
+(1995).
+[35] E. M. Izhikevich, Polychronization:
+computation with
+spikes, Neural Comput. 18 245 (2006).
+[36] G. Buzs´aki and K. Mizuseki, The log-dynamic brain: how
+skewed distributions affect network operations, Nat. Rev.
+Neurosci. 15, 264 (2014).
+[37] H. Sakaguchi and Y. Kuramoto, A soluble active rotater
+model showing phase transitions via mutual entertain-
+ment, Prog. Theor. Phys.. 76, 576 (1986).
+

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+Electronic character of charge order in square planar low valence nickelates
+Y. Shen,1, ∗ J. Sears,1 G. Fabbris,2 J. Li,3 J. Pelliciari,3 M. Mitrano,4 W. He,1 Junjie
+Zhang,5, 6 J. F. Mitchell,5 V. Bisogni,3 M. R. Norman,5 S. Johnston,7, 8 and M. P. M. Dean1, †
+1Condensed Matter Physics and Materials Science Department,
+Brookhaven National Laboratory, Upton, New York 11973, USA
+2Advanced Photon Source, Argonne National Laboratory, Lemont, Illinois 60439, USA
+3National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA
+4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
+5Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA
+6Institute of Crystal Materials, Shandong University, Jinan, Shandong 250100, China
+7Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37966, USA
+8Institute of Advanced Materials and Manufacturing, The University of Tennessee, Knoxville, Tennessee 37996, USA
+(Dated: January 12, 2023)
+Charge order is a central feature of the physics of cuprate superconductors and is known to arise
+from a modulation of holes with primarily oxygen character. Low-valence nickelate superconductors
+also host charge order, but the electronic character of this symmetry breaking is unsettled. Here,
+using resonant inelastic x-ray scattering at the Ni L2-edge, we identify intertwined involvements of
+Ni 3dx2−y2, 3d3z2−r2, and O 2pσ orbitals in the formation of diagonal charge order in an overdoped
+low-valence nickelate La4Ni3O8. The Ni 3dx2−y2 orbitals, strongly hybridized with planar O 2pσ,
+largely shape the spatial charge distribution and lead to Ni site-centered charge order. The 3d3z2−r2
+orbitals play a small, but non-negligible role in the charge order as they hybridize with the rare-
+earth 5d orbitals. Our results reveal that the low-energy physics and ground-state character of these
+nickelates are more complex than those in cuprates.
+I.
+INTRODUCTION
+One of the common threads linking different classes
+of unconventional superconductors is their propensity to
+host proximate competing orders such as charge and spin
+stripes [1, 2].
+For example, the cuprate superconduc-
+tors exhibit diagonal (with respect to the Cu-O bonds)
+spin stripes when underdoped [3–5], while Cu-O bond
+oriented (parallel) charge order dominates the rest of the
+phase diagram [6, 7]. The detection of superconductivity
+and charge order in the square-planar low-valence fam-
+ily of nickelates therefore presents a fascinating oppor-
+tunity to study the degree of similarity between differ-
+ent unconventional superconducting families [8–17]. In-
+triguingly, different nickelates within the structural se-
+ries of Rn+1NinO2n+2 (R stands for a rare earth and n is
+the number of neighboring NiO2 layers) also host differ-
+ent charge ordered phases. Underdoped materials with
+n = ∞ and R = La, Nd exhibit parallel charge order
+[15–17], whereas n = 3 material La4Ni3O8, which is ef-
+fectively 1/3 overdoped, manifests diagonal charge order
+[14]. Many researchers have emphasized that charge or-
+der plays an important role in the physics of cuprates
+[18–21].
+In particular, there is good evidence showing
+that charge/spin order is a fundamental feature of min-
+imal Hubbard model descriptions of the cuprates [22–
+24].
+Some researchers have suggested that charge and
+spin order can intertwine with superconductivity to form
+pair density waves [25, 26], or that dynamic charge/spin
+∗ [email protected]
+† [email protected]
+fluctuations might promote superconductivity [27–29].
+Others have associated charge order fluctuations with
+the anomalous “strange metal” electronic transport in
+cuprates [30].
+Understanding the electronic states in-
+volved in charge order formation is a prerequisite to test-
+ing all these scenarios in low-valence nickelates and is also
+important more generally for understanding charge order
+as a prevalent feature of correlated quantum materials.
+Here, we use Ni L2-edge RIXS to determine the elec-
+tronic character of the charge order in La4Ni3O8. We
+find that both the Ni 3dx2−y2 and 3d3z2−r2 orbitals are
+involved in charge order formation.
+The former con-
+tributes most of the charge modulation while the latter
+dominates the RIXS spectra in the post-edge regime and
+so plays a less important role.
+As the charge-transfer
+energy of these nickelates is larger than that of cuprates
+but comparable to the on-site Coulomb interaction, the
+holes involved in the charge modulation reside predomi-
+nately on Ni sites, despite an appreciable amount of holes
+occupying the O orbitals. Our results indicate that the
+low-energy electronic structure and charge order of low-
+valence nickelates is largely shaped by hybridized 3dx2−y2
+and planar O 2pσ orbitals, similar to cuprates, while
+some differences exist due to the multi-band physics in-
+troduced by Ni 3d3z2−r2 orbitals hybridized with rare-
+earth 5d states.
+II.
+RESULTS
+The La4Ni3O8 nickelate samples studied here were pre-
+pared by reducing single crystals synthesized via the
+floating zone method (see the Appendix A for details),
+arXiv:2301.04184v1  [cond-mat.str-el]  10 Jan 2023
+
+2
+tpd
+tpp
+Ni1
+Ni2
+Ni3
+a
+b
+c
+σ
+Ni L2-edge
+�
+θ
+(a)
+Nickel
+Oxygen
+NiO2 plane
+La4Ni3O8 sample
+(b)
+(c)
+(d)
+(e)
+(f)
+-0.2
+-0.1
+0
+0.1
+0.2
+Energy loss (eV)
+0.32
+0.36
+0.34
+0.32
+0.34
+0.32
+0.34
+0.32
+0.34
+0.32
+0.34
+(H, H) (r.l.u.)
+(H, H) (r.l.u.)
+(H, H) (r.l.u.)
+(H, H) (r.l.u.)
+(H, H) (r.l.u.)
+T = 40 K
+T = 50 K
+T = 70 K
+T = 90 K
+T = 110 K
+×10
+0
+25
+50
+75
+100
+Intensity (arb. units)
+(g)
+(h)
+40 K
+50 K
+70 K
+90 K
+110 K
+0.32
+0.33
+0.34
+0.35
+Q║=(H, H) (r.l.u.)
+40
+60
+80
+100 120
+Temperature (K)
+0
+2
+4
+6
+Intensity (arb. units)
+FIG. 1. Charge order transition in La4Ni3O8. (a) Schematic of the Ni L2-edge RIXS experimental setup. A single NiO2 layer
+is presented with stripes running vertically. A Ni3O10 cluster composed of Ni 3dx2−y2 and planar O 2pσ orbitals is embedded
+in it, tracing the charge order motif, in which hole poor Ni1 and Ni3 sites, shown in red, flank the hole rich Ni2 site depicted
+in purple. (b)–(f) RIXS intensity maps with σ polarized incident photons at the indicated temperatures obtained by changing
+the in-plane sample angle θ. (g) Quasi-elastic-line amplitudes extracted from the data presented in (b)–(f) as a function of
+in-plane momentum transfer in reciprocal lattice units (r.l.u.). The solid lines are fitting curves with pseudo-Voigt profiles. (h)
+Temperature dependence of the fitted peak amplitudes. The bold gray line is a guide to the eye.
+and will be indexed in terms of scattering vector Q =
+(2π/a, 2π/a, 2π/c) with a = b = 3.97 ˚A, c = 26.092 ˚A.
+As the n = 3 member of the low-valence nickelate family,
+it possesses a trilayer structure with a nominal 3d8+2/3
+valence.
+This leads to a 1/3-hole self-doping with re-
+spect to the undoped 3d9 state, putting it in the over-
+doped regime of the phase diagram [13, 31]. It shares
+the same structural motif as infinite-layer nickelates with
+square-planar NiO2 layers stacked without apical oxy-
+gens, leading to dominant Ni 3dx2−y2 character near
+the Fermi energy. Although La4Ni3O8 has two inequiv-
+alent NiO2 layers, they are expected to show similar
+electronic structure as indicated by theoretical calcula-
+tions [32, 33], which is further supported by the obser-
+vation that the same charge order pattern is formed in
+both layers [14]. We study their properties using Ni L2-
+edge RIXS in order to avoid interference from the La
+M4-edge, which overlaps the Ni L3-edge (see the Ap-
+pendix B for details). As shown in Fig. 1(a), charge or-
+der in La4Ni3O8 is quasi-two-dimensional in nature and
+occurs at Q∥ = QCO = (1/3, 1/3), where a strong peak
+is observed in the quasi-elastic region of the RIXS inten-
+sity map at 40 K [see Fig. 1(b)]. The in-plane correlation
+length is larger than 100 nm, which might be limited by
+the sample mosaic, suggesting the long range nature of
+the charge order [14]. This charge order peak persists up
+to 90 K and disappears above 110 K, indicating a tran-
+sition temperature of around 100 K [see Figs. 1(c)–(h)],
+consistent with the reported charge order from hard x-ray
+diffraction measurements [14]. No indication of charge
+order is apparent in equivalent measurements of metallic
+Pr4Ni3O8 samples prepared in the same way (Supple-
+mental Material Sec. I [34]).
+We begin by identifying the active electronic states
+in La4Ni3O8 using x-ray spectroscopy. Figure 2(a) and
+2(b) show the L2-edge RIXS energy maps taken with
+σ x-ray polarization in the ab-plane and π x-ray po-
+larization approximately parallel to the c-axis, respec-
+tively. The RIXS maps mainly comprise dd and charge-
+transfer excitations that are predominantly localized and
+resonate at the Ni L2-edge, and diagonal fluorescence
+features (Supplemental Material Sec. II [34]). To distin-
+guish among these contributions, we integrated the RIXS
+spectra along the incident energy axis and show the re-
+sult in Fig. 2(c). With σ polarization, the spectra above
+4 eV energy loss are dominated by mostly featureless flu-
+orescence originating from particle-hole excitations that
+can be understood from an itinerant framework involv-
+ing transitions from extended electronic bands spanning
+many unit cells [35]. Charge transfer excitations are also
+
+3
+CT
+dd
+FL
+dd
+FL
+(a)
+(b)
+0
+1
+2
+3
+4
+5
+6
+Energy loss (eV)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0
+1
+2
+3
+4
+5
+6
+Intensity (arb. units)
+σ-pol.
+�-pol.
+0
+2
+4
+6
+8
+Energy loss (eV)
+0.0
+0.2
+0.4
+0.6
+Intensity (arb. units)
+0.8
+dd
+CT
++
+FL
+(c)
+σ-pol.
+�-pol.
+Integral over
+incident energy
+0.00
+0.01
+0.02
+0.03
+Intensity (arb. units)
+865
+870
+875
+880
+Incident energy (eV)
+CT
+FL
+(d)
+5.5 ≤ Eloss ≤ 6
+σ-pol.
+�-pol.
+x0.1, CO
+CO
+866
+868
+870
+872
+874
+Incident energy (eV)
+866
+868
+870
+872
+874
+Incident energy (eV)
+-0.10
+-0.05
+0.00
+0.05
+0.10
+Energy loss (eV)
+0
+4
+8
+12
+16
+-0.10
+-0.05
+0.00
+0.05
+0.10
+0
+4
+8
+12
+16
+Intensity (arb. units)
+(e)
+(f)
+FIG. 2.
+RIXS energy maps and the resonant behaviors of the charge order (CO) peak.
+(a, b) RIXS intensity maps as a
+function of incident photon energy with (a) σ x-ray polarization in the ab plane of the sample and (b) π x-ray polarization
+approximately parallel to the c-axis. Several components can be identified: charge transfer excitations (CT), dd excitations (dd)
+and constant-emission-energy fluorescence (FL). (c) Integral of the RIXS spectra along the incident energy axis. The dashed
+lines are guides to the eye. (d) Incident energy dependence of the integrated RIXS spectra between 5.5 and 6 eV energy loss.
+(e, f) RIXS intensity maps around the quasi-elastic regime with Q fixed at QCO. Note that the intensity in (e) is multiplied
+by 0.1 for clarity in visualizing the signal.
+visible above 4 eV but only at resonance. Below 4 eV,
+prominent dd excitations emerge that dominate over the
+featureless fluorescence (dashed lines). With π polariza-
+tion, the fluorescence contributes most of the spectral
+weight and the dd excitations are much weaker.
+The
+strong dichroism of dd excitations reflects the dominant
+Ni 3dx2−y2 orbital character near the Fermi energy in
+low-valence nickelates.
+To further distinguish between charge-transfer excita-
+tions and fluorescence, we inspect the RIXS spectra be-
+tween 5.5 and 6 eV energy loss, well above the dd excita-
+tion threshold. As shown in Fig. 2(d), the charge-transfer
+excitations and fluorescence are separated in the incident
+energy axis, with the former stronger in the σ polariza-
+tion channel, indicating appreciable dx2−y2-pσ hybridiza-
+tion where pσ indicates O orbitals that are parallel to the
+Ni-O bonds. In contrast, the fluorescence is stronger in
+the π polarization channel, suggesting that states involv-
+ing Ni 3d3z2−r2 orbitals dominate the fluorescence for a
+broad range of energy losses above ∼3 eV. The broadness
+of these states is in contrast with cuprates, and suggests
+that although the Ni 3d3z2−r2 orbitals are mostly oc-
+cupied and localized, their unoccupied components are
+hybridized with the rare earth 5d orbitals and thus con-
+tribute to dispersive states. This conclusion is consistent
+with density functional theory (DFT)+dynamical mean
+field theory (DMFT) calculations [32], as well as RIXS
+simulations for RNiO2 that studied the effect of switching
+on and off the rare-earth hybridization [36]. Meanwhile,
+the Ni 3dx2−y2 orbitals exhibit less hybridization with
+the rare earth 5d orbitals and are more localized. Here,
+since we are measuring at the Ni L edge and the Ni t2g
+orbitals are expected to lie well below the Fermi energy,
+we only consider Ni eg orbitals [34].
+Based on the resonant behavior of the different states
+identified, we now examine how the 3dx2−y2 and 3d3z2−r2
+orbitals participate in the charge order. Figure 2(e) and
+2(f) show the RIXS energy maps around the quasi-elastic
+regime at QCO, i.e. the resonant elastic x-ray scatter-
+ing (REXS) signals.
+The peak intensity strongly res-
+onates at the Ni L2-edge in the σ polarization channel
+[see Fig. 2(e)], confirming that the (1/3, 1/3) Bragg peak
+in La4Ni3O8 involves a charge modulation and is not
+purely structural. Surprisingly, the charge order peak in
+the π polarization channel, although much weaker, res-
+onates at the pre- and post-edge regimes but not at the
+main edge [see Fig. 2(f)], distinct from that in cuprates
+[37–39]. First, this observation indicates that both the
+3dx2−y2 and 3d3z2−r2 orbitals are involved in charge order
+formation with the latter much less prominent. Second,
+
+4
+(a)
+868
+870
+872
+Incident energy (eV)
+σ-pol.
+(b)
+868
+870
+872
+Incident energy (eV)
+�-pol.
+0
+1
+2
+3
+Energy loss (eV)
+Intensity (arb. units)
+0
+0.25
+FIG. 3. Low-energy electronic states in La4Ni3O8. Calcula-
+tions of the RIXS energy maps at the Ni L2-edge for (a) σ and
+(b) π incident x-ray polarization. The calculations reproduce
+the experimental energy-scale and polarization of the dd exci-
+tations evincing an appropriate minimal model for La4Ni3O8.
+the charge order peak in the post-edge regime with π po-
+larization suggests that the states far above the Fermi
+energy also show charge modulation, which is mostly
+contributed by 3d3z2−r2 orbitals. Considering that the
+3d3z2−r2 density of states in the post-edge regime is likely
+caused by hybridization with the rare-earth 5d orbitals,
+this indicates potential involvement of rare-earth orbitals
+in the charge order formation. Similarly, the weak pre-
+edge charge order peak with π polarization indicates that
+the 3d3z2−r2 density of states near the Fermi energy is
+nonzero but small.
+Having established the involvement of Ni orbitals in
+the charge order formation, now we look at the role of
+oxygen states. To do this, we use exact diagonalization
+(ED) methods which allow us to solve the resonant cross-
+section and break down the contributions from different
+states. Since the charge order is commensurate with a
+period of three Ni sites and there is a strong hybridiza-
+tion between the Ni and O orbitals, the smallest cluster
+one can use to describe the charge-ordered state involves
+three Ni-O plaquettes, which we label 1, 2, & 3.
+We
+choose a bond-oriented cluster, as illustrated in Fig. 1(a),
+given that the Ni-O hopping dominates the kinetic en-
+ergy.
+In order to compute REXS we use the atomic
+scattering factors from the cluster and add these am-
+plitudes to simulate an effective two-dimensional NiO2
+plane as shown in Fig. 1(a). The appropriate parame-
+ters for this cluster, and in particular the charge-transfer
+energy ∆ = 5.6 eV and the on-site Coulomb repulsion
+Udd = 6.5 eV, have been empirically determined by prior
+x-ray measurements of this material at the O K-edge
+[40]. We use open boundary conditions and construct the
+Hamiltonian in the hole language (see the Appendix C
+for details).
+Four holes are introduced to the cluster,
+which is appropriate for the d9−1/3 electronic configu-
+ration of La4Ni3O8. Without any additional constraints,
+the holes will be evenly distributed among different NiO4
+plaquettes with minimal charge disproportionation and
+no symmetry breaking is expected. To realize the charge
+order observed in La4Ni3O8, we manually introduce a
+potential difference [41], ∆ϵd, for different Ni sites by
+lowering the orbital energies of Ni2 by 2∆ϵd/3 and rais-
+ing those of Ni1 and Ni3 by ∆ϵd/3. Based on the sim-
+ilar magnetic exchange of charge ordered La4Ni3O8 and
+metallic Pr4Ni3O8 [42], ∆ϵd must be significantly smaller
+than the charge-transfer energy. Thus, we choose it to
+be ∆ϵd = 0.8 eV while noting that apart from modu-
+lating the intensity of the charge order peak, the results
+are similar provided ∆ϵd is not made unfeasibly large
+(Supplemental Material Fig. S5 [34]). This choice leads
+to a charge disproportionation of ∆n = 0.32, which is
+of a similar order of magnitude as that in cuprates [37].
+This value is much smaller than the fully disproportion-
+ate limit ∆n = 1, consistent with DFT calculations that
+indicate a small charge modulation upon charge ordering
+in this system [31]. When examining the electronic con-
+figuration of the cluster, we find that the ground state is
+a singlet, and the first excited state is a triplet, which is
+around 70 meV above the ground state, consistent with
+the magnetic excitations found in La4Ni3O8 [42].
+Figure 3 shows the calculated Ni L2-edge RIXS en-
+ergy maps with all the Ni 3d and O 2p orbitals included,
+which qualitatively reproduce the localized dd excitations
+observed experimentally. Note that the small cluster size
+means that we can only capture a limited number of dis-
+crete states. For this reason, fluorescence features are not
+fully captured, which would require a continuous distri-
+bution of states. This can be seen more clearly in the
+π polarization channel where the fluorescence dominates
+the spectra in experimental data [see Fig. 2(b)] but only
+the weak dd excitations are present in our cluster calcu-
+lations [see Fig. 3(b)].
+Having verified the relevant parameters via the RIXS
+maps, we computed the x-ray absorption spectrum
+(XAS) and REXS response of La4Ni3O8 using a simi-
+lar ED approach and identical parameters and plot the
+results in Fig. 4 (Supplemental Material Sec. V [34]).
+The charge disproportionation in the cluster implies a
+REXS response at QCO. The predicted REXS resonance
+shown in Fig. 4(c) nicely captures the main two peak
+structure of the experimental REXS resonance shown in
+Fig. 4(b). The same applies for the XAS as shown in
+Fig. 4(a). In fact, the lineshape of the resonant profile of
+the charge order peak is sensitive to the charge-transfer
+energy, and neither the pure charge-transfer nor Mott-
+Hubbard scenarios can describe the observed resonant
+behaviors (Supplemental Material Fig. S6 [34]), demon-
+
+5
+strating the mixed charge-transfer/Mott-Hubbard char-
+acters of charge order in this material. To understand
+the nature of the two resonant features, we projected
+the wavefunctions of the RIXS intermediate states onto
+the Fock basis which specifies the location of the holes.
+Two main manifolds are seen for each Ni site. The first
+manifold is primarily attributed to transitions resonant
+with d10L0 states, where L stands for ligand holes on
+the four oxygen σ orbitals surrounding the Ni site. The
+second manifold is mainly resonant with d9L0 and d10L1
+states caused by the doped holes, similar to the cuprates
+[43, 44]. With nonzero ∆ϵd, the manifolds of different
+Ni sites split along the incident energy axis, as shown in
+Fig. 4(c). The successful description of the charge order
+in La4Ni3O8 using our cluster model indicates that about
+70% of the holes participating in the charge modulation
+are on Ni, with the remaining 30% on oxygen, as depicted
+in the inset to Fig. 4(b).
+III.
+DISCUSSION
+Our Ni-dominant charge order distribution is quite dif-
+ferent from cuprates, in which the charge order has dom-
+inant oxygen character [37, 45]. This difference mainly
+arises from the larger charge transfer energy in nicke-
+lates compared to cuprates. Another difference is that in
+cuprates, the 3d3z2−r2 orbitals are strongly localized at
+energies more than 1.5 eV away from the 3dx2−y2 orbitals
+[46], and thus not involved in the low-energy physics. For
+square-planar nickelates, our analysis of La4Ni3O8 indi-
+cates that the 3d3z2−r2 density of states, though small, is
+spread out over an extended energy range, likely due to
+hybridization with the rare earth 5d orbitals. It should be
+noted that although the 3d3z2−r2 orbital involvement in
+the charge order formation is nonzero, its contribution is
+much less than the hybridized 3dx2−y2 and 2pσ orbitals,
+as indicated by the stronger charge order peak in the σ
+polarization channel. These factors mean that minimal
+theoretical models of charge order in nickelates must ex-
+plicitly include both Ni and O states alongside strong
+correlations.
+Another result of our model is that the
+doped sites in charge ordered nickelates are much closer
+to a low-spin S = 0 state than to a high-spin S = 1 state,
+unlike La2−xSrxNiO4, whose high-spin physics drives in-
+sulating behavior across the vast majority of its phase
+diagram [47].
+Recently, RIXS measurements in infinite-layer nicke-
+late films have discovered and studied charge order at
+Q∥ = (1/3, 0) in undoped and underdoped samples [15–
+17], resembling the charge order in cuprates, but differing
+from the diagonal charge order in La4Ni3O8. In terms
+of these differing wavevectors, theoretical model studies
+in the cuprates have shown that charge order at (Q, 0)
+and (Q, Q) are close in energy, the eventual choice of the
+charge order wavevector being sensitive to details of the
+electronic structure and correlations [48, 49]. This idea is
+supported by the experimental observation that the dop-
+868
+869
+870
+871
+872
+873
+Incident energy (eV)
+2
+1
+0
+Intensity (arb. units)
+Ni1: d 10L0
+Ni1: d 10L1
+Ni1: d 9L0
+Ni2: d 10L0
+Ni2: d 10L1
+Ni2: d 9L0
+REXS calculation
+(c)
+6
+4
+2
+0
+Intensity (arb. units)
+Ni1
+Ni3
+Ni2
+O
+σ-pol.
+�-pol.
+REXS data
+(b)
+(a)
+1.2
+0.8
+0.4
+0
+Intensity (arb. units)
+XAS, σ-pol.
+Data
+Ni1+Ni3
+Ni2
+Calculation
+FIG. 4. Electronic character of charge order. (a) x-ray ab-
+sorption spectrum (XAS) data at the Ni L2 edge in the σ
+polarization channel along with the calculation results with
+∆ϵd = 0.8 eV. Note that Ni1 and Ni3 are symmetry-related.
+(b) Fitted peak amplitudes of the quasi-elastic intensities pre-
+sented in Fig. 2(e)&(f), representing the resonant behaviors of
+the charge order peak. Inset is a schematic of the electronic
+character of the charge order showing a dominant modula-
+tion of Ni orbitals along with an appreciable modulation of
+the oxygen orbitals.
+(c) Simulation of the incident energy
+dependence of the charge order peak intensity with σ inci-
+dent polarization and ∆ϵd = 0.8 eV. The vertical bars are
+weights of different configurations of the RIXS intermediate
+states, the total height of which is normalized according to
+the simulated charge order peak intensity of each state. The
+accurate simulation of the Ni 3d and O 2p components of the
+resonance verifies our model, which is used to extract the elec-
+tronic character of the charge order illustrated in the inset to
+panel (b).
+ing dependent charge order wavevector varies in different
+cuprate families [20], similar to what has been seen more
+recently in the infinite-layer nickelates [15]. In view of
+this, the difference in wavevector probably does not re-
+flect a difference in the mechanisms at play in charge
+order formation. It should, however, be noted that the
+parallel charge order seen in infinite-layer materials oc-
+curs at a lower hole concentration.
+More information can be obtained by comparing the
+
+6
+states involved in charge order formation for different
+low-valence nickelates [15–17].
+All these recent works
+support an appreciable role for Ni in charge order forma-
+tion. However, controversy exists regarding whether the
+rare-earth-Ni hybridization is crucial for charge order for-
+mation [16], or whether the charge modulation on rare-
+earth states only plays a secondary parasitic role [15].
+Our results support the latter scenario in La4Ni3O8. Re-
+garding the involvement of oxygen states, we provide the
+first spectroscopic modeling that allows this question to
+be addressed quantitatively. We deduce a mixed charge-
+transfer/Mott-Hubbard picture for the charge order and
+70%/30% split of Ni vs. O contributions to the charge
+modulation. This contradicts some of the previous sug-
+gestions for infinite-layer nickelates, which propose a neg-
+ligible role for oxygen in charge order formation and that
+in-plane and out-of-plane Ni states contribute roughly
+equally [16].
+These differences are puzzling consider-
+ing that different members of the Rn+1NinO2n+2 family
+share similar Ni-O bonding, magnetic exchange [42, 50],
+superconducting transition temperatures [12, 13, 51, 52],
+and calculated electronic structures [53]. Part of the chal-
+lenge of making this comparison is that RIXS maps of
+infinite-layer films, as well as their charge order proper-
+ties, vary substantially between different samples of nom-
+inally the same composition [15–17]. In this regard, our
+quantitative spectroscopic analysis on single crystals is
+valuable considering that these samples show more con-
+sistent spectral properties than films of infinite layer ma-
+terials [15–17].
+IV.
+CONCLUSION
+In summary, we have used RIXS measurements at
+the Ni L2-edge to study the character of the electronic
+structure and charge order in the low-valence nickelate
+La4Ni3O8. Our work is unique in providing a realistic
+quantitative empirical model for charge order and vali-
+dating it using Q-resolved spectroscopy at the charge or-
+der wavevector. Different from cuprates where the spatial
+charge modulation dominantly resides on ligand orbitals,
+the charge order in La4Ni3O8 is mostly contributed by
+the Ni sites due to the larger charge transfer energy in
+low-valence nickelates. In addition to the dominant role
+of in-plane Ni 3dx2−y2 and O 2pσ orbitals, the out-of-
+plane Ni 3d3z2−r2 orbitals also participate in the charge
+order, this being enabled by their hybridization with
+the rare-earth 5d orbitals. Thus, our results reveal that
+the overall low-energy physical properties of low-valence
+nickelates are shaped by Ni 3dx2−y2 and O 2pσ orbitals,
+while the detailed electronic structure is fine tuned by
+Ni 3d3z2−r2 and rare-earth 5d orbitals. This reveals that
+multi-orbital physics is crucial to low-valence nickelates,
+indicating that several different ground states are close
+in energy. This observation points to a more complex,
+and perhaps an even richer, phenomenology than their
+cuprate cousins, while charge order remains an intrinsic
+character of these strongly correlated materials.
+The RIXS data generated in this study have been de-
+posited in the Zenodo database under accession code [to
+be assigned].
+ACKNOWLEDGMENTS
+Work at Brookhaven and the University of Tennessee
+(RIXS measurements and the interpretation and model
+Hamiltonian calculations) was supported by the U.S. De-
+partment of Energy, Office of Science, Office of Basic
+Energy Sciences, under Award Number DE-SC0022311.
+Work at Argonne was supported by the U.S. DOE, Of-
+fice of Science, Basic Energy Sciences, Materials Science
+and Engineering Division (nickelate sample synthesis and
+first principles calculations).
+Work performed at Har-
+vard University (data interpretation and paper writing)
+was supported by the US Department of Energy, Division
+of Materials Science, under Contract No. DESC0012704.
+This research used resources at the SIX beamline of the
+National Synchrotron Light Source II, a U.S. DOE Office
+of Science User Facility operated for the DOE Office of
+Science by Brookhaven National Laboratory under Con-
+tract No. DE-SC0012704.
+Appendix A: Sample synthesis
+Parent Ruddlesden-Popper La4Ni3O10 and Pr4Ni3O10
+were prepared using the high-pressure optical floating
+zone method. Sample reduction was performed by cleav-
+ing small crystals from the boules and heating them
+in a flowing H2/Ar gas mixture as described previously
+[31]. We adopt the tetragonal notation with space group
+I4/mmm and lattice constants of a = b = 3.97 ˚A,
+c = 26.092 ˚A to describe reciprocal space.
+Using this
+notation, the samples had a c-axis surface normal. The
+high quality of these samples is confirmed by prior stud-
+ies [40, 42]. Single crystals of La4Ni3O8 are particularly
+suitable for this study as they exhibit more consistent
+XAS spectra and charge order properties than thin films
+of infinite-layer nickelates [15–17]].
+Appendix B: RIXS measurements
+High-energy-resolution RIXS measurements were per-
+formed at the SIX beamline at the NSLS-II. Although
+the sample geometry and the energy of the Ni L2-edge
+resonance limits reciprocal space access, charge order in
+La4Ni3O8 has a c-axis correlation length of less than one
+unit cell, which means that the charge order Bragg peaks
+are accessible for a wide range of L values [14]. We chose
+to measure at the Ni L2-edge instead of the L3 edge to
+avoid contamination from the La M-edge which is very
+close to the Ni L3-edge and can strongly distort the reso-
+nant process [54]. In view of this, we fixed the spectrome-
+
+7
+ter angle at its maximum value of 2Θ = 153◦ throughout
+the measurements of the charge order peak. The sam-
+ples were aligned with the crystalline [0, 0, L] and [H,
+H, 0] directions lying in the horizontal scattering plane
+to access the charge order peak with momentum transfer
+QCO = (1/3, 1/3, L) where L ≈ 1.75. In this geometry,
+the x-ray intensity is dominated by charge, rather than
+spin, scattering (Supplemental Material Sec. IV [34]).
+Spectra designed to study the charge order resonance
+in the σ polarization channel, such as Fig. 2(e), were
+taken with 24 meV energy resolution.
+For the charge
+order in the π polarization channel, such as Fig. 2(f),
+a relaxed energy resolution of 32 meV was used to in-
+crease throughput. Whenever the energy was changed,
+the sample was rotated in order to remain at the same
+in-plane scattering vector. In order to study the high-
+energy features, as done in Fig. 2(a) and 2(b), the en-
+ergy resolution was further relaxed to 48 meV and the
+sample and spectrometer were slightly offset from the
+diffraction condition with a sample angle of 14.3◦ and
+a spectrometer angle of 2Θ = 147◦ to avoid saturating
+the detector. Note that the strong elastic intensity over-
+whelms the low-energy inelastic signals such as that from
+the magnetic excitations studied previously [42]. Data
+collected with different energy-resolution configurations
+were normalized by the dd excitations measured with the
+same sample geometry.
+Upon illumination by very strong elastic scattering
+from charge order, a weak periodic error was identified in
+the spectrometer grating which created the weak feature
+in the energy gain side of Fig. 2(a). This was confirmed
+by measuring reference elastic scattering.
+Appendix C: Exact diagonalization calculations
+The RIXS spectra and REXS responses presented here
+were calculated using the Kramers-Heisenberg formula in
+the dipole approximation through the EDRIXS software
+[55, 56]. The eigenstates for the initial/final and interme-
+diate states are obtained from exact diagonalization of a
+Ni3O10 cluster with four holes and open boundary con-
+ditions. To fully take into account the many-body and
+multi-orbital effects, we explicitly include the Coulomb
+interactions and nearest-neighbor inter-atomic hoppings
+in our model, and construct the Hamiltonian in hole lan-
+guage. We use the same parameters as those used in the
+O K-edge calculations which are proved to well describe
+the RIXS data [40].
+By doing so, the charge-transfer
+energy ∆ is set to 5.6 eV and the on-site Coulomb repul-
+sion to 6.5 eV, locating the material in the mixed charge-
+transfer/Mott-Hubbard regime of the Zaanen-Sawatzky-
+Allen (ZSA) scheme. We also include the spin-orbit cou-
+pling for the Ni 3d electrons, which is very small and is
+expected to play a minimal role. For simplicity, the scat-
+tering angle 2Θ is kept at 150◦ and the sample angle is
+fixed to θ = 15◦.
+The total RIXS scattering amplitude is calculated via
+F =
+�
+i
+FieiQ·ri
+(C1)
+where Fi and ri are the scattering amplitude and position
+of each Ni site, respectively. The charge order peak was
+then calculated by combining the atomic scattering am-
+plitudes with the phases appropriate for tiling the cluster
+into the NiO2 plane as shown in Fig. 1(a).
+[1] M. R. Norman, The challenge of unconventional super-
+conductivity, Science 332, 196 (2011).
+[2] D. J. Scalapino, A common thread: The pairing inter-
+action for unconventional superconductors, Reviews of
+Modern Physics 84, 1383 (2012).
+[3] K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. Waki-
+moto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shi-
+rane, R. J. Birgeneau, M. Greven, M. A. Kastner, and
+Y. J. Kim, Doping dependence of the spatially modulated
+dynamical spin correlations and the superconducting-
+transition temperature in La2−xSrxCuO4, Physical Re-
+view B 57, 6165 (1998).
+[4] S. Wakimoto, J. M. Tranquada, T. Ono, K. M. Kojima,
+S. Uchida, S. H. Lee, P. M. Gehring, and R. J. Birgeneau,
+Diagonal static spin correlation in the low-temperature
+orthorhombic
+Pccn
+phase
+of
+La1.55Nd0.4Sr0.05CuO4,
+Physical Review B 64, 174505 (2001).
+[5] S. R. Dunsiger, Y. Zhao, Z. Yamani, W. J. L. Buy-
+ers, H. A. Dabkowska, and B. D. Gaulin, Incommen-
+surate spin ordering and fluctuations in underdoped
+La2−xBaxCuO4, Physical Review B 77, 224410 (2008).
+[6] R. Arpaia, S. Caprara, R. Fumagalli, G. De Vecchi,
+Y. Y. Peng, E. Andersson, D. Betto, G. M. De Luca,
+N. B. Brookes, F. Lombardi, M. Salluzzo, L. Braicovich,
+C. Di Castro, M. Grilli, and G. Ghiringhelli, Dynamical
+charge density fluctuations pervading the phase diagram
+of a Cu-based high-Tc superconductor, Science 365, 906
+(2019).
+[7] H. Miao, G. Fabbris, R. J. Koch, D. G. Mazzone, C. S.
+Nelson, R. Acevedo-Esteves, G. D. Gu, Y. Li, T. Yil-
+imaz, K. Kaznatcheev, E. Vescovo, M. Oda, T. Kuro-
+sawa, N. Momono, T. Assefa, I. K. Robinson, E. S. Bozin,
+J. M. Tranquada, P. D. Johnson, and M. P. M. Dean,
+Charge density waves in cuprate superconductors beyond
+the critical doping, npj Quantum Materials 6, 31 (2021).
+[8] D. Li, B. Y. Wang, K. Lee, S. P. Harvey, M. Osada, B. H.
+
+8
+Goodge, L. F. Kourkoutis, and H. Y. Hwang, Supercon-
+ducting dome in Nd1−xSrxNiO2 infinite layer films, Phys-
+ical Review Letters 125, 027001 (2020).
+[9] S. Zeng, C. S. Tang, X. Yin, C. Li, M. Li, Z. Huang,
+J. Hu, W. Liu, G. J. Omar, H. Jani, Z. S. Lim, K. Han,
+D. Wan, P. Yang, S. J. Pennycook, A. T. S. Wee, and
+A. Ariando, Phase diagram and superconducting dome of
+infinite-layer Nd1−xSrxNiO2 thin films, Physical Review
+Letters 125, 147003 (2020).
+[10] M. Osada, B. Y. Wang, K. Lee, D. Li, and H. Y. Hwang,
+Phase diagram of infinite layer praseodymium nickelate
+Pr1−xSrxNiO2 thin films, Physical Review Materials 4,
+121801 (2020).
+[11] M. Osada, B. Y. Wang, B. H. Goodge, S. P. Harvey,
+K. Lee, D. Li, L. F. Kourkoutis, and H. Y. Hwang, Nick-
+elate superconductivity without rare-earth magnetism:
+(La,Sr)NiO2, Advanced Materials 33, 2104083 (2021).
+[12] S. Zeng, C. Li, E. Chow Lin, Y. Cao, Z. Zhang,
+S. Tang Chi, X. Yin, S. Lim Zhi, J. Hu, P. Yang, and
+A. Ariando, Superconductivity in infinite-layer nickelate
+La1−xCaxNiO2 thin films, Science Advances 8, eabl9927
+(2022).
+[13] G. A. Pan, D. Ferenc Segedin, H. LaBollita, Q. Song,
+E. M. Nica, B. H. Goodge, A. T. Pierce, S. Doyle, S. No-
+vakov, D. C´ordova Carrizales, et al., Superconductivity
+in a quintuple-layer square-planar nickelate, Nature Ma-
+terials 21, 160 (2022).
+[14] J. Zhang, Y.-S. Chen, D. Phelan, H. Zheng, M. R. Nor-
+man, and J. F. Mitchell, Stacked charge stripes in the
+quasi-2D trilayer nickelate La4Ni3O8, Proceedings of the
+National Academy of Sciences 113, 8945 (2016).
+[15] M. Rossi, M. Osada, J. Choi, S. Agrestini, D. Jost,
+Y. Lee, H. Lu, B. Y. Wang, K. Lee, A. Nag, et al., A
+broken translational symmetry state in an infinite-layer
+nickelate, Nature Physics 18, 869–873 (2022).
+[16] C. C. Tam, J. Choi, X. Ding, S. Agrestini, A. Nag,
+B. Huang, H. Luo, M. Garc´ıa-Fern´andez, L. Qiao, and K.-
+J. Zhou, Charge density waves in infinite-layer NdNiO2
+nickelates, Nature Materials 10.1038/s41563-022-01330-1
+(2022).
+[17] G. Krieger, L. Martinelli, S. Zeng, L. E. Chow, K. Kum-
+mer, R. Arpaia, M. Moretti Sala, N. B. Brookes, A. Ar-
+iando,
+N. Viart,
+M. Salluzzo,
+G. Ghiringhelli, and
+D. Preziosi, Charge and spin order dichotomy in NdNiO2
+driven by the capping layer, Physical Review Letters 129,
+027002 (2022).
+[18] J. M. Tranquada, Spins, stripes, and superconductivity in
+hole-doped cuprates, AIP Conference Proceedings 1550,
+114 (2013).
+[19] R. Comin and A. Damascelli, Resonant x-ray scattering
+studies of charge order in cuprates, Annual Review of
+Condensed Matter Physics 7, 369 (2016).
+[20] A. Frano, S. Blanco-Canosa, B. Keimer, and R. J. Birge-
+neau, Charge ordering in superconducting copper oxides,
+Journal of Physics: Condensed Matter 32, 374005 (2020).
+[21] J. M. Tranquada, M. P. M. Dean, and Q. Li, Supercon-
+ductivity from charge order in cuprates, Journal of the
+Physical Society of Japan 90, 111002 (2021).
+[22] E. W. Huang, C. B. Mendl, S. Liu, S. Johnston, H.-C.
+Jiang, B. Moritz, and T. P. Devereaux, Numerical evi-
+dence of fluctuating stripes in the normal state of high-Tc
+cuprate superconductors, Science 358, 1161 (2017).
+[23] B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P.
+Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and
+G. K.-L. Chan, Stripe order in the underdoped region of
+the two-dimensional Hubbard model, Science 358, 1155
+(2017).
+[24] P. Mai, S. Karakuzu, G. Balduzzi, S. Johnston, and T. A.
+Maier, Intertwined spin, charge, and pair correlations in
+the two-dimensional Hubbard model in the thermody-
+namic limit, Proceedings of the National Academy of Sci-
+ences 119, e2112806119 (2022).
+[25] Q. Li, M. H¨ucker, G. D. Gu, A. M. Tsvelik, and J. M.
+Tranquada, Two-dimensional superconducting fluctua-
+tions in stripe-ordered La1.875Ba0.125CuO4, Physical Re-
+view Letters 99, 067001 (2007).
+[26] E. Berg,
+E. Fradkin,
+E.-A. Kim,
+S. A. Kivelson,
+V. Oganesyan, J. M. Tranquada, and S. C. Zhang,
+Dynamical layer decoupling in a stripe-ordered high-
+Tc superconductor, Physical Review Letters 99, 127003
+(2007).
+[27] V. J. Emery, S. A. Kivelson, and O. Zachar, Spin-gap
+proximity effect mechanism of high-temperature super-
+conductivity, Physical Review B 56, 6120 (1997).
+[28] S. A. Kivelson, E. Fradkin, and V. J. Emery, Electronic
+liquid-crystal phases of a doped Mott insulator, Nature
+393, 550 (1998).
+[29] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Theory
+of intertwined orders in high temperature superconduc-
+tors, Reviews of Modern Physics 87, 457 (2015).
+[30] C. Castellani, C. Di Castro, and M. Grilli, Singular quasi-
+particle scattering in the proximity of charge instabilities,
+Physical Review Letters. 75, 4650 (1995).
+[31] J. Zhang, A. Botana, J. Freeland, D. Phelan, H. Zheng,
+V. Pardo, M. Norman, and J. Mitchell, Large orbital po-
+larization in a metallic square-planar nickelate, Nature
+Physics 13, 864 (2017).
+[32] J. Karp, A. Hampel, M. Zingl, A. S. Botana, H. Park,
+M. R. Norman, and A. J. Millis, Comparative many-body
+study of Pr4Ni3O8 and NdNiO2, Physical Review B 102,
+245130 (2020).
+[33] E. M. Nica, J. Krishna, R. Yu, Q. Si, A. S. Botana, and
+O. Erten, Theoretical investigation of superconductivity
+in trilayer square-planar nickelates, Physical Review B
+102, 020504(R) (2020).
+[34] See Supplemental Material at [URL will be inserted by
+publisher] for measurements of Pr4Ni3O8, discussion of
+the RIXS process, consideration of spin order, and fur-
+ther calculations. This also includes reference [57–59].
+[35] G. Fabbris, D. Meyers, L. Xu, V. M. Katukuri, L. Hozoi,
+X. Liu, Z.-Y. Chen, J. Okamoto, T. Schmitt, A. Uldry,
+B. Delley, G. D. Gu, D. Prabhakaran, A. T. Boothroyd,
+J. van den Brink, D. J. Huang, and M. P. M. Dean, Dop-
+ing dependence of collective spin and orbital excitations
+in the spin-1 quantum antiferromagnet La2−xSrxNiO4
+observed by x rays, Physical Review Lett. 118, 156402
+(2017).
+[36] K. Higashi, M. Winder, J. Kuneˇs, and A. Hariki, Core-
+level x-ray spectroscopy of infinite-layer nickelate: LDA+
+DMFT study, Physical Review X 11, 041009 (2021).
+[37] P. Abbamonte, A. Rusydi, S. Smadici, G. D. Gu, G. A.
+Sawatzky, and D. L. Feng, Spatially modulated ‘Mot-
+tness’ in La2−xBaxCuO4, Nature Physics 1, 155 (2005).
+[38] Y.
+Y.
+Peng,
+R.
+Fumagalli,
+Y.
+Ding,
+M.
+Minola,
+S. Caprara, D. Betto, M. Bluschke, G. M. De Luca,
+K. Kummer, E. Lefran¸cois, M. Salluzzo, H. Suzuki,
+M. Le Tacon, X. J. Zhou, N. B. Brookes, B. Keimer,
+L. Braicovich, M. Grilli, and G. Ghiringhelli, Re-entrant
+
+9
+charge order in overdoped (Bi,Pb)2.12Sr1.88CuO6+δ out-
+side the pseudogap regime, Nature Materials 17, 697
+(2018).
+[39] J. Li, A. Nag, J. Pelliciari, H. Robarts, A. Walters,
+M. Garcia-Fernandez, H. Eisaki, D. Song, H. Ding,
+S. Johnston, R. Comin, and K.-J. Zhou, Multiorbital
+charge-density wave excitations and concomitant phonon
+anomalies in Bi2Sr2LaCuO6+δ, Proceedings of the Na-
+tional Academy of Sciences 117, 16219 (2020).
+[40] Y. Shen, J. Sears, G. Fabbris, J. Li, J. Pelliciari, I. Jar-
+rige, X. He, I. Boˇzovi´c, M. Mitrano, J. Zhang, J. F.
+Mitchell, A. S. Botana, V. Bisogni, M. R. Norman,
+S. Johnston, and M. P. M. Dean, Role of oxygen states
+in the low valence nickelate La4Ni3O8, Physical Review
+X 12, 011055 (2022).
+[41] A. J. Achkar, F. He, R. Sutarto, J. Geck, H. Zhang, Y.-
+J. Kim, and D. G. Hawthorn, Resonant x-ray scattering
+measurements of a spatial modulation of the Cu 3d and
+O 2p energies in stripe-ordered cuprate superconductors,
+Physical Review Letters 110, 017001 (2013).
+[42] J. Q. Lin, P. Villar Arribi, G. Fabbris, A. S. Botana,
+D. Meyers, H. Miao, Y. Shen, D. G. Mazzone, J. Feng,
+S. G. Chiuzb˘aian, A. Nag, A. C. Walters, M. Garc´ıa-
+Fern´andez, K.-J. Zhou, J. Pelliciari, I. Jarrige, J. W.
+Freeland, J. Zhang, J. F. Mitchell, V. Bisogni, X. Liu,
+M. R. Norman, and M. P. M. Dean, Strong superex-
+change in a d9−δ nickelate revealed by resonant inelastic
+x-ray scattering, Physical Review Letters 126, 087001
+(2021).
+[43] C. T. Chen, L. H. Tjeng, J. Kwo, H. L. Kao, P. Rudolf,
+F. Sette, and R. M. Fleming, Out-of-plane orbital charac-
+ters of intrinsic and doped holes in La2−xSrxCuO4, Phys-
+ical Review Letters 68, 2543 (1992).
+[44] M. Schneider, R. S. Unger, R. Mitdank, R. M˘uller,
+A. Krapf, S. Rogaschewski, H. Dwelk, C. Janowitz,
+and R. Manzke, Evolution of the density of states
+at the Fermi level of Bi2−yPbySr2−xLaxCuO6+δ and
+Bi2Sr2−xLaxCuO6+δ cuprates with hole doping, Physi-
+cal Review B 72, 014504 (2005).
+[45] F. C. Zhang and T. M. Rice, Effective Hamiltonian for
+the superconducting Cu oxides, Physical Review B 37,
+3759 (1988).
+[46] M. Moretti Sala, V. Bisogni, C. Aruta, G. Balestrino,
+H. Berger, N. B. Brookes, G. M. d. Luca, D. Di Castro,
+M. Grioni, M. Guarise, P. G. Medaglia, F. Miletto Gra-
+nozio, M. Minola, P. Perna, M. Radovic, M. Salluzzo,
+T. Schmitt, K. J. Zhou, L. Braicovich, and G. Ghir-
+inghelli, Energy and symmetry of dd excitations in un-
+doped layered cuprates measured by Cu L3 resonant
+inelastic x-ray scattering, New Journal of Physics 13,
+043026 (2011).
+[47] H. Ulbrich and M. Braden, Neutron scattering studies on
+stripe phases in non-cuprate materials, Physica C: Su-
+perconductivity 481, 31 (2012), stripes and Electronic
+Liquid Crystals in Strongly Correlated Materials.
+[48] A. Melikyan and M. R. Norman, Symmetry of the charge
+density wave in cuprates, Physical Review B 89, 024507
+(2014).
+[49] P. Corboz, T. M. Rice, and M. Troyer, Competing states
+in the t-J model:
+Uniform d-wave state versus stripe
+state, Physical Review Letters 113, 046402 (2014).
+[50] H. Lu, M. Rossi, A. Nag, M. Osada, D. F. Li, K. Lee,
+B. Y. Wang, M. Garcia-Fernandez, S. Agrestini, Z. X.
+Shen, E. M. Been, B. Moritz, T. P. Devereaux, J. Zaa-
+nen, H. Y. Hwang, K.-J. Zhou, and W. S. Lee, Magnetic
+excitations in infinite-layer nickelates, Science 373, 213
+(2021).
+[51] D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R.
+Lee, Y. Cui, Y. Hikita, and H. Y. Hwang, Supercon-
+ductivity in an infinite-layer nickelate, Nature 572, 624
+(2019).
+[52] M. Osada, B. Y. Wang, B. H. Goodge, K. Lee, H. Yoon,
+K. Sakuma, D. Li, M. Miura, L. F. Kourkoutis, and H. Y.
+Hwang, A superconducting praseodymium nickelate with
+infinite layer structure, Nano Letters 20, 5735 (2020).
+[53] Low valence nickelate electronic structure are rather sim-
+ilar provided they are compared at similar effective dop-
+ings [32, 60].
+[54] C. Sch¨ußler-Langeheine, J. Schlappa, A. Tanaka, Z. Hu,
+C.
+F.
+Chang,
+E.
+Schierle,
+M.
+Benomar,
+H.
+Ott,
+E. Weschke, G. Kaindl, O. Friedt, G. A. Sawatzky, H.-J.
+Lin, C. T. Chen, M. Braden, and L. H. Tjeng, Spec-
+troscopy of stripe order in La1.8Sr0.2NiO4 using resonant
+soft x-ray diffraction, Physical Review Letters 95, 156402
+(2005).
+[55] EDRIXS
+website,
+https://github.com/NSLS-II/
+edrixs, accessed: 2022-05-19.
+[56] Y. Wang, G. Fabbris, M. Dean, and G. Kotliar, EDRIXS:
+An open source toolkit for simulating spectra of resonant
+inelastic x-ray scattering, Computer Physics Communi-
+cations 243, 151 (2019).
+[57] M. W. Haverkort, M. Zwierzycki, and O. K. Ander-
+sen, Multiplet ligand-field theory using wannier orbitals,
+Physical Review B 85, 165113 (2012).
+[58] M. W. Haverkort, Theory of resonant inelastic x-ray scat-
+tering by collective magnetic excitations, Physical Re-
+view Letters 105, 167404 (2010).
+[59] A. J. Achkar, R. Sutarto, X. Mao, F. He, A. Frano,
+S.
+Blanco-Canosa,
+M.
+Le
+Tacon,
+G.
+Ghiringhelli,
+L. Braicovich, M. Minola, M. Moretti Sala, C. Mazzoli,
+R. Liang, D. A. Bonn, W. N. Hardy, B. Keimer, G. A.
+Sawatzky, and D. G. Hawthorn, Distinct charge orders in
+the planes and chains of ortho-III-ordered YBa2Cu3O6+δ
+superconductors identified by resonant elastic x-ray scat-
+tering, Physical Review Letters 109, 167001 (2012).
+[60] H. LaBollita and A. S. Botana, Electronic structure and
+magnetic properties of higher-order layered nickelates:
+Lan+1NinO2n+2(n = 4 − 6), Physical Review B 104,
+035148 (2021).
+
+Supplemental Material: Electronic character of charge order in square planar low
+valence nickelates
+Y. Shen,∗ J. Sears, G. Fabbris, J. Li, J. Pelliciari, M. Mitrano, W. He, Junjie
+Zhang, J. F. Mitchell, V. Bisogni, M. R. Norman, S. Johnston, and M. P. M. Dean†
+(Dated: January 12, 2023)
+I.
+ABSENCE OF DIAGONAL CHARGE ORDER IN Pr4Ni3O8
+To confirm the absence of diagonal charge order in metallic Pr4Ni3O8 [1], we performed resonant inelastic x-ray
+scattering (RIXS) measurements near Q∥ = (1/3, 1/3) in Pr4Ni3O8. Figure S1 shows the RIXS spectra in the quasi-
+elastic regime with σ polarized incident photons. No superlattice peaks are found but only background evolving
+smoothly with the in-plane sample angle θ, which is primarily caused by the self-absorption effect. Note that despite
+the absence of long-range or short-range stripe order indicated here, stripe related spin fluctuations are distinguished
+in the inelastic regime [2].
+0.2
+0.1
+0.0
+0.1
+0.2
+Energy loss (eV)
+(a)
+10
+15
+20
+25
+30
+ (deg)
+0.0
+0.2
+0.4
+0.6
+Intensity (arb. units)
+(b)
+0
+2
+4
+6
+8
+10
+Intensity (arb. units)
+FIG. S1. Absence of charge order in Pr4Ni3O8 at 40 K. (a) RIXS intensity map around Q∥ = (1/3, 1/3) in the quasi-elastic
+regime at the Ni L3-edge. The experimental configuration is the same as that for La4Ni3O8. (b) Quasi-elastic amplitudes
+extracted from (a).
+∗ [email protected]
+† [email protected]
+arXiv:2301.04184v1  [cond-mat.str-el]  10 Jan 2023
+
+2
+II.
+RIXS PROCESS FOR DIFFERENT EXCITATIONS
+Here we discuss the RIXS process for different excitations. Due to the presence of the strong core-hole potential in
+the RIXS intermediate states, the electron that is excited from the core level is constrained to a few unit cells near the
+Ni site where the x-ray absorption takes place. This effect competes with the kinetic energy of the electron and leads
+to intertwined excitations in the RIXS spectra. In a simplified picture, the orbital states during the RIXS process can
+be divided into three categories based on how they are affected by the core-hole potential [see Fig. S2(a)]. The first
+one involves the Ni 3d orbitals that are strongly localized at the core-hole site. The second one involves the ligand
+orbitals that surround the Ni site and strongly hybridize with the Ni 3d orbitals. They are largely localized but could
+show a finite bandwidth. The third one involves continuous electronic bands that are mostly unperturbed by the
+core-hole potential and behave itinerantly with an appreciable bandwidth. The localized Ni 3d orbitals can hybridize
+with the continuous bands in an orbital dependent fashion. At the Ni L-edge, the core electron is predominantly
+excited to the unoccupied localized Ni 3d orbitals [see Fig. S2(b)]. During the photon emission process, either an
+electron from another 3d orbital deexcites to fill the core hole, leading to dd multiplet excitations [see Fig. S2(d)], or
+an electron from the ligand orbitals hops to the Ni site, resulting in charge-transfer excitations [Fig. S2(e)]. Since the
+ligand orbitals normally lie at a lower energy, the charge-transfer excitations usually occur with a larger energy loss
+than the dd excitations and are much weaker at the Ni L-edge as they are made possible through hybridization. In the
+post-edge regime, the core electron is excited to the unoccupied states in the continuous bands through hybridization
+in the intermediate state [see Fig. S2(c)], and during the photon emission an electron below the Fermi level deexcites
+to fill the core hole, leading to the fluorescence [see Fig. S2(f)]. As the deexciting process is dominated by electrons
+near the Fermi energy, fluorescence tends to present a constant emission photon energy. Note that at the Ni L-edge
+RIXS process, contributions from rare earth and oxygen states are seen via their hybridization with atomic Ni 3d
+orbitals. In real materials, there are no clear boundaries between the localized orbitals and continuous bands. Thus,
+different excitations are also intertwined but the weights are quite different, which helps us distinguish them in the
+RIXS spectra.
+electron
+hole
+Ni site
+Core level
+Multiplets
+Ligand
+Continuous bands
+Energy
+Distance
+Fermi
+level
+Final states
+Intermediate states
+fluorescence
+Post-edge
+Charge-transfer excitations
+dd excitations
+At edge
+(a)
+(b)
+(d)
+(e)
+(c)
+(f)
+FIG. S2. RIXS process for different excitations. (a) Legend for each symbol. (b, c) Photon absorption process and corresponding
+intermediate states. (d)–(f) Photon emission process and corresponding final states. For the fluorescence excitation scenario,
+the multiplets are not well defined so they are replaced by dashed lines.
+
+3
+0
+30
+60
+90
+120
+150
+180
+ (deg)
+0
+1
+2
+3
+4
+Absorption ratio for  vs  polarization
+d3z2
+r2
+dxz/dyz
+dx2
+y2/dxy
+FIG. S3. Intensity ratio for dipole absorption between π and σ polarization channels as a function of sample angle θ. The
+vertical dashed line indicates the experimental configuration we used for charge order measurements while the horizontal dotted
+line denotes a unit ratio.
+III.
+POLARIZATION DEPENDENCE OF FLUORESCENCE
+The polarization dependence of dd excitations in cuprates and nickelates has been widely discussed [3, 4]. Here,
+we focus on fluorescence to show how the orbital information can be extracted by comparing the RIXS intensity in
+different polarization channels. Since we are measuring at the Ni L edge, the RIXS signal can only arise from either
+Ni orbitals or Ni states hybridized with other orbitals. For the fluorescence features, the photon emission process
+is quite similar, with electrons from the crystalline environment surrounding the Ni site deexciting to fill the core
+hole. Hence, the main intensity difference between these two polarization channels comes from the photon absorption
+process, the cross-section of which can be simulated in the dipole approximation. Fig. S3 presents the x-ray dipole
+absorption intensity ratio between π and σ polarization channels as a function of sample angle θ. For the experimental
+configuration we used (vertical dashed line), the biggest contribution to the π over σ polarization intensity ratio is
+the Ni 3d3z2−r2 orbitals while 3dx2−y2 and 3dxy contribute equally to σ over π polarization intensity ratio. Since the
+3dx2−y2 orbitals dominate near the Fermi level and are expected to show stronger hybridization with oxygen orbitals
+[5], 3dxy orbitals are expected to play a less important role. Moreover, the t2g states do not make any significant
+contribution to the unoccupied states. Thus, we focus on Ni eg orbitals during the discussion in the main text, which
+are the subject of most of the debates over the appropriate theoretical models.
+IV.
+MINIMAL CONTRIBUTION OF SPIN ORDER TO THE REXS SIGNAL
+In La4Ni3O8, spin order takes place concomitantly with the charge order and shares the same Q∥. Hence we need
+to invoke cross-section considerations to separate the possible contribution of charge and spin order [6].
+With π incident x-ray polarization, charge order contributes to the measured signal in the π-π′ scattering channel
+while the spin order is responsible for the π-σ′ channel. The resonant elastic x-ray scattering (REXS) intensity ratio
+between these channels can be estimated by (ki · kf)2/(ϵi × ϵ′
+f · M)2 = cos2 2Θ/ sin2 θ ≈ 11.8, where ki (kf) is the
+initial (final) x-ray wavevector, ϵi (ϵ′
+f) is the initial (final) x-ray polarization, and M is the spin direction, which is
+parallel to the c-axis in this case. Based on this formula we can see that the REXS signal with π incident polarization
+is dominantly of charge order origin.
+Regarding the spin order contribution with σ incident x-ray polarization, we can compare the peak intensity with
+grazing-in and grazing-out conditions. Since the charge order composes the σ-σ′ channel, its intensity is expected to
+be the same in these two geometries. For spin order signal that is only observable in the σ-π′ channel, the intensity
+
+4
+0.32
+0.33
+0.34
+0.35
+(H, H) (r.l.u.)
+0
+2
+4
+6
+8
+Intensity (arb. units)
+grazing-in
+0.32
+0.33
+0.34
+0.35
+grazing-out
+FIG. S4. Comparison of the superlattice peak intensity with grazing-in and grazing-out conditions. The scattering angle 2Θ
+was fixed to 153◦ and the data were collected in σ polarization channel at 40 K. The solid lines are guides to the eye. Both
+peaks are found to have essentially the same intensity, which confirms that the peak arises from charge, rather than spin, order.
+TABLE S1. Full list of parameters used for the ED calculations. The on-site orbital energies, hopping integrals and Coulomb
+interactions are kept the same as those used in the O K-edge calculations [7], and Vpdπ = −Vpdσ/2, Vppπ = −Vppσ/4. The
+potential difference, ∆ϵd, only applies to the Ni 3d orbitals. Note that the crystal field splitting that is instead used in a Ni
+atomic model is a combination of point charge potential and orbital hybridization, which can be estimated through ligand field
+theory [5]. The resulting effective crystal field splitting gives 10Dq = 0.971, ∆eg = 1.041, ∆t2g = 0.342 eV, which are of a
+similar energy scale as the dd excitations observed in the RIXS measurements. ζi and ζn are spin-orbit coupling parameters of
+the Ni 3d electrons for the initial and intermediate states, respectively, and ζc is the spin-orbit coupling strength for the Ni 2p
+core electrons. The core-hole lifetime is set to be 0.6 eV. All parameters are in units of eV.
+On-site orbital energies
+Hopping integrals
+ϵdx2−y2
+ϵd3z2−r2
+ϵdxy
+ϵdxz/yz
+ϵpσ
+ϵpπ/pz
+Vpdσ
+Vppσ
+0
+0.2
+0.1
+0.3
+5.6
+6.1
+1.57
+0.6
+Spin-orbit coupling
+On-site Coulomb interactions
+ζi
+ζn
+ζc
+F 0
+dd
+F 2
+dd
+F 4
+dd
+F 0
+pp
+F 2
+pp
+0.083
+0.102
+11.507
+5.58
+6.89
+4.31
+3.3
+5
+Inter-site Coulomb interactions
+Core-hole potential
+F 0
+dp
+F 2
+dp
+G1
+dp
+G3
+dp
+F 0
+dp
+F 2
+dp
+G1
+dp
+G3
+dp
+1
+0
+0
+0
+7.869
+5.405
+4.051
+2.304
+ratio between grazing-in and grazing-out conditions is (kf, grazing−in ·M)2/(kf, grazing−out ·M)2 ≈ 5.6, indicating that
+the spin order signal should be strongly suppressed with grazing-out condition. Figure S4 shows the Q dependence
+of the superlattice peak with both conditions, which are comparable with each other, proving that the superlattice
+peak observed with σ incident x-ray polarization is also dominantly of charge order origin.
+V.
+CHARGE ORDER IN ED CALCULATIONS.
+We use cluster ED to study the charge order in the low-valence nickelate La4Ni3O8. The full list of the parameters
+used is presented in Table S1. The validity of our cluster model and parameters has been verified by calculating
+the RIXS energy maps and confirming that they capture the main features of the measurements as shown in the
+main text. In the calculations, we include all the Ni 3d and O 2p orbitals, which leads to a large Hilbert space and
+correspondingly only a limited number of states can be solved for. Fortunately, the accessible energy range covers
+the dd excitations so that we can make a direct comparison with the experimental data. The calculated results are
+broadened using a Gaussian profile with a full width at half maximum of 0.3 eV and are shown in Fig. 3 of the main
+text.
+To fully explore the charge order character in the ED calculations, we need to cover a large incident energy range
+but only the ground state is needed to calculate the REXS signals. Thus, we only include the Ni 3dx2−y2 and O 2pσ
+
+5
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+Hole occupation
+(a)
+Ni1
+Ni1L
+Ni2
+Ni2L
+0.0
+0.4
+0.8
+1.2
+1.6
+2.0
+d (eV)
+0.00
+0.25
+0.50
+0.75
+Charge disproportionation
+(b)
+(Ni2+Ni2L)-(Ni1+Ni1L)
+10
+12
+14
+16
+18
+Incident energy (eV)
+0.0
+0.5
+1.0
+1.5
+2.0
+Intensity (arb. units)
+(c)
+REXS@QCO, -pol.
+d=0.0
+d=0.1
+d=0.2
+d=0.3
+d=0.4
+d=0.5
+d=0.6
+d=0.7
+d=0.8
+d=0.9
+d=1.0
+FIG. S5. The emergence of charge order by introducing the potential difference term ∆ϵd. (a) Hole occupations of different
+sites as a function of ∆ϵd. Ni1L stands for the ligand orbitals for Ni1 (the surrounding four oxygens). Correspondingly, one
+oxygen is shared by Ni1L and Ni2L. (b) Charge disproportionation defined as the hole occupation difference of Ni1+Ni1L and
+Ni2+Ni2L. (c) Calculated REXS signals at QCO with different ∆ϵd. All the calculations are performed with ∆ = 5.6 eV.
+orbitals during the calculations of the charge order, which dominate the ground state, so that a tractable basis size
+is realized. To trigger charge order in the Ni3O10 cluster, we introduce a potential difference ∆ϵd as described in the
+main text. In a microscopic model like we use here, the onsite energy shift and charge occupation are intrinsically
+coupled, which is different from a phenomenological model where these two factors can be tuned independently [8, 9].
+As shown in Fig. S5, when ∆ϵd is zero, the hole occupations on different Ni sites are almost the same while the hole
+occupations of ligand orbitals are slightly imbalanced since Ni2 shares oxygens with both Ni1 and Ni3, leading to a
+small charge disproportionation. With increasing ∆ϵd, the charge imbalances on both the Ni and ligand orbitals are
+enhanced with the former much more prominent, indicating that most of the spatial charge modulation resides on
+the Ni sites, leading to a Ni site-centered charge order. Correspondingly, a charge-order peak emerge in the REXS
+calculations, the intensity of which increases with increasing charge disproportionation while the lineshape only evolves
+by a little.
+After testing the effect of ∆ϵd, here we compare results with different charge-transfer energy ∆ in addition to the
+calculated results presented in the main text. As shown in Fig. S6, in the charge-transfer regime (∆ ≪ Udd), a
+sharp resonant peak is obtained, resembling the experimental observations in cuprates. With increasing ∆, the REXS
+lineshape evolves correspondingly. In the Mott-Hubbard limit (∆ ≫ Udd), the charge order peak becomes broader and
+shows multiple peak features. Compared with the data presented in the main text, we conclude that a charge-transfer
+energy with an intermediate strength (∆ ≈ Udd) matches the experimental results the best.
+[1] Junjie Zhang, A.S. Botana, J.W. Freeland, D. Phelan, Hong Zheng, V. Pardo, M.R. Norman, and J.F. Mitchell, “Large
+orbital polarization in a metallic square-planar nickelate,” Nature Physics 13, 864–869 (2017).
+[2] J. Q. Lin, P. Villar Arribi, G. Fabbris, A. S. Botana, D. Meyers, H. Miao, Y. Shen, D. G. Mazzone, J. Feng, S. G. Chiuzb˘aian,
+A. Nag, A. C. Walters, M. Garc´ıa-Fern´andez, Ke-Jin Zhou, J. Pelliciari, I. Jarrige, J. W. Freeland, Junjie Zhang, J. F.
+Mitchell, V. Bisogni, X. Liu, M. R. Norman, and M. P. M. Dean, “Strong superexchange in a d9−δ nickelate revealed by
+resonant inelastic x-ray scattering,” Physical Review Letters 126, 087001 (2021).
+[3] M. Rossi, H. Lu, A. Nag, D. Li, M. Osada, K. Lee, B. Y. Wang, S. Agrestini, M. Garcia-Fernandez, J. J. Kas, Y.-D. Chuang,
+Z. X. Shen, H. Y. Hwang, B. Moritz, Ke-Jin Zhou, T. P. Devereaux, and W. S. Lee, “Orbital and spin character of doped
+carriers in infinite-layer nickelates,” Physical Review B 104, L220505 (2021).
+[4] M. Moretti Sala, V. Bisogni, C. Aruta, G. Balestrino, H. Berger, N. B. Brookes, G. M. de Luca, D. Di Castro, M. Grioni,
+
+6
+10
+12
+14
+16
+18
+Incident energy (eV)
+0
+1
+2
+3
+4
+5
+Intensity (arb. units)
+REXS@QCO, -pol.
+=2.5
+=3.5
+=4.5
+=5.6
+=6.7
+=7.8
+=8.9
+FIG. S6.
+Calculated REXS signals at the charge order wavevector QCO with different charge-transfer energy ∆.
+All the
+calculations are performed with ∆ϵd = 0.8 eV and U = 6.5 eV.
+M. Guarise, P. G. Medaglia, F. Miletto Granozio, M. Minola, P. Perna, M. Radovic, M. Salluzzo, T. Schmitt, K. J. Zhou,
+L. Braicovich, and G. Ghiringhelli, “Energy and symmetry of dd excitations in undoped layered cuprates measured by Cu
+L3 resonant inelastic x-ray scattering,” New Journal of Physics 13, 043026 (2011).
+[5] M. W. Haverkort, M. Zwierzycki,
+and O. K. Andersen, “Multiplet ligand-field theory using wannier orbitals,” Physical
+Review B 85, 165113 (2012).
+[6] M. W. Haverkort, “Theory of resonant inelastic x-ray scattering by collective magnetic excitations,” Physical Review Letters
+105, 167404 (2010).
+[7] Y. Shen, J. Sears, G. Fabbris, J. Li, J. Pelliciari, I. Jarrige, Xi He, I. Boˇzovi´c, M. Mitrano, Junjie Zhang, J. F. Mitchell,
+A. S. Botana, V. Bisogni, M. R. Norman, S. Johnston,
+and M. P. M. Dean, “Role of oxygen states in the low valence
+nickelate La4Ni3O8,” Physical Review X 12, 011055 (2022).
+[8] A. J. Achkar, R. Sutarto, X. Mao, F. He, A. Frano, S. Blanco-Canosa, M. Le Tacon, G. Ghiringhelli, L. Braicovich,
+M. Minola, M. Moretti Sala, C. Mazzoli, Ruixing Liang, D. A. Bonn, W. N. Hardy, B. Keimer, G. A. Sawatzky, and D. G.
+Hawthorn, “Distinct charge orders in the planes and chains of ortho-III-ordered YBa2Cu3O6+δ superconductors identified
+by resonant elastic x-ray scattering,” Physical Review Letters 109, 167001 (2012).
+[9] A. J. Achkar, F. He, R. Sutarto, J. Geck, H. Zhang, Y.-J. Kim, and D. G. Hawthorn, “Resonant x-ray scattering measure-
+ments of a spatial modulation of the Cu 3d and O 2p energies in stripe-ordered cuprate superconductors,” Physical Review
+Letters 110, 017001 (2013).
+

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+arXiv:2301.12145v1  [math.PR]  28 Jan 2023
+Normal approximation of subgraph counts in
+the random-connection model
+Qingwei Liu∗
+Nicolas Privault†
+Division of Mathematical Sciences
+School of Physical and Mathematical Sciences
+Nanyang Technological University
+21 Nanyang Link, Singapore 637371
+January 31, 2023
+Abstract
+This paper derives normal approximation results for subgraph counts written as
+multiparameter stochastic integrals in a random-connection model based on a Pois-
+son point process. By combinatorial arguments we express the cumulants of general
+subgraph counts using sums over connected partition diagrams, after cancellation of
+terms obtained by M¨obius inversion. Using the Statuleviˇcius condition, we deduce con-
+vergence rates in the Kolmogorov distance by studying the growth of subgraph count
+cumulants as the intensity of the underlying Poisson point process tends to infinity.
+Our analysis covers general subgraphs in the dilute and full random graph regimes,
+and tree-like subgraphs in the sparse random graph regime.
+Keywords: Random-connection model, subgraph count, normal approximation, Kolmogorov
+distance, cumulant method, Poisson point process, random graphs.
+Mathematics Subject Classification: 60F05, 60D05, 05C80, 60G55.
+1
+Introduction
+This paper treats the asymptotic behavior of random subgraph counts in the random-
+connection model (RCM), which is used to model physical systems in e.g. wireless networks,
+complex networks, and statistical mechanics. Our approach relies on the study of cumulant
+growth rates as the intensity of the underlying Poisson point process tends to infinity.
+The distributional approximation of subgraph counts has attracted significant interest
+in the random graph literature.
+In [Ruc88], conditions for the asymptotic normality of
+∗[email protected]
+†[email protected]
+1
+
+renormalized subgraph counts have been obtained in the Erd˝os-R´enyi random graph model
+[ER59, Gil59]. Those results have been made more precise in [BKR89] by the derivation
+of convergence rates in the Wasserstein distance via Stein’s method. They have also been
+strengthened in [KRT17] using the Kolmogorov distance in the case of triangle counts, and
+in [PS20] in the case of general subgraphs G. The case of triangles has also been treated in
+[R¨ol22] by the Stein-Tikhomirov method, which has been extended to general subgraphs in
+[ER21]. In [Kho08], the counts of line (X-model) and cycles (Y -model) in discrete Erd˝os-
+R´enyi models have been analyzed via the asymptotic behavior of their cumulants. In compar-
+ison with [Kho08], we derive Kolmogorov convergence rates and our results are not restricted
+to line and cycle graphs, as they cover more general subgraphs.
+The random connection-model is a natural generalization of the Erd˝os-R´enyi random
+graph in which vertices are randomly located and can be connected with position-dependent
+probabilities. Studying the random-connection model and obtaining normal approximation
+error bounds is more difficult due to the additional layer of complexity coming from the
+randomness of vertex locations. In [LNS21], a central limit theorem and Kolmogorov con-
+vergence rates have been presented for the number of components isomorphic to a given
+finite connected graph in the random-connection model, together with a study of first mo-
+ments and covariances. Recently, a Central Limit Theorem has been derived in [CT22] for
+the counts of induced subgraphs in the random-connection model under certain stabilization
+and moment conditions.
+In this paper, we derive normal approximation rates under a relatively mild condition
+on the connection function of the random-connection model, by deriving growth rates of
+cumulants written as sums over connected partitions. To the best of our knowledge, this is
+the first time that the normal approximation of subgraph counts with convergence rates is
+established in the random-connection model. Furthermore, various random graph regimes
+are discussed.
+A number of probabilistic conclusions can be derived from the behavior of cumulants
+of random variables using the Statuleviˇcius condition, including convergence rates in the
+Kolmogorov distance and moderate deviation principles, see [SS91], [DE13], [DJS22]. In
+[GT18a, GT18b], this method has been used to derive concentration inequalities, normal
+approximation with error bounds, and moderate deviation principles for random polytopes.
+Given µ a finite diffuse measure on Rd, we consider a random-connection model based on
+an underlying Poisson point process Ξ on Rd with intensity of the form λµ(dx), in which any
+2
+
+two vertices x, y in Ξ are connected with the probability Hλ(x, y) := cλH(x, y) ∈ [0, 1], where
+Hλ is the connection function of the model. Here, we investigate the limiting behavior of the
+count NG of a given subgraph G as the intensity λ of the underlying Poisson point process
+on Rd tends to infinity. To this end, we use the combinatorics of the cumulants κn(NG)
+based on moment expressions obtained in [Pri19] for multiparameter stochastic integrals in
+the random-connection model.
+Using partition diagrams and dependency graph arguments, we start by showing in
+Proposition 3.3 that the (virtual) cumulants of a random functional admitting a certain con-
+nectedness factorization property (3.1) can be expressed as sums over connected partition
+diagrams, generalizing Lemma 2 in [MM91]. A related result has been obtained in [Jan19]
+in the particular case of two-parameter Poisson stochastic integrals, in relation to cluster
+expansions for Gibbs point processes in statistical mechanics. In Proposition 4.3, we apply
+Proposition 3.3 to express the cumulants of multiparameter stochastic integrals, for which
+this factorization property can be checked from the moment formulas for multiparameter
+stochastic integrals computed in Proposition 4.1.
+Such expressions allow us to determine the dominant terms in the growth of cumulants
+as the intensity λ of the underlying point process tends to infinity, by estimating the counts
+of vertices and edges in connected partition diagrams as in [Kho08]. We work under a mild
+condition (6.3) which is satisfied by e.g. any translation-invariant continuous connection
+function H : Rd × Rd → [0, 1] non vanishing at 0, such as the Rayleigh connection function
+given by H(x, y) = e−β∥x−y∥2, x, y ∈ Rd, for some β > 0.
+For our analysis of cumulant behavior we identify the leading terms in the sum (5.3)
+over connected partition diagrams. When G is a connected graph with r := |V (G)| vertices,
+satisfying Assumption 6.1 in the dilute regime (6.1) where λ−1 ≪ cλ ≤ 1, the dominant
+terms correspond to connected partition diagrams with the highest number of blocks, as
+found in [Pri22] in the case of k-hop counting in the one-dimensional random-connection
+model. In Theorem 6.1 this yields the cumulant bounds
+(n − 1)!cn|E(G)|
+λ
+(K1λ)1+(r−1)n ≤ κn(NG) ≤ n!r−1cn|E(G)|
+λ
+(K2λ)1+(r−1)n,
+λ > 0,
+for some constants K1, K2 > 0 independent of λ, n ≥ 1. From the Statuleviˇcius condition
+(A.1) below, see [RSS78, DJS22], we deduce the Kolmogorov distance bound
+sup
+x∈R
+��P
+� �
+NG ≤ x
+�
+− P(Z ≤ x)
+�� ≤
+C
+λ1/(4r−6) ,
+λ → ∞,
+3
+
+see Corollary 7.1, and a moderate deviation principle by Theorem 1.1 of [DE13].
+In the sparse regime (6.2) where cλ ≤ λ−α for some α ≥ 1, the maximal rate λα−(α−1)r is
+attained for G a tree-like graph, and in Theorem 6.2 we obtain the cumulant bounds
+(K1)(r−1)nλα−(α−1)r ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλα−(α−1)r,
+λ > 0,
+if G is a tree, and
+(K1)rλr−α|E(G)| ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλr−α|E(G)|,
+λ > 0,
+if G is a not a tree, such as e.g.
+a cycle graph.
+As a consequence of the Statuleviˇcius
+condition (A.1), when G is a tree we find the Kolmogorov distance bound
+sup
+x∈R
+��P
+� �
+NG ≤ x
+�
+− P(Z ≤ x)
+�� ≤ Cλ−(α−(α−1)r)/(4r−6),
+λ → ∞,
+provided that 1 ≤ α < r/(r − 1), see Corollary 7.2.
+Convergence rates in the Kolmogorov distances may be improved into classical Berry-
+Esseen rates when the connection function H(x, y) is {0, 1}-valued, e.g.
+in disk models
+as in [Pri22], by representing subgraph counts as multiple Poisson stochastic integrals and
+using the fourth moment theorem for U-statistics and sums of multiple stochastic integrals
+Corollary 4.10 in [ET14], see also Theorem 3 in [LRR16] or Theorem 6.3 in [PS22] for
+Hoeffding decompositions. In the general case where H(x, y) is [0, 1]-valued this method no
+longer applies, this is why we rely on the Statuleviˇcius condition which in turn may yield
+suboptimal convergence rates.
+The paper is organized as follows. Sections 2 and 3 introduce the preliminary frame-
+work and notations on connected partition diagrams and combinatorics of virtual cumulants
+that will be used for the expression of cumulants of multiparameter stochastic integrals in
+Section 4 and for subgraph counts in Section 5. Those expressions are applied in Section 6
+to derive cumulant growth rates in the random-connection model, with application to Kol-
+mogorov rates in subgraph counting via the Statuleviˇcius condition in Section 7.
+2
+Set partitions and diagram connectivity
+Given η a finite set, we denote by Π(η) the collection of its set partitions, and we let |σ|
+denote the number of blocks in any partition σ ∈ Π(η). Given ρ, σ two set partitions, we
+4
+
+say that σ is coarser than ρ, or that ρ is finer than σ, and we write ρ ⪯ σ, if every block
+in σ is a combination of blocks in ρ. We also denote by ρ ∨ σ the finest partition which is
+coarser than ρ and σ, and by ρ∧σ the coarsest partition that is finer than ρ and σ. We let �0
+be the finest partition, which is made of a single element in each block, and we let �1 be the
+coarsest (one-block) partition. In general, given any graph G we denote by V (G) the set of
+its vertices, and by E(G) the set of its edges.
+Our study of cumulants and moments of functionals of random fields relies on partition
+diagrams, see [MM91, Kho08, PT11] and references therein for additional background. In
+what follows we let [n] := {1, 2, . . . , n} for n ≥ 1.
+Definition 2.1 Let n, r ≥ 1.
+1. Given η ⊂ [n] we let Π(η × [r]) denote the set of all partitions of the set
+η × [r] :=
+�
+(k, l) : k ∈ η, l = 1, . . . , r
+�
+.
+2. We also let πη := (πi)i∈η ∈ Π(η × [r]) denote the partition made of the |η| blocks of size r
+given by
+πk := {(k, 1), . . . , (k, r)},
+k ∈ η.
+Next, we introduce the definition of partition diagrams.
+Definition 2.2 Let n, r ≥ 1. Given η ⊂ [n] and ρ ∈ Π(η × [r]) a partition of η × [r], we
+denote by Γ(ρ, πη) the diagram, or graphical representation of the partition ρ, constructed
+by:
+1. arranging the elements of η × [r] into an array of |η| rows and r columns, and
+2. adding edges connecting neighbors within a same block in ρ.
+In addition, we say that the partition diagram Γ(ρ, π) is connected when ρ ∨ πη = �1.
+For example, taking η := {2, 3, 5, 8, 10}, given the partitions
+ρ =
+�
+{(2, 1), (3, 1), (3, 2), (3, 3)}, {(2, 2), (2, 3), (2, 4), (3, 4)}, {(5, 1)}, {(5, 2), (8, 2)},
+{(5, 3)}, {(5, 4), (8, 3)}, {(8, 1), (10, 1)}, {(8, 4)}, {(10, 2), (10, 3), (10, 4)}
+�
+and
+σ =
+�
+{(2, 1), (3, 1)}, {(2, 2)}, {(2, 3), (3, 4)}, {(2, 4)}, {(3, 2), (5, 2), (8, 2)},
+5
+
+{(3, 3), (5, 4), (8, 3), (10, 2)}, {(5, 1)}, {(5, 3)}, {(8, 1), (10, 1)}, {(8, 4)}, {(10, 3)}, {(10, 4)}
+�
+,
+of η × [4], Figure 1−a) presents an example of a non-connected partition diagram Γ(ρ, π),
+and Figure 1−b) presents an example of a connected partition diagram Γ(σ, π),
+2
+3
+5
+8
+10
+1
+2
+3
+4
+(a) Non-connected partition diagram Γ(ρ, π).
+2
+3
+5
+8
+10
+1
+2
+3
+4
+(b) Connected partition diagram Γ(σ, π).
+Figure 1: Two examples of partition diagrams with η = {2, 3, 5, 8, 10}, n = 10, r = 4.
+Note that the above notion of connected partition diagram is distinct from that of irreducible
+partition, see, e.g., [BOR85].
+Definition 2.3 Let n ≥ 1, G a connected graph with |V (G)| = r ≥ 1 vertices, and consider
+G1, . . . , Gn copies of G respectively built on π1, . . . , πn. Let also ρ ∈ Π(η × [r]) be a partition
+of η × [r].
+1. We let �ρG be the multigraph constructed on the blocks of ρ by adding an edge between two
+blocks ρ1, ρ2 of the partition ρ whenever there exists (k, l1) ∈ ρ1 and (k, l2) ∈ ρ2 such that
+(l1, l2) is an edge in Gk.
+2. We let ρG be the graph constructed on the blocks of ρ by removing redundant edges in �ρG,
+so that at most one edge remains between any two blocks ρ1, ρ2 ∈ ρ.
+Figure 2-b) presents an illustration of the multigraph �ρG and graph ρG on the blocks of ρ
+when G is the line graph {(1, 2), (2, 4), (3, 4)} on {1, 2, 3, 4}.
+6
+
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) and multigraph �ρG in blue.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(b) Diagram Γ(ρ, π) and graph ρG in red.
+Figure 2: Diagram and graphs G, ρG, �ρG with n = 5, r = 4.
+Definition 2.4 Let n, r ≥ 1, and let ρ ∈ Π([n] × [r]) be a partition of [n] × [r].
+1. For b ⊂ [n], we let ρb ⊂ ρ be defined as
+ρb := {c ∈ ρ : c ⊂ b × [r]}.
+2. Given η ⊂ [n] we split any partition ρ of η × [r] into the equivalence classes deduced from
+the connected components of the diagram ρG, as
+ρ =
+�
+b×[r]∈ρ∨π
+b⊂[n]
+ρb,
+(2.1)
+As an example, in Figure 3-a), when b = {1, 2} we have
+ρ{1,2} =
+�
+{(1, 1), (2, 1), (2, 2), (2, 3)}, {(1, 2), (1, 3), (1, 4), (2, 4)}
+�
+,
+and the partition (2.1) is illustrated in Figure 3-b) with b1 = {1, 2} and b2 = {3, 4, 5}.
+7
+
+1
+2
+3
+4
+5
+1
+2
+3
+4
+ρ{1,2}
+(a) Diagram Γ(ρ, π) and block ρ{1,2}.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+ρb1
+ρb2
+(b) Splitting {ρb1, ρb2} of ρ according to ρG.
+Figure 3: Splitting of the partition ρ with ρ ∨ π = {π1 ∪ π2, π3 ∪ π4 ∪ π5} and n = 5, r = 4.
+Definition 2.5 Let n, r ≥ 1. Given σ ∈ Π([n]) a partition of [n], we let Πσ([n]×[r]) denote
+the set of partitions ρ of [n] × [r] such that
+ρ ∨ π = {b × [r] : b ∈ σ},
+and we partition Π([n] × [r]) as
+Π([n] × [r]) =
+�
+σ∈Π([n])
+Πσ([n] × [r]).
+(2.2)
+We note that given η ⊂ [n], the set Π�1(η × [r]) consists of the partitions ρ of η × [r] for
+which the diagram ρG is connected, as in Figure 4. In what follows, we also will use non-flat
+partition diagrams Γ(ρ, π) such that ρ ∧ π = �0, see Chapter 4 of [PT11] and Figure 4.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) and multigraph �ρG in blue.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(b) Diagram Γ(ρ, π) and graph ρG in red.
+Figure 4: Connected non-flat partition diagram with G a cycle graph and n = 5, r = 4.
+8
+
+Lemma 2.6 a) Let n, r ≥ 1. The cardinality of the set
+C(n, r) := {ρ ∈ Π�1([n] × [r]) : ρ ∧ π = �0}
+of connected non-flat partition diagrams on [n] × [r] satisfies
+|C(n, r)| ≤ n!r−1rn−1r!n−1,
+n ≥ 1.
+(2.3)
+b) Let n ≥ 1 and r ≥ 2. The cardinality of the set
+Mn := {ρ ∈ C(n, r) : |ρ| = 1 + (r − 1)n}
+of maximal connected non-flat partition diagrams on [n] × [r] satisfies
+((r − 1)r)n−1(n − 1)! ≤ |Mn| ≤ ((r − 1)r)n−1n!2,
+n ≥ 1.
+(2.4)
+Proof.
+a) We have |C(1, r)| = 1. Given a connected partition diagram Γ(ρ, π) in C(n+1, r),
+we construct a connected undirected graph �ρ on [n + 1] as in Figure 5-a), and note that
+�ρ contains a spanning tree ρ, see e.g. Theorem 4.2.3 in [BR12], as shown in Figure 5-b).
+In addition, the tree ρ has at most r leaves, because after removing any of root of ρ, the
+remaining partition can be reconnected using no more than r vertices from the root. Then,
+starting for any leaf in the tree ρ, ρ must be made from a connected partition diagram
+in C(n, r), completed by a choice of at most (n + 1)r−1r! allocations of r − 1 vertices into
+existing or new blocks. Indeed, note that at least one out of r vertices in the leaf is used for
+an existing connection.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) and graph �ρ.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(b) Diagram Γ(ρ, π) and spanning tree ρ.
+Figure 5: Example of graph �ρ and its spanning tree subgraph.
+9
+
+This yields the induction inequality
+|C(n + 1, r)| ≤ r(n + 1)r−1r!|C(n, r)|,
+from which we conclude to (2.3).
+b) Proceeding similarly to part (a), we have |M1| = 1 and the recursion
+r × (1 + (r − 1)n) × |Mn| ≤ |Mn+1| ≤ (n + 1)r × (1 + (r − 1)n) × |Mn|,
+n ≥ 1,
+which yields
+((r − 1)r)n−1
+n−1
+�
+i=1
+�
+i +
+1
+r − 1
+�
+≤ |Mn| ≤ n!((r − 1)r)n−1
+n−1
+�
+i=1
+�
+i +
+1
+r − 1
+�
+,
+n ≥ 1,
+from which (2.4) follows.
+□
+3
+Virtual cumulants
+The following definition uses the concept of independence of a virtual field with respect to
+graph connectedness, see Relation (17) in [MM91, p. 34].
+Definition 3.1 Let n, r ≥ 1. We say that a mapping F defined on partitions of [n] × [r]
+admits the connectedness factorization property if it decomposes according to the partition
+(2.1) as
+F(ρ) =
+�
+b×[r]∈ρ∨π
+F(ρb),
+ρ ∈ Π([n] × [r]).
+(3.1)
+In what follows, given F a mapping defined on the partitions of [n] × [r], we will use the
+M¨obius transform �F of F, defined as
+�F(η) :=
+�
+ρ∈Π(η×[r])
+F(ρ),
+η ⊂ [n],
+with �F(∅) := 0, see [Rot64] and § 2.5 of [PT11]. We refer to [MM91, p. 33] for the following
+definition.
+Definition 3.2 Let n, r ≥ 1. The virtual cumulant G of a mapping F on �
+η⊂[n] Π(η × [r])
+is defined by letting CF(η) := �F(η) when |η| = 1, and then recursively by
+CF(η) := �F(η) −
+�
+σ∈Π(η)
+|σ|≥2
+�
+b∈σ
+CF(b),
+η ⊂ [n],
+|η| ≥ 2.
+(3.2)
+10
+
+In the particular case r = 1, we note that when (X1, . . . , Xn) is a sequence of random
+variables, letting
+F(ρ) := E
+��
+b∈ρ
+�
+i∈b
+Xi
+�
+= E
+� n
+�
+i=1
+Xi
+�
+,
+Relation (3.2) shows that
+CF(η) =
+�
+σ∈Π[η]
+(−1)|σ|−1(|σ| − 1)!
+�
+b∈σ
+F({b}) =
+�
+σ∈Π[η]
+(−1)|σ|−1(|σ| − 1)!
+�
+b∈ρ
+E
+��
+i∈b
+Xi
+�
+,
+coincides with the actual joint cumulant of (Xi)t∈η, η ⊂ [n].
+The following proposition is an extension of the classical Lemma 2 in [MM91, p. 34], see
+also Lemma 3.1 in [Kho08].
+Proposition 3.3 Let n, r ≥ 1. Let F be a mapping defined on �
+η⊂[n] Π(η × [r]) and admit-
+ting the connectedness factorization property (3.1). Then, for η ⊂ [n] with η ̸= ∅, the virtual
+cumulant of F is given by the sum
+CF(η) =
+�
+σ∈Π�1(η×[r])
+(connected)
+F(σ)
+(3.3)
+over connected partition diagrams on η × [r].
+Proof.
+The claim is true when |η| = 1. Assume that it is true for all η ⊂ [n] for some n ≥ 1,
+and let η be such that |η| = n + 1. By (2.2) and (3.1), we have
+�F(η)
+=
+�
+ρ∈Π(η×[r])
+F(ρ)
+=
+�
+σ∈Π(η)
+�
+ρ∈Πσ(η×[r])
+F(ρ)
+=
+�
+σ∈Π(η)
+�
+ρ∈Πσ(η×[r])
+�
+b∈σ
+F(ρb)
+=
+�
+σ∈Π(η)
+�
+b∈σ
+�
+ρ∈Π�1(b×[r])
+(connected)
+F(ρ)
+=
+�
+ρ∈Π�1(η×[r])
+(connected)
+F(ρ) +
+�
+σ∈Π(η)
+|σ|≥2
+�
+b∈σ
+CF(b),
+where the last equality follows from the induction hypothesis (3.3) when |η| ≤ n. The proof
+is completed by subtracting the last term on both sides.
+□
+11
+
+In the particular case r = 1, we note that when (X1, . . . , Xn) is a sequence of independent
+random variables, the functional
+F(ρ) := E
+��
+b∈ρ
+�
+i∈b
+Xi
+�
+=
+�
+i∈[n]
+E[Xi]
+satisfies the connectedness factorization property (3.1), and Proposition 3.3 recovers the
+vanishing of the joint cumulants of (Xi)i∈η when |η| ≥ 2, as the set Π�1(η × [1]) of connected
+partition diagrams on η × [1] is empty in this case.
+4
+Cumulants of multiparameter stochastic integrals
+Consider a Poisson point process Ξ on Rd, d ≥ 1, with intensity measure Λ on Rd, constructed
+on the space
+Ω =
+�
+ω = {xi}i∈I ⊂ Rd : #(A ∩ ω) < ∞ for all compact A ∈ B(Rd)
+�
+of locally finite configurations on Rd, whose elements ω ∈ Ω are identified with the Radon
+point measures ω =
+�
+x∈ω
+ǫx, where ǫx denotes the Dirac measure at x ∈ Rd.
+By [LP18,
+Corollary 6.5], almost every element ω of Ω can be represented as ω = {Vi}1≤i≤N, where
+(Vi)i≥1 is a random sequence in Rd and a N ∪ {∞}-valued random variable N.
+In this section, using sums over partitions we express the moments of the multiparameter
+stochastic integral
+�
+V1,...,Vr∈Ξ
+uG(V1, . . . , Vr) =
+�
+(Rd)r uG(x1, . . . , xr)ω(dx1) · · · ω(dxr),
+(4.1)
+where uG(x1, . . . , xr) is a measurable process of the form
+uG(x1, . . . , xr) :=
+�
+(i,j)∈E(G)
+vi,j(xi, xj),
+and vi,j(x, y), (i, j) ∈ E(G), are random processes v(x, y) independent of the underlying
+Poisson point process Ξ. The next proposition is a consequence of Proposition 2 in [Pri19],
+which relies on Proposition 3.1 of [Pri12] and Lemma 2.1 of [BRSW17].
+Proposition 4.1 Let n ≥ 1 and r ≥ 2. The n-th moment of the multiparameter stochastic
+integral (4.1) is given by the summation
+�
+ρ∈Π([n]×[r])
+�
+(Rd)|ρ| E
+
+
+n
+�
+k=1
+�
+(i,j)∈E(Gk)
+v
+�
+xρ
+k,i, xρ
+k,j
+�
+
+
+�
+η∈V (ρG)
+Λ(dxη),
+(4.2)
+12
+
+where we let xρ
+k,l := xη whenever (k, l) ∈ η, for ρ ∈ Π([n] × [r]) and η ∈ ρ.
+The next proposition rewrites the product in (4.2) as a product on the edges of the graph ρG
+similarly to Proposition 4 of [Pri19] when v(x, y) vanishes on the diagonal, and it generalizes
+Proposition 2.4 of [Jan19] from two-parameter Poisson stochastic integrals to multiparameter
+integrals of higher orders.
+Proposition 4.2 Let n ≥ 1, r ≥ 2, and assume that the process v(x, y) vanishes on diag-
+onals, i.e. v(x, x) = 0, x ∈ Rd. Then, the n-th moment of the multiparameter stochastic
+integral (4.1) is given by the summation
+�
+ρ∈Π([n]×[r])
+ρ∧π=�0
+(non−flat)
+�
+(Rd)|ρ|
+�
+(η1,η2)∈E(ρG)
+E
+�
+v(xη1, xη2)m(η1,η2)�
+�
+η∈V (ρG)
+Λ(dxη),
+over connected non-flat diagrams, where m(η1, η2) represents the multiplicity of the edge
+(η1, η2) in the multigraph �ρG.
+The next proposition is a consequence of Propositions 3.3 and 4.2, and it also extends
+Proposition 2.5 of [Jan19] from the two-parameter case to the multiparameter case. Note
+that in our setting, the two-parameter case only applies to the edge counting.
+Proposition 4.3 Let n ≥ 1, r ≥ 2, and assume that the process v(x, y) vanishes on diag-
+onals, i.e. v(x, x) = 0, x ∈ Rd. Then, the n-th cumulant of the multiparameter stochastic
+integral (4.1) is given by the summation
+�
+ρ∈Π�1([n]×[r])
+ρ∧π=�0
+(non−flat connected)
+�
+(Rd)|ρ|
+�
+(η1,η2)∈E(ρG)
+E
+�
+v(xη1, xη2)m(η1,η2)�
+�
+η∈V (ρG)
+Λ(dxη)
+(4.3)
+over connected non-flat partition diagrams.
+Proof.
+The functional
+F(ρ) :=
+�
+ρ∈Π([n]×[r])
+ρ∧π=�0
+(non−flat)
+�
+(Rd)|ρ|
+�
+(η1,η2)∈E(ρG)
+E
+�
+v(xη1, xη2)m(η1,η2)�
+�
+η∈V (ρG)
+Λ(dxη)
+satisfies the connectedness factorization property (3.1), as for σ = b × [r] ∈ ρ ∨ π and
+σ′ = b′ ×[r] ∈ ρ∨π with b ̸= b′, the variables (xη)η∈ρb are distinct from the variables (xη)η∈ρb′
+in the above integration. Hence, (4.3) follows from Proposition 3.3.
+□
+13
+
+5
+Cumulants of subgraph counts
+Let H : Rd × Rd → [0, 1] denote a measurable connection function such that
+0 <
+�
+Rd H(x, y)Λ(dx) < ∞,
+for all y ∈ R. Given ω ∈ Ω, for any x, y ∈ ω with x ̸= y, an edge connecting x and y
+is added with probability H(x, y), independently of the other pairs, and in this case we
+write x ↔ y. The resulting random graph, together with the point process Ξ, is called the
+random-connection model and denoted by GH(Ξ).
+In the case where the connection function H is given by H(x, y) := 1{∥x−y∥≤R} for some
+R > 0, the resulting graph is completely determined by the geometric of the underlying
+point process Ξ, and is called a random geometric graph, which is included as a special case
+in this paper.
+Given G a connected graph with |V (G)| = r vertices, we denote NG the count of sub-
+graphs isomorphic to G in the random-connection model GH(Ξ), which can be represented
+as the multiparameter stochastic integral
+NG :=
+�
+V1,...,Vr∈Ξ
+�
+(i,j)∈E(G)
+1{Vi↔Vj} =
+�
+(Rd)r
+�
+(i,j)∈E(G)
+1{xi↔xj} ω(dx1) · · ·ω(dxr),
+up to division by the number of automorphisms of G. Here, we have 1{Vi↔Vj} = 1 or 0
+depending whether Vi and Vj are connected or not by an edge in GH(Ξ), with
+1{x↔x} = 0,
+x ∈ Rd.
+(5.1)
+The following result is a direct consequence of Proposition 4.3 by taking v(x, y) := 1{x↔y}
+in (4.3) and by using non-flat partition diagrams Γ(ρ, π) such that ρ ∧ π = �0, to take into
+account condition (5.1).
+Proposition 5.1 Let n ≥ 1 and r ≥ 2. The moments and cumulants of NG are given by
+the summation
+E[(NG)n] =
+�
+ρ∈Π([n]×[r])
+ρ∧π=�0
+(non−flat)
+�
+(Rd)|��|
+�
+�
+(η1,η2)∈E(ρG)
+H(xη1, xη2)
+�
+�
+η∈V (ρG)
+Λ(dxη),
+(5.2)
+over non-flat partition diagrams, and by the summation
+κn(NG) =
+�
+ρ∈Π�1([n]×[r])
+ρ∧π=�0
+(non−flat connected)
+�
+(Rd)|ρ|
+�
+�
+(η1,η2)∈E(ρG)
+H(xη1, xη2)
+�
+�
+η∈V (ρG)
+Λ(dxη),
+(5.3)
+14
+
+over connected non-flat partition diagrams.
+Proof.
+Relations (5.2)-(5.3) are consequence of Proposition 4.3, after taking vi,j(xi, xj) :=
+1{xi↔xj}, (i, j) ∈ E(G). The summations are restricted to non-flat partition diagrams due
+to condition (5.1) as in Section 2 of [Pri19].
+□
+6
+Asymptotic growth of subgraph count cumulants
+We assume that the intensity measure of the Poisson point process Ξ on Rd has the form
+Λλ(dx) = λµ(dx),
+λ > 0,
+where µ is a finite diffuse measure on Rd. We investigate the asymptotic behaviour of the
+cumulants κn(NG) as the intensity λ tends to infinity, as a consequence of the partition
+diagram representation of cumulant. For this, we consider the subgraph count in GH(Ξ)
+obtained by replacing H(x, y) with Hλ(x, y) := cλH(x, y), in which case every term in (5.3)
+contributes a factor c|E(ρG)|
+λ
+λ|V (ρG)|.
+In what follows, given two positive functions f and g on (1, ∞) we write f(λ) ≪ g(λ) if
+limλ→∞ g(λ)/f(λ) = ∞, and we consider the following regimes.
+• Dilute regime: for some constant K > 0 we have
+1
+λ ≪ cλ ≤ K,
+λ → ∞.
+(6.1)
+• Sparse regime: for some constants K > 0 and α ≥ 1 we have
+cλ ≤ K
+λα,
+λ → ∞.
+(6.2)
+In case cλ = K for all λ > 0 we also say that we are in the full random graph regime, and
+in the sequel we take K = 1 for simplicity.
+Assumption 6.1 Let r ≥ 2. There exist two constants c, C > 0 such that for any connected
+non-flat partition diagram Γ(ρ, π), ρ ∈ Π�1([n] × [r]), n ≥ 1, we have
+c|E(ρG)|C|V (ρG)| ≤
+�
+Rd · · ·
+�
+Rd
+�
+�
+(i,j)∈E(ρG)
+H(xi, xj)
+�
+�
+k∈V (ρG)
+µ(dxk).
+(6.3)
+15
+
+We note that (6.3) is satisfied by e.g. any translation-invariant continuous kernel function
+H : Rd×Rd → [0, 1] non vanishing at 0, including the standard Rayleigh connection function
+given by H(x, y) = e−β∥x−y∥2, x, y ∈ Rd, for some β > 0. Indeed, for those kernels there
+exists c > 0 and a Borel set B ⊂ Rd such that µ(B) > 0 and
+H(x, y) = H(x − y, 0) ≥ c1B(x)1B(y),
+x, y ∈ Rd,
+hence
+c|E(ρG)|(µ(B))|V (ρG)|
+=
+c|E(ρG)|
+�
+B
+· · ·
+�
+B
+�
+k∈V (ρG)
+µ(dxk)
+≤
+�
+Rd · · ·
+�
+Rd
+�
+�
+(i,j)∈E(ρG)
+H(xi, xj)
+�
+�
+k∈V (ρG)
+µ(dxk).
+In what follows, we consider the centered and normalized subgraph count cumulants defined
+as
+�
+NG := NG − κ1(NG)
+�
+κ2(NG)
+,
+n ≥ 1.
+The following result shows that for n ≥ 3 the normalized cumulant κn( �NG) tends to zero
+in (6.5), hence �
+NG converges in distribution to the normal distribution by Theorem 1 in
+[Jan88].
+Theorem 6.1 (Dilute regime) Let r ≥ 2 and consider G a connected graph with |V (G)| =
+r vertices, satisfying Assumption 6.1 in the dilute regime (6.1). We have the cumulant bounds
+(n − 1)!cn|E(G)|
+λ
+(K1λ)1+(r−1)n ≤ κn(NG) ≤ n!r−1cn|E(G)|
+λ
+(K2λ)1+(r−1)n
+(6.4)
+for some constants K1, K2 > 0 independent of λ, n ≥ 1, and
+��κn
+� �NG
+��� ≤ n!r−1(Kλ)−(n/2−1),
+λ ≥ 1,
+n ≥ 2,
+(6.5)
+where K > 0 is a constant independent of λ > 0 and n ≥ 1.
+Proof.
+We identify the leading terms in the sum (5.3) over connected partition diagrams,
+knowing that every vertex in ρG contributes a factor λ, and that every edge contributes a
+factor cλ, therefore every summand in (5.3) contributes a factor c|E(ρG)|
+λ
+λ|V (ρG)|.
+Modifying ρ ∈ Π�1([n] × [r]) by splitting a block in two means adding a vertex to ρG, and
+therefore a adding factor λ to the corresponding term in (5.3). At the same time, this entails
+16
+
+no loss of edge but possibly the addition of an edge to ρG, which results into an additional
+factor cλ with λcλ ≫ 1 by (6.1). Hence, the leading terms in (5.3) are those associated with
+the connected partition diagrams Γ(ρ, π) having the highest block count, i.e. which have
+1 + (r − 1)n blocks, see Figure 6 for a sample of such partition diagram.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) and graph �ρG in blue.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(b) Diagram Γ(ρ, π) and graph ρG in red.
+Figure 6: Example of maximal connected partition diagram with n = 5 and r = 4.
+We note that any maximal partition ρ satisfies |E(ρG)| = n × |E(G)|, as can be checked in
+Figure 6. Therefore, by (2.3)-(2.4), (5.3) and (6.3), we obtain
+cn|E(G)|C1+(r−1)ncn|E(G)|
+λ
+((r − 1)r)n−1(n − 1)!λ1+(r−1)n
+≤
+λ1+(r−1)ncn|E(G)|
+λ
+�
+ρ∈Mn
+�
+(Rd)1+(r−1)n
+�
+�
+(η1,η2)∈E(ρG)
+Hλ(xη1, xη2)
+�
+�
+η∈V (ρG)
+µ(dxη),
+≤
+κn(NG)
+≤
+n!r−1rn−1r!n−1(µ(Rd))1+(r−1)ncn|E(G)|
+λ
+λ1+(r−1)n,
+which yields (6.4). Regarding (6.5), we have, for n ≥ 2,
+��κn( �NG)
+�� ≤
+n!r−1cn|E(G)|
+λ
+(K2λ)1+(r−1)n
+�
+(2 − 1)!c2|E(G)|
+λ
+(K1λ)1+2(r−1)�n/2 = K2
+�(K2/K1)r−1
+√K1
+�n
+n!r−1λ−(n/2−1).
+□
+The following result yields a positive cumulant growth of order α − (α − 1)r > 0 in (6.6) for
+trees in the sparse regime with α ∈ [1, r/(r − 1)), while in the case of non-tree graphs such
+as cycle graphs the growth rate r − α|E(G)| ≤ (1 − α)r ≤ 0 is negative or zero in (6.8) and
+(6.10). In addition, the normalized cumulant κn( �NG) tends to zero for n ≥ 3 in (6.7) only
+17
+
+when G is a tree, in which case �
+NG converges in distribution to the normal distribution by
+Theorem 1 in [Jan88]. We note that when α = 1, (6.7) is consistent with (6.5).
+Theorem 6.2 (Sparse regime) Let G be a connected graph with |V (G)| = r vertices,
+r ≥ 2, satisfying Assumption 6.1 in the sparse regime (6.2).
+a) If G is a tree, i.e. |E(G)| = r − 1, we have the cumulant bounds
+(K1)(r−1)nλα−(α−1)r ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλα−(α−1)r,
+(6.6)
+for some constants K1 > 0, K2 > 1 independent of λ, n ≥ 1, and
+��κn( �
+NG)
+�� ≤ (K3)nn!r−1λ−(α−(α−1)r)(n/2−1),
+λ ≥ 1,
+n ≥ 2,
+(6.7)
+where K3 := (K2/K1)r−1.
+b) If G is not a tree, i.e. |E(G)| ≥ r, we have the cumulant bounds
+(K1)rλr−α|E(G)| ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλr−α|E(G)|,
+(6.8)
+for some constants K1 > 0, K2 > 1 independent of λ, n ≥ 1, and
+��κn( �
+NG)
+�� ≤ n!r−1(K3)nλ(α|E(G)|−r)(n/2−1),
+λ ≥ 1,
+n ≥ 2,
+(6.9)
+for some K3 > 0.
+c) If G is a cycle, i.e. |E(G)| = r, we have the cumulant bounds
+(K1)rλ−(α−1)r ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλ−(α−1)r,
+(6.10)
+for some constants K1 > 0, K2 > 1 independent of λ, n ≥ 1, and
+��κn( �
+NG)
+�� ≤ n!r−1(K3)nλ(α−1)(n/2−1)r,
+λ ≥ 1,
+n ≥ 2,
+(6.11)
+for some K3 > 0.
+Proof.
+In the sparse regime (6.2), every edge in the graph ρG contributes a power λ−α and
+every vertex contributes a power λ, hence every term in (5.3) contributes a power
+λ|V (ρG)|−α|E(ρG)| = λα−(α−1)|V (ρG)|+(|V (ρG)|−|E(ρG)|−1)α ≤ λα−(α−1)|V (ρG)|
+(6.12)
+18
+
+since |V (ρG)| − |E(ρG)| − 1 ≤ 0. In addition, for any connected partition diagram Γ(ρ, π)
+with ρ ∈ Π�1([n] × [r]), we have
+r ≤ |V (ρG)| ≤ 1 + (r − 1)n.
+a) When G is a tree and the graph ρG is also a tree, i.e. |V (ρG)| − |E(ρG)| − 1 = 0, the
+maximal order λα−(α−1)|V (ρG)| is attained in (6.12), see Figure 7 for an example.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) and multigraph �ρG in blue.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(b) Diagram Γ(σ, π) and graph ρG in red.
+Figure 7: Example of connected partition diagram with ρG a tree and n = 5, r = 4.
+In this case, the corresponding term in (5.3) contributes a power
+λ|V (ρG)|−α|E(ρG)| = λα−(α−1)|V (ρG)|,
+λ ≥ 1.
+In this case, since |V (ρG)| ≥ r and α ≥ 1, the optimal rate λα−(α−1)r is attained by the
+partition diagrams Γ(ρ, π) such that |V (ρG)| = r, as illustrated in Figure 8.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) and multigraph �ρG in blue.
+1
+2
+3
+4
+5
+1
+2
+3
+4
+(b) Diagram Γ(ρ, π) and graph ρG in red.
+Figure 8: Tree diagram ρG with G a tree with |V (ρG)| = r and n = 5, r = 4.
+19
+
+We conclude to (6.6) using Lemma 2.6 as in the proof of Theorem 6.1, by upper bounding
+the count of connected partitions from (2.3). Regarding (6.7), we have
+��κn( �
+NG)
+��
+≤
+n!r−1K(r−1)n
+2
+((K1)2(r−1)λα−(α−1)r)n/2λα−(α−1)r
+=
+�K2
+K1
+�(r−1)n
+n!r−1λ−(α−(α−1)r)(n/2−1),
+n ≥ 2.
+b) When G is not a tree it contains at least one cycle, and for any partition ρ ∈ Π�1([n] × [r])
+the same holds for the graph ρG. In this case, the highest order contribution in (5.3) is
+attained by connected non-flat partition diagrams Γ(ρ, π), ρ ∈ Π�1([n] × [r]), such that ρG
+has |V (ρG)| = r vertices, and their contribution is given by a power of order λr−α|E(G)|. An
+example of such partition diagram ρ is given in Figure 9, with G a cycle.
+1
+2
+3
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) and multigraph �ρG in blue.
+1
+2
+3
+1
+2
+3
+4
+(b) Diagram Γ(σ, π) and graph ρG in red.
+Figure 9: Cycle diagram ρG with G a cycle graph and n = 5, r = 4.
+Indeed, in order to remain non flat, the partition diagram ρ can only be modified into a
+partition diagram σ by splitting a block of ρG in two, which entails the addition of a number
+q of edges, q ≥ 1, resulting into an additional factor λ1−qα ≤ 1 that may only lower the order
+of the contribution, see Figures 10-13 for an example where G is a graph with one cycle.
+1
+2
+3
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) with order λ4−4α.
+1
+2
+3
+1
+2
+3
+4
+(b) Diagram Γ(σ, π) with order λ5−5α = λ4−4αλ−(α−1).
+Figure 10: Splitting of a vertex with addition of one edge and n = 3, r = 4.
+20
+
+1
+2
+3
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) with order λ4−4α.
+1
+2
+3
+1
+2
+3
+4
+(b) Diagram Γ(σ, π) with order λ5−6α = λ4−4αλ1−2α.
+Figure 11: Splitting of a vertex with addition of three edges and n = 3, r = 4.
+1
+2
+3
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) with order λ4−4α.
+1
+2
+3
+1
+2
+3
+4
+(b) Diagram Γ(σ, π) with order λ5−6α = λ4−4αλ1−2α.
+Figure 12: Splitting of a vertex with addition of two edges and n = 3, r = 4.
+1
+2
+3
+1
+2
+3
+4
+(a) Diagram Γ(ρ, π) with order λ4−4α.
+1
+2
+3
+1
+2
+3
+4
+(b) Diagram Γ(σ, π) with order λ5−6α = λ4−4αλ1−2α.
+Figure 13: Splitting of a vertex with addition of two edges and n = 3, r = 4.
+When G is a triangle with n = 2 and r = 3, the above procedure can be reversed by first
+merging a vertex and then gluing edges, see Figure 14, which results into “overlapping” all
+copies of the graph G.
+21
+
+(a) Merging one vertex.
+(b) Gluing one edge.
+(c) Gluing three edges.
+Figure 14: Diagram patterns with G a triangle and n = 2, r = 3.
+As in part-(b) above, we lower bound κn(NG) using a single partition, and we upper bound
+using the total count of connected non-flat partition diagrams using Lemma 2.6-b) to obtain
+(6.8). Regarding (6.9), we have
+��κn( �
+NG)
+�� ≤ n!r−1(K2)(r−1)nλr−α|E(G)|
+((K1)rλr−α|E(G)|)n/2
+= n!r−1(K2)(r−1)n
+(K1)nr/2 λ−(r−α|E(G)|)(n/2−1),
+n ≥ 2.
+c) is a direct consequence of part b) above.
+□
+7
+Asymptotic normality of subgraph counts
+The cumulant bound (6.5) shows that the centered and normalized subgraph count �
+NG
+satisfies the Statuleviˇcius condition (A.1) below, see [RSS78, DJS22], with γ := r − 2. As a
+consequence, we have the following result, in which the Berry-Esseen rate is obtained when
+r = 2.
+Corollary 7.1 (Dilute regime) Let G be a connected graph with |V (G)| = r vertices,
+r ≥ 2, satisfying Assumption 6.1 in the dilute regime (6.1).
+We have the Kolmogorov
+distance bound
+sup
+x∈R
+��P
+� �
+NG ≤ x
+�
+− P(Z ≤ x)
+�� ≤ Cλ−1/(4r−6),
+(7.1)
+with rate 1/(4r − 6) as λ tends to infinity, where C > 0 depends only on H and G.
+In addition, by Theorem 1.1 of [DE13], �NG satisfies a moderate deviation principle with
+speed a2
+λ = o(λ1/(2r−3)) and rate function x2/2, see Lemma A.1-iii) in appendix. The cu-
+mulant bounds (6.7), (6.9), (6.11) show that the centered and normalized subgraph count
+�
+NG satisfies the Statuleviˇcius condition (A.1) below, see [RSS78, DJS22], with γ := r − 2.
+As a consequence, we have the following result, in which (7.2) is consistent with (7.1) when
+α = 1.
+22
+
+Corollary 7.2 (Sparse regime) Let G be a tree with |V (G)| = r ≥ 2 vertices, satisfying
+Assumption 6.1 in the sparse regime (6.2) with α ∈ [1, r/(r − 1)). We have the Kolmogorov
+distance bound
+sup
+x∈R
+��P
+� �
+NG ≤ x
+�
+− P(Z ≤ x)
+�� ≤ Cλ−(α−(α−1)r)/(4r−6),
+(7.2)
+as λ tends to infinity, where C > 0 depends only on H and G.
+In addition, by Theorem 1.1 of [DE13], �NG satisfies a moderate deviation principle with
+speed a2
+λ = o(λ(α−(α−1)r)/(2r−3)) and rate function x2/2, see Lemma A.1-iii) in appendix.
+Remark 7.3 We note that up to division by 2r − 3, the rate in (7.2) is consistent with
+the rate (α − (α − 1)r)/2 obtained for the counting of trees in the Erd˝os-R´enyi graph, cf.
+Corollary 4.10 of [PS20].
+Remark 7.4 Since (α|E(G)| − r)(n/2 − 1) ≥ (α − 1)(n/2 − 1)r ≥ 0, no significant Kol-
+mogorov bounds are derived from (6.9) and (6.11) for cycle and other non-tree graphs in the
+sparse regime, which is consistent with Corollaries 4.8-4.9 of [PS20].
+A
+Appendix
+The following results are summarized from the “main lemmas” in Chapter 2 of [SS91] and
+[DE13], and are tailored to our RCM applications. We let Φ denote the cumulative distri-
+bution function of the standard normal distribution.
+Lemma A.1 Let (Xλ)λ≥1 be a family of random variables with mean zero and unit variance
+for all λ > 0. Suppose that for all λ ≥ 1, all moments of the random variable Xλ exist and
+that the cumulants of Xλ satisfy
+|κj(Xλ)| ≤ (j!)1+γ
+(∆λ)j−2,
+j ≥ 3,
+(A.1)
+where γ ≥ 0 is a constant not depending on λ, while ∆λ ∈ (0, ∞) may depend on λ. Then,
+the following assertions hold.
+i) (Kolmogorov bound). One has
+sup
+x∈R
+|P(Xλ ≤ x) − Φ(x)| ≤
+C
+(∆λ)1/(1+2γ) ,
+(A.2)
+for some constant C only depending on γ, see [SS91, Corollary 2.1] and [DJS22, Theo-
+rem 2.4].
+23
+
+ii) (Concentration inequality). For any x ≥ 0 and sufficiently large λ,
+P(|Xλ| ≥ x) ≤ 2 exp
+�
+−1
+4 min
+�
+x2
+21/(1+γ), (x∆λ)1/(1+γ)
+��
+.
+(A.3)
+See the corollary to [SS91, Lemma 2.4].
+iii) (Moderate deviation principle). Let (aλ)λ>0 be a sequence of real numbers tending to
+infinity, and such that
+lim
+λ→∞
+aλ
+(∆λ)1/(1+2γ) = 0.
+Then, (a−1
+λ Xλ)λ>0 satisfies a moderate deviation principle with speed a2
+λ and rate func-
+tion x2/2, see [DE13, Theorem 1.1].
+iv) (Normal approximation with Cram´er corrections). There exists a constant c > 0 such
+that for all λ ≥ 1 and x ∈ (0, c(∆λ)1/(1+2γ)) we have
+P(Xλ ≥ x)
+1 − Φ(x)
+=
+�
+1 + O
+�
+x + 1
+(∆λ)1/(1+2γ)
+��
+exp
+�˜L(x)
+�
+,
+P(Xλ ≤ −x)
+Φ(−x)
+=
+�
+1 + O
+�
+x + 1
+(∆λ)1/(1+2γ)
+��
+exp
+�˜L(−x)
+�
+,
+where ˜L(x) is related to the Cram´er-Petrov series, see [SS91, Lemma 2.3].
+References
+[BKR89]
+A.D. Barbour, M. Karo´nski, and A. Ruci´nski. A central limit theorem for decomposable random
+variables with applications to random graphs. J. Combin. Theory Ser. B, 47(2):125–145, 1989.
+[BOR85]
+E.A. Bender, A.M. Odlyzko, and L.B. Richmond. The asymptotic number of irreducible parti-
+tions. European J. Combin., 6(1):1–6, 1985.
+[BR12]
+R. Balakrishnan and K. Ranganathan. A textbook of graph theory. Universitext. Springer, New
+York, second edition, 2012.
+[BRSW17] K. Bogdan, J. Rosi´nski, G. Serafin, and L. Wojciechowski. L´evy systems and moment formulas
+for mixed Poisson integrals. In Stochastic analysis and related topics, volume 72 of Progr. Probab.,
+pages 139–164. Birkh¨auser/Springer, Cham, 2017.
+[CT22]
+V. H. Can and K. D. Trinh. Random connection models in the thermodynamic regime: central
+limit theorems for add-one cost stabilizing functionals. Electron. J. Probab., 27:1–40, 2022.
+[DE13]
+H. D¨oring and P. Eichelsbacher. Moderate deviations via cumulants. J. Theoret. Probab., 26:360–
+385, 2013.
+[DJS22]
+H. D¨oring, S. Jansen, and K. Schubert. The method of cumulants for the normal approximation.
+Probab. Surv., 19:185–270, 2022.
+[ER59]
+P. Erd˝os and A. R´enyi. On random graphs. I. Publ. Math. Debrecen, 6:290–297, 1959.
+[ER21]
+P. Eichelsbacher and B. Rednoß. Kolmogorov bounds for decomposable random variables and
+subgraph counting by the Stein-Tikhomirov method. Preprint arXiv:2107.03775, 2021.
+24
+
+[ET14]
+P. Eichelsbacher and C. Th¨ale. New Berry-Esseen bounds for non-linear functionals of Poisson
+random measures. Electron. J. Probab., 19:no. 102, 25, 2014.
+[Gil59]
+E.N. Gilbert. Random graphs. Ann. Math. Statist, 30(4):1141–1144, 1959.
+[GT18a]
+J. Grote and C. Th¨ale. Concentration and moderate deviations for Poisson polytopes and poly-
+hedra. Bernoulli, 24:2811–2841, 2018.
+[GT18b]
+J. Grote and C. Th¨ale. Gaussian polytopes: a cumulant-based approach. Journal of Complexity,
+47:1–41, 2018.
+[Jan88]
+S. Janson. Normal convergence by higher semiinvariants with applications to sums of dependent
+random variables and random graphs. Ann. Probab., 16(1):305–312, 1988.
+[Jan19]
+S. Jansen. Cluster expansions for Gibbs point processes. Adv. in Appl. Probab., 51(4):1129–1178,
+2019.
+[Kho08]
+O. Khorunzhiy. On connected diagrams and cumulants of Erd˝os-R´enyi matrix models. Comm.
+Math. Phys., 282:209–238, 2008.
+[KRT17]
+K. Krokowski, A. Reichenbachs, and C. Th¨ale. Discrete Malliavin-Stein method: Berry-Esseen
+bounds for random graphs and percolation. Ann. Probab., 45(2):1071–1109, 2017.
+[LNS21]
+G. Last, F. Nestmann, and M. Schulte. The random connection model and functions of edge-
+marked Poisson processes: second order properties and normal approximation.
+Ann. Appl.
+Probab., 31(1):128–168, 2021.
+[LP18]
+G. Last and M.D. Penrose. Lectures on the Poisson process, volume 7 of Institute of Mathematical
+Statistics Textbooks. Cambridge University Press, Cambridge, 2018.
+[LRR16]
+R. Lachi`eze-Rey and M. Reitzner. U-statistics in stochastic geometry. In G. Peccati and M. Re-
+itzner, editors, Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itˆo
+Chaos Expansions and Stochastic Geometry, volume 7 of Bocconi & Springer Series, pages 229–
+253. Springer, Berlin, 2016.
+[MM91]
+V.A. Malyshev and R.A. Minlos. Gibbs random fields, volume 44 of Mathematics and its Appli-
+cations (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991.
+[Pri12]
+N. Privault. Moments of Poisson stochastic integrals with random integrands. Probability and
+Mathematical Statistics, 32(2):227–239, 2012.
+[Pri19]
+N. Privault.
+Moments of k-hop counts in the random-connection model.
+J. Appl. Probab.,
+56(4):1106–1121, 2019.
+[Pri22]
+N. Privault. Asymptotic analysis of k-hop connectivity in the 1D unit disk random graph model.
+Preprint arXiv:2203.14535, 40 pages, 2022.
+[PS20]
+N. Privault and G. Serafin. Normal approximation for sums of discrete U-statistics - application
+to Kolmogorov bounds in random subgraph counting. Bernoulli, 26(1):587–615, 2020.
+[PS22]
+N. Privault and G. Serafin. Berry-Esseen bounds for functionals of independent random variables.
+Electron. J. Probab., 27:1–37, 2022.
+[PT11]
+G. Peccati and M. Taqqu. Wiener Chaos: Moments, Cumulants and Diagrams: A survey with
+Computer Implementation. Bocconi & Springer Series. Springer, 2011.
+[R¨ol22]
+A. R¨ollin.
+Kolmogorov bounds for the normal approximation of the number of triangles in
+the Erd˝os-R´enyi random graph.
+Probability in the Engineering and Informational Sciences,
+36(3):747–773, 2022.
+[Rot64]
+G.-C. Rota. On the foundations of combinatorial theory. I. Theory of M¨obius functions. Z.
+Wahrscheinlichkeitstheorie und Verw. Gebiete, 2:340–368, 1964.
+[RSS78]
+R. Rudzkis, L. Saulis, and V.A. Statuljaviˇcus. A general lemma on probabilities of large devia-
+tions. Litovsk. Mat. Sb., 18(2):99–116, 217, 1978.
+25
+
+[Ruc88]
+A. Ruci´nski.
+When are small subgraphs of a random graph normally distributed?
+Probab.
+Theory Related Fields, 78:1–10, 1988.
+[SS91]
+L. Saulis and V.A. Statuleviˇcius. Limit theorems for large deviations, volume 73 of Mathematics
+and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991.
+26
+

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+arXiv:2301.02392v1  [physics.plasm-ph]  6 Jan 2023
+Moment-Fourier approach to ion parallel fluid closures and
+transport for a toroidally confined plasma
+Jeong-Young Ji,∗ Eric D. Held, and J. Andrew Spencer
+Department of Physics, Utah State University, Logan, Utah 84322, USA
+Yong-Su Na
+Department of Nuclear Engineering,
+Seoul National University, Seoul 08826, South Korea
+Abstract
+A general method of solving the drift kinetic equation is developed for an axisymmetric magnetic
+field. Expanding a distribution function in general moments a set of ordinary differential equa-
+tions are obtained.
+Successively expanding the moments and magnetic-field involved quantities
+in Fourier series, a set of linear algebraic equations is obtained. The set of full (Maxwellian and
+non-Maxwellian) moment equations is solved to express the density, temperature, and flow veloc-
+ity perturbations in terms of radial gradients of equilibrium pressure and temperature. Closure
+relations that connect parallel heat flux density and viscosity to the radial gradients and parallel
+gradients of temperature and flow velocity, are also obtained by solving the non-Maxwellian mo-
+ment equations. The closure relations combined with the linearized fluid equations reproduce the
+same solution obtained directly from the full moment equations. The method can be generalized
+to derive closures and transport for an electron-ion plasma and a multi-ion plasma in a general
+magnetic field.
+∗ [email protected]
+1
+
+I.
+INTRODUCTION
+For magnetically confined plasmas, neoclassical transport theory describes particle, heat,
+and momentum transport of a steady-state plasma due to Coulomb collisions in an inhomo-
+geneous magnetic field [1–7]. The neoclassical transport is obtained by solving the first order
+drift kinetic equation [8, 9] assuming a zeroth order background distribution (see Ref. [10, 11]
+for reviews). Due to difficulty in treating the integro-differential collision operator in veloc-
+ity space, modified collision operators have been adopted for analytical work. Numerical
+work may adopt the Landau (Fokker-Planck) collision operator with desired accuracy by in-
+creasing velocity space resolution. Numerous transport codes have been developed to solve
+the continuum drift kinetic equation with a modified [12, 13] or an exact Landau collision
+operator [14–19].
+For describing a macroscopic state of a tokamak plasma, the fluid variables are of primary
+importance and solving fluid equations instead of the kinetic equation may be sufficient. Due
+to significantly lower dimensionality of position space compared to phase space, numerically
+solving fluid equations has a great advantage over solving the kinetic equation [20–24]. The
+key issue is to obtain proper closures to capture desired physics effects. Even though the
+heat flux density is derived in neoclassical transport theory, it cannot serve as one of closures
+for the temperature equation because it is derived from the fluid equations, and hence,
+expressed in terms of the zeroth-order density and temperature instead of the (first-order)
+fluid variables whose evolution equations are to be closed. That is, the heat flux derived
+from the divergence free condition plays no role for the divergence term in the temperature
+equation.
+In this work, we introduce an analytic method to solve the drift kinetic equation to obtain
+closures and transport. For a magnetized plasma, the parallel moment equations are derived
+in Ref. [25]. One advantage of the moment approach is the availability of the exact collisional
+moments of the linearized Landau operator [26]. The moment-based collision operator can
+be utilized for the linear and nonlinear gyrokinetic Coulomb collision operator [27–29]. For
+slab geometry where the magnetic field strength does not change along a magnetic field
+line, the drift-kinetic equation can be converted to a linear system of ordinary differential
+equations with constant coefficients. This linear system can be analytically solved for the
+2
+
+parallel moments using the eigenvector method [30].
+On the other hand, for an inhomogeneous magnetic field of a tokamak, the drift kinetic
+equations becomes a linear system of ordinary differential equations with varying coefficients.
+This means that the eigenvector method used in the integral closure [30] does not work. For
+a system of linear differential equations with varying coefficients, we can Fourier-expand
+the varying coefficients and moments to build a system of linear algebraic equations. While
+truncation both in the moments and Fourier modes is inevitable, the solution of the truncated
+system is equivalent to that of the drift kinetic equation when convergence is achieved by
+increasing the number of moments and Fourier modes. The solution moments can then be
+used to construct the distribution function that is the solution of the drift kinetic equation.
+Therefore the moment solution can be used for benchmarking numerous fluid and kinetic
+codes.
+In Sec. II, we present the parallel moment equations which are equivalent to the first order
+drift kinetic equation. In Sec. III, we use the Fourier expansion to solve the general moment
+equations for fluid quantities in Fourier series.
+The convergent solution is presented as
+the numbers of moments and Fourier modes increase. In Sec. IV, we derive closures and
+incorporate them into fluid equations to reproduce the fluid quantities. In Sec.V, we conclude
+and discuss possible extensions of the work to more general plasmas.
+II.
+DRIFT KINETIC EQUATION AND MOMENT EQUATIONS
+In standard neoclassical transport theory (see Ref. [11] for a general review), drift kinetic
+equations are solved for ion and electron transport. An analytic solution can be obtained
+for an axisymmetric magnetic field
+B = I∇ζ + ∇ζ × ∇ψ
+(1)
+where 2πψ is the poloidal flux, 2πI/µ0 is the poloidal current, µ0 is the magnetic perme-
+ability, and ζ is the toroidal angle.
+For simplicity, we assume a circular magnetic field
+B =
+B0
+1 + ǫ cos θ
+(2)
+3
+
+where θ is the poloidal angle, B0 is a constant reference field, ǫ = r/R0 is the inverse aspect
+ratio, and R0 and r respectively are the major and minor radii of a circular-shape flux
+surface.
+For ion transport, the ion-electron collisions are often ignored and the reduced ion drift
+kinetic equation for the first-order distribution function f1 becomes
+v∥∂∥(f1 − F) = C(f1)
+(3)
+with
+F = −Iv∥
+Ω
+df0
+dψ = −Iv∥
+Ω
+�d ln p0
+dψ
++
+�
+s2 − 5
+2
+� d ln T0
+dψ
+�
+f0
+(4)
+and
+f0(ψ, w) =
+n0(ψ)
+[2πmT0(ψ)]3/2e−w/T0(ψ) =
+n0
+π3/2v3
+0
+e−s2
+(5)
+in the (ψ, θ, w = mv2/2, µ = mv2
+⊥/2B) coordinates, where ∂∥ = b·∇ = (B/B)·∇, v∥ = b·v,
+Ω = qB/m, v0 =
+�
+2T0/m, and s = v/v0. Note that flux surfaces can be labeled by the
+lowest-order density n0, temperature T0, or pressure p0 = n0T0. The collision operator is a
+Landau operator linearized with respect to a static Maxwellian distribution function f0,
+C(f1) = C(f1, f0) + C(f0, f1).
+(6)
+One difficulty of solving the kinetic equation (3) is in treating the collision operator, an
+integro-differential operator in velocity space. In standard analytical neoclassical theory,
+the Landau operator is often approximated as the Lorentz pitch-angle scattering operator
+with an additional momentum restoring term for an analytical treatment. In the moment
+approach, the linearized collision operator can be analytically calculated and explicitly rep-
+resented by a matrix of collision coefficients. In this work, we solve a system of parallel
+moment equations introduced in Ref. [25, 26]. The moment equations can also be derived
+from the drift kinetic equation as shown below.
+In the moment method of this work, a gyro-averaged distribution function f1 is expanded
+as
+f1 = f0
+�
+l,k
+ˆP lk ˆ
+Mlk
+(7)
+4
+
+with orthonormal polynomials
+ˆP lk =
+1
+√¯σlk
+P lk =
+1
+√¯σlk
+slPl(v∥/v)L(l+1/2)
+k
+(s2),
+where Pl is a Legendre polynomial, L(l+1/2)
+k
+is an associated Laguerre (Sonine) polynomial,
+and the normalization constants are
+¯σlk = ¯σlλlk, ¯σl =
+1
+2l + 1, λlk = (l + k + 1/2)!
+k!(1/2)!
+.
+(8)
+Several lowest order moments of f1 are:
+ˆ
+M00 = n1/n0 (density),
+ˆ
+M01 = −
+�
+3/2T1/T0
+(temperature),
+ˆ
+M10 =
+√
+2u/v0 (parallel flow velocity u = V1∥),
+ˆ
+M11 = −
+�
+4/5h∥/v0p0
+(parallel heat flux density), and ˆ
+M20 =
+�
+3/4π∥/p0 (parallel viscosity), where p0 = n0T0.
+The neoclassical thermodynamic drive term can also be expanded as
+v∥∂∥F =v0∂∥ ln B
+B/B0
+f0
+��
+2 ˆP 00 − 2
+�
+2
+3
+ˆP 01 + 1
+√
+3
+ˆP 20
+�
+ˆp0,ψ
++
+�
+−5
+�
+2
+3
+ˆP 01 + 2
+�
+10
+3
+ˆP 02 + 1
+√
+3
+ˆP 20 −
+�
+7
+6
+ˆP 21
+�
+ˆT0,ψ
+�
+,
+(9)
+where
+ˆp0,ψ =
+I
+qv0B0n0
+dp0
+dψ ,
+(10)
+ˆT0,ψ =
+I
+qv0B0
+dT0
+dψ .
+(11)
+Taking the ˆP jp moment of Eq. (3) yields
+�
+lk
+ψjp,lk∂∥ ˆ
+Mlk + ψjp,lk
+B
+(∂∥ ln B) ˆ
+Mlk = 1
+λC
+cjp,lk ˆ
+Mlk + ∂∥ ln B
+B/B0
+�
+gjp
+p ˆp0,ψ + gjp
+T ˆT0,ψ
+�
+, (12)
+where λC = v0τii (the ion mean free path). Note that eliminating (j, p) = (0, 0), (0, 1), and
+(1,0) moment equations from Eq. (12) yields a set of closure moment equations, similar to
+the closure moment equations in slab geometry Ref. [31]. The constant coefficients ψjp,lk,
+ψjp,lk
+B
+, and cjp,lk are defined by
+�
+d3vv∥ ˆP jp ˆP lkf0 = n0v0ψjp,lk,
+(13)
+�
+d3vv∥ ˆP jp(∂∥ ˆP lk)f0 = n0v0(∂∥ ln B)ψjp,lk
+B
+,
+(14)
+5
+
+�
+d3v ˆP jpC(f0 ˆP lk) = n0
+τii
+cjp,lk = n0
+τii
+δjlcj
+pk.
+(15)
+The nonvanishing gjp in Eq. (9) are
+g0,0
+p
+= 2, g0,1
+p
+= −2
+�
+2
+3, g2,0
+p
+= 1
+√
+3
+(16)
+and
+g0,1
+T
+= −5
+�
+2
+3, g0,2
+T
+= 2
+�
+10
+3 , g2,0
+T
+= 1
+√
+3
+, g2,1
+T
+= −
+�
+7
+6.
+(17)
+Noting that ψjp,lk = δj,j±1ψj±
+lk , ψjp,j+1,k
+B
+= −(j + 2)ψjp,j+1,k/2, and ψjp,j−1,k
+B
+= (j −
+1)ψjp,j−1,k/2 (see Ref. [25]) and defining
+∂j+
+∥
+= ∂∥ − j + 2
+2
+∂∥ ln B,
+∂j−
+∥
+= ∂∥ + j − 1
+2
+∂∥ ln B,
+(18)
+we can combine the ψ and ψB terms to rewrite Eq. (12) as
+�
+k
+ψj−
+pk ∂j−
+∥
+ˆ
+Mj−1,k +
+�
+k
+ψj+
+pk ∂j+
+∥
+ˆ
+Mj+1,k = 1
+λC
+�
+k
+cj
+pk ˆ
+Mjk + ∂∥ ln B
+B/B0
+�
+gjp
+p ˆpψ + gjp
+T ˆTψ
+�
+. (19)
+Although Eq. (12) for j = 0, 1, · · · , L − 1 and k = 0, 1, · · · , K − 1 is a truncated system,
+there exist L and K such that the solution does not change when increasing the number of
+moments higher than L and K. In other words, there exists a convergent solution of Eq. (12)
+which can be considered as a solution of Eq. (3). Therefore Eq. (12) for the truncated set
+of moments is quantitatively equivalent to Eq. (3).
+III.
+FOURIER METHOD OF SOLVING MOMENT EQUATIONS
+In the axisymmetric magnetic field (1), physical quantities on a flux surface depends on θ
+only. Using ∂∥ = (B · ∇θ/B)∂/∂θ = (Bθ/B)∂θ and dividing Eq. (12) by Bθ/B yields a
+system of ordinary differential equations
+�
+lk
+ψjp,lk∂θ ˆ
+Mlk + ψjp,lk
+B
+(∂θ ln B) ˆ
+Mlk =
+B
+BθλC
+cjp,lk ˆ
+Mlk + ∂θ ln B
+B/B0
+�
+gjp
+p ˆpψ + gjp
+t ˆTψ
+�
+. (20)
+6
+
+Since the coefficient ∂θ ln B is θ-dependent, the eigenvector method used in deriving integral
+closures [30] does not work. Instead, we adopt the Fourier method to convert the system of
+differential equations to a system of algebraic equations. Note that Eq. (20) forms a linear
+system of ordinary differential equations for the parallel moments
+ˆ
+Mlk and the Fourier
+expansion of coefficients, moments, and drive terms will convert the differential system to a
+linear algebraic system.
+In the Fourier method, all physical quantities are expanded in Fourier series. For A = ˆ
+Mlk(θ)
+and ∂θ ln B/(B/B0),
+A(θ) = A(0)+A(1−) sin θ+A(1+) cos θ+A(2−) sin 2θ+A(2+) cos 2θ+· · · =
+�
+m
+A(m)ϕ(m), (21)
+with Fourier modes
+ϕ(0) = 1, ϕ(1) = ϕ(1−) = sin θ, ϕ(2) = ϕ(1+) = cos θ, · · · ,
+ϕ(2n−1) = ϕ(n−) = sin nθ, ϕ(2n) = ϕ(n+) = cos nθ, · · ·
+(22)
+where the Fourier index is denoted in the parentheses. The Fourier coefficient for A(θ) can
+be obtained by
+A(m) =
+1
+σ(m)
+�
+dθϕ(m)A(θ),
+(23)
+where σ(0) = 2π and σ(m) = π for m > 0. The derivative ∂θ and the θ-dependent coefficients
+in Eq. (20) become matrices in Fourier representation. For O = ∂θ, ∂θ ln B, and B/BθλC,
+the Fourier matrix elements O(i,j) are obtained by
+O(i,j) =
+1
+σ(i)
+�
+dθϕ(i)Oϕ(j),
+(24)
+and the Fourier representation of O ˆ
+Mlk becomes
+�
+O ˆ
+Mlk�
+(i) =
+1
+σ(i)
+�
+dθϕ(i)O
+�
+j
+ˆ
+Mlk
+(j)ϕ(j) =
+�
+j
+O(i,j) ˆ
+Mlk
+(j).
+(25)
+Then the (m)th Fourier component of Eq. (20) becomes a system of algebraic equations
+ψjp,lk (∂θ)(m,n) ˆ
+Mlk
+(n) + ψjp,lk
+B
+(∂θ ln B)(m,n) ˆ
+Mlk
+(n) =
+cjp,lk
+�
+B
+BθλC
+�
+(m,n)
+ˆ
+Mlk
+(n) +
+�∂θ ln B
+B/B0
+�
+(m)
+�
+gjp
+p ˆp0,ψ + gjp
+T ˆT0,ψ
+�
+,
+(26)
+7
+
+-0.05
+0
+0.05
+-0.05
+0
+0.05
+-3
+-2
+-1
+0
+1
+2
+3
+0.3
+0.4
+0.5
+0.6
+0.7
+Figure 1. First-order density, temperature, and parallel flow velocity for ǫ = 0.1, K0 = 100, nF = 4,
+and for LK = 10×20 (red, dotted), 20×40 (green, dash-dotted), 40×80 (blue solid), and 80×160
+(cyan, dashed). The ratios n1/n0, T1/T0, and u/v0 are plotted in units of ˆT0,ψ.
+where summation over l, k, and n is implied. The system of algebraic equations can be
+written in matrix form,
+�ψ∂θ�
+�
+ˆ
+M
+�
++�ψB∂θ ln B�
+�
+ˆ
+M
+�
+=
+�
+cB/BθλC
+� �
+ˆ
+M
+�
++
+�
+(gpˆp0 + gT ˆT0)(B0/B)(∂θ ln B)
+�
+, (27)
+where �ψ∂θ� = [ψ] ⊗ (∂θ)F, �ψB∂θ ln B� = [ψB] ⊗ (∂θ ln B)F, and
+�
+cB/BθλC
+�
+= [c] ⊗
+�
+B/BθλC
+�
+F with ⊗ denoting a tensor product of two matrices. The ith row and jth column
+of a Fourier matrix (O)F is O(i,j), and the dimension of the linear system is N = LKF =
+(the number of Legendre polynomials)(the number of Laguerre polynomials)(the number of
+Fourier modes).
+8
+
+-0.05
+0
+0.05
+-0.05
+0
+0.05
+-3
+-2
+-1
+0
+1
+2
+3
+0.2
+0.4
+0.6
+0.8
+Figure 2. First-order density, temperature, and parallel flow velocity for ǫ = 0.1, K0 = 100, LK =
+40 × 80, and for nF = 1 (red, dotted), 2 (green, dash-dotted), 4 (blue solid), and 7 (cyan, dashed).
+The ratios n1/n0, T1/T0, and u/v0 are plotted in units of ˆT0,ψ.
+The solution
+�
+ˆ
+M
+�
+can be obtained by inverting or singular-value-decomposing the matrix,
+�
+ˆ
+M
+�
+=
+�
+�ψ∂θ� + �ψB∂θ ln B� −
+�
+cB/BθλC
+��−1
+ns
+�
+(gpˆp0,ψ + gT ˆT0,ψ)(B0/B)(∂θ ln B)
+�
+, (28)
+where the subscript ‘ns’ denotes the nonsingular part of the matrix. It is found that elimi-
+nating n(0) and T(0) components makes the matrix nonsingular [see also remarks in relation
+to Eqs. (48) and (50)]. Then the Fourier components of the first order fluid quantities can
+9
+
+-0.05
+0
+0.05
+-0.05
+0
+0.05
+-3
+-2
+-1
+0
+1
+2
+3
+0.3
+0.4
+0.5
+0.6
+0.7
+Figure 3. First-order density, temperature, and parallel flow velocity for ǫ = 0.3, K0 = 100, nF = 4,
+and for LK = 10×20 (red, dotted), 20×40 (green, dash-dotted), 40×80 (blue solid), and 80×160
+(cyan, dashed).
+be read from the solution
+�
+ˆ
+M
+�
+,
+N = ˆp0,ψNp0 + ˆT0,ψNT0,
+T = ˆp0,ψTp0 + ˆT0,ψTT0,
+(29)
+U = ˆp0,ψUp0 + ˆT0,ψUT0,
+where N = (ˆn)F = (n1/n0)F, T = ( ˆT)F = (T1/T0)F, U = (ˆu)F = (u/v0)F, Nα, Tβ, and Uβ
+(β = p0, T0) are column vectors of Fourier components. With the Fourier components, the
+first-order fluid quantities can be constructed from Eq. (21). For example, the density due
+to ˆp0,ψ and ˆT0,ψ, respectively, are ˆn = �
+m Np0
+(m)ϕ(m)ˆp0,ψ and ˆn = �
+m NT0
+(m)ϕ(m) ˆT0,ψ, where
+Nβ
+(m) is the (m)th Fourier component of the column vector Nβ.
+10
+
+-0.05
+0
+0.05
+-0.05
+0
+0.05
+-3
+-2
+-1
+0
+1
+2
+3
+0.2
+0.4
+0.6
+0.8
+Figure 4. First-order density, temperature, and parallel flow velocity for ǫ = 0.3, K0 = 100, N =
+LK = 40 × 80, and for nF = 1 (red, dotted), 5 (green, dash-dotted), 9 (blue solid), and 13 (cyan,
+dashed). The ratios n1/n0, T1/T0, and u/v0 are plotted in units of ˆT0,ψ.
+The inverse collisionality of the system is characterized by a Knudsen number, the ratio of
+the mean free path to the gradient scale length. Defining a basic Knudsen number for a
+tokamak K0 = B/BθλC, the effective Knudsen number would be roughly K0∂θ ln B ∼ mK0
+where m is the typical Fourier mode of the system.
+Although the solution (28) can be
+obtained for an arbitrary axisymmetric magnetic field, circular magnetic fields [see Eq. (2)]
+are considered in this work. For the circular magnetic field (2), the basic Knudsen number
+is given by K0 ∼ λC/qR0 where q is the safety factor and the Fourier mode m is determined
+by the inverse aspect ratio ǫ = r/R0. In general, the effective Knudsen number increases as
+λC and ǫ increase.
+11
+
+Figure 5. The first-order distribution function f1 at θ = −π/3 in the s⊥-s∥ plane for ǫ = 0.3
+and K0 = 100. The white dashed lines indicate the passing/trapped boundary. The ratio f1/f0 is
+plotted in units of ˆp0,ψ in (a), (c), and (d) and in units of ˆT0,ψ in (b).
+Figure 6. The first-order distribution function f1 at θ = π/3 on the s⊥-s∥ plane for ǫ = 0.3 and
+K0 = 100. The white dashed lines indicate the passing/trapped boundary. The ratio f1/f0 is plotted
+in units of ˆp0,ψ in (a), (c), and (d) and in units of ˆT0,ψ in (b).
+12
+
+0.2
+0.1
+0
+-0.1
+-0.2
+0.1
+0.05
+0
+-0.05
+-0.13
+3
+0.15
+2.5
+2.5
+0.1
+2
+0.05
+2
+1.5
+0
+1.5
+1
+0.05
+1
+-0.1
+0.5
+0.5
+-0.15
+0
+0
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+s1
+(c) fi/ fo due to dpo /db and dTo /dab for To,b = po,b
+(d) fi/ fo due to dpo /db and dTo/db for To, = 0.3po,b
+3
+3
+2.5
+0.05
+2.5
+2
+2
+0
+1.5
+1.5
+1
+1
+-0.05
+0.5
+0.5
+0
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+s1(a) fi/ fo due to dpo/db
+(b) fi/ fo due to dTo/db0.2
+0.1
+0
+-0.1
+-0.2
+0.1
+0.05
+0
+-0.05
+-0.13
+3
+0.15
+2.5
+2.5
+0.1
+2
+0.05
+2
+1.5
+0
+1.5
+1
+0.05
+1
+-0.1
+0.5
+0.5
+-0.15
+0
+0
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+s1
+(c) fi/fo due to dpo /db and dTo /dab for To,b = po,b
+(d) fi/ fo due to dpo /db and dTo/db for To, = 0.3po,b
+3
+3
+2.5
+2.5
+0.05
+2
+2
+1.5
+1.5
+0
+1
+1
+0.5
+-0.05
+0.5
+0
+0
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+s1Figure 7. The first-order distribution function f1 at s = 0.7 on the θ-µ plane for ǫ = 0.3 and
+K0 = 100.
+The white dashed line indicates the passing/trapped boundary.
+The ratio f1/f0 is
+plotted in units of ˆp0,ψ in (a), (c), and (d) and in units of ˆT0,ψ in (b).
+The solution responding to the radial pressure gradient dp0/dψ shows that Np0 = 0, Tp0 = 0,
+and Up0 = −(1, 0, ǫ, · · · )T = −(B0/B)F. This means that the ˆp0,ψ drive contributes only to
+the flow velocity as ˆu = −ˆp0,ψB0/B + γuB/B0, consistent with the continuity equation
+∇ · (n0V1) = 0. Here γu is an integration constant that can be determined by temperature
+and flow velocity equations. It turns out that γu is proportional to ˆT0,ψ as verified from the
+solution and as discussed in Sec. IV.
+For the solution responding to the radial temperature gradient dT0/dψ, the density, tem-
+perature, and parallel flow velocity are shown in Fig. 1 in the case of ǫ = 0.1, K0 = 100, and
+nF = 4 (F = 2nF + 1 = 9). A convergence study increases the number of moments to show
+that the LK = 40 × 80 moment solution converges and can be considered practically exact.
+Note that the polynomials ˆP lk in Eq. (7) form a complete set. The necessary number of
+moments for convergence increases as K0 increases. A convergence study that increases the
+13
+
+0.2
+0.15
+0.1
+0.05
+0
+0.01
+0
+-0.01
+0.02
+-0.03
+-0.04
+-0.05
+-0.06bm /odm on ann of/lf (e)
+@n /on o ann of/lc (a)
+0
+0.6
+0.6
+-0.02
+ To/ Bol
+0.5
+0.5
+0.04
+0.4
+0.4
+[units of 
+-0.06
+JOS
+0.3
+[units
+-0.08
+0.3
+0.2
+0.1
+0.2
+0.1
+-0.12
+0.1
+0
+0.14
+0
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+0
+0
+(c) fi/ fo due to dpo /dab and dTo/db for To,b = po,b
+(d) fi/ fo due to dpo/dab and dTo/db for To, = 0.3po,b
+0.1
+0.6
+0.6
+0.08
+0.5
+0.5
+0.06
+0.4
+0.4
+0.04
+JO
+JO
+units
+[units
+0.3
+0.02
+0.3
+0.2
+0
+0.2
+0.1
+0.02
+0.1
+-0.04
+0
+0
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+0
+0number of Fourier modes from 1 to 7 (see Figure 2) shows that the nF = 4 mode solution
+converges and may be considered to be very accurate. The necessary number of Fourier
+modes for convergence increases as ǫ increases.
+Figures 3 and 4 show the density, temperature, and parallel flow velocity for ǫ = 0.3, a
+larger inverse aspect ratio, and K0 = 100. The LK = 40 × 80 moment solution, while not as
+accurate as in the ǫ = 0.1 case, is still very accurate for practical use, and the LK = 80×160
+solution is expected to be accurate. This is because ǫ = 0.3 requires more Fourier modes
+than ǫ = 0.1 for an accurate expansion of the magnetic field. Higher Fourier modes make the
+effective Knudsen number larger. The necessary number of Fourier modes for convergence
+is nF = 13.
+The moment solution can be used to construct the distribution function that is a solution
+of the kinetic equation (3). Since all fluid quantities relevant to physical observables involve
+several lowest order of moments, the reconstruction of the distribution function from the
+moments may be redundant. Nevertheless, the distribution function itself is important for
+understanding the kinetic behavior of a plasma. In the moment expansion, the high order
+moments near truncation of the moment expansion could be inaccurate and may adversely
+affect the convergence of the distribution function. However we find that those moments
+near truncation are several orders smaller than the fluid moments, making the truncation
+errors ignorable once the convergence is achieved. Figures 5 and 6 show the distribution
+functions constructed from the moment solution on the s⊥-s∥ plane at θ = −π/3 and π/3,
+respectively. Figure 7 shows the distribution function at s = 0.7 on the θ-µ plane.
+IV.
+FLUID EQUATIONS AND CLOSURES
+In neoclassical transport theory, one solves Eq. (3) to express f1 in terms of f0 (or F) and take
+moments of the solution f1 to express u in terms of dp0/dψ and dT0/dψ. These expressions
+can be directly obtained by solving Eq. (12). In this section we derive closure relations
+that can be used for closing and advancing (nonlinear) fluid equations for density, flow
+velocity, and temperature. They can also be incorporated into linearized fluid equations to
+reproduce the expressions of n1, T1 and u that are obtained in Sec. III. Although the closures
+14
+
+are represented in the Fourier basis, the formalism developed here can be applied to any
+basis such as a finite element basis or finite difference basis in numerical methods.
+The linearized fluid equations for n1, u, and T1 can be obtained from the original fluid
+equations with n = n0 + n1, T = T0 + T1, V = ub + b × ∇p0/n0qB, h = h∥b + 5p0b ×
+∇T0/2qB, and π = (3π∥/2)(bb − b2I/3) where b = B/B.
+They are equivalent to the
+{P 00, mv0P 10, −T0P 01} moments of Eq. (3) and can be read from Eq. (20) for (j, p) = (0, 0),
+(1, 0), and (0, 1):
+∂0+
+θ ˆu = 2ˆp0,ψ
+∂θ ln B
+B/B0
+,
+(30)
+∂0+
+θ ˆu + ∂0+
+θ ˆh = (2ˆp0,ψ + 5 ˆT0,ψ)∂θ ln B
+B/B0
+,
+(31)
+∂1−
+θ ˆn + ∂1−
+θ
+ˆT + ∂1+
+θ ˆπ = 0,
+(32)
+where ˆu = u/v0, ˆh = h∥/v0p0, ˆπ = π∥/p0, and ∂l±
+θ
+is defined by Eq. (18) with ∂∥ replaced by
+∂θ. For this fluid system to be closed, closure quantities ˆh and ˆπ should relate to first-order
+(ˆn, ˆu, and ˆT) and equilibrium (ˆp0,ψ and ˆT0,ψ) fluid quantities.
+In order to obtain the closure relations, the rows corresponding to fluid equations need to
+be removed from Eq. (20). Then the corresponding columns appear as drives (sources) [gθ]
+in the system:
+[ψ′]
+�
+∂θ ˆ
+M′�
++[ψ′
+B] (∂θ ln B)
+�
+ˆ
+M′�
+=
+B
+BθλC
+[c′]
+�
+ˆ
+M′�
++[gθ]+ ∂θ ln B
+B/B0
+��
+g′
+p
+�
+ˆp0,ψ + [g′
+T] ˆT0,ψ
+�
+,
+(33)
+where ′ denotes the removal of fluid columns and rows. For example,
+�
+ˆ
+M′�
+is a column vec-
+tor ( ˆ
+M0,2, · · · ˆ
+M0,K+1, ˆ
+M1,1, · · · , ˆ
+M1,K, ˆ
+M2,0, · · · , ˆ
+M2,K−1, · · · , ˆ
+ML−1,0, · · · , ˆ
+ML−1,K−1). The
+nonvanishing elements of [gθ] are
+g1,1
+θ
+=
+√
+5
+2 ∂θ ˆT,
+(34)
+g2,0
+θ
+= −
+√
+3
+2 Wθ, Wθ = 4
+3∂2−
+∥ ˆu.
+(35)
+From Fourier representation of Eq. (33),
+�ψ′∂θ�
+�
+ˆ
+M′�
++�ψ′
+B∂θ ln B�
+�
+ˆ
+M′�
+=
+�
+cB/BθλC
+� �
+ˆ
+M′�
++�gθ�+
+�
+(g′
+pˆp0 + g′
+T ˆT0)(B0/B)(∂θ ln B)
+�
+,
+15
+
+(36)
+the solution can be obtained,
+�
+ˆ
+M′�
+=
+�
+�ψ′∂θ� + �ψ′
+B∂θ ln B� −
+�
+cB/BθλC
+��−1 �
+gθ + (gpˆp0,ψ + gT ˆT0,ψ)(B0/B)(∂∥ ln B)
+�
+.
+(37)
+Fourier components of closures ˆh = −
+√
+5 ˆ
+M1,1/2 and ˆπ = 2 ˆ
+M2,0/
+√
+3 can be read from the
+solution and expressed in terms of ˆp0,ψ, and ˆT0,ψ, ˆT, and ˆu:
+H = ˆp0,ψHp0 + ˆT0,ψHT0 + KhhDT + KhπW,
+(38)
+S = ˆp0,ψSp0 + ˆT0,ψST0 + KπhDT + KππW,
+(39)
+where H = (ˆh)F, S = (ˆπ)F, and W = (Wθ)F = (4/3)D2−U ≡ DWU, Hβ, and Sβ (β = p0, T0)
+are column vectors, and D = (∂θ)F , Dl± = (∂l±
+θ )F, and Kαβ (α, β = h, π) are matrices. Here a
+column vector Hβ and Sβ connects the closures h∥ and π∥ to a radial gradient of zeroth-order
+pressure (β = p0) or temperature (β = T0), and a matrix Kαβ connects closures α = h and
+π to a parallel gradient of first-order temperature (β = h) or parallel flow velocity (β = π).
+The closures in the position space can be constructed from the solution vector, for example,
+ˆh(θ) = �
+i ϕ(i){Hp0
+(i)ˆp0,ψ +HT0
+(i) ˆT0,ψ +�
+j[Khh
+(i,j)(DT)(j) +Khπ
+(i,j)W(j)]ϕ(j)}, where Hβ
+(i) is the (i)th
+Fourier component of the column vector Hβ and Kαβ
+(i,j) is the (i)th row and (j)th column
+of the matrix Kαβ. Figures 8 and 9, respectively, show the parallel heat flux density and
+viscosity due to ˆp0,ψ, ˆT0,ψ, and several Fourier modes of ∂θ ˆT and Wθ. As the Fourier mode
+of the thermodynamic drives increases, the contribution to the closure quantity decreases.
+By combining closure relations with the time-independent, linear fluid equations, we can
+reproduce the fluid variables of Sec. III. Using (B0/B)∂θ ln B = −∂θ(B0/B) and eliminating
+Eq. (30) from Eq. (31), we write the Fourier representation of Eqs. (30)-(32),
+D0+U = −2ˆp0,ψDB−1,
+(40)
+D0+H = −5 ˆT0,ψDB−1,
+(41)
+DN + DT + D1+S = 0,
+(42)
+16
+
+-2
+-1.5
+-1
+-0.5
+0
+-4
+-2
+0
+2
+-0.2
+-0.1
+0
+0.1
+0.2
+-2
+0
+2
+4
+-0.5
+0
+0.5
+1
+Figure 8.
+Parallel heat flux density due to (a) dp0/dψ and dT0/dψ, (b) (∂θ ˆT)(m+) cos mθ, (c)
+(∂θ ˆT)(m−) sin mθ, (d) (Wθ)(m+) cos mθ, and (e) (Wθ)(m−) sin mθ.
+The dimensionless heat flux,
+h∥/v0p0, is plotted in units of (a) ˆp0,ψ and ˆT0,ψ, (b) (∂θ ˆT)(m+), (c) (∂θ ˆT)(m−), (d) (Wθ)(m+), and
+(e) (Wθ)(m−).
+17
+
+-0.02
+-0.01
+0
+0.01
+0.02
+-0.1
+-0.05
+0
+0.05
+0.1
+-1
+0
+1
+2
+-80
+-60
+-40
+-20
+0
+-0.2
+0
+0.2
+Figure 9.
+Parallel viscosity due to (a) dp0/dψ and dT0/dψ,
+(b) (∂θ ˆT)(m+) cos mθ,
+(c)
+(∂θ ˆT)(m−) sin mθ, (d) (Wθ)(m+) cos mθ, and (e) (Wθ)(m−) sin mθ. The dimensionless viscosity π∥/p0
+is plotted in units of (a) ˆp0,ψ and ˆT0,ψ, (b) (∂θ ˆT)(m+), (c) (∂θ ˆT)(m−), (d) (Wθ)(m+), and (e)
+(Wθ)(m−).
+18
+
+where B−1 = (B0/B)F. Then we combine with closures (38) and (39) to write
+L
+
+
+
+
+
+N
+T
+U
+
+
+
+
+ = Rp0 ˆp0,ψ + RT0 ˆT0,ψ.
+(43)
+where
+L =
+
+
+
+
+
+0
+0
+D0+
+0
+D0+KhhD
+D0+KhπDW
+D D + D1+KπhD D1+KππDW
+
+
+
+
+ ,
+(44)
+Rp0 = −
+
+
+
+
+
+2DB−1
+D0+Hp0
+D1+Sp0
+
+
+
+
+ , RT0 = −
+
+
+
+
+
+0
+5DB−1
+D1+ST0
+
+
+
+
+ .
+(45)
+Using the singular value decomposition, we can invert the nonsingular part of L and obtain
+the solution vector (N, T, U) in terms of ˆp0,ψ and ˆT0,ψ.
+The solution vector reproduces
+Eq. (29) with the column vector (Nβ, Tβ, Uβ) = (L−1
+ns ) Rβ for β = p0 and T0.
+Now we discuss how to obtain the parallel flow velocity and heat flux density when not
+using the singular value decomposition but instead, analytically calculating the integration
+constants. From Eqs. (40) and (41), we have
+U = −ˆp0,ψB−1 + γuB,
+(46)
+H = −5
+2
+ˆT0,ψB−1 + γhB,
+(47)
+where γu and γh are expansion coefficients for the null space of D0+ (D0+B = 0), and
+B = (B/B0)F. Combining Eq. (38) with (47), we have
+DT = γuFu + γhFh + ˆp0,ψFp + ˆT0,ψFT,
+(48)
+where
+Fu = −Khh,−1KhπDWB,
+Fh = Khh,−1B,
+Fp = −Khh,−1 �
+Hp0 − KhπDWB−1
+�
+,
+FT = −Khh,−1
+�
+HT0 + 5
+2B−1
+�
+,
+(49)
+19
+
+Combining Eq. (39) with Eq. (42) and using Eqs. (46) and (48), we have
+DN + DT = γuGu + γhGh + ˆp0,ψGp + ˆT0,ψGT
+(50)
+where
+Gu = −D1+ �
+KπhFu + KππDWB
+�
+,
+Gh = −D1+KπhFh,
+Gp = −D1+ �
+Sp0 + KπhFp − KππDWB−1
+�
+,
+GT = −D1+ �
+ST0 + KπhFT�
+.
+(51)
+The temperature and density can be obtained by inverting the nonsingular part of D in
+Eqs. (48) and (50). The null space of D is spanned by [ϕ(0)]F, which corresponds to the
+constant term in the Fourier series. Since the lowest-order density (n0) and temperature
+(T0) are constant, we set n(0) = 0 and T(0) = 0 without loss of generality. From the first row
+corresponding to the constant (0) Fourier mode,
+0 = γuFu
+(0) + γhFh
+(0) + ˆp0,ψFp
+(0) + ˆT0,ψFT
+(0),
+(52)
+0 = γuGu
+(0) + γhGh
+(0) + ˆp0,ψGp
+(0) + ˆT0,ψGT
+(0),
+(53)
+we can determine the integration constants γu and γh,
+
+ γu
+γh
+
+ = −
+
+ Fu
+(0) Fh
+(0)
+Gu
+(0) Gh
+(0)
+
+
+−1 
+ Fp
+(0) FT
+(0)
+Gp
+(0) GT
+(0)
+
+
+
+ ˆp0,ψ
+ˆT0,ψ
+
+ .
+(54)
+Then Eqs. (46) and (47) with the constants obtained in Eq. (54) agree with the corresponding
+column vectors of the solution (28). Note that the heat flux obtained here is not a closure
+and satisfies ∇ · h = 0.
+Before concluding this section, a few remarks are in order. First, Eqs. (40) and (41) are
+equivalent to ∇ · (n0V1) = 0 and ∇ · h = 0. Inserting the lowest order solutions V1⊥ =
+(1/qB2)B × ∇p0 and h⊥ = (5p0/2qB2)B × ∇T0 obtained from ∇p0 − n0qV1 × B/m =
+0 and (5/2)p0∇T0 − qh × B = 0, one can derive ˆu = −ˆp0,ψB0/B + γuB/B0 and ˆh =
+−5 ˆT0,ψB0/2B + γhB/B0 where γu and γh are integration constants. Second, Fp and Gp
+vanish when ion-electron collisions are ignored. By setting f1 = g + F, Eq. (3) becomes
+v∥∂∥g = C(g) + C(F). Note that the ˆp0,ψ term in C(F) = C(F, f 0) + C(f 0, F) vanishes due
+20
+
+to momentum conservation and does not affect g. Therefore the term ˆp0,ψ contributes only
+to the flow velocity moment of f1 and hence Fp in Eq. (48) and Gp in Eq. (50) must vanish.
+Third, in the closure calculation, the ˆp0,ψ drive appears in Wθ of the viscosity equation and
+affects closure quantities. However, the ˆp0,ψ term in V1∥ of Wθ exactly cancels the ˆp0,ψ term
+in V1⊥ of Wθ making n1 and T1 independent of the ˆp0,ψ drive. Fourth, for an electron-ion
+plasma (a, b) = (e, i) and (i, e), the ˆpa0,ψ and ˆpb0,ψ drives do not vanish in the collision
+operator C(Fa, f 0
+b ) + C(f 0
+a, Fb) for the ga equation and do affect ga unless V1a∥ = V1b∥.
+V.
+CONCLUSION AND FUTURE WORK
+We have demonstrated how to solve the drift kinetic equation using the general moment
+equations to obtain transport and closure relations.
+Using the moment-Fourier method
+developed here, one can directly solve a full set of parallel moment equations equivalent to
+the drift kinetic equation for fluid variables (density, flow velocity, and temperature) and/or
+fluxes (particle flux, electric current, heat flux, etc.). The solution moments can be used to
+construct the distribution function that is the solution of the drift kinetic equation. One can
+also solve the non-Maxwellian moment equations to express parallel closures in terms of fluid
+variables. The closures can be combined with linearized fluid equations to reproduce the
+fluid variables and/or fluxes obtained from the full set of parallel moment equations. More
+importantly, the closures can be utilized to advance a system of fluid equations in numerical
+simulations with nonlinear terms kept when nonlinear effects are significant. Note that the
+drift kinetic equation yields only linearized fluid equations by nature, e.g. Eqs. (30)-(32),
+and hence cannot capture the nonlinear effects.
+While the formalism developed here is only applied in the case of a single component plasma
+in a circular axisymmetric magnetic field, it can be generalized to a multi-component plasma
+in a tokamak with arbitrarily shaped nested flux surfaces. As long as the magnetic field
+is Fourier-expandable, the moment-Fourier approach developed here is applicable. For a
+multi-component plasma, the collisional heating and friction terms, respectively, will modify
+Eqs. (31) and (32). The collision terms introduce couplings of temperatures and flow veloc-
+ities between unlike species and, as a result, the dp0/dψ term will affect all other fluid and
+closure moments as remarked at the end of Sec. IV. Although ion-electron collisions in the
+21
+
+ion theory are ignored based on the small-mass-ratio approximation in the existing theories
+(including this work), the momentum and energy conservations require those terms in the
+ion fluid equations. These effects can be investigated by solving coupled moment equations
+with the Fourier method. The transport and closure relations for an electron-ion plasma
+will be presented in the near future.
+The moment-Fourier method developed here is applicable to a plasma with an arbitrary
+Knudsen number in a general magnetic field, as long as convergence can be achieved by
+increasing the number of moments and Fourier modes.
+In the high-collisionality limit,
+B/BθλC ≪ 1, the closure coefficients Kαβ in Eqs. (38) and (39) reproduce the corresponding
+Braginskii closure coefficient [32, 33]. In the small inverse aspect ratio limit, ǫ ≪ 1, the Kαβ
+reproduce the corresponding integral closure [31]. In principle, the moment-Fourier solutions
+are practically exact once convergence is achieved. The necessary numbers of moments and
+Fourier modes, respectively, increase as the Knudsen number and the inverse aspect ratio
+increase. In practice, the moment approach is limited by the accuracy of the inverse matrix
+in Eqs. (28) and (37). For low collisionality nFK0 ≳ 104, the required matrix dimension
+for convergence is LKF ≳ 106, and the inverse matrix becomes inaccurate due to a large
+condition number, even with the exact null space eliminated in the case of Eq. (28). For low
+collisionality, the drift kinetic equation may be solved numerically. However, in the collision-
+less limit, we find that the drift kinetic equation should be solved analytically for accurate
+closure and transport relations. The results in the collisionless limit will be presented in the
+near future, too. It is also notable that the finite element basis used in Refs. [18] and [19]
+makes the convergence faster than the Legendre polynomial basis.
+Since the computational effort to calculate the convergent closures is tremendous when
+the effective collisionality is low, it may be impractical to compute the closures during a
+fluid simulation. For practical applications, we plan to develop explicit formulas of closures
+which can be expressed in terms of magnetic field parameters, ǫ for a circular geometry
+or Fourier components for a general magnetic field. The explicit expressions of closures
+can be developed for practical values of ǫ ≲ 0.4 (at the edge of the ITER tokamak) and
+nFK0 ≲ 104 (at the core of ITER). Once the closures have been obtained for the magnetic field
+parameters, they can be conveniently used without time-consuming moment calculations.
+Furthermore, calculating γu in Eq. (46) will be performed for general ǫ and collisionality of
+22
+
+interest for a quantitative analysis of convergence depending on the number of moments and
+Fourier modes.
+DATA AVAILABILITY STATEMENT
+The data that support the findings of this study are available upon request from the authors.
+ACKNOWLEDGMENTS
+The research was supported by the U.S. DOE under Grant Nos. DE-SC0022048 and DE-
+FG02-04ER54746 and by National R&D Program through the National Research Foundation
+of Korea (NRF) funded by Ministry of Science and ICT (2021M3F7A1084419).
+[1] A. A. Galeev and R. Z. Sagdeev, Soviet Physics JETP 26, 233 (1968).
+[2] M. N. Rosenbluth, R. D. Hazeltine, and F. L. Hinton, Phys. Fluids 15, 116 (1972).
+[3] R. D. Hazeltine, F. L. Hinton, and M. N. Rosenbluth, The Physics of Fluids 16, 1645 (1973),
+https://aip.scitation.org/doi/pdf/10.1063/1.1694191.
+[4] F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 (1976).
+[5] S. P. Hirshman and D. J. Sigmar, Nucl. Fusion 21, 1079 (1981).
+[6] C.
+S.
+Chang
+and
+F.
+L.
+Hinton,
+The Physics of Fluids 25, 1493 (1982),
+https://aip.scitation.org/doi/pdf/10.1063/1.863934.
+[7] M. Taguchi, Plasma Physics and Controlled Fusion 30, 1897 (1988).
+[8] R. D. Hazeltine, Plasma Phys. 15, 77 (1973).
+[9] R. D. Hazeltine and J. D. Meiss, Plasma Confinement (Dover Pub., Inc., New York, 2003).
+[10] R. Balescu, Transport Processes in Plasmas (North-Holland, Amsterdam, 1988) vols. 1 and 2.
+[11] P. Helander and D. J. Sigmar, Collisional Transport in Magnetized Plasmas (Cambridge Uni-
+versity Press, Cambridge, 2002).
+[12] E. A. Belli and J. Candy, Plasma Physics and Controlled Fusion 50, 095010 (2008).
+23
+
+[13] E. A. Belli and J. Candy, Plasma Physics and Controlled Fusion 51, 075018 (2009).
+[14] E. A. Belli and J. Candy, Plasma Physics and Controlled Fusion 54, 015015 (2011).
+[15] M. Landreman and D. R. Ernst, Plasma Physics and Controlled Fusion 54, 115006 (2012).
+[16] M. Landreman and D. R. Ernst, Journal of Computational Physics 243, 130 (2013).
+[17] E.
+D.
+Held,
+S.
+E.
+Kruger,
+J.-Y.
+Ji,
+E.
+A.
+Belli,
+and
+B.
+C.
+Lyons,
+Physics of Plasmas 22, 032511 (2015), https://doi.org/10.1063/1.4914165.
+[18] J.
+R.
+Jepson,
+C.
+C.
+Hegna,
+E.
+D.
+Held,
+J.
+A.
+Spencer,
+and
+B.
+C.
+Lyons,
+Physics of Plasmas 28, 082503 (2021), https://doi.org/10.1063/5.0054978.
+[19] J.
+A.
+Spencer,
+B.
+Adair,
+E.
+D.
+Held,
+J.-Y.
+Ji,
+and
+J.
+R.
+Jepson,
+Journal of Computational Physics 450, 110862 (2022).
+[20] C. Sovinec, A. Glasser, T. Gianakon, D. Barnes, R. Nebel, S. Kruger, D. Schnack, S. Plimpton,
+A. Tarditi, and M. Chu, Journal of Computational Physics 195, 355 (2004).
+[21] S. C. Jardin, N. Ferraro, X. Luo, J. Chen, J. Breslau, K. E. Jansen,
+and M. S. Shephard,
+Journal of Physics: Conference Series 125, 012044 (2008).
+[22] J.
+Breslau,
+N.
+Ferraro,
+and
+S.
+Jardin,
+Physics of Plasmas 16, 092503 (2009),
+https://doi.org/10.1063/1.3224035.
+[23] B.
+Dudson,
+M.
+Umansky,
+X.
+Xu,
+P.
+Snyder,
+and
+H.
+Wilson,
+Computer Physics Communications 180, 1467 (2009).
+[24] M. Hoelzl, G. Huijsmans, S. Pamela, M. Bécoulet, E. Nardon, F. Artola, B. Nkonga, C. Atana-
+siu, V. Bandaru, A. Bhole, D. Bonfiglio, A. Cathey, O. Czarny, A. Dvornova, T. Fehér, A. Fil,
+E. Franck, S. Futatani, M. Gruca, H. Guillard, J. Haverkort, I. Holod, D. Hu, S. Kim,
+S. Korving, L. Kos, I. Krebs, L. Kripner, G. Latu, F. Liu, P. Merkel, D. Meshcheriakov,
+V. Mitterauer, S. Mochalskyy, J. Morales, R. Nies, N. Nikulsin, F. Orain, J. Pratt, R. Ra-
+masamy, P. Ramet, C. Reux, K. Särkimäki, N. Schwarz, P. S. Verma, S. Smith, C. Sommariva,
+E. Strumberger, D. van Vugt, M. Verbeek, E. Westerhof, F. Wieschollek,
+and J. Zielinski,
+Nuclear Fusion 61, 065001 (2021).
+[25] J.-Y. Ji and E. D. Held, Phys. Plasmas 21, 042102 (2014).
+[26] J.-Y. Ji and E. D. Held, Phys. Plasmas 13, 102103 (2006).
+[27] J.-Y. Ji and E. D. Held, Phys. Plasmas 16, 102108 (2009).
+[28] R. Jorge, P. Ricci, and N. F. Loureiro, Journal of Plasma Physics 83, 905830606 (2017).
+[29] R. Jorge, B. J. Frei, and P. Ricci, Journal of Plasma Physics 85, 905850604 (2019).
+24
+
+[30] J.-Y. Ji, E. D. Held, and C. R. Sovinec, Phys. Plasmas 16, 022312 (2009).
+[31] J.-Y. Ji, H. Q. Lee, and E. D. Held, Phys. Plasmas 24, 022127 (2017).
+[32] S. I. Braginskii, in Reviews of Plasma Physics, Vol. 1, edited by M. A. Leontovich (Consultants
+Bureau, New York, 1965) p. 205.
+[33] J.-Y. Ji and E. D. Held, Phys. Plasmas 22, 062114 (2015).
+25
+

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+PHOTONIC SPATIAL-EULER ISING MACHINE FOR SOLVING
+20000-NODE MAX-CUT PROBLEM ∗
+Xin Ye, Wenjia Zhang, Shaomeng Wang, Xiaoxuan Yang, Zuyuan He
+State Key Laboratory of Advanced Optical Communication Systems and Networks
+Shanghai Jiao Tong University
+Shanghai 200240, China
+{Wenjia Zhang}[email protected]
+ABSTRACT
+To tackle challenging combinatorial optimization problems, analog computing machines based on
+the nature-inspired Ising model are attracting increasing attentions in order to disruptively overcome
+the impending limitations on conventional electronic computers. Photonic spatial Ising machine has
+become an unique and primitive solution with all-to-all connections to solve large-scale Max-cut
+problems. However, spin configuration and flipping requires two independent sets of spatial light
+modulators (SLMs) for amplitude and phase modulation, which will lead to tremendous engineering
+difficulty of optical alignment and coupling. We report a novel quadrature photonic spatial-Euler
+Ising machine to realize large-scale and flexible spin-interaction configuration and spin-flip in a
+single spatial light modulator, and develop a noise enhancement approach by adding digital white
+noise onto detected optical signals. We experimentally show that such proposal accelerates solving
+(un)weighted, (non)fully connected, 20736-node Max-cut problems, which offers obvious advantages
+over simulation and heuristic algorithm results in digital computers.
+1
+Introduction
+Complex systems related research has progressed at a rapid pace due to high-throughput data acquisition techniques
+[1, 2, 3]. Contrarily, comprehensive processing and optimization of big data with complex structures and correlations is
+a prerequisite for the vast applications and spectacular advancement in bioinformatics [4, 5], pharmaceutical medicine
+[6, 7], finance [8, 9], cryptography [10, 11], and artificial intelligence (AI) [12, 13]. Therefore, powerful mathematical
+models and hardware processors are critically utilised to analyse high-dimensional data sets and complex systems. The
+Ising model, depicting Markov chains of interacting binary units, is a typical model used to study complex systems
+[14, 15]. Various artificial Ising machines developed based on this model accelerate conventional electronic computers
+in performing optimization tasks involving non-deterministic polynomial time (NP)-hard problems and combinatorial
+optimisation tasks, such as the Max-cut, protein folding, number partition and travelling salesman problem(TSP)
+[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
+Among these Ising solutions, the photonic Ising machine, by leveraging light interference to emulate spin interaction in
+ferromagnets, offers substantial benefits of high connectivity and speed in ground state search [37]. Recently, there
+propose various innovative photonic constructions for Ising model, such as optical coherent Ising machines (CIM)
+[19, 24, 20, 21, 22, 23], photonic recurrent Ising sampler (PRIS) [34, 35], and spatial photonic Ising machines (SPIM)
+[28, 29, 30, 31, 32, 33, 38]. These proposals, originated from Ising model given by H = − �
+<l,k> Jl,kxlxk where
+Jl,k is the interaction between spins and spin binary state xl ∈ {1, −1}, are designed to search for ground state of
+Ising model with the minimum Hamiltonian by either iterative sampling or directly evolving the ensemble energy
+regarding the established mapping of a particular combinatorial problem. Although the coherent Ising machines
+performs comparably to the quantum annealing, it lacks the advantages of parallel processing in optical computing since
+it requires an extremely long fiber cavity to simulate spins through temporal multiplexing [19]. Chip-level photonic
+Ising samplers are embedded with specialised heuristic method to provide sample solutions to the ground state of Ising
+∗
+arXiv:2301.04651v1  [cs.ET]  11 Jan 2023
+
+Figure 1: Architecture of the quadrature photonic spatial-Euler Ising machine. (a)The schematic and principle
+of Euler-SIM. (b) Images of the initial and target intensity. The white bar corresponds to the length of 20 µm. (c) Initial
+and final phase masks encoding on SLM in one experiment.
+models, but currently fail to scale up [34] and heuristic algorithms are difficult to converge into optimum point for
+a large-scale problem. In contrast, spatial photonic Ising machines encoding the spins as a phase matrix in spatial
+light modulators (SLMs), can implement spin scales up to tens of thousands [28, 31]. This approach, using spatial
+Fourier transformation as basic building block, can be expressed by H = − �
+<l,k> εlεkxlxk, which indicates that the
+interaction coefficient Jl,k is set by the amplitude modulation εl and εk. This scheme is compatible with an Ising model
+with fully connected interactions (or an equivalent quadratic unconstrained binary optimization (QUBO) problem) due
+to its high connectivity and scalability [31].
+However, the proposed Ising machine still need external spatial amplitude modulator and thereby spin configuration
+and flipping will require two independent sets of spatial light modulators (SLMs) for amplitude and phase modulation,
+which will lead to tremendous engineering difficulty of optical alignment and coupling [31]. In our previous work, we
+proposed quadrature spatial Ising machine to provide flexibility for interaction configuration by introducing spatial
+spins interference with quadrature phase design [32, 39]. However, the proposed Ising machine still need external
+spatial amplitude modulator and thereby spin configuration and flipping will require two independent sets of spatial
+light modulators (SLMs) for amplitude and phase modulation, which will lead to tremendous engineering difficulty of
+optical alignment and coupling.
+In this paper, we propose a novel quadrature photonic spatial-Euler Ising machine (Euler-SIM) where intensity
+modulation is performed based on Euler’s Formula by extending quadrature phase configuration. To estimate the
+performance of Euler-SIM, we conduct experiments and simulations on the Max-cut problem with over 20000 nodes.
+The max cut value in experiment is improved by 32% over simulation results and 34% over Sahni-Gonzales (SG)
+algorithm with a hundredfold speedup. The results demonstrate the superiority of our structure in terms of result
+yield and speed of solving NP-hard problems beyond the traditional von Neumann processor. Furthermore, we also
+investigate noise enhancement approach through experiments, finding that up to 8% performance enhancement by
+adding external Gaussian white noise on the detected optical amplitude.
+2
+Principle of quadrature photonic spatial-Euler Ising machine
+Fig.1(a) shows the architecture design of Euler-SIM. An extended coherent light source shines on the SLM screen.
+The phase mask of SLM is configured by four parts to encode both the interaction coefficients and the spin states. In
+this case, a spin with amplitude information will consist of four parts ei(φl−αl),ei(θl−βl),ei(φl+αl),ei(θl+βl). On the
+one hand, the spin state xl is encoded by the modulated phase φl ∈ {0, π}, and the corresponding yl is encoded by
+the quadrature phase θl ∈ { π
+2 , 3π
+2 }. There satisfies a specific transformation relation between y and x determined by
+the interaction matrix, y = Ax [32]. On the other hand, arbitrary amplitudes scaled down to the range (−1, 1) can be
+converted into phase. According to the corollary to Euler’s Formula, the cosine functions can be interpreted as weighted
+sums of the exponential functions,as
+cos αl = ℜ(eiαl) = eiαl + e−iαl
+2
+(1)
+2
+
+Initial I(u,v)
+Initial Phase Mask
+ei(t-βi)
+20μm
+ei(pi+an)
+ntyi
+Target I(u,v)
+ Final Phase Mask
+H=
+(EIEXiXkNnyiyk)
+,k
+1x13
+15
+20μm
+feedbackFigure 2: Experimental and simulation results of Max-cut problem. (a) Graph division of a 100-node Max-cut
+problem obtained by Euler-SIM. (b) Graph division of a 100-node Max-cut problem obtained with the SG algorithm.
+(c) Experimental searching for max cut value of 20736 nodes. (d) Simulated searching for max cut value of 20736
+nodes. (e) Experimental and simulation results for Max-cut problems with graph densities of [0.5, 1.0].
+Thus, phase and amplitude information can then be encoded simultaneously according to extra phases αl, as
+εlxl = 1
+2[ei(φl−αl) + ei(φl+αl)]
+(2)
+The modulated wave is passed through a lens to achieve spatial Fourier Transform and result in a superimposed effect at
+the centre intensity
+I(0, 0) = (xT ε + yT η)(εT x + ηT y)
+(3)
+which followed an opposite trend to the corresponding Hamiltonian
+H = −
+�
+<l,k>
+(εlεkxlxk ± ηlηkylyk)
+(4)
+Therefore, we can search for the ground state of Ising model by maximising the central light intensity during the
+experiment.
+Based on the above architecture design, we construct the experimental setup. An incident beam with λ = 632.8nm is
+injected into the beam expander with a rectangular aperture to produce a 12.5 × 7.1 mm2 rectangular light spot, which
+completely covers the phase-only reflective SLM (HOLOEYE LETO-3-CFS-127) plane to activate all the 1920×1080
+pixels with a pixel pitch of 6.4 µm. Here, a full-coverage spot is essential to maximise pixel utilisation while minimizing
+modulation-related pixel alignment issues. In the rear, a lens (focal length f = 150 mm) is arranged to perform a
+two-dimensional Fourier transform on the beam with modulated wavefront. Finally, we probe the field intensity by the
+charge-coupled device (CCD) camera on the back focal plane of the lens. The 2758×2208 sensor in the CCD (QSI 600)
+has 800 kHz read rate and 16 bit digital resolution, providing extremely high resolution. The intensity images are loaded
+into the central processing unit (CPU) for subsequent calculations in the electrical domain and feed back. Fig.1(b)
+3
+
+(a)
+(b)
+(c)
+×104
+×108
+.285
+2
+1.8
+3.28
+1.6
+ Distance
+Euclidean Distance
+1.4
+3.275
+Cut Value
+1.2 
+Value
+Euclidean J
+3.27
+1
+0.8
+3.265
+0.6
+0.4
+3.26
+0.2
+Euler-SIM
+SG algorithm
+3.255
+0
+(100 nodes)
+(100 nodes)
+1
+11
+21
+31
+41
+51
+61
+71
+81
+91
+Iteration
+(d)
+(e)
+×108
+×108
+×108
+5
+1.2
+4.5
+2
+4
+Hamiltonian
+3.5
+0.8
+1.5
+Hamiltonian
+Cut Value
+ Value,
+３
+2.5
+Cut
+2
+ Experimental results
+0.4
+1.5
+ Simulated results
+0.5
+1
+0.2
+ SG algorithm
+0.5
+0
+0
+0
+1
+11
+21
+31
+41
+51
+61
+71
+81
+91
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Iteration
+Graph DensityFigure 3: Experimental and simulation results of Max-cut problem with added digital noise. (a) Initial image
+acquired by CCD. (b) Gaussian white noise matrices with noise level of 0.1. (c) Polluted image for computation. (d)
+Simulated results of Max-cut problems with added digital noise of 0.02 0.08. The black dashed line represents the
+unnoised results as a reference line. (e) Experimental results for Max-cut problems with added digital noise. The ideal
+noise levels related to different graph densities are marked at the top of the orange bars.
+exhibits the images of the initial detected intensityI and target intensityIT , which is focused by a uniform beam without
+any modulation. Here, we calculate the Euclidean distance ∥IT − I∥2 as a cost function of the simulated annealing (SA)
+algorithm, thus generating a new phase mask to refresh SLM screen. And the initial and final phase masks describing
+the spin states are illustrated in Fig.1(c). This procedure is continuously cycled to govern the Hamiltonian evolution
+until the system stabilises to the ground state.
+3
+Experiments and Discussion
+3.1
+Experimental performances and numerical simulations
+The Max-cut problem, requiring to find the cut of the given graph into two subsets with the maximum value of their
+connecting weighted edges, can be formulated into an equivalent Ising model without local fields [40]. An unweighted
+and all-to-all connection max-cut problem make it easier by assuming Jl,k taking values of ±1, whereas many NP
+problems can only be converted into weighted sparse max-cut problems for solution [41]. Our proposed scheme
+perfectly implements the mapping of the latter. For each cut, the cut value is denoted as
+W = 1
+2
+�
+<l,k>
+wl,k(1 − xlxk)
+(5)
+where wl,k is the weight between the l-th vertex and the k-th vertex. The related Hamiltonian we use is H =
+�
+<l,k> wl,kxlxk and the weight can be expressed as
+wl,k = cos αl cos αk ± cos βl cos βk
+(6)
+Thus, we can maximise the cut value by looking for the minimum Hamiltonian.
+Given that it is too complex to be solved precisely with a large scale problem as the exact solvers generally fail with
+1000 vertexes [34, 42]. Before the experiments we need to perform a reference calculation on the conventional electrical
+computing platform. Usually, the Goemans-Williamson SDP (GW-SDP) algorithm is one of the most popular methods
+to solve the Max-cut problem with a guarantee of solution quality. However, it fails to solve large-scale problems owing
+to the inordinately long time consumption [43]. Therefore, for large instances, we choose to employ another classical
+greedy heuristic algorithm called the Sahni-Gonzales (SG) method, which is known to find approximate solutions
+to large Max-cut problems in polynomial time, comparable to the GW-SDP [24, 36]. Using this method, a set of
+4
+
+(a)
+(b)
+(c)
+(e)
+×108
+1.85
+0.8
+0.08
+0.6
+1.8
+ Spontaneous Noise Only
+0.4
+0.07
+0.2
+1.75
+ Added Digital Noise
+(d)
+×108
+1.4
+1.7
+-0.02
+0.04
+-0
+1.35
+lue
+1.65
+-0.03
+0.04
+Val
+1.3
+Value
+0.06
+0.08
+0.06
+1.25
+1.55
+0.05
+0.07
+1.2
+1.5
+1.15
+1.45
+1.1
+1.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Graph Density
+Graph DensityTable 1: Performance comparison between Euler-SIM and other Ising machines for solving Max-cut problems.
+Ising machine
+Implementation
+Problem Type
+Problem Scale
+Time to Resolution
+8-FPGA SB machine [44]
+Easy
+All-to-all,Weighted
+16,384-node
+1.2 ms
+PRIS [34]
+Easy
+All-to-all,Unweighted
+100-node
+63 ns per-step
+CIM with DOPO [24]
+Very hard
+All-to-all,Unweighted
+100000-node
+785 µs
+CIM with OEPO [45]
+Hard
+Sparse,Unweighted
+56-node
+4.5 µs
+D-wave 2000Q [46, 36]
+Very hard
+Sparse,Weighted
+2500-node
+> 104 s (for 55 nodes)
+Euler-SIM
+Easy
+All-to-all,Weighted
+20000-node
+325 s
+20736-node Max-cut problems with varies graph densities is implemented on CPUs (Intel i9-13900K, 5.8 GHz) to
+derive the max cut values, consuming 11 hours on average.
+The division of 20736 points is too convoluted to be plotted. We present the division of the 100-node Max-cut problem
+solved by the Euler-SIM in Fig. 2 (a) and the SG algorithm in Fig. 2 (b), respectively. Fig. 2 (c) plots the results of five
+experiments on the weighted Max-cut problem with 20736 fully connected nodes. During the 100 iterations, the cut
+value increases as the Euclidean distance decreases and stabilizes at around 1.759 × 108, with a 122 times speedup
+compared to the SG algorithm.
+Furthermore, we carried out five simulations of the same problem in MATLAB to approximate the operation of the
+photonic Ising machine. As shown in Fig. 2 (d), the cut value remarkably increases to 1.076 × 108 as the Hamiltonian
+of the Ising model converges rapidly. An interesting finding is that the simulation results are inferior to the experimental
+results. Finally, we extended our experiments and simulations for the Max-cut problem with graph densities of 0.5-1.0
+compared with the SG algorithm. The results are shown as Fig. 2 (e), which statistically demonstrates that our
+Euler-SIM offers compelling advantages for handling large-scale Max-cut problem that outweighs electronic computers,
+in comparison with both simulation results and SG algorithm. The experimental max cut values exceed the SG algorithm
+by an average of 34% and achieve a maximum of 49% with graph density of 1.0, which precisely captures the inherent
+advantage of fully connected systems. Additionally, the experimental results routinely outperform the simulated results
+by roughly 32%. The reasons for this occurrence will be discussed in the next section.
+3.2
+Noise enhancement approach
+The detection susceptible to noise may cause some uncertainty in experiments and we speculate that it is the discrepancy
+that makes it easier to jump out of the local optimum and fit better with the SA algorithm, resulting in a better solution.
+In fact, several related works have reported that noise-accelerated or noise-enhanced photonic Ising machines can be
+used to solve large-scale combinatorial optimization problems [29, 35]. Considering that the spontaneous noise of the
+system is challenging to gauge, we develop a noise enhancement approach by adding digital white noise onto detected
+optical signals. More specifically, we generate a group of white noise matrices with different variances and add to
+the CCD acquired images (after normalization) separately to provide a group of polluted images for computation and
+feedback, as shown in Fig. 3(a)-(c).
+Since large noise can blur the image and prevent the algorithm from converging, we pre-simulate to clarify a suitable
+range of noise levels, defined as variance. The noise level is eventually lowered to less than 0.1, ensuring the correct
+execution of the algorithm to obtain a feasible solution. And the simulated results in Fig. 3(d) show that Gaussian noise
+with a noise level of 0.02 to 0.03 may enhance the outcomes, which will be more striking for higher graph densities, up
+to 2.7%. In subsequent experiments shown as Fig. 3(e), we find that the added digital noise do improve the experimental
+results with an average increase of 2.9%. Similar to the numerical simulation results, a more significant improvement
+still appears in the larger graph density, up to 8.6%. Note that the ideal noise level fluctuates with the graph density
+rather than in a fixed threshold, which is different from numerical simulations.
+3.3
+Discussion on the Euler-SIM performance
+We also compare the performance of the Euler-SIM in solving Max-Cut problems to other Ising machines in Table
+1. We evaluated relevant metrics, such as implementation, problem type and scale, time to resolution (or speed) and
+obtained the following conclusions:
+1. Efficiently solving large-scale Max-cut problems. Compared to most solutions [44, 34, 45, 46], we comfortably
+solve Max-cut problems with size over 20,000, approaching the highest reported record so far [24]. In fact, we
+take the adjacent 10×10 pixels as an operation unit for the same encoding to ensure the consistency of the
+5
+
+Ising system in our experiments, thus do not maximise the use of all pixel points. With further optimisation of
+the alignment and detection capabilities, it is feasible to scale up the problem hundredfold.
+2. Flexible mapping of (non)fully connected Max-cut problems with arbitrary amplitude. Considering experi-
+mental setups, many schemes prefer to demonstrate the process of solving the benchmark sparse unweighted
+Max-cut problem [34, 24, 45]. Obviously, being unweighted reduces the complexity, and fully connected
+problems are of more practical value and harder to implement than sparse ones [44]. As a result, many designs
+take great efforts to achieve fully connection. Quantum annealers sacrifice scale, and CIMs also address this
+deficiency by various schemes [15]. And achieving weighted is even harder. However, it is where SIM excel
+and our design further magnifies this advantage by liberal switching between fully and non-fully connected,
+weighted and unweighted problems.
+3. Simple and cost-effective experimental construction. Large power consumption and high costs are required
+by quantum annealers because of the cryogenic environment. Even CIMs impose rigorous experimental
+requirements. Fiber oscillators of tens of kilometres are applied to keep optical loss and optical gain within
+thresholds and thus guarantee spin-to-spin coupling, bringing fairly large roundtrip loss [15]. In contrast, our
+approach based on a simple SLM is superior in terms of experimental cost and manoeuvrability.
+Despite the fact that different Ising machines demonstrate their respective attractions in tackling Max-cut problems, such
+as ultra-large scale [24], ultra-high speed [34], high stability [45], and arbitrary Max-cut problem mapping [44, 34], our
+design, by adopting a more economical experimental architecture, achieves the magnitude adjacent to the largest scale
+and free mapping of (non)fully connected, (un)weighted Max-cut problems, which has greater practical implications for
+solving NP-hard problems. Although the design leaves much to be desired in terms of computational speed, which
+is constrained by the optoelectronic transmission of data and the refresh frequency of the SLM, it still exhibits speed
+advantages over electrical computation and even quantum annealing.
+4
+Conclusion
+In summary, our proposed Euler-SIM utilises Euler’s Formula to achieve amplitude-phase integrated modulation
+and solves the Max-cut problem with 20,736 nodes. The experimental results present around 32% max cut value
+improvement over simulations and 34% over SG algorithm running on the electronic computer, validating better
+optimization performance and fast speed of the optical computing paradigm. Additionally, it is also a noise-friendly
+Ising machine that not only exhibits a large tolerance for system noise, but even rationalizes noise as a potential boost
+to system performance. Thus, this Euler-SIM will become a large-scale optical stochastic computing architecture for
+solving optimization problems of various complex systems.
+Acknowledgments
+This work is supported by the National Key Research and Development Program of China under Grant
+2019YFB1802903, National Natural Science Foundation of China under Grant 62175146 and 62235011 and Major Key
+Project of PCL (PCL2021A14).
+References
+[1] Giorgio Parisi. Complex systems: a physicist’s viewpoint. arXiv preprint cond-mat/0205297, 2002.
+[2] Marc Mézard, Giorgio Parisi, and Miguel Angel Virasoro. Spin glass theory and beyond: An Introduction to the
+Replica Method and Its Applications, volume 9. World Scientific Publishing Company, 1987.
+[3] Miguel Aguilera, S Amin Moosavi, and Hideaki Shimazaki. A unifying framework for mean-field theories of
+asymmetric kinetic ising systems. Nature communications, 12(1):1–12, 2021.
+[4] Qinghua Liu, Liman Wang, Anthony G Frutos, Anne E Condon, Robert M Corn, and Lloyd M Smith. Dna
+computing on surfaces. Nature, 403(6766):175–179, 2000.
+[5] Andrea Degasperi, Dirk Fey, and Boris N Kholodenko. Performance of objective functions and optimisation
+procedures for parameter estimation in system biology models. NPJ systems biology and applications, 3(1):1–9,
+2017.
+[6] Lars-Petter Granan. The ising model applied on chronification of pain. Pain Medicine, 17(1):5–9, 2016.
+[7] Douglas B Kitchen, Hélène Decornez, John R Furr, and Jürgen Bajorath. Docking and scoring in virtual screening
+for drug discovery: methods and applications. Nature reviews Drug discovery, 3(11):935–949, 2004.
+6
+
+[8] Rosario N Mantegna and H Eugene Stanley. Introduction to econophysics: correlations and complexity in finance.
+Cambridge university press, 1999.
+[9] Stan Van Hoesel and Rudolf Müller. Optimization in electronic markets: examples in combinatorial auctions.
+Netnomics, 3(1):23–33, 2001.
+[10] Baonan Wang, Feng Hu, Haonan Yao, and Chao Wang. Prime factorization algorithm based on parameter
+optimization of ising model. Scientific reports, 10(1):1–10, 2020.
+[11] Andrew D King, Juan Carrasquilla, Jack Raymond, Isil Ozfidan, Evgeny Andriyash, Andrew Berkley, Mauricio
+Reis, Trevor Lanting, Richard Harris, Fabio Altomare, et al.
+Observation of topological phenomena in a
+programmable lattice of 1,800 qubits. Nature, 560(7719):456–460, 2018.
+[12] Masanao Yamaoka, Chihiro Yoshimura, Masato Hayashi, Takuya Okuyama, Hidetaka Aoki, and Hiroyuki Mizuno.
+Ising computer. Hitachi Review, 65(6):157, 2016.
+[13] Qiang Zhang, Dandan Deng, Wenting Dai, Jixin Li, and Xinwen Jin. Optimization of culture conditions for
+differentiation of melon based on artificial neural network and genetic algorithm. Scientific Reports, 10(1):1–8,
+2020.
+[14] Andrew Lucas. Ising formulations of many np problems. Frontiers in physics, page 5, 2014.
+[15] Naeimeh Mohseni, Peter L McMahon, and Tim Byrnes. Ising machines as hardware solvers of combinatorial
+optimization problems. Nature Reviews Physics, 4(6):363–379, 2022.
+[16] Kihwan Kim, M-S Chang, Simcha Korenblit, Rajibul Islam, Emily E Edwards, James K Freericks, G-D Lin,
+L-M Duan, and Christopher Monroe. Quantum simulation of frustrated ising spins with trapped ions. Nature,
+465(7298):590–593, 2010.
+[17] Colin D Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M Sage. Trapped-ion quantum computing:
+Progress and challenges. Applied Physics Reviews, 6(2):021314, 2019.
+[18] DL—Bixby Applegate and V RE—Chvátal. Cook, wj: The traveling salesman problem: A computational study,
+2007.
+[19] Fuxi Cai, Suhas Kumar, Thomas Van Vaerenbergh, Xia Sheng, Rui Liu, Can Li, Zhan Liu, Martin Foltin, Shimeng
+Yu, Qiangfei Xia, et al. Power-efficient combinatorial optimization using intrinsic noise in memristor hopfield
+neural networks. Nature Electronics, 3(7):409–418, 2020.
+[20] Masoud Babaeian, Dan T Nguyen, Veysi Demir, Mehmetcan Akbulut, Pierre-A Blanche, Yushi Kaneda, Saikat
+Guha, Mark A Neifeld, and N Peyghambarian. A single shot coherent ising machine based on a network of
+injection-locked multicore fiber lasers. Nature communications, 10(1):1–11, 2019.
+[21] Takahiro Inagaki, Yoshitaka Haribara, Koji Igarashi, Tomohiro Sonobe, Shuhei Tamate, Toshimori Honjo, Alireza
+Marandi, Peter L McMahon, Takeshi Umeki, Koji Enbutsu, et al. A coherent ising machine for 2000-node
+optimization problems. Science, 354(6312):603–606, 2016.
+[22] Peter L McMahon, Alireza Marandi, Yoshitaka Haribara, Ryan Hamerly, Carsten Langrock, Shuhei Tamate,
+Takahiro Inagaki, Hiroki Takesue, Shoko Utsunomiya, Kazuyuki Aihara, et al. A fully programmable 100-spin
+coherent ising machine with all-to-all connections. Science, 354(6312):614–617, 2016.
+[23] Ryan Hamerly, Takahiro Inagaki, Peter L McMahon, Davide Venturelli, Alireza Marandi, Tatsuhiro Onodera,
+Edwin Ng, Carsten Langrock, Kensuke Inaba, Toshimori Honjo, et al. Experimental investigation of performance
+differences between coherent ising machines and a quantum annealer. Science advances, 5(5):eaau0823, 2019.
+[24] Toshimori Honjo, Tomohiro Sonobe, Kensuke Inaba, Takahiro Inagaki, Takuya Ikuta, Yasuhiro Yamada, Takushi
+Kazama, Koji Enbutsu, Takeshi Umeki, Ryoichi Kasahara, et al. 100,000-spin coherent ising machine. Science
+advances, 7(40):eabh0952, 2021.
+[25] Chihiro Yoshimura, Masato Hayashi, Takuya Okuyama, and Masanao Yamaoka. Implementation and evaluation
+of fpga-based annealing processor for ising model by use of resource sharing. International Journal of Networking
+and Computing, 7(2):154–172, 2017.
+[26] Igor Gershenzon, Geva Arwas, Sagie Gadasi, Chene Tradonsky, Asher Friesem, Oren Raz, and Nir Davidson. Exact
+mapping between a laser network loss rate and the classical xy hamiltonian by laser loss control. Nanophotonics,
+9(13):4117–4126, 2020.
+[27] Peter L McMahon, Alireza Marandi, Yoshitaka Haribara, Ryan Hamerly, Carsten Langrock, Shuhei Tamate,
+Takahiro Inagaki, Hiroki Takesue, Shoko Utsunomiya, Kazuyuki Aihara, et al. A fully programmable 100-spin
+coherent ising machine with all-to-all connections. Science, 354(6312):614–617, 2016.
+[28] D Pierangeli, G Marcucci, and C Conti. Large-scale photonic ising machine by spatial light modulation. Physical
+review letters, 122(21):213902, 2019.
+7
+
+[29] Davide Pierangeli, Giulia Marcucci, Daniel Brunner, and Claudio Conti. Noise-enhanced spatial-photonic ising
+machine. Nanophotonics, 9(13):4109–4116, 2020.
+[30] Davide Pierangeli, Giulia Marcucci, and Claudio Conti. Adiabatic evolution on a spatial-photonic ising machine.
+Optica, 7(11):1535–1543, 2020.
+[31] Yisheng Fang, Junyi Huang, and Zhichao Ruan. Experimental observation of phase transitions in spatial photonic
+ising machine. Physical Review Letters, 127(4):043902, 2021.
+[32] Wenchen Sun, Wenjia Zhang, Yuanyuan Liu, Qingwen Liu, and Zuyuan He. Quadrature photonic spatial ising
+machine. Optics Letters, 47(6):1498–1501, 2022.
+[33] Jiayi Ouyang, Yuxuan Liao, Zhiyao Ma, Deyang Kong, Xue Feng, Xiang Zhang, Xiaowen Dong, Kaiyu Cui,
+Fang Liu, Wei Zhang, et al. An on-demand photonic ising machine with simplified hamiltonian calculation by
+phase-encoding and intensity detection. arXiv preprint arXiv:2207.05072, 2022.
+[34] Charles Roques-Carmes, Yichen Shen, Cristian Zanoci, Mihika Prabhu, Fadi Atieh, Li Jing, Tena Dubˇcek, Chenkai
+Mao, Miles R Johnson, Vladimir ˇCeperi´c, et al. Heuristic recurrent algorithms for photonic ising machines. Nature
+communications, 11(1):1–8, 2020.
+[35] Mihika Prabhu, Charles Roques-Carmes, Yichen Shen, Nicholas Harris, Li Jing, Jacques Carolan, Ryan Hamerly,
+Tom Baehr-Jones, Michael Hochberg, Vladimir ˇCeperi´c, et al. Accelerating recurrent ising machines in photonic
+integrated circuits. Optica, 7(5):551–558, 2020.
+[36] Suryendy Dutta, Abhishek Khanna, AS Assoa, Hanjong Paik, Darrell G Schlom, Zoltán Toroczkai, Arijit
+Raychowdhury, and Suman Datta. An ising hamiltonian solver based on coupled stochastic phase-transition
+nano-oscillators. Nature Electronics, 4(7):502–512, 2021.
+[37] Ernst Ising. Beitrag zur theorie des ferro-und paramagnetismus. PhD thesis, Grefe & Tiedemann, 1924.
+[38] Santosh Kumar, He Zhang, and Yu-Ping Huang. Large-scale ising emulation with four body interaction and
+all-to-all connections. Communications Physics, 3(1):1–9, 2020.
+[39] Wenchen Sun. In OFC 2022, page M2G.4, 2022.
+[40] Michel X Goemans and David P Williamson.
+Improved approximation algorithms for maximum cut and
+satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6):1115–1145, 1995.
+[41] Michael R Garey. A guide to the theory of np-completeness. Computers and intractability, 1979.
+[42] Sera Kahruman, Elif Kolotoglu, Sergiy Butenko, and Illya V Hicks. On greedy construction heuristics for the
+max-cut problem. International Journal of Computational Science and Engineering, 3(3):211–218, 2007.
+[43] Takuya Okuyama, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Masanao Yamaoka. Binary optimization by
+momentum annealing. Physical Review E, 100(1):012111, 2019.
+[44] Kosuke Tatsumura, Masaya Yamasaki, and Hayato Goto. Scaling out ising machines using a multi-chip architecture
+for simulated bifurcation. Nature Electronics, 4(3):208–217, 2021.
+[45] Qizhuang Cen, Hao Ding, Tengfei Hao, Shanhong Guan, Zhiqiang Qin, Jiaming Lyu, Wei Li, Ninghua Zhu, Kun
+Xu, Yitang Dai, et al. Large-scale coherent ising machine based on optoelectronic parametric oscillator. Light:
+Science & Applications, 11(1):1–10, 2022.
+[46] D-Wave Systems Inc.
+Hybrid solvers for quadratic optimization.
+https://www.dwavesys.com/media/
+soxph512/hybrid-solvers-for-quadratic-optimization.pdf, April 2022.
+8
+

8NE3T4oBgHgl3EQfqQrl/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf,len=396
+page_content='PHOTONIC SPATIAL-EULER ISING MACHINE FOR SOLVING  20000-NODE MAX-CUT PROBLEM ∗  Xin Ye, Wenjia Zhang, Shaomeng Wang, Xiaoxuan Yang, Zuyuan He  State Key Laboratory of Advanced Optical Communication Systems and Networks  Shanghai Jiao Tong University  Shanghai 200240, China  {Wenjia Zhang}wenjia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='zhang@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='cn  ABSTRACT  To tackle challenging combinatorial optimization problems, analog computing machines based on  the nature-inspired Ising model are attracting increasing attentions in order to disruptively overcome  the impending limitations on conventional electronic computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Photonic spatial Ising machine has  become an unique and primitive solution with all-to-all connections to solve large-scale Max-cut  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' However, spin configuration and flipping requires two independent sets of spatial light  modulators (SLMs) for amplitude and phase modulation, which will lead to tremendous engineering  difficulty of optical alignment and coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' We report a novel quadrature photonic spatial-Euler  Ising machine to realize large-scale and flexible spin-interaction configuration and spin-flip in a  single spatial light modulator, and develop a noise enhancement approach by adding digital white  noise onto detected optical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' We experimentally show that such proposal accelerates solving  (un)weighted, (non)fully connected, 20736-node Max-cut problems, which offers obvious advantages  over simulation and heuristic algorithm results in digital computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  1  Introduction  Complex systems related research has progressed at a rapid pace due to high-throughput data acquisition techniques  [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Contrarily, comprehensive processing and optimization of big data with complex structures and correlations is  a prerequisite for the vast applications and spectacular advancement in bioinformatics [4, 5], pharmaceutical medicine  [6, 7], finance [8, 9], cryptography [10, 11], and artificial intelligence (AI) [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Therefore, powerful mathematical  models and hardware processors are critically utilised to analyse high-dimensional data sets and complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The  Ising model, depicting Markov chains of interacting binary units, is a typical model used to study complex systems  [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Various artificial Ising machines developed based on this model accelerate conventional electronic computers  in performing optimization tasks involving non-deterministic polynomial time (NP)-hard problems and combinatorial  optimisation tasks, such as the Max-cut, protein folding, number partition and travelling salesman problem(TSP)  [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Among these Ising solutions, the photonic Ising machine, by leveraging light interference to emulate spin interaction in  ferromagnets, offers substantial benefits of high connectivity and speed in ground state search [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Recently, there  propose various innovative photonic constructions for Ising model, such as optical coherent Ising machines (CIM)  [19, 24, 20, 21, 22, 23], photonic recurrent Ising sampler (PRIS) [34, 35], and spatial photonic Ising machines (SPIM)  [28, 29, 30, 31, 32, 33, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' These proposals, originated from Ising model given by H = − �  <l,k> Jl,kxlxk where  Jl,k is the interaction between spins and spin binary state xl ∈ {1, −1}, are designed to search for ground state of  Ising model with the minimum Hamiltonian by either iterative sampling or directly evolving the ensemble energy  regarding the established mapping of a particular combinatorial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Although the coherent Ising machines  performs comparably to the quantum annealing, it lacks the advantages of parallel processing in optical computing since  it requires an extremely long fiber cavity to simulate spins through temporal multiplexing [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Chip-level photonic  Ising samplers are embedded with specialised heuristic method to provide sample solutions to the ground state of Ising  ∗  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='04651v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='ET]  11 Jan 2023  Figure 1: Architecture of the quadrature photonic spatial-Euler Ising machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (a)The schematic and principle  of Euler-SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (b) Images of the initial and target intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The white bar corresponds to the length of 20 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (c) Initial  and final phase masks encoding on SLM in one experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  models, but currently fail to scale up [34] and heuristic algorithms are difficult to converge into optimum point for  a large-scale problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' In contrast, spatial photonic Ising machines encoding the spins as a phase matrix in spatial  light modulators (SLMs), can implement spin scales up to tens of thousands [28, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' This approach, using spatial  Fourier transformation as basic building block, can be expressed by H = − �  <l,k> εlεkxlxk, which indicates that the  interaction coefficient Jl,k is set by the amplitude modulation εl and εk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' This scheme is compatible with an Ising model  with fully connected interactions (or an equivalent quadratic unconstrained binary optimization (QUBO) problem) due  to its high connectivity and scalability [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  However, the proposed Ising machine still need external spatial amplitude modulator and thereby spin configuration  and flipping will require two independent sets of spatial light modulators (SLMs) for amplitude and phase modulation,  which will lead to tremendous engineering difficulty of optical alignment and coupling [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' In our previous work, we  proposed quadrature spatial Ising machine to provide flexibility for interaction configuration by introducing spatial  spins interference with quadrature phase design [32, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' However, the proposed Ising machine still need external  spatial amplitude modulator and thereby spin configuration and flipping will require two independent sets of spatial  light modulators (SLMs) for amplitude and phase modulation, which will lead to tremendous engineering difficulty of  optical alignment and coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  In this paper, we propose a novel quadrature photonic spatial-Euler Ising machine (Euler-SIM) where intensity  modulation is performed based on Euler’s Formula by extending quadrature phase configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' To estimate the  performance of Euler-SIM, we conduct experiments and simulations on the Max-cut problem with over 20000 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  The max cut value in experiment is improved by 32% over simulation results and 34% over Sahni-Gonzales (SG)  algorithm with a hundredfold speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The results demonstrate the superiority of our structure in terms of result  yield and speed of solving NP-hard problems beyond the traditional von Neumann processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Furthermore, we also  investigate noise enhancement approach through experiments, finding that up to 8% performance enhancement by  adding external Gaussian white noise on the detected optical amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  2  Principle of quadrature photonic spatial-Euler Ising machine  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='1(a) shows the architecture design of Euler-SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' An extended coherent light source shines on the SLM screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  The phase mask of SLM is configured by four parts to encode both the interaction coefficients and the spin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' In  this case, a spin with amplitude information will consist of four parts ei(φl−αl),ei(θl−βl),ei(φl+αl),ei(θl+βl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' On the  one hand, the spin state xl is encoded by the modulated phase φl ∈ {0, π}, and the corresponding yl is encoded by  the quadrature phase θl ∈ { π  2 , 3π  2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' There satisfies a specific transformation relation between y and x determined by  the interaction matrix, y = Ax [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' On the other hand, arbitrary amplitudes scaled down to the range (−1, 1) can be  converted into phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' According to the corollary to Euler’s Formula, the cosine functions can be interpreted as weighted  sums of the exponential functions,as  cos αl = ℜ(eiαl) = eiαl + e−iαl  2  (1)  2  Initial I(u,v)  Initial Phase Mask  ei(t-βi)  20μm  ei(pi+an)  ntyi  Target I(u,v)  Final Phase Mask  H=  (EIEXiXkNnyiyk)  ,k  1x13  15  20μm  feedbackFigure 2: Experimental and simulation results of Max-cut problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (a) Graph division of a 100-node Max-cut  problem obtained by Euler-SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (b) Graph division of a 100-node Max-cut problem obtained with the SG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  (c) Experimental searching for max cut value of 20736 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (d) Simulated searching for max cut value of 20736  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (e) Experimental and simulation results for Max-cut problems with graph densities of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' phase and amplitude information can then be encoded simultaneously according to extra phases αl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' as  εlxl = 1  2[ei(φl−αl) + ei(φl+αl)]  (2)  The modulated wave is passed through a lens to achieve spatial Fourier Transform and result in a superimposed effect at  the centre intensity  I(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 0) = (xT ε + yT η)(εT x + ηT y)  (3)  which followed an opposite trend to the corresponding Hamiltonian  H = −  �  <l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='k>  (εlεkxlxk ± ηlηkylyk)  (4)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' we can search for the ground state of Ising model by maximising the central light intensity during the  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Based on the above architecture design, we construct the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' An incident beam with λ = 632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8nm is  injected into the beam expander with a rectangular aperture to produce a 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='1 mm2 rectangular light spot, which  completely covers the phase-only reflective SLM (HOLOEYE LETO-3-CFS-127) plane to activate all the 1920×1080  pixels with a pixel pitch of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Here, a full-coverage spot is essential to maximise pixel utilisation while minimizing  modulation-related pixel alignment issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' In the rear, a lens (focal length f = 150 mm) is arranged to perform a  two-dimensional Fourier transform on the beam with modulated wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Finally, we probe the field intensity by the  charge-coupled device (CCD) camera on the back focal plane of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The 2758×2208 sensor in the CCD (QSI 600)  has 800 kHz read rate and 16 bit digital resolution, providing extremely high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The intensity images are loaded  into the central processing unit (CPU) for subsequent calculations in the electrical domain and feed back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='1(b)  3  (a)  (b)  (c)  ×104  ×108  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='285  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='6  Distance  Euclidean Distance  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='275  Cut Value  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='2  Value  Euclidean J  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='27  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='265  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='2  Euler-SIM  SG algorithm  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='255  0  (100 nodes)  (100 nodes)  1  11  21  31  41  51  61  71  81  91  Iteration  (d)  (e)  ×108  ×108  ×108  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  2  4  Hamiltonian  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  Hamiltonian  Cut Value  Value,  ３  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  Cut  2  Experimental results  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  Simulated results  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='2  SG algorithm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  0  0  0  1  11  21  31  41  51  61  71  81  91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='9  1  Iteration  Graph DensityFigure 3: Experimental and simulation results of Max-cut problem with added digital noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (a) Initial image  acquired by CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (b) Gaussian white noise matrices with noise level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (c) Polluted image for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (d)  Simulated results of Max-cut problems with added digital noise of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The black dashed line represents the  unnoised results as a reference line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (e) Experimental results for Max-cut problems with added digital noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The ideal  noise levels related to different graph densities are marked at the top of the orange bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  exhibits the images of the initial detected intensityI and target intensityIT , which is focused by a uniform beam without  any modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Here, we calculate the Euclidean distance ∥IT − I∥2 as a cost function of the simulated annealing (SA)  algorithm, thus generating a new phase mask to refresh SLM screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' And the initial and final phase masks describing  the spin states are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' This procedure is continuously cycled to govern the Hamiltonian evolution  until the system stabilises to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  3  Experiments and Discussion  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='1  Experimental performances and numerical simulations  The Max-cut problem, requiring to find the cut of the given graph into two subsets with the maximum value of their  connecting weighted edges, can be formulated into an equivalent Ising model without local fields [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' An unweighted  and all-to-all connection max-cut problem make it easier by assuming Jl,k taking values of ±1, whereas many NP  problems can only be converted into weighted sparse max-cut problems for solution [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Our proposed scheme  perfectly implements the mapping of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' For each cut, the cut value is denoted as  W = 1  2  �  <l,k>  wl,k(1 − xlxk)  (5)  where wl,k is the weight between the l-th vertex and the k-th vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The related Hamiltonian we use is H =  �  <l,k> wl,kxlxk and the weight can be expressed as  wl,k = cos αl cos αk ± cos βl cos βk  (6)  Thus, we can maximise the cut value by looking for the minimum Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Given that it is too complex to be solved precisely with a large scale problem as the exact solvers generally fail with  1000 vertexes [34, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Before the experiments we need to perform a reference calculation on the conventional electrical  computing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Usually, the Goemans-Williamson SDP (GW-SDP) algorithm is one of the most popular methods  to solve the Max-cut problem with a guarantee of solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' However, it fails to solve large-scale problems owing  to the inordinately long time consumption [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Therefore, for large instances, we choose to employ another classical  greedy heuristic algorithm called the Sahni-Gonzales (SG) method, which is known to find approximate solutions  to large Max-cut problems in polynomial time, comparable to the GW-SDP [24, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Using this method, a set of  4  (a)  (b)  (c)  (e)  ×108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8  Spontaneous Noise Only  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='75  Added Digital Noise  (d)  ×108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='04  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='35  lue  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='04  Val  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='3  Value  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='9  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='9  1  Graph Density  Graph DensityTable 1: Performance comparison between Euler-SIM and other Ising machines for solving Max-cut problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Ising machine  Implementation  Problem Type  Problem Scale  Time to Resolution  8-FPGA SB machine [44]  Easy  All-to-all,Weighted  16,384-node  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='2 ms  PRIS [34]  Easy  All-to-all,Unweighted  100-node  63 ns per-step  CIM with DOPO [24]  Very hard  All-to-all,Unweighted  100000-node  785 µs  CIM with OEPO [45]  Hard  Sparse,Unweighted  56-node  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5 µs  D-wave 2000Q [46, 36]  Very hard  Sparse,Weighted  2500-node  > 104 s (for 55 nodes)  Euler-SIM  Easy  All-to-all,Weighted  20000-node  325 s  20736-node Max-cut problems with varies graph densities is implemented on CPUs (Intel i9-13900K, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='8 GHz) to  derive the max cut values, consuming 11 hours on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  The division of 20736 points is too convoluted to be plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' We present the division of the 100-node Max-cut problem  solved by the Euler-SIM in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 2 (a) and the SG algorithm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 2 (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 2 (c) plots the results of five  experiments on the weighted Max-cut problem with 20736 fully connected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' During the 100 iterations, the cut  value increases as the Euclidean distance decreases and stabilizes at around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='759 × 108, with a 122 times speedup  compared to the SG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Furthermore, we carried out five simulations of the same problem in MATLAB to approximate the operation of the  photonic Ising machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 2 (d), the cut value remarkably increases to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='076 × 108 as the Hamiltonian  of the Ising model converges rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' An interesting finding is that the simulation results are inferior to the experimental  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Finally, we extended our experiments and simulations for the Max-cut problem with graph densities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='0  compared with the SG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The results are shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 2 (e), which statistically demonstrates that our  Euler-SIM offers compelling advantages for handling large-scale Max-cut problem that outweighs electronic computers,  in comparison with both simulation results and SG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The experimental max cut values exceed the SG algorithm  by an average of 34% and achieve a maximum of 49% with graph density of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='0, which precisely captures the inherent  advantage of fully connected systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Additionally, the experimental results routinely outperform the simulated results  by roughly 32%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The reasons for this occurrence will be discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='2  Noise enhancement approach  The detection susceptible to noise may cause some uncertainty in experiments and we speculate that it is the discrepancy  that makes it easier to jump out of the local optimum and fit better with the SA algorithm, resulting in a better solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  In fact, several related works have reported that noise-accelerated or noise-enhanced photonic Ising machines can be  used to solve large-scale combinatorial optimization problems [29, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Considering that the spontaneous noise of the  system is challenging to gauge, we develop a noise enhancement approach by adding digital white noise onto detected  optical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' More specifically, we generate a group of white noise matrices with different variances and add to  the CCD acquired images (after normalization) separately to provide a group of polluted images for computation and  feedback, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 3(a)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Since large noise can blur the image and prevent the algorithm from converging, we pre-simulate to clarify a suitable  range of noise levels, defined as variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The noise level is eventually lowered to less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='1, ensuring the correct  execution of the algorithm to obtain a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' And the simulated results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 3(d) show that Gaussian noise  with a noise level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='02 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='03 may enhance the outcomes, which will be more striking for higher graph densities, up  to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' In subsequent experiments shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 3(e), we find that the added digital noise do improve the experimental  results with an average increase of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Similar to the numerical simulation results, a more significant improvement  still appears in the larger graph density, up to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Note that the ideal noise level fluctuates with the graph density  rather than in a fixed threshold, which is different from numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='3  Discussion on the Euler-SIM performance  We also compare the performance of the Euler-SIM in solving Max-Cut problems to other Ising machines in Table  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' We evaluated relevant metrics, such as implementation, problem type and scale, time to resolution (or speed) and  obtained the following conclusions:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Efficiently solving large-scale Max-cut problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Compared to most solutions [44, 34, 45, 46], we comfortably  solve Max-cut problems with size over 20,000, approaching the highest reported record so far [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' In fact, we  take the adjacent 10×10 pixels as an operation unit for the same encoding to ensure the consistency of the  5  Ising system in our experiments, thus do not maximise the use of all pixel points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' With further optimisation of  the alignment and detection capabilities, it is feasible to scale up the problem hundredfold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Flexible mapping of (non)fully connected Max-cut problems with arbitrary amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Considering experi-  mental setups, many schemes prefer to demonstrate the process of solving the benchmark sparse unweighted  Max-cut problem [34, 24, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Obviously, being unweighted reduces the complexity, and fully connected  problems are of more practical value and harder to implement than sparse ones [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' As a result, many designs  take great efforts to achieve fully connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Quantum annealers sacrifice scale, and CIMs also address this  deficiency by various schemes [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' And achieving weighted is even harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' However, it is where SIM excel  and our design further magnifies this advantage by liberal switching between fully and non-fully connected,  weighted and unweighted problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Simple and cost-effective experimental construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Large power consumption and high costs are required  by quantum annealers because of the cryogenic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Even CIMs impose rigorous experimental  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Fiber oscillators of tens of kilometres are applied to keep optical loss and optical gain within  thresholds and thus guarantee spin-to-spin coupling, bringing fairly large roundtrip loss [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' In contrast, our  approach based on a simple SLM is superior in terms of experimental cost and manoeuvrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Despite the fact that different Ising machines demonstrate their respective attractions in tackling Max-cut problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' such  as ultra-large scale [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' ultra-high speed [34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' high stability [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' and arbitrary Max-cut problem mapping [44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' our  design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' by adopting a more economical experimental architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' achieves the magnitude adjacent to the largest scale  and free mapping of (non)fully connected,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' (un)weighted Max-cut problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' which has greater practical implications for  solving NP-hard problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Although the design leaves much to be desired in terms of computational speed, which  is constrained by the optoelectronic transmission of data and the refresh frequency of the SLM, it still exhibits speed  advantages over electrical computation and even quantum annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  4  Conclusion  In summary, our proposed Euler-SIM utilises Euler’s Formula to achieve amplitude-phase integrated modulation  and solves the Max-cut problem with 20,736 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The experimental results present around 32% max cut value  improvement over simulations and 34% over SG algorithm running on the electronic computer, validating better  optimization performance and fast speed of the optical computing paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Additionally, it is also a noise-friendly  Ising machine that not only exhibits a large tolerance for system noise, but even rationalizes noise as a potential boost  to system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Thus, this Euler-SIM will become a large-scale optical stochastic computing architecture for  solving optimization problems of various complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Acknowledgments  This work is supported by the National Key Research and Development Program of China under Grant  2019YFB1802903, National Natural Science Foundation of China under Grant 62175146 and 62235011 and Major Key  Project of PCL (PCL2021A14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  References  [1] Giorgio Parisi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Complex systems: a physicist’s viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' arXiv preprint cond-mat/0205297, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [2] Marc Mézard, Giorgio Parisi, and Miguel Angel Virasoro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Spin glass theory and beyond: An Introduction to the  Replica Method and Its Applications, volume 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' World Scientific Publishing Company, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [3] Miguel Aguilera, S Amin Moosavi, and Hideaki Shimazaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' A unifying framework for mean-field theories of  asymmetric kinetic ising systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature communications, 12(1):1–12, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [4] Qinghua Liu, Liman Wang, Anthony G Frutos, Anne E Condon, Robert M Corn, and Lloyd M Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Dna  computing on surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature, 403(6766):175–179, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [5] Andrea Degasperi, Dirk Fey, and Boris N Kholodenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Performance of objective functions and optimisation  procedures for parameter estimation in system biology models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' NPJ systems biology and applications, 3(1):1–9,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [6] Lars-Petter Granan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' The ising model applied on chronification of pain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Pain Medicine, 17(1):5–9, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [7] Douglas B Kitchen, Hélène Decornez, John R Furr, and Jürgen Bajorath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Docking and scoring in virtual screening  for drug discovery: methods and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature reviews Drug discovery, 3(11):935–949, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  6  [8] Rosario N Mantegna and H Eugene Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Introduction to econophysics: correlations and complexity in finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Cambridge university press, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [9] Stan Van Hoesel and Rudolf Müller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Optimization in electronic markets: examples in combinatorial auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Netnomics, 3(1):23–33, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [10] Baonan Wang, Feng Hu, Haonan Yao, and Chao Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Prime factorization algorithm based on parameter  optimization of ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Scientific reports, 10(1):1–10, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [11] Andrew D King, Juan Carrasquilla, Jack Raymond, Isil Ozfidan, Evgeny Andriyash, Andrew Berkley, Mauricio  Reis, Trevor Lanting, Richard Harris, Fabio Altomare, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Observation of topological phenomena in a  programmable lattice of 1,800 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature, 560(7719):456–460, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [12] Masanao Yamaoka, Chihiro Yoshimura, Masato Hayashi, Takuya Okuyama, Hidetaka Aoki, and Hiroyuki Mizuno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Ising computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Hitachi Review, 65(6):157, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [13] Qiang Zhang, Dandan Deng, Wenting Dai, Jixin Li, and Xinwen Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Optimization of culture conditions for  differentiation of melon based on artificial neural network and genetic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Scientific Reports, 10(1):1–8,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [14] Andrew Lucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Ising formulations of many np problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Frontiers in physics, page 5, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [15] Naeimeh Mohseni, Peter L McMahon, and Tim Byrnes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Ising machines as hardware solvers of combinatorial  optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature Reviews Physics, 4(6):363–379, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [16] Kihwan Kim, M-S Chang, Simcha Korenblit, Rajibul Islam, Emily E Edwards, James K Freericks, G-D Lin,  L-M Duan, and Christopher Monroe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Quantum simulation of frustrated ising spins with trapped ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature,  465(7298):590–593, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [17] Colin D Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M Sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Trapped-ion quantum computing:  Progress and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Applied Physics Reviews, 6(2):021314, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [18] DL—Bixby Applegate and V RE—Chvátal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Cook, wj: The traveling salesman problem: A computational study,  2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [19] Fuxi Cai, Suhas Kumar, Thomas Van Vaerenbergh, Xia Sheng, Rui Liu, Can Li, Zhan Liu, Martin Foltin, Shimeng  Yu, Qiangfei Xia, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Power-efficient combinatorial optimization using intrinsic noise in memristor hopfield  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature Electronics, 3(7):409–418, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [20] Masoud Babaeian, Dan T Nguyen, Veysi Demir, Mehmetcan Akbulut, Pierre-A Blanche, Yushi Kaneda, Saikat  Guha, Mark A Neifeld, and N Peyghambarian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' A single shot coherent ising machine based on a network of  injection-locked multicore fiber lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature communications, 10(1):1–11, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [21] Takahiro Inagaki, Yoshitaka Haribara, Koji Igarashi, Tomohiro Sonobe, Shuhei Tamate, Toshimori Honjo, Alireza  Marandi, Peter L McMahon, Takeshi Umeki, Koji Enbutsu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' A coherent ising machine for 2000-node  optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Science, 354(6312):603–606, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [22] Peter L McMahon, Alireza Marandi, Yoshitaka Haribara, Ryan Hamerly, Carsten Langrock, Shuhei Tamate,  Takahiro Inagaki, Hiroki Takesue, Shoko Utsunomiya, Kazuyuki Aihara, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' A fully programmable 100-spin  coherent ising machine with all-to-all connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Science, 354(6312):614–617, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [23] Ryan Hamerly, Takahiro Inagaki, Peter L McMahon, Davide Venturelli, Alireza Marandi, Tatsuhiro Onodera,  Edwin Ng, Carsten Langrock, Kensuke Inaba, Toshimori Honjo, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Experimental investigation of performance  differences between coherent ising machines and a quantum annealer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Science advances, 5(5):eaau0823, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [24] Toshimori Honjo, Tomohiro Sonobe, Kensuke Inaba, Takahiro Inagaki, Takuya Ikuta, Yasuhiro Yamada, Takushi  Kazama, Koji Enbutsu, Takeshi Umeki, Ryoichi Kasahara, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' 100,000-spin coherent ising machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Science  advances, 7(40):eabh0952, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [25] Chihiro Yoshimura, Masato Hayashi, Takuya Okuyama, and Masanao Yamaoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Implementation and evaluation  of fpga-based annealing processor for ising model by use of resource sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' International Journal of Networking  and Computing, 7(2):154–172, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [26] Igor Gershenzon, Geva Arwas, Sagie Gadasi, Chene Tradonsky, Asher Friesem, Oren Raz, and Nir Davidson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Exact  mapping between a laser network loss rate and the classical xy hamiltonian by laser loss control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nanophotonics,  9(13):4117–4126, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [27] Peter L McMahon, Alireza Marandi, Yoshitaka Haribara, Ryan Hamerly, Carsten Langrock, Shuhei Tamate,  Takahiro Inagaki, Hiroki Takesue, Shoko Utsunomiya, Kazuyuki Aihara, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' A fully programmable 100-spin  coherent ising machine with all-to-all connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Science, 354(6312):614–617, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [28] D Pierangeli, G Marcucci, and C Conti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Large-scale photonic ising machine by spatial light modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Physical  review letters, 122(21):213902, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  7  [29] Davide Pierangeli, Giulia Marcucci, Daniel Brunner, and Claudio Conti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Noise-enhanced spatial-photonic ising  machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nanophotonics, 9(13):4109–4116, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [30] Davide Pierangeli, Giulia Marcucci, and Claudio Conti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Adiabatic evolution on a spatial-photonic ising machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Optica, 7(11):1535–1543, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [31] Yisheng Fang, Junyi Huang, and Zhichao Ruan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Experimental observation of phase transitions in spatial photonic  ising machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Physical Review Letters, 127(4):043902, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [32] Wenchen Sun, Wenjia Zhang, Yuanyuan Liu, Qingwen Liu, and Zuyuan He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Quadrature photonic spatial ising  machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Optics Letters, 47(6):1498–1501, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [33] Jiayi Ouyang, Yuxuan Liao, Zhiyao Ma, Deyang Kong, Xue Feng, Xiang Zhang, Xiaowen Dong, Kaiyu Cui,  Fang Liu, Wei Zhang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' An on-demand photonic ising machine with simplified hamiltonian calculation by  phase-encoding and intensity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='05072, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [34] Charles Roques-Carmes, Yichen Shen, Cristian Zanoci, Mihika Prabhu, Fadi Atieh, Li Jing, Tena Dubˇcek, Chenkai  Mao, Miles R Johnson, Vladimir ˇCeperi´c, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Heuristic recurrent algorithms for photonic ising machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature  communications, 11(1):1–8, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [35] Mihika Prabhu, Charles Roques-Carmes, Yichen Shen, Nicholas Harris, Li Jing, Jacques Carolan, Ryan Hamerly,  Tom Baehr-Jones, Michael Hochberg, Vladimir ˇCeperi´c, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Accelerating recurrent ising machines in photonic  integrated circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Optica, 7(5):551–558, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [36] Suryendy Dutta, Abhishek Khanna, AS Assoa, Hanjong Paik, Darrell G Schlom, Zoltán Toroczkai, Arijit  Raychowdhury, and Suman Datta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' An ising hamiltonian solver based on coupled stochastic phase-transition  nano-oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature Electronics, 4(7):502–512, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [37] Ernst Ising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Beitrag zur theorie des ferro-und paramagnetismus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' PhD thesis, Grefe & Tiedemann, 1924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [38] Santosh Kumar, He Zhang, and Yu-Ping Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Large-scale ising emulation with four body interaction and  all-to-all connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Communications Physics, 3(1):1–9, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [39] Wenchen Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' In OFC 2022, page M2G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='4, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [40] Michel X Goemans and David P Williamson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  Improved approximation algorithms for maximum cut and  satisfiability problems using semidefinite programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Journal of the ACM (JACM), 42(6):1115–1145, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [41] Michael R Garey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' A guide to the theory of np-completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Computers and intractability, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [42] Sera Kahruman, Elif Kolotoglu, Sergiy Butenko, and Illya V Hicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' On greedy construction heuristics for the  max-cut problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' International Journal of Computational Science and Engineering, 3(3):211–218, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [43] Takuya Okuyama, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Masanao Yamaoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Binary optimization by  momentum annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Physical Review E, 100(1):012111, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [44] Kosuke Tatsumura, Masaya Yamasaki, and Hayato Goto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Scaling out ising machines using a multi-chip architecture  for simulated bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Nature Electronics, 4(3):208–217, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [45] Qizhuang Cen, Hao Ding, Tengfei Hao, Shanhong Guan, Zhiqiang Qin, Jiaming Lyu, Wei Li, Ninghua Zhu, Kun  Xu, Yitang Dai, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Large-scale coherent ising machine based on optoelectronic parametric oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content=' Light:  Science & Applications, 11(1):1–10, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
+page_content='  [46] D-Wave Systems Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'}
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+arXiv:2301.00784v1  [math.RT]  2 Jan 2023
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR
+REPRESENTATIONS
+LÉA BITTMANN
+Abstract. We interpret a formula established by Lapid-Mínguez on real regular rep-
+resentations of GLn over a local non-archimedean field as a matrix determinant. We
+use the Lewis Carroll determinant identity to prove new relations between real regular
+representations. Through quantum affine Schur-Weyl duality, these relations generalize
+Mukhin-Young’s Extended T -systems, for representations of the quantum affine algebra
+Uqppslkq, which are themselves generalizations of the celebrated T -system relations.
+Contents
+1.
+Introduction
+1
+2.
+Preliminaries
+3
+3.
+Good segments
+7
+4.
+Determinant formula
+10
+5.
+Extended T-system formula
+11
+6.
+Relation to quantum affine algebras representations
+17
+Appendix A.
+Ferrers boards
+19
+References
+20
+1. Introduction
+The context of this work is the representation theory of GLnpFq (where F is a non-
+archimedean local field), or equivalently of the type A quantum affine algebra Uqppslkq
+(where q P Cˆ is not a root of unity). Indeed, through Chari-Pressley’s quantum affine
+Schur-Weyl duality [CP95], the category of complex smooth finite-length representations of
+GLnpFq is equivalent to the category of (level n) finite-dimensional Uqppslkq-modules, when
+k ě n. Since both contexts are equivalent, we will work with the category C of GLnpFq
+representations in most of this paper. Both these categories have been intensively (and
+independently) studied, but some important natural questions remain open.
+The normalized parabolic induction, denoted by ˆ, endows this category with a ring
+category structure, and its Grothendieck group R with a ring structure. Irreducible rep-
+resentations in the category C have been classified by Zelevinsky [Zel80] using multiseg-
+ments (formal sums of segments). For m “ ∆1 ` ∆2 ` ¨ ¨ ¨ ` ∆N a multisegment, the
+corresponding irreducible representation Zpmq is obtained as the unique irreducible sub-
+representation of the standard representation ζpmq :“ Zp∆1q ˆ Zp∆2q ˆ ¨ ¨ ¨ ˆ Zp∆Nq. The
+classes of the irreducible representations and the standard representations form two bases
+of the Grothendieck ring R, the change of basis matrix between them is unitriangular, with
+coefficients which can be expressed in terms of Kazhdan-Lusztig polynomials (see [Zel81],
+[CG97]). A similar story was established for finite-dimensional representations of Uqppslkq
+(see the work of Nakajima [Nak01]). This gives an algorithm to compute the classes of
+1
+
+2
+LÉA BITTMANN
+the simple representations from the classes of the standard representations. However, in
+practice the actual computation of the coefficients can be very difficult.
+For some specific classes of irreducible representations, remarkable formulas have been
+established to compute their classes as linear combinations of classes of standard represen-
+tations. The work of Tadić [Tad95], and then Chenevier-Renard [CR08], established such
+a formula for Speh representations. Cleverly, this formula can be seen as the computation
+of the determinant of a matrix, and it was then proved using the Lewis Carroll identity
+(also called Dodgson’s rule of determinant). In [LM14], Lapid-Mínguez generalized Tadić’s
+formula to a larger class of representations called ladder representations. Then, in [LM18]
+the same authors established an even more general formula (see (4.1) below), for regular
+representations which are real - Zpmq such that Zpmq ˆ Zpmq is irreducible.
+Furthermore, in [LM14], Lapid-Mínguez used the Lewis Carroll identity to obtain a
+remarkable relation between the classes of some of these ladder representations.
+For
+Zpmq “ Zp∆1 ` ¨ ¨ ¨ ` ∆N) a ladder representation, we have the following relation in
+R [LM14, Corollary 12]:
+(1.1)
+Zp∆1 ` ¨ ¨ ¨ ` ∆N´1q ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆Nq “ Zpmq ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆N´1q
+` Zpm1q ˆ Zpm2q,
+where Zpm1q, Zpm2q are also ladders (see Theorem 5.5). Through quantum affine Schur-
+Weyl duality, relation (1.1) has been established independently by Mukhin-Young in [MY12,
+Theorem 4.1] for representations of the quantum affine algebra Uqppslkq, under the name
+Extended T-systems.
+The extended T-systems are generalizations of the famous T-system relations, which
+are sets of recurrence relations of crucial importance in the study of certain integrable
+systems (see review [KNS11]).
+For representations of quantum affine algebras, the T-
+systems are relations in the Grothendieck ring R between classes of Speh representations
+(called Kirillov-Reshetikhin modules there).
+These relations were proved in all simply-
+laced types (A, D or E) by Nakajima [Nak01] and in all types by Hernandez [Her06].
+Additionally, the T-systems, and their extended version, can be interpreted as short exact
+sequences between irreducible finite-dimensional Uqppgq-modules.
+More recently, the T-systems gained a new interpretation as exchange relations in a
+Fomin-Zelevinsky cluster algebra [FZ02]. Indeed, in [HL16] Hernandez-Leclerc proved this
+interpretation of T-systems as cluster transformations and used it to the prove that the
+Grothendieck ring of the category of finite-dimensional Uqppgq-modules (in all Dynkin types)
+had the structure of a cluster algebra. Note that Duan-Li-Luo obtained in [DLL19] another
+generalization of the T-systems, different from Mukhin-Young extended T-systems, which
+they also interpreted as exchange relations in the cluster algebra structure.
+In the present work, we establish formulas generalizing the extended T-systems of
+Mukhin-Young, for some real regular representations. Regular representations have a per-
+mutation associated to them and in [LM18], Lapid-Mínguez gave a sufficient condition for
+a regular representation to be real, as a pattern avoidance condition on the permutation
+associated to the representation. We show, using the notion of Ferres boards and the work
+of Sjostrand [Sjo07] that under the same pattern avoidance condition, Lapid-Mínguez’s
+formula [LM18, Theorem 1.2 (9)] can be written as a matrix determinant. Our relations
+are then obtained using some choice of Lewis Carroll identities. As our main result, we
+prove the following (Theorem 5.1 and Corollary 5.3): for Zpmq “ Zp∆1 ` ¨ ¨ ¨ ` ∆N) a
+regular representation such that the associated permutation σ avoids the patterns 3412
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+3
+and 4231, we have the following relations in R ((5.1) and (5.2)):
+Zpmz∆Nq ˆ Zpmz∆σpNqq “ Zpmq ˆ Zpmz∆N, ∆σpNqq ` Zpm1
+1q ˆ Zpm2
+1q,
+(1.2)
+Zpmz∆1q ˆ Zpmz∆σp1qq “ Zpmq ˆ Zpmz∆1, ∆σp1qq ` Zpm1
+2q ˆ Zpm2
+2q,
+(1.3)
+where m1
+1, m2
+1, m1
+2 and m2
+2 are real regular representations.
+As part of Theorem 5.1 and Corollary 5.3, we also prove that, as the extended T-systems,
+these relations correspond to a decomposition of a module of length 2, i.e. the two terms in
+the right hand side of (1.2) and (1.3) are irreducible representations. We prove this using
+Lapid-Mínguez’s [LM16] combinatorial irreducibility criteria, as well as a newly introduced
+notion of good segments in a mutlisegment, which enables us to prove by induction that
+some parabolic induction of irreducible representations are irreducible.
+The paper is organized as follows. We start with some reminders about segments, multi-
+segments, p-adic representations of GLnpFq and the Zelevinsky classification in Section 2.
+We also recall Lapid-Mínguez’s [LM16] irreducibility criteria for a parabolic induction of
+two representations, using socles and cosocles. In Section 3, we introduce the notion of
+good segments and use it to obtain some combinatorial criteria to prove that certain para-
+bolic inductions Zp∆q ˆ Zpmq, where Zpmq is a regular representation are irreducible. We
+also prove an existence result for good segments (Proposition 3.7). In particular, we obtain
+that every regular representation whose permutation avoids the patterns 3412 and 4231
+has at least two good segments, from which we can recover that such representations are
+real. In Section 4, we use the notion of Ferres boards and results from Sjostrand [Sjo07] and
+Chepuri–Sherman-Bennett [CSB21] to write existing relations as determinants of matrices.
+The main result is stated and then proved in Section 5, in which we also give examples.
+Finally, in Section 6 we translate our results to the context of quantum affine algebra
+representations, and give some perspective, in particular in relation to cluster algebras.
+Acknowledgements. We would like to thank Alberto Mínguez for providing inspiration
+for this work.
+The author was partially supported by the European Research Council
+(ERC) under the European Union’s Horizon 2020 research and innovation programme
+under grant agreement No 948885 and by the Royal Society University Research Fellowship.
+2. Preliminaries
+2.1. Segments and multisegments.
+Definition 2.1. A segment is a pair of integers a ď b P Z, denoted by ra; bs.
+Let Seg denote the set of segments.
+The extremities of the segment ∆ “ ra; bs P Seg are denoted by bp∆q “ a and ep∆q “ b.
+We also write ÐÝ
+∆ “ ra ´ 1; b ´ 1s.
+Definition 2.2. Two segments ∆ “ ra; bs and ∆1 “ rc; ds are linked if
+a ă c
+and
+c ´ 1 ď b ă d,
+or
+c ă a
+and
+a ´ 1 ď d ă b.
+In the first case, we say that ∆ precedes ∆1 and write ∆ ă ∆1.
+Example 2.3. A few examples of linked and unlinked pairs of segments:
+1
+3
+4
+5
+are linked.
+1
+2
+4
+5
+are not linked.
+1
+4
+3
+5
+are linked.
+1
+5
+2
+4
+are not linked.
+
+4
+LÉA BITTMANN
+Definition 2.4. A multisegment m is a finite formal sum of segments of Seg (with possible
+multiplicities), m “ ∆1 ` ∆2 ` ¨ ¨ ¨ ` ∆N. Let Mult denote the set of multisegments.
+A sequence of segments p∆1, . . . , ∆Nq is said to be ordered if, for all 1 ď i ă j ď N, ∆i
+does not precedes ∆j. If m P Mult, and p∆1, . . . , ∆Nq is an ordered sequence of segments
+such that m “ ��1 ` ¨ ¨ ¨ ` ∆N, we say that p∆1, . . . , ∆Nq is an ordered form of m.
+2.2. Representations. Let F be a non-archimedean local field with a normalized absolute
+value | ¨ | and let D be a finite-dimensional central division F-algebra. For n P Zě1, let
+CpGLnq be the category of complex, smooth representations of GLnpDq of finite length
+and IrrpGLnq the set of equivalence classes of irreducible objects of CpGLnq.
+For πi P
+CpGLniq, i “ 1, 2, denote by π1ˆπ2 P CpGLn1`n2q the representation which is parabolically
+induced from π1 b π2. The parabolic induction endows the category À
+Ně0 CpGLnq with
+the structure of a tensor category.
+For any supercuspidal representation ρ P Ť
+nPZě0 IrrpGLnq, there exists a unique positive
+real number sρ such that ρ|¨|sρˆρ is reducible. Let νρ “ |¨|sρ, we write ÝÑρ “ ρνρ, ÐÝρ “ ρν´1
+ρ .
+A cuspidal line is an equivalence class on Ť
+nPZě0 IrrpGLnq for the equivalence relation given
+by ρ „ ÝÑρ .
+For a fixed cuspidal line L, consider CL the Serre ring subcategory of À
+Ně0 CpGLnq
+consisting of the representations whose supercuspidal support is contained in L. Then all
+categories CL are equivalent as ring categories and the study of À
+Ně0 CpGLnq amounts to
+the study of one CL. From now on, we fix a cuspidal line, drop the subscript and consider
+the category C, its set of equivalence classes of irreducible objects Irr and its Grothendieck
+ring R.
+For ∆ “ ra; bs P Seg, consider the induced representation
+Ira; bs :“ ρνa
+ρ ˆ ρνa`1
+ρ
+ˆ ¨ ¨ ¨ ˆ ρνb
+ρ.
+Definition 2.5. We consider the socle and cosocle of this representation:
+Zra; bs :“ socpIra; bsq,
+maximal semi-simple submodule,
+Lra; bs :“ cospIra; bsq,
+maximal semi-simple quotient.
+The following is known (see for example [Zel80]).
+Proposition 2.6. For ∆1, . . . , ∆N P Seg, Zp∆1qˆ¨ ¨ ¨ˆZp∆Nq (resp. Lp∆1qˆ¨ ¨ ¨ˆLp∆Nq)
+is irreducible if and only if the segments ∆1, . . . , ∆N are pairwise unlinked.
+For m P Mult and p∆1, . . . , ∆Nq an ordered form of m, define the standard module:
+ζpmq :“ Zp∆1q ˆ ¨ ¨ ¨ ˆ Zp∆Nq.
+From the previous proposition, ζpmq does not depend on the chosen order.
+Theorem 2.7. [Zel80][Zelevinsky Classification] The map
+m ÞÑ Zpmq :“ socpζpmqq,
+defines a bijection
+Mult „
+ÝÑ Irr .
+2.3. Families of representations. We are interested in some particular families of rep-
+resentations. Let Zpmq be an irreducible representation, with m “ ∆1 ` ¨ ¨ ¨ ` ∆N P Mult.
+Definition 2.8. The irreducible representation Zpmq is a Speh representation if ∆i`1 “ ÐÝ
+∆i,
+for all 1 ď i ď N ´ 1.
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+5
+Example 2.9. The representations corresponding to the multisegments
+r3; 3s ` r2; 2s ` r1; 1s ` r0; 0s “
+‚0
+‚1
+‚2
+‚3
+and
+r2; 4s ` r1; 3s ` r0; 2s “
+4
+2
+1
+0
+3
+2
+are Speh representations.
+Definition 2.10. The irreducible representation Zpmq is a ladder representation if, for all
+1 ď i ď N ´ 1, ep∆i`1q ă ep∆iq and bp∆i`1q ă bp∆iq.
+Example 2.11. All Speh representations are particular cases of ladder representations.
+The representations corresponding to the multisegments
+r2; 5s ` r1; 3s ` r0; 0s “
+‚0
+1
+2
+5
+3
+,
+r4; 7s ` r1; 2s “
+4
+7
+2
+1
+are ladder representations.
+Definition 2.12. The irreducible representation Zpmq is a regular representation if, for
+all 1 ď i ‰ j ď N, ep∆jq ‰ ep∆iq and bp∆jq ‰ bp∆iq. By extension, the multisegment m
+is also called regular.
+Example 2.13. All ladder representations are particular cases of regular representations.
+The representation
+Zpr1; 5s ` r0; 4s ` r2; 3sq “
+1
+5
+0
+4
+2
+3
+is a regular representation.
+Definition 2.14. If Zpmq is a regular representation, then one can define a corresponding
+permutation σm as follows.
+Write m “ ra1; b1s ` ra2; b2s ` ¨ ¨ ¨ ` raN; bNs, and assume
+b1 ą b2 ą ¨ ¨ ¨ ą bN, then σm P SN is such that
+aσmp1q ă aσmp2q ă ¨ ¨ ¨ ă aσmpNq.
+Remark 2.15. If Zpmq is a ladder representation, then the associated permutation is w0,
+the longest element of SN.
+Definition 2.16. An irreducible representation π is said to be real if π ˆ π is also irre-
+ducible.
+Remark 2.17. Real representations are usually called square-irreducible representations in
+this context, but we use real here, which is the terminology coming from the work of Kang-
+Kashiwara-Kim-Oh [KKKO15] on representations of quantum affine algebras, where the
+notion appeared in a crucial way (see Section 6.2).
+The following is one of the main results of [LM18].
+Theorem 2.18. The regular representation Zpmq is real if and only if there does not exists
+a sequence 1 ď j1 ă ¨ ¨ ¨ ă jr ď N, r ě 4 such that if a1
+i “ aji and b1
+i “ bji, then either
+a1
+i`1 ă a1
+i ď b1
+i`1 ` 1, i “ 3, . . . , r ´ 1, a1
+3 ă a1
+1 ď b1
+3 ` 1, and a1
+r ă a1
+2 ă a1
+r´1,
+or
+a1
+i`1 ă a1
+i ď b1
+i`1 ` 1, i “ 4, . . . , r ´ 1, a1
+4 ă a1
+2 ď b1
+4 ` 1, and a1
+3 ă a1
+r ă a1
+1 ă a1
+ℓ,
+
+6
+LÉA BITTMANN
+where ℓ “ 2 if r “ 4 and ℓ “ r ´ 1 otherwise.
+If the permutation σm avoids the patterns 4231 and 3412, then the condition of Theo-
+rem 2.18 is satisfied. We will call these representations pattern avoiding regular.
+The same patterns avoidance condition correspond to the smoothness condition of the
+Schubert variety Xσm (see [LS90]).
+Note that in particular, all ladder representations are real.
+2.4. Irreducibility criteria. The following result will be much used.
+Lemma 2.19. [MS14] Let π1 and π2 be irreducible representations, and π be a represen-
+tation such that
+(a) π is a subrepresentation of π1 ˆ π2,
+(b) π is a quotient of π2 ˆ π1,
+(c) π1 b π2 has multiplicity 1 in the Jordan-Hölder sequence of π1 ˆ π2,
+Then π is irreducible.
+Definition 2.20. Given π1 “ Zpm1q and π2 “ Zpm2q, we write LIpπ1, π2q (resp. RIpπ1, π2q)
+for the condition
+Zpm1 ` m2q “ socpπ1 ˆ π2q
+(resp.
+Zpm1 ` m2q “ cospπ1 ˆ π2qq.
+Lemma 2.21. Let m be a multisegment and ∆ a segment. Then we have the following
+equivalences:
+LIpZp∆q, Zpmqq
+ðñ
+Zpm ` ∆q ãÑ Zp∆q ˆ Zpmq,
+RIpZp∆q, Zpmqq
+ðñ
+Zpm ` ∆q ãÑ Zpmq ˆ Zp∆q.
+Proof. The first statement follows from the fact that the segment representation Zp∆q is
+a left multiplier (see [LM16, Definition 4.3]), thus Zp∆q ˆ Zpmq has a unique irreducible
+submodule, which appears with multiplicity 1 in the Jordan-Hölder sequence of Zp∆q ˆ
+Zpmq.
+The second statement can be deduced from the first by the use of the contragredient,
+or more precisely [LM16, Lemma 3.9].
+□
+Proposition 2.22. [LM16] π1 ˆ π2 is irreducible if and only if LIpπ1, π2q and RIpπ1, π2q.
+In [LM16], Lapid-Minguez introduced a combinatorial setup in order to determine
+whether the conditions RIpZp∆q, Zpmqq and LIpZp∆q, Zpmqq where satisfied, for ∆ P Seg
+and m P Mult. Let us recall it here.
+Write m “ ∆1 ` ¨ ¨ ¨ ` ∆N, and consider the sets
+X∆,m “ ti | ∆ ă ∆iu ,
+˜X∆,m “ ti | ∆i ă ∆u ,
+Y∆,m “
+!
+i | ÐÝ
+∆ ă ∆i
+)
+,
+˜Y∆,m “
+!
+i | ÐÝ
+∆i ă ∆
+)
+.
+Definition 2.23. Let LCp∆, mq be the condition that there exists an injective function
+f : X∆,m Ñ Y∆,m such that for all 1 ď i ď N, ∆fpiq ă ∆i.
+Let RCp∆, mq be the condition that there exists an injective function f : ˜X∆,m Ñ ˜Y∆,m
+such that for all 1 ď i ď N, ∆i ă ∆fpiq.
+The function of Definition 2.23 are called matching functions.
+Proposition 2.24. [LM16] The conditions LCp∆, mq and LIpZp∆q, Zpmqq (resp. RCp∆, mq
+and RIpZp∆q, Zpmqq) are equivalent.
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+7
+Combining this result with Proposition 2.22, we get the following.
+Corollary 2.25. The parabolic induction Zp∆qˆZpmq is irreducible if and only if LCp∆, mq
+and RCp∆, mq.
+3. Good segments
+3.1. Definition.
+Definition 3.1. A segment ∆ in a multisegment m P Mult is called a good segment if
+(i) Zp∆q ˆ Zpmq is irreducible.
+(ii)
+#
+Zpmq ãÑ Zp∆q ˆ Zpm´q,
+or Zpmq ãÑ Zpm´q ˆ Zp∆q,
+, where m´ “ mzt∆u.
+If the first (resp. second) subcase of (ii) is satisfied, ∆ is called a good left (resp. right)
+segment of m.
+Using Lemma 2.21 as well as Proposition 2.4, we have the following equivalences:
+∆ is a good left segment of m
+ðñ
+LCp∆, mq, RCp∆, mq,
+and LCp∆, m´q.
+,
+(3.1)
+∆ is a good right segment of m
+ðñ
+LCp∆, mq, RCp∆, mq,
+and RCp∆, m´q.
+(3.2)
+3.2. Combinatorial criteria.
+Lemma 3.2. If m “ ∆1 ` ¨ ¨ ¨ ` ∆N is a multisegment and ∆0 is a segment such that
+∆0 ` m is a regular multisegment, then
+LCp∆0, mq ô Ei, ∆0 ă ∆i,
+RCp∆0, mq ô Ei, ∆i ă ∆0.
+Proof. We will prove the first equivalence, the second being exactly analog.
+From the
+definition of the condition LC, the implication
+LCp∆0, mq ð Ei, ∆0 ă ∆i
+is clear.
+Let i P Y∆0,m. If i R X∆0,m, then either bp∆0q “ bp∆iq or ep∆0q “ ep∆iq, which is
+a contradiction. Thus Y∆0,m Ă X∆,m. Now, if X∆0,m ‰ H, then LCp∆0, mq can not be
+satisfied (by Hall’s marriage theorem).
+□
+Lemma 3.3. If m “ ∆1 ` ¨ ¨ ¨ ` ∆N is an ordered regular multisegment with σ “ σm the
+associated permutation, then for all 1 ď i ď N,
+• the condition LCp∆i, mq is equivalent to σ´1 is strictly decreasing on X∆i,m,
+• the condition RCp∆i, mq is equivalent to σ´1 is strictly decreasing on ˜X∆i,m.
+Proof. As before, we only prove the first statement. Fix 1 ď i ď N. If X∆i,m “ H, the
+equivalence is trivial.
+Suppose X∆i,m ‰ H, then with the same reasoning as in the proof of Lemma 3.2,
+Y∆i,m Ă X∆i,m Y tiu.
+Suppose Y∆i,m “ X∆i,m Y tiu. Let X∆i,m “ tj1 ą j2 ą ¨ ¨ ¨ ą jmu. Then, since m is
+ordered, ep∆j1q ă ep∆j2q ă ¨ ¨ ¨ ă ep∆jmq.
+
+8
+LÉA BITTMANN
+If σ´1 is strictly decreasing on X∆i,m, then bp∆j1q ă bp∆j2q ă ¨ ¨ ¨ ă bp∆jmq. Since all
+jk P X∆i,m, we have ∆jℓ ă ∆jℓ`1 for all 1 ď ℓ ď m ´ 1. Thus the function
+(3.3)
+f :
+X∆,m
+Ñ Y∆,m,
+j1
+ÞÑ i,
+jℓ`1
+ÞÑ jℓ,
+1 ď ℓ ď m ´ 1,
+is a matching function from X∆i,m to Y∆i,m. Thus LCp∆i, mq.
+If Y∆i,m Ĺ X∆i,m Y tiu, then there exists j P X∆i,m such that bp∆jq “ ep∆iq ` 1, and
+Y∆i,m “ pX∆i,mztjuq Y tiu. If σ´1 is strictly decreasing on X∆,m, then necessarily j “ jm
+and the function f from (3.3) is a matching function from X∆i,m to Y∆i,m, as jm does not
+appear in the image of f.
+Conversely, suppose LCp∆i, mq and let f be a matching function from X∆i,m to Y∆i,m.
+Necessarily, fpj1q “ i, as ∆i is the only segment considered which precedes ∆j1. Recur-
+sively, we see that f is the function from (3.3). As it is a matching function, we deduce
+that ∆jℓ ă ∆jℓ`1 for all 1 ď ℓ ď m ´ 1, and thus σ´1 is strictly decreasing on X∆i,m.
+□
+Remark 3.4. From Lemma 3.2, if m is a regular multisegment, for all 1 ď i ď k, LCp∆i, m´
+∆iq (resp. RCp∆i, m ´ ∆iq) is equivalent to the fact that ∆i precedes (resp. is preceded
+by) no segment in m.
+Combining with Lemma 3.3, we have the following equivalences:
+∆ is a good left segment of m
+ðñ
+∆ precedes no other segment of m and ∆
+forms a ladder with the segments which pre-
+cede it.
+∆ is a good right segment of m
+ðñ
+∆ is preceded by no other segment of m and
+∆ forms a ladder with the segments which are
+preceded by it.
+The following result is clear using this criteria.
+Lemma 3.5. If m1 is a sub-multisegment of m and ∆ P m1 is a good segment for m, then
+it is a good segment for m1.
+Remark 3.6. Note that the converse is not true. For example, any segment ∆ is a good
+segment for itself, but not necessarily a good segment for any multisegment containing it.
+3.3. Existence results.
+Proposition 3.7. For N ě 2, let m “ ∆1 ` ∆2 ` ¨ ¨ ¨ ` ∆N be a regular multisegment
+such that for all i, ∆i “ rai, bis with b1 ą b2 ą ¨ ¨ ¨ ą bN and the associated permutation
+σ avoids the patterns 4231 and 3412, and π “ Zpmq is a prime irreducible representation.
+Then
+either ∆1
+or
+∆σp1q,
+and
+either ∆N
+or
+∆σpNq
+correspond to good segments of m.
+Moreover, if σpNq “ 1 (resp. σp1q “ N), then ∆N is a good right segment (resp. ∆1
+is a good left segment) of m. If σpNq “ 1 and σp1q “ N then m is a ladder.
+Proof. First of all, if σp1q “ 1, then ∆1 is not linked with any other segment of m, and it
+is both a good left and a good right segment of m.
+Let i0 “ σp1q and suppose i0 ą 1. Suppose neither ∆1 nor ∆i0 are good segments.
+From Lemma 3.3, σ´1
+m
+is neither decreasing on X∆1,m nor on ˜X∆i0,m. We consider different
+cases.
+If there exists i ă j ă i0 such that ∆i ă ∆1 and ∆j ă ∆1, or ∆i0 ă ∆i and ∆i0 ă ∆j
+and ∆j ć ∆j, the configuration is the following:
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+9
+∆i0
+∆j
+∆i
+∆1
+The pattern 4231 appears in this configura-
+tion, which is a contradiction.
+Otherwise, there exists at least one 1 ă i ă i0 such that ∆i0 ă ∆i and ∆i ć ∆1 and
+one i0 ă j such that ∆j ă ∆1 and ∆j ć ∆i0. The configuration is the following:
+∆j
+∆i0
+∆i
+∆1
+The pattern 3412 appears in this configura-
+tion, which is a contradiction.
+The proof for ∆N and ∆σpNq is exactly symmetric.
+Now, suppose σpNq “ 1. We know that either ∆1 or ∆N is a good segment of m. If
+∆1 is a good segment and ∆N is not a good segment, then we are necessarily in the first
+configuration drawn above, which features the pattern 4231. Similarly, if σp1q “ N, then
+either ∆1 or ∆N is a good segment of m. The same pattern avoidance condition implies
+that ∆1 is necessarily good.
+If both σpNq “ 1 and σp1q “ N then any pair of segments between ∆1 and ∆N which
+does not form a ladder would create a 4231 pattern. Thus m is a ladder.
+□
+This result has the following consequence.
+Corollary 3.8. Let π “ Zpmq be a regular representation avoiding the patterns 4231 and
+3412, then π has at least two good segments.
+Remark 3.9. This criteria allows us to recover the implication (which is established by
+Theorem 2.18 [LM18]):
+m avoids the patterns 4231 and 3412
+ùñ
+Zpmq is real.
+This can be proved by induction on N, the number of segments in the multisegment m.
+For completeness, let us detail the reasoning.
+If N “ 1, then m “ ∆ is just a segment, and Zp∆q is real (for example as an application
+of Proposition 2.6).
+If N ě 2 and m avoids the patterns 4231 and 3412, then from Corollary 3.8, m has
+at least one good segment ∆. Suppose without loss of generality that it is a good left
+segment. Then
+Zpmq ˆ Zpmq ãÑ Zpmq ˆ Zp∆q
+looooooomooooooon
+irreducible
+ˆZpm´q,
+ãÑ Zp∆q ˆ Zpmq ˆ Zpm´q ãÑ Zp∆q ˆ Zp∆q
+looooooomooooooon
+irreducible
+ˆZpm´q ˆ Zpm´q.
+However, m´ has N ´ 1 segments, and satisfy the pattern avoidance condition.
+Thus
+Zpm´q ˆ Zpm´q is irreducible by induction hypothesis.
+Similarly, as Zpmq և Zpm´q ˆ Zp∆q,
+Zpmq ˆ Zpmq և Zpm´q ˆ Zpm´q ˆ Zp∆q ˆ Zp∆q.
+Then the irreducibility of Zpmq ˆ Zpmq is obtained through Lemma 2.19.
+Notice we only used the existence of one good segment in the proof, although there is
+two from Corollary 3.8.
+
+10
+LÉA BITTMANN
+4. Determinant formula
+4.1. Alternate sum formula. One of the results of [LM18] is an alternate sum formula
+for every regular real representation using standard representations. Let π “ Zpmq be a
+regular real representation, with m “ ra1; b1s ` ¨ ¨ ¨ ` raN; bNs such that b1 ą ¨ ¨ ¨ ą bN.
+In the Grothendieck ring,
+(4.1)
+π “
+ÿ
+σ1PSN
+σ0ďσ1ďσ
+sgnpσ1σqZpraσp1q; bσ1p1qsq ˆ Zpraσp2q; bσ1p2qsq ˆ ¨ ¨ ¨ ˆ ZpraσpNq; bσ1pNqsq,
+where σ “ σm and for all i,
+σ´1
+0 piq “ max
+␣
+j ď xi | j R σ´1
+0 pti ` 1, . . . , Nuq
+(
+,
+with xi “ #tj | aj ď bi ` 1u.
+Remark 4.1. The permutation σ0 satisfies
+σ0 ď σ1
+ô
+@i P t1, . . . , Nu, aσpiq ď bσ1piq ` 1.
+We deduce that equation (4.1) is equivalent to
+(4.2)
+π “
+ÿ
+σ1PSN
+σ1ďσ
+sgnpσ1σqZpraσp1q; bσ1p1qsq ˆ Zpraσp2q; bσ1p2qsq ˆ ¨ ¨ ¨ ˆ ZpraσpNq; bσ1pNqsq.
+Indeed, for σ1 ą σ0, at least one of the Zpraσpiq, bσ1piqsq is not defined, and the term does
+not contribute to the sum in (4.2).
+For all i P t1, . . . , Nu, set a1
+i “ aσpiq, then equation (4.2) can be rewritten
+(4.3)
+π “ sgnpσq
+ÿ
+σ1PrId,σs
+sgnpσ1qZpra1
+1; bσ1p1qsq ˆ Zpra1
+2; bσ1p2qsq ˆ ¨ ¨ ¨ ˆ Zpra1
+N; bσ1pNqsq,
+where rId, σs denotes the Bruhat interval of permutations in SN lower than σ.
+4.2. Matrix determinant. Equation (4.3) is similar to the determinant of a matrix, with
+some entries replaced by zeros to account for the missing permutations σ1. More precisely,
+for σ1, σ2 permutations in SN, let
+Γrσ1, σ2s :“ tpi, σpiqq | σ P rσ1, σ2s, 1 ď i ď Nu,
+then permutations whose graph is contained in ΓrId, σs form the right convex hull, from
+the work of Sjöstrand [Sjo07]. The following is obtained using [Sjo07, Theorem 4].
+Proposition 4.2. [CSB21, Proposition 3.3] If the permutation σ P SN avoids the patterns
+4231 and 34121, and M “ pmi,jq1ďi,jďN is a square N ˆ N-matrix, then
+(4.4)
+detpM|ΓrId,σsq “
+ÿ
+σ1PrId,σs
+sgnpσ1qm1,σ1p1qm2,σ1p2q ¨ ¨ ¨ mN,σ1pNq.
+Remark 4.3. Using Ferrers boards (see Appendix A), the determinant in equation (4.4)
+can be computed placing the coefficient mi,j in the box pi, jq of rNs2 if it is coloured and
+0 if it is not. Note that the dots are placed on the Zprai; bisq.
+Combining Proposition 4.2 with (4.3), and assuming σ avoids the patterns 4231 and
+3412, we obtain the following:
+(4.5)
+π “ sgnpσq det
+`
+pZpa1
+i; bjqq1ďi,jďN|ΓrId,σs
+˘
+.
+1in [Sjo07, CSB21], the pattern avoidance condition is weaker, permutations are assumed to avoid the
+patterns 4231, 35142, 42513, and 351624
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+11
+4.3. Lewis Carroll’s identity. The following result is usually called the Lewis Carroll’s
+identity or the Desnanot–Jacobi identity.
+Proposition 4.4. For M a square N ˆ N-matrix, and A, B Ă t1, . . . , Nu, let MB
+A be the
+matrix obtained from M by removing all rows indexed by elements of A and all columns
+indexed by elements of B. Then, for all 1 ď a ă a1 ď N and 1 ď b ă b1 ď N,
+(4.6)
+detpMq detpMb,b1
+a,a1q “ detpMb
+aq detpMb1
+a1q ´ detpMb1
+a q detpMb
+a1q.
+We can use Proposition 4.4 with equation (4.5) to write relations in the Grothendieck
+ring R involving π. However, if M “
+`
+pZpa1
+i; bjqq1ďi,jďN|ΓrId,σs
+˘
+, the determinant of the
+submatrix Mj
+i does not necessarily realize (up to a sign) the class of an irreducible repre-
+sentation Zpm1q in R, for all 1 ď i, j ď N.
+Example 4.5. Let m “ r1; 4s`r0; 3s`r2; 2s, the corresponding permutation is the reflection
+σ “ p12q P S3. The alternate sum formula for the class of the irreducible representation is
+Zpmq “ Zpr1; 4sq ˆ Zpr0; 3sq ˆ Zpr2; 2sq ´ Zpr0; 4sq ˆ Zpr1; 3sq ˆ Zpr2; 2sq,
+“ ´
+������
+Zpr0; 4sq
+Zpr0; 3sq
+0
+Zpr1; 4sq
+Zpr1; 3sq
+0
+0
+0
+Zpr2; 2sq
+������
+.
+Let M be the above matrix, then
+detpM1
+2 q “ Zpr0; 3sq ˆ Zpr2; 2sq “ Zpr0; 3s ` r2; 2sq
+P R,
+detpM1
+3 q “ 0 ‰ Zpm1q.
+Nevertheless, it is possible to write explicit formulas in the Grothendieck ring R in some
+interesting cases.
+We will use the following key result.
+Proposition 4.6. [CSB21, Proposition 4.17] Let σ be a permutation in SN avoiding the
+patterns 4231 and 3412, and choose i P rNs. Let σi P SN´1 be the "flatten" permutation
+obtained from σ by removing pi, σpiqq and shifting the remaining numbers appropriately.
+Then for M a pN ´ 1q ˆ pN ´ 1q-matrix,
+(4.7)
+detpM|ΓrId,σsσpiq
+i
+q “ detpM|ΓrId,σisq.
+Remark 4.7. Note for M a N ˆ N-matrix and for 1 ď i, j ď N,
+`
+M|ΓrId,σs
+˘j
+i “ Mj
+i |ΓrId,σsj
+i .
+5. Extended T-system formula
+Our main result is the following, which will be proven in Section 5.2 and 5.3.
+Theorem 5.1. Let m “ ∆1`∆2`¨ ¨ ¨`∆N be a regular multisegment, such that b1 ą b2 ą
+¨ ¨ ¨ ą bN, where for all 1 ď i ď N, ∆i “ rai; bis. Assume the corresponding permutation σ
+avoids the patterns 4231 and 3412, and that σpNq ‰ N. Let
+I “
+"
+i | aN ď ai
+bi ď bσpNq
+*
+“ ti1 ă i2 ă ¨ ¨ ¨ iru.
+Then, we have the following relation, in the Grothendieck ring R:
+(5.1)
+Zpmz∆Nq ˆ Zpmz∆σpNqq “ Zpmq ˆ Zpmz∆N, ∆σpNqq ` Zpm1q ˆ Zpm2q,
+
+12
+LÉA BITTMANN
+where
+m1 “
+ÿ
+jRI
+∆j `
+r´1
+ÿ
+k“1
+raik; bik`1s,
+m2 “
+ÿ
+iRI
+∆i `
+r´1
+ÿ
+k“1
+raik`1; biks.
+Moreover, the products in both terms on the right hand side of (5.1) are irreducible.
+Remark 5.2.
+(1) If σpNq “ N, then the segment ∆N is not linked to any other segment
+of m. In that case
+Zpmq “ Zpmz∆Nq ˆ Zp∆Nq.
+(2) As σ avoids the pattern 4231, the segments ∆i, with i P I form a ladder.
+Corollary 5.3. Let us assume the permutation σ avoids the patterns 4231 and 3412 and
+satisfies σp1q ‰ 1. Let
+J “
+"
+j | aj ď a1
+bσp1q ď bj
+*
+“ tj1 ă j2 ă ¨ ¨ ¨ jsu.
+The following relation in satisfied in the Grothendieck ring R:
+(5.2)
+Zpmz∆1q ˆ Zpmz∆σp1qq “ Zpmq ˆ Zpmz∆1, ∆σp1qq ` Zpm1q ˆ Zpm2q,
+where
+m1 “
+ÿ
+iRJ
+∆i `
+s´1
+ÿ
+k“1
+rajk; bjk`1s,
+m2 “
+ÿ
+iRJ
+∆i `
+s´1
+ÿ
+k“1
+rajk`1; bjks.
+Proof. The result is obtained by applying Theorem 5.1 to the irreducible representation
+Zpm1q, with m1 “ r´bN; ´aNs ` ¨ ¨ ¨ ` r´b1; ´a1s.
+□
+Example 5.4.
+(1) Let m “ r2; 3s`r0; 2s`r1; 1s. The corresponding regular represen-
+tation Zpmq is real, since its associated permutation is σ “ 231. It has two good
+right segments, which are r0; 2s and r1; 1s (∆σp1q and ∆3). Applying Theorem 5.1
+gives the following relation:
+Zpr2; 3s ` r0; 2sq ˆ Zpr0; 2s ` r1; 1sq “ Zpmq ˆ Zpr0; 2sq ` Zpr0; 2sq ˆ Zpr1; 3s ` r0; 2sq.
+Note that Zpr0; 2s`r1; 1sq – Zpr0; 2sqˆZpr1; 1sq, and in this case the above relation
+can be simplified by Zpr0; 2sq.
+(2) Let m “ r1; 6s`r3; 5s`r0; 4s`r2; 3s. The corresponding regular representation Zpmq
+is real, since its associated permutation is σ “ 3142. It has two good segments,
+which are r1; 6s (left) and r2; 3s (right). Applying Theorem 5.1 gives the following
+relation:
+Zpr1; 6s ` r3; 5s ` r0; 4sq ˆ Zpr1; 6s ` r0; 4s ` r2; 3sq “ Zpmq ˆ Zpr1; 6s ` r0; 4sq
+` Zpr1; 6s ` r0; 4s ` r3; 3sq ˆ Zpr1; 6s ` r0; 4s ` r2; 5sq.
+Whereas applying Corollary 5.3 gives the following relation:
+Zpr3; 5s ` r0; 4s ` r2; 3sq ˆ Zpr1; 6s ` r3; 5s ` r2; 3sq “ Zpmq ˆ Zpr3; 5s ` r2; 3sq
+` Zpr3; 5s ` r2; 3s ` r1; 4sq ˆ Zpr3; 5s ` r2; 3s ` r0; 6sq.
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+13
+5.1. Ladder case. If m is a ladder, then the corresponding permutation is the longest
+permutation w0. Thus ΓrId, w0s “ rNs. In that case, the result of Theorem 5.1 is al-
+ready known, as Corollary 12 of [LM14], or Theorem 4.1 in [MY12], in the language of
+representations of quantum affine algebras.
+Theorem 5.5. Let m “ ∆1 ` ¨ ¨ ¨ ` ∆N be a ladder multisegment, with ∆i “ rai; bis, then
+(5.3)
+Zp∆1 ` ¨ ¨ ¨ ` ∆N´1q ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆Nq “ Zpmq ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆N´1q
+` Zpra1; b2s ` ¨ ¨ ¨ ` raN´1; bNsq ˆ Zpra2; b1s ` ¨ ¨ ¨ ` raN; bN´1sq.
+In this case, the result comes from the application of the Lewis Carroll identity (4.6)
+to the matrix pZprai; bjsqq1ďi,jďN, on lines and columns 1 and N. However, in order to
+understand better the general case, let us consider in more details the application of the
+Lewis Carroll identity to the matrix M “ pZpra1
+i; bjsqq1ďi,jďN (recall that a1
+i “ aN´i`1).
+One can look at what happens to the Ferrers boards (see Appendix A) in this case . The
+permutation w0 is represented by an anti-diagonal, and the Ferrers board is the full grid.
+Taking out row 1 and column N, one gets exactly the grid corresponding to the longest
+element of SN´1.
+S5 Q p15qp24q “
+‚
+‚
+‚
+‚
+‚
+ÝÑ
+‚
+‚
+‚
+‚
+“ p14qp23q P S4
+As the signature of the longest permutation in SN is p´1qt N
+2 u, one has
+p´1qt N´1
+2
+u detpM1
+Nq “ Zp∆1 ` ¨ ¨ ¨ ` ∆N´1q,
+p´1qt N´1
+2
+u detpMN
+1 q “ Zp∆2 ` ¨ ¨ ¨ ` ∆Nq,
+p´1qt N
+2 u´1 detpM1,N
+1,N q “ Zp∆2 ` ¨ ¨ ¨ ` ∆N´1q.
+Now, taking out row 1 and column 1, or row N and column N, one gets again the grid
+corresponding to the longest element of SN´1, but the dots have moved. For example,
+if one does a cyclic permutation of the columns by shifting them to the left and placing
+column 1 at the end, then taking out row 1 and column 1 gives the same result as taking
+out row 1 and column N in the shifted board.
+‚
+‚
+‚
+‚
+‚
+ÝÑ
+‚
+‚
+‚
+‚
+‚
+‚
+‚
+‚
+‚
+‚
+ÝÑ
+‚
+‚
+‚
+‚
+‚
+‚
+‚
+The same operation can be applied to the matrix M. Note that the new dots are placed
+on the coefficients Zpra1; b2sq, . . . , ZpraN´1; bNsq. The permutation of the columns does
+not change the sign of the determinant because the columns on which the determinant is
+computed are not permuted with respect to one another.
+detpM1
+1 q “ detpshiftpMqN
+1 q “ p´1qt N´1
+2
+uZpra1; b2s ` ¨ ¨ ¨ ` raN´1; bNsq.
+Similarly,
+detpMN
+N q “ p´1qt N´1
+2
+uZpra2; b1s ` ¨ ¨ ¨ ` raN; bN´1sq.
+
+14
+LÉA BITTMANN
+Finally, the Lewis Carroll identity (4.6) gives relation (5.1) of Theorem 5.5.
+Moreover, the irreductibility of the terms Zpmq ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆N´1q and Zpra1; b2s `
+¨ ¨ ¨ ` raN´1; bNsq ˆ Zpra2; b1s ` ¨ ¨ ¨ ` raN; bN´1sq is proven in [BLM13, Exemple 4.5].
+5.2. Proof of relation (5.1). Let us apply Proposition 4.4 (Lewis Carroll’s identity) to the
+matrix ˜
+M “ M|ΓrId,σs, where M “ ppZpa1
+i; bjqq1ďi,jďNq, on rows σ´1pNq, N and columns
+σpNq, N:
+(5.4)
+detp ˜
+Mq detp ˜
+MσpNq,N
+σ´1pNq,Nq “ detp ˜
+MσpNq
+σ´1pNqq detp ˜
+MN
+N q ´ detp ˜
+MN
+σ´1pNqq detp ˜
+MσpNq
+N
+q.
+Using Proposition 4.6,
+det
+´
+˜
+MσpNq
+N
+¯
+“ det
+´
+MσpNq
+N
+|ΓrId,σNs
+¯
+,
+det
+´
+˜
+MN
+σ´1pNq
+¯
+“ det
+´
+MN
+σ´1pNq|ΓrId,σσ´1pNqs
+¯
+.
+Since σN and σσ´1pNq satisfy the pattern avoidance condition, using (4.5), one has
+det
+´
+MσpNq
+N
+|ΓrId,σNs
+¯
+“ sgnpσNqZp∆1 ` ¨ ¨ ¨ ` {
+∆σpNq ` ¨ ¨ ¨ ` ∆Nq,
+det
+´
+MN
+σ´1pNq|ΓrId,σσ´1pNqs
+¯
+“ sgnpσσ´1pNqqZp∆1 ` ∆2 ` ¨ ¨ ¨ ` ∆N´1q.
+Note that for i P rNs, Mσpiq
+i
+is obtained by taking out the row and column containing the
+coefficient Zpra1
+i; bσpiqsq “ Zp∆σpiqq.
+Similarly,
+det
+´
+˜
+MσpNq,N
+σ´1pNq,N
+¯
+“ sgnpσσ´1pNqN
+qZp∆1 ` ¨ ¨ ¨ ` {
+∆σpNq ` ¨ ¨ ¨ ` ∆N´1q.
+Let us now consider the coefficients detp ˜
+MN
+N q and detp ˜
+MσpNq
+σ´1pNqq. Using Lemma A.3, we
+know that either the two columns σpNq and N or the two rows σ´1pNq and N of ˜
+M are
+similar (the zeros are at the same place). Assume that the two columns σpNq and N of ˜
+M
+are similar, meaning that there are as many zeros above the coefficient Zpra1
+σ´1pNq; bσpNqsq
+as above Zpra1
+σ´1pNq; bNsq. Note that since σ avoids the pattern 4231, the dots in the lower
+right corner of the Ferrers board form a ladder. In particular, one can apply the same
+reasoning as in the ladder case.
+Let us cyclically permute the columns σpNq, . . . , N in order to obtain column N in
+position σpNq, and take the determinant of the minor shiftp ˜
+MN
+N q “ shiftp ˜
+MqσpNq
+N
+instead
+of the minor ˜
+MN
+N . Because theses columns have the same zero block in their upper part,
+these determinants are equal. The resulting permutation is σN (same permutation as if
+we had taken out row N and column σpNq).
+‚
+‚
+‚
+‚
+‚
+ÝÑ
+‚
+‚
+‚
+‚
+‚
+‚
+‚
+‚
+ÝÑ
+‚
+‚
+‚
+‚
+‚
+For j R I, there are still dots placed on the Zpraj; bjsq, and the new dots are placed on
+the Zpraik`1; biksq, for 1 ď k ď r.
+Hence,
+detp ˜
+MN
+N q “ detpshiftp ˜
+MqσpNq
+N
+q “ sgnpσNqZpm2q.
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+15
+Similarly,
+detp ˜
+MσpNq
+σ´1pNqq “ sgnpσσ´1pNqqZpm1q.
+Let us now consider the signatures of the permutations, using the criteria of Lemma A.4:
+sgnpσq “
+ÿ
+‚
+7
+"
+‚
+‚
+*
+.
+Let us separate the grid (or matrix) in 4 blocks, A, B, C and the upper right being empty
+(or filled with zeros).
+p0q
+A
+B
+C
+‚
+pN, σpNqq
+‚
+pσ´1pNq, Nq
+Note that zone C contains r dots, where r is the cardinal of I. Going from σ to σN, we
+take out the dot on the last line. It is clear that all dots in zones A, B and C except the
+bottom one will have the same contribution to the sum ℓ. The difference is thus equal to
+the contribution of the bottom dot pN, σpNqq, which is r ´ 1. Hence
+sgnpσNq “ sgnpσqp´1qr´1.
+Now, going from σ to σσ´1pNq, we take out the dot pσ´1pNq, Nq. All dots is block A will
+still have the same contribution, but the dots in block B will count one less dot in their
+upper right corner. The remaining dots in block C will also count one less dot. Thus,
+sgnpσσ´1pNqq “ sgnpσqp´1qr´1`7B.
+Going from σ to σσ´1pNqN
+, we take out both these dots, and the signature of the resulting
+permutation is
+sgnpσσ´1pNqN
+q “ sgnpσqp´1q7B`1.
+Simplifying the signs in (5.4), we get the desired relation (5.1).
+Finally, in the case where it is not the two columns but the two rows σ´1pNq and N of
+ΓrId, σs which are identical, one can apply the same procedure of cyclically permuting the
+rows of the matrix ˜
+M. As a result,
+detp ˜
+MN
+N q “ sgnpσσ´1pNqqZpm2q,
+detp ˜
+MσpNq
+σ´1pNqq “ sgnpσNqZpm1q.
+But a symmetric reasoning on the signatures, we also get the desired relation, which
+concludes the proof of (5.1).
+5.3. Proof of irreducibility.
+5.3.1. Irreductibility of ZpmqˆZpmz∆N, ∆σpNqq: As in Remark 3.9, let us prove this result
+by induction on N ě 3, the number of segments in m. For N “ 3, assuming σp3q ‰ 3, then
+Zpmz∆3, ∆σp3qq “ Zp∆q, where ∆ is necessarily a good segment of m by Proposition 3.7.
+By definition, Zpmq ˆ Zp∆q is irreducible.
+Let N ě 4, from Proposition 3.7, as σ avoids the patterns 4231 and 3412, we know that
+either m is a ladder or it has at least one good segment ∆ which is different from ∆N and
+∆σpNq. The ladder case has been considered in Section 5.1. Otherwise, using Lemma 3.5,
+we know that ∆ is also a good segment of m1 :“ mz∆N, ∆σpNq (on the same side).
+
+16
+LÉA BITTMANN
+We can assume without loss of generality that ∆ is a good left segment of m and m1.
+Then, as in the proof of Corollary 3.8,
+Zpmq ˆ Zpm1q ãÑ Zpmq ˆ Zp∆q
+looooooomooooooon
+irreducible
+ˆZpm1z∆q – Zp∆q ˆ Zpmq ˆ Zpm1z∆q
+ãÑ Zp∆q ˆ Zp∆q
+looooooomooooooon
+irreducible
+ˆZpmz∆q ˆ Zpm1z∆q.
+By induction, Zpmz∆q ˆ Zpm1z∆q is irreducible.
+Similarly,
+Zpmq ˆ Zpm1q և Zpmz∆q ˆ Zpm1z∆q ˆ Zp∆q ˆ Zp∆q.
+We conclude that Zpmq ˆ Zpm1q is irreducible by Lemma 2.19.
+5.3.2. Irreducibility of Zpm1q ˆ Zpm2q: As before, we prove this by induction, this time on
+N ´ r, where r “ |J|. If r “ N, then m is a ladder, and the result was proven in [BLM13,
+Exemple 4.5]. If N ą r, then m is not a ladder. In that case, either ∆1 or ∆σp1q is a good
+segment of m and does not form a ladder with ∆N and ∆σpNq. This good segment is not
+one of the ∆ik and thus it is a segment of m1 and m2. Let us prove it is a common good
+segment of m1 and m2.
+Let us assume that σpNq ‰ 1 and that ∆1 is a good left segment of m. Clearly, ∆1 does
+not precede any segment of m1 or m2.
+Suppose ∆1 does not form a ladder with the segments which precedes it in m1. Thus
+there exists ∆, ∆1 such that ∆ ă ∆1, ∆1 ă ∆1 and ∆1 Ĺ ∆. As ∆1 is a good segment of
+m, necessarily exactly one of ∆, ∆1 is in m while the other has be shifted. If ∆1 P m, then
+∆ “ raik; bik`1s for some k. In that case, ∆ik`1 ă ∆1 and ∆1 ć ∆ik`1, which contradicts
+the fact that ∆1 is a good segment of m.
+∆1
+bik`1
+∆1
+aik
+aik`1
+If ∆ “ rai; bis P m, then there is k such that ∆1 “ raik; bik`1s. If both bik ą bi and
+aik`1 ă ai then i P I, which contradicts the fact that ∆ has not been shifted.
+∆1
+∆
+bik`1
+aik
+aik`1
+bik
+aik
+If bik ă bi, then ∆ik, ∆, ∆1 do not form a ladder in m, if aik`1 ą ai then ∆, ∆ik`1, ∆1 do
+not form a ladder in m. In both cases, it contradicts the fact that ∆1 is a good segment of
+m.
+∆1
+∆
+bik`1
+aik
+bik
+aik
+∆1
+∆
+bik`1
+aik
+aik`1
+Hence by the criteria in section 3.2, ∆1 is good segment of m1. We show in a similar way
+that ∆1 is good segment of m2.
+Then,
+Zpm1q ˆ Zpm2q ãÑ Zp∆1q ˆ Zp∆1q
+loooooooomoooooooon
+irreducible
+ˆZpm1z∆1q ˆ Zpm2z∆1q.
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+17
+By induction, Zpm1z∆1q ˆ Zpm2z∆1q is irreducible.
+Similarly,
+Zpm1q ˆ Zpm2q և Zpm1z∆1q ˆ Zpm2z∆1q ˆ Zp∆1q ˆ Zp∆1q.
+We conclude that Zpm1q ˆ Zpm2q is irreducible by Lemma 2.19.
+6. Relation to quantum affine algebras representations
+6.1. Translation of results. As mentioned above, the result of Theorem 5.1 has a quan-
+tum affine analog through quantum affine Schur-Weyl duality. Indeed, when q is not a
+root of unity, Chari-Pressley [CP96] have established an equivalence of categories between
+the category of finite-dimensional representations of the affine Hecke algebra 9Hq2pnq and
+the category of (level n) finite-dimensional representations of the quantum affine algebra
+Uqppslkq, when k ě n. Moreover, through type theory (see for example [Hei11]), it is known
+that finite-dimensional representations of the affine Hecke algebra 9Hq2pnq are equivalent
+to finite length representations of GLnpFq.
+This equivalence is monoidal, in the sense that the parabolic induction of two represen-
+tations in C is translated into the tensor product of the corresponding Uqppslkq-modules.
+Instead of multisegments, finite-dimensional irreducible Uqppslkq-modules have been clas-
+sified [CP95] using Drinfeld polynomials, which correspond to their highest-weights. By a
+process similar to the reduction to cuspidal lines described in the beginning of Section 2.2,
+the study of the category of finite-dimensional Uqppslkq-modules amounts to the study of a
+skeleton Serre subcategory C , introduced by Hernandez-Leclerc (see [HL10, Section 3.7]),
+in relation to cluster algebras. Let R denote the Grothendieck ring of the monoidal cate-
+gory C .
+Simple objects in the category C are then parametrized, up to isomorphism, by mono-
+mials in the formal variables Yi,p, pi, pq P ˆI :“ tt1, . . . , k ´ 1u ˆ Z | i ` p ` 1 P 2Zu. The
+correspondence between segments and formal variables is as follows:
+(6.1)
+ra; bs
+ÞÑ
+Yb´a`1,´a´b,
+r1´i´p
+2
+; i´p´1
+2
+s
+�
+Yi,p.
+Since we are using the Zelevinsky classification for the representations of GLnpFq, from
+now on irreducible Uqppslkq-modules will be denoted LpMq, with M their highest loop-
+weight, in the set of dominant loop-weights:
+ˆPℓ :“
+# N
+ź
+j“1
+Yij,pj | @ 1 ď j ď N, pij, pjq P ˆI
++
+.
+Through this correspondence, ladder representations are usually called snake modules
+in the context of quantum affine algebras. For completeness, recall the definition of snakes
+modules by Mukhin-Young. For M “ śN
+j“1 Yij,pj P ˆPℓ, the simple module LpMq is a snake
+module if and only if, for all 1 ď j ď N,
+pj`1 ´ pj ě |ij`1 ´ ij| ` 2.
+It clearly translates to the definition of ladders, as in Definition 2.10. Note that a definition
+of snake modules for type B quantum affine algebras was also introduced by Mukhin-Young.
+Moreover, as stated above, Theorem 5.5 from [LM14] was previously established by
+Mukhin-Young in terms of snake modules.
+
+18
+LÉA BITTMANN
+For M “ śN
+j“1 Yij,pj P ˆPℓ such that LpMq is a snake module, we have the following
+relation, in the Grothendieck ring R [MY12, Theorem 4.1]:
+(6.2)
+«
+L
+˜N´1
+ź
+j“1
+Yij,pj
+¸ff
+¨
+«
+L
+˜ N
+ź
+j“2
+Yij,pj
+¸ff
+“
+«
+L
+˜N´1
+ź
+j“2
+Yij,pj
+¸ff
+¨ rLpMqs
+`
+“
+LpM1q
+‰
+¨
+“
+LpM2q
+‰
+,
+where M1, M2 are called the neighboring snakes of M, and correspond to m1 and m2 in
+this case. Note that relation (6.2) was established in [MY12] also in type B. Moreover, as
+in our result, both terms on the left hand side of (6.2) correspond to irreducible modules.
+For these reasons, our theorem 5.1 is a generalization of [MY12, Theorem 4.1], and we
+have established some new relations between irreducible representations of Uqppslkq.
+Example 6.1. Let us translate the relations obtained in Example 5.4:
+(1) For k ě 4, let M “ Y2,´5Y3,´2Y1,´2 P ˆPℓ. As before, the corresponding regular
+representation LpMq is real. Applying Theorem 5.1 gives the following relation:
+LpY2,´5Y3,´2q ¨ LpY3,´2Y1,´2q “ LpMq ¨ LpY3,´2q ` LpY3,´2q ˆ LpY3,´4Y3,´2q.
+(2) For k ě 7, let M “ Y6,´7Y3,´8Y5,´4Y2,´5 P ˆPℓ. The corresponding regular repre-
+sentation LpMq is real and applying Theorem 5.1 gives the following relation:
+LpY6,´7Y3,´8Y5,´4q ¨ LpY6,´7Y5,´4Y2,´5q “ LpMq ¨ LpY6,´7Y5,´4q
+` LpY6,´7Y5,´4Y1,´6q ¨ LpY6,´7Y5,´4Y4,´7q.
+Whereas applying Corollary 5.3 gives the following relation:
+LpY3,´8Y5,´4Y2,´5q ¨ LpY6,´7Y3,´8Y2,´5q “ LpMq ¨ LpY3,´8Y2,´5q
+` LpY3,´8Y2,´5Y4,´5q ¨ LpY3,´8Y2,´5Y7,´6q.
+Note that when k “ 7, the right hand side of the last relation simplifies as
+LpY3,´8Y2,´5Y4,´5q ¨ LpY3,´8Y2,´5q.
+6.2. Relation to cluster algebras. In [HL16], Hernandez and Leclerc proved that the
+Grothendieck ring R had a cluster algebra structure for which the initial cluster variables
+are Kirillov-Reshetikhin modules (or Speh representations, as in Definition 2.8). Moreover,
+one of the key ingredients used for this result is the fact that the T-system relations (of
+which the Mukhin-Young extended T-systems are generalizations) correspond to exchange
+relations in the cluster algebra structure. The same authors also conjectured [HL16, Con-
+jecture 5.2] that the cluster variables were in bijection with the prime real simple modules.
+Part of this conjecture was proven by Kashiwara-Kim-Oh-Park in [KKOP21], where they
+proved that all cluster variables correspond to prime real simple modules.
+In [DLL19] Duan-Li-Luo proved that prime snake modules correspond to cluster vari-
+ables, thus proving Hernandez-Leclerc’s conjecture for snake modules, and for that purpose
+introduced new relations in the Grothendieck ring R, which they interpreted as exchange
+relations. However, it is unclear whether (some of) the Mukhin-Young extended T-systems
+can be interpreted as exchange relations.
+One of the motivations behind this work was to obtain more generalizations of the T-
+system relations, which could be interpreted as exchange relations. We conjecture that,
+equipped with more explicit relations such as (5.1) and (5.2), one could prove that all prime
+real regular representations (for which there exists the criterion of Theorem 2.18 [LM18])
+correspond to cluster variables.
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+19
+However, we already observe that not all relations (5.1) and (5.2) have the form of an
+exchange relation. For example, in the relation in Example 6.1 (1), one of the factors in
+the left hand side is not prime LpY3,´2Y1,´2q – LpY3,´2q¨LpY1,´2q. Thus the left hand side
+is a product of three prime irreducible modules, and the relation cannot be an exchange
+relation.
+Appendix A. Ferrers boards
+Permutations in SN can be represented by placing dots in an N ˆ N-grid.
+For all
+1 ď i ď N, place a dot in the box pi, σpiqq. Then the set ΓrId, σs Ă rNs2 can be represented
+in the grid by colouring the boxes pi, σ1piqq, for σ1 ď σ.
+Example A.1. The grid corresponding to the permutation σ “ 152463 is
+‚
+‚
+‚
+‚
+‚
+‚
+Remark A.2. By the study of Sjöstrand [Sjo07], the set ΓrId, σs is a right-aligned Skew
+Ferrers board, in particular it is the union of the rectangles
+‚
+‚
+for all pairs of (not
+necessarily distinct) dots ���.
+Lemma A.3. The σ be a permutation in SN which avoids the pattern 3412, then either
+the columns σpNq and N or the rows σ´1pNq and N of ΓrId, σs are identical.
+Proof. If the columns σpNq and N of ΓrId, σs are different, then there is a dot above
+pσ´1pNq, Nq and to the right of pN, σpNqq.
+‚
+‚
+‚
+pσ´1pNq, Nq
+pN, σpNqq
+Similarly, if the rows σ´1pNq and N are different, then there is a dot to the left of pN, σpNqq
+and below pσ´1pNq, Nq.
+Thus if both the columns σpNq and N and the rows σ´1pNq and N are different, then
+there a 3412 configuration, which is a contradiction.
+‚
+‚
+‚
+‚
+□
+The following is clear from the definition of the signature.
+Lemma A.4. The signature of the permutation σ is equal to p´1qℓ, where ℓ is the sum
+over all dots ‚ of the number of dots strictly above and to the right of ‚.
+
+20
+LÉA BITTMANN
+References
+[BLM13] I. Badulescu, E. Lapid, and A. Mínguez.
+Une condition suffisante pour
+l’irréductibilité d’une induite parabolique de GLpm, Dq.
+Ann. Inst. Fourier
+(Grenoble), 63(6):2239–2266, 2013.
+[CG97] N. Chriss and V. Ginzburg. Representation Theory and Complex Geometry.
+Birkhäuser Boston, MA, first edition, 1997.
+[CP95] V. Chari and A. Pressley. Quantum affine algebras and their representations. In
+Representations of groups (Banff, AB, 1994), volume 16 of CMS Conf. Proc.,
+pages 59–78. Amer. Math. Soc., Providence, RI, 1995.
+[CP96] V Chari and A Pressley. Quantum affine algebras and affine Hecke algebras.
+Pacific J. Math., 174(2):295–326, 1996.
+[CR08] G. Chenevier and D. Renard. Characters of Speh representations and Lewis-
+Caroll identity. Represent. Theory, 12:447–452, 2008.
+[CSB21] S. Chepuri and M. Sherman-Bennett. 1324- and 2143-avoiding kazhdan–lusztig
+immanants and k-positivity.
+Canadian Journal of Mathematics, page 1–31,
+2021.
+[DLL19] B. Duan, J.-R. Li, and Y.-F. Luo. Cluster algebras and snake modules. Journal
+of Algebra, 519:325 – 377, 2019.
+[FZ02] S. Fomin and A. Zelevinsky. Cluster algebras. I. Foundations. J. Amer. Math.
+Soc., 15(2):497–529, 2002.
+[Hei11] V. Heiermann.
+Opérateurs d’entrelacement et algèbres de Hecke avec
+paramètres d’un groupe réductif p-adique: le cas des groupes classiques. Selecta
+Math. (N.S.), 17(3):713–756, 2011.
+[Her06] D. Hernandez. The Kirillov-Reshetikhin conjecture and solutions of T-systems.
+J. Reine Angew. Math., 596:63–87, 2006.
+[HL10] D. Hernandez and B. Leclerc. Cluster algebras and quantum affine algebras.
+Duke Math. J., 154(2):265–341, 2010.
+[HL16] D. Hernandez and B. Leclerc. A cluster algebra approach to q-characters of
+Kirillov-Reshetikhin modules. J. Eur. Math. Soc. (JEMS), 18(5):1113–1159,
+2016.
+[KKKO15] S-J. Kang, M. Kashiwara, M. Kim, and S. Oh. Simplicity of heads and socles
+of tensor products. Compos. Math., 151(2):377–396, 2015.
+[KKOP21] M. Kashiwara, M. Kim, S.-J. Oh, and E. Park. Monoidal categorification and
+quantum affine algebras II. arXiv e-prints, page arXiv:2103.10067, March 2021.
+[KNS11] A. Kuniba1, T. Nakanishi, and J. Suzuki. T-systems and Y-systems in inte-
+grable systems. J. Phys. A: Math. Theor., 44, 2011.
+[LM14] E. Lapid and A. Mínguez. On a determinantal formula of Tadić. Amer. J.
+Math., 136(1):111–142, 2014.
+[LM16] E. Lapid and A. Mínguez. On parabolic induction on inner forms of the gen-
+eral linear group over a non-archimedean local field.
+Selecta Math. (N.S.),
+22(4):2347–2400, 2016.
+[LM18] E. Lapid and A. Mínguez. Geometric conditions for ˝-irreducibility of certain
+representations of the general linear group over a non-archimedean local field.
+Adv. Math., 339:113–190, 2018.
+[LS90] V. Lakshmibai and B. Sandhya. Criterion for smoothness of Schubert varieties
+in Slpnq{B. Proc. Indian Acad. Sci. Math. Sci., 100(1):45–52, 1990.
+[MS14] A. Mínguez and V. Sécherre.
+Représentations lisses modulo ℓ de GLmpDq.
+Duke Math. J., 163(4):795–887, 2014.
+
+ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS
+21
+[MY12] E. Mukhin and C. A. S. Young. Extended T-systems. Selecta Math. (N.S.),
+18(3):591–631, 2012.
+[Nak01] H. Nakajima. Quiver varieties and finite-dimensional representations of quan-
+tum affine algebras. J. Amer. Math. Soc., 14(1):145–238, 2001.
+[Sjo07] J. Sjostrand. Bruhat intervals as rooks on skew Ferrers boards. J. Combin.
+Theory Ser. A, 114(7):1182–1198, 2007.
+[Tad95] M. Tadić. On characters of irreducible unitary representations of general linear
+groups. Abh.Math.Semin.Univ.Hambg., 65:341–363, 1995.
+[Zel80] A. Zelevinsky. Induced representations of reductive p-adic groups. II. On irre-
+ducible representations of GLpnq. Ann. Sci. École Norm. Sup. (4), 13(2):165–
+210, 1980.
+[Zel81] A. Zelevinsky. The p-adic analog of the Kazhdan-Lusztig hypothesis. Funct
+Anal Its Appl, 15:83–92, 1981.
+Léa Bittmann, Hodge Institute, University of Edinburgh, United Kingdom.
+Email address: [email protected]
+

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+Augmented Reality’s Potential for Identifying and
+Mitigating Home Privacy Leaks
+Stefany Cruz1, Logan Danek1, Shinan Liu2, Christopher Kraemer6, Zixin Wang3
+Nick Feamster2, Danny Yuxing Huang4, Yaxing Yao5, Josiah Hester6
+1Northwestern University, 2University of Chicago, 3Zhejiang University
+4New York University, 5University of Maryland, Baltimore County, 6Georgia Institute of Technology
+Abstract—Users face various privacy risks in smart homes, yet
+there are limited ways for them to learn about the details of such
+risks, such as the data practices of smart home devices and their
+data flow. In this paper, we present Privacy Plumber, a system
+that enables a user to inspect and explore the privacy “leaks” in
+their home using an augmented reality tool. Privacy Plumber
+allows the user to learn and understand the volume of data
+leaving the home and how that data may affect a user’s privacy—
+in the same physical context as the devices in question, because
+we visualize the privacy leaks with augmented reality. Privacy
+Plumber uses ARP spoofing to gather aggregate network traffic
+information and presents it through an overlay on top of the
+device in an smartphone app. The increased transparency aims to
+help the user make privacy decisions and mend potential privacy
+leaks, such as instruct Privacy Plumber on what devices to block,
+on what schedule (i.e., turn off Alexa when sleeping), etc. Our
+initial user study with six participants demonstrates participants’
+increased awareness of privacy leaks in smart devices, which
+further contributes to their privacy decisions (e.g., which devices
+to block).
+I.
+INTRODUCTION
+The increasing adoption of Internet-connected smart de-
+vices has brought huge improvements to our lives. Yet, these
+devices also raise significant privacy concerns from their users,
+such as sensitive data collection [53], [51], data sharing [51],
+and data misuse [22], [23], [27]. Literature has suggested
+many types of privacy risks associated with smart devices.
+For example, some seemingly innocent data, such as the
+network traffic shapes and patterns of smart devices, may
+reveal sensitive personal information, such as users’ daily
+schedule, their gender, date of birth, social security number,
+location, and behaviors [5], [3].
+However, many risks are not obvious to users due to the
+opaque nature of the data practices of smart devices; the
+average users lack an understanding of how their data is
+collected, processed, and shared [51], [50], [21]. Prior research
+has proposed various ways to increase users’ awareness of the
+data practices in smart homes, such as data dashboards, mobile
+phone apps, ambient light and sounds, and so on [44], [15],
+[9], [16]. Some other mechanisms (e.g., IoT Inspector [15])
+focus on specific aspects of the data practices and present
+network traffic data to users so that they can access first-
+hand data of the data flow in/out of smart devices. Yet,
+most mechanisms we know decouple such transparency from
+the device themselves—i.e., users need to learn about the
+data practices separately from the smart devices—making
+the information less intuitive to consume, especially for the
+average user. In addition, these mechanisms do not provide
+users with the ability to take action if they notice unexpected
+data practices (e.g., blocking the data from being sent out to
+third parties).
+In this paper, we focus on the data flow in and out
+of smart devices. We build a proof-of-concept smartphone-
+based augmented reality system called Privacy Plumber to
+increase users’ awareness of the data flows of smart devices
+and provide them with controls to block certain data flow if
+needed. We focus on data flow rather than other aspects of
+data practices (e.g., types of data being collected) mostly due to
+practicality and feasibility reason, as we can reasonably capture
+data flow and identify its source and destination using ARP
+spoofing [15]. In addition, from the smart devices’ perspective,
+these devices have multiple tiers of software, all of which entail
+some type of tracking. Such tracking is generally embodied
+in the data flow. We use augmented reality to visualize data
+flows in the same physical environment as the devices in
+question; this method could potentially help users establish
+a connection between the devices and their data flows in the
+same context. Users’ proper understanding of data flow may
+help them understand the privacy implications of devices such
+as smart TVs [28], voice assistants [15], children’s toys [10],
+security cameras [24], [35], and smart light bulbs [8].
+The development of Privacy Plumber is inspired by the
+following three gaps in the literature. First, the data flows of
+smart devices are opaque and not visible to users. Second,
+existing tools to monitor network traffic of smart devices (e.g.,
+IoT Inspector [15], open.Dash [9]) require a certain level of
+technical knowledge to be able to interpret the results—not to
+mention that the results are often decoupled from the physical
+environment where the smart devices are situated. Oftentimes,
+the results are presented on, for instance, dashboards on
+computers or phones, where there is a disconnection between
+the visualization of data flows and the smart devices that
+create the data flow. Third, existing tools or mechanisms do
+not provide users with the ability to control unnecessary or
+unexpected data flows. With Privacy Plumber, we aim to bridge
+the gaps and increase users’ awareness and control of the data
+flow in smart devices.
+arXiv:2301.11998v1  [cs.CR]  27 Jan 2023
+
+Fig. 1: Privacy Plumber lets a user find and mitigate potential
+privacy violations in the smart home. The figure shows a
+user walking around the smart home and inspecting the traffic
+and trackers coming out of a Samsung Smart Fridge using
+the Augmented Reality enabled app. Furthermore, (not shown
+in the picture above) users can use built-in, infrastructure-
+free controls to limit traffic of devices to times of day—
+without requiring any additional hardware or modifications
+to the network. The graph shows the actual network traffic
+as the user interacted with the Smart Fridge: A: turning on
+the ice maker; B: browsing recipe; C: browsing goods; D:
+interacting with the Bixby voice assistant of the fridge; E:
+opening the fridge door; F: adding items to the shopping list.
+During these interactions, the Smart Fridge communicated with
+various advertising and tracking services, such as DoubleClick
+and Tapad.
+Privacy Plumber uses augmented reality (AR) techniques
+and visualizes real-time network traffic flowing in and out
+of smart devices through an overlay. It allows users to find
+potential privacy leaks in their homes by pointing the AR-
+based app at smart devices. As shown in Figure 1, the app adds
+an overlay on top of the smart devices in which it displays
+a real-time data flow based on the network traffic with the
+necessary information for users to understand it. We chose
+to use AR because, as privacy is highly contextual [32], it
+can provide strong contextual connections between the actual
+real-time privacy leaks, and the user actions (or inaction).
+This allows the smartphone to function as a viewfinder into
+the invisible world of data flow and identify potential privacy
+violations. The smartphone application relies on a companion
+software tool hosted on a laptop or desktop on the same home
+network. This tool discovers smart devices in a user’s home,
+intercepts their traffic via ARP spoofing [48], and analyzes the
+data flow (e.g., what traffic is leaving the home over time)—
+without requiring the user to modify their network settings
+0
+1
+Nest Camera
+Live streaming
+0
+2
+Samsung Fridge
+Door opening
+Recipe browsing
+0
+10
+20
+30
+40
+50
+60
+70
+80
+time [s]
+0
+10
+Amazon Echo
+Weather reporting
+Radio playing
+traffic [mbps]
+Fig. 2: Outbound network traffic from various smart home IoT
+devices: a Nest Camera, an Amazon Echo, and a Samsung
+Smart Fridge. Traffic increases or provides a fingerprint for
+many types of seemingly benign actions, creating a privacy
+leak. Current systems do not provide real-time context or
+ability to experiment with these devices, nor control their
+leakage.
+or install additional hardware. When users would like to take
+action and block certain data flow, ARP-spoofing is used again
+to jam specific devices’ traffic (thereby blocking the device)
+at the time of day set by the user.
+We build a proof-of-concept prototype and conducted a
+pilot study with 6 participants in our lab to collect their
+feedback on the prototype. Our initial findings have suggested
+that Privacy Plumber helped participants understand the net-
+work traffic, increased their awareness of potential privacy
+violations, and helped them make more informed decisions
+on how to handle IoT devices.
+This paper makes three contributions. First, to the best of
+our knowledge, Privacy Plumber is the first mechanism that
+provides users with real-time information on the data flow of
+their smart devices. This paper proves the possibility of using
+AR-based technology as a viable option to increase users’
+awareness of the data flows of smart devices. Second, our
+initial evaluation shows promising results, indicating users’ po-
+tential acceptance of these technologies. Third, we summarized
+lessons learned from the pilot user study to inform the design
+and development of future systems that aim to improve users’
+awareness of data practices in smart homes.
+II.
+BACKGROUND AND RELATED WORK
+In this section we discuss related work seeking to under-
+stand or discover privacy leaks, and the tools that exist to
+help users understand and mitigate them. Privacy Plumber is
+meant to to provide a handheld and zero-cost inspection and
+experimentation tool for privacy leaks of nearby smart devices
+in the home, and a straightforward and low burden method for
+mitigating those leaks.
+A. Privacy Issues in Smart Home
+Over the decades, privacy issues have been deeply dis-
+closed in smart home, such as transparency of data collection,
+data sharing, and accessibility [20], [50], [49], [16], [53], [30],
+[50]. Some smart home devices have always-on sensors that
+capture users’ offline activities in their homes and transmit
+relevant information outside of the home, especially for cloud
+services run by device manufacturers [6].
+2
+
+02:006Privacy
+PLumbe
+X
+LiveDeviceTraffic
+720
+540
+360
+180
+10
+CurrentDeviceTraffic:984.3bytesinthelast10seconds
+Sendingdatato17differentdestinations,including2
+advertisingservice(s)
+Thisdataisequivalentto536.9wordsoftextor0.5
+picturesperminute
+DeviceDetails&Control>DoubleClick
+pedel
+DoubleClick
+Tapad
+traffic [mbps]
+6
+A
+E
+C
+4
+B
+2
+0
+10
+20
+30
+40
+50
+60
+70
+80
+time [s]In the meantime, users are concerned about leaks of
+sensitive information [23], [51], [25], such as visual and
+auditory information which they see as private [23], [25].
+Thus, users have a strong desire to protect themselves against
+such recordings being accessed without their permission [30],
+[19]. However, some information users perceived as not very
+sensitive also lead to privacy leaks. For example, the home
+temperature could be used to determine whether a house is
+occupied or not, as a precursor to burglary [20].
+In fact, smart devices give off digital exhaust which can
+be used by third parties including a user’s Internet Service
+Provider, advertisers, device manufacturers, and others, to
+fingerprint activities and get sensitive information. Shown in
+Figure 2 is the network traffic and trackers of various smart
+home devices.This network traffic forms the basis of most
+leaks.
+B. Tools for Enhancing Smart Home Privacy
+Most related to Privacy Plumber are tools that watch or
+monitor network traffic in the home and provide something
+of use to the user, whether visualization and information,
+education, or a mechanism for control.
+Sophisticated, technically literate users can use systems that
+block advertising and tracking domains (e.g., PiHole [38] and
+pfSense [34]), but these methods are bespoke and often require
+additional or dedicated hardware (e.g., Raspberry Pi for Pi-
+Hole, and a supporting custom router for pfSense). Other tools
+have provided insight into what might be exposed from web-
+browsing activities, including WiFi privacy Ticker [11], but do
+not consider or scale to the new problems of connected devices
+with physical sensors and abilities in a space. Aretha [40]
+explores this tool space and proposed (but did not deploy)
+a simple firewall-based control mechanism. Aretha presents
+data in aggregate instead of contextually and in real-time.
+None of these techniques investigate a range of IoT devices,
+usually constrained by studies with participants in their own
+homes, in a time when smart home adoption is low (Aretha
+had three participants, and only one had more than a phone,
+tablet, and Alexa). None of these techniques develop a scalable
+(no additional hardware required) way to interpret privacy
+leaks and control them. Emerging smart devices are highly
+contextual and location sensitive, an Alexa in the bedroom
+versus the kitchen has different privacy exposure (i.e. the
+former gives sleep times, the latter exposes eating habits).
+Moreover, tracking these devices’ privacy exposures presents
+a technical challenge because the traffic is not centralized
+through a web browser or laptop. A tool is needed to visualize
+privacy leaks from smart devices in real-time and in context,
+educate users on the consequences of these leaks, and provide
+control mechanisms for partially mitigating these leaks.
+Wifi Privacy Ticker [11] demonstrated a first method for
+improving the awareness of users in terms of privacy by
+providing a count of the amount of sensitive data that was
+being transmitted unencrypted over the network awareness.
+By seeing this in real-time, users could adjust their behavior
+or find encrypted means to browse the web. Of course, this
+ticker was developed well before the current generation of
+smart devices, however the underlying concept of surfacing the
+invisible privacy leaks remains the same for Privacy Plumber,
+but for smart devices. Xray-refine [46], [47] provided smart
+phone users a means to visualize their exposure profile, based
+on the duration of app use. This method was an educational
+solution, but users had to adjust behavior to work around the
+constraints of the apps they were using, in some cases, opting
+out of apps to reduce privacy exposure.
+Finally, recent work like Aretha [40], PriView [36], Lu-
+mos [41], and IoT Inspector [15] look at making usable
+visualizations and mechanisms to understand and interpret data
+coming from smart devices in the home. IoT Inspector is a
+simple-to-install desktop application that uses ARP spoofing
+to gather network traffic on the Wifi network of the desk-
+top/laptop. This information is sent and collated at a server,
+and then viewed online by the user, listing different trackers
+and websites that are attached to smart device usage. Because
+of the ease of installation and no extra hardware requirement,
+IoT Inspector was deployed by thousands of users.
+In comparison, Aretha is a part research tool, part ex-
+ploratory users tool for exploring a design space of privacy
+tools and controls. Aretha helps users become aware of the net-
+work traffic flows in their homes while also educating users to
+regain their privacy in the connected home. Aretha suggests the
+use of firewall mechanisms controllable by the user, but does
+not implement them. Aretha, owing to a hardware requirement
+(a device must be attached to the Wifi router in the home) was
+only deployed in three homes, compared to the massive scale
+deployment of IoT Inspector. Similarly, PriView also has a
+hardware requirement; its users need to have dedicated external
+thermal cameras (e.g., FLIR One [36]) attached to their phones.
+For Lumos, there is no special hardware environment, although
+the focus is more on identifying hidden smart devices rather
+than analyzing the network traffic for privacy leaks.
+Privacy Plumber is not meant as a research tool or a design
+space exploration tool. It is meant as an actual, real world
+system with a focus on scalability and ease of deployment in
+any home, similar to IoT inspector. Unlike both IoT inspector
+and Aretha, Privacy Plumber provides real-time and contextual
+visualizations of privacy leaks, real-time ability to plug those
+leaks (as well as automated rule setting for plugging leaks),
+and enables experimentation in real-time.
+Finally, other significant measurement campaigns on in-
+home traffic have been conducted, focusing on the Wifi net-
+work itself or devices in the home [39], [18]. These have
+usually been for research purposes and need finding and are
+useful for informing the design of Privacy Plumber, but are not
+necessarily tools for controlling smart home device privacy.
+C. Determining Home Activities from Network Traffic
+Complementary to Privacy Plumber are other works which
+demonstrate the ability to infer activities from network traffic:
+whether on a phone, smart device, or laptop [4]. By analyzing
+the patterns of network traffic in the home, occupancy, habits
+such as sleeping, watching TV, listening to music, and some-
+times preferences, can all be determined. HomeSnitch [33],
+Peek-a-boo [1], and HoMonit [52] all utilize machine learning
+with varying degrees of success to identify activities in the
+home from network traffic. Other tools utilized for monitoring
+Internet connected smart devices in the home, IoT Sentinel [26]
+and IoT Sense [7], have shown that particular devices can be
+3
+
+fingerprinted by their traffic patterns. Enabling another way
+for an ISP or third party to determine the activity in the
+home. Each of these systems and methods are complementary
+with Privacy Plumber; inferred activities from traffic would be
+useful to surface in Privacy Plumber for the user to understand
+privacy exposure and know when to mitigate it, and device
+fingerprinting provides a way for zero-registration or setup of
+Privacy Plumber in a home.
+D. Challenges: Contextual, Real-time Privacy Understanding
+and Control in the Home
+Despite the diverse work in the smart home privacy space,
+significant gaps and challenges remain, which we detail below.
+C1: Users can’t model what devices are doing, especially
+without context. With tools like IoT Inspector, a user might be
+able to count the number of trackers and advertisers contacted
+in a day from the sum of their interactions with smart devices.
+But how can a user know that turning on the NPR podcast
+on their smart fridge will send thousands of bytes of informa-
+tion to Bloomberg News for advertising purposes? How can
+they know that turning on the device sends a short burst of
+traffic? Users know that data captured will often be used for
+advertising, which often generates an adverse reaction [45].
+However, with smart devices, it is not always clear what
+actions or contexts trigger data being transmitted [13]. Things
+like Privacy labels for websites and smart devices are meant to
+give a method for scoring devices privacy [42], [17]. However,
+these are static representations of the privacy exposure of a
+device. With tools like IoT inspector and Aretha, aggregate
+views of data are seen (as opposed to real-time views), not
+associated with very fine user actions: like the turn on the light,
+say command to Alexa, or open the fridge door. Because of
+this granularity, the mental models of what devices are doing,
+and what they are sharing, are very perplexing. Privacy tools
+must address this lack of action mapping to network traffic,
+enabling contextual integrity [32] in real-time.
+C2: Users don’t have intuitive methods to control the
+privacy “valve”. Users want devices that provide helpful
+features, but they do not know the cost of this ease. One option
+is to just unplug the device; however, this is all or nothing.
+Users need a way to valve the privacy flow to something they
+are comfortable with, or to at least be able to analyze the
+tradeoffs [43]. Making privacy more ”tangible” [2] is one way
+this can be done; where the privacy leaks are more visceral.
+Selective firewalls (such as pfSense [34]), or other more fine
+grained network mechanisms may provide a means to control
+the privacy valve, but this must be intuitive and understandable
+to the user, and they must be able to actually ”see” the effect
+of turning this valve.
+C3: Smart devices are context (location, time, action)
+dependent. Smart devices are necessarily scattered around
+the home; and this will continue as more devices become
+intelligent, and more applications are explored. Watching a
+desktop or laptop traffic meter and figuring out which device
+in which room is doing what at which times, is mentally trying
+for the user and disassociates the device from the physical
+space that defines its context and use. Just like when trying to
+find leaks in pipes, physical proximity is required. Handheld
+inspection tools provide mobility, and enable in-situ fixing and
+experimentation.
+C4: Users can’t experiment. Indeed, because of contextual
+changes in how private information is leaked, experimentation
+is difficult with existing tools that generally provide traffic
+summaries. Interactions with smart devices can last only a few
+seconds. Enabling a user to experiment with different actions
+and uses of a smart device, and then see the associated network
+traffic in real-time, would provide a powerful way to build a
+mental model. However, providing an ability to experiment is
+challenging with the current suite of tools.
+C5: Technical challenge of scalability and deployment. If
+a privacy tool is to be useful and translate to the general
+public, it must be hardware free, or at least trivially easy to
+deploy to enable scalability and broad adoption. Commercial
+products like fing.com embed all functionality in a single
+phone application. Large scale deployments like with IoT
+Inspector are enabled through a desktop application that is
+easy to install. However, these methods do not provide controls
+since that is technically difficult to do without custom hardware
+put between the Wifi endpoint and the user. On the other
+hand, hardware requirements or custom install procedures
+reduce the deployment size of tools like Aretha, or narrow
+the user base by requiring technical ability, as with PiHole.
+It is not clear how to implement mechanisms of control
+without changing the Wifi network and infrastructure. To
+create scalable, user-centered, novice friendly privacy tools,
+mechanisms for enabling control of smart device traffic without
+hardware intervention must be developed.
+III.
+SYSTEM DESIGN
+We present Privacy Plumber as a proof-of-concept and
+end-to-end system to address the challenges listed of scalable
+and general population serving privacy tools for emerging
+smart homes. Privacy Plumber is inspired by various handheld
+tools for identifying and fixing faults in large and complex
+systems. For example, acoustic leak finding has been used for
+decades to localize leaks in gas and water pipelines. Handheld
+oscilloscopes, multimeters, and RF Spectrum Analyzers have
+helped engineers debug problems in large electrical systems.
+These handheld devices make the invisible signals visible and
+interactive. They allow real-time experimentation and debug-
+ging. Inspired by these devices, Privacy Plumber is designed
+to offer a general user a level of insight and control of the
+invisible privacy leaks that are rampant in Internet-connected
+smart devices in the home. Privacy Plumber is composed of
+two pieces as shown in Figure 3:
+(1) the IoT Network Analyzer, a desktop application
+which collects real-time data on smart devices on the shared
+Wifi network, and provides an infrastructure and hardware free
+mechanism to block arbitrary devices traffic, and;
+(2) the Privacy Plumber phone application, which serves
+as a viewfinder or inspector for any smart devices in view, and
+presents data from the desktop application, including device
+network traffic and potential privacy leaks to the users, along
+with educational content matched to what is known about the
+device, all in real time.
+4
+
+ARP
+ Spoofer
+Viewfinder
+Educational
+Views
+Packet
+Analyzer
+Traffic Analyzer
+Controls
+IoT Network Analyzer
+Router
+Privacy Plumber App
+Traffic  and Control
+Visualizer
+IoT Devices
+Network
+Traffic
+ARP
+Spoofing
+Leak
+Information
+Fig. 3: System diagram of Privacy Plumber including the IoT Network Analyzer and Privacy Plumber mobile application. IoT
+Network Analyzer runs on a computer that is connected to a user’s router. IoT Network Analyzer automatically discovers and
+captures IoT devices on the same network using ARP spoofing. Privacy Plumber connects with IoT Network Analyzer to present
+the network analysis in AR. The user can then examine their devices’ network traffic and control when they want their devices
+to be on or off.
+Overview of Usage. A user would first download, install,
+and run IoT Network Analyzer on their computer and the
+Privacy Plumber app on their mobile phone, such that both
+the computer and the phone are on the same local area
+network. Let us assume that the user is interested in inspecting
+a smart device like an Amazon Echo. While running the
+Privacy Plumber app, the user points the phone camera to
+Echo and speaks a voice command (e.g., “Alexa, what is
+the weather?”) IoT Network Analyzer captures all network
+traffic between Echo and the Internet, analyzes the packets,
+and identifies destinations that are third-party advertising and
+tracking companies. The Privacy Plumber app extracts this
+information from IoT Network Analyzer and visualizes key
+statistics for the user—such as real-time bandwidth usage of
+the device and the number of advertising and tracking services
+contacted—as an overlay in the AR view.
+When the user points the phone camera at a device, the
+Privacy Plumber app does not recognize devices with computer
+vision algorithms. Instead, for the purpose of this prototype,
+we print a QR code on each IoT device. The QR code includes
+the device’s MAC address, its name, and the manufacturer. The
+app uses the phone’s camera to scan for the QR code, identifies
+the device based on the QR code, and displays the device with
+a dial menu around it (see Figure 4a). The options in the menu
+allow the user to see the outbound traffic from the device as
+well as read a brief article stating what types of information
+the device may be tracking. The user may also use the Device
+Control menu (Figure 4c) to manually block or allow traffic
+from the device. Future versions of the app will use computer
+vision to recognize devices; see the discussion in Section V.
+Privacy Threat Model and Assumptions. We assume that
+a user’s privacy may be potentially violated if an IoT device
+exhibits either or both of the following behaviors. In Threat 1,
+an IoT device could contact an advertising and tracking service
+on the Internet. In Threat 2, an IoT device could be sending
+out network traffic to hosts on the Internet when the user does
+not expect any network activities—for example, when the user
+is not interacting with the device.
+We design both the Privacy Plumber app and IoT Network
+Analyzer with this privacy threat model in mind. IoT Network
+Analyzer captures packets, analyzes the headers, identifies the
+destination hosts (based on the IP addresses, domain names,
+and the TLS Server Name Indication fields), and determines
+if a destination host is an advertising and tracking company.
+The Privacy Plumber app displays the number of advertising
+and tracking services (e.g., the red text below the graph in
+Figure 4b), thereby helping users toward identifying Threat 1.
+Based on the byte counters from IoT Network Analyzer, the
+Privacy Plumber app also shows a bandwidth graph that plots
+the bytes sent per second over time (e.g., the time-series graph
+in Figure 4b). This graph could help users correlate network
+activities with human interactions—or the lack thereof—with
+given IoT devices and thus identify possible instances of
+Threat 2. Note that IoT Network Analyzer does not parse the
+payload of packets to discover sensitive information within the
+traffic, as the network traffic is likely encrypted.
+A. Design Goals
+Privacy Plumber must make the underlying behavior of the
+devices in the home apparent, and enable forms of fine-grained
+(informed) control of the leakage of sensitive information for
+the user. Towards this end, and addressing the challenges
+described in Section II-D, we are guided by the following
+design goals.
+(1) Handheld and Mobile. Smart devices are scattered
+throughout the home. Phone adoption is nearly universal.
+Using a phone as a window into the information world gives
+5
+
+Privacy
+Jagwnnd
+X
+Live Device Traffic
+Bytes/seo
+1500
+0
+1125
+0
+7500
+3750
+-10
+Current Device Traffic: 26954.1 bytes in the last 10
+seconds
+Sending data to 15 different destinations
+This data is equivalent to 14702.2 words of text or 13.5
+pictures per minute
+Amazon Details
+Status: Allowed
+BLOCK
+ALLOW
+Device
+Device
+Traffic
+Traffic
+Custom Time Rule
+ApplyRule
+Time (ex. 8:00AM)
+Turn On
+Turn Off:
+Time (ex. 8:00PM)
+RemoveRulecontext and a sense of place. The phone form factor increases
+the likelihood of adoption and allows for inspection on the go;
+users can trigger or interact with devices and easily watch the
+movement of data, instead of having to return to the desktop.
+(2) Real-time. Seeing statistics after the fact, as in most
+systems, is not as impactful or helpful when developing a
+model of how devices operate. Moreover, real-time analysis
+enables experimentation, providing users with a mechanism for
+exploring limitless scenarios and quickly associating triggers
+with outcomes.
+(3) Infrastructure/Hardware Free. Many other meth-
+ods require custom hardware. This increases cost and raises
+the barrier to entry. We hope to enable anyone, especially
+those that may have limited autonomy over infrastructure (i.e.
+renters, low-resourced communities) to be able to inspect the
+devices put in their living space.
+(4) Intuitive Controls. Complex mechanisms to control or
+limit the flow of privacy are not interpretable by users, and are
+possibly frustrating. Configuring a firewall is not a task most
+people would enjoy. Straightforward controls, with visible
+results, once those controls are put in place, are essential for
+users to trust the capability of the system.
+(5) Educational. The ever-changing landscape of devices
+and the security/privacy arms is impossible to keep up with for
+privacy tools. Assisting users in understanding what makes cer-
+tain devices leakier (e.g., always-on microphone) is essential.
+To realize these design goals, we build the Privacy Plumber
+app—i.e., the handheld form factor—and IoT Network An-
+alyzer as a two-part architecture working in tandem. Both
+systems must be running on the same local area network.
+IoT Network Analyzer, running on a computer, captures and
+analyzes network traffic between smart devices on the network
+and the Internet. IoT Network Analyzer acts as a server and
+provides the above information over an HTTP REST API. The
+Privacy Plumber app, acting as a client, regularly polls the
+REST API and presents the analysis as an AR overlay to users.
+In the following sections, we detail the pieces of the system
+and how they interact to enable understanding and control of
+smart devices in the home. In Section III-B we discuss the
+IoT Network Analyzer and its role in capturing and curating
+privacy leak information; in Section III-D we describe the
+phone app design; in Section III-C we detail the mechanisms
+we use for controlling devices on a schedule, and finally, in
+Section III-E we describe a few ways to use Privacy Plumber.
+B. Low Burden Home Network Traffic Capture
+To use the Privacy Plumber app, the user must also
+have IoT Network Analyzer running on a computer (macOS,
+Windows, or Linux) that is on the same local area network
+as the phone. For our study, we run IoT Network Analyzer
+on a Raspberry Pi 3 Model B that is connected to the lab’s
+network via Ethernet. We based IoT Network Analyzer’s code
+on the open-source project, IoT Inspector [15], and made
+modifications according to our needs. In particular, whereas
+the original IoT Inspector constantly sends captured traffic’s
+metadata to the researchers’ servers, IoT Network Analyzer
+runs without the Internet; it processes the captured traffic
+locally and exposes the traffic via a REST API. Furthermore,
+whereas the original IoT Inspector runs on users’ computers
+and visualizes the traffic in a browser-based dashboard, IoT
+Network Analyzer uses an AR-based app, Privacy Plumber,
+to visualize the network traffic; the mobile app reads the
+processed traffic through the abovementioned REST API and
+presents the results as an AR overlay.
+Once running, IoT Network Analyzer automatically discov-
+ers IoT devices on the network, captures their network traffic
+via ARP spoofing, produces traffic statistics (e.g., bandwidth
+usage and identifying advertising and tracking services) over
+a local HTTP REST API, and blocks select devices (if desired
+by the user). We explain each of these features below.
+Discovering IoT devices. Upon launch, IoT Network An-
+alyzer automatically broadcasts ARP packets to the local
+area network and discovers active devices. To identify IoT
+devices, Huang et al. [15] describe an algorithm that infers
+the likely identities of IoT devices based on MAC OUI (i.e.,
+Organizationally Unique Identifier, basically the first three
+octets of a MAC address), DNS, and UPnP messages. For the
+prototype in this paper, we only use the MAC OUI. Within the
+code of IoT Network Analyzer, we have already hard-coded
+the mapping between OUIs and names of five IoT devices in
+our lab (which we can find out beforehand). In this way, IoT
+Network Analyzer can instantaneously identify the IoT device
+in our lab without relying on the device identification algorithm
+in Huang et al. [15].
+Capturing network traffic. Once IoT Network Analyzer iden-
+tifies a known IoT device on the lab’s network, it automatically
+starts intercepting network traffic between the device and the
+Internet via ARP spoofing, a technique used in the original IoT
+Inspector implementation and which incurs an overhead of 3.4
+Kbps, given that we have five IoT devices in the lab [15].1
+Obtaining traffic statistics. All traffic to and from IoT devices
+in our lab is redirected through IoT Network Analyzer. In do-
+ing so, IoT Network Analyzer is able to obtain statistics about
+the network traffic for every device, including the device’s
+MAC address (from which to extract the OUI and determine
+the device’s identity based on our hard-coded mapping); the
+number and size of packets (from which to infer the bandwidth
+usage); the remote IP addresses, DNS requests and responses,
+and the Server Name Indication field within TLS packets
+(from which to infer the remote hostname and whether the
+hostname is associated with a known advertising and tracking
+company, based on the Disconnect block list [12]. IoT Network
+Analyzer presents all these statistics and information via an
+HTTP REST API that the Privacy Plumber app can access over
+the local area network. For example, if the computer running
+IoT Network Analyzer has a local IP address of Ii, then the
+Privacy Plumber app (on the same local network) can access
+the traffic information via http://[Ii]/get traffic.
+Phone
+Application:
+App
+Implementation. The Privacy
+Plumber mobile app was implemented in Unity using C#
+and is cross-platform, tested on Android and iPhone. The
+1Per Huang et al. [15], our setup includes N = 5 devices. It follows that
+N(N + 1) = 30 spoofed ARP packets are sent every two seconds. As each
+ARP packet has 28 bytes, the overhead is 28×30/2∗8 = 3, 360 Bits/second
+or 3.4 Kbps.
+6
+
+(a) View finder
+(b) Traffic view
+(c) Controls
+(d) Education
+Fig. 4: Illustration of mobile application design. (a) Device recognition with interactive menu. (b) Live traffic inspection. (c)
+Rule-based device traffic control (i.e., blocking and unblocking). (d) Educational material on privacy details.
+app works by communicating with IoT Network Analyzer via
+HTTP GET requests, as described in the previous paragraph,
+to obtain JSON-encoded information about the devices on the
+network and their traffic. Parsing these JSON objects, the app
+visualizes the information as charts and text on the AR display
+(e.g., Figure 4b). The app also shows an interface where users
+could block an IoT device’s traffic, e.g., Figure 4c. Once the
+user confirms, the app sends the corresponding request to IoT
+Network Analyzer via the HTTP REST API, and IoT Network
+Analyzer would subsequently block the device by jamming the
+device with corrupt ARP packets.
+C. User Control of Privacy Leaks from a Phone
+With Privacy Plumber we also want to help the user feel
+more empowered by allowing them to take control of their
+devices with the ability to block device traffic. Users can
+manually block or allow device traffic indefinitely, or they can
+set rules to govern when they want their device to be on or off
+and for how long (Figure 4c). Users are also given the option
+to physically power off their device altogether. In this way,
+Privacy Plumber provides a closed-loop system where users
+can analyze the information flow out of a given device, then
+immediately apply direct control over that device in response
+and receive immediate feedback via the traffic view.
+To illustrate how a user might control an IoT device’s
+traffic, let us say that a user feels uncomfortable with an IoT
+device communicating with the Internet. The user can use the
+Privacy Plumber app to block Internet access on the device.
+As shown in Figure 4c, the user can click “Block Traffic”
+on the app to indefinitely block the device, or specify when
+to block and unblock the device. Moreover, the app sends
+an HTTP request to IoT Network Analyzer, using the REST
+API 2 (where Ii is the IP address of the running instance
+of IoT Network Analyzer). During the period of blocking,
+IoT Network Analyzer jams the communication of the device
+by using a corrupt source MAC address in the spoofed ARP
+packets (as described in Section III-B). IoT Network Analyzer
+stops this process at [unblock time], upon which IoT Network
+Analyzer sends out spoofed ARP packets without the corrupt
+source MAC address. This gives users the ability to control
+the times of day when they want their devices to be on or off.
+Privacy Plumber’s software-based device blocking offers
+several advantages over simply turning off or disconnecting
+a device. First, users do not need physical access to the
+device; for instance, many surveillance cameras are mounted
+on ceilings and are difficult to power off. Second, users can
+temporarily disable a device when they are feeling uncom-
+fortable, e.g., blocking Amazon Echo for an hour during a
+sensitive phone call or conversation, through Privacy Plumber.
+Such temporary blocking is difficult to achieve through Echo’s
+app (no such feature) or manually (e.g., the user has to
+remind themselves to re-connect Echo again). Third, though
+not currently implemented, Privacy Plumber, with the help
+of IoT Network Analyzer, can block based on the context
+(i.e., future work). For example, when IoT Network Analyzer
+detects the presence of a user’s phone on the network (e.g., by
+checking if the phone responds to periodic ARP requests), IoT
+Network Analyzer automatically blocks all indoor cameras;
+when the phone leaves the network (e.g., when the user is
+out), IoT Network Analyzer could automatically unblock all
+indoor cameras.
+Technical Mechanism for Blocking Devices. A major differ-
+ence with respect to IoT Inspector’s original implementation is
+2http://[Ii]/block/[device id]/[block time]/[unblock time]
+7
+
+Privacy
+Lagwnd
+(A)
+amagon
+M
+Device Details & Control >Privacy
+PLumber
+X
+Live Device Traffic
+Bytes/sec
+720
+540
+360
+180
+-10
+.9
+CurrentDeviceTraffic:984.3bytes inthe last10seconds
+Sending data to 17 different destinations, including 2
+advertising service(s)
+This data is equivalent to 536.9 words of text or 0.5
+pictures per minute
+M
+amazon
+Device Details & Control >Privacy
+Lagwd
+(A)
+M
+cef:co.ob:
+Amazon Details
+Status:Allowed
+BLOCK
+ALLOW
+Device
+Device
+Traffic
+Traffic
+CustomTime Rule
+ApplyRule
+:uo ni
+Time (ex. 8:00AM)
+Tum Off:
+Time (ex. 8:00PM)
+Remove RulePrivacy
+Laqwnd
+X
+Device Privacy Details
+Echo Smart Speaker
+This voice assistant is like a butler, providing voice access and
+control of music and other media, enabling voice based
+purchases, making phone calls, and helping to keep track of
+things,among many other functions.
+Potential Privacy Leaks
+Location
+Your physical location is leaked when interacting with this
+device. Because motion can be detected, your near exact
+locationwithinaroomcanbedetermined.
+讠 Activity
+(A)
+amazo
+三
+DeviceDetails&Control>that we have added the capability of blocking devices in IoT
+Network Analyzer. Using the HTTP REST API 3, the Privacy
+Plumber app can request IoT Network Analyzer to block a
+certain device at a particular time (for instance, because the
+user does not want the device to be communicating to the
+Internet). Upon receiving this request, IoT Network Analyzer
+jams the network communication of the device by sending it
+spoofed ARP packets with corrupt MAC addresses.
+To illustrate this process, let us assume that the computer
+running IoT Network Analyzer has a MAC address Mi and IP
+address Ii. Let us also assume that IoT Network Analyzer is
+about to intercept the communication from the gateway (with
+MAC address Mg and IP address Ig) to a particular device
+(with MAC Md and IP Id) without blocking. To do so, every
+two seconds, IoT Network Analyzer sends an ARP packet to
+the device, such that the ARP packet has a source MAC of
+Mi and a source IP of Ig, along with a destination MAC of
+Md and a destination IP of Id. In contrast, let us assume that
+IoT Network Analyzer is to block the device. It sends the
+same ARP packet to the device, except that the ARP packet’s
+source MAC is a series of random numbers (instead of Mi)
+that represent an unreachable MAC address on the local area
+network.
+D. Visualizing and Understanding Traffic in Real-Time
+One of the goals of Privacy Plumber is to show users
+contextualized network activities of IoT devices to help them
+pinpoint the potential privacy risks. In this section, we discuss
+how Privacy Plumber utilizes Augmented Reality to help users
+contextually visualize their devices’ network traffic informa-
+tion in real-time, provide a chart of network traffic in real-
+time, and provide links to other research in which the privacy
+concerns of the inspected device have been studied (including
+home behavior inference, sleeping behaviors, and personal
+data). Lastly, users are able to send feedback and bug reports.
+Use of Augmented Reality. The use of AR visualization
+makes the interaction with the device the user is inspecting
+more tangible and contextual. While IoT Inspector [15] and
+IoT Network Analyzer are text-only data-driven analyzers that
+can only be accessed using browser HTTP requests, Privacy
+Plumber is a fully-fledged interactive application due to the
+utilization of AR. By pointing their camera at the device
+being inspected, the user can see, in their environment, the
+traffic coming out of the device that they are physically
+inspecting. Users can interact with their devices and receive
+immediate feedback about data output and communication
+with advertisers. Combined with manual device control, this is
+intended to help the user feel informed and in control of the
+IoT devices that physically surround them, similar to the use
+of a TV remote control.
+Learning About Privacy Threats. We aim to educate and
+inform users on how their IoT devices expose their network
+traffic information to third parties. In Figures 4d and 5, the
+app shows icons surrounding the IoT device. When any of
+these icons are pressed, they provide links to other research
+materials—which we have manually curated in advance—
+where the privacy concerns of the inspected device have been
+3http://[Ii]/block/[device id]/[block time]
+studied. Depending on the device, Privacy Plumber provides
+the following categories of potential privacy violations repre-
+sented by icons:
+•
+Location: Your physical location either roughly (your
+address) or fine-grained (room you are in).
+•
+Activity: Your physical activity such as walking, talk-
+ing, sleeping.
+•
+Screen: Your online activity, such as when you browse
+videos on YouTube or surf the web.
+•
+Identity: Attributes that can identify you such as your
+face or voice.
+•
+Shopping: Data on your usage of money or products.
+•
+Health: Can infer different health markers without
+consent (heart rate, breathing, and others).
+E. Privacy Plumber Example Use Cases
+In this section, we illustrate two example use cases of
+the Privacy Plumber app. We focus on the ability of Privacy
+Plumber to enable experimentation and the usefulness of a
+real-time inspector. We will describe the users’ reactions in
+Section 4.3.
+Example 1: Is Echo Always Listening?
+A user may use the Privacy Plumber app to correlate net-
+work activities on an Amazon Echo device with the user’s
+interactions—or the lack thereof—with it. While pointing the
+AR camera at the device, the user could invoke a voice com-
+mand, such as “Alexa, what is the weather”, while observing
+the device’s bandwidth usage graph on the Privacy Plumber
+app. Afterward, the user may physically press the mute button
+on Echo, repeat the same voice command, and observe the
+bandwidth usage graph on the app.
+Example 2: What is this App on My Smart Fridge?
+Many smart fridges have built-in touch-screen panels. For
+example, the Samsung Smart Fridge has a tablet-like touch-
+screen panel to control various settings of the fridge (such as
+temperature). The panel also allows users to access various
+built-in apps, such as checking recipes or ordering ingredients
+online. A user who is concerned with the privacy of such
+apps may point the AR camera at the fridge, interact with
+an app, and observe the advertising and tracking services
+counter on the app. This counter shows, in real-time, the total
+number of advertising and tracking services that the fridge has
+communicated with, based on the Disconnect block list [12].
+IV.
+PILOT USER STUDY
+To test how users react to Privacy Plumber and inform its
+future iteration, we conducted a pilot study with 6 participants
+to experiment with, understand, and control the potential
+privacy violations of IoT devices. It should be noted that the
+pilot study would be best conducted in participants’ homes.
+However, due to University research restrictions, the COVID-
+19 pandemic has made it difficult for us to recruit real users,
+distribute hardware (e.g., phones powerful enough for AR
+and Raspberry Pi’s for running IoT Network Analyzer), and
+conduct a free-living study.
+8
+
+We conducted a one-day controlled lab study in our IoT
+Lab with 6 participants. Participants were invited to use
+the Privacy Plumber app while interacting with several IoT
+devices in the lab, including Samsung Smart Fridge, Amazon
+Echo, Google Home, Samsung Smart TV, and Google’s Nest
+Cam. Our goal is to assess whether using augmented reality
+to display network traffic (i.e., by using Privacy Plumber)
+influenced the participants’ awareness of privacy and changed
+their behaviors.
+In the following sections, we present the details of the pilot
+study and discuss some highlights in the results as well as
+lessons learned to inform the next iterative of Privacy Plumber.
+A. Participants Recruitment and Demographics
+We recruited 6 graduate students from our institution
+through our university mailing list. We did so rather than
+recruiting from a broader population sample because of the
+constraints our university implemented during the COVID-19
+pandemic (i.e., external members were not permitted to enter
+our buildings). Our sample consisted of four males and two
+females. Three of the participants were between the ages of
+18-24, two participants were between the ages of 25-34, and
+one participant was between the ages of 35-44.
+B. Study Procedure and Data Collection
+For safety reasons and to implement social distancing
+procedures, only two people were allowed in the IoT Lab
+during the study. Aside from the participant, one of the co-
+authors in this paper served as the research coordinator. They
+were present throughout the user study to help guide the
+participants or troubleshoot any technical difficulties that could
+arise during the study procedure.
+Before the study began, each participant filled out a back-
+ground pre-survey on a computer in the IoT lab. We asked
+questions about their demographics, how technically savvy
+they are, their smart device experiences, their general under-
+standing of privacy, and their concerns about their information
+being exposed to third parties.
+After completing the survey, our research coordinator
+handed each participant a script and an Android mobile phone
+that had Privacy Plumber installed. Following the script, each
+participant opened the Privacy Plumber app, kept it running in
+the foreground, and interacted with one IoT device at a time.
+Regardless of the IoT device, each interaction consists of the
+following steps, as prescribed in the script:
+1)
+Using the Privacy Plumber app, the participant
+scanned the QR code that we had placed on the
+IoT device. The QR code encodes the device’s MAC
+address, device name, and manufacturer. Based on the
+QR code, Privacy Plumber shows the corresponding
+device’s AR model on the screen.
+2)
+The participant used the app to inspect the device’s
+traffic, while not doing anything to the device.
+3)
+The participant interacted with the device (which we
+will describe in detail). During the interactions, the
+participants observed the network traffic graph on the
+app.
+Fig. 5: This screen on the phone application describes the
+different categories of privacy leaks that different devices have,
+based on a database that we manually curated in advance.
+4)
+Using the app, the participant clicked on any of the
+icons surrounding the AR model of the device and
+read the educational material.
+After interacting with all the IoT devices, participants
+returned the phone to the research coordinator and responded
+to a post-survey that asked the same questions as in the pre-
+survey, along with a usability survey. We discuss the results in
+more depth in Sections IV-C and IV-D. We also include our
+pre- and post-surveys in the Appendix.
+Below, we describe each participant’s scripted interactions
+with each device—i.e., showing Step 3 in detail. During the
+interactions with the devices, users can access the educational
+content which is summarized from Mozilla’s “privacy not
+included” handout [29] and academic literature. Each device
+is described by the categories of privacy exposure they create,
+those categories are shown in Figure 5.
+Samsung Smart Fridge. The fridge has a built-in touchscreen
+on the door. Through the touchscreen, users have the ability
+to interact with several built-in apps, such as managing the
+shopping list, checking what is inside the fridge, and searching
+for recipes online. Users can also interact with the touchscreen
+using voice commands, using the trigger word, “Bixby.”
+Per the script, the research coordinator instructed the
+participant to follow the following three tasks. (i) Once the
+participant scanned the QR code of the smart fridge, they
+said the voice command, “Hey Bixby, do we have mangoes?”
+Bixby, the fridge’s voice assistant, would say “no,”. The
+participant then said, “Hey Bixby, can you add mangos to
+my shopping list?” Immediately, the participant looked at the
+9
+
+Privacy Leak Categories
+Smart devices can collect private information about you
+intentionally or incidentally in the following ways.
+Location
+Your physical location either roughly (your address) or fine-
+grained (room you are in).
+Activity
+Your physical activity, such as walking, talking, sleeping.
+Screen
+Your online activity, like when you watch videos on YouTube
+or surf the web.
+Identity
+Collects attributes that can identify you such as your face or
+voice.
+Shopping
+Collects data on your usage of money or products.
+Health
+Can infer different health markers without consent
+(heartrate, breathing and others).-3
+-2
+-1
+0
+1
+2
+3
+Strongly Disagree                                                                                                            
+Strongly Agree                
+I think about what information I may be exposing to 3rd parties 
+when I interact smart devices.
+I am not concerned over the 
+information I may be exposing to 3rd 
+parties when I interact with smart 
+devices.
+Privacy Plumber has made me more aware of what information I may 
+be exposing to 3rd parties when I interact with smart devices.
+Privacy Plumber has made me more aware of privacy concerns 
+regarding smart devices.
+Fig. 6: Representation of participants’ average agreement ratings relating to statements about information being exposed to third
+parties and privacy concerns caused by interacting with IoT devices. Participants rated the first two statements before and after
+the study, while the last two statements were rated at the end of the study. The results show that after the study, participants
+displayed an increase in awareness and concern about how their information is being handled when interacting with IoT devices.
+Privacy Plumber app and observed the network traffic emitting
+from the fridge for about 30 seconds. (ii) The participant
+said, “Hey Bixby, find me a Ramen recipe.” The recipe app
+popped up on the touchscreen. Using the finger, the participant
+browsed through the available recipes on the screen, while
+observing the network traffic on Privacy Plumber for 30
+seconds. (iii) The participant opened the fridge door and then
+closed it. Once again, they inspected the fridge’s network
+traffic through the Privacy Plumber app for 30 seconds.
+Amazon Echo. Interactions with Echo consists of the follow-
+ing 3 tasks. (i) The participant said the voice command, “Alexa,
+play a thunderstorm sound.” Immediately, the participant ob-
+served the network traffic on the app for 30 seconds. (ii) The
+participant physically pressed the “mute” button on the Echo
+and watch the device’s network traffic for 15 seconds. (iii) The
+participant said the same voice command as in Task (i) and
+observed the traffic in the app.
+Google Home. The participant said a voice command, “Hey
+Google, what was the final score in the Super Bowl last year?”
+The participant immediately started observing the network
+traffic on the app for 30 seconds.
+Samsung Smart TV. The participant used the TV’s remote
+control to navigate to the Bloomberg app on the home screen.
+They then started streaming a live video on the Bloomberg app
+for one minute while they observed the network traffic with
+the Privacy Plumber app.
+Nest Cam. Interactions with the camera consists of the follow-
+ing 2 tasks. (i) The participant walked into the field of view of
+the camera and stay there for five seconds, walked out of the
+camera’s field of view, and observed inspect the network traffic
+with the Privacy Plumber app. They repeated this task as many
+times as they liked. (ii) The participant blocked the network
+traffic to and from the camera using the built-in feature on the
+Privacy Plumber app. The participant observed the network
+traffic for 10 seconds, walked in front of the camera’s field
+of view, waited for another ten seconds, and unblocked the
+device using the Privacy Plumber functionality.
+C. Analysis of Pre-Study and Post-Study Surveys
+We asked each participant to complete two surveys: (i) a
+pre-Study Survey that they filled out on a dedicated computer
+at the beginning of the study, i.e., before the participants
+interacted with the Privacy Plumber app or the IoT devices;
+(ii) a post-Study Survey that the participants filled out on
+the dedicated computer after interacting with all the five IoT
+devices. We present the results below.
+In Figure 6 we present the participants’ agreement rating
+responses for two statements that were asked in the pre-study
+survey and post-study survey. We observe that for those two
+statements participants seemed less concerned by how their
+information is exposed to third parties when they interact with
+IoT devices before they performed the activities in the study.
+After participants completed the study, they were more aware
+and concerned about how their information was disclosed to
+third parties. The last two statements of Figure 6 were only
+given in the post-study survey, which asked participants to rate
+whether Privacy Plumber was useful in helping them become
+more aware of privacy concerns and how their information is
+being shared with third parties. On average, participants some-
+what agreed that Privacy Plumber helped raise their awareness
+and privacy concerns. Participants found that Privacy Plumber
+was helpful in that it helped them visualize what information
+was being shared.
+Additionally, we discuss the results of participants’ re-
+sponses with the IoT devices before and after the study. We
+show that after the study participants felt less safe with how
+IoT devices handle their data. Participants were presented with
+three statements and were asked to rate whether they agree or
+disagree with these statements on a scale of one to five, where
+a 1 meant they strongly agree and a 5 represents a strongly
+disagree rating. Table I demonstrates the average change in
+attitudes participants had before the study and after the study.
+We note that before the study, on average participants neither
+agreed nor disagreed with the statements presented in Table I.
+After completing the study, the average rating agreement score
+increased to “somewhat agree” on the last two statements on
+all IoT devices. The exception was in the first statement, the
+scores for the Amazon Echo and Google Home. This indicates
+10
+
+Survey Question
+Smart Fridge
+Amazon Echo
+Google Home
+Smart TV
+Nest Cam
+pre
+post
+pre
+post
+pre
+post
+pre
+post
+pre
+post
+I think this device could be (or is) useful or
+valuable to my daily life and routine.
+3
+3.17
+2.86
+2.5
+2.71
+2.5
+2.43
+2.33
+2.71
+3
+I am comfortable having this device in
+my house and always on.
+2.29
+3.5
+3.86
+4.17
+3.86
+4.17
+2.29
+3.5
+3.43
+4.17
+I am comfortable having this device in
+my house if I can automatically control
+when it is on, or off.
+1.29
+2.17
+2.29
+2.5
+2
+2.33
+1.14
+2
+2.29
+2.83
+Strongly Disagree (5) to Strongly Agree (1)
+TABLE I: Results of the survey on user awareness and comfort with smart devices, before and after using Privacy Plumber to
+inspect those devices. Scores are listed for both pre- and post-study surveys for each device. The higher the scores, the more
+strongly the participant disagreed with the survey question statement.
+that after using Privacy Plumber in the study, participants felt
+that the Amazon Echo and Google would still find it useful to
+use in their households.
+We also observe that the Smart Fridge, Smart TV, and
+the Nest cam had the most significant change in attitude. We
+gathered a few quotes from participants in which they describe
+how they felt about interacting with these IoT devices and
+using Privacy Plumber to inspect their network traffic:
+IoT devices provide more information to third par-
+ties than people thought. I think apps like Privacy
+Plumber can help people to make better decisions
+when using IoT devices — (P1)
+Cool to see when and how much traffic each device
+sends at any given moment! — (P5)
+I think the app does make me more aware about
+how the traffic is associated with the behavior of the
+device. Having some control over the traffic is nice.
+That being said, if I do have the device in my home,
+I probably would like to use it, and in that case, I
+have to allow traffic, which I have no control about
+what could pass or could not pass. In that sense, I
+can only accept certain privacy risks. — (P2)
+It was interesting to see the potential privacy leaks
+shown next to the device. Some leaks/ privacy im-
+plications were surprising. Liked the ability to al-
+low/block traffic, it was also cool to see the real-
+time traffic including communication with third-party
+advertisers. Liked the app interface. —(P6)
+These quotes, along with results from Figure 6 and Table I,
+suggest that Privacy Plumber helped participants understand
+the network traffic, increased their awareness of potential
+privacy violations, and helped them make more informed
+decisions on how to handle IoT devices.
+D. Analysis of the Usability Survey
+At the end of the study, each participant completed the
+usability survey. Overall, most participants indicated that they
+would use Privacy Plumber in their home network, found it
+easy to use and user-friendly, and agreed that most people
+would learn to use Privacy Plumber quickly. We summarize
+the results below:
+•
+When asked if they would use the Privacy Plumber
+mobile app to inspect the data the IoT devices in their
+homes, two participants said they strongly agreed with
+the statement and four participants said they somewhat
+agreed to use Privacy Plumber.
+•
+When participants were asked if they found Privacy
+Plumber easy to use, four of them somewhat agreed,
+one participant strongly agreed, and one participant
+somewhat disagreed.
+•
+When presented the statement “I imagine that most
+people would learn to use Privacy Plumber very
+quickly”, the responses were across the board spec-
+trum. Three participants rated that comment as
+strongly agreed, one participant rated the statement
+with a somewhat agree, one participant responded
+that they felt neither agreed or disagreed with the
+statement, and one participant somewhat disagreed.
+•
+When participants were asked to rate the overall user-
+friendliness’s of Privacy Plumber, four participants
+rated the Privacy Plumber app as good and two
+participants rated Privacy Plumber as fair.
+We gave participants an open-ended question if they would
+improve the usability of Privacy Plumber, and if so, how.
+We show their responses in Appendix B. All in all, partici-
+pants seemed to respond somewhat positively towards Privacy
+Plumber. It shows that Privacy Plumber may have the potential
+to be distributed to the general public after further studies.
+We hope to build off our current platform and implement the
+suggestions our participants gave us in future work.
+E. Performance: System Overhead and Battery Life Impact
+Network Overhead. IoT Network Analyzer intercepts the
+network traffic of select IoT devices via ARP spoofing, a
+technique that could introduce network overhead especially
+to the targeted IoT devices. This overhead comes from two
+sources. First, the spoofed ARP packets consume extra band-
+width, although the overhead is relatively small—i.e., less than
+60 Kilobytes/second even if 50 IoT devices are under ARP
+11
+
+spoofing [15]). The second source of overhead comes from
+the Raspberry Pi 3 Model B, where we run IoT Network
+Analyzer in the lab. The Raspberry Pi is connected to the
+lab’s network via Ethernet. For all IoT devices to which IoT
+Network Analyzer sends spoofed ARP packets, all inbound
+(i.e., download) and outbound (i.e., upload) traffic to and
+from the IoT devices has to first go through the Raspberry
+Pi before IoT Network Analyzer forwards the traffic to the
+targeted device and to the Internet respectively. Effectively,
+the Raspberry Pi introduces a bottleneck for the ARP-spoofed
+devices.
+To measure the overhead as a result of the Raspberry Pi
+bottleneck, we conduct the following experiment. We install
+the Ookla Speed Test app on an Android phone that is con-
+nected to the the lab’s WiFi network. We have the Ookla app
+run 15 back-to-back speed tests, which measure the inbound
+and outbound traffic rates with respect to a server in our city,
+as well as the latency of packets. Using the same setup, we
+repeat the same experiment, except that we have IoT Network
+Analyzer inspect the phone’s traffic via ARP spoofing.
+We find significant overhead as a result of IoT Network
+Analyzer. Without ARP spoofing, the app achieves, on average,
+an inbound rate of 293.6 ± 15.4 Mbps, an outbound rate of
+94.1±0.2 Mbps, and a latency of 5.7±0.5 milliseconds. With
+ARP spoofing by IoT Network Analyzer, the app achieves, on
+average, an inbound rate of 41.4±74.6 Mbps, an outbound rate
+of 72.8 ± 14.1 Mbps, and a latency of 5.9 ± 0.5 milliseconds.
+Compared with the case without ARP spoofing, IoT Network
+Analyzer reduces the inbound rate by 85.9% and outbound rate
+by 22.6%, while increasing the latency by 3.5%.
+Despite the seemingly significant reduction in bandwidth,
+we argue that IoT Network Analyzer is unlikely to degrade
+usability, as the network analyzer is not always running (only
+when inspecting, or blocking a specific device). Additionally,
+the overhead can be reduced with improved hardware. Ac-
+cording to Netflix, 25 Mbps of inbound rate is sufficient to
+stream Ultra HD contents [31]. A user who inspects a smart
+TV using IoT Network Analyzer is likely to enjoy Ultra HD
+streaming given the reduced inbound rate of 41.4±74.6 Mbps
+under ARP spoofing. If a user desires to reduce the network
+overhead, the user could upgrade the computer that runs IoT
+Network Analyzer, as Raspberry Pi 3 is anecdotally known for
+its poor networking performance [37], [38]. Possible upgrade
+option could include a computer—or ODroid if the user needs
+the compact form factor [14]—that is shipped with a fast CPU
+and a Gigabit Ethernet card.
+Battery Lifetime. We used AccuBattery on android, to try to
+understand the energy cost. This does not hold across phones,
+so we compare the energy cost against YouTube and TikTok
+for ten minutes of streaming video. With all the background
+application killed, 10 minutes of Privacy Plumber impacts
+3.98% (159mAh) of the battery lifetime, while YouTube costs
+2.63% (105mAh) and TikTok costs 3.9% (156mAh). Privacy
+Plumber is only meant for point inspection and short usage to
+analyze new devices in the home, or experiment with different
+setups, so it should not impact battery lifetime too much since
+it is not always on. Moreover, the battery lifetime cost is
+similar to that of streaming videos online, a normal function,
+therefore users should not expect significant battery lifetime
+loss due to usage of Privacy Plumber.
+V.
+DISCUSSION ON LIMITATIONS AND FUTURE WORK
+Comparing users’ mental models against actual contents
+of IoT network traffic. Our results show that users’ mental
+model of how IoT devices communicate with the Internet may
+be inconsistent with how devices appear to behave, but it is
+unclear whether this mental model is consistent with the actual
+contents of the communication. For example, two participants
+in our study did not expect network traffic from Amazon Echo
+when the device’s microphone was on mute. Presumably, the
+participants expected Amazon not to send any audio data back
+to Amazon during mute. In this case, Echo’s apparent behavior
+was the communication with the Internet on mute; in contrast,
+whether Echo actually sent out audio data was unknown. Our
+system did not extract the contents of the communication,
+which could be encrypted based on previous results [4].
+Despite the encrypted contents, man-in-the-middling is
+possible (e.g., per Moghaddam et al. [28]). In future in-lab
+studies, we plan to modify IoT Network Analyzer to intercept
+and decrypt IoT traffic, assuming that devices do not validate
+certificates and/or do not use certificate pinning. We hope to
+extract the payload from some of the TLS connections, identify
+exactly what devices are sending to the Internet, and compare
+it against users’ mental models.
+Automated, contextualized blocking of devices. The current
+prototype allows users to set a block/unblock schedule for IoT
+devices. Although this feature provides users with fine-grained
+control, it requires manual effort from the user both in setting
+what devices to block and when to block.
+We plan to augment this feature with automated device
+blocking based on contextualized information that IoT Net-
+work Analyzer already collects. For example, a user could
+create a rule on IoT Network Analyzer that would automat-
+ically block surveillance cameras if IoT Network Analyzer
+detects the presence of mobile phones (based on ARP and
+pings) in the home network (which could suggest that the
+residents are home); otherwise, it can unblock the cameras to
+capture, say, unauthorized entry into the property. As another
+example, let’s say a user has an Amazon Echo and a smart
+TV in the living room. The user could create another rule that
+lets IoT Network Analyzer automatically block Amazon Echo
+if it detects active streaming traffic from the smart TV, as the
+user may not want Echo to capture any conversations while
+the family is watching TV in the living room. In short, by
+leveraging the IoT traffic that IoT Network Analyzer already
+collects, users could create automated, contextualized rules to
+block IoT devices from collecting sensitive data.
+Deployment roadmap and challenges. We plan to deploy the
+Privacy Plumber app and IoT Network Analyzer to real-world
+users at scale. Based on our current prototype, we plan to make
+the following modifications.
+Operating system support. Once deployed, our system will
+have the same two-component architecture, although we will
+expand the Privacy Plumber app to both iOS and Android
+(current prototype), and IoT Network Analyzer to all major
+non-mobile operating systems including macOS, Windows,
+12
+
+and Linux (current prototype). This process will likely be
+straightforward, as we developed both components with cross-
+OS platforms (Unity for the app and pure Python for IoT
+Network Analyzer).
+Network-based device identification. We will develop
+network-based device identification mechanisms to help users
+distinguish among their devices and identify the device(s)
+of interest. The current prototype identifies devices based
+on a hard-coded mapping between MAC OUIs and device
+names, because we already know the inventory of IoT devices
+in the lab. For real-world deployment, we will incorporate
+IoT Inspector’s device identification algorithm [15], so that
+our system will dynamically infer device names based on
+the network signature, which includes not only OUIs, but
+also DNS queries, UPnP banners, mDNS names, and DHCP
+hostnames. We will also use information in the 802.11 frames
+to discover and locate devices [41].
+Image-based device identification. To complement the
+network-based approach, we will also develop image-based
+device identification mechanisms for the AR camera. Cur-
+rently, the Privacy Plumber app identifies devices based on
+printed QR codes on or near select IoT devices, such that
+the QR codes encode the MAC addresses and the names of
+devices. For real-world deployment, we will use computer
+vision to train a model of common IoT device types, such as
+voice assistants, smart TVs, and surveillance cameras (where
+security and privacy issues are commonly found in the litera-
+ture). This model will help the AR app recognize possible IoT
+devices (e.g., “likely a smart TV”). The app will then refine
+the recognition with the network-based device identification
+algorithm (e.g., “whether the device is indeed a smart TV based
+on the network signatures”) and manual user input if necessary.
+Both the network- and image-based approaches will hopefully
+help the app identify IoT devices in real-world settings.
+Expanded user study. The user study, as a pilot, has a small
+sample size and is limited to graduate students, who may
+be more inquisitive or technically-inclined than the general
+population. We hope to scale out the testing to a larger
+userbase, both in lab and in real homes, in future work. We will
+also compare the participants’ changes in privacy awareness
+against other visualization tools (e.g., IoT Inspector [15] and
+Aretha [40]). Finally, we will conduct in-depth studies on
+various ways to visualize privacy leaks in AR (e.g., icon
+overlays and animations).
+VI.
+SUMMARY
+This paper presented Privacy Plumber, an end-to-end sys-
+tem demonstrating how a general population of end users can
+potentially have insight into the network traffic of smart home
+IoT devices, and how these users can control when these smart
+devices could communicate with the Internet with one click of
+a button. Designed after the concept of a leak detector, Privacy
+Plumber is a phone app with a tethered desktop application—
+IoT Network Analyzer—that provides an inspect and correct
+interface supported by network traffic analysis (inspect) and
+automated and timed network traffic jamming (correct).
+Privacy Plumber is the first real-world inspection and
+control system that can be deployed in any home without new
+hardware or router modifications. Using AR, the tool aims to
+help users model IoT device activities within the context of
+the physical environment and of user interactions (addressing
+challenges C1 and C3, per Section II-D); it gives users the
+option to block IoT devices and control the privacy “valve”
+(C2); it provides users with an interface to visualize IoT device
+activities as users interact with devices (C4); and it requires
+a modern AR-supported phone and computer, without any
+dedicated or specialized hardware (C5).
+We evaluated Privacy Plumber inside an instrumented smart
+home space with a variety of devices not previously evaluated
+for any privacy-enhancing tool, including a smart fridge, a
+smart TV, voice assistants, and Internet-connected surveillance
+cameras. We found that using Privacy Plumber improved users’
+awareness of potential privacy violations of devices and that
+the system was generally easy to use and afforded useful
+controls. In the future, we hope tools like Privacy Plumber will
+give mechanisms back to the user for stymieing the flow of
+private information outside the home, especially as our homes
+and living spaces become smarter, often without our consent.
+ACKNOWLEDGMENT
+This research is based upon work supported by the National
+Science Foundation under award numbers CNS-2219867,
+CNS-1739809, and CNS-1915847. Any opinions, findings, and
+conclusions or recommendations expressed in this material are
+those of the authors and do not necessarily reflect the views of
+the National Science Foundation. The research is also based
+on work supported by gifts from Consumer Reports and Meta.
+REFERENCES
+[1]
+Abbas Acar, Hossein Fereidooni, Tigist Abera, Amit Kumar Sikder,
+Markus Miettinen, Hidayet Aksu, Mauro Conti, Ahmad-Reza Sadeghi,
+and Selcuk Uluagac. Peek-a-boo: I see your smart home activities, even
+encrypted! In Proceedings of the 13th ACM Conference on Security and
+Privacy in Wireless and Mobile Networks, pages 207–218, 2020.
+[2]
+Imtiaz Ahmad, Rosta Farzan, Apu Kapadia, and Adam J Lee. Tangible
+privacy: Towards user-centric sensor designs for bystander privacy. Pro-
+ceedings of the ACM on Human-Computer Interaction, 4(CSCW2):1–
+28, 2020.
+[3]
+Omar Alrawi, Chaz Lever, Manos Antonakakis, and Fabian Monrose.
+Sok: Security evaluation of home-based iot deployments. In 2019 IEEE
+symposium on security and privacy (sp), pages 1362–1380. IEEE, 2019.
+[4]
+Noah Apthorpe, Danny Yuxing Huang, Dillon Reisman, Arvind
+Narayanan, and Nick Feamster. Keeping the smart home private with
+smart (er) iot traffic shaping.
+Proceedings on Privacy Enhancing
+Technologies, 2019(3):128–148, 2019.
+[5]
+Noah Apthorpe, Dillon Reisman, and Nick Feamster. A smart home is
+no castle: Privacy vulnerabilities of encrypted iot traffic. arXiv preprint
+arXiv:1705.06805, 2017.
+[6]
+Noah
+Apthorpe,
+Dillon
+Reisman,
+Srikanth
+Sundaresan,
+Arvind
+Narayanan, and Nick Feamster.
+Spying on the smart home: Pri-
+vacy attacks and defenses on encrypted iot traffic.
+arXiv preprint
+arXiv:1708.05044, 2017.
+[7]
+Bruhadeshwar Bezawada, Maalvika Bachani, Jordan Peterson, Hossein
+Shirazi, Indrakshi Ray, and Indrajit Ray. Iotsense: Behavioral finger-
+printing of iot devices. arXiv preprint arXiv:1804.03852, 2018.
+[8]
+Patrick Bombik, Tom Wenzel, Jens Grossklags, and Sameer Patil. A
+multi-region investigation of the perceptions and use of smart home
+devices.
+Proceedings on Privacy Enhancing Technologies, 3:6–32,
+2022.
+13
+
+[9]
+Nico Castelli, Corinna Ogonowski, Timo Jakobi, Martin Stein, Gunnar
+Stevens, and Volker Wulf. What happened in my home? an end-user de-
+velopment approach for smart home data visualization. In Proceedings
+of the 2017 CHI Conference on Human Factors in Computing Systems,
+pages 853–866, 2017.
+[10]
+Gordon Chu, Noah Apthorpe, and Nick Feamster. Security and privacy
+analyses of internet of things children’s toys. IEEE Internet of Things
+Journal, 6(1):978–985, 2018.
+[11]
+Sunny Consolvo, Jaeyeon Jung, Ben Greenstein, Pauline Powledge,
+Gabriel Maganis, and Daniel Avrahami.
+The wi-fi privacy ticker:
+improving awareness & control of personal information exposure on
+wi-fi.
+In Proceedings of the 12th ACM international conference on
+Ubiquitous computing, pages 321–330, 2010.
+[12]
+Disconnect, Inc. Disconnect tracking protection, 2021.
+[13]
+Yuanyuan Feng, Yaxing Yao, and Norman Sadeh. A design space for
+privacy choices: Towards meaningful privacy control in the internet of
+things. In Proceedings of the 2021 CHI Conference on Human Factors
+in Computing Systems, pages 1–16, 2021.
+[14]
+HardKernel. ODROID-XU4, 2021.
+[15]
+Danny Yuxing Huang, Noah Apthorpe, Frank Li, Gunes Acar, and Nick
+Feamster. Iot inspector: Crowdsourcing labeled network traffic from
+smart home devices at scale. Proceedings of the ACM on Interactive,
+Mobile, Wearable and Ubiquitous Technologies, 4(2):1–21, 2020.
+[16]
+Haojian Jin, Boyuan Guo, Rituparna Roychoudhury, Yaxing Yao,
+Swarun Kumar, Yuvraj Agarwal, and Jason I Hong. Exploring the needs
+of users for supporting privacy-protective behaviors in smart homes. In
+CHI Conference on Human Factors in Computing Systems, pages 1–19,
+2022.
+[17]
+Patrick Gage Kelley, Joanna Bresee, Lorrie Faith Cranor, and Robert W
+Reeder.
+A” nutrition label” for privacy.
+In Proceedings of the 5th
+Symposium on Usable Privacy and Security, pages 1–12, 2009.
+[18]
+Christian Kreibich, Nicholas Weaver, Boris Nechaev, and Vern Paxson.
+Netalyzr: Illuminating the edge network. In Proceedings of the 10th
+ACM SIGCOMM conference on Internet measurement, pages 246–259,
+2010.
+[19]
+Hosub Lee and Alfred Kobsa. Understanding user privacy in internet
+of things environments. In 2016 IEEE 3rd World Forum on Internet of
+Things (WF-IoT), pages 407–412. IEEE, 2016.
+[20]
+Huichen Lin and Neil W Bergmann. Iot privacy and security challenges
+for smart home environments. Information, 7(3):44, 2016.
+[21]
+Heather Richter Lipford, Madiha Tabassum, Paritosh Bahirat, Yaxing
+Yao, and Bart P Knijnenburg. Privacy and the internet of things. Modern
+Socio-Technical Perspectives on Privacy, page 233, 2022.
+[22]
+Nathan Malkin, Julia Bernd, Maritza Johnson, and Serge Egelman.
+“what can’t data be used for?” privacy expectations about smart tvs
+in the us. In Proceedings of the 3rd European Workshop on Usable
+Security (EuroUSEC), London, UK, 2018.
+[23]
+Nathan Malkin, Joe Deatrick, Allen Tong, Primal Wijesekera, Serge
+Egelman, and David Wagner. Privacy attitudes of smart speaker users.
+Proceedings on Privacy Enhancing Technologies, 2019(4), 2019.
+[24]
+Shrirang Mare, Franziska Roesner, and Tadayoshi Kohno. Smart devices
+in airbnbs: Considering privacy and security for both guests and hosts.
+Proc. Priv. Enhancing Technol., 2020(2):436–458, 2020.
+[25]
+Emily McReynolds, Sarah Hubbard, Timothy Lau, Aditya Saraf, Maya
+Cakmak, and Franziska Roesner. Toys that listen: A study of parents,
+children, and internet-connected toys. In Proceedings of the 2017 CHI
+conference on human factors in computing systems, pages 5197–5207,
+2017.
+[26]
+Markus Miettinen, Samuel Marchal, Ibbad Hafeez, N Asokan, Ahmad-
+Reza Sadeghi, and Sasu Tarkoma.
+Iot sentinel: Automated device-
+type identification for security enforcement in iot. In 2017 IEEE 37th
+International Conference on Distributed Computing Systems (ICDCS),
+pages 2177–2184. IEEE, 2017.
+[27]
+Phoebe Moh, Noel Warford, Pubali Datta, Nathan Malkin, Adam Bates,
+and Michelle L Mazurek. Characterizing misuse and snooping in home
+iot devices.
+[28]
+Hooman Mohajeri Moghaddam, Gunes Acar, Ben Burgess, Arunesh
+Mathur, Danny Yuxing Huang, Nick Feamster, Edward W Felten, Pra-
+teek Mittal, and Arvind Narayanan. Watching you watch: The tracking
+ecosystem of over-the-top tv streaming devices.
+In Proceedings of
+the 2019 ACM SIGSAC Conference on Computer and Communications
+Security, pages 131–147, 2019.
+[29]
+Mozilla. Privacy Not Included., 2021.
+[30]
+Pardis Emami Naeini, Sruti Bhagavatula, Hana Habib, Martin Degeling,
+Lujo Bauer, Lorrie Faith Cranor, and Norman Sadeh. Privacy expecta-
+tions and preferences in an {IoT} world. In Thirteenth Symposium on
+Usable Privacy and Security (SOUPS 2017), pages 399–412, 2017.
+[31]
+Netflix. Internet Connection Speed Recommendations, 2021.
+[32]
+Helen Nissenbaum.
+Privacy in context: Technology, policy, and the
+integrity of social life. Stanford University Press, 2009.
+[33]
+TJ OConnor, Reham Mohamed, Markus Miettinen, William Enck,
+Bradley Reaves, and Ahmad-Reza Sadeghi.
+Homesnitch: behavior
+transparency and control for smart home iot devices. In Proceedings of
+the 12th Conference on Security and Privacy in Wireless and Mobile
+Networks, pages 128–138, 2019.
+[34]
+pfSense.org. World’s MOST Trusted Open Source Firewall, 2021.
+[35]
+James Pierce, Richmond Y Wong, and Nick Merrill. Sensor illumi-
+nation: Exploring design qualities and ethical implications of smart
+cameras and image/video analytics. In Proceedings of the 2020 CHI
+Conference on Human Factors in Computing Systems, pages 1–19,
+2020.
+[36]
+Sarah Prange, Ahmed Shams, Robin Piening, Yomna Abdelrahman, and
+Florian Alt. Priview– exploring visualisations to support users’ privacy
+awareness.
+In Proceedings of the 2021 CHI Conference on Human
+Factors in Computing Systems, CHI ’21, New York, NY, USA, 2021.
+Association for Computing Machinery.
+[37]
+Raspberry Pi Discussion Forum. RPi 3B+ gigabit ethernet bad download
+speeds, 2018.
+[38]
+Raspberry Pi Dramble. Networking Benchmarks, 2021.
+[39]
+Abbas Razaghpanah, Rishab Nithyanand, Narseo Vallina-Rodriguez,
+Srikanth Sundaresan, Mark Allman, Christian Kreibich, and Phillipa
+Gill.
+Apps, trackers, privacy, and regulators: A global study of the
+mobile tracking ecosystem. 2018.
+[40]
+William Seymour, Martin J Kraemer, Reuben Binns, and Max
+Van Kleek. Informing the design of privacy-empowering tools for the
+connected home. In Proceedings of the 2020 CHI Conference on Human
+Factors in Computing Systems, pages 1–14, 2020.
+[41]
+Rahul Anand Sharma, Elahe Soltanaghaei, Anthony Rowe, and Vyas
+Sekar. Lumos: Identifying and localizing diverse hidden IoT devices
+in an unfamiliar environment.
+In 31st USENIX Security Symposium
+(USENIX Security 22), pages 1095–1112, Boston, MA, August 2022.
+USENIX Association.
+[42]
+Yun Shen and Pierre-Antoine Vervier. Iot security and privacy labels.
+In Annual Privacy Forum, pages 136–147. Springer, 2019.
+[43]
+Madiha Tabassum, Tomasz Kosinski, and Heather Richter Lipford. ” i
+don’t own the data”: End user perceptions of smart home device data
+practices and risks. In Fifteenth Symposium on Usable Privacy and
+Security ({SOUPS} 2019), 2019.
+[44]
+Parth Kirankumar Thakkar, Shijing He, Shiyu Xu, Danny Yuxing
+Huang, and Yaxing Yao. “it would probably turn into a social faux-pas”:
+Users’ and bystanders’ preferences of privacy awareness mechanisms
+in smart homes. In CHI Conference on Human Factors in Computing
+Systems, pages 1–13, 2022.
+[45]
+Blase Ur, Pedro Giovanni Leon, Lorrie Faith Cranor, Richard Shay,
+and Yang Wang. Smart, useful, scary, creepy: perceptions of online
+behavioral advertising.
+In proceedings of the eighth symposium on
+usable privacy and security, pages 1–15, 2012.
+[46]
+Max Van Kleek, Reuben Binns, Jun Zhao, Adam Slack, Sauyon Lee,
+Dean Ottewell, and Nigel Shadbolt.
+X-ray refine: Supporting the
+exploration and refinement of information exposure resulting from
+smartphone apps.
+In Proceedings of the 2018 CHI Conference on
+Human Factors in Computing Systems, pages 1–13, 2018.
+[47]
+Max Van Kleek, Ilaria Liccardi, Reuben Binns, Jun Zhao, Daniel J
+Weitzner, and Nigel Shadbolt. Better the devil you know: Exposing
+the data sharing practices of smartphone apps. In Proceedings of the
+2017 CHI Conference on Human Factors in Computing Systems, pages
+5208–5220, 2017.
+[48]
+Sean Whalen.
+An introduction to arp spoofing.
+Node99 [Online
+Document], 2001.
+14
+
+[49]
+Peter Worthy, Ben Matthews, and Stephen Viller. Trust me: doubts and
+concerns living with the internet of things. In Proceedings of the 2016
+ACM Conference on Designing Interactive Systems, pages 427–434,
+2016.
+[50]
+Yaxing Yao, Justin Reed Basdeo, Smirity Kaushik, and Yang Wang.
+Defending my castle: A co-design study of privacy mechanisms for
+smart homes. In Proceedings of the 2019 CHI conference on human
+factors in computing systems, pages 1–12, 2019.
+[51]
+Eric Zeng, Shrirang Mare, and Franziska Roesner. End user security
+and privacy concerns with smart homes. In Thirteenth Symposium on
+Usable Privacy and Security (SOUPS 2017), pages 65–80, 2017.
+[52]
+Wei Zhang, Yan Meng, Yugeng Liu, Xiaokuan Zhang, Yinqian Zhang,
+and Haojin Zhu. Homonit: Monitoring smart home apps from encrypted
+traffic.
+In Proceedings of the 2018 ACM SIGSAC Conference on
+Computer and Communications Security, pages 1074–1088, 2018.
+[53]
+Serena Zheng, Noah Apthorpe, Marshini Chetty, and Nick Feamster.
+User perceptions of smart home iot privacy. Proceedings of the ACM
+on human-computer interaction, 2(CSCW):1–20, 2018.
+APPENDIX
+SURVEY QUESTIONS
+All questions require responses in Likert scales, ranging
+from “Strongly Agree” (1) to “Strongly Disagree” (5).
+A. Pre-Study Survey Questions
+1)
+When I am in a smart home, I think about what in-
+formation I may be exposing to vendors, companies,
+and 3rd parties when I interact with or sit in the same
+space with smart devices in the home.
+2)
+I am not concerned about the information I may be
+exposing to 3rd parties when I interact with or sit in
+the same space as smart devices in a smart home.
+3)
+I think this device could be (or is) useful or valuable
+to my daily life and routine.
+•
+Smart Fridge
+•
+Google Home
+•
+Amazon Echo
+•
+Smart TV
+•
+Nest Cam
+4)
+I am comfortable having this device in my house and
+always on.
+•
+Smart Fridge
+•
+Google Home
+•
+Amazon Echo
+•
+Smart TV
+•
+Nest Cam
+B. Post-Study Survey Questions
+1)
+When I am in a smart home, I think about what in-
+formation I may be exposing to vendors, companies,
+and 3rd parties when I interact with or sit in the same
+space with smart devices in the home.
+2)
+I am not concerned about the information I may be
+exposing to 3rd parties when I interact with or sit in
+the same space as smart devices in a smart home.
+3)
+Privacy Plumber has made me more aware of what
+information I may be exposing to 3rd parties when I
+interact with smart devices in the home.
+4)
+I feel Privacy Plumber has made me more aware
+of privacy and security concerns surrounding IoT
+devices.
+5)
+I think this device could be (or is) useful or valuable
+to my daily life and routine.
+•
+Smart Fridge
+•
+Google Home
+•
+Amazon Echo
+•
+Smart TV
+•
+Nest Cam
+6)
+I am comfortable having this device in my house and
+always on.
+•
+Smart Fridge
+•
+Google Home
+•
+Amazon Echo
+•
+Smart TV
+•
+Nest Cam
+7)
+Finally, please provide any other thoughts or obser-
+vations from participating in this experiment with
+Privacy Plumber (open ended).
+ADDITIONAL RESPONSES FROM THE USABILITY SURVEY
+We gave participants an open-ended question if they would
+improve the usability if privacy plumber, if so how. We
+obtained the following responses from each participant.
+I would include more guidance or instructions in the app
+for first-time users. (P1)
+I think the app is generally easy-to-use, although I might
+want more functionalities in the app. There are certain laten-
+cies in the app, which can be annoying. It would be more
+helpful if I can know if the device is not sending any traffic,
+or it is just simply late (e.g., adding a loading icon). (P2)
+Make it possible to view past trends (a la net microscope)
+and scroll backwards in time, so I can get the context of how
+much traffic is regularly sent. Give me a global view of the
+worst offenders. Still some work to do on basic stability. It
+only works on devices that people have obviously ALREADY
+DECIDED TO BUY, which is a weird sample. Obviously, I
+don’t have QR codes printed out on all of my household
+electronics. (P3)
+I had difficulties trying to access the buttons, and the
+images seemed lagged a little. But the info was very useful
+overall. (P4)
+Fix where the traffic and ‘learn more about the device’
+buttons once you’ve scanned the QR code. It’s a bit awkward
+to have to hold the phone back up to the device. Maybe add
+the units (byte/kB) to the left hand side of the graph instead
+of above it for the traffic visualization. (P5)
+The plots are not super-intuitive but I liked the representa-
+tions in terms of text/pictures which is easier to comprehend.
+I would also be interested to see what advertisers the infor-
+mation is being leaked to. While the AR thing is cool, I would
+also like the option to just scroll through a list of devices.
+That ways I do not have to be close to the device and would
+also be able to monitor its activity when I am not close to
+the device. In fact, I would be interested in seeing the device
+communication (including interaction w/ advertisers) in that
+case. (P6)
+15
+

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+A Survey on Protein Representation Learning: Retrospect and Prospect
+Lirong Wu 1,2 ∗ , Yufei Huang 1,2 ∗ , Haitao Lin 1,2 , Stan Z. Li 1†
+1 AI Lab, Research Center for Industries of the Future, Westlake University
+2 College of Computer Science and Technology, Zhejiang University
+{wulirong,huangyufei,linhaitao,stan.zq.li}@westlake.edu.cn
+Abstract
+Proteins are fundamental biological entities that
+play a key role in life activities. The amino acid
+sequences of proteins can be folded into stable 3D
+structures in the real physicochemical world, form-
+ing a special kind of sequence-structure data. With
+the development of Artificial Intelligence (AI) tech-
+niques, Protein Representation Learning (PRL) has
+recently emerged as a promising research topic
+for extracting informative knowledge from mas-
+sive protein sequences or structures. To pave the
+way for AI researchers with little bioinformatics
+background, we present a timely and comprehen-
+sive review of PRL formulations and existing PRL
+methods from the perspective of model architec-
+tures, pretext tasks, and downstream applications.
+We first briefly introduce the motivations for pro-
+tein representation learning and formulate it in a
+general and unified framework. Next, we divide
+existing PRL methods into three main categories:
+sequence-based, structure-based, and sequence-
+structure co-modeling. Finally, we discuss some
+technical challenges and potential directions for
+improving protein representation learning. The lat-
+est advances in PRL methods are summarized in
+a GitHub repository https://github.com/LirongWu/
+awesome-protein-representation-learning.
+1
+Introduction
+Proteins perform specific biological functions that are essen-
+tial for all living organisms and therefore play a key role when
+investigating the most fundamental questions in the life sci-
+ences. The proteins are composed of one or several chains of
+amino acids that fold into a stable 3D structure to enable vari-
+ous biological functionalities. Therefore, understanding, pre-
+dicting, and designing proteins for biological processes are
+critical for medical, pharmaceutical, and genetic research.
+Previous approaches on protein modeling are mostly driven
+by biological or physical priors, and they explore com-
+plex sequence-structure-function relationships through en-
+ergy minimization [Rohl et al., 2004; Xu and Zhang, 2011],
+∗Equal contribution, † Corresponding author
+dynamics simulations [Hospital et al., 2015; Karplus and
+Petsko, 1990], etc.
+With the development of artificial in-
+telligence and low-cost sequencing technologies, data-driven
+Protein Representation Learning (PRL) [Jumper et al., 2021;
+Rao et al., 2019; Rives et al., 2021; Hermosilla and Ropinski,
+2022; Jing et al., 2020] has made remarkable progress due to
+its superior performance in modeling complex nonlinear rela-
+tionships. The primary goal of protein representation learning
+is to extract transferable knowledge from protein data with
+well-designed model architectures and pretext tasks, and then
+generalize the learned knowledge to various protein-related
+downstream applications, ranging from structure prediction
+to sequence design. Despite their great progress, it is still
+tricky for AI researchers without bioinformatics background
+to get started with protein representation learning, and one
+obstacle is the vast amount of physicochemical knowledge in-
+volved behind the proteins. Therefore, a survey on PRL meth-
+ods that is friendly to the AI community is urgently needed.
+Existing surveys related to PRL [Iuchi et al., 2021; Unsal
+et al., 2020; Hu et al., 2021; Torrisi et al., 2020] are mainly
+developed from the perspective of biological applications, but
+do not go deeper into other important aspects, such as model
+architectures and pretext tasks. Overall, our contributions can
+be summarized as follows: (1) Comprehensive review. Our
+survey provides a comprehensive and up-to-date review of
+existing PRL methods from the perspective of the model ar-
+chitectures and pretext tasks. (2) New taxonomy. We divide
+existing PRL methods into three categories: sequence-based,
+structure-based, and sequence-structure co-modeling. (3) De-
+tailed Implementations. We summarize the paper lists and
+open-source codes in a public GitHub repository, setting the
+stage for the development of more future works. (4) Future
+directions. We point out the technical limitations of current
+research and discuss several promising directions.
+2
+Notation and Problem Statement
+The sequence of amino acids can be folded into a stable 3D
+structure, forming a special kind of sequence-structure data,
+which determines its properties and functions. Therefore, we
+can model each protein as a graph G = (V, E, X, F), where V
+is the ordered set of N nodes in the graph representing amino
+acid residues and E ∈ V × V is the set of edges that connects
+the nodes. Each node u ∈ V in graph G can be attributed with
+a scalar-vector tuple xu = (su, Vu), where su ∈ RO and
+arXiv:2301.00813v1  [cs.LG]  31 Dec 2022
+
+Vu ∈ R3×P . Each edge e ∈ E can be attributed with a scalar-
+vector tuple fe = (se, Ve), where se ∈ RT and Ve ∈ R3×D.
+Given a model architecture fθ(·) and a set of K losses
+of pretext tasks {L(1)
+pre(θ, η1), L(2)
+pre(θ, η2), · · · , L(K)
+pre (θ, ηK)}
+with projection heads {gηk(·)}K
+k=1, Protein Representation
+Learning (PRL) usually works in a two-stage manner: (1)
+Pre-training the model fθ(·) with pretext tasks; and (2) Fine-
+tuning the pre-trained model fθinit(·) with a projection head
+gω(·) under the supervision of a specific downstream task
+Ltask(θ, ω). The learning objective can be formulated as
+θ∗, ω∗ = arg min
+(θ,ω) Ltask(θinit, ω),
+s.t. θinit, {η∗
+k}K
+k=1 = arg min
+θ,{ηk}K
+k=1
+K
+�
+k=1
+λkL(k)
+pre(θ, ηk)
+(1)
+where {λk}K
+k=1 are trade-off task hyperparameters. A high-
+level overview of the PRL framework is shown in Fig. 1.
+In practice, if we set K = 1, ω = η1, i.e., L(1)
+pre(θ, η1) =
+Ltask(θ, ω), it is equivalent to learning task-specific repre-
+sentations directly under downstream supervision, which in
+this survey can be considered as a special case of Eq. (1).
+Pretext Tasks
+Prediction 
+Head        
+Prediction 
+Head        
+Encoder 
+Downstream Task
+Encoder 
+Step 2 
+Fine-tune 
+Step 1 
+Pre-train 
+Figure 1: A general framework for protein representation learning.
+In this survey, we mainly focus on the model architecture
+fθ(·) and pretext tasks {L(k)
+pre(θ, ηk)}K
+k=1 for protein repre-
+sentation learning, and defer the discussion on downstream
+applications until Sec. 5. A high-level overview of this sur-
+vey with some representative examples is shown in Fig. 2.
+3
+Model Architectures
+In this section, we summarize some commonly used model
+architectures for learning protein sequences or structures.
+3.1
+Sequence-based Encoder
+The sequence encoder takes as input (V, X) and then aims to
+capture the dependencies between amino acids. [Wang et al.,
+2019] treats protein sequences as a special “biological lan-
+guage” and then establishes an analogy between such “bio-
+logical language” and natural (textual) language. Inspired by
+this, many classical model architectures developed for natural
+language processing can be directly extended to handle pro-
+tein sequences [Asgari et al., 2019]. Depending on whether
+a single sequence or multiple sequences are to be encoded,
+there are a variety of different sequence-based encoders.
+Single Sequences
+The commonly used sequence encoders for modeling single
+sequences include Variational Auto-Encoder (VAE) [Sinai et
+al., 2017; Ding et al., 2019], Recurrent Neural Networks
+(RNNs) [Armenteros et al., 2020], Long Short-Term Memory
+(LSTM) [Hochreiter and Schmidhuber, 1997], BERT [Devlin
+et al., 2018], Transformer [Vaswani et al., 2017]. Based on
+the vanilla Transformer, [Wu et al., 2022] proposes a novel
+geometry-inspired transformer (Geoformer) to further distill
+the structural and physical pairwise relationships between
+amino acids into the learned protein representation. If we
+do not consider the ordering of amino acids in the sequences,
+we can also directly apply Convolutional Neural Networks
+(CNNs) [LeCun et al., 1995] or ResNet [He et al., 2016] to
+capture the local dependencies between adjacent amino acids.
+MSA Sequences
+The long-standing practices in computational biology are to
+make inferences from a family of evolutionarily related se-
+quences [Weigt et al., 2009; Thomas et al., 2005; Lapedes
+et al., 1999].
+Therefore, there have been several multi-
+ple sequences encoders proposed to capture co-evolutionary
+information by taking as input a set of sequences in the
+form of multiple sequence alignment (MSA). For exam-
+ple, MSA Transformer [Rao et al., 2021] extends the self-
+attention mechanism to the MSA setting, which interleaves
+self-attention across rows and columns to capture dependen-
+cies between amino acids and between sequences. As a cru-
+cial component of AlphaFold2, Evoformer [Jumper et al.,
+2021] alternatively updates MSA and Pair representations in
+each block, which encode co-evolutionary information in se-
+quences and relations between residues, respectively.
+3.2
+Structure-based Encoder
+Despite the effectiveness of sequence-based encoders, the
+power of pre-training with protein structures has been rarely
+explored, even though protein structures are known to be de-
+terminants of protein functions. To better utilize this critical
+structural information, a large number of structure-based en-
+coders have been proposed to model structural information,
+which can be mainly divided into three categories: feature
+map-based, message-passing GNNs, and geometric GNNs.
+Feature map-based Methods
+The use of deep learning to model protein 3D structures could
+be traced back to a decade ago [Zhang and Zhang, 2010;
+Schaap et al., 2001]. Early methods directly extracted sev-
+eral hand-crafted feature maps from protein structures and
+then applied 3D CNNs to model the geometric information
+of proteins [Derevyanko et al., 2018; Amidi et al., 2018;
+Townshend et al., 2019]. Later work extended 3D CNNs to
+spherical convolution for identifying interaction patterns on
+protein surfaces [Sverrisson et al., 2021; Gainza et al., 2020].
+Message-passing GNNs
+To further capture the geometric relationships and biomedi-
+cal interactions between amino acids, it has been proposed
+to first construct a graph from the extracted feature maps by
+thresholding or k Nearest Neighbors (kNN) [Preparata and
+Shamos, 2012]. Then, many existing message-passing Graph
+Neural Networks (GNNs) can be directly applied to model
+protein structures, including Graph Convolutional Network
+(GCN) [Kipf and Welling, 2016], Graph Isomorphism Net-
+work (GIN) [Xu et al., 2018], and GraphSAGE [Hamilton
+
+PRL
+Preliminaries
+Notation and Problem Statement
+Architectures
+Sequence-based
+Single Sequence
+LSTM [Hochreiter and Schmidhuber, 1997], Transformer [Vaswani et al., 2017], CNNs [LeCun et al., 1995]
+MSA Sequence
+MSA Transformer [Rao et al., 2021], Evoformer [Jumper et al., 2021]
+Structure-based
+Feature map-based
+3D CNNs [Derevyanko et al., 2018], Spherical CNNs [Sverrisson et al., 2021]
+Message-passing GNNs
+GCNs [Kipf and Welling, 2016], IEConv [Hermosilla et al., 2020], GearNet [Zhang et al., 2022]
+Geometric GNNs
+GVP [Jing et al., 2020], GBP [Aykent and Xia, 2022], DWP [Li et al., 2022]
+Sequence-structure Co-modeling
+DeepFRI [Gligorijevi´c et al., 2021], LM-GVP [Wang et al., 2021]
+Pretext Tasks
+Sequence-based
+Supervised
+PLUS [Min et al., 2021], Profile Prediction [Sturmfels et al., 2020], Progen [Madani et al., 2020]
+Self-Supervised
+MLM [Rao et al., 2019], PMLM [He et al., 2021], NAP [Alley et al., 2019], CPC [Lu et al., 2020]
+Structure-based
+Contrative
+Multiview Contrast [Hermosilla and Ropinski, 2022; Zhang et al., 2022]
+Predictive
+Distance and Angle Prediction [Chen et al., 2022], Dihedral Prediction [Hermosilla and Ropinski, 2022]
+Sequence-structure Co-modeling
+Full-atomic Structure Prediction [Jumper et al., 2021; Hu et al., 2022]
+Applications
+Property Prediction
+Stability [Rao et al., 2019], Fold Quality [Baldassarre et al., 2021], Mutation Effect [Meier et al., 2021], PPI [Wang et al., 2019]
+Structure Prediction
+Full-atomic or Backbone Prediction [Hiranuma et al., 2021; Wu et al., 2022], Structure Inpainting [McPartlon and Xu, 2022]
+Protein Design
+Template-based [Ingraham et al., 2019], De Novo [Huang et al., 2016; Koepnick et al., 2019]
+Structure-Based Drug Design
+Auto-regressive [Liu et al., 2022a; Peng et al., 2022], Diffusion [Lin et al., 2022; Schneuing et al., 2022]
+Figure 2: A high-level overview of this survey with representative examples.
+et al., 2017]. However, the edges in the protein graph may
+have some key properties, such as dihedral angles and direc-
+tions, which determine the biological function of proteins.
+With this in mind, there have been several structure-based
+encoders proposed to simultaneously leverages the node and
+edge features of the protein graph. For example, [Hermosilla
+et al., 2020] proposes IE convolution (IEconv) to simultane-
+ously capture the primary, secondary and tertiary structures
+of proteins by incorporating intrinsic and extrinsic distances
+between nodes. Besides, [Hermosilla and Ropinski, 2022]
+adopts a similar architecture to IEConv, but introduces seven
+additional edge features to efficiently describe the relative po-
+sition and orientation of neighboring nodes.
+Furthermore,
+GearNet [Zhang et al., 2022] proposes a simple structure en-
+coder, which encodes spatial information by adding different
+types of sequential or structural edges and then performs both
+node-level and edge-level message passing simultaneously.
+Geometric GNNs
+The above message-passing GNNs incorporate the 3D geom-
+etry of proteins by encoding the vector features Vu/Ve into
+rotation-invariant scalars su/se. However, reducing this vec-
+tor information directly to scalars may not fully capture com-
+plex geometry. Therefore, geometric-aware neural networks
+are proposed to bake 3D rigid transformations into network
+operations, leading to SO(3)-invariant and equivariant GNNs.
+For example, [Jing et al., 2020] introduces Geometric Vector
+Perceptrons (GVPs), which replace standard multi-layer per-
+ceptrons (MLPs) in feed-forward layers and operate directly
+on both scalar and vector features under a global coordinate
+system. Besides, [Aykent and Xia, 2022] proposes Geometric
+Bottleneck Perceptron (GBPs) to integrate geometric features
+and capture complex geometric relations in the 3D structure,
+based on which a new SO(3)-equivariant message passing
+neural network is proposed to support a variety of geomet-
+ric representation learning tasks. To achieve more sensitive
+geometric awareness in both global transformations and local
+relations, [Li et al., 2022] proposes Directed Weight Percep-
+trons (DWPs) by extending not only the hidden neurons but
+the weights from scalars to 2D/3D vectors, naturally saturat-
+ing the network with 3D structures in the Euclidean space.
+3.3
+Sequence-structure Encoder
+Compared to sequence- and structure-based encoders, com-
+paratively less work has focused on the co-encoding of pro-
+tein sequences and structures. The mainstream model archi-
+tecture is to extract amino acid representations as node fea-
+tures by a language model and then capture the dependencies
+between amino acids using a GNN module. For example,
+[Gligorijevi´c et al., 2021] introduces DeepFRI, a Graph Con-
+volutional Network (GCN) for predicting protein functions
+by leveraging sequence representations extracted from a pro-
+tein language model (LSTM) and protein structures. Besides,
+LM-GVP [Wang et al., 2021] is composed of a protein lan-
+guage model (composed of Transformer blocks) and a GVP
+network, where the protein LM takes protein sequences as
+input to compute amino acid embeddings and the GVP net-
+work is used to make predictions about protein properties on a
+graph derived from the protein 3D structure. Moreover, [You
+
+and Shen, 2022] applies the hierarchical RNN and GAT to
+encode both protein sequences and structures and proposes a
+cross-interaction module to enforce a learned relationship be-
+tween the encoded embeddings of the two protein modalities.
+4
+Pretext Task
+The pretext tasks are designed to extract meaningful repre-
+sentations from massive data through optimizing some well-
+designed objective functions. In this section, we summarize
+some commonly used pretext tasks for learning on proteins.
+4.1
+Sequence-based Pretext Task
+There have been many pretext tasks proposed for pre-training
+language models, including Masked Language Modeling
+(MLM) and Next Sentence Prediction (NSP) [Devlin et al.,
+2018], which can be naturally extended to pre-train protein
+sequences. We divide existing sequence-based pretext tasks
+into two main categories: self-supervised and supervised.
+Self-supervised Pretext Task
+The self-supervised pretext tasks utilize the training data itself
+as supervision signals without the need for additional annota-
+tions. If we consider an amino acid in a sequence as a word
+in a sentence, we can naturally extend masked language mod-
+eling to protein sequences. For example, we can statically or
+dynamically mask out a single or a set of contiguous amino
+acids and then predict the masked amino acids from the re-
+maining sequences [Rao et al., 2019; Elnaggar et al., 2020;
+Rives et al., 2021; Rao et al., 2021; Nambiar et al., 2020;
+Xiao et al., 2021]. Besides, [McDermott et al., 2021] com-
+bines adversarial training with MLM and proposes to mask
+amino acids in a learnable manner. Taking into account the
+dependence between masked amino acids, Pairwise MLM
+(PMLM) [He et al., 2021] proposes to model the probabil-
+ity of a pair of masked amino acids instead of predicting the
+probability of a single amino acid. Besides, Next Amino acid
+Prediction (NAP) [Alley et al., 2019; Elnaggar et al., 2020;
+Strodthoff et al., 2020] aims to predict the type of the next
+amino acid based on a set of given sequence fragments. Dif-
+ferent from the above methods, Contrastive Predictive Cod-
+ing (CPC) [Lu et al., 2020] applies different augmentation
+transformations on the input sequence to generate different
+views, and then maximizes the agreement of two jointly sam-
+pled pairs against that of two independently sampled pairs.
+Supervised Pretext Task
+The supervised pretext tasks use additional labels as auxiliary
+information to guide the model to learn knowledge relevant
+to downstream tasks. For example, PLUS [Min et al., 2021]
+devises a protein-specific pretext task, namely Same-Family
+Prediction (SFP), which trains a model to predict whether a
+given protein pair belongs to the same protein family. The
+protein family labels provide weak structural information and
+help the model learn structurally contextualized representa-
+tions. Besides, [Sturmfels et al., 2020] proposes to use HMM
+profiles derived from MSA as labels and then take Profile Pre-
+diction as a pretext task to help the model learn information
+about protein structures. In addition, to leverage the exponen-
+tially growing protein sequences that lack costly structural
+annotations, Progen [Madani et al., 2020] trains a language
+model with conditioning tags that encode various annotations,
+such as taxonomic, functional, and locational information.
+4.2
+Structure-based Pretext Task
+Despite the great progress in the design of structure-based
+encoders and graph-based pretext tasks [Wu et al., 2021;
+Xie et al., 2022; Liu et al., 2022b], there are few efforts focus-
+ing on the structure-based pre-training of proteins. Existing
+structure-based pretext tasks for proteins can be mainly clas-
+sified into two branches: contrastive and predictive methods.
+Contrastive Pretext Task
+The primary goal of contrastive methods is to maximize the
+agreement of two jointly sampled positive pairs. For example,
+Multiview Contrast [Hermosilla and Ropinski, 2022] pro-
+poses to randomly sample two sub-structures from each pro-
+tein, encoder them into two representations, and finally max-
+imize the similarity between representations from the same
+protein while minimizing the similarity between representa-
+tions from different proteins. Besides, [Zhang et al., 2022]
+adopts almost the same architecture as Multiview Contrast,
+but replaces GearNet with IEConv as the structure encoder.
+Predictive Pretext Task
+The contrastive methods deal with the inter-data information
+(data-data pairs). In contrast, the predictive methods aim to
+self-generate informative labels from the data as supervision
+and handle the data-label relationships. Categorized by dif-
+ferent types of pseudo labels, the predictive methods have
+different designs that can capture different levels of struc-
+tural protein information. For example, [Chen et al., 2022]
+proposes two predictive tasks, namely Distance Prediction
+and Angle Prediction, which take hidden representations of
+residues as input and aim to predict the relative distance be-
+tween pairwise residues and the angle between two edges,
+respectively, which helps to learn structure-aware protein rep-
+resentations. Furthermore, [Hermosilla and Ropinski, 2022]
+propose Residue Type Prediction and Dihedral Prediction
+based on geometric or biochemical properties. Specifically,
+Residue Type Prediction randomly masks the node features
+of some residues and then lets the structure-based encoders
+predict these masked residue types. Instead, Dihedral Pre-
+diction constructs a learning objective by predicting the di-
+hedral angle between three consecutive edges. Besides, [You
+and Shen, 2022] proposes graph completion (GraphComp),
+which takes as input a protein graph with partially masked
+residues and then makes predictions for those masked tokens.
+4.3
+Sequence-structure Pretext Task
+Most of the existing methods design pretext tasks for a single
+modality but ignore the dependencies between sequences and
+structures. If we can design the pretext task based on both
+protein sequences and structures, it should capture richer in-
+formation than using single modality data. In practice, there
+is no clear boundary between pretext tasks and downstream
+tasks. For example, AlphaFold2 [Jumper et al., 2021] takes
+full-atomic structure prediction as a downstream task. How-
+ever, if we are concerned with protein property prediction,
+structure prediction can also be considered as a pretext task
+
+Table 1: Summary of representative protein representation learning methods.
+Method
+Category
+Architecture
+Pretext Task
+Year
+Bio2Vec-CNN [Wang et al., 2019]
+Sequence-based
+CNN
+-
+2019
+TAPE [Rao et al., 2019]
+Sequence-based
+ResNet, LSTM, Transformer
+Masked Language Modeling,
+Next Amino Acid Prediction
+2019
+UniRep [Alley et al., 2019]
+Sequence-based
+Multiplicative LSTM
+Next Amino Acid Prediction
+2019
+TripletProt [Nourani et al., 2020]
+Sequence-based
+Siamese Networks
+Contrastive Predictive Coding
+2020
+PLP-CNN [Shanehsazzadeh et al., 2020]
+Sequence-based
+CNN
+-
+2020
+CPCProt [Lu et al., 2020]
+Sequence-based
+GRU, LSTM
+Contrastive Predictive Coding
+2020
+MuPIPR [Zhou et al., 2020]
+Sequence-based
+GRU, LSTM
+Next Amino Acid Prediction
+2020
+ProtTrans [Elnaggar et al., 2020]
+Sequence-based
+Transformer, Bert, XLNet
+Masked Language Modeling
+2020
+DMPfold [Kandathil et al., 2020]
+Sequence-based
+GRU, ResNet
+-
+2020
+Profile Prediction [Sturmfels et al., 2020]
+Sequence-based
+Transformer
+HMM Profile Prediction
+2020
+PRoBERTa [Nambiar et al., 2020]
+Sequence-based
+Transformer
+Masked Language Modeling
+2020
+UDSMProt [Strodthoff et al., 2020]
+Sequence-based
+LSTM
+Next Amino Acid Prediction
+2020
+ESM-1b [Rives et al., 2021]
+Sequence-based
+Transformer
+Masked Language Modeling
+2021
+PMLM [He et al., 2021]
+Sequence-based
+Transformer
+Pairwise Masked Language Modeling
+2021
+MSA Transformer [Rao et al., 2021]
+Sequence-based
+MSA Transformer
+Masked Language Modeling
+2021
+ProteinLM [Xiao et al., 2021]
+Sequence-based
+BERT
+Masked Language Modeling
+2021
+PLUS [Min et al., 2021]
+Sequence-based
+Bidirectional RNN
+Masked Language Modeling,
+Same-Family Prediction
+2021
+Adversarial MLM [McDermott et al., 2021]
+Sequence-based
+Transformer
+Masked Language Modeling,
+Adversarial Training
+2021
+ProteinBERT [Brandes et al., 2022]
+Sequence-based
+BERT
+Masked Language Modeling
+2022
+CARP [Yang et al., 2022a]
+Sequence-based
+CNN
+Masked Language Modeling
+2022
+3DCNN [Derevyanko et al., 2018]
+Structure-based
+3DCNN
+-
+2018
+IEConv [Hermosilla et al., 2020]
+Structure-based
+IEConv
+-
+2020
+GVP-GNN [Jing et al., 2020]
+Structure-based
+GVP
+-
+2020
+GraphMS [Cheng et al., 2021]
+Structure-based
+GCN
+Multiview Contrast
+2021
+DL-MSFM [Gelman et al., 2021]
+Structure-based
+GCN
+-
+2021
+PG-GNN [Xia and Ku, 2021]
+Structure-based
+PG-GNN
+-
+2021
+CRL [Hermosilla and Ropinski, 2022]
+Structure-based
+IEConv
+Multiview Contrast
+2022
+DW-GNN [Li et al., 2022]
+Structure-based
+DWP
+-
+2022
+GBPNet [Aykent and Xia, 2022]
+Structure-based
+GBP
+-
+2022
+GearNet [Zhang et al., 2022]
+Structure-based
+GearNet
+Multiview Contrast,
+Distance and Dihedral Prediction,
+Residue Type Prediction
+2022
+ATOMRefine [Wu and Cheng, 2022]
+Structure-based
+SE(3) Transformer
+-
+2022
+STEPS [Chen et al., 2022]
+Structure-based
+GIN
+Distance and Dihedral Prediction
+2022
+GraphCPI [Quan et al., 2019]
+Co-Modeling
+CNN, GNN
+-
+2019
+MT-LSTM [Bepler and Berger, 2019]
+Co-Modeling
+Bidirectional LSTM
+Contact prediction,
+Pairwise Similarity Prediction
+2019
+LM-GVP [Wang et al., 2021]
+Co-Modeling
+Transformer, GVP
+-
+2021
+AlphaFold2 [Jumper et al., 2021]
+Co-Modeling
+Evoformer
+Masked Language Modeling,
+Full-atomic Structure Prediction
+2021
+DeepFRI [Gligorijevi´c et al., 2021]
+Co-Modeling
+LSTM, GCN
+-
+2021
+HJRSS [Mansoor et al., 2021]
+Co-Modeling
+SE(3) Transformer
+Masked Language Modeling,
+Graph Completion
+2021
+GraSR [Xia et al., 2022]
+Co-Modeling
+LSTM, GCN
+Momentum Contrast
+2022
+CPAC [You and Shen, 2022]
+Co-Modeling
+Hierarchical RNN, GAT
+Masked Language Modeling,
+Graph Completion
+2022
+MIF-ST [Yang et al., 2022b]
+Co-Modeling
+CNN, GNN
+Masked Inverse Folding
+2022
+OmegaFold [Wu et al., 2022]
+Co-Modeling
+Geoformer
+Masked Language Modeling,
+Full-atomic Structure Prediction
+2022
+that enables the learned sequence representations to contain
+sufficient structural information. It was found by [Hu et al.,
+2022] that the representations from AlphFold2’s Evoformer
+could work well on various protein-related downstream tasks,
+including fold classification, stability prediction, etc. More-
+over, [Yang et al., 2022b] proposes a novel pre-training pre-
+text task, namely Masked Inverse Folding (MIF), which trains
+a model to reconstruct the original amino acids conditioned
+on the corrupted sequence and the backbone structure.
+5
+Downstream Tasks (Applications)
+In the above, we have presented a variety of commonly used
+model architectures and pretext tasks for protein representa-
+
+tion learning, based on which we summarized the surveyed
+works in Table. 1, listing their categories, model architec-
+tures, pretext tasks, and publication years. In this section, we
+can divide existing downstream tasks for protein representa-
+tion learning into the following four main categories: protein
+property prediction, protein (complex) structure prediction,
+protein design, and structure-based drug design.
+It is worth noting that some downstream tasks have labels
+(i.e., model outputs) that do not change with rigid body trans-
+formations of the inputs (if they can, e.g., protein structures).
+For example, various protein property prediction tasks take
+a transformable protein structure as input and output a con-
+stant prediction, usually modeled as a simple multi-label clas-
+sification problem or multiple binary classification problem.
+However, the labels of some downstream tasks will change
+equivariantly with the inputs, and these tasks are getting more
+and more attention. Typically, the learning objectives of these
+tasks are structure-related, and they usually have higher re-
+quirements on the model architecture, requiring the model to
+be SE(3)-equivariant. We believe that from the perspective
+of protein representation learning, the approaches to different
+downstream tasks can also learn from each other.
+5.1
+Protein Property Prediction
+The protein property prediction aims to regress or classify
+some important properties from protein sequences or struc-
+tures that are closely related to biological functions, such
+as the types of secondary structure, the strength of connec-
+tions between amino acids, types of protein folding, fluo-
+rescence intensity, protein stability, etc. [Rao et al., 2019].
+Besides, several protein-specific prediction tasks can also be
+grouped into this category, including quality evaluation of
+protein folding [Baldassarre et al., 2021], predicting the ef-
+fect of mutations on protein function [Meier et al., 2021], and
+predicting protein-protein interactions [Wang et al., 2019].
+5.2
+Protein (Complex) Structure Prediction
+The primary goal of protein structure prediction is to pre-
+dict the structural coordinates from a given set of amino
+acid sequences. Some approaches aim to predict only back-
+bone coordinates [Baek et al., 2021; Si et al., 2020], while
+others focus on the more challenging full-atomic coordi-
+nate predictions [Jumper et al., 2021; Wu et al., 2022;
+Rao et al., 2021]. On the other hand, protein structure refine-
+ment [Hiranuma et al., 2021; Wu and Cheng, 2022] proposes
+to update a coarse protein structure to generate a more fine-
+grained structure in an iterative manner. Besides, the task of
+protein structure inpainting aims to reconstruct the complete
+protein structure from a partially given sub-structure [McPart-
+lon and Xu, 2022] or distance map [Lee and Kim, 2022].
+5.3
+Protein Design
+Deep learning-based protein design has made tremendous
+progress in recent years, and the major works can be di-
+vided into three categories. The first one is to pre-train the
+model with a large number of sequences from the same pro-
+tein family, and then use it to generate new homologous se-
+quences [Smith and Smith, 1990]. The structure-based meth-
+ods aim to directly generate the protein sequences under the
+condition of a given protein structure [Ingraham et al., 2019].
+The last and most challenging one is the de novo protein de-
+sign [Huang et al., 2016; Korendovych and DeGrado, 2020;
+Koepnick et al., 2019], which aims to generate both protein
+sequences and structures conditioned on taxonomic and key-
+word tags such as molecular function and cellular component.
+5.4
+Structure-Based Drug Design
+Structure-Based Drug Design (SBDD) is a promising direc-
+tion for fast and cost-efficient compound discovery. Specif-
+ically, SBDD designs inhibitors or activators (usually small
+molecules, i.e., drugs) directly against protein targets of inter-
+est, which means a high success rate and efficiency [Kuntz,
+1992; Drews, 2000]. In the past two years, a line of auto-
+regressive methods have been proposed for SBDD [Liu et al.,
+2022a; Peng et al., 2022; Masuda et al., 2020], which gener-
+ate molecule atoms one by one conditioned on given structure
+context of protein targets. Recently, there are some works
+based on Denoising Diffusion Probabilistic Model (DDPM)
+[Lin et al., 2022; Schneuing et al., 2022]. Targeting on spe-
+cific protein pockets, the diffusion-based methods generate
+molecule atoms as a whole from random gaussian noise.
+The above methods are all dependent on a proper repre-
+sentation module of protein, especially the protein structure.
+The early attempt of deep generative models in this field [Luo
+et al., 2021] uses 3D CNN as the protein structure context
+encoder to get meaningful and roto-translation invariant fea-
+tures. With the development of protein structure representa-
+tion methods, particularly the geometric-aware models, sub-
+sequent methods widely use geometric-(equi/in)variant net-
+works, such as EGNN [Gong and Cheng, 2019], GVP [Jing
+et al., 2020], and IPA [Jumper et al., 2021], as the backbones.
+It is worth noting that protein representation models are not
+only common in various protein structure context encoders,
+but many generative decoders can also adopt its architectural
+design. From this example, we can see that protein represen-
+tation is a very fundamental problem and that many down-
+stream tasks involving proteins can benefit from advances of
+protein representation research in various aspects, including
+better embeddings and more excellent model architectures.
+6
+Deep Insights and Future Outlooks
+6.1
+Deeper Insights
+On the basis of a detailed review of the model architectures,
+pretext tasks, and downstream tasks, we would like to provide
+some deeper insights into protein representation learning.
+Insights 1: PRL is the core of deep protein modeling
+With the development of deep learning, deep protein mod-
+eling is becoming a popular research topic, and one of its
+core is how to learn “meaningful” representations for pro-
+teins. This involves three key issues: (1) Feature Extraction:
+model architectures; (2) Pre-training: pretext tasks; and (3)
+Application: downstream tasks. An in-depth investigation of
+the above three key issues is of great importance for the de-
+velopment of more deep protein modeling methods.
+
+Insights 2: Task-level convertibility
+Throughout this survey, one of the main points we have em-
+phasized is the convertibility between downstream tasks and
+pretext tasks. We believe we are the first to explain the role
+of pretext tasks from this perspective, which seems to have
+been rarely involved in previous work. For example, we di-
+rectly categorize some well-known downstream tasks, such as
+full-atomic structure prediction, as a specific kinds of pretext
+tasks. The motivation behind such an understanding lies in
+the fact that the definition of a task is itself a relative concept
+and that different tasks can help the model extract different
+aspects of information, which may be complementary to each
+other. For example, full-atomic structure prediction helps the
+model capture rich structural information, which is also ben-
+eficial for various protein property prediction tasks, such as
+folding prediction, since it is known that protein structure of-
+ten determines protein function. This suggests that whether
+a specific task is a downstream task or a pretext task usually
+depends on what we are concerned about, and the role of a
+task may keep changing from application to application.
+Insights 3: Data-specific criterion for design selections
+It is tricky to discuss the advantages and disadvantages of dif-
+ferent methods or designs because the effectiveness of differ-
+ent methods depends heavily on the size, format, and com-
+plexity of the data. For example, for simple small-scale data,
+Transformer is not necessarily more effective than traditional
+LSTM for sequence modeling, and the situation may be com-
+pletely opposite for large-scale complex data.
+Therefore,
+there is no “optimal” architecture or pretext task that works
+for all data types and downstream tasks, and the criterion for
+the selection of architecture and pretext task is data-specific.
+6.2
+Future Outlooks
+Despite the great progress of existing methods, challenges
+still exist due to the complexity of proteins. In this section,
+we suggest some promising directions for future work.
+Direction 1: Broader application scenarios
+The biological research topics on proteins are diverse, but
+most of the existing work has delved into only a small subset
+of them, due to the fact that these topics have been well for-
+malized by some representative works, such as AlphaFlod2
+[Jumper et al., 2021] for protein structure prediction and
+TAPE [Rao et al., 2019] for protein property prediction. As
+a result, it is more worthwhile to explore the role of protein
+representation learning in a wider range of biological applica-
+tion scenarios than to design some overly complex modules
+for subtle performance gains in a well-formalized application.
+Direction 2: Unified evaluation protocols
+Research in protein representation learning is now in an era of
+barbarism. While a great deal of new works are emerging ev-
+ery day, most of them are on unfair comparisons, such as with
+different datasets, architectures, metrics, etc. For example,
+some MSA-based works on structure prediction have been
+blatantly compared with those single-sequence-based works
+and claimed to be better. To promote the health of the field,
+there is an urgent need to establish unified evaluation proto-
+cols in various downstream tasks to provide fair comparisons.
+Direction 3: Protein-specific designs
+Previous PRL methods directly take mature architectures and
+pretext tasks from the natural language processing field to
+train proteins. For example, modeling protein sequences us-
+ing LSTM may be a major innovation, but replacing LSTM
+with Bi-LSTM for stuble performance improvements makes
+little sense. Now, it is time to step out of this comfort zone
+of scientific research, and we should no longer be satisfied
+with simply extending techniques from other domains to the
+protein domain. PRL is not only a machine learning problem
+but also a biological problem, so we should consider design-
+ing more protein-specific architectures and pretext tasks by
+incorporating protein-related domain knowledge. In particu-
+lar, most of the existing work on PRL is based on unimodal
+protein sequences or structures, and it requires more work ex-
+ploring sequence-structure co-modeling to fully explore the
+correspondence between 1D sequences and 3D structures.
+Direction 4: Margin from pre-training to fine-tuning
+Currently, tremendous efforts are focusing on protein pre-
+training strategies.
+However, how to fine-tune these pre-
+trained models to specific downstream tasks is still under-
+explored. Though numerous strategies have been proposed
+to address this problem in the fields of computer vision and
+natural language processing [Zhuang et al., 2020], they are
+difficult to be directly applied to proteins. One obstacle to
+knowledge transfer is the huge variability between different
+protein datasets, both in terms of sequence length and struc-
+tural complexity. The second one is poor generalization of
+pre-trained models especially for various tasks where collect-
+ing labeled data is laborious. Therefore, it is an important
+issue to design protein-specific techniques to minimize the
+margin between pre-training and downstream tasks.
+Direction 5: Lack of explainability
+While existing protein representation learning methods have
+achieved promising results on a variety of downstream tasks,
+we still know little about what the model has learned from
+protein data. Which of the feature patterns, sequence frag-
+ments, or sequence-structure relationships has been learned?
+These are important issues for understanding and interpret-
+ing model behavior, especially for those privacy-secure tasks
+such as drug design, but are missing in current PRL works.
+Overall, the interpretability of PRL methods remains to be
+explored further in many respects, which helps us understand
+how the model works and provides a guide for better usage.
+7
+Conclusions
+A comprehensive survey of the literature on protein repre-
+sentation learning is conducted in this paper. We develop a
+general unified framework for PRL methods. Moreover, we
+systematically divide existing PRL methods into three main
+categories: sequence-based, structure-based, and sequence-
+structure co-modeling from three different perspectives, in-
+cluding model architectures, pretext tasks, and downstream
+applications. Finally, we point out the technical limitations
+of the current research and provide promising directions for
+future work on PRL. We hope this survey to pave the way for
+follow-up AI researchers with no bioinformatics background,
+setting the stage for the development of more future works.
+
+References
+[Alley et al., 2019] Ethan C Alley, Grigory Khimulya, Suro-
+jit Biswas,
+Mohammed AlQuraishi,
+and George M
+Church.
+Unified rational protein engineering with
+sequence-based deep representation learning.
+Nature
+methods, 16(12):1315–1322, 2019.
+[Amidi et al., 2018] Afshine Amidi, Shervine Amidi, Dim-
+itrios Vlachakis, Vasileios Megalooikonomou, Nikos Para-
+gios, and Evangelia I Zacharaki. Enzynet: enzyme classi-
+fication using 3d convolutional neural networks on spatial
+representation. PeerJ, 6:e4750, 2018.
+[Armenteros et al., 2020] Jose Juan Almagro Armenteros,
+Alexander Rosenberg Johansen, Ole Winther, and Henrik
+Nielsen. Language modelling for biological sequences–
+curated datasets and baselines. BioRxiv, 2020.
+[Asgari et al., 2019] Ehsaneddin Asgari, Nina Poerner, Al-
+ice C McHardy, and Mohammad RK Mofrad.
+Deep-
+prime2sec: deep learning for protein secondary structure
+prediction from the primary sequences.
+BioRxiv, page
+705426, 2019.
+[Aykent and Xia, 2022] Sarp Aykent and Tian Xia.
+Gbp-
+net: Universal geometric representation learning on pro-
+tein structures. In Proceedings of the 28th ACM SIGKDD
+Conference on Knowledge Discovery and Data Mining,
+pages 4–14, 2022.
+[Baek et al., 2021] Minkyung Baek, Frank DiMaio, Ivan
+Anishchenko,
+Justas Dauparas,
+Sergey Ovchinnikov,
+Gyu Rie Lee, Jue Wang, Qian Cong, Lisa N Kinch,
+R Dustin Schaeffer, et al. Accurate prediction of protein
+structures and interactions using a three-track neural net-
+work. Science, 373(6557):871–876, 2021.
+[Baldassarre et al., 2021] Federico
+Baldassarre,
+David
+Men´endez Hurtado, Arne Elofsson, and Hossein Az-
+izpour.
+Graphqa:
+protein model quality assessment
+using graph convolutional networks.
+Bioinformatics,
+37(3):360–366, 2021.
+[Bepler and Berger, 2019] Tristan
+Bepler
+and
+Bonnie
+Berger.
+Learning
+protein
+sequence
+embeddings
+using information from structure.
+arXiv preprint
+arXiv:1902.08661, 2019.
+[Brandes et al., 2022] Nadav Brandes, Dan Ofer, Yam Peleg,
+Nadav Rappoport, and Michal Linial. Proteinbert: A uni-
+versal deep-learning model of protein sequence and func-
+tion. Bioinformatics, 38(8):2102–2110, 2022.
+[Chen et al., 2022] Can Chen, Jingbo Zhou, Fan Wang, Xue
+Liu, and Dejing Dou.
+Structure-aware protein self-
+supervised learning.
+arXiv preprint arXiv:2204.04213,
+2022.
+[Cheng et al., 2021] Shicheng Cheng, Liang Zhang, Bo Jin,
+Qiang Zhang, Xinjiang Lu, Mao You, and Xueqing Tian.
+Graphms: Drug target prediction using graph represen-
+tation learning with substructures.
+Applied Sciences,
+11(7):3239, 2021.
+[Derevyanko et al., 2018] Georgy Derevyanko, Sergei Gru-
+dinin, Yoshua Bengio, and Guillaume Lamoureux. Deep
+convolutional networks for quality assessment of protein
+folds. Bioinformatics, 34(23):4046–4053, 2018.
+[Devlin et al., 2018] Jacob Devlin, Ming-Wei Chang, Ken-
+ton Lee, and Kristina Toutanova.
+Bert: Pre-training of
+deep bidirectional transformers for language understand-
+ing. arXiv preprint arXiv:1810.04805, 2018.
+[Ding et al., 2019] Xinqiang Ding,
+Zhengting Zou,
+and
+Charles L Brooks III. Deciphering protein evolution and
+fitness landscapes with latent space models. Nature com-
+munications, 10(1):1–13, 2019.
+[Drews, 2000] J¨urgen Drews. Drug discovery: A historical
+perspective. Science, 287(5460):1960–1964, 2000.
+[Elnaggar et al., 2020] Ahmed
+Elnaggar,
+Michael
+Heinzinger, Christian Dallago, Ghalia Rihawi, Yu Wang,
+Llion Jones, Tom Gibbs, Tamas Feher, Christoph Angerer,
+Martin Steinegger, et al.
+Prottrans: towards cracking
+the language of life’s code through self-supervised deep
+learning and high performance computing. arXiv preprint
+arXiv:2007.06225, 2020.
+[Gainza et al., 2020] Pablo Gainza, Freyr Sverrisson, Fred-
+erico Monti, Emanuele Rodola, D Boscaini, MM Bron-
+stein, and BE Correia.
+Deciphering interaction finger-
+prints from protein molecular surfaces using geometric
+deep learning. Nature Methods, 17(2):184–192, 2020.
+[Gelman et al., 2021] Sam Gelman, Sarah A Fahlberg, Pete
+Heinzelman, Philip A Romero, and Anthony Gitter. Neu-
+ral networks to learn protein sequence–function relation-
+ships from deep mutational scanning data. Proceedings of
+the National Academy of Sciences, 118(48):e2104878118,
+2021.
+[Gligorijevi´c et al., 2021] Vladimir Gligorijevi´c, P Douglas
+Renfrew, Tomasz Kosciolek, Julia Koehler Leman, Daniel
+Berenberg, Tommi Vatanen, Chris Chandler, Bryn C Tay-
+lor, Ian M Fisk, Hera Vlamakis, et al.
+Structure-based
+protein function prediction using graph convolutional net-
+works. Nature communications, 12(1):1–14, 2021.
+[Gong and Cheng, 2019] Liyu Gong and Qiang Cheng. Ex-
+ploiting edge features for graph neural networks. In Pro-
+ceedings of the IEEE/CVF conference on computer vision
+and pattern recognition, pages 9211–9219, 2019.
+[Hamilton et al., 2017] Will Hamilton, Zhitao Ying, and Jure
+Leskovec.
+Inductive representation learning on large
+graphs. In Neural information processing systems, pages
+1024–1034, 2017.
+[He et al., 2016] Kaiming He, Xiangyu Zhang, Shaoqing
+Ren, and Jian Sun. Deep residual learning for image recog-
+nition. In Proceedings of the IEEE conference on computer
+vision and pattern recognition, pages 770–778, 2016.
+[He et al., 2021] Liang He, Shizhuo Zhang, Lijun Wu,
+Huanhuan Xia, Fusong Ju, He Zhang, Siyuan Liu, Yingce
+Xia, Jianwei Zhu, Pan Deng, et al.
+Pre-training co-
+evolutionary protein representation via a pairwise masked
+language model. arXiv preprint arXiv:2110.15527, 2021.
+[Hermosilla and Ropinski, 2022] Pedro
+Hermosilla
+and
+Timo Ropinski.
+Contrastive representation learning for
+
+3d protein structures. arXiv preprint arXiv:2205.15675,
+2022.
+[Hermosilla et al., 2020] Pedro Hermosilla, Marco Sch¨afer,
+Matˇej Lang, Gloria Fackelmann, Pere Pau V´azquez,
+Barbora Kozl´ıkov´a, Michael Krone, Tobias Ritschel, and
+Timo Ropinski. Intrinsic-extrinsic convolution and pool-
+ing for learning on 3d protein structures. arXiv preprint
+arXiv:2007.06252, 2020.
+[Hiranuma et al., 2021] Naozumi
+Hiranuma,
+Hahnbeom
+Park, Minkyung Baek, Ivan Anishchenko, Justas Dau-
+paras, and David Baker.
+Improved protein structure
+refinement guided by deep learning based accuracy
+estimation. Nature communications, 12(1):1–11, 2021.
+[Hochreiter and Schmidhuber, 1997] Sepp Hochreiter and
+J¨urgen Schmidhuber. Long short-term memory. Neural
+computation, 9(8):1735–1780, 1997.
+[Hospital et al., 2015] Adam Hospital, Josep Ramon Go˜ni,
+Modesto Orozco, and Josep L Gelp´ı. Molecular dynamics
+simulations: advances and applications. Advances and ap-
+plications in bioinformatics and chemistry: AABC, 8:37,
+2015.
+[Hu et al., 2021] Lun Hu, Xiaojuan Wang, Yu-An Huang,
+Pengwei Hu, and Zhu-Hong You. A survey on computa-
+tional models for predicting protein–protein interactions.
+Briefings in Bioinformatics, 22(5):bbab036, 2021.
+[Hu et al., 2022] Mingyang Hu, Fajie Yuan, Kevin K Yang,
+Fusong Ju, Jin Su, Hui Wang, Fei Yang, and Qiuyang
+Ding. Exploring evolution-based &-free protein language
+models as protein function predictors.
+arXiv preprint
+arXiv:2206.06583, 2022.
+[Huang et al., 2016] Po-Ssu Huang, Scott E Boyken, and
+David Baker. The coming of age of de novo protein de-
+sign. Nature, 537(7620):320–327, 2016.
+[Ingraham et al., 2019] John Ingraham, Vikas Garg, Regina
+Barzilay, and Tommi Jaakkola.
+Generative models for
+graph-based protein design. Advances in neural informa-
+tion processing systems, 32, 2019.
+[Iuchi et al., 2021] Hitoshi Iuchi, Taro Matsutani, Keisuke
+Yamada, Natsuki Iwano, Shunsuke Sumi, Shion Hosoda,
+Shitao Zhao, Tsukasa Fukunaga, and Michiaki Hamada.
+Representation learning applications in biological se-
+quence analysis. Computational and Structural Biotech-
+nology Journal, 19:3198–3208, 2021.
+[Jing et al., 2020] Bowen Jing, Stephan Eismann, Patricia
+Suriana, Raphael JL Townshend, and Ron Dror. Learning
+from protein structure with geometric vector perceptrons.
+arXiv preprint arXiv:2009.01411, 2020.
+[Jumper et al., 2021] John Jumper, Richard Evans, Alexan-
+der Pritzel, Tim Green, Michael Figurnov, Olaf Ron-
+neberger, Kathryn Tunyasuvunakool, Russ Bates, Au-
+gustin ˇZ´ıdek, Anna Potapenko, et al.
+Highly accu-
+rate protein structure prediction with alphafold. Nature,
+596(7873):583–589, 2021.
+[Kandathil et al., 2020] Shaun M Kandathil, Joe G Greener,
+Andy M Lau, and David T Jones. Deep learning-based
+prediction of protein structure using learned representa-
+tions of multiple sequence alignments.
+Biorxiv, pages
+2020–11, 2020.
+[Karplus and Petsko, 1990] Martin Karplus and Gregory A
+Petsko. Molecular dynamics simulations in biology. Na-
+ture, 347(6294):631–639, 1990.
+[Kipf and Welling, 2016] Thomas N Kipf and Max Welling.
+Semi-supervised classification with graph convolutional
+networks. arXiv preprint arXiv:1609.02907, 2016.
+[Koepnick et al., 2019] Brian Koepnick, Jeff Flatten, Tamir
+Husain, Alex Ford, Daniel-Adriano Silva, Matthew J
+Bick, Aaron Bauer, Gaohua Liu, Yojiro Ishida, Alexander
+Boykov, et al. De novo protein design by citizen scientists.
+Nature, 570(7761):390–394, 2019.
+[Korendovych and DeGrado, 2020] Ivan
+V
+Korendovych
+and William F DeGrado.
+De novo protein design, a
+retrospective. Quarterly reviews of biophysics, 53, 2020.
+[Kuntz, 1992] Irwin D. Kuntz.
+Structure-based strategies
+for drug design and discovery. Science, 257(5073):1078–
+1082, 1992.
+[Lapedes et al., 1999] Alan S Lapedes, Bertrand G Giraud,
+LonChang Liu, and Gary D Stormo. Correlated mutations
+in models of protein sequences: phylogenetic and struc-
+tural effects. Lecture Notes-Monograph Series, pages 236–
+256, 1999.
+[LeCun et al., 1995] Yann LeCun, Yoshua Bengio, et al.
+Convolutional networks for images, speech, and time se-
+ries. The handbook of brain theory and neural networks,
+3361(10):1995, 1995.
+[Lee and Kim, 2022] Jin Sub Lee and Philip M Kim. Pro-
+teinsgm: Score-based generative modeling for de novo
+protein design. bioRxiv, 2022.
+[Li et al., 2022] Jiahan Li, Shitong Luo, Congyue Deng,
+Chaoran Cheng, Jiaqi Guan, Leonidas Guibas, Jian Peng,
+and Jianzhu Ma.
+Directed weight neural networks for
+protein structure representation learning. arXiv preprint
+arXiv:2201.13299, 2022.
+[Lin et al., 2022] Haitao Lin, Yufei Huang, Meng Liu, Xu-
+anjing Li, Shuiwang Ji, and Stan Z Li. Diffbp: Generative
+diffusion of 3d molecules for target protein binding. arXiv
+preprint arXiv:2211.11214, 2022.
+[Liu et al., 2022a] Meng Liu, Youzhi Luo, Kanji Uchino,
+Koji Maruhashi, and Shuiwang Ji.
+Generating 3d
+molecules for target protein binding. In International Con-
+ference on Machine Learning, 2022.
+[Liu et al., 2022b] Yixin Liu, Ming Jin, Shirui Pan, Chuan
+Zhou, Yu Zheng, Feng Xia, and Philip Yu. Graph self-
+supervised learning: A survey.
+IEEE Transactions on
+Knowledge and Data Engineering, 2022.
+[Lu et al., 2020] Amy X Lu, Haoran Zhang, Marzyeh Ghas-
+semi, and Alan Moses. Self-supervised contrastive learn-
+ing of protein representations by mutual information max-
+imization. BioRxiv, 2020.
+
+[Luo et al., 2021] Shitong Luo, Jiaqi Guan, Jianzhu Ma, and
+Jian Peng. A 3D generative model for structure-based drug
+design. In Thirty-Fifth Conference on Neural Information
+Processing Systems, 2021.
+[Madani et al., 2020] Ali Madani, Bryan McCann, Nikhil
+Naik, Nitish Shirish Keskar, Namrata Anand, Raphael R
+Eguchi, Po-Ssu Huang, and Richard Socher. Progen: Lan-
+guage modeling for protein generation.
+arXiv preprint
+arXiv:2004.03497, 2020.
+[Mansoor et al., 2021] Sanaa Mansoor,
+Minkyung Baek,
+Umesh Madan, and Eric Horvitz. Toward more general
+embeddings for protein design: Harnessing joint represen-
+tations of sequence and structure. bioRxiv, 2021.
+[Masuda et al., 2020] Tomohide Masuda, Matthew Ragoza,
+and David Ryan Koes. Generating 3d molecular structures
+conditional on a receptor binding site with deep generative
+models. arXiv preprint arXiv:2010.14442, 2020.
+[McDermott et al., 2021] Matthew
+McDermott,
+Brendan
+Yap, Harry Hsu, Di Jin, and Peter Szolovits. Adversarial
+contrastive pre-training for protein sequences.
+arXiv
+preprint arXiv:2102.00466, 2021.
+[McPartlon and Xu, 2022] Matthew McPartlon and Jinbo
+Xu. Attnpacker: An end-to-end deep learning method for
+rotamer-free protein side-chain packing. bioRxiv, 2022.
+[Meier et al., 2021] Joshua Meier,
+Roshan Rao,
+Robert
+Verkuil, Jason Liu, Tom Sercu, and Alex Rives. Language
+models enable zero-shot prediction of the effects of muta-
+tions on protein function. Advances in Neural Information
+Processing Systems, 34:29287–29303, 2021.
+[Min et al., 2021] Seonwoo Min, Seunghyun Park, Siwon
+Kim, Hyun-Soo Choi, Byunghan Lee, and Sungroh Yoon.
+Pre-training of deep bidirectional protein sequence rep-
+resentations with structural information.
+IEEE Access,
+9:123912–123926, 2021.
+[Nambiar et al., 2020] Ananthan Nambiar, Maeve Heflin,
+Simon Liu, Sergei Maslov, Mark Hopkins, and Anna Ritz.
+Transforming the language of life: transformer neural net-
+works for protein prediction tasks. In Proceedings of the
+11th ACM International Conference on Bioinformatics,
+Computational Biology and Health Informatics, pages 1–
+8, 2020.
+[Nourani et al., 2020] Esmaeil Nourani, Ehsaneddin Asgari,
+Alice C McHardy, and Mohammad RK Mofrad. Triplet-
+prot: Deep representation learning of proteins based on
+siamese networks. Biorxiv, 2020.
+[Peng et al., 2022] Xingang Peng, Shitong Luo, Jiaqi Guan,
+Qi Xie, Jian Peng, and Jianzhu Ma. Pocket2mol: Efficient
+molecular sampling based on 3d protein pockets. In Inter-
+national Conference on Machine Learning, 2022.
+[Preparata and Shamos, 2012] Franco
+P
+Preparata
+and
+Michael I Shamos.
+Computational geometry:
+an
+introduction. Springer Science & Business Media, 2012.
+[Quan et al., 2019] Zhe Quan, Yan Guo, Xuan Lin, Zhi-Jie
+Wang, and Xiangxiang Zeng.
+Graphcpi: Graph neural
+representation learning for compound-protein interaction.
+In 2019 IEEE International Conference on Bioinformatics
+and Biomedicine (BIBM), pages 717–722. IEEE, 2019.
+[Rao et al., 2019] Roshan Rao, Nicholas Bhattacharya, Neil
+Thomas, Yan Duan, Peter Chen, John Canny, Pieter
+Abbeel, and Yun Song. Evaluating protein transfer learn-
+ing with tape. Advances in neural information processing
+systems, 32, 2019.
+[Rao et al., 2021] Roshan M Rao, Jason Liu, Robert Verkuil,
+Joshua Meier, John Canny, Pieter Abbeel, Tom Sercu, and
+Alexander Rives. Msa transformer. In International Con-
+ference on Machine Learning, pages 8844–8856. PMLR,
+2021.
+[Rives et al., 2021] Alexander Rives, Joshua Meier, Tom
+Sercu, Siddharth Goyal, Zeming Lin, Jason Liu, Demi
+Guo, Myle Ott, C Lawrence Zitnick, Jerry Ma, et al.
+Biological structure and function emerge from scal-
+ing unsupervised learning to 250 million protein se-
+quences.
+Proceedings of the National Academy of Sci-
+ences, 118(15):e2016239118, 2021.
+[Rohl et al., 2004] Carol A Rohl,
+Charlie EM Strauss,
+Kira MS Misura, and David Baker. Protein structure pre-
+diction using rosetta. In Methods in enzymology, volume
+383, pages 66–93. Elsevier, 2004.
+[Schaap et al., 2001] Marcel G Schaap, Feike J Leij, and
+Martinus Th Van Genuchten. Rosetta: A computer pro-
+gram for estimating soil hydraulic parameters with hi-
+erarchical pedotransfer functions. Journal of hydrology,
+251(3-4):163–176, 2001.
+[Schneuing et al., 2022] Arne
+Schneuing,
+Yuanqi
+Du,
+Charles Harris, Arian Jamasb, Ilia Igashov, Weitao Du,
+Tom Blundell, Pietro Li´o, Carla Gomes, Max Welling,
+et al.
+Structure-based drug design with equivariant
+diffusion models. arXiv preprint arXiv:2210.13695, 2022.
+[Shanehsazzadeh et al., 2020] Amir Shanehsazzadeh, David
+Belanger, and David Dohan. Is transfer learning neces-
+sary for protein landscape prediction?
+arXiv preprint
+arXiv:2011.03443, 2020.
+[Si et al., 2020] Dong Si, Spencer A Moritz, Jonas Pfab, Jie
+Hou, Renzhi Cao, Liguo Wang, Tianqi Wu, and Jianlin
+Cheng. Deep learning to predict protein backbone struc-
+ture from high-resolution cryo-em density maps. Scientific
+reports, 10(1):1–22, 2020.
+[Sinai et al., 2017] Sam Sinai,
+Eric Kelsic,
+George M
+Church, and Martin A Nowak. Variational auto-encoding
+of protein sequences. arXiv preprint arXiv:1712.03346,
+2017.
+[Smith and Smith, 1990] Randall F Smith and Temple F
+Smith. Automatic generation of primary sequence patterns
+from sets of related protein sequences. Proceedings of the
+National Academy of Sciences, 87(1):118–122, 1990.
+[Strodthoff et al., 2020] Nils Strodthoff,
+Patrick Wagner,
+Markus Wenzel, and Wojciech Samek. Udsmprot: univer-
+sal deep sequence models for protein classification. Bioin-
+formatics, 36(8):2401–2409, 2020.
+
+[Sturmfels et al., 2020] Pascal Sturmfels, Jesse Vig, Ali
+Madani, and Nazneen Fatema Rajani. Profile prediction:
+An alignment-based pre-training task for protein sequence
+models. arXiv preprint arXiv:2012.00195, 2020.
+[Sverrisson et al., 2021] Freyr
+Sverrisson,
+Jean
+Feydy,
+Bruno E Correia, and Michael M Bronstein. Fast end-to-
+end learning on protein surfaces. In Proceedings of the
+IEEE/CVF Conference on Computer Vision and Pattern
+Recognition, pages 15272–15281, 2021.
+[Thomas et al., 2005] John Thomas, Naren Ramakrishnan,
+and Chris Bailey-Kellogg. Graphical models of residue
+coupling in protein families.
+In Proceedings of the 5th
+international workshop on Bioinformatics, pages 12–20,
+2005.
+[Torrisi et al., 2020] Mirko Torrisi, Gianluca Pollastri, and
+Quan Le.
+Deep learning methods in protein structure
+prediction. Computational and Structural Biotechnology
+Journal, 18:1301–1310, 2020.
+[Townshend et al., 2019] Raphael Townshend, Rishi Bedi,
+Patricia Suriana, and Ron Dror. End-to-end learning on
+3d protein structure for interface prediction. Advances in
+Neural Information Processing Systems, 32, 2019.
+[Unsal et al., 2020] Serbulent Unsal, Heval Atas¸, Muam-
+mer Albayrak, Kemal Turhan, Aybar C Acar, and Tunca
+Do˘gan. Evaluation of methods for protein representation
+learning: a quantitative analysis. bioRxiv, 2020.
+[Vaswani et al., 2017] Ashish Vaswani, Noam Shazeer, Niki
+Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez,
+Łukasz Kaiser, and Illia Polosukhin. Attention is all you
+need. Advances in neural information processing systems,
+30, 2017.
+[Wang et al., 2019] Yanbin Wang, Zhu-Hong You, Shan
+Yang, Xiao Li, Tong-Hai Jiang, and Xi Zhou. A high ef-
+ficient biological language model for predicting protein–
+protein interactions. Cells, 8(2):122, 2019.
+[Wang et al., 2021] Zichen Wang, Steven A Combs, Ryan
+Brand, Miguel Romero Calvo, Panpan Xu, George Price,
+Nataliya Golovach, Emannuel O Salawu, Colby J Wise,
+Sri Priya Ponnapalli, et al. Lm-gvp: A generalizable deep
+learning framework for protein property prediction from
+sequence and structure. bioRxiv, 2021.
+[Weigt et al., 2009] Martin Weigt, Robert A White, Hendrik
+Szurmant, James A Hoch, and Terence Hwa. Identification
+of direct residue contacts in protein–protein interaction by
+message passing. Proceedings of the National Academy of
+Sciences, 106(1):67–72, 2009.
+[Wu and Cheng, 2022] Tianqi
+Wu
+and
+Jianlin
+Cheng.
+Atomic
+protein
+structure
+refinement
+using
+all-atom
+graph representations and se (3)-equivariant graph neural
+networks. bioRxiv, 2022.
+[Wu et al., 2021] Lirong Wu,
+Haitao Lin,
+Cheng Tan,
+Zhangyang Gao, and Stan Z Li. Self-supervised learning
+on graphs: Contrastive, generative, or predictive. IEEE
+Transactions on Knowledge and Data Engineering, 2021.
+[Wu et al., 2022] Ruidong Wu, Fan Ding, Rui Wang, Rui
+Shen, Xiwen Zhang, Shitong Luo, Chenpeng Su, Zuo-
+fan Wu, Qi Xie, Bonnie Berger, et al. High-resolution de
+novo structure prediction from primary sequence. bioRxiv,
+2022.
+[Xia and Ku, 2021] Tian Xia and Wei-Shinn Ku. Geomet-
+ric graph representation learning on protein structure pre-
+diction. In Proceedings of the 27th ACM SIGKDD Con-
+ference on Knowledge Discovery & Data Mining, pages
+1873–1883, 2021.
+[Xia et al., 2022] Chunqiu Xia, Shi-Hao Feng, Ying Xia, Xi-
+aoyong Pan, and Hong-Bin Shen. Fast protein structure
+comparison through effective representation learning with
+contrastive graph neural networks. PLoS computational
+biology, 18(3):e1009986, 2022.
+[Xiao et al., 2021] Yijia Xiao, Jiezhong Qiu, Ziang Li,
+Chang-Yu Hsieh, and Jie Tang.
+Modeling protein us-
+ing large-scale pretrain language model. arXiv preprint
+arXiv:2108.07435, 2021.
+[Xie et al., 2022] Yaochen Xie, Zhao Xu, Jingtun Zhang,
+Zhengyang Wang, and Shuiwang Ji. Self-supervised learn-
+ing of graph neural networks: A unified review.
+IEEE
+Transactions on Pattern Analysis and Machine Intelli-
+gence, 2022.
+[Xu and Zhang, 2011] Dong Xu and Yang Zhang. Improv-
+ing the physical realism and structural accuracy of protein
+models by a two-step atomic-level energy minimization.
+Biophysical journal, 101(10):2525–2534, 2011.
+[Xu et al., 2018] Keyulu Xu, Weihua Hu, Jure Leskovec, and
+Stefanie Jegelka.
+How powerful are graph neural net-
+works? arXiv preprint arXiv:1810.00826, 2018.
+[Yang et al., 2022a] Kevin K Yang, Alex X Lu, and Nicolo K
+Fusi. Convolutions are competitive with transformers for
+protein sequence pretraining. bioRxiv, 2022.
+[Yang et al., 2022b] Kevin K Yang, Niccol`o Zanichelli, and
+Hugh Yeh. Masked inverse folding with sequence transfer
+for protein representation learning. bioRxiv, 2022.
+[You and Shen, 2022] Yuning You and Yang Shen. Cross-
+modality and self-supervised protein embedding for
+compound–protein affinity and contact prediction. Bioin-
+formatics, 38(Supplement 2):ii68–ii74, 2022.
+[Zhang and Zhang, 2010] Jian Zhang and Yang Zhang.
+A
+novel side-chain orientation dependent potential derived
+from random-walk reference state for protein fold selec-
+tion and structure prediction.
+PloS one, 5(10):e15386,
+2010.
+[Zhang et al., 2022] Zuobai Zhang, Minghao Xu, Arian Ja-
+masb, Vijil Chenthamarakshan, Aurelie Lozano, Payel
+Das, and Jian Tang.
+Protein representation learn-
+ing by geometric structure pretraining.
+arXiv preprint
+arXiv:2203.06125, 2022.
+[Zhou et al., 2020] Guangyu Zhou, Muhao Chen, Chelsea JT
+Ju, Zheng Wang, Jyun-Yu Jiang, and Wei Wang. Muta-
+tion effect estimation on protein–protein interactions us-
+
+ing deep contextualized representation learning. NAR ge-
+nomics and bioinformatics, 2(2):lqaa015, 2020.
+[Zhuang et al., 2020] Fuzhen Zhuang, Zhiyuan Qi, Keyu
+Duan, Dongbo Xi, Yongchun Zhu, Hengshu Zhu, Hui
+Xiong, and Qing He. A comprehensive survey on transfer
+learning. Proceedings of the IEEE, 109(1):43–76, 2020.
+

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+1
+Counterfactual Explanations for Land Cover
+Mapping in a Multi-class Setting
+Cassio F. Dantas, Diego Marcos, Dino Ienco
+Abstract—Counterfactual explanations are an emerging tool
+to enhance interpretability of deep learning models. Given a
+sample, these methods seek to find and display to the user similar
+samples across the decision boundary. In this paper, we propose a
+generative adversarial counterfactual approach for satellite image
+time series in a multi-class setting for the land cover classification
+task. One of the distinctive features of the proposed approach is
+the lack of prior assumption on the targeted class for a given
+counterfactual explanation. This inherent flexibility allows for the
+discovery of interesting information on the relationship between
+land cover classes. The other feature consists of encouraging the
+counterfactual to differ from the original sample only in a small
+and compact temporal segment. These time-contiguous perturba-
+tions allow for a much sparser and, thus, interpretable solution.
+Furthermore, plausibility/realism of the generated counterfactual
+explanations is enforced via the proposed adversarial learning
+strategy.
+I. INTRODUCTION
+Deep learning techniques have gained widespread popu-
+larity in the remote sensing field due to impressive results
+on a variety of tasks such as image super-resolution, image
+restoration, biophysical variables estimation and land cover
+classification from satellite image time series (SITS) data [1].
+Of particular importance, this last task provides useful knowl-
+edge to support many downstream geospatial analyses [2].
+Despite the high performances achieved by recent deep learn-
+ing frameworks on this task, they remain black-box models
+with limited understanding on their internal behavior. Due
+to this limitation, there is a growing need for improving the
+interpretability of deep learning models in remote sensing with
+the objective to raise up their acceptability and usefulness, as
+their decision-making processes are often not transparent [3]–
+[5]. Counterfactual explanation methods have recently received
+increasing attention as a means to provide some level of
+interpretability [6]–[8] to these black-box models. Counter-
+factual explanations aim to describe the behaviour of a model
+by providing minimal changes to the input data that would
+result in realistic samples that result in the model predicting
+a different class.
+For these perturbations to be more easily interpretable it is
+desirable that they are sparse and that they can be identified
+with some semantic element of the input data. In the case
+of time series, this would require to perturb a short and
+contiguous section of the timeline [9].
+Cassio F. Dantas and Dino Ienco are with UMR-TETIS laboratory, IN-
+RAE, University of Montpellier, France (email: [email protected];
+[email protected]).
+Diego Marcos is with Inria, University of Montpellier, France (email:
+[email protected])
+Related work: Most papers on counterfactual explana-
+tions focus on image data, while much fewer concentrate on
+time series [9]–[15]. To the best of our knowledge, this is the
+first paper focusing more specifically on counterfactuals for
+remote sensing time series data. While [9], [10] also generate
+time-contiguous perturbations, counterfactual plausibility is
+achieved by replacing an interval of the time series by a portion
+of another sample from the dataset [9] or shapelet motifs [10]
+(also used in [12]). In contrast, we use an adversarial approach
+to learn a counterfactual generator. In a multivariate setting,
+the approach in [11] replaces entire variables (not just a time
+section) with variables from another multivariate sample in
+the dataset. Related adversarial approaches are proposed in
+[13], [14], but time localization is not enforced. Finally, in
+many existing approaches only the binary classification case
+is considered [10], [14], [15], and when applied to the multi-
+class case, it usually requires explicitly picking a target class
+for every counterfactual explanation [11], [13]–[15].
+Contributions: Here, we propose a counterfactual genera-
+tion approach in a multi-class land cover classification setting
+for satellite image time series data. The proposed approach
+generates counterfactual explanations that are plausible (i.e.
+belong as much as possible to the data distribution) and close
+to the original data (modifying only a limited and contiguous
+set of time entries by a small amount). Finally, it is not
+necessary to pre-determine a target class for the generated
+counterfactual.
+Paper outline: In Section II we describe the considered
+study case with the associated remote sensing data. After
+detailing the proposed method in Section III, we present the
+experimental results in Section IV. Concluding remarks and
+future works are outlined in Section V.
+II. STUDY AREA
+The study site covers an area around the town of Koumbia,
+in the Province of Tuy, Hauts-Bassins region, in the south-
+west of Burkina Faso. This area has a surface of about 2338
+km2, and is situated in the sub-humid sudanian zone. The
+surface is covered mainly by natural savannah (herbaceous and
+shrubby) and forests, interleaved with a large portion of land
+(around 35%) used for rainfed agricultural production (mostly
+smallholder farming). The main crops are cereals (maize,
+sorghum and millet) and cotton, followed by oleaginous and
+leguminous crops. Several temporary watercourses constitute
+the hydrographic network around the city of Koumbia. Fig-
+ure 1 presents the study site with the reference data (ground
+truth) superposed on a Sentinel-2 image.
+arXiv:2301.01520v1  [cs.LG]  4 Jan 2023
+
+2
+Fig. 1: Location of the Koumbia study site. The corresponding
+ground truth is shown on the right.
+Fig. 2: Acquisition dates of the Sentinel-2 Satellite Image Time
+Series on the year 2020.
+Concerning the satellite data, we collected a time series
+of Sentinel-2 images spanning the year 2020 from January
+to December. All images were provided by the THEIA Pole
+platform1 at level-2A, which consist of atmospherically cor-
+rected surface reflectances (cf. MAJA processing chain [16])
+and relative cloud/shadow masks. A standard pre-processing
+was performed over each band to replace cloudy pixel values
+as detected by the available cloud masks based on the method
+proposed in [17]. Figure 2 depicts the acquisition dates of the
+Sentinel-2 satellite image time series. Finally, from the spectral
+raw bands at 10-m of spatial resolution the NDVI (Normalized
+Differential Vegetation Index) was derived.
+The GT (ground truth) data for the study site is a collection
+of (i) digitized plots from a GPS field mission performed in
+October 2020 and mostly covering classes within cropland and
+(ii) additional reference plots on non-crop classes obtained by
+photo-interpretation by an expert. Finally, the polygons have
+been rasterized at the S2 spatial resolution (10-m), resulting
+in 79961 labeled pixels. The statistics related to the GT are
+reported in Table I.
+Class
+Label
+Pixels
+1
+Cereals
+9 731
+2
+Cotton
+6 971
+3
+Oleaginous
+7 950
+4
+Grassland
+12 998
+5
+Shrubland
+22 546
+6
+Forest
+17 435
+7
+Bare Soil/Built-up
+1 125
+8
+Water
+1 205
+Total
+79 961
+TABLE I: Koumbia study site Ground Truth statistics.
+Classi er
+Real
+Counterfactual
+(frozen)
+Noiser
+Class A
+Class B
+Discriminator
+Fig. 3: Schematic representation of the proposed approach.
+III. PROPOSED METHOD
+A. Architecture overview
+For the counterfactual generation, we propose a GAN
+(generative adversarial network) inspired architecture which
+is summarized in Fig. 3.
+A counterfactual xCF is obtained for each input sample x
+by adding a perturbation δ to the original signal:
+xCF = x + δ
+(1)
+The perturbation δ is generated by a Noiser module which is
+learned with the goal to swap the prediction of the Classifier.
+Finally, a Discriminator module is leveraged to ensure the
+generation of realistic counterfactual examples.
+B. Networks implementation and training
+Regarding the different components on which our frame-
+work is built on, we get inspiration by state of the art
+literature in the field of satellite image time series land cover
+mapping. For the Classifier network we leverage the Temporal
+Convolutional Neural Network (TempCNN) model proposed
+in [18]. This architecture has an encoder based on several
+one-dimensional convolutional layers to explicitly cope with
+the temporal dimension of the time series data followed by
+two fully connected layers and a final output layer to provide
+the multi-class decision.
+For the Discriminator network we adopt the same archi-
+tecture as the Classifier network and we replace the output
+layer with a single neuron with sigmoid activation function
+as commonly done for discriminator networks in adversarial
+learning [19].
+Concerning the Noiser module, it is implemented as a multi-
+layer perceptron network with two hidden layers (each with
+128 neurons) and an output layer with the same dimensionality
+of the time series data. For each of the hidden layers, batch
+normalization, tangent activation function and a drop-out reg-
+ularization are employed in this order while for the output
+layer only the tangent activation function is used. The tangent
+activation function allows us to restrict the output domain
+between -1 and +1 thus, facilitating the learning process of
+the different networks.
+The Classifier model is pre-trained on the training set and,
+successively, frozen during the adversarial learning stage since
+this stage is devoted to learn the model weights associated to
+the Noiser and the Discriminator (see section III-D).
+1http://theia.cnes.fr
+
+Legend:
+000000
+Cereals
+Cotton
+Oleag./Legum
+Grassland
+Shrubland
+Forest
+B. Soil/Built-up
+WaterDD
+DD
+DDD
+B
+2020-01
+2020-03
+2020-05
+2020-07
+2020-09
+2020-11
+2021-013
+The Noiser module is updated with respect to a composite
+loss made of three parts detailed in sections III-C to III-E.
+Lnoiser = Lcl + λgenLgen + λw-ℓ1Lw-ℓ1
+(2)
+C. Class-swapping loss
+To generate counterfactuals that effectively change the pre-
+dicted class for a given input we use the following loss:
+Lcl = − 1
+n
+n
+�
+i=1
+y(i) log(1 − p(y(i)))
+(3)
+It enforces the reduction of the classifier’s softmax output for
+the original label y(i), here denoted p(y(i)), eventually leading
+to a change on the predicted class.
+Note that, conversely to standard literature [13], [15] in
+which a target class for the counterfactual example is chosen
+a priori, here we purposely do not enforce the prediction of
+a predefined target class. Instead, we let the Noiser free to
+generate a perturbation δ that will change the classifier output
+to any other class different from yi.
+D. GAN-based regularization for plausibility
+Counterfactual plausibility is enforced via a GAN-inspired
+architecture, where a discriminator is trained to identify unreal-
+istic counterfactuals while, simultaneously, the Noiser module
+acts as a generator with the goal to fool the discriminator in
+a two player game.
+The Discriminator is updated with respect to a standard
+GAN loss classifying real versus fake (counterfactual) sam-
+ples:
+Ldsc = − 1
+n
+n
+�
+i=1
+�
+log D(x(i)) + log
+�
+1 − D(x(i)
+CF)
+��
+(4)
+where D(x(i)) denotes the discriminator’s output for a real
+input x(i) (with expected output 1) and D(x(i)
+CF) its output for
+a fake input x(i)
+CF (with expected output 0).
+The following non-saturating generator loss is used in the
+Noiser update:
+Lgen = − 1
+n
+n
+�
+i=1
+log
+�
+D(x(i)
+CF)
+�
+(5)
+Lgen is minimized when the discriminator wrongly identifies
+the counterfactuals as real inputs.
+E. Unimodal regularization for time-contiguity
+To generate perturbations concentrated around a contiguous
+time frame we employ a weighted L1-norm penalization,
+with weights growing quadratically around a central time ˜t(i)
+chosen independently for each sample i ∈ {1, . . . , n}:
+Lw-ℓ1 = 1
+n
+n
+�
+i=1
+T
+�
+t=1
+d(t, ˜t(i))2|δ(i)
+t |
+(6)
+where, for the i-th sample, ˜t(i) is chosen as the time step with
+the highest absolute value perturbation ˜t(i) = argmaxt |δ(i)
+t |.
+To avoid biasing ˜t towards the center, we use the modulo
+distance d(t, ˜t) = min
+�
+(t − ˜t)%T, (˜t − t)%T
+�
+which treats
+the time samples as a circular list.
+This regularization also brings a degree of sparsity to the
+generated perturbation δ, since its entries will tend to vanish
+when getting far away from ˜t. Finally, penalizing the entries
+of δ enforces the proximity (similarity) between xCF and x.
+IV. RESULTS
+In this section we inspect the behaviour of the proposed
+method considering the study case introduced in Section II.
+More precisely, we first provide a general analysis of the class
+transitions induced by the counterfactual generation process.
+Secondly, we discuss per-class average perturbations generated
+by our framework as well as specific counterfactual examples.
+Then, we assess the plausibility of the generated counterfactual
+examples via anomaly detection strategies as suggested in [15].
+Finally, we perform an ablation analysis to assess the role of
+the different loss functions involved in the learning process of
+our framework.
+A. Experimental setup
+The Koumbia study case described in Section II was split
+into training, validation and test sets containing respectively
+50-17-33% of the 79961 samples. Each data sample cor-
+responds to a (univariate) NDVI time series with 24 time
+samples (cf. Fig. 2).
+First, the Classifier was trained over 1000 epochs with batch
+size 32 and Adam optimizer with learning rate 10−4 and
+weight decay of same value. The model weights corresponding
+to the best obtained F1-score on the validation set were kept.
+Then, with the classifier weights frozen, the Noiser and
+Discriminator modules are simultaneously trained over 100
+epochs with batch size 128 and Adam optimizer.
+Regularization parameters: we set λgen = 5 · 10−1 and
+λw-ℓ1 = 5 · 10−2 on the reported results. In practice, in-
+creasing these weights implies in further constraining the
+set of admissible perturbations which, in turn, leads to a
+smaller rate of successful counterfactual samples –i.e., those
+that actually change the classifier’s prediction (see details in
+section IV-E). The chosen values lead to a success rate of
+about 50%. Naturally, by further relaxing these constraints
+(reducing λgen and λw-ℓ1) would lead to higher success rates,
+but the generated counterfactual samples would be of lesser
+quality in terms of plausibility (due to λgen) as well as time
+localization and proximity (due to λw-ℓ1).
+B. Visualizing class relationships
+The class transitions induced by the counterfactual samples
+are summarized in Fig.
+4. The left (resp. right) graph was
+generated by feeding the obtained network with each of the
+training (resp. test) data samples. They present very similar
+behavior, which attests the fact that the proposed method
+generalizes well to previously unseen data. We recall that the
+class transitions are to no extent pre-defined on our approach;
+on the contrary, our method allows input samples from the
+
+4
+CEREALS
+COTTON
+OLEAGINOUS
+GRASSLAND
+SHRUBLAND
+FOREST
+B.
+W.
+CEREALS
+COTTON
+OLEAGINOUS
+GRASSLAND
+SHRUBLAND
+FOREST
+B.
+W.
+Fig. 4: Summary of class transitions induced by the counter-
+factuals. Training data (left) and test data (right), where B.
+stands for Bare Soil and W. for Water classes.
+Fig. 5: Examples of average counterfactual perturbations be-
+tween classes Cereals and Grassland on both ways. Shaded
+area corresponds to the standard deviation.
+same class to freely split-up into multiple target classes.
+Transitions obtained in such a way thus bring up valuable
+insights on the relation between classes.
+The obtained transitions are very much in line with the
+intuitive relation between the different classes. For instance,
+the three crop-related classes (Cereals, Cotton and Oleaginous)
+form a very coherent cluster, with almost all transitions staying
+within the sub-group. The vegetation classes Shrubland and
+Forest are most often sent to one another, while Grassland
+remains much closer to the crop classes (especially Oleagi-
+nous). The Bare Soil class is also most often transformed into
+Oleaginous. Finally, the Water class is very rarely modified
+by the counterfactual learning process, which is somewhat ex-
+pected due to its very distinct characteristic (NDVI signature)
+compared to the other classes.
+The ratio of successful class-swapping counterfactual sam-
+ples –i.e., those that actually change the classifier’s prediction–
+was 52.7% (17947 over 34066) for the training data and 43.8%
+(8765 over 20006) for the test data, considering only the
+samples that were correctly classified before counterfactuals.
+C. Counterfactual examples
+Examples of average perturbation profiles for two different
+class transitions are depicted in Fig 5.
+It is interesting to notice how the perturbations correspond
+roughly to the opposite of each other, which is quite suitable
+since they correspond to opposite transitions between the same
+two classes.
+Fig. 6: Examples of original time series with corresponding
+counterfactual from classes Shrubland (4) and Forest (5) on
+both ways.
+Two illustrative examples of counterfactual explanations are
+shown in Fig. 6. It is interesting to observe the similarity
+between the generated counterfactual and a real data example
+from the same class (on the neighboring plot).
+To transform a Shrubland sample into a Forest one, NDVI is
+added between the months of July and October. The opposite
+is done to obtain the reverse transition, which matches the
+general knowledge of such land cover classes on the consid-
+ered study area. Also note that the NDVI peak is slight shifted
+from one class to another.
+From the provided examples, one can verify that the ob-
+tained counterfactual do look realistic (this aspect is further
+evaluated in section IV-D) besides differing from the real
+signal only on a contiguous time window. These two properties
+have been explicitly enforced via the losses in eqs. (5) and (6).
+D. Plausibility analysis
+In this section, we quantify to what extent the proposed
+counterfactual explanations fit the original data distribution.
+To do so, we run an anomaly detection method, Isolation
+Forest [20], on both the original data and corresponding
+counterfactuals. To attest the importance of the proposed
+adversarial training for the generation of realistic/plausible
+counterfactuals, we perform an ablation study confronting
+the proposed model trained with and without the generator
+loss in Eq. (5). Fig. 7 shows contingency matrices relating
+the isolation forest outputs on the original data (rows) and
+on the corresponding counterfactual explanations (columns).
+Two counterfactual generation approaches are investigated: the
+proposed method (left matrix) and its non-adversarial variant
+(right matrix). In the figures, diagonal entries correspond
+to matching isolation forest outputs –i.e., same prediction
+(inlier/outlier) for both real and counterfactual data. Later,
+in Table II we compute some metrics on such contingency
+matrices to further quantify and summarize the behaviour of
+the compared methods. The proposed counterfactual model
+achieves impressive results, even leading to more samples
+identified as inliers than the real data itself (23806 against
+23755), since proposed approach converts less inliers into
+outliers (164) than the other way around (215).
+The non-adversarial variant, on the other hand, obtains
+considerably more degraded results, as it converts as many
+as 4338 real inlier samples into outliers (about 20 times
+more). Such a gap becomes evident when looking at the
+
+Cereals -→Grassland(876 CFs)
+0.3
+0.2
+0.1
+0.0
+0.1
+-0.20.9
+0.8
+0.7 -
+NDVI
+0.6
+0.5
+0.4
+Real (Shrubland)
+CF (Forest)
+0.3
+20-
+-03
+2020-
+2020-
+-09
+2020-
+-01
+2020-
+2020-
+2021Grassland-→Cereals(1394CFs)
+0.1
+0.0
+0.1-
+0.2
+心0.9 
+0.8 -
+0.7 -
+0.6
+0.5
+0.4 -
+Real (Forest)
+0.3
+CF (Shrubland)
+2020-
+2020-
+-09
+-03
+2020-
+2020-
+2020-
+2021-5
+Inlier
+Outlier
+Counterfactual
+Inlier
+Outlier
+Real
+99.3%
+(23591)
+0.7%
+(164)
+7.1%
+(215)
+92.9%
+(2820)
+Proposed model
+Inlier
+Outlier
+Counterfactual
+Inlier
+Outlier
+Real
+81.7%
+(19417)
+18.3%
+(4338)
+1.2%
+(35)
+98.8%
+(3000)
+Non-adversarial
+Fig. 7: Isolation forest results on real (rows) and counterfactual
+data (columns). Proposed model with (left) and without (right)
+adversarial loss during training. Row-normalized percentages.
+corresponding accuracy and normalized mutual information
+(NMI) computed w.r.t. the isolation forest results on the
+original data (cf. Table II). Such scores measure to what degree
+the inlier/outlier partitioning obtained on the counterfactual
+samples (for each of the two compared variants) matches the
+one obtained on the original data. The higher they are the
+better the two partitions match. The obtained results clearly
+show that counterfactual plausibility is achieved thanks to the
+adversarial training process.
+Method
+Accuracy
+NMI
+Inliers ratio
+Proposed
+98.6%
+0.808
+88.9%
+Non-adversarial
+83.7%
+0.337
+72.6%
+TABLE II: Plausibility analysis using different performance
+metrics. Isolation Forest results on the real data were used as
+ground truth for the accuracy and NMI scores.
+E. Other ablation studies
+In Table III we compare the number of successful class-
+swapping counterfactual samples as well as the average ℓ2
+and ℓ1 norms of the perturbations δ generated by the proposed
+model and two variants ignoring the generator loss (Lgen) and
+the weighted-ℓ1 loss (Lw-ℓ1), respectively.
+One can see that the removal of the auxiliary losses signif-
+icantly bumps the class-swapping rate, but it happens at the
+expense of either: 1) counterfactual plausibility, as shown in
+the Section IV-D for the removal of Lgen; 2) counterfactual
+proximity/similarity, as demonstrated by the dramatic increase
+on the norm of the generated perturbations (or, equivalently,
+the distance between x and xCF) upon removal of Lw-ℓ1.
+Method
+Class-swap CF
+Average ∥δ∥2
+Average ∥δ∥1
+Proposed
+43.8%
+0.24 ± 0.18
+0.76 ± 0.54
+Without Lgen
+83.7%
+0.97 ± 0.47
+1.69 ± 0.99
+Without Lw-ℓ1
+99.6%
+4.79 ± 0.07
+23.3 ± 0.53
+TABLE III: Ablation study on test data.
+V. CONCLUSION
+In this letter we have presented a new framework to generate
+counterfactual SITS samples of vegetation indices (i.e. NDVI)
+for the land cover classification task. The proposed method
+overcomes the restriction to apriori define the source and the
+target classes for the counterfactual generation process while
+it exploits adversarial learning to ensure realistic counterfac-
+tual samples. As possible future work, we would extend the
+framework to the case of multivariate time series satellite data
+as well as leverage the feedback provided by the generated
+counterfactual samples to improve the robustness of the land
+cover classifier regarding the most frequent class confusions.
+REFERENCES
+[1] Q. Yuan, H. Shen, T. Li, Z. Li, S. Li, Y. Jiang, H. Xu, W. Tan, Q. Yang,
+J. Wang, J. Gao, and L. Zhang, “Deep learning in environmental remote
+sensing: Achievements and challenges,” Remote Sensing of Environment,
+vol. 241, p. 111716, 2020.
+[2] J. Inglada, A. Vincent, M. Arias, B. Tardy, D. Morin, and I. Rodes,
+“Operational high resolution land cover map production at the country
+scale using satellite image time series,” Remote. Sens., vol. 9, no. 1,
+p. 95, 2017.
+[3] A. Adadi and M. Berrada, “Peeking inside the black-box: a survey on
+explainable artificial intelligence (XAI),” IEEE access, vol. 6, 2018.
+[4] R. Guidotti, A. Monreale, S. Ruggieri, F. Turini, F. Giannotti, and
+D. Pedreschi, “A survey of methods for explaining black box models,”
+ACM Comput. Surv., vol. 51, no. 5, Sep. 2019.
+[5] A. B. Arrieta, N. D´ıaz-Rodr´ıguez, J. Del Ser, A. Bennetot, S. Tabik,
+A. Barbado, S. Garc´ıa, S. Gil-L´opez, D. Molina, R. Benjamins et al.,
+“Explainable Artificial Intelligence (XAI): Concepts, taxonomies, op-
+portunities and challenges toward responsible AI,” Information fusion,
+vol. 58, pp. 82–115, 2020.
+[6] S. Wachter, B. Mittelstadt, and C. Russell, “Counterfactual explanations
+without opening the black box: Automated decisions and the gdpr,”
+Harv. JL & Tech., vol. 31, p. 841, 2017.
+[7] S. Verma, J. Dickerson, and K. Hines, “Counterfactual explanations for
+machine learning: A review,” arXiv preprint arXiv:2010.10596, 2020.
+[8] R. Guidotti, “Counterfactual explanations and how to find them: litera-
+ture review and benchmarking,” Data Mining and Knowledge Discovery,
+pp. 1–55, 2022.
+[9] E. Delaney, D. Greene, and M. T. Keane, “Instance-based counterfactual
+explanations for time series classification,” in International Conference
+on Case-Based Reasoning.
+Springer, 2021, pp. 32–47.
+[10] P. Li, S. F. Boubrahimi, and S. M. Hamd, “Motif-guided time series
+counterfactual explanations,” arXiv preprint arXiv:2211.04411, 2022.
+[11] E. Ates, B. Aksar, V. J. Leung, and A. K. Coskun, “Counterfactual
+explanations for multivariate time series,” in International Conference
+on Applied Artificial Intelligence (ICAPAI), 2021, pp. 1–8.
+[12] R. Guidotti, A. Monreale, F. Spinnato, D. Pedreschi, and F. Giannotti,
+“Explaining any time series classifier,” in IEEE International Conference
+on Cognitive Machine Intelligence (CogMI), 2020, pp. 167–176.
+[13] J. Lang, M. Giese, W. Ilg, and S. Otte, “Generating sparse coun-
+terfactual explanations for multivariate time series,” arXiv preprint
+arXiv:2206.00931, 2022.
+[14] A. Van Looveren, J. Klaise, G. Vacanti, and O. Cobb, “Conditional
+generative models for counterfactual explanations,” arXiv preprint
+arXiv:2101.10123, 2021.
+[15] S. Filali Boubrahimi and S. M. Hamdi, “On the mining of time series
+data counterfactual explanations using barycenters,” in ACM CIKM.
+ACM, 2022, p. 3943–3947.
+[16] O. Hagolle, M. Huc, D. Villa Pascual, and G. Dedieu, “A multi-temporal
+and multi-spectral method to estimate aerosol optical thickness over
+land, for the atmospheric correction of formosat-2, landsat, venµs and
+sentinel-2 images,” Rem. Sens., vol. 7, no. 3, pp. 2668–2691, 2015.
+[17] J. Inglada, A. Vincent, M. Arias, and B. Tardy, “iota2-a25386,” Jul.
+2016. [Online]. Available: https://doi.org/10.5281/zenodo.58150
+[18] C. Pelletier, G. I. Webb, and F. Petitjean, “Temporal convolutional neural
+network for the classification of satellite image time series,” Remote.
+Sens., vol. 11, no. 5, p. 523, 2019.
+[19] A. Creswell, T. White, V. Dumoulin, K. Arulkumaran, B. Sengupta, and
+A. A. Bharath, “Generative adversarial networks: An overview,” IEEE
+Signal Process. Mag., vol. 35, no. 1, pp. 53–65, 2018.
+[20] O. Li, H. Liu, C. Chen, and C. Rudin, “Deep learning for case-
+based reasoning through prototypes: A neural network that explains its
+predictions,” AAAI Conference on Artificial Intelligence, vol. 32, no. 1,
+Apr. 2018.
+

ItAzT4oBgHgl3EQfjv0x/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf,len=434
+page_content='1  Counterfactual Explanations for Land Cover  Mapping in a Multi-class Setting  Cassio F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Dantas, Diego Marcos, Dino Ienco  Abstract—Counterfactual explanations are an emerging tool  to enhance interpretability of deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Given a  sample, these methods seek to find and display to the user similar  samples across the decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' In this paper, we propose a  generative adversarial counterfactual approach for satellite image  time series in a multi-class setting for the land cover classification  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' One of the distinctive features of the proposed approach is  the lack of prior assumption on the targeted class for a given  counterfactual explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' This inherent flexibility allows for the  discovery of interesting information on the relationship between  land cover classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The other feature consists of encouraging the  counterfactual to differ from the original sample only in a small  and compact temporal segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' These time-contiguous perturba-  tions allow for a much sparser and, thus, interpretable solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Furthermore, plausibility/realism of the generated counterfactual  explanations is enforced via the proposed adversarial learning  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' INTRODUCTION  Deep learning techniques have gained widespread popu-  larity in the remote sensing field due to impressive results  on a variety of tasks such as image super-resolution, image  restoration, biophysical variables estimation and land cover  classification from satellite image time series (SITS) data [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Of particular importance, this last task provides useful knowl-  edge to support many downstream geospatial analyses [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Despite the high performances achieved by recent deep learn-  ing frameworks on this task, they remain black-box models  with limited understanding on their internal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Due  to this limitation, there is a growing need for improving the  interpretability of deep learning models in remote sensing with  the objective to raise up their acceptability and usefulness, as  their decision-making processes are often not transparent [3]–  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Counterfactual explanation methods have recently received  increasing attention as a means to provide some level of  interpretability [6]–[8] to these black-box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Counter-  factual explanations aim to describe the behaviour of a model  by providing minimal changes to the input data that would  result in realistic samples that result in the model predicting  a different class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  For these perturbations to be more easily interpretable it is  desirable that they are sparse and that they can be identified  with some semantic element of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' In the case  of time series, this would require to perturb a short and  contiguous section of the timeline [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Cassio F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Dantas and Dino Ienco are with UMR-TETIS laboratory, IN-  RAE, University of Montpellier, France (email: cassio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='fraga-dantas@inrae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='fr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  dino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='ienco@inrae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Diego Marcos is with Inria, University of Montpellier, France (email:  diego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='marcos@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='fr)  Related work: Most papers on counterfactual explana-  tions focus on image data, while much fewer concentrate on  time series [9]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' To the best of our knowledge, this is the  first paper focusing more specifically on counterfactuals for  remote sensing time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' While [9], [10] also generate  time-contiguous perturbations, counterfactual plausibility is  achieved by replacing an interval of the time series by a portion  of another sample from the dataset [9] or shapelet motifs [10]  (also used in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' In contrast, we use an adversarial approach  to learn a counterfactual generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' In a multivariate setting,  the approach in [11] replaces entire variables (not just a time  section) with variables from another multivariate sample in  the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Related adversarial approaches are proposed in  [13], [14], but time localization is not enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Finally, in  many existing approaches only the binary classification case  is considered [10], [14], [15], and when applied to the multi-  class case, it usually requires explicitly picking a target class  for every counterfactual explanation [11], [13]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Contributions: Here, we propose a counterfactual genera-  tion approach in a multi-class land cover classification setting  for satellite image time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The proposed approach  generates counterfactual explanations that are plausible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  belong as much as possible to the data distribution) and close  to the original data (modifying only a limited and contiguous  set of time entries by a small amount).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Finally, it is not  necessary to pre-determine a target class for the generated  counterfactual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Paper outline: In Section II we describe the considered  study case with the associated remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' After  detailing the proposed method in Section III, we present the  experimental results in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Concluding remarks and  future works are outlined in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' STUDY AREA  The study site covers an area around the town of Koumbia,  in the Province of Tuy, Hauts-Bassins region, in the south-  west of Burkina Faso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' This area has a surface of about 2338  km2, and is situated in the sub-humid sudanian zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The  surface is covered mainly by natural savannah (herbaceous and  shrubby) and forests, interleaved with a large portion of land  (around 35%) used for rainfed agricultural production (mostly  smallholder farming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The main crops are cereals (maize,  sorghum and millet) and cotton, followed by oleaginous and  leguminous crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Several temporary watercourses constitute  the hydrographic network around the city of Koumbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Fig-  ure 1 presents the study site with the reference data (ground  truth) superposed on a Sentinel-2 image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='01520v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='LG]  4 Jan 2023  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 1: Location of the Koumbia study site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The corresponding  ground truth is shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 2: Acquisition dates of the Sentinel-2 Satellite Image Time  Series on the year 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Concerning the satellite data, we collected a time series  of Sentinel-2 images spanning the year 2020 from January  to December.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' All images were provided by the THEIA Pole  platform1 at level-2A, which consist of atmospherically cor-  rected surface reflectances (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' MAJA processing chain [16])  and relative cloud/shadow masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' A standard pre-processing  was performed over each band to replace cloudy pixel values  as detected by the available cloud masks based on the method  proposed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Figure 2 depicts the acquisition dates of the  Sentinel-2 satellite image time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Finally, from the spectral  raw bands at 10-m of spatial resolution the NDVI (Normalized  Differential Vegetation Index) was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  The GT (ground truth) data for the study site is a collection  of (i) digitized plots from a GPS field mission performed in  October 2020 and mostly covering classes within cropland and  (ii) additional reference plots on non-crop classes obtained by  photo-interpretation by an expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Finally, the polygons have  been rasterized at the S2 spatial resolution (10-m), resulting  in 79961 labeled pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The statistics related to the GT are  reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Class  Label  Pixels  1  Cereals  9 731  2  Cotton  6 971  3  Oleaginous  7 950  4  Grassland  12 998  5  Shrubland  22 546  6  Forest  17 435  7  Bare Soil/Built-up  1 125  8  Water  1 205  Total  79 961  TABLE I: Koumbia study site Ground Truth statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Classi er  Real  Counterfactual  (frozen)  Noiser  Class A  Class B  Discriminator  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 3: Schematic representation of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' PROPOSED METHOD  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Architecture overview  For the counterfactual generation, we propose a GAN  (generative adversarial network) inspired architecture which  is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  A counterfactual xCF is obtained for each input sample x  by adding a perturbation δ to the original signal:  xCF = x + δ  (1)  The perturbation δ is generated by a Noiser module which is  learned with the goal to swap the prediction of the Classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Finally, a Discriminator module is leveraged to ensure the  generation of realistic counterfactual examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Networks implementation and training  Regarding the different components on which our frame-  work is built on, we get inspiration by state of the art  literature in the field of satellite image time series land cover  mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' For the Classifier network we leverage the Temporal  Convolutional Neural Network (TempCNN) model proposed  in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' This architecture has an encoder based on several  one-dimensional convolutional layers to explicitly cope with  the temporal dimension of the time series data followed by  two fully connected layers and a final output layer to provide  the multi-class decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  For the Discriminator network we adopt the same archi-  tecture as the Classifier network and we replace the output  layer with a single neuron with sigmoid activation function  as commonly done for discriminator networks in adversarial  learning [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Concerning the Noiser module, it is implemented as a multi-  layer perceptron network with two hidden layers (each with  128 neurons) and an output layer with the same dimensionality  of the time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' For each of the hidden layers, batch  normalization, tangent activation function and a drop-out reg-  ularization are employed in this order while for the output  layer only the tangent activation function is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The tangent  activation function allows us to restrict the output domain  between -1 and +1 thus, facilitating the learning process of  the different networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  The Classifier model is pre-trained on the training set and,  successively, frozen during the adversarial learning stage since  this stage is devoted to learn the model weights associated to  the Noiser and the Discriminator (see section III-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  1http://theia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='cnes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='fr  Legend:  000000  Cereals  Cotton  Oleag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='/Legum  Grassland  Shrubland  Forest  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Soil/Built-up  WaterDD  DD  DDD  B  2020-01  2020-03  2020-05  2020-07  2020-09  2020-11  2021-013  The Noiser module is updated with respect to a composite  loss made of three parts detailed in sections III-C to III-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Lnoiser = Lcl + λgenLgen + λw-ℓ1Lw-ℓ1  (2)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Class-swapping loss  To generate counterfactuals that effectively change the pre-  dicted class for a given input we use the following loss:  Lcl = − 1  n  n  �  i=1  y(i) log(1 − p(y(i)))  (3)  It enforces the reduction of the classifier’s softmax output for  the original label y(i), here denoted p(y(i)), eventually leading  to a change on the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Note that, conversely to standard literature [13], [15] in  which a target class for the counterfactual example is chosen  a priori, here we purposely do not enforce the prediction of  a predefined target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Instead, we let the Noiser free to  generate a perturbation δ that will change the classifier output  to any other class different from yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' GAN-based regularization for plausibility  Counterfactual plausibility is enforced via a GAN-inspired  architecture, where a discriminator is trained to identify unreal-  istic counterfactuals while, simultaneously, the Noiser module  acts as a generator with the goal to fool the discriminator in  a two player game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  The Discriminator is updated with respect to a standard  GAN loss classifying real versus fake (counterfactual) sam-  ples:  Ldsc = − 1  n  n  �  i=1  �  log D(x(i)) + log  �  1 − D(x(i)  CF)  ��  (4)  where D(x(i)) denotes the discriminator’s output for a real  input x(i) (with expected output 1) and D(x(i)  CF) its output for  a fake input x(i)  CF (with expected output 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  The following non-saturating generator loss is used in the  Noiser update:  Lgen = − 1  n  n  �  i=1  log  �  D(x(i)  CF)  �  (5)  Lgen is minimized when the discriminator wrongly identifies  the counterfactuals as real inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Unimodal regularization for time-contiguity  To generate perturbations concentrated around a contiguous  time frame we employ a weighted L1-norm penalization,  with weights growing quadratically around a central time ˜t(i)  chosen independently for each sample i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' , n}:  Lw-ℓ1 = 1  n  n  �  i=1  T  �  t=1  d(t, ˜t(i))2|δ(i)  t |  (6)  where, for the i-th sample, ˜t(i) is chosen as the time step with  the highest absolute value perturbation ˜t(i) = argmaxt |δ(i)  t |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  To avoid biasing ˜t towards the center, we use the modulo  distance d(t, ˜t) = min  �  (t − ˜t)%T, (˜t − t)%T  �  which treats  the time samples as a circular list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  This regularization also brings a degree of sparsity to the  generated perturbation δ, since its entries will tend to vanish  when getting far away from ˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Finally, penalizing the entries  of δ enforces the proximity (similarity) between xCF and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' RESULTS  In this section we inspect the behaviour of the proposed  method considering the study case introduced in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  More precisely, we first provide a general analysis of the class  transitions induced by the counterfactual generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Secondly, we discuss per-class average perturbations generated  by our framework as well as specific counterfactual examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Then, we assess the plausibility of the generated counterfactual  examples via anomaly detection strategies as suggested in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Finally, we perform an ablation analysis to assess the role of  the different loss functions involved in the learning process of  our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Experimental setup  The Koumbia study case described in Section II was split  into training, validation and test sets containing respectively  50-17-33% of the 79961 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Each data sample cor-  responds to a (univariate) NDVI time series with 24 time  samples (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  First, the Classifier was trained over 1000 epochs with batch  size 32 and Adam optimizer with learning rate 10−4 and  weight decay of same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The model weights corresponding  to the best obtained F1-score on the validation set were kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Then, with the classifier weights frozen, the Noiser and  Discriminator modules are simultaneously trained over 100  epochs with batch size 128 and Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Regularization parameters: we set λgen = 5 · 10−1 and  λw-ℓ1 = 5 · 10−2 on the reported results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' In practice, in-  creasing these weights implies in further constraining the  set of admissible perturbations which, in turn, leads to a  smaller rate of successful counterfactual samples –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=', those  that actually change the classifier’s prediction (see details in  section IV-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The chosen values lead to a success rate of  about 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Naturally, by further relaxing these constraints  (reducing λgen and λw-ℓ1) would lead to higher success rates,  but the generated counterfactual samples would be of lesser  quality in terms of plausibility (due to λgen) as well as time  localization and proximity (due to λw-ℓ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Visualizing class relationships  The class transitions induced by the counterfactual samples  are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' right) graph was  generated by feeding the obtained network with each of the  training (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' test) data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' They present very similar  behavior, which attests the fact that the proposed method  generalizes well to previously unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' We recall that the  class transitions are to no extent pre-defined on our approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  on the contrary, our method allows input samples from the  4  CEREALS  COTTON  OLEAGINOUS  GRASSLAND  SHRUBLAND  FOREST  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  CEREALS  COTTON  OLEAGINOUS  GRASSLAND  SHRUBLAND  FOREST  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 4: Summary of class transitions induced by the counter-  factuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Training data (left) and test data (right), where B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  stands for Bare Soil and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' for Water classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 5: Examples of average counterfactual perturbations be-  tween classes Cereals and Grassland on both ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Shaded  area corresponds to the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  same class to freely split-up into multiple target classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Transitions obtained in such a way thus bring up valuable  insights on the relation between classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  The obtained transitions are very much in line with the  intuitive relation between the different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' For instance,  the three crop-related classes (Cereals, Cotton and Oleaginous)  form a very coherent cluster, with almost all transitions staying  within the sub-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The vegetation classes Shrubland and  Forest are most often sent to one another, while Grassland  remains much closer to the crop classes (especially Oleagi-  nous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The Bare Soil class is also most often transformed into  Oleaginous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Finally, the Water class is very rarely modified  by the counterfactual learning process, which is somewhat ex-  pected due to its very distinct characteristic (NDVI signature)  compared to the other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  The ratio of successful class-swapping counterfactual sam-  ples –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=', those that actually change the classifier’s prediction–  was 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='7% (17947 over 34066) for the training data and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='8%  (8765 over 20006) for the test data, considering only the  samples that were correctly classified before counterfactuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Counterfactual examples  Examples of average perturbation profiles for two different  class transitions are depicted in Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  It is interesting to notice how the perturbations correspond  roughly to the opposite of each other, which is quite suitable  since they correspond to opposite transitions between the same  two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 6: Examples of original time series with corresponding  counterfactual from classes Shrubland (4) and Forest (5) on  both ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Two illustrative examples of counterfactual explanations are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' It is interesting to observe the similarity  between the generated counterfactual and a real data example  from the same class (on the neighboring plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  To transform a Shrubland sample into a Forest one, NDVI is  added between the months of July and October.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The opposite  is done to obtain the reverse transition, which matches the  general knowledge of such land cover classes on the consid-  ered study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Also note that the NDVI peak is slight shifted  from one class to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  From the provided examples, one can verify that the ob-  tained counterfactual do look realistic (this aspect is further  evaluated in section IV-D) besides differing from the real  signal only on a contiguous time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' These two properties  have been explicitly enforced via the losses in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Plausibility analysis  In this section, we quantify to what extent the proposed  counterfactual explanations fit the original data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  To do so, we run an anomaly detection method, Isolation  Forest [20], on both the original data and corresponding  counterfactuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' To attest the importance of the proposed  adversarial training for the generation of realistic/plausible  counterfactuals, we perform an ablation study confronting  the proposed model trained with and without the generator  loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 7 shows contingency matrices relating  the isolation forest outputs on the original data (rows) and  on the corresponding counterfactual explanations (columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Two counterfactual generation approaches are investigated: the  proposed method (left matrix) and its non-adversarial variant  (right matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' In the figures, diagonal entries correspond  to matching isolation forest outputs –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=', same prediction  (inlier/outlier) for both real and counterfactual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Later,  in Table II we compute some metrics on such contingency  matrices to further quantify and summarize the behaviour of  the compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The proposed counterfactual model  achieves impressive results, even leading to more samples  identified as inliers than the real data itself (23806 against  23755), since proposed approach converts less inliers into  outliers (164) than the other way around (215).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  The non-adversarial variant, on the other hand, obtains  considerably more degraded results, as it converts as many  as 4338 real inlier samples into outliers (about 20 times  more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Such a gap becomes evident when looking at the  Cereals -→Grassland(876 CFs)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='7 -  NDVI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='4  Real (Shrubland)  CF (Forest)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='3  20-  03  2020-  2020-  09  2020-  01  2020-  2020-  2021Grassland-→Cereals(1394CFs)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='1-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='2  心0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='8 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='7 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='4 -  Real (Forest)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='3  CF (Shrubland)  2020-  2020-  09  03  2020-  2020-  2020-  2021-5  Inlier  Outlier  Counterfactual  Inlier  Outlier  Real  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='3%  (23591)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='7%  (164)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='1%  (215)  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='9%  (2820)  Proposed model  Inlier  Outlier  Counterfactual  Inlier  Outlier  Real  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='7%  (19417)  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='3%  (4338)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='2%  (35)  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='8%  (3000)  Non-adversarial  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 7: Isolation forest results on real (rows) and counterfactual  data (columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Proposed model with (left) and without (right)  adversarial loss during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Row-normalized percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  corresponding accuracy and normalized mutual information  (NMI) computed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' the isolation forest results on the  original data (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Such scores measure to what degree  the inlier/outlier partitioning obtained on the counterfactual  samples (for each of the two compared variants) matches the  one obtained on the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The higher they are the  better the two partitions match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The obtained results clearly  show that counterfactual plausibility is achieved thanks to the  adversarial training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Method  Accuracy  NMI  Inliers ratio  Proposed  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='6%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='808  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='9%  Non-adversarial  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='7%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='337  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='6%  TABLE II: Plausibility analysis using different performance  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Isolation Forest results on the real data were used as  ground truth for the accuracy and NMI scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Other ablation studies  In Table III we compare the number of successful class-  swapping counterfactual samples as well as the average ℓ2  and ℓ1 norms of the perturbations δ generated by the proposed  model and two variants ignoring the generator loss (Lgen) and  the weighted-ℓ1 loss (Lw-ℓ1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  One can see that the removal of the auxiliary losses signif-  icantly bumps the class-swapping rate, but it happens at the  expense of either: 1) counterfactual plausibility, as shown in  the Section IV-D for the removal of Lgen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 2) counterfactual  proximity/similarity, as demonstrated by the dramatic increase  on the norm of the generated perturbations (or, equivalently,  the distance between x and xCF) upon removal of Lw-ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Method  Class-swap CF  Average ∥δ∥2  Average ∥δ∥1  Proposed  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='8%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='54  Without Lgen  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='7%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='47  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='99  Without Lw-ℓ1  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='6%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='07  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='53  TABLE III: Ablation study on test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' CONCLUSION  In this letter we have presented a new framework to generate  counterfactual SITS samples of vegetation indices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' NDVI)  for the land cover classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' The proposed method  overcomes the restriction to apriori define the source and the  target classes for the counterfactual generation process while  it exploits adversarial learning to ensure realistic counterfac-  tual samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' As possible future work, we would extend the  framework to the case of multivariate time series satellite data  as well as leverage the feedback provided by the generated  counterfactual samples to improve the robustness of the land  cover classifier regarding the most frequent class confusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  REFERENCES  [1] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Yuan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Shen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Tan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Yang,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Gao, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Zhang, “Deep learning in environmental remote  sensing: Achievements and challenges,” Remote Sensing of Environment,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 241, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 111716, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Inglada, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Vincent, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Arias, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Tardy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Morin, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Rodes,  “Operational high resolution land cover map production at the country  scale using satellite image time series,” Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 1,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 95, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Adadi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Berrada, “Peeking inside the black-box: a survey on  explainable artificial intelligence (XAI),” IEEE access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 6, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Guidotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Monreale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Ruggieri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Turini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Giannotti, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Pedreschi, “A survey of methods for explaining black box models,”  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 5, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Arrieta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' D´ıaz-Rodr´ıguez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Del Ser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Bennetot, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Tabik,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Barbado, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Garc´ıa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Gil-L´opez, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Molina, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Benjamins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=',  “Explainable Artificial Intelligence (XAI): Concepts, taxonomies, op-  portunities and challenges toward responsible AI,” Information fusion,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 58, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 82–115, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Wachter, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Mittelstadt, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Russell, “Counterfactual explanations  without opening the black box: Automated decisions and the gdpr,”  Harv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' JL & Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 31, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 841, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Verma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Dickerson, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Hines, “Counterfactual explanations for  machine learning: A review,” arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='10596, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Guidotti, “Counterfactual explanations and how to find them: litera-  ture review and benchmarking,” Data Mining and Knowledge Discovery,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 1–55, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [9] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Delaney, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Greene, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Keane, “Instance-based counterfactual  explanations for time series classification,” in International Conference  on Case-Based Reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Springer, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 32–47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Boubrahimi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Hamd, “Motif-guided time series  counterfactual explanations,” arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='04411, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Ates, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Aksar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Leung, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Coskun, “Counterfactual  explanations for multivariate time series,” in International Conference  on Applied Artificial Intelligence (ICAPAI), 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Guidotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Monreale, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Spinnato, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Pedreschi, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Giannotti,  “Explaining any time series classifier,” in IEEE International Conference  on Cognitive Machine Intelligence (CogMI), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 167–176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Lang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Giese, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Ilg, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Otte, “Generating sparse coun-  terfactual explanations for multivariate time series,” arXiv preprint  arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='00931, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Van Looveren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Klaise, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Vacanti, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Cobb, “Conditional  generative models for counterfactual explanations,” arXiv preprint  arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='10123, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Filali Boubrahimi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Hamdi, “On the mining of time series  data counterfactual explanations using barycenters,” in ACM CIKM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  ACM, 2022, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 3943–3947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [16] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Hagolle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Huc, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Villa Pascual, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Dedieu, “A multi-temporal  and multi-spectral method to estimate aerosol optical thickness over  land, for the atmospheric correction of formosat-2, landsat, venµs and  sentinel-2 images,” Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 2668–2691, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Inglada, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Vincent, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Arias, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Tardy, “iota2-a25386,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='58150  [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Pelletier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Webb, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Petitjean, “Temporal convolutional neural  network for the classification of satellite image time series,” Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 523, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Creswell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' White, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Dumoulin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Arulkumaran, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Sengupta, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Bharath, “Generative adversarial networks: An overview,” IEEE  Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 53–65, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content='  [20] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Chen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' Rudin, “Deep learning for case-  based reasoning through prototypes: A neural network that explains its  predictions,” AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 1,  Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'}

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+1
+Modeling Sequential Recommendation as
+Missing Information Imputation
+Yujie Lin, Zhumin Chen, Zhaochun Ren, Chenyang Wang, Qiang Yan, Maarten de Rijke, Xiuzhen Cheng,
+Fellow, IEEE, and Pengjie Ren
+Abstract—Side information is being used extensively to improve the effectiveness of sequential recommendation models. It is said to
+help capture the transition patterns among items. Most previous work on sequential recommendation that uses side information models
+item IDs and side information separately. This can only model part of relations between items and their side information. Moreover, in
+real-world systems, not all values of item feature fields are available. This hurts the performance of models that rely on side
+information. Existing methods tend to neglect the context of missing item feature fields, and fill them with generic or special values,
+e.g., unknown, which might lead to sub-optimal performance. To address the limitation of sequential recommenders with side
+information, we define a way to fuse side information and alleviate the problem of missing side information by proposing a unified task,
+namely the missing information imputation (MII), which randomly masks some feature fields in a given sequence of items, including
+item IDs, and then forces a predictive model to recover them. By considering the next item as a missing feature field, sequential
+recommendation can be formulated as a special case of MII. We propose a sequential recommendation model, called missing
+information imputation recommender (MIIR), that builds on the idea of MII and simultaneously imputes missing item feature values and
+predicts the next item. We devise a dense fusion self-attention (DFSA) for MIIR to capture all pairwise relations between items and
+their side information. Empirical studies on three benchmark datasets demonstrate that MIIR, supervised by MII, achieves a
+significantly better sequential recommendation performance than state-of-the-art baselines.
+Index Terms—Sequential recommendation, side information fusion, missing information imputation
+!
+1
+INTRODUCTION
+S
+EQUENTIAL recommendation models transition patterns
+among items and generates a recommendation for the
+next item [1]. Traditional sequential recommendation so-
+lutions use the item ID as the only item feature field [2,
+3, 4, 5, 6, 7, 8]. In real-world cases, however, there is
+rich side information in the form of multiple types of
+structural feature fields, such as categories and brands,
+and unstructured feature fields, e.g., titles and descriptions,
+that can help to better model transitions between items.
+In recent years, several publications have exploited side
+information to improve sequential recommendation perfor-
+mance [9, 10, 11, 12, 13, 14, 15]. Most focus on designing
+different mechanisms to fuse side information into rec-
+ommendation models. For example, Hidasi et al. [9] use
+parallel recurrent neural networks (RNNs) [16] to encode
+•
+Yujie Lin, School of Computer Science and Technology, Shandong Univer-
+sity, Qingdao, China, E-mail: [email protected]
+•
+Zhumin Chen, School of Computer Science and Technology, Shandong
+University, Qingdao, China, E-mail: [email protected]
+•
+Zhaochun Ren, School of Computer Science and Technology, Shandong
+University, Qingdao, China, E-mail: [email protected]
+•
+Chenyang Wang, School of Computer Science and Technology, Shandong
+University, Qingdao, China, E-mail: [email protected]
+•
+Qiang
+Yan,
+WeChat,
+Tencent,
+Guangzhou,
+China,
+E-mail:
+[email protected]
+•
+Maarten de Rijke, Informatics Institute, University of Amsterdam, Ams-
+terdam, The Netherlands, E-mail: [email protected]
+•
+Xiuzhen Cheng, School of Computer Science and Technology, Shandong
+University, Qingdao, China, E-mail: [email protected]
+•
+Pengjie Ren, School of Computer Science and Technology, Shandong
+University, Qingdao, China, E-mail: [email protected]
+the information in item IDs and attributes, respectively, and
+then combine the outputs of RNNs for item recommenda-
+tion. Zhang et al. [10] employ two groups of self-attention
+blocks [17] for modeling items and features, and fuse them
+in the final stage.
+Importantly, previous work for sequential recommenda-
+tion with side information usually regards side information
+as an auxiliary representation of the item, so models item
+IDs and side information separately. As a result, such meth-
+ods only encode partial relations in item sequences, e.g., the
+relation between an item and its side information, while the
+relation between an item and the side information of other
+items in the sequence is not well captured.
+Even more importantly, previous studies often assume
+that all side information is available, which is rarely the
+case in real-world scenarios. As illustrated in Fig. 1(a),
+i.e., the second and third items lack category and title
+information, respectively. Previous work has proposed to
+fill such gaps with special values, such as a general category
+and a padding text, to make models trainable and produce
+outputs. However, for different items and item sequences,
+these special values are the same: they do not provide useful
+and specific information for recommendations and might
+introduce biases into the model learning instead [18]. As a
+result, as illustrated in Fig. 1(b), a model might recommend
+the wrong item. Instead, we propose to impute the missing
+side information, so that the recommendation model can use
+information from missing feature fields based on contexts,
+as illustrated in Fig. 1(c).
+Some recent studies address the probem of missing
+side information in recommendation data. Wang et al. [19]
+arXiv:2301.01762v1  [cs.IR]  4 Jan 2023
+
+2
+(a) Original sequence.
+(b) Existing work without imputation.
+(c) Our work with imputation.
+Fig. 1. Sequential recommendation of items with side information. Gray
+blocks represent missing information. “[PAD]” (in (b)) indicates padding
+with generic or special values as often done in existing work. “[Impute]”
+(in (c)) indicates imputation with actual values for missing feature fields.
+employ an auto-encoder (AE) with a modality dropout
+to recover the missing rating and side information. Shi
+et al. [18] propose an adaptive feature sampling strategy
+to introduce more missing feature fields into the training
+process, which increases the robustness of the recommen-
+dation model against missing side information. Wu et al.
+[20] define item recommendation and attribute inference in
+a user-item bipartite graph with attributes, and propose a
+graph convolutional network (GCN) [21] based model to
+join these two tasks. However, the work just listed mainly
+targets non-sequential recommendation. Moreover, it treats
+item recommendation and side information imputation as
+different tasks.
+In this work, we seek to design a sequential recommen-
+dation model that can handle missing feature fields of items
+in items sequences. The main challenge is how to adap-
+tively impute missing information, including missing side
+information and the next item, according to the information
+available in the item sequence. First, we propose a task,
+the missing information imputation (MII) task that randomly
+masks some non-missing feature fields, including item IDs,
+in the input sequence, and then asks the model to recover
+them in the output. Since the next item to be recommended
+can also be seen as a missing feature field in the sequence,
+MII unifies the missing side information imputation task
+with the next item prediction task. MII can be considered
+as the extension of the masked item prediction task [22]
+that only considers and masks item IDs. Based on the
+MII task, we propose a sequential recommendation model,
+called missing information imputation recommender (MIIR),
+that jointly imputes missing side information and predicts
+the next item for the given item sequence. MIIR employs
+a dense fusion self-attention (DFSA) mechanism to fuse the
+information in IDs and other feature fields for predicting
+both missing side information and the next item. DFSA
+captures the relation between any pair of feature fields in
+the input sequence, allowing it to fully fuse various types
+of (side) information to impute missing feature values and
+address the main recommendation challenge.
+We conduct extensive experiments on three public
+datasets and show that MIIR significantly outperforms
+state-of-the-art sequential recommendation baselines. We
+also confirm that (i) imputing missing side information
+and (ii) DFSA both help to improve the performance of
+sequential recommendation.
+The main contributions of this work are as follows:
+• We propose to unify the missing side information imputa-
+tion task and the sequential recommendation task through
+missing information imputation (MII). To the best of our
+knowledge, this is the first work of its kind in sequential
+recommendation.
+• We present a novel sequential recommendation model,
+missing information imputation recommender (MIIR),
+that employs MII to provide the signal for simultaneously
+imputing the missing item side information and predict-
+ing the next item and dense fusion self-attention (DFSA)
+to fuse various information.
+• We conduct extensive experiments to verify the effective-
+ness of MII, MIIR, and DFSA in sequential recommenda-
+tion.
+2
+RELATED WORK
+2.1
+Sequential recommendation with side information
+Side information fusion has been widely used in sequential
+recommendation because it can help to capture transition
+patterns among items. We classify existing work into work
+that uses self-attention and work that does not.
+As to work that does not use self-attention, Hidasi et al.
+[9] employ parallel RNNs to extract the information from
+ID sequences of item IDs and sequences of features; they
+then examine different ways of combining the outputs of
+the RNNs. Zhou et al. [23] propose self-supervised tasks
+to maximize the mutual information between an item and
+its attributes or between a sequence of item IDs and the
+sequence of their attributes. Yuan et al. [24] construct a het-
+erogeneous graph to aggregate different types of categorical
+attributes, then aggregate the representations of attribute
+types to get item representations.
+Inspired by the success of self-attention mechanisms
+[25, 26, 27], some work uses self-attention to fuse items and
+side information. Zhang et al. [10] first use a vanilla atten-
+tion mechanism to fuse different types of side information
+on each item, and then use two branches of self-attention
+blocks to model transition patterns between IDs and side
+information; they then concatenate the hidden states of the
+two blocks for item recommendation. Liu et al. [28] pro-
+pose a non-invasive self-attention mechanism that uses pure
+item ID representations as values and representations that
+integrate side information as queries and keys to calculate
+the attention. Xie et al. [15] decouple the non-invasive self-
+attention of different types of side information to get fused
+attention matrices for items.
+Although many methods have been proposed for se-
+quential recommendation with side information, they (i) ne-
+glect the missing information problem, and use fixed special
+values to fill missing feature fields, which might harm the
+performance, and (ii) hardly explore the relation between
+an item and the side information of other items in the same
+sequence. These are aspects that we contribute on top of
+prior work.
+
+Title
+Title
+Truth
+Title
+Item
+Item
+Item
+Item
+Category
+Category
+CategoryTitle
+Title
+[PAD]
+Title
+Predict
+Item
+Item
+Item
+Item
+Category
+[PAD]
+Category
+CategoryTitle
+Title
+[Impute]
+Title
+Predict
+Item
+Item
+Item
+Item
+Category
+Category
+[Impute]
+Category3
+2.2
+Missing side information in recommendation
+In real-world applications, the side information of users and
+items may be incomplete or missing, which may hurt the
+performance of recommendation models that rely on side
+information.
+The traditional way to solve the problem of missing side
+information is to fill the missing feature fields with heuristic
+values [29, 30, 18], such as the most frequent feature values,
+average values, randomized values, the value unknown,
+or padding. As some studies have reported, these special
+values are independent of the context, and using them
+may lead to biased parameter estimation and prediction
+[31, 32]. Another way to deal with missing feature fields
+is to impute their missing values. Early approaches use
+KNN-based methods [33] or auto-encoders (AEs) [34, 35]
+to predict the missing data. Wang et al. [19] propose an AE-
+based model with modality dropout, which randomly drops
+representations of user or item information of different
+modalities in hidden states and reconstructs them by an AE.
+Cao et al. [36] present a translation-based recommendation
+model that models preferences as translations from users to
+items, and jointly trains it with a knowledge graph (KG)
+completion model that predicts the missing relations in the
+KG for incorporating knowledge into the recommendation
+model. Instead of imputing the missing side information,
+Shi et al. [18] propose an adaptive feature sampling strategy,
+which employs layer-wise relevance propagation [37] to
+calculate the importance of different features and samples
+features to make the model more robust against unknown
+features. Wu et al. [20] propose a GCN-based model to
+jointly predict users’ preferences to items and predict the
+missing attribute values of users or items.
+What we add on top of prior work on missing infor-
+mation in recommendation is that we focus on missing
+information in the context of sequential recommendation.
+3
+METHOD
+3.1
+Overview
+Before going into details of the proposed MII task and
+MIIR model, we introduce notation used in this paper.
+We denote the item set as I = {i1, . . . , iNi}, where Ni
+is the number of items and each item ID ik ∈ RNi is
+represented as a one-hot vector. In addition to IDs, items
+have other feature fields corresponding to their side infor-
+mation. In this work, we consider categorical feature fields,
+including category and brand, and textual feature fields,
+including title and description. We denote the category set
+as C = {c1, . . . , cNc}, where Nc is the number of categories
+and each category ck ∈ RNc is a one-hot vector. Similarly,
+we denote the brand set as B = {b1, . . . , bNb}, where Nb
+is the number of brands and each brand bk ∈ RNb. For
+titles and descriptions of items, we employ BERT [38] to
+encode them into fixed-length vectors of size 768. We denote
+all titles and all descriptions as T = {t1, . . . , tNi} and
+D = {d1, . . . , dNi}, respectively, where tk and dk ∈ R768.
+We use S = [s1, . . . , sn] to denote a sequence with n items,
+where sk = [si
+k, sc
+k, sb
+k, st
+k, sd
+k] is the sequence of features
+fields of the k-th item, si
+k ∈ I, sc
+k ⊆ C, sb
+k ∈ B, st
+k ∈ T,
+and sd
+k ∈ D. As an item may have multiple categories,
+(a) Sequential recommendation task.
+(b) Missing information imputation task.
+Fig. 2. Comparing the sequential recommendation task and the missing
+information imputation task. (Same visual conventions as in Fig. 1.)
+we let sc
+k be a subset of C, which can be represented as
+a multi-hot vector sc
+k ∈ RNc. For missing item IDs, cate-
+gories and brands, we have special one-hot vectors denoted
+as imiss ∈ I, cmiss ∈ C and bmiss ∈ B, respectively.
+For missing titles and descriptions, we use the vector of
+“[CLS][SEP]” encoded by BERT to represent them, which
+are denoted as tmiss ∈ T and dmiss ∈ D, respectively. These
+missing representations will be used in both MIIR and the
+baselines. It is worth noting that other feature fields can be
+formalized and modeled in a similar way.
+The
+missing
+information
+imputation
+task
+is
+to
+im-
+pute
+the
+values
+of
+the
+missing
+feature
+fields
+in S.
+The sequential recommendation task is to predict the next
+item sn+1 for S. By appending a new item sn+1
+=
+[imiss, cmiss, bmiss, tmiss, dmiss] to the end of S and im-
+puting the imiss of sn+1, we can formulate the next item
+prediction task as a special case of missing information
+imputation task. In Fig. 2, we compare the sequential rec-
+ommendation task and the missing information imputation
+task. In the sequential recommendation task, the next item is
+not considered as a missing data. In the missing information
+imputation task, the next item is simply a missing feature
+field. A model for the missing information imputation task
+that follows a unified way to impute both the next item
+and the other missing side information can be used for
+sequential recommendation.
+To unify the missing side information imputation and
+next item recommendation tasks, we propose a sequential
+recommendation model called missing information imputa-
+tion recommender (MIIR). As we illustrate in Fig. 3, MIIR
+consists of three main components: (i) an embedding layer,
+(ii) a dense fusion self-attention (DFSA) mechanims, and
+
+Fusion
+T
+ID
+Side information808
+Missing information imputation
+0808
+8.084
+Fig. 3. Architecture of the missing information imputation recommender
+(MIIR). MIIR takes a sequence of randomly masked feature fields as
+input. It transforms the input sequence into embeddings using the em-
+bedding layer. Then it employs a dense fusion self-attention mechanism
+to fuse information in the sequence. Finally, MIIR uses an output layer
+to reconstruct the input sequence and calculate the MII loss on masked
+feature fields. (Same visual conventions as in Fig. 1.)
+(iii) an output layer. First, the embedding layer translates
+the input sequence into a series of embeddings. Then, the
+DFSA mechanism employs several transformer [17] layers
+to model the relation between any pair of feature fields in
+the sequence and fuse side information into the model for
+both imputation and recommendation. Finally, the output
+layer imputes the missing feature values including item IDs
+in the sequence based on the output of DFSA. Next, we will
+introduce the details of these main components.
+3.2
+Embedding layer
+The embedding layer projects all item feature fields in the
+input sequence into low-dimensional dense vectors with a
+unified length.
+For the k-th item sk = [si
+k, sc
+k, sb
+k, st
+k, sd
+k] in the given
+sequence S, the embedding layer uses different ways to
+translate different feature fields. For the high-dimensional
+sparse vectors of si
+k, sc
+k and sb
+k, we follow Eq. 1 to get the
+item embedding ei
+k ∈ Re, the category embedding ec
+k ∈ Re,
+and the brand embedding eb
+k ∈ Re:
+ei
+k = Eisi
+k,
+ec
+k = Ecsc
+k,
+eb
+k = Ebsb
+k,
+(1)
+where Ei ∈ Re×Ni is the item embedding matrix, Ec ∈
+Re×Nc is the category embedding matrix, Eb ∈ Re×Nb is
+the brand embedding matrix, and e is the embedding size.
+For the high-dimensional dense vectors of st
+k and sd
+k, we
+project them into low-dimensional embeddings, i.e., the title
+embedding et
+k ∈ Re and the description embedding ed
+k ∈
+Re, respectively, using Eq. 2:
+et
+k = Etst
+k,
+ed
+k = Edsd
+k,
+(2)
+where Et ∈ Re×768 and Ed ∈ Re×768 are the projection
+matrices.
+In order to distinguish different types of feature fields
+in the same item, we learn a field embedding for each
+type of feature fields. We denote the field embeddings of
+ID, category, brand, title and description as f i, f c, f b, f t
+and f d ∈ Re, respectively. To distinguish different items
+in different positions in the same sequence, we also inject
+the position information into the model by learning position
+embeddings, where the k-th position embedding is denoted
+as pk ∈ Re. Finally, we add each field embedding to the
+corresponding item or feature embedding of sk, and add pk
+to all embeddings of sk, as shown in Eq. 3:
+Hk =
+�
+�����
+hi
+k
+hc
+k
+hb
+k
+ht
+k
+hd
+k
+�
+�����
+=
+�
+�����
+ei
+k + f i + pk
+ec
+k + f c + pk
+eb
+k + f b + pk
+et
+k + f t + pk
+ed
+k + f d + pk
+�
+�����
+,
+(3)
+where hi
+k, hc
+k, hb
+k, ht
+k, hd
+k ∈ Re, and Hk ∈ R5×e is the hidden
+state of sk that is the stack of all embeddings of its feature
+fields in order.
+3.3
+Dense fusion self-attention
+The dense fusion self-attention (DFSA) mechanism follows a
+unified way to impute missing feature fields, both item IDs
+and side information. To exploit the information in a given
+context for imputation, we need to model the relations be-
+tween different feature fields and fuse the representations of
+various feature fields. DFSA calculates the attention values
+between any pair of feature fields and fuses the information
+of other feature fields based on the attention value. By
+calculating the attention value, DFSA captures all possible
+(hence dense) pairwise relations between feature fields to
+facilitate missing information imputation.
+Specifically, we first stack the hidden states of all items
+in S in order by Eq. 4:
+H =
+�
+����
+H1
+H2
+...
+Hn
+�
+���� ,
+(4)
+where H ∈ R5n×e is the hidden state matrix of S. Then,
+DFSA employs a transformer with L layers to update H.
+Each transformer layer Trm(·) is composed of two sub-
+layers: (i) multi-head self-attention MH(·) and (ii) position–
+wise feed-forward PFFN(·), as defined in Eq. 5:
+Hl+1 = Trm(Hl) = LN( �Hl + Dropout(PFFN( �Hl)))
+�Hl = LN(Hl + Dropout(MH(Hl)))
+MH(Hl) = [head1; . . . ; headh]WH
+headi = Attn(HlWQ
+i , HlWK
+i , HlWV
+i )
+Attn(Q, K, V) = softmax(QK⊤/√e + M)V
+PFFN( �Hl) = GELU( �HlWF
+1 + bF
+1 )WF
+2 + bF
+2 ,
+(5)
+
+808
+Output sequence
+Output layer
+王
+Dense fusion self-attention
+Position embedding
++
+Field embedding
++
+Item/feature embedding
+Embedding layer
+0808
+08
+Randomly masked
+Mask
+Input sequence5
+where LN is layer normalization [39], Dropout is dropout
+[40], Attn is attention, GELU is a Gaussian error linear unit
+activation [41], [. . . ; . . .] is the concatenation operation, h
+is the number of heads, WH ∈ Re×e, WQ
+i , WK
+i , WV
+i
+∈
+Re×e/h, WF
+1 ∈ Re×4e, WF
+2 ∈ R4e×e, bF
+1 ∈ R4e and bF
+2 ∈
+Re are trainable parameters, Hl and Hl+1 ∈ R5n×e are the
+output hidden state matrices in the l-th layer and the (l+1)-
+th layer, and H0 = H.
+The matrix M ∈ R5n×5n in Eq. 5 is the attention mask
+which is defined as:
+Mj,y
+i,x =
+� 0,
+allow to attend,
+−∞,
+prevent from attending,
+(6)
+where i and j ∈ {1, . . . , n}, x and y ∈ {i, c, b, t, d}, Mj,y
+i,x ∈
+M is the mask to control whether the feature field sy
+j can
+attend to the feature field sx
+i . We set all Mj,y
+i,x = 0,1 which
+means we allow to attend between any pair of feature fields
+in the sequence. Therefore, the DFSA can model relations
+and fuse information between all possible pairs of feature
+fields to facilitate both imputation and recommendation.
+3.4
+Output layer
+The output layer reconstructs the input feature fields based
+on the output hidden states of DFSA. First, we split the final
+output hidden state matrix HL of DFSA by Eq. 7:
+HL = �E =
+�
+�����
+�E1
+�E2
+...
+�En
+�
+�����
+,
+where �Ek =
+�
+�����
+ˆei
+k
+ˆec
+k
+ˆeb
+k
+ˆet
+k
+ˆed
+k
+�
+�����
+,
+(7)
+and ˆei
+k, ˆec
+k, ˆeb
+k, ˆet
+k, ˆed
+k ∈ Re. Similar to the embedding layer,
+the output layer takes different ways to reconstruct different
+types of feature fields. Specifically, for the categorical feature
+fields, we calculate the probability distributions pi
+k ∈ RNi,
+pc
+k ∈ RNc and pb
+k ∈ RNb of the item ID, category and brand
+of the k-th item sk as follows:
+pi
+k = softmax(Ei⊤ˆei
+k)
+pc
+k = sigmoid(Ec⊤ˆec
+k)
+pb
+k = softmax(Eb⊤ˆeb
+k),
+(8)
+where Ei ∈ Re×Ni, Ec ∈ Re×Nc, Eb ∈ Re×Nb are the re-
+used item embedding matrix, category embedding matrix
+and brand embedding matrix in the embedding layer, re-
+spectively. Note that we see the category prediction as a
+series of binary classifications, because an item may contain
+multiple categories. Then we get the reconstructed item ID
+ˆsi
+k ∈ RNi, category ˆsc
+k ∈ RNc and brand ˆsb
+k ∈ RNb based on
+the probability distributions, as shown in Eq. 9:
+ˆsi
+k = argmax(pi
+k)
+ˆsc
+k = 1(pc
+k > 0.5)
+ˆsb
+k = argmax(pb
+k),
+(9)
+where 1(α) is the indicator function that equals 1 if α is true
+and 0 otherwise. Meanwhile, for the textual feature fields,
+1Here we neglect the padding items.
+we follow Eq. 10 to get the reconstructed title ˆst
+k ∈ R768 and
+description ˆsd
+k ∈ R768 directly:
+ˆst
+k = Otˆet
+k,
+ˆsd
+k = Odˆed
+k,
+(10)
+where Ot ∈ R768×e and Od ∈ R768×e are the projection
+matrices.
+3.5
+Missing information imputation loss
+We train MIIR with MII. MII first randomly masks feature
+fields in the sequence with probability p, i.e., replacing a
+non-missing feature value with the corresponding missing
+feature value imiss, cmiss, bmiss, tmiss or dmiss. For the
+k-th item sk in the sequence S, we use mi
+k, mc
+k, mb
+k, mt
+k
+and md
+k ∈ {true, false} to denote whether its ID, category,
+brand, title and description are masked. Then, MIIR learns
+to recover the masked feature fields by MII and impute the
+missing feature values based on the context.
+Specifically, there are differences in the calculation of
+the missing information imputation loss for different types
+of feature fields. For the categorical feature fields (i.e., ID,
+category and brand), our goal is to minimize the cross-
+entropy loss:
+Li
+k = −1(mi
+k)si
+k
+⊤ log(pi
+k)
+Lc
+k = −1(mc
+k)(sc
+k
+⊤ log(pc
+k) + (1 − sc
+k
+⊤) log(1 − pc
+k))/Nc
+Lb
+k = −1(mb
+k)sb
+k
+⊤ log(pb
+k),
+(11)
+where Li
+k, Lc
+k and Lb
+k are the imputation loss for the item
+ID, category and brand of sk, respectively. For the textual
+feature fields (i.e., title and description), our goal is to
+minimize the mean square error loss:
+Lt
+k = 1(mt
+k)∥st
+k − ˆst
+k∥2
+Ld
+k = 1(md
+k)∥sd
+k − ˆsd
+k∥2,
+(12)
+where Lt
+k and Ld
+k are the imputation loss for the title and
+description of sk. The missing information imputation objective
+of the entire model on S is shown in Eq. 13:
+Lmii
+S
+= 1/n
+n
+�
+k=1
+Lmii
+k
+Lmii
+k
+= Li
+k + Lc
+k + Lb
+k + Lt
+k + Ld
+k.
+(13)
+Note that since the item ID is one of the feature fields and
+the next item prediction is a MII task, MIIR trained by MII
+can directly be applied to sequential recommendation.
+In experiments, we also consider to further fine-tune
+MIIR or directly train MIIR with the masked item prediction
+loss to make the model only focus on the item prediction
+task. Specifically, we randomly mask some items with their
+all feature fields in the given sequence, and then let MIIR
+predict the masked item IDs only. The recommendation loss
+(i.e., the masked item prediction loss) on S is defined as:
+Lrec
+S
+= 1/n
+n
+�
+k=1
+Lrec
+k
+Lrec
+k
+= Li
+k = −1(mi
+k)si
+k
+⊤log(pi
+k),
+(14)
+where Lrec
+k
+is the recommendation loss for sk.
+
+6
+TABLE 1
+Summary of the datasets. The missing rate is the percentage of
+missing feature fields in all feature fields. Especially, “Missing rate D” is
+the missing rate on the dataset after discarding side information.
+Dataset
+Beauty Sports and Outdoors Toys and Games
+#items
+121,291
+194,715
+164,978
+#sequences
+52,374
+84,368
+58,314
+Average length
+8.97
+8.50
+8.99
+#categories
+656
+3,035
+957
+#brands
+13,188
+14,163
+14,135
+Missing rate
+12.54%
+20.11%
+11.20%
+Missing rate D
+56.32%
+60.12%
+55.51%
+4
+EXPERIMENTAL SETUP
+4.1
+Research questions
+In this paper, we seek to answer the following research
+questions:
+(RQ1) How does MIIR perform on the sequential recom-
+mendation task compared to state-of-the-art meth-
+ods?
+(RQ2) What are the benefits of training MIIR with MII?
+(RQ3) Does modeling the relation between any pair of
+feature fields in item sequences help sequential rec-
+ommendation?
+(RQ4) How about the performance of MIIR on imputing
+the missing side information?
+(RQ5) What can we find about MIIR by the case study?
+4.2
+Datasets
+There are many public datasets for experimenting with
+sequential recommendation; see [1]. However, we need
+sequential recommendation datasets that come with side
+information. We conduct experiments on three public
+datasets: “Beauty”, “Sports and Outdoors” and “Toys and
+Games” [42], as they have rich item side information, in-
+cluding category, brand, title and description.
+We follow common practices [10, 28] to process the
+datasets. We sort each user’s records in chronological order
+to construct an item sequence. We filter out item sequences
+whose length is less than 5 to avoid noise from the cold-
+start problem. For each item sequence, we use the last
+item for test, the second last item for validation, and the
+rest items for training. For each test or validation item,
+we randomly sample 99 negative items for ranking. We
+randomly discard side information of items with probability
+0.5. We use “Beauty D”, “Sports and Outdoors D” and
+“Toys and Games D” to denote the datasets after discarding
+side information. The statistics of the datasets after pre-
+processing are summarized in Table 1.
+4.3
+Baselines
+We compare MIIR with the following recommendation
+baselines, which can be grouped into (i) methods without
+side information fusion, (ii) methods with side information
+fusion and (iii) methods with missing feature values.
+• Methods without side information fusion:
+– GRU4Rec employs RNNs to capture sequential pat-
+terns between items for sequential recommendation [2].
+– SASRec uses the self-attention mechanism to model
+item sequences for next item recommendations [6].
+– BERT4Rec uses a bidirectional self-attention network
+train-ed by a masked item prediction task for sequential
+recommendation [8].
+• Methods with side information fusion:
+– PRNN employs parallel RNNs to process items and
+their side information respectively, then combines the
+hidden states of the RNNs for next item prediction [9].
+– FDSA leverages two separate self-attention networks
+to model the ID transition patterns and the feature
+transition patterns respectively, then concatenates the
+outputs of two networks for next item prediction [10].
+– NOVA adopts a non-invasive self-attention mecha-
+nism to leverage side information under the BERT4Rec
+framework for sequential recommendation [28].
+• Methods with missing feature values:
+– RFS randomly samples feature fields to introduce more
+missing information when training [18]. RFS is to make
+the model more robust with missing feature values
+instead of imputing missing feature fields. We combine
+RFS with FDSA and NOVA, and denote the variants as
+FDSA+RFS and NOVA+RFS.
+– LRMM designs an auto-encoder with the modality
+dropout to impute both user ratings and missing side
+information for each item [19]. Note that LRMM is not
+a sequential model. Furthermore, we use the imputed
+missing side information by LRMM to train FDSA
+and NOVA, and denote them as FDSA+LRMM and
+NOVA+LRMM.
+Other methods with side information fusion, such as [23,
+24], can only model categorical item side information; for
+a fair comparison, we do not consider them as baselines.
+In addition to the baselines listed above, we compare MIIR
+against four variants, namely MIIR-F, MIIR-R, MIIR-M, and
+Sparse-MIIR, to be defined in Section 5.1, 5.2 and 5.3.
+We unify the sequential recommendation loss in all base-
+lines, MIIR, and its variants to the cross-entropy loss, rather
+than the pairwise loss [43], to avoid noise due to negative
+sampling in the pairwise loss.
+4.4
+Metrics and implementation
+To evaluate the performance of sequential recommendation
+methods, we employ two widely used evaluation metrics:
+HR@k (hit ratio) and MRR (mean reciprocal rank) [1], where
+k ∈ {5, 10}.
+• HR measures the proportion of the sequences whose
+ground-truth items are amongst the top ranked items in
+all test sequences.
+• MRR is the average of reciprocal ranks of the ground-
+truth items.
+For all baselines and our proposed model, we initialize
+the trainable parameters randomly with the Xavier method
+[44]. We train all methods with the Adam optimizer [45]
+for 100 epochs, with a batch size of 128 and a learning rate
+of 0.0001. We also apply gradient clipping [46] with range
+[−5, 5] during training. According to the average length in
+Table 1, we set the maximum sequence length to 20 for both
+datasets for all methods.
+
+7
+TABLE 2
+Performance comparison of MIIR, variants, and the baselines on the
+“Beauty” dataset. MIIR-F is a variant of MIIR that is fine-tuned using the
+recommendation loss (see Section 5.1) and MIIR-R is a variant trained
+using the recommendation loss only (see Section 5.2). The highest
+overall performance is denoted in bold face. The highest performance
+among the baselines is underlined. Impr. (%) is the performance gain
+of MIIR against the best baseline method. ∗ indicates that an
+improvement is statistically significant based on a two-sided paired
+t-test with p < 0.05.
+Beauty
+Beauty D
+Method
+HR@5 HR@10 MRR HR@5 HR@10 MRR
+GRU4Rec
+31.58
+42.50
+21.47
+31.58
+42.50
+21.47
+SASRec
+32.83
+43.61
+23.16
+32.83
+43.61
+23.16
+BERT4Rec
+33.22
+43.77
+23.58
+33.22
+43.77
+23.58
+PRNN
+32.27
+42.70
+23.08
+31.80
+42.55
+22.23
+FDSA
+35.22
+44.83
+25.39
+35.02
+44.68
+25.33
+NOVA
+34.99
+45.07
+25.02
+34.21
+44.38
+24.80
+FDSA+RFS
+35.45
+45.40
+25.68
+34.73
+44.56
+25.17
+NOVA+RFS
+35.57
+45.61
+25.74
+34.26
+44.24
+24.97
+LRMM
+22.74
+32.95
+17.09
+18.04
+26.94
+13.96
+FDSA+LRMM
+35.35
+45.15
+25.62
+35.10
+44.73
+25.52
+NOVA+LRMM
+35.35
+45.31
+25.50
+34.31
+44.53
+25.01
+MIIR
+38.92
+48.61
+29.46
+37.30
+46.85
+27.90
+MIIR-F
+38.73
+48.01
+29.28
+37.12
+46.48
+27.95
+MIIR-R
+35.59
+45.60
+25.85
+34.92
+44.96
+25.41
+Impr. (%)
++3.35∗
++3.00∗ +3.72∗ +2.20∗
++2.12∗ +2.38∗
+TABLE 3
+Performance comparison of MIIR, variants, and the baselines on the
+“Sports and Outdoors” dataset.
+Sports and Outdoors
+Sports and Outdoors D
+Method
+HR@5 HR@10 MRR HR@5 HR@10
+MRR
+GRU4Rec
+33.54
+44.57
+23.70
+33.54
+44.57
+23.70
+SASRec
+34.46
+44.69
+25.41
+34.46
+44.69
+25.41
+BERT4Rec
+35.12
+45.24
+26.11
+35.12
+45.24
+26.11
+PRNN
+37.41
+47.25
+27.23
+36.01
+46.18
+26.12
+FDSA
+39.16
+48.08
+29.27
+37.30
+46.74
+27.20
+NOVA
+37.95
+47.54
+28.08
+36.15
+45.96
+26.90
+FDSA+RFS
+38.18
+47.18
+28.31
+37.17
+46.65
+27.01
+NOVA+RFS
+37.63
+47.41
+27.33
+35.86
+45.52
+26.84
+LRMM
+28.65
+41.36
+20.50
+19.79
+30.34
+15.13
+FDSA+LRMM
+39.48
+48.52
+29.41
+38.46
+47.67
+28.24
+NOVA+LRMM
+38.18
+47.76
+28.30
+37.28
+46.78
+27.32
+MIIR
+43.66
+52.63
+32.66
+40.55
+49.80
+30.04
+MIIR-F
+42.66
+51.49
+32.01
+39.98
+48.98
+29.86
+MIIR-R
+40.01
+49.70
+29.40
+38.07
+47.82
+27.77
+Impr. (%)
++4.18∗
++4.11∗ +3.25∗ +2.09∗
++2.13∗
++1.80∗
+All hyper-parameters of the baselines are set following
+the suggestions from the original papers. For the hyper-
+parameters of MIIR, we set the embedding size e to 64, the
+number of heads h to 4, and the number of layers L to 3. We
+set the dropout rate in DFSA and the mask probability p in
+MII to 0.5.
+5
+EXPERIMENTAL RESULTS
+5.1
+Overall performance
+To answer RQ1, we compare MIIR against the recommenda-
+tion models listed in Section 4.3 on the three datasets from
+TABLE 4
+Performance comparison of MIIR, variants, and the baselines on the
+“Toys and Games” dataset.
+Toys and Games
+Toys and Games D
+Method
+HR@5 HR@10 MRR HR@5 HR@10 MRR
+GRU4Rec
+31.19
+42.15
+21.90
+31.19
+42.15
+21.90
+SASRec
+31.74
+41.22
+24.51
+31.74
+41.22
+24.51
+BERT4Rec
+31.45
+41.22
+23.25
+31.45
+41.22
+23.25
+PRNN
+34.00
+44.25
+24.32
+32.71
+42.98
+23.23
+FDSA
+34.44
+43.89
+26.03
+32.70
+42.33
+24.69
+NOVA
+34.50
+44.34
+25.86
+34.00
+43.74
+25.06
+FDSA+RFS
+34.81
+44.62
+26.30
+33.41
+43.64
+25.22
+NOVA+RFS
+35.33
+45.29
+26.27
+33.39
+43.26
+24.73
+LRMM
+29.88
+40.96
+21.87
+19.85
+29.83
+15.15
+FDSA+LRMM
+35.20
+44.50
+26.49
+33.43
+42.94
+25.18
+NOVA+LRMM
+35.65
+45.50
+26.61
+34.51
+44.47
+25.51
+MIIR
+40.11
+49.80
+29.64
+39.01
+48.89
+28.74
+MIIR-F
+39.00
+47.76
+29.57
+38.25
+47.45
+28.75
+MIIR-R
+35.80
+45.37
+26.00
+34.69
+44.30
+24.81
+Impr. (%)
++4.46∗
++4.30∗ +3.03∗ +4.50∗
++4.42∗ +3.23∗
+Section 4.2. Table 2, 3 and 4 list the evaluation results of
+all methods on each dataset, respectively. Based on these
+results, we have the following observations.
+First, on all datasets, MIIR performs significantly better
+than all baselines by a large margin despite the different
+missing rates, in terms of HR@5, HR@10 and MRR. MIIR
+has two major advantages: (i) MIIR trains the model using
+MII to enhance its ability to deal with missing side infor-
+mation in sequential recommendation (see detailed analysis
+in Section 5.2), and (ii) MIIR employs DFSA to improve the
+side information fusion in the model (see Section 5.3 for
+further analysis).
+Second, the item side information can help sequential
+recommender systems to more accurately model the tran-
+sition patterns among items. To verify this, we divide all
+methods into three groups: (i) GRU4Rec and PRNN that
+are based on RNNs; (ii) SASRec and FDSA that are based
+on left-to-right self-attention networks; and (iii) BERT4Rec,
+NOVA, and MIIR that employ bidirectional self-attention
+networks and the masked item prediction task. In each
+group, we see that methods that fuse side information
+outperform methods that only rely on item IDs, which
+illustrates that item side information does help.
+Third, the performance of PRNN, FDSA, NOVA and
+MIIR on the “Beauty”, “Sports and Outdoors” and “Toys
+and Games” datasets is higher than that on the discarded
+versions of the datasets (i.e., “Beauty D”, “Sports and Out-
+doors D” and “Toys and Games D”). We see two reasons for
+this difference: (i) the “Beauty D”, “Sports and Outdoors D”
+and “Toys and Games D” datasets discard some side infor-
+mation, so the available side information becomes less, and
+(ii) using the special values (i.e., imiss, cmiss, bmiss, tmiss
+and dmiss) to fill missing feature fields may be harmful to
+PRNN, FDSA and NOVA.
+Fourth, by comparing FDSA+RFS and NOVA+RFS with
+FDSA and NOVA, we see RFS cannot consistently improve
+the performance of FDSA and NOVA on all datasets. What’s
+worse, RFS would degrade the performance of FDSA and
+NOVA in some cases. Because RFS is to introduce more
+
+8
+TABLE 5
+Performance comparison of whether to exploit missing feature fields on
+the “Beauty” dataset. MIIR-M and MIIR-R-M are the variants of MIIR
+and MIIR-R respectively that mask missing feature fields in
+self-attention (see Section 5.2).
+Beauty
+Beauty D
+Method
+HR@5 HR@10 MRR HR@5 HR@10 MRR
+MIIR
+38.92
+48.61
+29.46
+37.30
+46.85
+27.90
+MIIR-R
+35.59
+45.60
+25.85
+34.92
+44.96
+25.41
+MIIR-M
+39.16
+48.67
+29.45
+37.12
+46.58
+27.83
+MIIR-R-M
+36.40
+46.31
+27.11
+34.71
+45.01
+25.42
+TABLE 6
+Performance comparison of whether to exploit missing feature fields on
+the “Sports and Outdoors” dataset.
+Sports and Outdoors
+Sports and Outdoors D
+Method
+HR@5 HR@10 MRR HR@5 HR@10
+MRR
+MIIR
+43.66
+52.63
+32.66
+40.55
+49.80
+30.04
+MIIR-R
+40.01
+49.70
+29.40
+38.07
+47.82
+27.77
+MIIR-M
+43.04
+52.12
+32.16
+40.36
+49.65
+29.81
+MIIR-R-M
+39.71
+48.98
+29.15
+38.33
+48.10
+28.12
+missing feature values into the model training instead of
+imputing missing feature fields, it cannot deal with the
+missing side information problem fundamentally.
+Fifth, the performance of LRMM is significantly worse
+than that of the sequential recommendation models with
+side
+information.
+LRMM
+even
+performs
+worse
+than
+GRU4Rec, SASRec and BERT4Rec that neglect the item
+side information. The main reason is that LRMM is not
+a sequential model, so it cannot exploit the relation and
+information in sequences to make recommendation and
+imputation, however it is essential in the sequential rec-
+ommendation task. We can also observe that FDSA+LRMM
+and NOVA+LRMM outperform FDSA and NOVA in exper-
+iments, which verifies the effectiveness of the imputation
+results of LRMM. It also proves imputing missing feature
+values is a better way to alleviate the missing side informa-
+tion problem than using fixed special values and RFS.
+Sixth, modeling sequential recommendation as missing
+information imputation is sufficient to train a recommen-
+dation model. To verify this, we conduct an experiment
+that first pre-trains MIIR using the missing information
+imputation loss (Eq. 13), and then fine-tunes it using the
+recommendation loss (Eq. 14). We use MIIR-F to denote this
+variant of MIIR. In Table 2 we see that MIIR-F performs
+worse than MIIR in most cases. Fine-tuning MIIR-F with the
+recommendation loss might lead to overfitting, resulting in
+performance decreases. This result supports the conclusion
+that with MII we can unify the sequential recommendation
+task as a particular type of missing information imputation
+task to train MIIR together with the other imputation task
+for missing item side information.
+5.2
+Benefits of MII
+To answer RQ2, we analyze how MIIR benefits from training
+with MII.
+TABLE 7
+Performance comparison of whether to exploit missing feature fields on
+the “Toys and Games” dataset.
+Toys and Games
+Toys and Games D
+Method
+HR@5 HR@10 MRR HR@5 HR@10 MRR
+MIIR
+40.11
+49.80
+29.64
+39.01
+48.89
+28.74
+MIIR-R
+35.80
+45.37
+26.00
+34.69
+44.30
+24.81
+MIIR-M
+39.33
+49.22
+28.97
+37.80
+47.58
+27.82
+MIIR-R-M
+35.22
+45.29
+26.28
+34.53
+44.47
+25.58
+In Table 2, 3 and 4, we report on results of a variant of
+MIIR that directly trains MIIR with the recommendation loss
+shown in Eq. 14. We write MIIR-R for this variant of MIIR
+without the supervised signal of MII. When we compare the
+performance of MIIR and MIIR-R, we see very substantial
+gaps. This confirms the effectiveness of training MIIR with
+MII, which accounts for the main part of the improvement
+of MIIR over other methods.
+To demonstrate that MIIR can mine useful information
+from missing feature fields by training with MII, we design
+a variant of MIIR called MIIR-M by masking missing feature
+fields. In MIIR-M, we revise the attention mask M used in
+Eq. 5, which is a null matrix in MIIR. The revision in M is
+defined as:
+Mj,y
+i,x =
+� −∞,
+if sx
+i or sy
+j ∈ {cmiss, bmiss, tmiss, dmiss},
+0,
+otherwise,
+(15)
+where the condition of sx
+i or sy
+j ∈ {cmiss, bmiss, tmiss,
+dmiss} depends on the original input sequence instead
+of the sequence after randomly masking. The purpose of
+the variant is to prevent the model from attending to the
+missing feature fields about item side information in the
+sequence. On the one hand, MIIR-M cannot mine and fuse
+any information in missing feature fields for sequential
+recommendation. On the other hand, MIIR-M is unable
+to exploit the information in non-missing feature fields to
+impute the missing side information. Besides, we mask
+missing feature fields for MIIR-R to analyze how missing
+feature values affects the performance of MIIR without MII,
+denoted as MIIR-R-M.
+In Table 5, 6 and 7, we compare MIIR and MIIR-R with
+MIIR-M and MIIR-R-M, respectively. We can find that MIIR
+outperforms MIIR-M in most cases, which illustrates that
+MIIR can extract useful information from missing feature
+fields to improve the sequential recommendation perfor-
+mance. We can also observe that MIIR-R-M performs better
+than MIIR-R in some cases. This phenomenon indicates that
+using fixed special values for filling missing feature fields
+can suffer the model performance, therefore masking miss-
+ing feature fields may be a better way without imputation.
+On the “Beauty” dataset, MIIR only achieves comparable
+performance with MIIR-M, and MIIR-R also performs worse
+than MIIR-R-M. However, the performance gap from MIIR
+to MIIR-M is smaller than that from MIIR-R to MIIR-R-
+M, and we have similar observations on other datasets.
+It illustrates that imputing missing feature values has the
+superiority over masking them for alleviating missing side
+information problem.
+
+9
+TABLE 8
+Performance comparison of dense and sparse attention on the
+“Beauty” dataset. Sparse-MIIR and Sparse-MIIR-R are variants of MIIR
+and MIIR-R, respectively, in which DFSA is replaced by SFSA (see
+Section 5.3).
+Beauty
+Beauty D
+Method
+HR@5 HR@10 MRR HR@5 HR@10 MRR
+MIIR
+38.92
+48.61
+29.46
+37.30
+46.85
+27.90
+MIIR-R
+35.59
+45.60
+25.85
+34.92
+44.96
+25.41
+Sparse-MIIR
+36.71
+46.60
+26.87
+36.04
+45.98
+26.34
+Sparse-MIIR-R
+34.95
+45.02
+25.35
+34.61
+44.84
+25.19
+TABLE 9
+Performance comparison of dense and sparse attention on the “Sports
+and Outdoors” dataset.
+Sports and Outdoors
+Sports and Outdoors D
+Method
+HR@5 HR@10 MRR HR@5 HR@10
+MRR
+MIIR
+43.66
+52.63
+32.66
+40.55
+49.80
+30.04
+MIIR-R
+40.01
+49.70
+29.40
+38.07
+47.82
+27.77
+Sparse-MIIR
+40.52
+50.04
+29.64
+39.24
+48.91
+28.67
+Sparse-MIIR-R
+38.61
+48.29
+28.25
+37.56
+47.72
+27.21
+MIIR-M also outperforms all baselines on the three
+datasets with different missing rates. Training MIIR with
+MII helps MIIR to make use non-missing feature fields.
+Imputing the masked non-missing feature values requires
+the model to capture the relations between different feature
+fields, so MII guides MIIR to better fuse side information
+into the model for improving the sequential recommenda-
+tion performance.
+5.3
+Effectiveness of DFSA
+To answer RQ3, we conduct an ablation study to analyze the
+effectiveness of DFSA in MIIR.
+We first compare MIIR-R, the variant of MIIR that is
+trained with recommendation loss only, with the baselines
+in Table 2, 3, 4. MIIR-R achieves better or comparable per-
+formance with the baselines on most evaluation metrics of
+all datasets, even without the help of MII. The main reason
+is that MIIR-R has dense fusion self-attention (DFSA) to
+better fuse information in the item sequence for improving
+sequential recommendation.
+In order to validate that it is important to model all pos-
+sible pairwise relations in an item sequence for sequential
+recommendation, we design another self-attention mecha-
+nism called sparse fusion self-attention (SFSA). SFSA modifies
+the attention mask M in Eq. 5 into:
+Mj,y
+i,x =
+� 0,
+if i == j or x == y,
+−∞,
+otherwise,
+(16)
+where the condition i == j or x == y means that SFSA
+only allows to attend between the pair of feature fields
+belonging to the same item or the same type. Therefore,
+SFSA only models the relation between different feature
+fields of the same item or the relation between the same type
+of feature fields of different items in the sequence. These
+relations are also modeled in some baselines, such as PRNN
+and FDSA.
+TABLE 10
+Performance comparison of dense and sparse attention on the “Toys
+and Games” dataset.
+Toys and Games
+Toys and Games D
+Method
+HR@5 HR@10 MRR HR@5 HR@10 MRR
+MIIR
+40.11
+49.80
+29.64
+39.01
+48.89
+28.74
+MIIR-R
+35.80
+45.37
+26.00
+34.69
+44.30
+24.81
+Sparse-MIIR
+37.61
+47.77
+27.23
+37.06
+47.27
+26.79
+Sparse-MIIR-R
+35.58
+45.66
+25.80
+34.46
+44.54
+24.53
+TABLE 11
+Performance comparison of LRMM and MIIR for the missing side
+information imputation on the “Beauty D”, “Sports and Outdoors D” and
+“Toys and Games D” datasets, where P: precision, R: recall, F1: F1
+score, ACC: accuracy, MSE: mean square error.
+Dataset
+Field
+Metric LRMM
+MIIR
+Beauty D
+Category
+P
+70.15
+79.64
+R
+48.41
+36.97
+F1
+52.96
+48.61
+Brand
+ACC
+7.84
+5.01
+Title
+MSE
+0.0871
+0.0514
+Description MSE
+0.1454
+0.0704
+Sports and
+Outdoors D
+Category
+P
+57.02
+74.97
+R
+51.31
+35.91
+F1
+46.38
+46.40
+Brand
+ACC
+6.06
+4.43
+Title
+MSE
+0.0927
+0.0534
+Description MSE
+0.1474
+0.0835
+Toys and
+Games D
+Category
+P
+72.32
+89.31
+R
+51.08
+42.31
+F1
+54.11
+55.50
+Brand
+ACC
+18.61
+14.49
+Title
+MSE
+0.0858
+0.0514
+Description MSE
+0.1427
+0.0777
+In Table 8, 9 and 10, we compare the performance of
+DFSA and SFSA as components of MIIR and MIIR-R. We
+write Sparse-MIIR and Sparse-MIIR-R for the variants of
+MIIR and MIIR-R, respectively, in which DFSA is replaced
+by SFSA. We can see that MIIR outperforms Sparse-MIIR
+on all datasets despite different missing rates. What’s more,
+MIIR-R outperforms Sparse-MIIR-R in most cases too. Mod-
+eling the relations between any pair of feature fields helps to
+make more effective use of item side information to improve
+sequential recommendation performance.
+By comparing MIIR with Sparse-MIIR, we also notice
+that the improvement by DFSA on the three datasets is
+higher than the improvement on the discarded versions of
+the datasets. We guess that DFSA may also model more
+noisy relations related to missing feature fields when the
+missing rate increases, which would make the performance
+degeneration.
+5.4
+Imputation performance (RQ4)
+To answer RQ4, we compare LRMM and MIIR based on
+their imputation results for the discarded side information
+of all datasets.
+For the test sequences from the “Beauty D”, “Sports and
+Outdoors D” and “Toys and Games D” datasets, we can
+compare the imputed results with the ground-truth before
+discard. For different types of feature fields, we consider
+
+10
+Fig. 4. (a) and (b) are two sequences with their imputed categories from the “Beauty D” dataset, (c) and (d) are two sequences with their imputed
+brands from the “Toys and Games D” dataset, (e) is the sequence with its imputed categories and brands from the “Sports and Outdoors D” dataset.
+different metrics: (i) for category that is corresponding to
+the multi-class classification task, we calculate the preci-
+sion, recall and F1 score for evaluation which bigger is
+better; (ii) for brand that is corresponding to the one-class
+classification task, we calculate the accuracy for evaluation
+which bigger is better; (iii) for title and description that are
+both corresponding to the multi-variable regression task,
+we calculate the mean square error (averaged by the length
+of title/description vector) for evaluation which smaller is
+better.
+In Table 11, we list the evaluation results for compari-
+son. We can observe that MIIR achieves better imputation
+performance than LRMM on the precision of category and
+the MSE of title and description. Whereas, LRMM outper-
+forms MIIR on the recall of category and the accuracy of
+brand. Both LRMM and MIIR can infer some discarded side
+information, so they can alleviate the missing side infor-
+mation problem. Comparing to LRMM, MIIR can exploit
+more information from the sequence to impute the missing
+side information. However, MIIR may also impute some
+inaccurate results due to the over-dependence on the given
+context.
+5.5
+Case Study (RQ5)
+Finally, to answer RQ5, we sample some test cases from
+datasets.
+As shown in Fig.4, we list some sequences with their
+imputed results. We can observe that MIIR can generate
+different feature values for missing feature fields according
+to different contexts (i.e., items and sequences), which is
+better than using fixed predefined values. What’s more,
+MIIR may infer the ground-truth missing value, including
+the side information of the target next item, to give the
+model with a more accurate guidance for recommendation.
+For example, MIIR imputes a part of the discarded cate-
+gories in the sequence (b) and the discarded brands in the
+sequence (d). We can also observe that the side information
+of items in the same sequence may be related, which is why
+MIIR can infer the ground-truth missing value in light of
+the given context. However, MIIR may be over-dependent
+on the information from the sequence, leading to impute
+inaccurate results. For instance, in the sequence (e), MIIR
+imputes the wrong categories and brand for the item 5401.
+Additionally, we visualize the attention weights from
+the missing item ID (i.e., the next item ID) to all feature
+fields in the given sequence in DFSA, as shown in Fig.5. We
+reshape the attention weights into a matrix with the shape
+of 5 × n, where 5 is the number of the feature field types
+and n is the sequence length. First, we can see that MIIR
+exploits the information from all feature fields of the given
+sequence to predict the next item, which emphasizes the
+necessity to model the relation between any pair of feature
+fields. Second, we can observe that different layers focus on
+different types of feature fields, where the first layer mainly
+attends to ID, and the third layer mainly attends to title
+and description. It illustrates MIIR gradually fuses different
+types of side information into the model by different layers.
+Because the information in textual feature fields is more
+difficult to extract, MIIR needs more deeper layers to fuse
+textual feature fields. Third, we can find different heads
+in the same layers have similar attention patterns, which
+means there may be some redundant parameters in MIIR.
+
+(a) 
+id: 22434
+id: 69550
+id: 52863
+id: 52866
+id: 52867
+id: 52868
+id: 65782
+id: 65790
+id: 83354
+Original
+Sequence
+category: 384, 487,
+category: 38, 271,
+category: 185, 384,
+category: 185, 384,
+category: 384, 487,
+category: 185, 384,
+category: 384, 487,
+category: 384, 487,
+category: 185, 384,
+489
+479
+487
+487
+489
+487
+489
+489
+487
+id: 22434
+id: 69550
+id: 52863
+id: 52866
+id: 52867
+id: 52868
+id: 65782
+id: 65790
+id: 83354
+Imputed
+Sequence
+category: 384, 487,
+category: 185, 384,
+category: 185, 384,
+category: 185, 384,
+category: 384, 487,
+category: 384, 487,
+category: 185, 384,
+category: 384, 487
+category: 384, 487
+489
+487
+487
+487
+489
+489
+487
+(b)
+id: 19904
+id: 101600
+id: 53133
+id: 25434
+id: 89249
+id: 44195
+id: 25437
+id: 103752
+Original
+id: 89561
+Sequence
+category: 114, 454,
+category: 108, 114,
+category: 114, 367,
+category: 114, 369,
+category: 114, 367,
+category: 43, 114,
+category: 41, 47, 114,
+category: 43, 114
+category: 15, 26, 56
+558
+493, 598
+598
+535, 598
+598
+267
+369, 598
+id: 19904
+id: 101600
+id: 53133
+id: 25434
+id: 44195
+id: 25437
+id: 89249
+id: 89561
+id: 103752
+Imputed
+Sequence
+category: 114, 454,
+category: 108, 114,
+category: 43, 114,
+category: 43, 114
+category: 114, 598
+category: 114, 598
+category: 114, 598
+category: 15, 26, 56
+category: 114, 598
+558
+493, 598
+267
+(c)
+id: 6177
+id: 96866
+id: 108189
+id: 117401
+id: 117402
+id: 134383
+id: 134384
+id: 157885
+Original
+Sequence
+brand: <missing>
+ brand: 3614
+brand: 3614
+brand: 3614
+brand: 3614
+ brand: <missing>
+brand: 7166
+brand: 3614
+id: 6177
+id: 96866
+id: 108189
+id: 117401
+id: 117402
+id: 134383
+id: 134384
+id: 157885
+Imputed
+Sequence
+ brand: 3614
+brand: 3614
+brand: 3614
+brand: 3614
+brand: 3614
+ brand: 3614
+brand: 7166
+brand: 3614
+(d)
+id: 27109
+id: 4139
+L096 :p!
+id: 63016
+id: 80413
+id: 16994
+id: 83174
+Original
+id: 23768
+Sequence
+brand: 11595
+brand: <missing>
+brand: 11595
+brand: 6844
+ brand: 12538
+brand: 2308
+brand: 11595
+brand: 5472
+id: 27109
+id: 4139
+id: 9607
+id: 63016
+id: 80413
+id: 16994
+id: 83174
+id: 23768
+Imputed
+Sequence
+brand: 11595
+brand: 11595
+brand: 5472
+brand: 11595
+brand: 6844
+ brand: 12538
+brand: 2308
+brand: 11595
+(e)
+id: 55944
+id: 1956
+id: 5401
+id: 22000
+id: 36936
+id: 173423
+id: 25085
+id: 101607
+Original
+Sequence
+category: 952, 1286,
+category: 632, 1562,
+category: 952, 1286,
+category: 952, 1286,
+category: 252, 1286,
+category: 296, 1286,
+category: 952, 1286,
+category: 2078, 2299
+ Discard
+2191, 2479
+2533
+2191, 2479
+2191, 2479
+2191, 2479
+2230, 2479
+2233, 2479
+brand: 4696
+brand: 4087
+brand: 6636
+brand: 4087
+brand: 4087
+ brand: <missing>
+brand: 9545
+brand: 10778
+Impute
+id: 55944
+id: 1956
+id: 5401
+id: 22000
+id: 36936
+id: 173423
+id: 25085
+id: 101607
+Imputed
+Sequence
+category: 952, 1286,
+category: 952, 1286,
+category: 952, 1286,
+category: 952, 1286,
+ Target Item
+category: 2078, 2299
+category: 1286, 2479
+category: 1286, 2479
+category: 1286, 2479
+2191, 2479
+2191, 2479
+2191, 2479
+2191, 2479
+brand: 4696
+ brand: 4087
+brand: 4087
+brand: 4087
+brand: 4087
+brand: 4087
+brand: 9545
+brand: 408711
+(a) A sequence from the “Beauty D” dataset.
+(b) A sequence from the “Sports and Outdoors D” dataset.
+(c) A sequence from the “Toys and Games D” dataset.
+Fig. 5. Visualization for the attention weights from the missing item ID field to all feature fields of all heads and layers in MIIR on three sequences
+from different datasets.
+6
+CONCLUSION
+We have studied the missing side information problem in
+sequential recommendation. We have proposed the missing
+information imputation (MII) task to unify the missing side
+information imputation task and the sequential recommen-
+dation task. We have presented a novel sequential recom-
+mendation model named missing information imputation
+recommender (MIIR) to simultaneously impute missing fea-
+ture values and predict the next item for a given sequence
+of items. We have proposed a dense fusion self-attention
+(DFSA) mechanism to model different relations in the item
+sequence and to fuse side information.
+Based on experiments and analyses on three datasets
+with different settings of the missing rates we have found
+that MIIR outperforms state-of-the-art methods for sequen-
+tial recommendation with side information. We have ver-
+ified that MIIR can identify useful side information from
+missing feature fields by training with the MII task, and
+that the DFSA mechanism improves the recommendation
+effectiveness of MIIR.
+As to broader implications of our work, we offer a new
+perspective by revealing a correlation between missing side
+information imputation and the sequential recommendation
+task. They both concern the prediction of missing infor-
+mation. The perspective operationalized with MIIR can be
+adopted as a foundational paradigm. Other prediction tasks
+related to recommendation, such as rating prediction, user
+profile prediction, and next basket recommendation can also
+be formulated as a MII task.
+Limitations of our work include the following: (i) since
+DFSA treats side information as part of the sequence (e.g.,
+in our case, the actual sequence length is 5x the number of
+items) and models all possible pairwise relations in an item
+sequence, it is computationally costly and not easy to scale
+to long sequences; and (ii) we have not optimized the MII
+losses on different types of feature fields in MIIR for the
+
+Layer 3 Head 2
+category
+category
+ category
+brand
+brand
+branc
+itle
+title
+description
+description
+ description
+Layer 1 Head 3
+Layer 1 Head 4 
+Layer 2 Head 3 
+Layer 2 Head 4
+Layer 3 Head 3 
+Layer 3 Head 4
+category
+category
+category
+brand
+ branc
+brand
+title
+itle
+title
+description
+description
+description
+1
+0.000 0.009
+0.090
+0.000
+0.0900.100
+0.000
+0.190
+0.081
+0.010Layer 1 Head 1
+Layer 1 Head 2
+Layer 2 Head 1
+Layer 2 Head 2
+category
+category
+ category
+brand
+brand
+brand
+title
+title
+description
+description
+ description
+Layer 2 Head 4
+Layer 1 Head 4
+Layer 2 Head 3
+Layer 3 Head 3 
+Layer 3 Head 4
+d
+category
+category
+category
+brand
+ brand
+brand
+title
+title
+description
+ description
+description
+97899800 1469157567833 540810595m1557
+0.190
+0.000
+0.000ayer 1 Head 2
+Layer 2 Head 2
+category
+category
+ category
+brand
+bran
+branc
+title
+title
+title
+description
+description
+ description
+ayer 1 Head 4
+Layer 2 Head 3 
+Layer 2 Head 4
+Layer 3 Head 3 
+Layer 3 Head 4
+category
+category
+category
+brand
+ brand
+brand
+title
+title
+description
+ description
+ description
+1699 83004 90253151526203262038 5850mis5716999 3004902531515220322038 5850m557
+16999 300402531515220322038850m5716999 3009025315152203620385850mi57
+16999 8300490253151526203262038 5850m55716999 3004902531515262032620385850ms57
++o-0.100
+0.000
+0.084
+0.000
+0.011
+0.05512
+recommendation task.
+We aim to further improve MIIR in different directions.
+We will assess the ability of the linear transformer [47, 48]
+to reduce the computational costs of DFSA and design a
+mechanism to filter out useless relations at an early stage.
+We also plan to design a tailored loss for MIIR by building
+on recent loss weighting methods [49, 50].
+REPRODUCIBILITY
+To facilitate reproducibility of the results reported in this
+paper, the code and data used in experiments are available
+at https://github.com/TempSDU/MIIR.
+REFERENCES
+[1]
+H. Fang, D. Zhang, Y. Shu, and G. Guo, “Deep learning for
+sequential recommendation: Algorithms, in��uential factors, and
+evaluations,” ACM Transactions on Information Systems, vol. 39,
+no. 1, pp. 1–42, 2020.
+[2]
+B. Hidasi, A. Karatzoglou, L. Baltrunas, and D. Tikk, “Session-
+based recommendations with recurrent neural networks,” in In-
+ternational Conference on Learning Representations, 2016.
+[3]
+J. Li, P. Ren, Z. Chen, Z. Ren, T. Lian, and J. Ma, “Neural attentive
+session-based recommendation,” in Conference on Information and
+Knowledge Management, 2017, pp. 1419–1428.
+[4]
+B. Hidasi and A. Karatzoglou, “Recurrent neural networks with
+top-k gains for session-based recommendations,” in Conference on
+Information and Knowledge Management, 2018, pp. 843–852.
+[5]
+J. Tang and K. Wang, “Personalized top-n sequential recommen-
+dation via convolutional sequence embedding,” in International
+Conference on Web Search and Data Mining, 2018, pp. 565–573.
+[6]
+W.-C. Kang and J. McAuley, “Self-attentive sequential recommen-
+dation,” in IEEE International Conference on Data Mining, 2018, pp.
+197–206.
+[7]
+S. Wu, Y. Tang, Y. Zhu, L. Wang, X. Xie, and T. Tan, “Session-based
+recommendation with graph neural networks,” in Proceedings of
+the AAAI Conference on Artificial Intelligence, vol. 33, no. 01, 2019,
+pp. 346–353.
+[8]
+F. Sun, J. Liu, J. Wu, C. Pei, X. Lin, W. Ou, and P. Jiang,
+“Bert4rec: Sequential recommendation with bidirectional encoder
+representations from transformer,” in Conference on Information and
+Knowledge Management, 2019, pp. 1441–1450.
+[9]
+B. Hidasi, M. Quadrana, A. Karatzoglou, and D. Tikk, “Parallel re-
+current neural network architectures for feature-rich session-based
+recommendations,” in ACM Conference on Recommender Systems,
+2016, pp. 241–248.
+[10] T. Zhang, P. Zhao, Y. Liu, V. S. Sheng, J. Xu, D. Wang, G. Liu,
+and X. Zhou, “Feature-level deeper self-attention network for
+sequential recommendation.” in International Joint Conference on
+Artificial Intelligence, 2019, pp. 4320–4326.
+[11] P. Wang, Y. Fan, L. Xia, W. X. Zhao, S. Niu, and J. Huang, “Kerl:
+A knowledge-guided reinforcement learning model for sequential
+recommendation,” in International ACM SIGIR Conference on Re-
+search and Development in Information Retrieval, 2020, pp. 209–218.
+[12] G. de Souza Pereira Moreira, S. Rabhi, J. M. Lee, R. Ak, and
+E. Oldridge, “Transformers4rec: Bridging the gap between nlp and
+sequential/session-based recommendation,” in ACM Conference on
+Recommender Systems, 2021, pp. 143–153.
+[13] R. Cai, J. Wu, A. San, C. Wang, and H. Wang, “Category-aware
+collaborative sequential recommendation,” in International ACM
+SIGIR Conference on Research and Development in Information Re-
+trieval, 2021, pp. 388–397.
+[14] U. Singer, H. Roitman, Y. Eshel, A. Nus, I. Guy, O. Levi, I. Hasson,
+and E. Kiperwasser, “Sequential modeling with multiple attributes
+for watchlist recommendation in e-commerce,” in International
+Conference on Web Search and Data Mining, 2022, pp. 937–946.
+[15] Y. Xie, P. Zhou, and S. Kim, “Decoupled side information fusion
+for sequential recommendation,” in International ACM SIGIR Con-
+ference on Research and Development in Information Retrieval, 2022.
+[16] Z. C. Lipton, J. Berkowitz, and C. Elkan, “A critical review of
+recurrent neural networks for sequence learning,” arXiv preprint
+arXiv:1506.00019, 2015.
+[17] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N.
+Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,”
+in Neural Information Processing Systems, vol. 30, 2017.
+[18] S. Shi, M. Zhang, X. Yu, Y. Zhang, B. Hao, Y. Liu, and S. Ma,
+“Adaptive feature sampling for recommendation with missing
+content feature values,” in Conference on Information and Knowledge
+Management, 2019, pp. 1451–1460.
+[19] C. Wang, M. Niepert, and H. Li, “Lrmm: Learning to recommend
+with missing modalities,” in Proceedings of the Conference on Empir-
+ical Methods in Natural Language Processing, 2018, pp. 3360–3370.
+[20] L. Wu, Y. Yang, K. Zhang, R. Hong, Y. Fu, and M. Wang, “Joint
+item recommendation and attribute inference: An adaptive graph
+convolutional network approach,” in International ACM SIGIR
+Conference on Research and Development in Information Retrieval,
+2020, pp. 679–688.
+[21] T. N. Kipf and M. Welling, “Semi-supervised classification with
+graph convolutional networks,” in International Conference on
+Learning Representations, 2017.
+[22] Z. Zeng, C. Xiao, Y. Yao, R. Xie, Z. Liu, F. Lin, L. Lin, and M. Sun,
+“Knowledge transfer via pre-training for recommendation: A re-
+view and prospect,” Frontiers in Big Data, p. 4, 2021.
+[23] K. Zhou, H. Wang, W. X. Zhao, Y. Zhu, S. Wang, F. Zhang, Z. Wang,
+and J.-R. Wen, “S3-rec: Self-supervised learning for sequential
+recommendation with mutual information maximization,” in Con-
+ference on Information and Knowledge Management, 2020, pp. 1893–
+1902.
+[24] X. Yuan, D. Duan, L. Tong, L. Shi, and C. Zhang, “Icai-sr: Item
+categorical attribute integrated sequential recommendation,” in
+International ACM SIGIR Conference on Research and Development
+in Information Retrieval, 2021, pp. 1687–1691.
+[25] X. Huang, S. Qian, Q. Fang, J. Sang, and C. Xu, “Csan: Contextual
+self-attention network for user sequential recommendation,” in
+Proceedings of the ACM International Conference on Multimedia, 2018,
+pp. 447–455.
+[26] G. Tang, M. M¨uller, A. R. Gonzales, and R. Sennrich, “Why self-
+attention? a targeted evaluation of neural machine translation
+architectures,” in Proceedings of the Conference on Empirical Methods
+in Natural Language Processing, 2018, pp. 4263–4272.
+[27] H. Zhao, J. Jia, and V. Koltun, “Exploring self-attention for image
+recognition,” in Proceedings of the IEEE/CVF Conference on Computer
+Vision and Pattern Recognition, 2020, pp. 10 076–10 085.
+[28] C. Liu, X. Li, G. Cai, Z. Dong, H. Zhu, and L. Shang, “Non-
+invasive self-attention for side information fusion in sequential
+recommendation,” in Proceedings of the AAAI Conference on Artificial
+Intelligence, vol. 35, no. 5, 2021, pp. 4249–4256.
+[29] Y. Lee, S.-W. Kim, S. Park, and X. Xie, “How to impute missing
+ratings? claims, solution, and its application to collaborative filter-
+ing,” in The Web Conference, 2018, pp. 783–792.
+[30] F. Biessmann, D. Salinas, S. Schelter, P. Schmidt, and D. Lange,
+““deep” learning for missing value imputation in tables with
+non-numerical data,” in Conference on Information and Knowledge
+Management, 2018, pp. 2017–2025.
+[31] B. M. Marlin and R. S. Zemel, “Collaborative prediction and
+ranking with non-random missing data,” in ACM Conference on
+Recommender Systems, 2009, pp. 5–12.
+[32] J. M. Hern´andez-Lobato, N. Houlsby, and Z. Ghahramani, “Prob-
+abilistic matrix factorization with non-random missing data,” in
+International Conference on Machine Learning, 2014, pp. 1512–1520.
+[33] R. Pan, T. Yang, J. Cao, K. Lu, and Z. Zhang, “Missing data
+imputation by k nearest neighbours based on grey relational
+structure and mutual information,” Applied Intelligence, vol. 43,
+no. 3, pp. 614–632, 2015.
+[34] B. K. Beaulieu-Jones and J. H. Moore, “Missing data imputation in
+the electronic health record using deeply learned autoencoders,”
+in Pacific Symposium on Biocomputing, 2017, pp. 207–218.
+[35] R. C. Pereira, M. S. Santos, P. P. Rodrigues, and P. H. Abreu,
+“Reviewing autoencoders for missing data imputation: Technical
+trends, applications and outcomes,” Journal of Artificial Intelligence
+Research, vol. 69, pp. 1255–1285, 2020.
+[36] Y. Cao, X. Wang, X. He, Z. Hu, and T.-S. Chua, “Unifying
+knowledge graph learning and recommendation: Towards a better
+understanding of user preferences,” in The Web Conference, 2019,
+pp. 151–161.
+[37] A. Binder, S. Bach, G. Montavon, K.-R. M¨uller, and W. Samek,
+“Layer-wise relevance propagation for deep neural network ar-
+chitectures,” International Conference on Information Science and
+Applications, pp. 913–922, 2016.
+
+13
+[38] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-
+training of deep bidirectional transformers for language under-
+standing,” in Proceedings of the Conference of the North American
+Chapter of the Association for Computational Linguistics, 2019, pp.
+4171–4186.
+[39] J. L. Ba, J. R. Kiros, and G. E. Hinton, “Layer normalization,” arXiv
+preprint arXiv:1607.06450, 2016.
+[40] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and
+R. Salakhutdinov, “Dropout: A simple way to prevent neural net-
+works from overfitting,” The Journal of Machine Learning Research,
+vol. 15, no. 1, pp. 1929–1958, 2014.
+[41] D. Hendrycks and K. Gimpel, “Gaussian error linear units
+(gelus),” arXiv preprint arXiv:1606.08415, 2016.
+[42] J. Ni, J. Li, and J. McAuley, “Justifying recommendations using
+distantly-labeled reviews and fine-grained aspects,” in Proceedings
+of the Conference on Empirical Methods in Natural Language Process-
+ing, 2019, pp. 188–197.
+[43] S. Rendle, C. Freudenthaler, Z. Gantner, and L. Schmidt-Thieme,
+“Bpr: Bayesian personalized ranking from implicit feedback,” in
+Proceedings of the Conference on Uncertainty in Artificial Intelligence,
+2009, pp. 452–461.
+[44] X. Glorot and Y. Bengio, “Understanding the difficulty of training
+deep feedforward neural networks,” in Proceedings of the Interna-
+tional Conference on Artificial Intelligence and Statistics, 2010, pp. 249–
+256.
+[45] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimiza-
+tion,” in International Conference on Learning Representations, 2015.
+[46] R. Pascanu, T. Mikolov, and Y. Bengio, “On the difficulty of
+training recurrent neural networks,” in International Conference on
+Machine Learning, 2013, pp. 1310–1318.
+[47] S. Wang, B. Z. Li, M. Khabsa, H. Fang, and H. Ma, “Linformer: Self-
+attention with linear complexity,” arXiv preprint arXiv:2006.04768,
+2020.
+[48] Y. Xiong, Z. Zeng, R. Chakraborty, M. Tan, G. Fung, Y. Li, and
+V. Singh, “Nystr¨omformer: A nystr¨om-based algorithm for ap-
+proximating self-attention,” in Proceedings of the AAAI Conference
+on Artificial Intelligence, vol. 35, no. 16, 2021, pp. 14 138–14 148.
+[49] Y. Du, W. M. Czarnecki, S. M. Jayakumar, M. Farajtabar, R. Pas-
+canu, and B. Lakshminarayanan, “Adapting auxiliary losses using
+gradient similarity,” arXiv preprint arXiv:1812.02224, 2018.
+[50] Y. Xu, X. Liu, Y. Shen, J. Liu, and J. Gao, “Multi-task learning
+with sample re-weighting for machine reading comprehension,”
+in Proceedings of the Conference of the North American Chapter of the
+Association for Computational Linguistics, 2019, pp. 2644–2655.
+

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