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-GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
-1
-Detecting Severity of Diabetic Retinopathy from
-Fundus Images using Ensembled Transformers
-Chandranath Adak, Senior Member, IEEE, Tejas Karkera, Soumi Chattopadhyay, Member, IEEE, and
-Muhammad Saqib
-Abstract— Diabetic Retinopathy (DR) is considered one
-of the primary concerns due to its effect on vision loss
-among most people with diabetes globally. The severity of
-DR is mostly comprehended manually by ophthalmologists
-from fundus photography-based retina images. This paper
-deals with an automated understanding of the severity
-stages of DR. In the literature, researchers have focused on
-this automation using traditional machine learning-based
-algorithms and convolutional architectures. However, the
-past works hardly focused on essential parts of the retinal
-image to improve the model performance. In this paper,
-we adopt transformer-based learning models to capture the
-crucial features of retinal images to understand DR sever-
-ity better. We work with ensembling image transformers,
-where we adopt four models, namely ViT (Vision Trans-
-former), BEiT (Bidirectional Encoder representation for im-
-age Transformer), CaiT (Class-Attention in Image Trans-
-formers), and DeiT (Data efficient image Transformers), to
-infer the degree of DR severity from fundus photographs.
-For experiments, we used the publicly available APTOS-
-2019 blindness detection dataset, where the performances
-of the transformer-based models were quite encouraging.
-Index Terms— Blindness Detection, Diabetic Retinopa-
-thy, Deep learning, Transformers.
-I. INTRODUCTION
-D
-IABETES Mellitus, also known as diabetes, is a disorder
-where the patient experiences increased blood sugar
-levels over a long period. Diabetic Retinopathy (DR) is a mi-
-crovascular complication of diabetes where the retina’s blood
-vessels get damaged, which can lead to poor vision and even
-blindness if untreated [1], [2]. Studies estimated that by twenty
-years after diabetes onset, about 99% (or 60%) of patients
-having type-I (or type-II) diabetes might have DR [1]. With
-a worldwide presence of DR patients of about 126.6 million
-in 2010, the current estimate is roughly around 191 million
-by 2030 [3], [4]. However, about 56% of new DR cases can
-be reduced by timely treatment and monitoring of the severity
-[5]. The ophthalmologist analyzes fundus images for lesion-
-based symptoms like microaneurysms, hard/ soft exudates, and
-hemorrhages to understand the severity stages of DR [1], [2].
-The positive DR is divided into the following stages [5]: (1)
-mild: the earliest stage that can contain microaneurysms, (2)
-C. Adak is with Dept. of CSE, IIT Patna, India-801106, T. Karkera is
-with Atharva College of Engineering, Mumbai, India-400095, S. Chat-
-topadhyay is with the Dept. of CSE, IIIT Guwahati, India-781015, and
-M. Saqib is with Data61, CSIRO, Australia-2122.
-Corresponding author: C. Adak (e-mail: [email protected])
-negative
-mild
-moderate
-severe
-proliferative
-Fig. 1. Fundus images with DR severity stages from APTOS-2019 [7].
-moderate: here, the blood vessels lose the ability to blood
-transportation, (3) severe: here, blockages in blood vessels
-can occur and gives a signal to grow new blood vessels, (4)
-proliferative: the advanced stage where new blood vessels start
-growing. Fig. 1 shows some fundus images representing the
-DR severity stages. Manual examining fundus images for DR
-severity stage grading may bring inconsistencies due to a high
-number of patients, less number of well-trained clinicians, long
-diagnosing time, unclear lesions, etc. Moreover, there may be
-disagreement among ophthalmologists in choosing the correct
-severity grade [6]. Therefore, computer-aided techniques have
-come into the scenario for better diagnosis and broadening the
-prospects of early-stage detection [2].
-Automated DR severity stage detection from fundus pho-
-tographs has been performed for the last two and half decades.
-Earlier, some image processing tools were used [8], [9], but
-the machine learning-based DR became popular in the early
-2000s. The machine learning-based techniques mostly relied
-on hand-engineered features that were carefully extracted from
-the fundus images and then fed to a classifier, e.g., Random
-Forest (RF) [10], KNN (K-Nearest Neighbors) [11], SVM
-(Support Vector Machine) [12], and ANN (Artificial Neural
-Network) [13]. Although SVM and ANN-based models were
-admired in the DR community, the hand-engineered feature-
-based machine learning models require efficient prior feature
-extraction, which may introduce errors for complex fundus
-images [1], [2]. On the other hand, deep learning-based models
-extract features automatically through convolution operations
-[14], [15]. Besides, from 2012, deep learning architectures
-rose to prominence in the computer vision community, which
-also influenced the DR severity analysis from fundus images
-[1]. The past deep learning-based techniques mostly employed
-CNN (Convolutional Neural Network) [1], [16]. However,
-the ability to give attention to certain regions/features and
-fade the remaining portions hardly exists in classical CNNs.
-For this reason, some contemporary methods incorporated
-attention mechanism [17], [18]. Although multiple research
-arXiv:2301.00973v1  [cs.CV]  3 Jan 2023
-
-LOGO2
-GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
-works are present in the literature [1], [2] and efforts were
-made to detect the existence of DR in the initial stages
-of its development, still there is a room for improving the
-performance by incorporating higher degrees of automated
-feature extraction using better deep learning models.
-In this paper, we employ the transformer model for leverag-
-ing its MSA (Multi-head Self-Attention) [19] to focus on the
-DR revealing region of the fundus image for understanding
-the severity. Moreover, the transformer model has shown high
-performance in recent days [19], [20]. Initially, we adopted
-ViT (Visual Transformer) [19] for detecting DR severity due to
-its outperformance on image classification tasks. ViT divides
-the input image into a sequence of patches and applies
-global attention [19]. Moreover, since standard ViT requires
-hefty amounts of data, we also adopted some other image
-transformer models, such as CaiT (Class-attention in image
-Transformers) [21], DeiT (Data-efficient image Transformer)
-[22], and BEiT (Bidirectional Encoder representation for im-
-age Transformer) [23]. CaiT is a modified version of ViT and
-employs specific class-attention [21]. DeiT uses knowledge
-distillation, which transfers the knowledge from one network
-to another and builds a teacher-student hierarchical network
-[22]. BEiT is inspired by BERT (Bidirectional Encoder Rep-
-resentations from Transformers) [24] to implement masking
-of image patches and to model the same for pre-training the
-ViT [23]. For experiments, we used the publicly available
-APTOS-2019 blindness detection dataset [7], where the in-
-dividual image transformers did not perform well. Therefore,
-we ensembled the image transformers to seek better predictive
-performance. The ensembled image transformer obtained quite
-encouraging results for DR severity stage detection. This is
-one of the earliest attempts to adopt and ensemble image
-transformers for DR severity stage detection, which is the main
-contribution of this paper.
-The rest of the paper is organized as follows. § II discusses
-the relevant literature about DR and § III presents the proposed
-methodology. Then § IV analyzes and discusses the experi-
-mental results. Finally, § V concludes this paper.
-II. RELATED WORK
-This section briefly presents the literature on DR severity
-detection from fundus images. The modern grading of DR
-severity stages can be traced in the report by ETDRS research
-group [25]. In the past, some image processing-based (e.g.,
-wavelet transform [8], radon transform [9]) strategies were
-published. For the last two decades, machine learning and
-deep learning-based approaches have shown dominance. We
-broadly categorize the related works into (a) hand-engineered
-feature-based models [11], [26], [27], and (b) deep feature-
-based models [2], which are discussed below.
-A. Hand-engineered Feature-based Models
-The hand-engineered feature-based models mostly em-
-ployed RF [26], KNN [28], SVM [27], ANN [29] for detecting
-DR severity stages. Acharya et al. [26] employed a decision
-tree with discrete wavelet/cosine transform-based features ex-
-tracted from retinal images. Casanova et al. [10] introduced RF
-for DR severity stage classification. In [30], RF was also used
-to assess DR risk. KNN classifier was employed in [11] to
-detect drusen, exudates, and cotton-wool spots for diagnosing
-DR. Tang et al. [28] used KNN for retinal hemorrhage detec-
-tion from fundus photographs. In [27], retinal changes due to
-DR was detected by using SVM. Akram et al. [12] used SVM
-and GMM (Gaussian Mixture Model) with enhanced features
-such as shape, intensity, and statistics of the affected region
-to identify microaneurysms for early detection of DR. ANN
-was employed in [13] to classify lesions for detecting DR
-severity. Osareh et al. [31] employed FCM (Fuzzy C-Means)-
-based segmentation and GA (Genetic algorithm)-based feature
-selection with ANN to detect exudates in DR. In [29], PSO
-(Particle Swarm Optimization) was used for feature selection,
-followed by ANN-based DR severity classification.
-B. Deep Feature-based Models
-The past deep architectures mostly used CNN for tackling
-DR severity. For example, Yu et al. [16] used CNN for
-detecting exudates in DR, Chudzik et al. [32] worked on
-microaneurysm detection using CNN with transfer learning
-and layer freezing, Gargeya and Leng [33] employed CNN-
-based deep residual learning to identify fundus images with
-DR. In [4], CNN was also used to identify DR severity stages
-and some related eye diseases, e.g., glaucoma and AMD (Age-
-related Macular Degeneration). In [34], some classical CNN
-architectures (e.g., AlexNet, VGG Net, GoogLeNet, ResNet)
-were employed for DR severity stage detection. Wang et al.
-[17] proposed Zoom-in-Net that combined CNN, attention
-mechanism, and a greedy algorithm to zoom in the region
-of interest for handling DR. A modified DenseNet169 ar-
-chitecture in conjunction with the attention mechanism was
-used in [18] to extract refined features for DR severity
-grading. In [35], a modified Xception architecture was em-
-ployed for DR classification. TAN (Texture Attention Net-
-work) was proposed in [36] by leveraging style (texture
-features) and content (semantic and contextual features) re-
-calibration mechanism. Tymchenko et al. [5] ensembled three
-CNN architectures (EfficientNet-B4 [37], EfficientNet-B5, and
-SE- ResNeXt50 [38]) for DR severity detection. Very recently,
-a few transformer-based models have come out, e.g., CoT-
-XNet [39] that combined contextual transformer and Xception
-architecture, SSiT [40] that employed self-supervised image
-transformers guided by saliency maps.
-III. METHODOLOGY
-This section first formalizes the problem statement, which
-is then followed by the proposal of solution architecture.
-A. Problem Formulation
-In this work, we are given an image I captured by the
-fundus photography, which is input to the architecture. The
-task is to predict the severity stage of diabetic retinopathy (DR)
-among negative, mild, moderate, severe, and proliferative,
-from I. We formulate the task as a multi-class classification
-problem [15]. Here, from I, features are extracted and fed to
-
-ADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS
-3
-a classifier to predict the DR severity class labels Á, where
-Á = {0, 1, 2, 3, 4} corresponds to {negative, mild, moderate,
-severe, proliferative}, respectively.
-B. Solution Architecture
-For detecting the severity stage of DR from a fundus
-photograph, we adopt image transformers, i.e., ViT (Vision
-Transformer) [19], BEiT (Bidirectional Encoder representation
-for image Transformer) [23], CaiT (Class-attention in image
-Transformers) [21], and DeiT (Data efficient image Trans-
-formers) [22], and ensemble them. However, we preprocess
-raw fundus images before feeding them into the transformers,
-which we discuss first.
-1) Preprocessing: The performance of deep learning mod-
-els is susceptible to the quality and quantity of data being
-passed to the model. Raw data as input can barely account for
-the best achievable performance of the model due to possible
-pre-existing noise and inconsistency in the images. Therefore,
-a definite flow of preprocessing is essential to train the model
-better [15].
-We now discuss various preprocessing and augmentation
-techniques [15], [41] applied to the raw fundus photographs
-for better learning. In a dataset, the fundus images may be of
-various sizes; therefore, we resize the image I into 256 × 256
-sized image Iz. We perform data augmentations on training set
-(DBtr), where we use centre cropping with central_fraction =
-0.5, horizontal/vertical flip, random rotations within a range
-of [0o, 45o], random brightness-change with max_delta =
-0.95, random contrast-change in the interval [0.1, 0.9]. We
-also apply CLAHE (Contrast Limited Adaptive Histogram
-Equalization) [42] on 30% samples of DBtr, which ensures
-over-amplification of contrast in a smaller region instead of
-the entire image.
-2) Transformer Networks: Deep learning models in com-
-puter vision tasks have long been dominated by CNN (Convo-
-lutional Neural Network) to extract high-level feature maps by
-passing the image through a series of convolution operations
-before feeding into the MLP (Multi-Layer Perceptron) for clas-
-sification [43]. In recent days, transformer models have shown
-a substantial rise in the NLP (Natural Language Processing)
-domain due to its higher performances [20]. In a similar quest
-to leverage high-level performance through transformers, it
-has been introduced in image classification and some other
-computer vision-oriented tasks [19]. Moreover, the transformer
-model has lesser image-specific inductive bias than CNN [19].
-To identify the severity stages of DR from fundus images,
-here we efficiently adopt and ensemble some image transform-
-ers, e.g., ViT [19], BEiT [23], CaiT [21], and DeiT [22].
-Before focusing on our ensembled transformer model, we
-discuss the adaptation of individual image transformers for
-our task, and start with ViT.
-a) Vision Transformer (ViT): The ViT model adopts the idea
-of text-based transformer models [44], where the idea is to take
-the input image as a series of image patches instead of textual
-words, and then extract features to feed it into an MLP [19].
-The pictorial representation of ViT is presented in Fig. 2.
-Here, the input image Iz is converted into a sequence of
-Fig. 2. Workflow of ViT.
-Fig. 3. Internal view of a transformer encoder (TE).
-flattened patches xi
-p (for i = 1, 2, . . . , np), each with size
-wp × wp × cp, where cp denotes the number of channels
-of Iz. Here, cp = 3, since Iz is an RGB fundus image. In
-our task, Iz is of size 256 × 256, and empirically, we choose
-wp = 64, which results np = ( 256
-64 )2 = 16. Each patch xi
-p is
-flattened further and mapped to a D-dimensional latent vector
-(i.e., patch embedding z0) through transformer layers using a
-trainable linear projection, as below.
-z0 = [xclass ; x1
-p E ; x2
-p E ; . . . ; xnp
-p E] + Epos
-(1)
-where,
-E
-is
-the
-patch
-embedding
-projection,
-E
-∈
-Rwp×wp×C×D; Epos is the position embeddings added to
-patch embeddings to preserve the positional information of
-patches, Epos ∈ R(np+1)×D; xclass = z0
-0 is a learnable
-embedding [24].
-After mapping patch images to the embedding space with
-positional information, we add a sequence of transformer
-encoders [19], [45]. The internal view of a transformer encoder
-can be seen in Fig. 3, which includes two blocks As and Fn.
-The As and Fn contain MSA (Multi-head Self-Attention) [19]
-and MLP [15] modules, respectively. LN (Layer Normaliza-
-tion) [46] and residual connection [15] are employed before
-and after each of these modules, respectively. This is shown
-in equation 2 with general semantics. Here, the MLP module
-comprises two layers having 4D and D neurons with GELU
-(Gaussian Error Linear Unit) non-linear activation function
-
-Patch + Position
-Embedding
-0
-MLP Head
-Softmax
-*
-Linear Projection of Flattened Patches
-1
-Transformer Encoder (TE)
-2
-3
-dn
-np
-pLx
-LN
-MSA
-LN
-MLP
-zi
-Zt4
-GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
-similar to [19].
-z′
-l = MSA(LN(zl−1)) + zl−1;
-zl = MLP(LN(z′
-l)) + z′
-l; l = 1, 2, . . . , L
-(2)
-where, L is the total number of transformer blocks. The core
-component of the transformer encoder is MSA with h heads,
-where each head includes SA (Scaled dot-product Attention)
-[19], [45]. Each head i ∈ {1, 2, ..., h} of MSA calculates a
-tuple comprising query, key, and value [19], i.e., (Qi, Ki, V i)
-as follows.
-Qi = XW i
-Q ; Ki = XW i
-K ; V i = XW i
-V
-(3)
-where, X is the input embedding, and WQ, WK, WV are the
-weight matrices used in the linear transformation. The tuple
-(Q, K, V ) is fed to SA that computes the attention required
-to pay to the input image patches, as below.
-SA(Q, K, V ) = ψ
-�QKT
-√Dh
-�
-V
-(4)
-where, ψ is softmax function, and Dh = D/h. The outcomes
-of SAs across all heads are concatenated in MSA, as follows.
-MSA(Q, K, V ) = [SA1 ; SA2 ; . . . ; SAh]WL
-(5)
-where WL is a weight matrix.
-After multiple transformer encoder blocks, the <class>
-token [24] enriches with the contextual information. The state
-of the learnable embedding at the outcome of the Transformer
-encoder (z0
-L) acts as the image representation y [19].
-y = LN(z0
-L)
-(6)
-Now, as shown in Fig. 2, we add an MLP head containing
-a hidden layer with 128 neurons. To capture the non-linearity,
-we use Mish [47] here. In the output layer, we keep five
-neurons with softmax activation function to obtain probability
-distribution s(j) in order to classify a fundus photograph into
-the abovementioned five severity stages of DR.
-b) Data efficient image Transformers (DeiT): For a lower
-amount of training data, ViT does not generalize well. In
-this scenario, DeiT can perform reasonably well and uses
-lower memory [22]. DeiT adopts the ViT-specific strategy and
-merges with the teacher-student scheme through knowledge
-distillation [48]. The crux of DeiT is the knowledge distillation
-mechanism, which is basically the knowledge transfer from
-one model (teacher) to another (student) [22]. Here, we use
-EfficientNet-B5 [37] as a teacher model that is trained apriori.
-The student model uses a transformer, which learns from the
-outcome of the teacher model through attention depending
-on a distillation token [22]. In this work, we employ hard-
-label distillation [22], where the hard decision of the teacher
-is considered as a true label, i.e., yt = argmaxcZt(c). The
-hard-label distillation objective is defined as follows.
-Lhard
-global = 0.5 LCE(ψ(Zs), y) + 0.5 LCE(ψ(Zs), yt)
-(7)
-where, LCE is the cross-entropy loss on ground-truth labels
-y, ψ is the softmax function, Zs and Zt are the student and
-teacher models’ logits, respectively. Using label smoothing,
-Fig. 4. The distillation procedure of DeiT.
-hard labels can be converted into soft ones [22].
-In Fig. 4, we present the distillation procedure of DeiT.
-Here, we add the <distillation> token to the transformer,
-which interacts with the <class> and <patch> tokens through
-transformer encoders. The transformer encoder used here is
-similar to the ViT’s one, which includes As and Fn blocks as
-shown in Fig. 3. The objective of the <distillation> token is to
-reproduce the teacher’s predicted label instead of the ground-
-truth label. The <distillation> and <class> tokens are learned
-by back-propagation [15].
-A linear classifier is used in DeiT instead of the MLP head
-of ViT [19], [22] to work efficiently with limited computa-
-tional resources.
-c) Class-attention in image Transformers (CaiT): CaiT
-usually performs better than ViT and DeiT with lesser FLOPs
-and learning parameters [15], when we need to increase the
-depth of the transformer [21]. CaiT is basically an upgraded
-version of ViT, which leverages layers with specific class-
-attention and LayerScale [21]. In Fig. 5, we show the workflow
-of CaiT.
-LayerScale aids CaiT to work at larger depths, where we
-separately multiply a diagonal matrix Mλ on the outputs of
-As and Fn blocks.
-z′
-l = Mλ(λl
-1, . . . , λl
-D) × MSA(LN(zl−1)) + zl−1;
-zl = Mλ(λ′l
-1, . . . , λ′l
-D) × MLP(LN(z′
-l)) + z′
-l
-(8)
-where, λl
-i and λ′l
-i are learning parameters, and other symbols
-denote the same as the above-mentioned ViT.
-In CaiT, the transformer layers dealing with self-attention
-Fig. 5. Workflow of CaiT.
-
-Self-attention
-lass-attentionO
-口ADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS
-5
-among patches are separated from class-attention layers that
-are introduced to dedicatedly extract the content of the patches
-into a vector, which can be sent to a linear classifier [21]. The
-<class> token is inserted in the latter stage, so that the initial
-layers can perform the self-attention among patches devotedly.
-In the class-attention stage, we alternatively use multi-head
-class-attention (Ac) [21] and Fn, as shown in Fig. 5, and
-update only the class embedding.
-d) Bidirectional Encoder representation for image Trans-
-former (BEiT): BEiT is a self-supervised model having its
-root in the BERT (Bidirectional Encoder Representations from
-Transformers) [23], [24]. In Fig. 6, we present the workflow
-of the pre-training of BEiT.
-The input image Iz is split into patches xi
-p and flattened
-into vectors, similar to the early-mentioned ViT. In BEiT, a
-backbone transformer is engaged, for which we use ViT [19].
-On the other hand, Iz is represented as a sequence of visual
-tokens vt = [vt1, vt2, . . . , vtnp] obtained by a discrete VAE
-(Variational Auto-Encoder) [49]. For visual token learning, we
-employ a tokenizer Tφ(vt | x) to map image pixels x to tokens
-vt, and decoder Dθ(x | vt) for reconstructing input image
-pixels x from vt [23].
-Here, a MIM (Masked Image Modeling) [23] task is per-
-formed to pre-train the image transformers, where some image
-patches are randomly masked, and the corresponding visual
-tokens are then predicted. The masked patches are replaced
-with a learnable embedding e[M]. We feed the corrupted image
-patches xM = {xi
-p : i /∈ M} �{e[M] : i ∈ M} to the
-transformer encoder. Here, M is the set of indices of masked
-positions.
-The encoded representation hL
-i
-is the hidden vector of
-the last transformer layer L for ith patch. For each masked
-Fig. 6. Workflow of BEiT pre-training.
-position, a softmax classifier ψ is used to predict the respective
-visual token, i.e., pMIM(vt′ | xM) = ψ(WMhL
-i + bM); where,
-WM and bM contain learning parameters for linear transfor-
-mation. The pre-training objective of BEiT is to maximize the
-log-likelihood of the correct token vti given xM, as below:
-max
-�
-x∈ DBtr
-EM
-� �
-i∈M
-log pMIM
-�
-vti | xM�
-�
-where, DBtr is the training dataset. The BEiT pre-training
-can be perceived as VAE training [23], [49], where we follow
-two stages, i.e., stage-1: minimizing loss for visual token
-reconstruction, stage-2: modeling masked image, i.e., learning
-prior pMIM by keeping Tφ and Dθ fixed. It can be written as
-follows:
-�
-(xi,xM
-i
-)
-∈ DBtr
-�
-�
-�
-�Evti∼Tφ(vt|xi) [log Dθ(xi|vti)]
-�
-��
-�
-stage-1
-+ log pMIM
-�
-ˆ
-vti|xM
-i
-�
-�
-��
-�
-stage-2
-�
-�
-�
-�
-where, ˆ
-vti = argmaxvt Tφ(vt | xi).
-3) Ensembled Transformers: The abovementioned four im-
-age transformers, i.e., ViT [19], DeiT [22], CaiT [21], and
-BEiT [23] are pre-trained on the training set DBtr. We now
-ensemble the transformers for predicting the severity stages
-from fundus images of the test set DBt, since ensembling
-multiple learning algorithms can achieve better performance
-than the constituent algorithms alone [50]. The pictorial rep-
-resentation of ensembled transformers is presented in Fig. 7.
-For an image sample from DBt, we obtain the softmax
-probability distribution s(j) : {P j
-1 , P j
-2 , . . . , P j
-nc} over jth
-transformer [15], for j = 1, 2, . . . , nT ; where, nc is the total
-number of classes (severity stages), and nT is count of the
-employed image transformers. Here, �nc
-i=1 P j
-i = 1, nc = 5
-(refer to § III-A), and nT = 4 since we use four separately
-trained distinct image transformers, as mentioned earlier.
-We obtain the severity stages/ class_labels Á|wm and Á|mv
-separately using two combination methods weighted mean and
-majority voting [50], respectively.
-Á|wm = argmaxi P µ
-i ; for i = 1, 2, . . . , nc ;
-P µ
-i =
-�nT
-j=1 αjP j
-i
-�nT
-j=1 αj
-(9)
-Fig. 7. Ensembled transformers.
-
-Masked
-Image
-Patches
-Original Image
-latten
-0
-*
-Tokenizer
-L
-1
-BEiT Encoder
-MIM Head
-2
-[M]
-h2
-3
-Unused at
-Pre-training
-Decoder
-Patch + Position
-Embedding
-Reconstructed ImageViT
-s(1)
-ative
-s(2)
-DeiT
-Mild
-Moderate
-Severe
-CaiT
-s(3)
-Proliferative
-c
-(4)
-BEiT6
-GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
-In this task, we choose �nT
-j=1 αj = 1.
-Á|mv = mode
-�
-argmaxi(P 1
-i ), argmaxi(P 2
-i ), . . . , argmaxi(P nT
-i
-)
-�
-= mode
-�
-argmaxi(s(1)), argmaxi(s(2)), . . . , argmaxi(s(nT ))
-�
-;
-for i = 1, 2, . . . , nc
-(10)
-In this task, we use cross-entropy as the loss function [41]
-in the employed image transformers. The AdamW optimizer
-is used here due to its weight decay regularization effect
-for tackling overfitting [51]. The training details with hyper-
-parameter tuning are mentioned in Section IV-B.
-IV. EXPERIMENTS AND DISCUSSIONS
-In this section, we present the employed database, followed
-by experimental results with discussions.
-A. Database Employed
-For our computational experiments, we used the publicly
-available training samples of Kaggle APTOS (Asia Pacific
-Tele-Ophthalmology Society) 2019 Blindness Detection dataset
-[7], i.e., APTOS-2019. This database (DB) contains fundus
-image samples of five severity stages of DR, i.e., negative,
-mild, moderate, severe, and proliferative. Fig. 1 shows some
-sample images from this dataset. In DB, a total of 3662 fundus
-images are available, which we divide into training (DBtr) and
-testing (DBt) datasets with a ratio of 7 : 3. As a matter of fact,
-DBtr and DBt sets are disjoint. The sample counts of different
-severity stages/ class_labels (Á) for DBtr and DBt are shown
-in Fig. 8 individually. Here, 49.3% samples are of negative
-DR (Á= 0). Among positive classes, most samples are from
-the moderate stage (Á= 2). From this figure, it can be seen
-DB is imbalanced due to containing a different number of
-samples corresponding to various severity stages. Therefore,
-we augmented the data during the training of our model as
-mentioned in § III-B.1. The data augmentation also helped in
-reducing the overfitting issue [15].
-Fig. 8. Count of samples in APTOS-2019 [7].
-B. Experimental Results
-This section discusses the performed experiments, analyzes
-the model outcome, and compares them with major state-of-
-the-art methods. We begin with discussing the experimental
-settings.
-1) Experiment Settings: We performed the experiments on
-the TensorFlow-2 framework having Python 3.7.13 over a
-machine with the following configurations: Intel(R) Xeon(R)
-CPU @ 2.00GHz with 52 GB RAM and Tesla T4 16 GB
-GPU. All the results shown here were obtained from DBt.
-The hyper-parameters of the framework were tuned and
-fixed during training with respect to the performance over
-some samples of DBt employed for hyper-parameter tuning.
-For all the image transformers used here (i.e., ViT, DeiT, CaiT,
-and BEiT), we empirically set the following hyper-parameters:
-transformer_layers (L) = 12, embedding_dimension (D) =
-384, num_heads (h) = 6. The following hyper-parameters
-were selected for AdamW [51]: initial_learning_rate = 10−3;
-exponential decay rates for 1st and 2nd moment estimates, i.e.,
-β1 = 0.9, β2 = 0.999; zero-denominator removal parameter
-(ε) = 10−8; and weight_decay = 10−3/4. For model training,
-the mini-batch size was fixed to 32.
-2) Model Performance: In Table I, we present the per-
-formance of our ensembled image transformer (EiT) using
-the combination schemes weighted mean (wm) and majority
-voting (mv), where we obtain 94.63% and 91.26% accuracy
-from EiTwm and EiTmv, respectively. We also ensembled
-multiple combinations of our employed transformers, and
-present their performances in this table. Here, the wm scheme
-performed better than mv. As evident from this table, ensem-
-bling various types of transformers improved the performance.
-Among single transformers (for nT = 1), CaiT performed
-the best. For nT = 2 and nT = 3, “BEiT + CaiT” and
-“DeiT + BEiT + CaiT” performed better than other respective
-combinations. Overall, EiTwm attained the best accuracy here.
-TABLE I
-PERFORMANCE OVER VARIOUS ENSEMBLING OF TRANSFORMERS
-nT
-Ensembled Transformers
-Accuracy (%)
-Weighted
-Majority
-mean
-voting
-1
-ViT
-82.21
-DeiT
-85.65
-BEiT
-86.74
-CaiT
-86.91
-2
-ViT + DeiT
-87.03
-86.55
-ViT + BEiT
-87.48
-87.03
-ViT + CaiT
-87.77
-87.21
-DeiT + BEiT
-88.18
-87.69
-DeiT + CaiT
-88.86
-87.93
-BEiT + CaiT
-89.28
-88.12
-3
-ViT + DeiT + BEiT
-90.53
-88.87
-ViT + DeiT + CaiT
-91.39
-89.56
-ViT + BEiT + CaiT
-92.14
-90.28
-DeiT + BEiT + CaiT
-93.46
-90.91
-4
-ViT + DeiT + BEiT + CaiT
-94.63
-91.26
-( EiT )
-In Fig. 10 of Appendix I, we present the coarse localization
-maps generated by Grad-CAM [52] from the employed indi-
-vidual image transformers to highlight the crucial regions for
-understanding the severity stages.
-a) Various Evaluation Metrics: Besides the accuracy, in
-Table II, we present the performance of EiT with respect to
-some other evaluation metrics, e.g., kappa score, precision,
-recall, F1 score, specificity, balanced accuracy [53]. Here,
-Cohen’s quadratic weighted kappa measures the agreement
-
-2000
-1800
- DBtr
- DBt
-1600
-541
-1400
-1200
- sampl
-1000
-800
-300
-#
-600
-1264
-400
-111
-699
-200
-88
-259
-135
-207
-0
-0
-1
-2
-3
-4
-severity stage/ class
-label (c)ADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS
-7
-between human-assigned scores (i.e., DR severity stages)
-and the EiT-predicted scores. Precision analyzes the true
-positive samples among the total positive predictions. Recall
-or sensitivity finds the true positive rate. Similarly, specificity
-computes the true negative rate. F1 score is the harmonic mean
-of precision and recall. Since the employed DB is imbalanced,
-we also compute the balanced accuracy, which is the arithmetic
-mean of sensitivity and specificity. In this table, we can see
-that for both EiTwm and EiTmv, the kappa scores are greater
-than 0.81, which comprehends the “almost perfect agreement”
-between the human rater and EiT [53]. Here, macro means
-the arithmetic mean of all per class precision/ recall/ F1 score.
-TABLE II
-PERFORMANCE OF EiT OVER VARIOUS EVALUATION METRICS
-Metric
-Weighted mean
-Majority voting
-(EiTwm)
-(EiTmv)
-Accuracy (%)
-94.63
-91.26
-Kappa score
-0.92
-0.87
-Macro Precision (%)
-90.55
-84.65
-Macro Recall (%)
-92.88
-88.81
-Macro F1-score (%)
-91.67
-86.55
-Macro Specificity (%)
-98.62
-97.74
-Balanced Accuracy (%)
-95.75
-93.27
-b) Individual Class Performance: Table III presents the
-individual performance of EiTwm and EiTmv for detecting
-every severity stage of DR. From this table, we can see our
-models produced the best precision and recall for negative DR
-(Á= 0), and the lowest for severe DR (Á= 3).
-TABLE III
-PERFORMANCE OF EiT ON EVERY DR SEVERITY STAGE
-class_label (Á)
-0
-1
-2
-3
-4
-EiTwm
-Precision (%)
-98.48
-86.67
-95.00
-83.61
-89.01
-Recall (%)
-95.75
-93.69
-95.00
-87.93
-92.05
-F1-score (%)
-97.09
-90.04
-95.00
-85.71
-90.50
-Specificity (%)
-98.56
-98.38
-98.12
-99.04
-99.01
-EiTmv
-Precision (%)
-96.74
-79.67
-94.14
-70.59
-82.11
-Recall (%)
-93.35
-88.29
-91.00
-82.76
-88.64
-F1-score (%)
-95.01
-83.76
-92.54
-76.19
-85.25
-Specificity (%)
-96.95
-97.47
-97.87
-98.08
-98.32
-In each row, the best result is marked bold, second-best is italic, and lowest is underlined.
-3) Comparison: In Table IV, we present a comparative
-analysis with some major contemporary deep learning archi-
-tectures, e.g., ResNet50 [54], InceptionV3 [55], MobileNetV2
-[56], Xception [57], DenseNet169 (Farag et al. [18]), Efficient-
-Net [37], and SE-ResNeXt50 [38]. We have also compared
-with recently published transformer-based models, i.e., CoT-
-XNet [39], and SSiT [40]. Comparison with some major
-related works [5], [35], [36] can also be seen in this table.
-Our EiTwm outperformed the major state-of-the-art methods
-with respect to accuracy, balanced accuracy, sensitivity, and
-specificity. Our EiTmv also performed quite well in terms of
-balanced accuracy.
-4) Impact of Hyper-parameters:
-We
-tuned
-the
-hyper-
-parameters and observed their impact on the experiment.
-a) MSA Head Count: We analyzed the performance im-
-pact of the number of heads (h) of MSA (Multi-head Self-
-Attention) in the transformer encoder and present in Fig. 9.
-As evident from this figure, the performance (accuracy) of
-TABLE IV
-COMPARATIVE STUDY
-Method
-Accuracy
-Sensitivity
-Specificity
-Balanced
-(%)
-(%)
-(%)
-Accuracy (%)
-ResNet50 [54]
-74.64
-56.52
-85.71
-71.12
-InceptionV3 [55]
-78.72
-63.64
-85.37
-74.51
-MobileNetV2 [56]
-79.01
-76.47
-84.62
-80.55
-Xception [57]
-79.59
-82.35
-86.32
-84.34
-Farag et al. [18]
-82.00
--
--
--
-Kassani et al. [35]
-83.09
-88.24
-87.00
-87.62
-TAN [36]
-85.10
-90.30
-92.00
--
-EfficientNet-B4 [37]
-90.30
-81.20
-97.60
-89.40
-EfficientNet-B5 [37]
-90.70
-80.70
-97.70
-89.20
-SE-ResNeXt50 [38]
-92.40
-87.10
-98.20
-92.65
-Tymchenko et al. [5]
-92.90
-86.00
-98.30
-92.15
-CoT-XNet [39]
-84.18
--
-95.74
--
-SSiT [40]
-92.97
--
--
--
-EiTmv [ours]
-91.26
-88.81
-97.74
-93.28
-EiTwm [ours]
-94.63
-92.88
-98.62
-95.75
-In each column, the best result is marked bold, and the second-best is underlined.
-Fig. 9. Impact of number of heads (h) in MSA on model performance.
-both EiTmv and EiTwm increased with the increment of h
-till h = 6, and started decreasing thereafter.
-b) Weights αj of EiTwm: We tuned the weights αj (refer
-to Eqn. 9) to see its impact on the performance of EiTwm.
-We obtained the best accuracy of 94.63% from EiTwm for
-α1 = α2 = 0.1, and α3 = α4 = 0.4. The performance of
-EiTwm during tuning of αj’s is shown in Table V.
-In Table VI, we also present the tuned αj’s that aided in
-obtaining the best performing ensembled transformers of Table
-I.
-5) Ablation Study: We here present the performed ablation
-study by ablating individual transformers. Our EiT is actually
-an ensembling of four different image transformers, i.e., ViT,
-DeiT, CaiT, and BEiT. We ablated each transformer and
-observed performance degradation than EiT. For example,
-considering the weighted mean scheme, when we ablated CaiT
-from EiT, the accuracy dropped by 4.1%. Similarly, ablating
-BEiT and CaiT deteriorated the accuracy by 7.6%. For our
-task, the best individual transformer (CaiT) attained 7.72%
-lower accuracy than EiTwm. More examples can be observed
-in Table I.
-6) Pre-training with Other Datasets: We checked the perfor-
-mance of our EiT model by pre-training with some other
-dataset. We took 1200 images of MESSIDOR [58] with
-adjudicated grades by [59] (say, DBM). From IDRiD [60],
-we also used “Disease Grading” dataset containing 516 images
-(say, DBI). Here, we made four training set setups from DBM,
-by taking 25%, 50%, 75%, and 100% of samples of DBM.
-
-96
-94.63
-EiTmv
-94
-92.34
-91.92
-92
-91
-90.28
-90
-89.43
-88.59
-87.67
-88
-87
-86
-84
-82
-80
-6
-8
-108
-GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
-TABLE V
-PERFORMANCE OF EiTwm BY TUNING WEIGHTS αj
-α1
-α2
-α3
-α4
-Accuracy (%)
-0.25
-0.25
-0.25
-0.25
-89.53
-0.85
-0.05
-0.05
-0.05
-82.29
-0.05
-0.85
-0.05
-0.05
-85.78
-0.05
-0.05
-0.85
-0.05
-86.92
-0.05
-0.05
-0.05
-0.85
-87.05
-0.7
-0.1
-0.1
-0.1
-82.35
-0.1
-0.7
-0.1
-0.1
-85.91
-0.1
-0.1
-0.7
-0.1
-87.04
-0.1
-0.1
-0.1
-0.7
-87.20
-0.5
-0.167
-0.167
-0.166
-82.88
-0.166
-0.5
-0.167
-0.167
-86.35
-0.167
-0.166
-0.5
-0.167
-87.62
-0.167
-0.167
-0.166
-0.5
-87.74
-0.3
-0.3
-0.2
-0.2
-88.16
-0.3
-0.2
-0.3
-0.2
-89.58
-0.3
-0.2
-0.2
-0.3
-90.27
-0.2
-0.3
-0.3
-0.2
-90.85
-0.2
-0.3
-0.2
-0.3
-91.67
-0.2
-0.2
-0.3
-0.3
-92.72
-0.4
-0.4
-0.1
-0.1
-91.18
-0.4
-0.1
-0.4
-0.1
-91.49
-0.4
-0.1
-0.1
-0.4
-92.15
-0.1
-0.4
-0.4
-0.1
-92.84
-0.1
-0.4
-0.1
-0.4
-93.47
-0.1
-0.1
-0.4
-0.4
-94.63
-TABLE VI
-TUNED WEIGHTS αj FOR TRANSFORMERS ENSEMBLED WITH
-WEIGHTED MEAN
-Transformerswm
-α1
-α2
-α3
-α4
-ViT + DeiT
-0.25
-0.75
--
--
-ViT + BEiT
-0.4
-0.6
--
--
-ViT + CaiT
-0.4
-0.6
--
--
-DeiT + BEiT
-0.4
-0.6
--
--
-DeiT + CaiT
-0.3
-0.7
--
--
-BEiT + CaiT
-0.5
-0.5
--
--
-ViT + DeiT + BEiT
-0.2
-0.3
-0.5
--
-ViT + DeiT + CaiT
-0.2
-0.3
-0.5
--
-ViT + BEiT + CaiT
-0.2
-0.4
-0.4
--
-DeiT + BEiT + CaiT
-0.3
-0.3
-0.4
--
-ViT + DeiT + BEiT + CaiT
-0.1
-0.1
-0.4
-0.4
-Similarly, four training setups were generated from DBI. As
-mentioned in § IV-A, we divided APTOS-2019 database (DB)
-in training (DBtr) and test (DBt) sets with a ratio of 7 : 3. In
-Table VII, we present the performance of EiT on DBt, while
-pre-training with DBM and DBI, and training with DBtr.
-It can be observed that the performance of EiT improved
-slightly when pre-trained with more data from other datasets.
-TABLE VII
-ACCURACY (%) OF EiT WITH PRE-TRAINING
-Pre-training data
-25%
-50%
-75%
-100%
-EiTwm
-DBM
-94.71
-94.78
-94.83
-94.88
-DBI
-94.65
-94.67
-94.7
-94.79
-DBM + DBI
-94.73
-94.85
-94.98
-95.13
-N.A.
-94.63
-EiTmv
-DBM
-91.35
-91.48
-91.56
-91.61
-DBI
-91.27
-91.32
-91.34
-91.35
-DBM + DBI
-91.42
-91.6
-91.68
-91.75
-N.A.
-91.26
-N.A.: without pre-training data
-V. CONCLUSION
-In this paper, we tackle the problem of automated severity
-stage detection of DR from fundus images. For this purpose,
-we propose two ensembled image transformers, EiTwm and
-EiTmv, by using weighted mean and majority voting combi-
-nation schemes, respectively. We here adopt four transformer
-models, i.e., ViT, DeiT, CaiT, and BEiT. For experimentation,
-we employed the publicly available APTOS-2019 blindness
-detection dataset, on which EiTwm and EiTmv attained
-accuracies of 94.63% and 91.26%, respectively. Although
-the employed dataset was imbalanced, our models performed
-quite well. Our EiTwm outperformed the major state-of-the-
-art techniques. We also performed an ablation study and
-observed the importance of the ensembling over the individual
-transformers.
-In the future, we will endeavor to improve the model perfor-
-mance with some imbalanced learning techniques. Currently,
-our model does not perform any lesion segmentation, which
-we will also attempt to explore some implicit characteristics
-of fundus images due to DR.
-APPENDIX I
-QUALITATIVE VISUALIZATION
-As mentioned in § IV-B.2, we present the Grad-CAM maps
-of the employed individual image transformers in Fig. 10.
-negative
-mild
-moderate
-severe
-proliferative
-Fig. 10.
-Fundus images (1st row) with Grad-CAM maps for ViT, DeiT,
-BEiT, CaiT as shown in 2nd, 3rd, 4th, 5th rows, respectively.
-REFERENCES
-[1] S. Stolte and R. Fang, “A survey on medical image analysis in diabetic
-retinopathy,” Medical image analysis, vol. 64, p. 101742, 2020.
-[2] N. Asiri et al., “Deep learning based computer-aided diagnosis systems
-for diabetic retinopathy: A survey,” Artificial intelligence in medicine,
-vol. 99, p. 101701, 2019.
-[3] Y. Zheng et al., “The worldwide epidemic of diabetic retinopathy,”
-Indian journal of ophthalmology, vol. 60, no. 5, p. 428, 2012.
-[4] D. S. W. Ting et al., “Development and validation of a deep learning
-system for diabetic retinopathy and related eye diseases using retinal
-images from multiethnic populations with diabetes,” JAMA, vol. 318,
-no. 22, pp. 2211–2223, 2017.
-[5] B. Tymchenko et al., “Deep learning approach to diabetic retinopathy
-detection,” in ICPRAM, 2020, pp. 501–509.
-
-SADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS
-9
-[6] J. Krause et al., “Grader variability and the importance of reference stan-
-dards for evaluating machine learning models for diabetic retinopathy,”
-Ophthalmology, vol. 125, no. 8, pp. 1264–1272, 2018.
-[7] APTOS 2019 Blindness Detection. [Online]. Available: https://www.
-kaggle.com/competitions/aptos2019-blindness-detection
-[8] G. Quellec et al., “Optimal wavelet transform for the detection of
-microaneurysms in retina photographs,” IEEE transactions on medical
-imaging, vol. 27, no. 9, pp. 1230–1241, 2008.
-[9] L. Giancardo et al., “Microaneurysm detection with radon transform-
-based classification on retina images,” in EMBS, 2011, pp. 5939–5942.
-[10] R. Casanova et al., “Application of random forests methods to diabetic
-retinopathy classification analyses,” PLOS one, vol. 9, no. 6, p. e98587,
-2014.
-[11] M. Niemeijer et al., “Automated detection and differentiation of drusen,
-exudates, and cotton-wool spots in digital color fundus photographs for
-diabetic retinopathy diagnosis,” Investigative ophthalmology & visual
-science, vol. 48, no. 5, pp. 2260–2267, 2007.
-[12] M. U. Akram et al., “Identification and classification of microaneurysms
-for early detection of diabetic retinopathy,” Pattern recognition, vol. 46,
-no. 1, pp. 107–116, 2013.
-[13] D. Usher et al., “Automated detection of diabetic retinopathy in digital
-retinal images: a tool for diabetic retinopathy screening,” Diabetic
-Medicine, vol. 21, no. 1, pp. 84–90, 2004.
-[14] J. D. Bodapati et al., “Deep convolution feature aggregation: an appli-
-cation to diabetic retinopathy severity level prediction,” Signal, Image
-and Video Processing, vol. 15, no. 5, pp. 923–930, 2021.
-[15] A. Zhang, Z. C. Lipton, M. Li, and A. J. Smola, “Dive into deep
-learning,” arXiv:2106.11342, 2021.
-[16] S. Yu et al., “Exudate detection for diabetic retinopathy with convolu-
-tional neural networks,” in EMBC, 2017, pp. 1744–1747.
-[17] Z. Wang et al., “Zoom-in-net: Deep mining lesions for diabetic retinopa-
-thy detection,” in MICCAI, 2017, pp. 267–275.
-[18] M. M. Farag et al., “Automatic severity classification of diabetic
-retinopathy based on densenet and convolutional block attention mod-
-ule,” IEEE Access, vol. 10, pp. 38 299–38 308, 2022.
-[19] A. Dosovitskiy et al., “An image is worth 16x16 words: Transformers
-for image recognition at scale,” arXiv:2010.11929, 2020.
-[20] A. M. Bra¸soveanu and R. Andonie, “Visualizing transformers for nlp:
-a brief survey,” in Int. Conf. on Information Visualisation (IV).
-IEEE,
-2020, pp. 270–279.
-[21] H. Touvron et al., “Going deeper with image transformers,” in ICCV,
-2021, pp. 32–42.
-[22] ——, “Training data-efficient image transformers & distillation through
-attention,” in ICML, 2021, pp. 10 347–10 357.
-[23] H. Bao, L. Dong, and F. Wei, “BEiT: BERT Pre-Training of Image
-Transformers,” in ICLR, arXiv:2106.08254, 2022.
-[24] J. Devlin et al., “BERT: Pre-training of Deep Bidirectional Transformers
-for Language Understanding,” arXiv:1810.04805, 2018.
-[25] “Grading diabetic retinopathy from stereoscopic color fundus pho-
-tographs—an extension of the modified airlie house classification: Etdrs
-report number 10,” Ophthalmology, vol. 98, no. 5, Supplement, pp. 786–
-806, 1991.
-[26] U. R. Acharya et al., “Automated diabetic macular edema (dme) grading
-system using dwt, dct features and maculopathy index,” Computers in
-biology and medicine, vol. 84, pp. 59–68, 2017.
-[27] K. M. Adal et al., “An automated system for the detection and clas-
-sification of retinal changes due to red lesions in longitudinal fundus
-images,” IEEE transactions on biomedical engineering, vol. 65, no. 6,
-pp. 1382–1390, 2017.
-[28] L. Tang et al., “Splat feature classification with application to retinal
-hemorrhage detection in fundus images,” IEEE Transactions on Medical
-Imaging, vol. 32, no. 2, pp. 364–375, 2013.
-[29] A. Herliana et al., “Feature selection of diabetic retinopathy disease
-using particle swarm optimization and neural network,” in CITSM.
-IEEE, 2018, pp. 1–4.
-[30] S. Sanromà et al., “Assessment of diabetic retinopathy risk with random
-forests.” in ESANN, 2016.
-[31] A. Osareh et al., “A computational-intelligence-based approach for
-detection of exudates in diabetic retinopathy images,” IEEE Transactions
-on Information Technology in Biomedicine, vol. 13, no. 4, pp. 535–545,
-2009.
-[32] P. Chudzik et al., “Microaneurysm detection using deep learning and
-interleaved freezing,” in Medical imaging 2018: image processing, vol.
-10574.
-SPIE, 2018, pp. 379–387.
-[33] R. Gargeya and T. Leng, “Automated identification of diabetic retinopa-
-thy using deep learning,” Ophthalmology, vol. 124, no. 7, pp. 962–969,
-2017.
-[34] S. Wan et al., “Deep convolutional neural networks for diabetic retinopa-
-thy detection by image classification,” Computers & Electrical Engineer-
-ing, vol. 72, pp. 274–282, 2018.
-[35] S. H. Kassani et al., “Diabetic retinopathy classification using a modified
-xception architecture,” in ISSPIT, 2019, pp. 1–6.
-[36] M. D. Alahmadi, “Texture attention network for diabetic retinopathy
-classification,” IEEE Access, vol. 10, pp. 55 522–55 532, 2022.
-[37] M. Tan and Q. Le, “Efficientnet: Rethinking model scaling for convo-
-lutional neural networks,” in ICML, 2019, pp. 6105–6114.
-[38] J. Hu et al., “Squeeze-and-excitation networks,” in CVPR, 2018, pp.
-7132–7141.
-[39] S. Zhao et al., “Cot-xnet: Contextual transformer with xception network
-for diabetic retinopathy grading,” Physics in Medicine & Biology, 2022.
-[40] Y. Huang et al., “Ssit: Saliency-guided self-supervised image trans-
-former for diabetic retinopathy grading,” arXiv:2210.10969, 2022.
-[41] I. Goodfellow et al., Deep learning.
-MIT press, 2016.
-[42] V. Stimper et al., “Multidimensional contrast limited adaptive histogram
-equalization,” IEEE Access, vol. 7, pp. 165 437–165 447, 2019.
-[43] D. Sarvamangala et al., “Convolutional neural networks in medical
-image understanding: a survey,” Evolutionary intel., pp. 1–22, 2021.
-[44] T. Wolf et al., “Transformers: State-of-the-Art Natural Language Pro-
-cessing,” in EMNLP: System Demonstrations, 2020, pp. 38–45.
-[45] A. Vaswani et al., “Attention is all you need,” Advances in neural
-information processing systems, vol. 30, 2017.
-[46] J. L. Ba, J. R. Kiros, and G. E. Hinton, “Layer normalization,”
-arXiv:1607.06450, 2016.
-[47] D. Misra, “Mish: A self regularized non-monotonic neural activation
-function,” BMVC, Paper #928, 2020.
-[48] G. Hinton, O. Vinyals, and J. Dean, “Distilling the Knowledge in a
-Neural Network,” in NIPS Deep Learning and Representation Learning
-Workshop, arXiv:1503.02531, 2015.
-[49] A. Ramesh et al., “Zero-shot text-to-image generation,” in ICML, 2021,
-pp. 8821–8831.
-[50] L. Rokach, “Ensemble-based Classifiers,” Artificial intelligence review,
-vol. 33, no. 1, pp. 1–39, 2010.
-[51] I. Loshchilov and F. Hutter, “Decoupled weight decay regularization,”
-ICLR, arXiv:1711.05101, 2019.
-[52] R. Selvaraju et al., “Grad-CAM: Visual Explanations from Deep Net-
-works via Gradient-based Localization,” in ICCV, 2017, pp. 618–626.
-[53] M. Grandini, E. Bagli, and G. Visani, “Metrics for multi-class classifi-
-cation: an overview,” arXiv preprint arXiv:2008.05756, 2020.
-[54] K. He et al., “Deep residual learning for image recognition,” in CVPR,
-2016, pp. 770–778.
-[55] C. Szegedy et al., “Rethinking the inception architecture for computer
-vision,” in CVPR, 2016, pp. 2818–2826.
-[56] M. Sandler et al., “Mobilenetv2: Inverted residuals and linear bottle-
-necks,” in CVPR, 2018, pp. 4510–4520.
-[57] F. Chollet, “Xception: Deep learning with depthwise separable convo-
-lutions,” in CVPR, 2017, pp. 1251–1258.
-[58] E. Decencière et al., “Feedback on a publicly distributed image database:
-the Messidor database,” Image Analysis & Stereology, vol. 33, no. 3, pp.
-231–234, 2014.
-[59] J. Krause et al., “Grader variability and the importance of reference stan-
-dards for evaluating machine learning models for diabetic retinopathy,”
-Ophthalmology, vol. 125, no. 8, pp. 1264–1272, 2018.
-[60] P. Porwal et al., “Indian Diabetic Retinopathy Image Dataset (IDRiD),”
-2018. [Online]. Available: https://dx.doi.org/10.21227/H25W98
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-arXiv:2301.13054v1  [cs.FL]  30 Jan 2023
-Monadic Expressions and their Derivatives
-Samira Attou1, Ludovic Mignot2, Clément Miklarz2, and Florent Nicart2
-1 Université Gustave Eiffel,
-5 Boulevard Descartes — Champs s/ Marne
-77454 Marne-la-Vallée Cedex 2
-2 GR2IF,
-Université de Rouen Normandie,
-Avenue de l’Université,
-76801 Saint-Étienne-du-Rouvray, France
-[email protected],
-{ludovic.mignot,clement.miklarz1, florent.nicart}@univ-rouen.fr
-Abstract. We propose another interpretation of well-known derivatives computations from regular expres-
-sions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch, in order to abstract the underlying data
-structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization
-advantage, we first introduce a new derivation technique based on the graded module monad and then show an
-application of this technique to generalize the parsing of expression with capture groups and back references.
-We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations
-(like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights).
-Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression,
-using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us
-to perform this construction in a unified way, whatever the underlying monad.
-Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques
-of functional programming, and we provide a web interface to manipulate concrete examples.
-1
-Introduction
-This paper is an extended version of [2].
-Regular expressions are a classical way to represent associations between words and value sets. As an example,
-classical regular expressions denote sets of words and regular expressions with multiplicities denote formal series.
-From a regular expression, solving the membership test (determining whether a word belongs to the denoted
-language) or the weighting test (determining the weight of a word in the denoted formal series) can be solved,
-following Kleene theorems [11,17] by computing a finite automaton, such as the position automaton [9,3,5,6].
-Another family of methods to solve these tests is the family of derivative computations, that does not require the
-construction of a whole automaton. The common point of these techniques is to transform the test for an arbitrary
-word into the test for the empty word, which can be easily solved in a purely syntactical way (i.e. by induction over
-the structure of expressions). Brzozowski [4] shows how to compute, from a regular expression E and a word w, a
-regular expression dw(E) denoting the set of words w′ such that ww′ belongs to the language denoted by E. Solving
-the membership test hence becomes the membership test for the empty word in the expression dw(E). Antimirov [1]
-modifies this method in order to produce sets of expressions instead of expressions, i.e. defines the partial derivatives
-∂w(E) as a set of expressions the sum of which denotes the same language as dw(E). If the number of derivatives
-is exponential w.r.t. the length |E| of E in the worst case3, the partial derivatives produce at most a linear number
-of expressions w.r.t. |E|. Lombardy and Sakarovitch [13] extends these methods to expressions with multiplicities.
-Finally, Sulzmann and Lu [18] apply these derivation techniques to parse POSIX expressions.
-It is well-known that these methods are based on a common operation, the quotient of languages. Furthermore,
-Antimirov’s method can be interpreted as the derivation of regular expression with multiplicities in the Boolean
-semiring. However, the Brzozowski computation does not produce the same expressions (i.e. equality over the syntax
-trees) as the Antimirov one.
-Main contributions: In this paper, we present a unification of these computations by applying notions of
-category theory to the category of sets, and show how to compute categorical automata as defined in [7], by reinter-
-preting the work started in [15]. We make use of classical monads to model well-known derivatives computations.
-Furthermore, we deal with extended expressions in a general way: in this paper, expressions can support extended
-3 as far as rules of associativity, commutativity and idempotence of the sum are considered, possibly infinite otherwise.
-
-operators like complement, intersection, but also any n-ary function (algebraic mean, extrema multiplications, etc.).
-The main difference with [15] is that we formally state the languages and series that the expressions denote in an
-inherent way w.r.t. the underlying monads.
-More precisely, this paper presents:
-– an extension of expressions to any n-ary function over the value set,
-– a monadic generalization of expressions,
-– a solution for the membership/weight test for these expressions,
-– a computation of categorical derivative automata,
-– a new monad that fits with the extension to n-ary functions,
-– an illustration implemented in Haskell using advanced functional programming,
-– an extension to capture groups and back references expressions.
-Motivation: The unification of derivation techniques is a goal by itself. Moreover, the formal tools used to
-achieve this unification are also useful: Monads offer both theoretical and practical advantages. Indeed, from a
-theoretical point of view, these structures allow the abstraction of properties and focus on the principal mechanisms
-that allow solving the membership and weight problems. Besides, the introduction of exotic monads can also facilitate
-the study of finiteness of derivated terms. From a practical point of view, monads are easy to implement (even in
-some other languages than Haskell) and allow us to produce compact and safe code. Finally, we can easily combine
-different algebraic structures or add some technical functionalities (capture groups, logging, nondeterminism, etc.)
-thanks to notions like monad transformers [10] that we consider in this paper.
-This paper is structured as follows. In Section 2, we gather some preliminary material, like algebraic structures
-or category theory notions. We also introduce some functions well-known to the Haskell community that can allow
-us to reduce the size of our equations. We then structurally define the expressions we deal with, the associated series
-and the weight test for the empty word in Section 3. In order to extend this test to any arbitrary word, we first state
-in Section 4 some properties required by the monads we consider. Once this so-called support is determined, we show
-in Section 5 how to compute the derivatives. The computation of derivative automata is explained in Section 6.
-A new monad and its associated derivatives computation is given in Section 7. An implementation is presented
-in Section 8. Finally, we show how to (alternatively to [18]) compute derivatives of capture group expressions in
-Section 9 and show that as far as the same operators are concerned, the derivative formulae are the same whatever
-the underlying monad is.
-2
-Preliminaries
-We denote by S → S′ the set of functions from a set S to a set S′. The notation λx → f(x) is an equivalent notation
-for a function f.
-A monoid is a set S endowed with an associative operation and a unit element. A semiring is a structure
-(S, ×, +, 1, 0) such that (S, ×, 1) is a monoid, (S, +, 0) is a commutative monoid, × distributes over + and 0 is an
-annihilator for ×. A starred semiring is a semiring with a unary function ⋆ such that
-k⋆ = 1 + k × k⋆ = 1 + k⋆ × k.
-A K-series over the free monoid (Σ∗, ·, ε) associated with an alphabet Σ, for a semiring K = (K, ×, +, 1, 0), is
-a function from Σ∗ to K. The set of K-series can be endowed with the structure of semiring as follows:
-1(w) =
-�
-1
-if w = ε,
-0
-otherwise,
-0(w) = 0,
-(S1 + S2)(w) = S1(w) + S2(w),
-(S1 × S2)(w) =
-�
-u·v=w
-S1(u) × S2(v).
-Furthermore, if S1(ε) = 0 (i.e. S1 is said to be proper), the star of S1 is the series defined by
-(S1)⋆(ε) = 1,
-(S1)⋆(w) =
-�
-n≤|w|,w=u1···un,uj̸=ε
-S1(u1) × · · · × S1(un).
-Finally, for any function f in Kn → K, we set:
-(f(S1, . . . , Sn))(w) = f(S1(w), . . . , Sn(w)).
-(1)
-A functor 4 F associates with each set S a set F(S) and with each function f in S → S′ a function F(f) from
-F(S) to F(S′) such that
-F(id) = id,
-F(f ◦ g) = F(f) ◦ F(g),
-4 More precisely, a functor over a subcategory of the category of sets.
-
-where id is the identity function and ◦ the classical function composition.
-A monad5 M is a functor endowed with two (families of) functions
-– pure, from a set S to M(S),
-– bind, sending any function f in S → M(S′) to M(S) → M(S′),
-such that the three following conditions are satisfied:
-bind(f)(pure(s)) = f(s),
-bind(pure) = id,
-bind(g)(bind(f)(m)) = bind(λx → bind(g)(f(x)))(m).
-Example 1. The Maybe monad associates:
-– any set S with the set Maybe(S) = {Just(s) | s ∈ S} ∪ {Nothing}, where Just and Nothing are two syntactic
-tokens allowing us to extend a set with one value;
-– any function f with the function Maybe(f) defined by
-Maybe(f)(Just(s)) = Just(f(s)),
-Maybe(f)(Nothing) = Nothing
-– is endowed with the functions pure and bind defined by:
-pure(s) = Just(s),
-bind(f)(Just(s)) = f(s),
-bind(f)(Nothing) = Nothing.
-Example 2. The Set monad associates:
-– with any set S the set 2S,
-– with any function f the function Set(f) defined by Set(f)(R) = �
-r∈R{f(r)},
-– is endowed with the functions pure and bind defined by:
-pure(s) = {s},
-bind(f)(R) =
-�
-r∈R
-f(r).
-Example 3. The LinComb(K) monad, for K = (K, ×, +, 1, 0), associates:
-– with any set S the set of K-linear combinations of elements of S, where a linear combination is a finite (formal,
-commutative) sum of couples (denoted by ⊞) in K × S where (k, s) ⊞ (k′, s) = (k + k′, s),
-– with any function f the function LinComb(K)(f) defined by
-LinComb(K)(f)(R) = ⊞
-(k,r)∈R
-(k, f(r)),
-– is endowed with the functions pure and bind defined by:
-pure(s) = (1, s),
-bind(f)(R) = ⊞
-(k,r)∈R
-k ⊗ f(r),
-where k ⊗ R = ⊞
-(k′,r)∈R
-(k × k′, r).
-To compact equations, we use the following operators for any monad M:
-f <$> s = M(f)(s),
-m >>= f = bind(f)(m).
-If <$> can be used to lift unary functions to the monadic level, >>= and pure can be used to lift any n-ary function
-f in S1 × · · · × Sn → S, defining a function liftn sending S1 × · · · × Sn → S to M(S1) × · · · × M(Sn) → M(S) as
-follows:
-liftn(f)(m1, . . . , mn) =m1 >>= (λs1 → . . .
-mn >>= (λsn → pure(f(s1, . . . , sn))) . . .)
-Let us consider the set
-1 = {⊤} with only one element. The images of this set by some previously defined monads
-can be evaluated as value sets classically used to weight words in association with classical regular expressions. As
-an example, Maybe(1) and Set(1) are isomorphic to the Boolean set, and any set LinComb(K)(1) can be converted
-into the underlying set of K. This property allows us to extend in a coherent way classical expressions to monadic
-expressions, where the type of the weights is therefore given by the ambient monad.
-5 More precisely, a monad over a subcategory of the category of sets.
-
-3
-Monadic Expressions
-As seen in the previous section, elements in M(1) can be evaluated as classical value sets for some particular
-monads. Hence, we use these elements not only for the weights associated with words by expressions, but also for
-the elements that act over the denoted series.
-In the following, in addition to classical operators (+, · and ∗), we denote:
-– the action of an element over a series by ⊙,
-– the application of a function by itself.
-Definition 1. Let M be a monad. An M-monadic expression E over an alphabet Σ is inductively defined as follows:
-E = a,
-E = ε,
-E = ∅,
-E = E1 + E2,
-E = E1 · E2,
-E = E∗
-1,
-E = α ⊙ E1,
-E = E1 ⊙ α,
-E = f (E1, . . . , En) ,
-where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
-function from (M(1))n to M(1).
-We denote by Exp(Σ) the set of monadic expressions over an alphabet Σ.
-Example 4. As an example of functions that can be used in our extension of classical operators, one can define the
-function ExtDist(x1, x2, x3) = max(x1, x2, x3) − min(x1, x2, x3) from N3 to N.
-Similarly to classical regular expressions, monadic expressions associate a weight with any word. Such a relation
-can be denoted via a formal series. However, before defining this notion, in order to simplify our study, we choose
-to only consider proper expressions. Let us first show how to characterize them by the computation of a nullability
-value.
-Definition 2. Let M be a monad such that the structure (M(1), +, ×, ⋆, 1, 0) is a starred semiring. The nullability
-value of an M-monadic expression E over an alphabet Σ is the element Null(E) of M(1) inductively defined as
-follows:
-Null(ε) = 1,
-Null(∅) = 0,
-Null(a) = 0,
-Null(E1 + E2) = Null(E1) + Null(E2),
-Null(E1 · E2) = Null(E1) × Null(E2),
-Null(E∗
-1) = Null(E1)⋆,
-Null(α ⊙ E1) = α × Null(E1),
-Null(E1 ⊙ α) = Null(E1) × α,
-Null(f(E1, . . . , En)) = f(Null(E1), . . . , Null(En)),
-where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
-function from (M(1))n to M(1).
-When the considered semiring is not a starred one, we restrict the nullability value computation to expressions
-where a starred subexpression admits a null nullability value. In order to compute it, let us consider the Maybe
-monad, allowing us to elegantly deal with such a partial function.
-Definition 3. Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring. The partial nullability
-value of an M-monadic expression E over an alphabet Σ is the element PartNull(E) of Maybe(M(1)) defined as
-follows:
-PartNull(ε) = Just(1),
-PartNull(∅) = Just(0),
-PartNull(a) = Just(0),
-PartNull(E1 + E2) = lift2(+)(PartNull(E1), PartNull(E2)),
-PartNull(E1 · E2) = lift2(×)(PartNull(E1), PartNull(E2)),
-PartNull(E∗
-1) =
-�
-Just(1)
-if PartNull(E1) = Just(0),
-Nothing
-otherwise,
-PartNull(α ⊙ E1) = (λE → α × E) <$> PartNull(E1),
-PartNull(E1 ⊙ α) = (λE → E × α) <$> PartNull(E1),
-PartNull(f(E1, . . . , En)) = liftn(f)(PartNull(E1), . . . , PartNull(En)),
-where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
-function from (M(1))n to M(1).
-
-An expression E is proper if its partial nullability value is not Nothing, therefore if it is a value Just(v); in this
-case, v is its nullability value, denoted by Null(E) (by abuse).
-Definition 4. Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring, and E be a M-monadic
-proper expression over an alphabet Σ. The series S(E) associated with E is inductively defined as follows:
-S(ε)(w) =
-�
-1
-if w = ε,
-0
-otherwise,
-S(∅)(w) = 0,
-S(a)(w) =
-�
-1
-if w = a,
-0
-otherwise,
-S(E1 + E2) = S(E1) + S(E2),
-S(E1 · E2) = S(E1) × S(E2),
-S(E∗
-1) = (S(E1))⋆,
-S(α ⊙ E1)(w) = α × S(E1)(w),
-S(E1 ⊙ α)(w) = S(E1)(w) × α,
-S(f(E1, . . . , En)) = f(S(E1), . . . , S(En)),
-where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
-function from (M(1))n to M(1).
-From now on, we consider the set Exp(Σ) of M-monadic expressions over Σ to be endowed with the structure of
-a semiring, and two expressions denoting the same series to be equal. The weight associated with a word w in Σ∗
-by E is the value weightw(E) = S(E)(w). The nullability of a proper expression is the weight it associates with ε,
-following Definition 3 and Definition 4.
-Proposition 1. Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring. Let E be an M-monadic
-proper expression over Σ. Then:
-Null(E) = weightε(E).
-The previous proposition implies that the weight of the empty word can be syntactically computed (i.e. inductively
-computed from a monadic expression). Now, let us show how to extend this computation by defining the computation
-of derivatives for monadic expressions.
-4
-Monadic Supports for Expressions
-A K-left-semimodule, for a semiring K = (K, ×, +, 1, 0), is a commutative monoid (S, ±, 0) endowed with a function
-⊲ from K × S to S such that:
-(k × k′) ⊲ s = k ⊲ (k′ ⊲ s),
-(k + k′) ⊲ s = k ⊲ s ± k′ ⊲ s,
-k ⊲ (s ± s′) = k ⊲ s ± k ⊲ s′,
-1 ⊲ s = s,
-0 ⊲ s = k ⊲ 0 = 0.
-A K-right-semimodule can be defined symmetrically.
-An operad [12,14] is a structure (O, (◦j)j∈N, id) where O is a graded set (i.e. O = �
-n∈N On), id is an element of
-O1, ◦j is a function defined for any three integers (i, j, k)6 with 0 < j ≤ k in Ok × Oi → Ok+i−1 such that for any
-elements p1 ∈ Om, p2 ∈ On, p3 ∈ Op:
-∀0 < j ≤ m, id ◦1 p1 = p1 ◦j id = p1,
-∀0 < j ≤ m, 0 < j′ ≤ n, p1 ◦j (p2 ◦j′ p3) = (p1 ◦j p2) ◦j+j′−1 p3,
-∀0 < j′ ≤ j ≤ m, (p1 ◦j p2) ◦j′ p3 = (p1 ◦j′ p3) ◦j+p−1 p2.
-Combining these compositions ◦j, one can define a composition ◦ sending Ok × Oi1 × · · · × Oik to Oi1+···+ik: for
-any element (p, q1, . . . , qk) in Ok × Ok,
-p ◦ (q1, . . . , qk) = (· · · ((p ◦k qk) ◦k−1 qk−1 · · · ) · · · ) ◦1 q1.
-Conversely, the composition ◦ can define the compositions ◦j using the identity element: for any two elements (p, q)
-in Ok × Oi, for any integer 0 < j ≤ k:
-p ◦j q = p ◦ (id, . . . , id
-�
-��
-�
-j−1 times
-, q, id, . . . , id
-�
-��
-�
-k−j times
-).
-As an example, the set of n-ary functions over a set, with the identity function as unit, forms an operad.
-A module over an operad (O, ◦, id) is a set S endowed with a function ⋇ from On × Sn to S such that
-f ⋇ (f1 ⋇ (s1,1, . . . , s1,i1), . . . , fn ⋇ (sn,1, . . . , sn,in))
-= (f ◦ (f1, . . . , fn)) ⋇ (s1,1, . . . , s1,i1, . . . , sn,1, . . . , sn,in).
-6 every couple (i, k) unambiguously defines the domain and codomain of a function ◦j
-
-The extension of the computation of derivatives could be performed for any monad. Indeed, any monad could
-be used to define well-typed auxiliary functions that mimic the classical computations. However, some properties
-should be satisfied in order to compute weights equivalently to Definition 4. Therefore, in the following we consider
-a restricted kind of monads.
-A monadic support is a structure (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇) satisfying:
-– M is a monad,
-– R = (M(1), +, ×, 1, 0) is a semiring,
-– M = (M(Exp(Σ)), ±, 0) is a monoid,
-– (M, ⋉) is a Exp(Σ)-right-semimodule,
-– (M, ⊲) is a R-left-semimodule,
-– (M, ⊳) is a R-right-semimodule,
-– (M(Exp(Σ)), ⋇) is a module for the operad of the functions over M(1).
-An expressive support is a monadic support (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇) endowed with a function toExp from
-M(Exp(Σ)) to Exp(Σ) satisfying the following conditions:
-weightw(toExp(m)) = m >>= weightw
-(2)
-toExp(m ⋉ F) = toExp(m) · F,
-(3)
-toExp(m ± m′) = toExp(m) + toExp(m′),
-(4)
-toExp(m ⊲ x) = toExp(m) ⊙ x,
-(5)
-toExp(x ⊳ m) = x ⊙ toExp(m),
-(6)
-toExp(f ⋇ (m1, . . . , mn)) = f(toExp(m1), . . . , toExp(mn)).
-(7)
-Let us now illustrate this notion with three expressive supports that will allow us to model well-known derivatives
-computations.
-Example 5 (The Maybe support).
-toExp(Nothing) = 0,
-toExp(Just(E)) = E,
-Nothing + m = m,
-m + Nothing = m,
-Just(⊤) + Just(⊤) = Just(⊤),
-Nothing × m = Nothing,
-m × Nothing = Nothing,
-Just(⊤) × Just(⊤) = Just(⊤),
-Nothing ± m = m,
-m ± Nothing = m,
-Just(E) ± Just(E′) = Just(E + E′),
-1 = Just(⊤),
-0 = Nothing,
-0 = Nothing,
-m ⋉ F = (λE → E · F) <$> m,
-m ⊲ m′ = m >>= (λx → m′),
-m ⊳ m′ = m′ >>= (λx → m),
-f ⋇ (m1, . . . , mn) = pure(f(toExp(m1), . . . , toExp(mn))).
-Example 6 (The Set support).
-toExp({E1, . . . , En}) = E1 + · · · + En,
-+ = ∪,
-× = ∩,
-± = ∪,
-1 = {⊤},
-0 = ∅,
-0 = ∅,
-m ⋉ F = (λE → E · F) <$> m,
-m ⊲ m′ = m >>= (λx → m′),
-m ⊳ m′ = m′ >>= (λx → m),
-f ⋇ (m1, . . . , mn) = pure(f(toExp(m1), . . . , toExp(mn))).
-Example 7 (The LinComb(K) support).
-toExp((k1, E1) ⊞ · · · ⊞ (kn, En)) = k1 ⊙ E1 + · · · + kn ⊙ En,
-+ = ⊞,
-(k, ⊤) × (k′, ⊤) = (k × k′, ⊤),
-1 = (1, ⊤),
-0 = (0, ⊤),
-± = ⊞,
-0 = (0, ⊤),
-m ⋉ F = (λE → E · F) <$> m,
-m ⊲ m′ = m >>= (λx → m′),
-m ⊳ k = (λE → E ⊙ k) <$> m,
-f ⋇ (m1, . . . , mn) = pure(f(toExp(m1), . . . , toExp(mn))).
-
-5
-Monadic Derivatives
-In the following, (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇, toExp) is an expressive support.
-Definition 5. The derivative of an M-monadic expression E over Σ w.r.t. a symbol a in Σ is the element da(E)
-in M(Exp(Σ)) inductively defined as follows:
-da(ε) = 0,
-da(∅) = 0,
-da(b) =
-�
-pure(ε)
-if a = b,
-0
-otherwise,
-da(E1 + E2) = da(E1) ± da(E2),
-da(E∗
-1) = da(E1) ⋉ E∗
-1,
-da(E1 · E2) = da(E1) ⋉ E2 ± Null(E1) ⊲ da(E2),
-da(α ⊙ E1) = α ⊲ da(E1),
-da(E1 ⊙ α) = da(E1) ⊳ α,
-da(f(E1, . . . , En)) = f ⋇ (da(E1), . . . , da(En))
-where b is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
-function from (M(1))n to M(1).
-The link between derivatives and series can be stated as follows, which is an alternative description of the classical
-quotient.
-Proposition 2. Let E be an M-monadic expression over an alphabet Σ, a be a symbol in Σ and w be a word in
-Σ∗. Then:
-weightaw(E) = da(E) >>= weightw.
-Proof. Let us proceed by induction over the structure of E. All the classical cases (i.e. the function operator left
-aside) can be proved following the classical methods ([1,4,13]). Therefore, let us consider this last case.
-da(f(E1, . . . , En)) >>= weightw
-= weightw(toExp(da(f(E1, . . . , En))))
-(Eq (2))
-= weightw(toExp(f ⋇ (da(E1), . . . , da(En)))
-(Def 5))
-= weightw(f(toExp(da(E1)), . . . , toExp(da(En))))
-(Eq (7))
-= f(weightw(toExp(da(E1))), . . . , weightw(toExp(da(En))))
-(Def 4, Eq (1))
-= f(da(E1) >>= weightw, . . . , da(En) >>= weightw)
-(Eq (2))
-= f(weightaw(E1), . . . , weightaw(En))
-(Ind. hyp.)
-= weightaw(f(E1, . . . , En))
-(Def 4, Eq (1))
-Let us define how to extend the derivative computation from symbols to words, using the monadic functions.
-Definition 6. The derivative of an M-monadic expression E over Σ w.r.t. a word w in Σ∗ is the element dw(E)
-in M(Exp(Σ)) inductively defined as follows:
-dε(E) = pure(E),
-da·v(E) = da(E) >>= dv,
-where a is a symbol in Σ and v a word in Σ∗.
-Finally, it can be easily shown, by induction over the length of the words, following Proposition 2, that the
-derivatives computation can be used to define a syntactical computation of the weight of a word associated with an
-expression.
-Theorem 1. Let E be an M-monadic expression over an alphabet Σ and w be a word in Σ∗. Then:
-weightw(E) = dw(E) >>= Null.
-Notice that, restraining monadic expressions to regular ones,
-– the Maybe support leads to the classical derivatives [4],
-– the Set support leads to the partial derivatives [1],
-– the LinComb support leads to the derivatives with multiplicities [13].
-Example 8. Let us consider the function ExtDist defined in Example 4 and the LinComb(N)-monadic expression
-E = ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗).
-da(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + a∗)
-daa(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 2 ⊙ a∗)
-
-daaa(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 3 ⊙ a∗)
-daab(E) = ExtDist(b∗, b∗, b∗a∗)
-weightaaa(E) = daaa(E) >>= Null
-= ExtDist(1 + 1, 1, 1 + 3) = 4 − 1 = 3
-weightaab(E) = daab(E) >>= Null = ExtDist(1, 1, 1) = 0
-In the next section, we show how to compute the derivative automaton associated with an expression.
-6
-Automata Construction
-A category C is defined by:
-– a class ObjC of objects,
-– for any two objects A and B, a set HomC(A, B) of morphisms,
-– for any three objects A, B and C, an associative composition function ◦C in HomC(B, C) −→ HomC(A, B) −→
-HomC(A, C),
-– for any object A, an identity morphism idA in HomC(A, A), such that for any morphisms f in HomC(A, B) and
-g in HomC(B, A), f ◦C idA = f and idA ◦C g = g.
-Given a category C, a C-automaton is a tuple (Σ, I, Q, F, i, δ, f) where
-– Σ is a set of symbols (the alphabet),
-– I is the initial object, in Obj(C),
-– Q is the state object, in Obj(C),
-– F is the final object, in Obj(C),
-– i is the initial morphism, in HomC(I, Q),
-– δ is the transition function, in Σ −→ HomC(Q, Q),
-– f is the value morphism, in HomC(Q, F).
-The function δ can be extended as a monoid morphism from the free monoid (Σ∗, ·, ε) to the morphism monoid
-(HomC(Q, Q), ◦C, idQ), leading to the following weight definition.
-The weight associated by a C-automaton A = (Σ, I, Q, F, i, δ, f) with a word w in Σ∗ is the morphism weight(w)
-in HomC(I, F) defined by
-weight(w) = f ◦C δ(w) ◦C i.
-If the ambient category is the category of sets, and if I =
-1, the weight of a word is equivalently an element of
-F. Consequently, a deterministic (complete) automaton is equivalently a Set-automaton with
-1 as the initial object
-and B as the final object.
-Given a monad M, the Kleisli composition of two morphisms f ∈ HomC(A, B) and g ∈ HomC(B, C) is the
-morphism (f >=> g)(x) = f(x) >>= g in HomC(A, C). This composition defines a category, called the Kleisli
-category K(M) of M, where:
-– the objects are the sets,
-– the morphisms between two sets A and B are the functions between A and M(B),
-– the identity is the function pure.
-Considering these categories:
-– a deterministic automaton is equivalently a K(Maybe)-automaton,
-– a nondeterministic automaton is equivalently a K(Set)-automaton,
-– a weighted automaton over a semiring K is equivalently a K(LinComb(K))-automaton,
-all with
-1 as both the initial object and the final object.
-Furthermore, for a given expression E, if i = pure(E), δ(a)(E′) = da(E′) and f = Null, we can compute
-the well-known derivative automata using the three previously defined supports, and the accessible part of these
-automata are finite ones as far as classical expressions are concerned [4,1,13].
-More precisely, extended expressions can lead to infinite automata, as shown in the next example.
-
-Example 9. Considering the computations of Example 8, it can be shown that
-dan(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗).
-Hence, there is not a finite number of derivated terms, that are the states in the classical derivative automaton.
-This infinite automaton is represented in Figure 1, where the final weights of the states are represented by double
-edges. The sink states are omitted.
-ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗)
-ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + a∗)
-ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 2 ⊙ a∗)
-ExtDist(b∗, b∗, b∗a∗)
-ExtDist(0, 0, a∗)
-ExtDist(b∗ + b∗a∗, b∗a∗b∗ + b∗, b∗a∗)
-ExtDist(b∗ + b∗a∗, b∗a∗b∗ + 2 ⊙ b∗, b∗a∗)
-ExtDist(a∗, a∗b∗, a∗)
-ExtDist(0, b∗, 0)
-ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗)
-ExtDist(b∗ + b∗a∗, b∗a∗b∗ + n ⊙ b∗, b∗a∗)
-1
-1
-2
-n
-1
-1
-2
-1
-n
-a
-b
-b
-a
-b
-a
-b
-a
-b
-a
-b
-a
-b
-a
-b
-a
-b
-a
-Fig. 1. The (infinite) derivative weighted automaton associated with E.
-In the following section, let us show how to model a new monad in order to solve this problem.
-7
-The Graded Module Monad
-Let us consider an operad O = (O, ◦, id) and the association sending:
-– any set S to �
-n∈N On × Sn,
-– any f in S → S′ to the function g in �
-n∈N On × Sn → �
-n∈N On × S′n:
-g(o, (s1, . . . , sn)) = (o, (f(s1), . . . , f(sn)))
-It can be checked that this is a functor, denoted by GradMod(O). Moreover, it forms a monad considering the two
-following functions:
-pure(s) = (id, s),
-(o, (s1, . . . , sn)) >>= f = (o ◦ (o1, . . . , on), (s1,1, . . . , s1,i1, . . . , sn,1, . . . , sn,in))
-where f(sj) = (oj, sj,1, . . . , sj,ij). However, notice that GradMod(O)(1) cannot be easily evaluated as a value space.
-Thus, let us compose it with another monad. As an example, let us consider a semiring K = (K, ×, +, 1, 0) and
-the operad O of the n-ary functions over K. Hence, let us define the functor7 GradComb(O, K) that sends S to
-GradMod(O)(LinComb(K)(S)).
-7 it is folk knowledge that the composition of two functors is a functor.
-
-To show that this combination is a monad, let us first define a function α sending GradComb(O, K)(S) to
-GradMod(O)(S). It can be easily done by converting a linear combination into an operadic combination, i.e. an
-element in GradMod(O)(S), with the following function toOp:
-toOp((k1, s1) ⊞ · · · ⊞ (kn, sn))
-= (λ(x1, . . . , xn) → k1 × x1 + · · · + kn × xn, (s1, . . . , sn)),
-α(o, (L1, . . . , Ln)) = (o ◦ (o1, . . . , on), (s1,1, . . . , s1,i1, . . . , sn,1, . . . , sn,in))
-where toOp(Lj) = (oj, (sj,1, . . . , sj,ij)).
-Consequently, we can define the monadic functions as follows:
-pure(s) = (id, (1, s)),
-(o, (L1, . . . , Ln)) >>= f = α(o, (L1, . . . , Ln)) >>= f
-where the second occurrence of >>= is the monadic function associated with the monad GradMod(O).
-Let us finally define an expressive support for this monad:
-toExp(o, (L1, . . . , Ln)) = o(toExp(L1), . . . , toExp(Ln)),
-(o, (L1, . . . , Ln)) + (o′, (L′
-1, . . . , L′
-n′)) = (o + o′, (L1, . . . , Ln, L′
-1, . . . , L′
-n′))
-(o, (L1, . . . , Ln)) × (o′, (L′
-1, . . . , L′
-n′)) = (o × o′, (L1, . . . , Ln, L′
-1, . . . , L′
-n′))
-± = +,
-1 = (id, (1, ⊤)),
-0 = (id, (0, ⊤)),
-0 = (id, (0, ⊤)),
-m ⋉ F = pure(toExp(m) · F),
-(o, (M1, . . . , Mk)) ⊲ (o′, (L1, . . . , Ln)) = (o(M1, . . . , Mk) × o′, (L1, . . . , Ln)),
-(o, (L1, . . . , Ln)) ⊳ (o′, (M1, . . . , Mk)) = (o × o′(M1, . . . , Mk), (L1, . . . , Ln))
-f ⋇ ((o1, (L1,1, . . . , L1,i1)), . . . , (on, (Ln,1, . . . , Ln,in)))
-= (f ◦ (o1, . . . , on), (L1,1, . . . , L1,i1, . . . , Ln,1, . . . , Ln,in))
-where (o + o′)(x1, . . . , xn+n′) = o(x1, . . . , xn) + o′(xn+1, . . . , xn+n′)
-(o × o′)(x1, . . . , xn+n′) = o(x1, . . . , xn) × o′(xn+1, . . . , xn+n′)
-Example 10. Let us consider that two elements in GradComb(O, K)(Exp(Σ)) are equal if they have the same image
-by toExp. Let us consider the expression E = ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗) of Example 8.
-da(E) = ExtDist ⋇ ((+, (a∗b∗, a∗)), (id, a∗b∗), (+, (a∗b∗a∗, a∗)))
-= (ExtDist ◦ (+, id, +), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗))
-daa(E) = (ExtDist ◦ (+, id, + ◦ (+, id)), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗, a∗))
-= (ExtDist ◦ (+, id, + ◦ (id, 2×)), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗))
-daaa(E) = (ExtDist ◦ (+, id, + ◦ (id, 3×)), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗))
-daab(E) = (ExtDist ◦ (+, id, +), (b∗, ∅, b∗, b∗a∗, ∅))
-= (ExtDist, (b∗, b∗, b∗a∗))
-weightaaa(E) = daaa(E) >>= Null
-= ExtDist ◦ (+, id, +)(1, 1, 1, 1, 3)
-= ExtDist(1 + 1, 1, 1 + 3) = 4 − 1 = 3
-weightaab(E) = daab(E) >>= Null = ExtDist(1, 1, 1) = 0
-Using this monad, the number of derivated terms, that is the number of states in the associated derivative automaton,
-is finite. Indeed, the computations are absorbed in the transition structure. This automaton is represented in
-Figure 2. Notice that the dashed rectangle represent the functions that are composed during the traversal associated
-with a word. The final weights are represented by double edges. The sink states are omitted. The state b∗ is duplicated
-to simplify the representation.
-
-ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗)
-ExtDist
-+
-+
-ExtDist
-+
-b∗a∗b∗
-+
-a∗b∗a∗
-+
-a∗b∗
-a∗
-b∗
-b∗a∗
-b∗
-1
-1
-1
-1
-1
-1
-1
-1
-a
-b
-a
-b
-b
-a
-b
-a
-a
-b
-b
-b
-Fig. 2. The Associated Derivative Automaton of ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗).
-However, notice that not every monadic expression produces a finite set of derivated terms, as shown in the next
-example.
-Example 11. Let us consider the expression E of Example 8 and the expression F = E · c∗. It can be shown that
-dan(F) = toExp(dan(E)) · c∗
-= ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗) · c∗.
-The study of the necessary and sufficient conditions of monads that lead to a finite set of derivated terms is one
-of the next steps of our work.
-8
-Haskell Implementation
-The notions described in the previous sections have been implemented in Haskell, as follows:
-– The notion of monad over a sub-category of sets is a typeclass using the Constraint kind to specify a sub-
-category;
-– n-ary functions and their operadic structures are implemented using fixed length vectors, the size of which is
-determined at compilation using type level programming;
-– The notion of graded module is implemented through an existential type to deal with unknown arities: Its
-monadic structure is based on an extension of heterogeneous lists, the graded vectors, typed w.r.t. the list of
-the arities of the elements it contains;
-– The parser and some type level functions are based on dependently typed programming with singletons [8],
-allowing, for example, determining the type of the monads or the arity of the functions involved at run-time;
-– An application is available here [16] illustrating the computations:
-• the backend uses servant to define an API;
-
-• the frontend is defined using Reflex, a functional reactive programming engine and cross compiled in
-JavaScript with GHCJS.
-As an example, the monadic expression of the previous examples can be entered in the web application as the
-input ExtDist(a*.b*+b*.a*,b*.a*.b*,a*.b*.a*).
-9
-Capture Groups
-Capture groups are a standard feature of POSIX regular expressions where parenthesis are used to memorize
-some part of the input string being matched in order to reuse either for substitution or matching. We give here
-an equivalent definition along with derivation formulae and a monadic definition. The semantic of this definition
-conforms to those of POSIX expressions. Precisely, when a capture group has been involved more than one time
-due to a stared subexpression, the value of the corresponding variable corresponds to the last capture.
-9.1
-Syntax of Expressions with Capture Groups
-A capture-group expression E over a symbol alphabet Σ and a variable alphabet Γ (or Σ, Γ-expression for short)
-is inductively defined as
-E = a,
-E = ε,
-E = ∅,
-E = F + G,
-E = F · G,
-E = F ∗,
-E = (F)x,
-E = x,
-where F and G are two Σ, Γ-expressions, a is a symbol in Σ, u is in Σ∗ and x is a variable in Γ. In the POSIX
-syntax, capture groups are implicitly mapped with variables respectively with the order of the opening parenthesis
-of a pair. Here, each capture group is associated explicitly to a variable by indexing the closing parenthesis with
-the name of this variable.
-9.2
-Contextual Expressions and their Contextual Languages
-In order to define the contextual language and the derivation of capture-group expressions, we need to extend the
-syntax of the expressions in order to attach to any capture group the current part of the input string captured
-during an execution.
-A contextual capture-group expression E over a symbol alphabet Σ and a variable alphabet Γ (or Σ, Γ-expression
-for short) is inductively defined as
-E = a,
-E = ε,
-E = ∅,
-E = F + G,
-E = F · G,
-E = F ∗,
-E = (F)u
-x,
-E = x,
-where F and G are two Σ, Γ-expressions, a is a symbol in Σ, u is in Σ∗ and x is a variable in Γ.
-Notice that a Σ, Γ-expression is equivalent to a contextual capture-group expression where u = ε for every
-occurrence of capture group.
-In the following, we consider that a context is a function from Γ to Maybe(Σ∗), modelling the possibility that
-a variable was initialized (or not) during the parsing. The set of contexts is denoted by Ctxt(Γ, Σ).
-Using these notions of contexts, let us now explain the semantics of contextual capture-group expressions. While
-parsing, a context is built to memorize the different affectations of words to variables. Therefore, a (contextual)
-language associated with an expression is a set of couples built from a language and the context that was used to
-compute it.
-The classic atomic cases (a symbol, the empty word or the empty set) are easy to define, preserving the context.
-Another one is the case of a variable x: the context is applied here to compute the associated word (if it exists) and
-is preserved.
-The recursive cases are interpreted as such:
-– The contextual language of a sum of two expressions is the union of their contextual languages, computed
-independently.
-– The contextual language of a catenation of two expressions F and G is computed in three steps. First, the
-contextual language of F is computed. Secondly, for each couple (L, ctxt) of this contextual language, the
-function ctxt is considered as the new context to compute the contextual language of G, leading to new couples
-(L′, ctxt′). Finally, for each of these combinations, a couple (L·L′, ctxt′) is added to form the resulting contextual
-language.
-
-– The contextual language of a starred expression is, classically, the infinite union of the powered contextual
-languages, computed by iterated catenations.
-– The contextual language of a captured expression (F)u
-x is computed in two steps. First, the contextual language
-of F is computed. Then, for each couple (L, ctxt) of it, a word w is chosen in L and the context ctxt must be
-updated coherently.
-More formally, the contextual language of a Σ, Γ-expression E associated with a context ctxt in Ctxt(Γ, Σ) is
-the subset Lctxt(E) of 2Σ∗ × Ctxt(Γ, Σ) inductively defined as follows:
-Lctxt(a) = {({a}, ctxt)},
-Lctxt(ε) = {({ε}, ctxt)},
-Lctxt(∅) = ∅,
-Lctxt(x) =
-�
-∅
-if ctxt(x) = Nothing,
-{({w}, ctxt)}
-otherwise if ctxt(x) = Just(w),
-Lctxt(F + G) = Lctxt(F) ∪ Lctxt(G),
-Lctxt(F · G) =
-�
-(L1,ctxt1)∈Lctxt(F ),
-(L2,ctxt2)∈Lctxt1(G)
-{(L1 · L2, ctxt2)},
-Lctxt(F ∗) =
-�
-n∈N
-(Lctxt(F))
-n,
-Lctxt((F)u
-x) =
-�
-(L1,ctxt1)∈Lctxt(F ),
-w∈L1
-{({w}, [ctxt1]x←uw)},
-where F and G are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ, u is in Σ∗, Ln is defined, for any
-set L of couples (language, context) by
-Ln =
-
-
-
-
-
-
-
-
-
-
-
-�
-(L,ctxt)∈L
-{({ε}, ctxt)}
-if n = 0,
-�
-(L1,ctxt1)∈L,
-(L2,ctxt2)∈Ln−1
-{(L1 · L2, ctxt2)}
-otherwise,
-and [ctxt]x←w is the context defined by
-[ctxt]x←w(y) =
-�
-Just(w)
-if x = y,
-ctxt(y)
-otherwise.
-The contextual language of an expression E is the set of couples obtained from an uninitialised context, where
-nothing is associated with any variable, that is the set
-Lλ_→Nothing(E).
-Finally, the language denoted by an expression E is the set of words obtained by forgetting the contexts, that is the
-set
-�
-(L,_)∈Lλ_→Nothing(E)
-L.
-Example 12. Let us consider the three following expressions over the symbol alphabet {a, b, c} and the variable
-alphabet {x}:
-E = E1 · E2,
-E1 = ((a∗)xbx)∗,
-E2 = cx.
-The language denoted by E2 is empty, since it is computed from the empty context, where nothing is associated
-with x. However, parsing E1 allows us to compute contexts that define word values to affect to x. Let us thus show
-how is defined the contextual language of E1:
-– the contextual language of (a∗)x is the set�
-n∈N
-{({an}, λx → Just(an))}
-where each word an is recorded in a context;
-– the contextual language of (a∗)xbx is the set
-�
-n∈N
-{({anban}, λx → Just(an))}
-where each word an is recorded in a context applied to evaluate the variable x;
-– the contextual language of E1 is the union of the two following sets S1 and S2:
-S1 = {({��}, λx → Nothing)}
-S2 = {({anban | n ∈ N}∗ · {ambam}, λx → Just(am)) | m ∈ N}
-where each iteration of the outermost star produces a new record for the variable x in the context; however,
-notice that only the last one is recorded at the end of the process.
-
-Finally, the language of E is obtained by considering the contexts obtained from the parsing of E1 to evaluate the
-occurrence of x in E2, leading to the set�
-m∈N
-({anban | n ∈ N}∗ · {ambamcam}).
-Obviously, some classical equations still hold with these computations:
-Lemma 1. Let E, F and G be three Σ, Γ-expressions and ctxt be a context in Ctxt(Γ, Σ). The two following
-equations hold:
-Lctxt(E · (F + G)) = Lctxt(E · F + E · G)
-Lctxt(F ∗) = Lctxt(ε + F · F ∗)
-Proof. Let us proceed by equality sequences:
-Lctxt(E · (F + G)) =
-�
-(L1,ctxt1)∈Lctxt(E),
-(L2,ctxt2)∈Lctxt1 (F +G)
-{(L1 · L2, ctxt2)}
-=
-�
-(L1,ctxt1)∈Lctxt(E),
-(L2,ctxt2)∈Lctxt1 (F )∪Lctxt1(G)
-{(L1 · L2, ctxt2)}
-=
-�
-(L1,ctxt1)∈Lctxt(E),
-(L2,ctxt2)∈Lctxt1 (F )
-{(L1 · L2, ctxt2)}
-∪
-�
-(L1,ctxt1)∈Lctxt(E),
-(L2,ctxt2)∈Lctxt1(G)
-{(L1 · L2, ctxt2)}
-= Lctxt(E · F) ∪ Lctxt(E · G)
-= Lctxt(E · F + E · G)
-Lctxt(F ∗) =
-�
-n∈N
-(Lctxt(F))
-n
-= (Lctxt(F))
-0 ∪
-�
-n∈N,n≥1
-(Lctxt(F))
-n
-= (Lctxt(F))
-0 ∪
-�
-n∈N
-Lctxt(F) · (Lctxt(F))
-n
-= (Lctxt(F))
-0 ∪ Lctxt(F) ·
-�
-n∈N
-(Lctxt(F))
-n
-= Lctxt(ε + F · F ∗)
-In order to solve the membership test for the contextual capture-group expressions, let us extend the classical
-derivation method. But first, let us show how to extend the nullability predicate, needed at the end of the process.
-9.3
-Nullability Computation
-The nullability predicate allows us to determine whether the empty word belongs to the language denoted by
-an expression. As far as capture groups are concerned, a context has to be computed. Therefore, the nullability
-predicate can be represented as a set of contexts the application of which produces a language that contains the
-empty word.
-As we have seen, the nullability depends on the current context. Given an expression and a context ctxt, the
-nullability predicate is a set in 2Ctxt(Γ,Σ), computed as follows:
-Nullctxt(ε) = {ctxt}
-Nullctxt(∅) = ∅
-Nullctxt(a) = ∅
-Nullctxt(x) =
-�
-{ctxt}
-if ctxt(x) = Just(ε)
-∅
-otherwise.
-Nullctxt(E + F) = Nullctxt(E) ∪ Nullctxt(F)
-Nullctxt(E · F) =
-�
-ctxt′∈Nullctxt(F ),
-ctxt′′∈Nullctxt′ (G)
-{ctxt′′}
-Nullctxt(E∗) = {ctxt}
-Nullctxt((E)u
-x) =
-�
-ctxt′∈Nullctxt(F )
-{[ctxt′]x←u}
-where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗.
-Example 13. Let us consider the three expressions of Example 12:
-E = E1 · E2,
-E1 = ((a∗)xbx)∗,
-E2 = cx.
-For any context ctxt,
-Nullctxt(E1) = {ctxt},
-Nullctxt(E2) = ∅,
-Nullctxt(E) = ∅.
-The nullability predicate allows us to determine whether there exists a couple in the contextual language of an
-expression such that its first component contains the empty word.
-
-Proposition 3. Let E be a Σ, Γ-expression and ctxt be a context in Ctxt(Γ, Σ). Then the two following conditions
-are equivalent:
-– Nullctxt(E) ̸= ∅,
-– ∃(L, _) ∈ Lctxt(E) | ε ∈ L.
-Proof. By induction over the structure of E:
-– If E = a ∈ Σ or E = ∅, the property holds since Nullctxt(E) is empty and since there is no couple (L, ctxt′) in
-Lctxt(E) with ε in L.
-– If E = ε, the following two conditions hold,
-Nullctxt(E) = {ctxt},
-Lctxt(E) = {({ε}, ctxt)},
-satisfying the stated condition.
-– If E = F + G, the following two conditions hold:
-Nullctxt(F + G) = Nullctxt(F) ∪ Nullctxt(G),
-Lctxt(F + G) = Lctxt(F) ∪ Lctxt(G).
-Since, by induction hypothesis, the following two conditions hold
-Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L,
-Nullctxt(G) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(G) | ε ∈ L,
-the proposition holds.
-– If E = F · G, the two following conditions hold:
-Nullctxt(F · G) =
-�
-ctxt′∈Nullctxt(F ),
-ctxt′′∈Nullctxt′(G),
-{ctxt′′},
-Lctxt(F · G) =
-�
-(L,ctxt′)∈Lctxt(F ),
-(L′,ctxt′′)∈Lctxt′(G),
-{(L · L′, ctxt′′)}.
-Since, by induction hypothesis, the two following conditions hold,
-Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L,
-Nullctxt′(G) ̸= ∅ ⇔ ∃(L, ctxt′′) ∈ Lctxt′(G) | ε ∈ L,
-the proposition holds.
-– If E = F ∗, since the two following conditions hold
-Nullctxt(F ∗) = {ctxt},
-Lctxt(F)
-0 = {({ε}, ctxt)} ∈ Lctxt(F ∗),
-the stated condition holds.
-– If E = (F)u
-x, both following conditions hold:
-Nullctxt((F)u
-x) =
-�
-ctxt′∈Nullctxt(F )
-{[ctxt′]x←u},
-Lctxt((F)u
-x) =
-�
-(L,ctxt′)∈Lctxt(F ),
-w∈L
-{({w}, [ctxt′]x←uw)}.
-Then, following induction hypothesis,
-Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L,
-the stated condition holds.
-– If E = x, both following conditions hold:
-Nullctxt(x) =
-�
-{ctxt}
-if ctxt(x) = Just(ε)
-∅
-otherwise,
-Lctxt(x) =
-�
-∅
-if ctxt(x) = Nothing,
-{({w}, ctxt)}
-otherwise if ctxt(x) = Just(w).
-Therefore, the proposition holds.
-9.4
-Derivation formulae
-Similarly to the nullability predicate, the derivation computation builds the context while parsing the expression.
-Therefore, the derivative of an expression with respect to a context is a set of couples (expression, context), induc-
-tively computed as follows, for any Σ, Γ-expression and for any context ctxt in Ctxt(Γ, Σ):
-dctxt
-a
-(ε) = ∅
-dctxt
-a
-(∅) = ∅
-
-dctxt
-a
-(b) =
-�
-∅
-if a ̸= b,
-{(ε, ctxt)}
-otherwise,
-dctxt
-a
-(x) =
-�
-dctxt
-a
-(w)
-if ctxt(x) = Just(w)
-∅
-otherwise
-dctxt
-a
-(F + G) = dctxt
-a
-(F) ∪ dctxt
-a
-(G)
-dctxt
-a
-(F · G) =
-�
-(ctxt′,F ′)∈dctxt
-a
-(F )
-{(F ′ · G, ctxt′)}
-∪
-�
-ctxt′∈Nullctxt(F )
-dctxt′
-a
-(G)
-dctxt
-a
-(F ∗) =
-�
-(ctxt′,F ′)∈dctxt
-a
-(F )
-{(F ′ · F ∗, ctxt′)}
-dctxt
-a
-((F)u
-x) =
-�
-(ctxt′,F ′)∈dctxt
-a
-(F )
-{((F ′)u·a
-x , ctxt′)}
-where F and G are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗.
-Example 14. Let us consider the three expressions of Example 12:
-E = E1 · E2,
-E1 = ((a∗)xbx)∗,
-E2 = cx.
-Then, for any context ctxt,
-dctxt
-a
-(E) = {((a∗)a
-xbx((a∗)xbx)∗cx, ctxt)},
-dctxt
-b
-(E) = {(x((a∗)xbx)∗cx, λx → ε)},
-dctxt
-c
-(E) = {(x, ctxt)}.
-The derivation of an expression allows us to syntactically express the computation of the quotient of the language
-components in contextual languages, where the quotient w−1(L) is the set {w′ | ww′ ∈ L}.
-Proposition 4. Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and a be a symbol in Σ. Then:
-�
-(E′,ctxt′)∈dctxt
-a
-(E)
-Lctxt′(E′) =
-�
-(L′,ctxt′)∈Lctxt(E)
-{(a−1(L′), ctxt′)}
-Proof. By induction over the structure of E, assimilating ∅ and {(∅, ctxt)} for any context ctxt.
-– If E = ε or E = ∅, the property vacuously holds.
-– If E = b ∈ Σ,
-�
-(E′,ctxt′)∈dctxt
-a
-(b)
-Lctxt′(E′) =
-�
-∅
-if b ̸= a,
-{({ε}, ctxt)}
-otherwise,
-= {(a−1({b}), ctxt)} =
-�
-(L′,ctxt′)∈Lctxt(b)
-{(a−1(L′), ctxt′)}.
-– If E = F + G,�
-(E′,ctxt′)∈dctxt
-a
-(F +G)
-Lctxt′(E′) =
-�
-(E′,ctxt′)∈dctxt
-a
-(F )∪dctxt
-a
-(G)
-Lctxt′(E′)
-=
-�
-(E′,ctxt′)∈dctxt
-a
-(F )
-Lctxt′(E′) ∪
-�
-(E′,ctxt′)∈dctxt
-a
-(G)
-Lctxt′(E′)
-=
-�
-(L′,ctxt′)∈Lctxt(F )
-{(a−1(L′), ctxt′)} ∪
-�
-(L′,ctxt′)∈Lctxt(G)
-{(a−1(L′), ctxt′)}
-=
-�
-(L′,ctxt′)∈Lctxt(F )∪Lctxt(G)
-{(a−1(L′), ctxt′)}
-=
-�
-(L′,ctxt′)∈Lctxt(F +G)
-{(a−1(L′), ctxt′)}.
-– If E = F · G,
-�
-(E′,ctxt′)∈dctxt
-a
-(F ·G)
-Lctxt′(E′) =
-�
-(ctxt′,F ′)∈dctxt
-a
-(F )
-Lctxt′(F ′ · G) ∪
-�
-ctxt′∈Nullctxt(F ),
-(G′,ctxt′′)∈dctxt′
-a
-(G)
-Lctxt′′(G′)
-=
-�
-(ctxt′,F ′)∈dctxt
-a
-(F ),
-(L1,ctxt1)∈Lctxt(F ′),
-(L2,ctxt2)∈Lctxt1(G)
-{(L1 · L2, ctxt2)} ∪
-�
-ctxt′∈Nullctxt(F ),
-(G′,ctxt′′)∈dctxt′
-a
-(G)
-Lctxt′′(G′)
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-(L2,ctxt2)∈Lctxt1 (G)
-{(a−1(L1) · L2, ctxt2)} ∪
-�
-ctxt1∈Nullctxt(F ),
-(L2,ctxt2)∈Lctxt1 (G)
-{(a−1(L2), ctxt2)}
-
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-(L2,ctxt2)∈Lctxt1 (G)
-{(a−1(L1) · L2, ctxt2)} ∪
-�
-∃(L,ctxt1)∈Lctxt(F )|ε∈L,
-(L2,ctxt2)∈Lctxt1(G)
-{(a−1(L2), ctxt2)}
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-(L2,ctxt2)∈Lctxt1 (G)
-{(a−1(L1) · L2, ctxt2)} ∪
-�
-(L1,ctxt1)∈Lctxt(F ),
-ε∈L1,
-(L2,ctxt2)∈Lctxt1 (G)
-{(a−1(L2), ctxt2)}
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-(L2,ctxt2)∈Lctxt1 (G)
-{(a−1(L1 · L2), ctxt2)}
-=
-�
-(L′,ctxt′)∈Lctxt(F ·G)
-{(a−1(L′), ctxt′)}.
-– If E = F ∗,
-�
-(E′,ctxt′)∈dctxt
-a
-(F ∗)
-Lctxt′(E′) =
-�
-(ctxt′,F ′)∈dctxt
-a
-(F )
-Lctxt′(F ′ · F ∗)
-=
-�
-(ctxt′,F ′)∈dctxt
-a
-(F ),
-(L1,ctxt1)∈Lctxt(F ′),
-(L2,ctxt2)∈Lctxt1(F ∗)
-{(L1 · L2, ctxt2)}
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-(L2,ctxt2)∈Lctxt1(F ∗)
-{(a−1(L1) · L2, ctxt2)}
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-(L2,ctxt2)∈Lctxt1(F ∗)
-{(a−1(L1 · L2), ctxt2)}
-=
-�
-(L′,ctxt′)∈Lctxt(F ·F ∗)
-{(a−1(L′), ctxt′)}
-=
-�
-(L′,ctxt′)∈Lctxt(ε+F ·F ∗)
-{(a−1(L′), ctxt′)}
-=
-�
-(L′,ctxt′)∈Lctxt(F ∗)
-{(a−1(L′), ctxt′)}
-– If E = (F)u
-x,
-�
-(E′,ctxt′)∈dctxt
-a
-((F )u
-x)
-Lctxt′(E′) =
-�
-(ctxt′,F ′)∈dctxt
-a
-(F )
-Lctxt′((F ′)u·a
-x )
-=
-�
-(ctxt′,F ′)∈dctxt
-a
-(F )
-(L1,ctxt1)∈Lctxt′(F ′),
-w∈L1
-{({w}, [ctxt1]x←uaw)}
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-w∈a−1(L1)
-{({w}, [ctxt1]x←uaw)}
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-aw∈L1
-{({w}, [ctxt1]x←uaw)}
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-aw∈L1
-{(a−1({aw}), [ctxt1]x←uaw)}
-=
-�
-(L1,ctxt1)∈Lctxt(F ),
-w∈L1
-{(a−1({w}), [ctxt1]x←uw)}
-=
-�
-(L′,ctxt′)∈Lctxt((F )u
-x)
-{(a−1(L′), ctxt′)}
-
-– If E = x,
-�
-(E′,ctxt′)∈dctxt
-a
-(x)
-Lctxt′(E′) =
-
-
-
-
-
-�
-(E′,ctxt′)∈dctxt
-a
-(w)
-Lctxt′(E′)
-if ctxt(x) = Just(w),
-∅
-otherwise,
-=
-
-
-
-
-
-�
-(w,ctxt)∈dctxt
-a
-(aw)
-Lctxt(w)
-if ctxt(x) = Just(aw),
-∅
-otherwise,
-=
-�
-{({w}, ctxt)}
-if ctxt(x) = Just(aw),
-∅
-otherwise,
-=
-�
-{(a−1({aw}), ctxt)}
-if ctxt(x) = Just(aw),
-∅
-otherwise,
-=
-�
-{(a−1({w}), ctxt)}
-if ctxt(x) = Just(w),
-∅
-otherwise,
-=
-�
-(L′,ctxt′)∈Lctxt(x)
-{(a−1(L′), ctxt′)}
-The derivation w.r.t. a word is, as usual, an iterated application of the derivation w.r.t. a symbol, recursively
-defined as follows, for any Σ, Γ-expression E, for any context ctxt in Ctxt(Γ, Σ), for any symbol a in Σ and for
-any word v in Σ∗:
-dctxt
-ε
-(E) = {(E, ctxt)},
-dctxt
-a·v (E) =
-�
-(E′,ctxt′)∈dctxt
-a
-(E)
-dctxt′
-v
-(E′).
-Example 15. Let us consider the three expressions of Example 14:
-E = E1 · E2,
-E1 = ((a∗)xbx)∗,
-E2 = cx.
-Then, for any context ctxt,
-dctxt
-ab (E) = dctxt
-b
-((a∗)a
-xbx((a∗)xbx)∗cx)
-= {(x((a∗)xbx)∗cx, λx → a)}
-dctxt
-aba (E) = dλx→a
-a
-(x((a∗)xbx)∗cx)
-= {(((a∗)xbx)∗cx, λx → a)}
-dctxt
-abac(E) = dλx→a
-c
-(((a∗)xbx)∗cx)
-= {(x, λx → a)}
-dctxt
-abaca(E) = dλx→a
-a
-(x)
-= {(ε, λx → a)}
-Such an operation allows us to syntactically compute the quotient.
-Proposition 5. Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and w be a word in Σ∗. Then:
-�
-(E′,ctxt′)∈dctxt
-w
-(E)
-Lctxt′(E′) =
-�
-(L′,ctxt′)∈Lctxt(E)
-{(w−1(L′), ctxt′)}
-Proof. By a direct induction over the structure of words.
-Finally, the membership test of a word w can be performed as usual by first computing the derivation w.r.t. w, and
-then by determining the existence of a nullable derivative, as a direct corollary of Proposition 3 and Proposition 5.
-Theorem 2. Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and w be a word in Σ∗. Then the two
-following conditions are equivalent:
-– ∃(L, _) ∈ Lctxt(E) | w ∈ L,
-– ∃(E′, ctxt′) ∈ dctxt
-w
-(E) | Nullctxt′(E′) ̸= ∅.
-We have shown how to compute the derivatives and solve the membership test in a classical way. Let us show how
-to embed the context computation in a convenient monad, in order to generalize the definitions to other structure
-than sets.
-
-9.5
-The StateT Monad Transformer
-Monads do not compose well in general. However, ones can consider particular combinations of these objects. Among
-those, well-known patterns are the monad transformers like the StateT Monad Transformer [10]. This combination
-allows us to mimick the use of global variables in a functional way. In our setting, it allows us to embed the context
-computation in an elegant way.
-Let S be a set and M be a monad. We denote by StateT(S, M) following the mapping:
-StateT(S, M)(A) = S → M(A × S).
-In other terms, StateT(S, M)(A) is the set of functions from S to the monadic structure M(A×S) based on couples
-in the cartesian product (A × S).
-The mapping StateT(S, M) can be equipped by a structure of functor, defined for any function f from a set A
-to a set B by
-StateT(S, M)(f)(state)(s) = M(λ(a, s) → (f(a), s))(state(s)).
-It can also be equipped with the structure of monad, defined for any function f from a set A to the set StateT(S, M)(B):
-pure(a) = λs → pure(a, s)
-bind(f)(state)(s) = state(s) >>= λ(a, s′) → f(a)(s′)
-9.6
-Monadic Definitions
-The previous definitions associated with capture-group expressions can be equivalently restated using the StateT
-monad transformer specialised with the Set monad.
-Let us first consider the following claims where M = StateT(Ctxt(Γ, Σ), Set), allowing us to bring closer M
-and the previous notion of monadic support:
-– R = (M(1), +, ×, 1, 0) is a semiring by setting:
-f1 + f2 = λs → f1(s) ∪ f2(s),
-f1 × f2 = f1 >>= λ_ → f2,
-1 = λs → {(⊤, s)} = pure(⊤),
-0 = λs → ∅,
-– M = (M(Exp(Σ)), ±, 0) is a monoid by setting:
-± = +,
-0 = 0,
-– (M, ⋉) is a Exp(Σ)-right-semimodule by setting:
-f ⋉ F = λs →
-�
-(E,ctxt)∈f(s)
-{(E · F, ctxt)},
-– (M, ⊲) is a R-left-semimodule by setting:
-f1 ⊲ f2 = f1 >>= λ_ → f2.
-Then, the nullable predicate formulae can be equivalently restated as an element in StateT(Ctxt(Γ, Σ), Set)(1),
-which is equal by definition to Ctxt(Γ, Σ) → Set(1 × Ctxt(Γ, Σ)), isomorphic to Ctxt(Γ, Σ) → Set(Ctxt(Γ, Σ)). It
-can inductively be computed as follows:
-Null(ε) = 1
-Null(∅) = 0
-Null(a) = 0
-Null(E + F) = Null(E) + Null(F)
-Null(E · F) = Null(E) × Null(F)
-Null(E∗) = 1
-Null(x)(ctxt) =
-�
-pure((⊤, ctxt))
-if ctxt(x) = Just(ε),
-∅
-otherwise,
-Null((E)u
-x)(ctxt) = Set(λ(⊤, ctxt′) → (⊤, [ctxt′]x←u))(Null(F)(ctxt)),
-where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗. Notice that
-these formulae are the same that the ones in Definition 2 as far as classical operators are concerned, and that these
-formulae can be easily generalized to other convenient monads than Set.
-Moreover, the derivative of an expression is an element in StateT(Ctxt(Γ, Σ), Set)(Exp(Σ, Γ)):
-da(ε) = 0
-da(∅) = 0
-da(b) =
-�
-0
-if a ̸= b,
-pure(ε)
-otherwise,
-da(E + F) = da(E) ± da(F)
-da(E · F) = da(E) ⋉ F + Null(E) ⊲ da(F)
-da(E∗) = da(E) ⋉ E∗
-da((E)u
-x) = StateT(Ctxt(Γ, Σ), Set)(λF → (F)ua
-x )(da(E))
-da(x)(ctxt) =
-�
-pure((w, ctxt))
-if ctxt(x) = Just(aw),
-∅
-otherwise,
-
-where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗. Once again, notice
-that these formulae are the same that the ones in Definition 5 as far as classical operators are concerned, and that
-these formulae can be easily generalized to other convenient monads than Set.
-Finally, the derivation w.r.t. a word is monadically defined as in previous sections:
-dε(E) = pure(E),
-dav(E) = da(E) >>= dv,
-and the membership test of a word w can be equivalently rewritten as follows:
-(dw(E) >>= Null)(λ_ → Nothing) ̸= ∅.
-10
-Conclusion and Perspectives
-In this paper, we achieved the first step of our plan to unify the derivative computation over word expressions.
-Monads are indeed useful tools to abstract the underlying computation structures and thus may allow us to consider
-some other functionalities, such as capture groups via the well-known StateT monad transformer [10]. We aim to
-study the conditions satisfying by monads that lead to finite set of derivated terms, and to extend this method
-to tree expressions using enriched categories. Finally, we plan to extend monadic derivation to other underlying
-monads for capture groups, linear combinations for example.
-References
-1. Antimirov, V.M.: Partial derivatives of regular expressions and finite automaton constructions. Theor. Comput. Sci.
-155(2) (1996) 291–319
-2. Attou, S., Mignot, L., Miklarz, C., Nicart, F.: Monadic expressions and their derivatives. In: NCMA. Volume 367 of
-EPTCS (2022) 49–64
-3. Berry, G., Sethi, R.:
-From regular expressions to deterministic automata.
-Theoretical computer science 48 (1986)
-117–126
-4. Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11(4) (1964) 481–494
-5. Caron, P., Flouret, M.: From glushkov wfas to k-expressions. Fundam. Informaticae 109(1) (2011) 1–25
-6. Champarnaud, J., Laugerotte, É., Ouardi, F., Ziadi, D.: From regular weighted expressions to finite automata. Int. J.
-Found. Comput. Sci. 15(5) (2004) 687–700
-7. Colcombet, T., Petrisan, D.: Automata and minimization. SIGLOG News 4(2) (2017) 4–27
-8. Eisenberg, R.A., Weirich, S.: Dependently typed programming with singletons. In: Haskell, ACM (2012) 117–130
-9. Glushkov, V.M.: The abstract theory of automata. Russian Mathematical Surveys 16(5) (1961) 1
-10. Jones, M.P.: Functional programming with overloading and higher-order polymorphism. In: Adv. Func. Prog. Volume
-925 of LNCS, Springer (1995) 97–136
-11. Kleene, S.: Representation of events in nerve nets and finite automata. Automata Studies Ann. Math. Studies 34
-(1956) 3–41 Princeton U. Press.
-12. Loday, J.L., Vallette, B.: Algebraic operads. Volume 346. Springer Science & Business Media (2012)
-13. Lombardy, S., Sakarovitch, J.: Derivatives of rational expressions with multiplicity. Theor. Comput. Sci. 332(1-3) (2005)
-141–177
-14. May, J.P.: The geometry of iterated loop spaces. Volume 271. Springer (2006)
-15. Mignot, L.: Une proposition d’implantation des structures d’automates, d’expressions et de leurs algorithmes associés
-utilisant les catégories enrichies (in french).
-Habilitation à diriger des recherches, Université de Rouen normandie
-(Décembre 2020) 212 pages.
-16. Mignot, L.: Monadic derivatives. https://github.com/LudovicMignot/MonadicDerivatives (2022)
-17. Schützenberger, M.P.: On the definition of a family of automata. Inf. Control. 4(2-3) (1961) 245–270
-18. Sulzmann, M., Lu, K.Z.M.: POSIX regular expression parsing with derivatives. In: FLOPS. Volume 8475 of Lecture
-Notes in Computer Science, Springer (2014) 203–220
-

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-AdvBiom: Adversarial Attacks on Biometric
-Matchers
-Debayan Deb, Vishesh Mistry, Rahul Parthe
-TECH5,
-Troy, MI, USA
-{debayan.deb, vishesh.mistry, rahul.parthe}@tech5-sa.com
-Abstract
-With the advent of deep learning models, face recognition systems have achieved
-impressive recognition rates. The workhorses behind this success are Convolutional
-Neural Networks (CNNs) and the availability of large training datasets. However,
-we show that small human-imperceptible changes to face samples can evade most
-prevailing face recognition systems. Even more alarming is the fact that the same
-generator can be extended to other traits in the future. In this work, we present how
-such a generator can be trained and also extended to other biometric modalities,
-such as fingerprint recognition systems.
-1
-Introduction
-The last decade has seen a massive influx of deep learning-based technologies that have tackled
-problems which were once thought to be unsolvable. Much of this progress can be attributed to
-Convolutional Neural Networks (CNNs) [1, 2] which are now deployed in a plethora of applications
-ranging from cancer detection to driving autonomous vehicles. Akin to the computer vision domain,
-the use of CNNs have completely changed the face of biometrics due to the availability of powerful
-computing devices (GPUs, TPUs) and deep architectures capable of learning rich features [3–5].
-Automated face recognition systems (AFR) have been proven to achieve accuracies as high as 99%
-True Accept Rate (TAR) @ 0.1% False Accept Rate (FAR) [6], majorly owing to publicly available
-large-scale face datasets.
-Unfortunately, studies have shown that CNN-based networks are vulnerable to adversarial pertur-
-bations1 [7–12]. It is not surprising that AFR systems too are not impervious to these attacks.
-Adversarial attacks to an AFR system can be classified into two categories - (i) impersonation attack
-where the hacker tries to perturb his face image to match it to a target victim, and (ii) obfuscation
-attack where the hacker’s face image is perturbed to match with a random identity. Both the above at-
-tacks involve the hacker adding targeted human-imperceptible perturbations to the face image. These
-adversarial attacks are different from face digital manipulation that include attribute manipulation
-and synthetic faces, and also from presentation attacks which involves the perpetrator wearing a
-physical artifact such as a mask or replaying a photograph/video of a genuine individual which may
-be conspicuous in scenarios where human operators are involved.
-Let us consider, as example, the largest deployment of fingerprint recognition systems - India’s
-Aadhaar Project [13], which currently has an enrolled gallery size of about 1.35 billion faces from
-nearly all of its citizens. In September 2022 alone, Aadhaar received 1.3 billion authentication
-requests2. In order to deny a citizen his/her rightful access to government benefits, healthcare, and
-financial services, an attacker can maliciously perturb enrolled face images such that they do not
-1Adversarial perturbations refer to altering an input image instance with small, human imperceptible changes
-in a manner that can evade CNN models.
-2https://bit.ly/3BzlpZJ
-arXiv:2301.03966v1  [cs.CV]  10 Jan 2023
-
-match to the genuine person during verification. In a typical AFR system, adversarial faces can
-be replaced with a captured face image in order to prevent the probe face from matching to any of
-its corresponding enrolled faces. Additionally, the attacker can compromise the entire gallery by
-inserting adversarial faces in the enrolled gallery, where no probe face will match to the correct
-identity’s gallery.
-Adversarial attacks can further be categorized into two types of attacks based on how the attack vector
-is trained and generated:
-1. White-box attack: Attacks in which the hacker has full knowledge of the recognition system,
-and iteratively perturbs every pixel by various optimization schemes are termed as white-box
-attacks [14–22].
-2. Black-box attack: With no information about the parameters of the recognition system,
-black-box attacks are deployed by either transferring attacks learned from an available AFR
-system [23–28], or querying the the target system for score [29–31] or decision [32, 33].
-3. Semi-whitebox attack: Here, a white-box model is utilized only during training and then ad-
-versarial examples are synthesized during inference without any knowledge of the deployed
-AFR model.
-We propose an automated adversarial synthesis method, named AdvBiom, which generates an ad-
-versarial image for a probe image and satisfies all the above requirements. The contributions of the
-paper are as follows:
-1. GAN-based AdvBiom that learns to generate visually realistic adversarial face images that
-are misclassified by state-of-the-art automated biometric systems.
-2. Adversarial images generated via AdvBiom are model-agnostic and transferable, and achieve
-high success rate on 5 state-of-the-art automated face recognition systems.
-3. Visualizing regions where pixels are perturbed and analyzing the transferability of AdvBiom .
-4. We show that AdvBiom achieves significantly higher attack success rate under current
-defense mechanisms compared to baselines.
-5. With the addition of the proposed Minutiae Displacement and Distortion modules, we show
-thatAdvBiom can also be extended to successfully evade automated fingerprint recognition
-systems.
-2
-Related Work
-2.1
-Adversarial Attacks
-As discussed earlier, adversarial attacks are broadly classified into white-box attacks and black-box
-attacks. A large number of white-box attacks are gradient-based where they analyze the gradients
-during the back-propagation of an available face recognition system and perform pixel-wise per-
-turbations to the target face image. While approaches such as FGSM [14] and PGD [17] exploit
-the high-dimensional space of deep networks to generate adversarial attacks, C&W [18] focuses on
-minimizing objective functions for optimal adversarial perturbations. However, the basic assump-
-tion in white-box attacks that the target recognition system will be available is not plausible. In
-real-life scenarios, the hacker will not have any information regarding the architecture, training and
-deployment of the recognition system.
-Black-box attacks can be classified into three major categories: transfer-based, score-based, and
-decision-based attacks. Transfer-based attacks train their adversarial attack generator using readily
-available recognition systems and then deploy the attacks onto a black-box target system. Dong et
-al. [23] proposed the use of momentum for efficient transferability of the adversarial samples. DI2-
-FGSM [24] suggested to increase input data diversity for improving transferability. Other approaches
-in this category include AI-FGSM [27] and TI-FGSM [28]. Score-based attacks [29–31] query the
-target system for scores and try to estimate its gradients. Decision-based attacks have the most
-challenging setting wherein only the decisions from the target system are queried. Some effective
-methods in this category include Evolutionary attack [32] and Boundary attack [33].
-2
-
-2.2
-Adversarial Attacks on Face Recognition
-Although adversarial attacks on face recognition systems have only been recently explored, there
-has been a significant number of effective approaches for evading AFR systems. Attacks on face
-recognition systems can be broadly categorized into physical attacks and digital attacks. Physical
-attacks involve generating adversarial physical artifacts which are ’worn’ on a face. Sharif et
-al. [34, 35] proposed generating adversarial eye-glass frames for attacking face recognition systems.
-In [36], adversarial printed stickers placed on a hat were generated. However, methods [34–36] are
-implemented in a white-box setting which is unrealistic. Additionally, Nguyen et al. [37] proposed
-an adversarial light projection attack using an on-premise projector. Yin et al. [38] generated and
-printed eye makeup patches to be stuck around the eyes. More recently, authors in [39] proposed an
-adversarial mask for impersonation attacks in a black-box setting. However, all the above methods
-suffer a major drawback of being unrealistic in an operational setting where a human operator is
-present.
-Digital attacks refer to manipulating and perturbing the pixels of a digital face image before being
-passed through a face recognition system. Early works [9, 18, 10, 8, 40] focused on gradient-based
-attacks for face recognition. However, these methods implement lp-norm perturbations to each pixel
-resulting in decreased attack transferability, and vulnerability to denoising models. Cauli et al. [41]
-implemented a backdoor attack where the target face recognition system’s training samples were
-manipulated. Apart from the fact that gaining access to the target AFR’s training samples is highly
-improbable, a thorough visual inspection of the samples can easily identify the digital artifacts. Other
-works employ more stealthy attack approaches against face recognition models. Dong et al. [32]
-proposed an evolutionary optimization method for generating adversarial faces in decision-based
-black-box settings. However, they require a minimum of 1,000 queries to the target face recognition
-system before a realistic adversarial face can be synthesized. [42] added a conditional variational
-autoencoder and attention modules to generate adversarial faces in a transfer-based black-box setting.
-However, they solely focused on impersonation attacks and require at least 5 image samples of the
-target subject for training and inference. Zhong et al. [43] implemented dropout [44] to improve
-the transferability of the adversarial examples. [38] perturbed the eye region of a face to produce
-adversarial eyeshadow artifacts. However, the artifacts are visibly conspicuous under close inspection.
-Deb et al. [25] used a GAN to generate minimal perturbations in salient facial regions. More recently,
-[45] and [46] have focused on manipulating facial attributes for targeted adversarial attacks.
-3
-Adversarial Faces
-3.1
-Preliminaries
-The goal of any attacker is to evade Automated Face Recognition (AFR) systems under either of the
-two settings:
-• Obfuscation Manipulate input face images in a manner such that they cannot be identified
-as the hacker, or
-• Impersonation Edit input face images such that they are identified as a target/desired
-individual (victim).
-While the manipulated face image evades the AFR system, a key requirement in a successful attack is
-such that the input face image should appear as a legitimate face photo of the attacker. In other words,
-the attacker desires an automated method of adding small and human-imperceptible changes to an
-input face image such that it can evade AFR systems while appear benign to human observers. These
-changes are denoted as adversarial perturbations and the manipulated image is hereby referred to as
-adversarial images3. In addition, the automated method of synthesizing adversarial perturbations is
-named as adversarial generator.
-Formally, given an input face image, x, an adversarial generator has two requirements under the
-obfuscation scenario:
-• synthesize an adversarial face image, xadv = x + δ, such that AFR systems fail to match
-xadv and x, and
-3We interchangeably use the terms adversarial images and adversarial faces in this paper.
-3
-
-• limit the magnitude of perturbation ||δ||p such that xadv appears very similar to x to humans.
-When the attack aims to impersonate a target individual, we need an image of the victim xtarget
-where the identity of x and xtarget are different. Therefore, constraints under the impersonation
-setting are as follows:
-• synthesize an adversarial face image, xadv = x + δ, such that AFR systems erroneously
-match xadv and xtarget, and
-• limit the magnitude of perturbation ||δ||p such that xadv appears very similar to x to humans.
-Obfuscation attempts (faces are perturbed such that they cannot be identified as the attacker) are gen-
-erally more effective [25], computationally efficient to synthesize [14, 17], and widely adopted [47]
-compared to impersonation attacks (perturbed faces can automatically match to a target subject).
-Therefore, this paper focuses on crafting obfuscation attacks, however, we will still show examples
-on synthesizing impersonation attacks.
-3.2
-Gradient-based Attacks
-In white-box attacks, the attacker is assumed to have the knowledge and access to the AFR system’s
-model and parameters. Naturally, we then expect a much better attack success rate under white-box
-settings since the attacker can carefully craft adversarial perturbations that necessarily evade the target
-AFR system. However, these white-box manipulations of face recognition models are impractical in
-real-world scenarios. For instance, assuming access to an airport’s already deployed AFR system
-may be extremely difficult.
-Nevertheless, it is advantageous to understand prevailing white-box methods. That is, if given access
-to a CNN-based AFR system, how could one utilize all of its model parameters to launch a successful
-adversarial attack?
-A common approach is to utilize gradients of the whitebox AFR models. Namely the attackers modify
-the image in the direction of the gradient of the loss function with respect to the input image. There
-are two prevailing approaches to perform such gradient-based attacks:
-• one-shot attacks, in which the attacker takes a single step in the direction of the gradient,
-and
-• iterative attacks where instead of a single step, several steps are taken until we obtain a
-successful adversarial pattern.
-3.2.1
-Fast Gradient Sign Method (FGSM)
-This method computes an adversarial image by adding a pixel-wide perturbation of magnitude in the
-direction of the gradient [14]. Under FGSM attack, we take a single step towards the direction of the
-gradient, and therefore, FGSM is very efficient in terms of computation time. Formally, given an
-input image x, we obtain an adversarial image xadv:
-xadv = x + ϵ · sign (▽xJ (x, y))
-where, J is the loss function used to train the AFR system (typically, softmax cross entropy loss), and
-y is the ground truth class label of x (typically, the subject ID of the identity in x).
-FGSM was first proposed for the object classification domain and therefore, utilizes softmax proba-
-bilities for crafting adversarial perturbations. Therefore, the number of object classes are assumed to
-be known during training and testing. However, face recognition systems do not utilize the softmax
-layer for classification (as the number of identities are not fixed during deployment) instead features
-from the last fully connected layer are used for comparing face images.
-We first modify FGSM appropriately in order to evade AFR systems rather than object classifiers.
-Instead of considering the softmax cross-entropy loss as J, we craft a new loss function that models
-real-world scenario4:
-LfeatureMatch = 1 − Ex
-�
-F(x) · F(xadv)
-||F(x)|| ||F(xadv)||
-�
-.
-4For brevity, we denote Ex ≡ Ex∈Pdata.
-4
-
-where, F is the matcher and F(x) is the feature representation of an input image x. The above feature
-matching loss function computes the cosine distance between a pair of images and ensures that the
-features between adversarial image xadv and input image x are as close as possible. Therefore, the
-gradient of the above loss ensures the features do not match and hence, can be considered as an
-obfuscation adversarial attack.
-In Fig. 1, we show the results of launching our modified FGSM attack on a state-of-the-art AFR
-system, namely ArcFace [3]. We see that with a single step and with minimal perturbations, the real
-and adversarial images of Tiger Woods does not match via ArcFace while humans can easily identity
-both images as pertaining to the same subject.
-(a) Real Input Image
-(b) Perturbation
-(c) FGSM [14]
-Figure 1: Adversarial face synthesized via FGSM [14]. A state-of-the-art face matcher, ArcFace [3], fails to
-match the adversarial and input image. Cosine similarity score (∈ [−1, 1]) between the two images is 0.27,
-while a score above 0.36 (threshold @ 0.1% False Accept Rate) indicates that two faces are of the same subject.
-3.2.2
-Projected Gradient Descent (PGD)
-An extreme case of white-box attacks is the PGD attack [17] where we assume that the attacker also
-has unlimited number of attempts to try and evade the deployed AFR system. Unlike FGSM, PGD is
-an iterative attack. PGD attempts to find the perturbation δ that maximises the loss of a model on a
-particular input while keeping the size of the perturbation smaller than a specified amount referred
-to as ϵ. We keep iterating until such a δ is obtained. Similar to FGSM, we modify the loss function
-of PGD to fit the requirements of AFR system by again considering LfeatureMatch as the loss. Fig.
-2 shows the results of PGD attack on ArcFace matcher. Note that due to multiple iterations, PGD
-attack on AFR systems is more powerful (lower cosine similarity) but also more visible to humans as
-compared to the single-step FGSM attack.
-(a) Real Input Image
-(b) Perturbation
-(c) PGD [17]
-Figure 2: Adversarial face synthesized via PGD [17]. A state-of-the-art face matcher, ArcFace [3], fails to match
-the adversarial and input image. Cosine similarity score (∈ [−1, 1]) between the two images is 0.12, while a
-score above 0.36 (threshold @ 0.1% False Accept Rate) indicates that two faces are of the same subject.
-3.3
-Geometric Perturbations (GFLM)
-Prior efforts in crafting adversarial faces have also tried non-linear deformations as a natural method
-for evading AFR systems [48]. Non-linear deformations are applied by performing geometric warping
-to the input face images.
-Unlike traditional adversarial perturbations that basically add an adversarial perturbation δ, authors
-in [48] propose a fast method of generating adversarial faces by altering the landmark locations of
-the input images. The resulting adversarial faces completely lie on the manifold of natural images,
-which makes it extremely difficult to detect any adversarial perturbations. Results of geometrically
-warped adversarial faces are presented in 3.
-5
-
-(a) Real Input Image
-(b) Perturbation
-(c) GFLM
-Figure 3: Adversarial face synthesized via GFLM [48]. A state-of-the-art face matcher, ArcFace [3], fails to
-match the adversarial and input image. Cosine similarity score (∈ [−1, 1]) between the two images is 0.33,
-while a score above 0.36 (threshold @ 0.1% False Accept Rate) indicates that two faces are of the same subject.
-3.4
-Attribute-based Perturbations
-Unlike geometric-warping and gradient-based attacks that may perturb every pixel in the image, a
-few studies propose manipulating only salient regions in faces, e.g., eyes, nose, and mouth.
-By restricting perturbations to only semantic regions of the face, SemanticAdv [46] generates
-adversarial examples in a more controllable fashion by editing a single semantic aspect through
-attribute-conditioned image editing. Fig. 4 shows results from adversarial manipulating semantic
-attributes. We can see while the attacks are indeed successful, it comes at the cost of altering the
-perceived identity as well as leads to degraded image quality.
-(a) Real Input Image
-(b) Blond
-(c) Bangs
-(d) Mouth Open
-(e) Eyeglasses
-(f) Makeup
-Figure 4: Adversarial face synthesized via manipulating semantic attributes [46]. All adversarial images (b-f)
-fail to match with the real image (a) via ArcFace [3].
-4
-AdvBiom: Learning to Synthesize Adversarial Attacks
-We find that majority of prior efforts on crafting adversarial attacks either degrade the visual quality
-where an observant human can still visually pick out the adversarial patterns. We also identify the
-following challenges with prior efforts:
-• Gradient-based attacks rely on white-box settings where the entire deployed CNN-based
-AFR system is available to the attacker to compute its gradients.
-• Geometrically-warping faces generally do not guarantee adversarial success and greatly
-distort the face image.
-• Semantic attribute manipulation can also degrade visual quality and may lead to greater
-conspicuous changes.
-Instead, we propose to train a network to “learn" the salient regions of the face that can be perturbed
-to evade AFR systems in a semi-whitebox setting. These leads to the following advantages over prior
-efforts:
-6
-
-• Perceptual Realism Given a large enough training dataset, a network can gradually learn to
-synthesize adversarial face images that are perceptually realistic such that a human observer
-can identify the image as a legitimate face image.
-• Higher Attack Success The faces can be learned to be perturbed in a manner such that they
-cannot be identified as the hacker (obfuscation at- tack) or automatically matched to a target
-subject (impersonation attack) by an AFR system.
-• Controllable The amount of perturbation can also be controllable by the attacker so that
-they can examine the success of the learning model as a function of amount of perturbation.
-• Transferability Due to the semi-whitebox setting: once the network learns to generate the
-perturbed instances based on a single face recognition system, attacks can be transferred to
-any black-box AFR systems.
-We propose an automated adversarial biometric synthesis method, named AdvBiom, which generates
-an adversarial image for a probe face image and satisfies all the above requirements.
-4.1
-Methodology
-Our goal is to synthesize a face image that visually appears to pertain to the target face, yet automatic
-face recognition systems either incorrectly matches the synthesized image to another person or does
-not match to target’s gallery images. AdvBiom comprises of a generator G, a discriminator D, and
-face matcher (see Figure 5).
-Probe
-ℒ"#$
-Synthesized
-+
-ℒ%&'()%)*
-ℒ+',)-,./)%0(
-Adversarial Mask
-1
-ℱ
-3
-Figure 5: Given a probe face image, AdvBiom automatically generates an adversarial mask that is then added to
-the probe to obtain an adversarial face image.
-Generator
-The proposed generator takes an input face image, x ∈ X, and outputs an image, G(x).
-The generator is conditioned on the input image x; for different input faces, we will get different
-synthesized images.
-Since our goal is to obtain an adversarial image that is metrically similar to the probe in the image
-space, x, it is not desirable to perturb all the pixels in the probe image. For this reason, we treat the
-output from the generator as an additive mask and the adversarial face is defined as x + G(x). If
-the magnitude of the pixels in G(x) is minimal, then the adversarial image comprises mostly of the
-probe x. Here, we denote G(x) as an “adversarial mask". In order to bound the magnitude of the
-adversarial mask, we introduce a perturbation loss during training by minimizing the L2 norm5:
-Lperturbation = Ex [max (ϵ, ∥G(x)∥2)]
-(1)
-where ϵ ∈ [0, ∞) is a hyperparameter that controls the minimum amount of perturbation allowed.
-5For brevity, we denote Ex ≡ Ex∈X .
-7
-
-In order to achieve our goal of impersonating a target subject’s face or obfuscating one’s own identity,
-we need a face matcher, F, to supervise the training of AdvBiom. For obfuscation attack, at each
-training iteration, AdvBiom tries to minimize the cosine similarity between face embeddings of the
-input probe x and the generated image x + G(x) via an identity loss function:
-Lidentity = Ex[F(x, x + G(x))]
-(2)
-For an impersonation attack, AdvBiom maximizes the cosine similarity between the face embeddings
-of a randomly chosen target’s probe, y, and the generated adversarial face x + G(x) via:
-Lidentity = Ex[1 − F(y, x + G(x))]
-(3)
-The perturbation and identity loss functions enforce the network to learn the salient facial regions
-that can be perturbed minimally in order to evade automatic face recognition systems.
-Discriminator
-Akin to previous works on GANs [49, 50], we introduce a discriminator in order
-to encourage perceptual realism of the generated images. We use a fully-convolution network as a
-patch-based discriminator [50]. Here, the discriminator, D, aims to distinguish between a probe, x,
-and a generated adversarial face image x + G(x) via a GAN loss:
-LGAN =
-Ex [log D(x)] +
-Ex[log(1 − D(x + G(x)))]
-(4)
-Finally, AdvBiom is trained in an end-to-end fashion with the following objectives:
-min
-D LD = −LGAN
-(5)
-min
-G LG = LGAN + λiLidentity + λpLperturbation
-(6)
-where λi and λp are hyper-parameters controlling the relative importance of identity and perturbation
-losses, respectively. Note that LGAN and Lperturbation encourage the generated images to be visually
-similar to the original face images, while Lidentity optimizes for a high attack success rate. After
-training, the generator G can generate an adversarial face image for any input image and can be tested
-on any black-box face recognition system.
-The overall algorithm describing the training procedure of AdvBiom can be found in Algorithm 1.
-4.2
-Experimental Results
-Evaluation Metrics
-We quantify the effectiveness of the adversarial attacks generated by Ad-
-vBiom and other state-of-the-art baselines via (i) attack success rate and (ii) structural similarity
-(SSIM).
-The attack success rate for obfuscation attack is computed as,
-Attack Success Rate = (No. of Comparisons < τ)
-Total No. of Comparisons
-(7)
-where each comparison consists of a subject’s adversarial probe and an enrollment image. Here, τ
-is a pre-determined threshold computed at, say, 0.1% FAR6. Attack success rate for impersonation
-attack is defined as,
-Attack Success Rate = (No. of Comparisons ≥ τ)
-Total No. of Comparisons
-(8)
-Here, a comparison comprises of an adversarial image synthesized with a target’s probe and matched
-to the target’s enrolled image. We evaluate the success rate for the impersonation setting via 10-fold
-cross-validation where each fold consists of a randomly chosen target.
-Similar to prior studies [42], in order to measure the similarity between the adversarial example and
-the input face, we compute the structural similarity index (SSIM) between the images. SSIM is a
-normalized metric between −1 (completely different image pairs) to 1 (identical image pairs).
-6For each face matcher, we pre-compute the threshold at 0.1% FAR on all possible image pairs in LFW.
-For e.g., threshold @ 0.1% FAR for ArcFace is 0.28.
-8
-
-Algorithm 1 Training AdvBiom. All experiments in this work use α = 0.0001, β1 = 0.5, β2 = 0.9,
-λi = 10.0, λp = 1.0, m = 32.
-We set ϵ = 3.0 (obfuscation), ϵ = 8.0 (impersonation).
-1: Input
-2:
-X
-Training Dataset
-3:
-F
-Cosine similarity between an image pair obtained by biometric matcher
-4:
-G
-Generator with weights Gθ
-5:
-D
-Discriminator with weights Dθ
-6:
-m
-Batch size
-7:
-α
-Learning rate
-8: for number of training iterations do
-9:
-Sample a batch of probes {x(i)}m
-i=1 ∼ X
-10:
-if impersonation attack then
-11:
-Sample a batch of target images y(i) ∼ X
-12:
-δ(i) = G((x(i), y(i))
-13:
-else if obfuscation attack then
-14:
-δ(i) = G(x(i))
-15:
-end if
-16:
-x(i)
-adv = x(i) + δ(i)
-17:
-Lperturbation = 1
-m
-��m
-i=1 max
-�
-ϵ, ||δ(i)||2
-��
-18:
-if impersonation attack then
-19:
-Lidentity = 1
-m
-��m
-i=1 F
-�
-x(i), x(i)
-adv
-��
-20:
-else if obfuscation attack then
-21:
-Lidentity = 1
-m
-��m
-i=1
-�
-1 − F
-�
-y(i), x(i)
-adv
-���
-22:
-end if
-23:
-LG
-GAN = 1
-m
-��m
-i=1 log
-�
-1 − D(x(i)
-adv)
-��
-24:
-LD = 1
-m
-�m
-i=1
-�
-log
-�
-D(x(i))
-�
-+ log
-�
-1 − D(x(i)
-adv)
-��
-25:
-LG = LG
-GAN + λiLidentity + λpLperturbation
-26:
-Gθ = Adam(▽GLG, Gθ, α, β1, β2)
-27:
-Dθ = Adam(▽DLD, Dθ, α, β1, β2)
-28: end for
-Datasets
-We train AdvBiom on CASIA-WebFace [51] and then test on LFW [52]7.
-• CASIA-WebFace [51] is comprised of 494,414 face images belonging to 10,575 different
-subjects. We removed 84 subjects that are also present in LFW and the testing images in
-this paper.
-• LFW [52] contains 13,233 web-collected images of 5,749 different subjects. In order to
-compute the attack success rate, we only consider subjects with at least two face images.
-After this filtering, 9,614 face images of 1,680 subjects are available for evaluation.
-All the testing images in this paper have no identity overlap with the training set, CASIA-
-WebFace [51].
-Data Preprocessing
-All face images are passed through MTCNN face detector [53] to detect five
-landmarks (two eyes, nose, and two mouth corners). Via similarity transformation, the face images
-are aligned. After transformation, the images are resized to 160 × 160. Prior to training and testing,
-each pixel in the RGB image is normalized by subtracting 127.5 and dividing by 128.
-Experimental Settings
-We use ADAM optimizers in Tensorflow with β1 = 0.5 and β2 = 0.9 for
-the entire network. Each mini-batch consists of 32 face images. We train AdvBiom for 200,000 steps
-with a fixed learning rate of 0.0001. Since our goal is to generate adversarial faces with high success
-7Training on CASIA-WebFace and evaluating on LFW is a common approach in face recognition literature [3,
-4]
-9
-
-rate, the identity loss is of utmost importance. We empirically set λi = 10.0 and λp = 1.0. We
-train two separate models and set ϵ = 3.0 and ϵ = 8.0 for obfuscation and impersonation attacks,
-respectively.
-Gallery
-Probe
-Proposed AdvBiom
-GFLM [48]
-PGD [17]
-FGSM [14]
-0.68
-0.14
-0.26
-0.27
-0.04
-0.38
-0.08
-0.12
-0.21
-0.02
-(a) Obfuscation Attack
-Target’s Gallery Target’s Probe
-Probe
-Proposed AdvBiom
-A3GN [42]
-FGSM [14]
-0.78
-0.10
-0.30
-0.29
-0.36
-0.80
-0.15
-0.34
-0.33
-0.42
-(b) Impersonation Attack
-Figure 6: Adversarial face synthesis results on LFW dataset in (a) obfuscation and (b) impersonation attack
-settings (cosine similarity scores obtained from ArcFace [3] with threshold @ 0.1% FAR= 0.28). The proposed
-method synthesizes adversarial faces that are seemingly inconspicuous and maintain high perceptual quality.
-Architecture
-Let c7s1-k be a 7 × 7 convolutional layer with k filters and stride 1. dk denotes a
-4 × 4 convolutional layer with k filters and stride 2. Rk denotes a residual block that contains two
-3 × 3 convolutional layers. uk denotes a 2× upsampling layer followed by a 5 × 5 convolutional
-layer with k filters and stride 1. We apply Instance Normalization and Batch Normalization to the
-generator and discriminator, respectively. We use Leaky ReLU with slope 0.2 in the discriminator
-and ReLU activation in the generator. The architectures of the two modules are as follows:
-• Generator:
-c7s1-64,d128,d256,R256,R256,R256, u128, u64, c7s1-3
-• Discriminator:
-d32,d64,d128,d256,d512
-A 1 × 1 convolutional layer with 3 filters and stride 1 is attached to the last convolutional layer of the
-discriminator for the patch-based GAN loss LGAN.
-We apply the tanh activation function on the last convolution layer of the generator to ensure
-that the generated image ∈ [−1, 1]. In the paper, we denoted the output of the tanh layer as an
-“adversarial mask”, G(x) ∈ [−1, 1] and x ∈ [−1, 1]. The final adversarial image is computed as
-10
-
-Obfuscation Attack
-Proposed AdvBiom
-GFLM [48]
-PGD [17]
-FGSM [14]
-Attack Success Rate (%) @ 0.1% FAR
-FaceNet [5]
-99.67
-23.34
-99.70
-99.96
-SphereFace [4]
-97.22
-29.49
-99.34
-98.71
-ArcFace [3]
-64.53
-03.43
-33.25
-35.30
-COTS-A
-82.98
-08.89
-18.74
-32.48
-COTS-B
-60.71
-05.05
-01.49
-18.75
-Structural Similarity
-0.95 ± 0.01
-0.82 ± 0.12
-0.29 ± 0.06
-0.25 ± 0.06
-Computation Time (s)
-0.01
-3.22
-11.74
-0.03
-Impersonation Attack
-Proposed AdvBiom
-A3GN [42]
-PGD [17]
-FGSM [14]
-Attack Success Rate (%) @ 0.1% FAR
-FaceNet [5]
-20.85 ± 0.40
-05.99 ± 0.19
-76.79 ± 0.26
-13.04 ± 0.12
-SphereFace [4]
-20.19 ± 0.27
-07.94 ± 0.19
-09.03 ± 0.39
-02.34 ± 0.03
-ArcFace [3]
-24.30 ± 0.44
-17.14 ± 0.29
-19.50 ± 1.95
-08.34 ± 0.21
-COTS-A
-20.75 ± 0.35
-15.01 ± 0.30
-01.76 ± 0.10
-01.40 ± 0.08
-COTS-B
-19.85 ± 0.28
-10.23 ± 0.50
-12.49 ± 0.24
-04.67 ± 0.16
-Structural Similarity
-0.92 ± 0.02
-0.69 ± 0.04
-0.77 ± 0.04
-0.48 ± 0.75
-Computation Time (s)
-0.01
-0.04
-11.74
-0.03
-White-box matcher (used for training)
-Black-box matcher (never used in training)
-Table 1: Attack success rates and structural similarities between probe and gallery images for obfus-
-cation and impersonation attacks. Attack rates for obfuscation comprises of 484,514 comparisons and
-the mean and standard deviation across 10-folds for impersonation reported. The mean and standard
-deviation of the structural similarities between adversarial and probe images along with the time
-taken to generate a single adversarial image (on a Quadro M6000 GPU) also reported.
-xadv = 2 × clamp
-�
-G(x) +
-� x+1
-2
-��1
-0 − 1. This ensures G(x) can either add or subtract pixels from
-x when G(x) ̸= 0. When G(x) → 0, then xadv → x.
-Face Recognition Systems
-For all our experiments, we employ 5 state-of-the-art face matchers8.
-Three of them are publicly available, namely, FaceNet [5], SphereFace [4], and ArcFace [3]. We also
-report our results on two commercial-off-the-shelf (COTS) face matchers, COTS-A and COTS-B9.
-We use FaceNet [5] as the white-box face recognition model, F, during training. All the testing
-images in this paper are generated from the same model (trained only with FaceNet) and tested on
-different matchers.
-4.2.1
-Comparison with Prevailing Adversarial Face Generators
-We compare our adversarial face synthesis method with state-of-the-art methods that have specifi-
-cally been implemented or proposed for faces, including GFLM [48], PGD [17], FGSM [14], and
-A3GN [42]10. In Table 1, we find that compared to the state-of-the-art, AdvBiom generates adversarial
-faces that are similar to the probe 6.
-Moreover, the adversarial images attain a high obfuscation attack success rate on 4 state-of-the-art
-black-box AFR systems in both obfuscation and impersonation settings. AdvBiom learns to perturb
-the salient regions of the face, unlike PGD [17] and FGSM [14], which alter every pixel in the
-image. GFLM [48], on the other hand, geometrically warps the face images and thereby, results
-in low structural similarity. In addition, the state-of-the-art matchers are robust to such geometric
-deformation which explains the low success rate of GFLM on face matchers. A3GN, another
-GAN-based method, however, fails to achieve a reasonable success rate in an impersonation setting.
-8All the open-source and COTS matchers achieve 99% accuracy on LFW under LFW protocol.
-9Both COTS-A and COTS-B utilize CNNs for face recognition. COTS-B is one of the top performers in the
-NIST Ongoing Face Recognition Vendor Test (FRVT) [54].
-10We train the baselines using their official implementations (detailed in the supplementary material).
-11
-
-4.2.2
-Ablation Study
-In order to analyze the importance of each module in our system, in Figure 7, we train three variants
-of AdvBiom for comparison by removing the discriminator (D), perturbation loss Lperturbation, and
-identity loss Lidentity, respectively.
-Input
-w/o D
-w/o Lprt
-w/o Lidt
-with all
-Figure 7: Variants of AdvBiom trained without the discriminator, perturbation loss, and identity loss, respectively.
-Every component of AdvBiom is necessary.
-The discriminator helps to ensure the visual quality of the synthesized faces are maintained. With
-the generator alone, undesirable artifacts are introduced. Without the proposed perturbation loss,
-perturbations in the adversarial mask are unbounded and therefore, leads to a lack in perceptual
-quality. The identity loss is imperative in ensuring an adversarial image is obtained. Without the
-identity loss, the synthesized image cannot evade state-of-the-art face matchers. We find that every
-component of AdvBiom is necessary in order to obtain an adversarial face that is not only perceptually
-realistic but can also evade state-of-the-art face matchers.
-4.2.3
-What is AdvBiom Learning?
-Via Lperturbation, during training, AdvBiom learns to perturb only the salient facial regions that can
-evade the face matcher, F (FaceNet [5] in our case). In Figure 8, AdvBiom synthesizes the adversarial
-masks corresponding to the probes. We then threshold the mask to extract pixels with perturbation
-magnitudes exceeding 0.40. It can be inferred that the eyebrows, eyeballs, and nose contain highly
-discriminative information that an AFR system utilizes to identify an individual. Therefore, perturbing
-these salient regions are enough to evade state-of-the-art face recognition systems.
-4.2.4
-Transferability of AdvBiom
-In Table 1, we find that attacks synthesized by AdvBiom when trained on a white-box matcher
-(FaceNet), can successfully evade 5 other face matchers that are not utilized during training in both
-obfuscation and impersonation settings. In order to investigate the transferability property of AdvBiom,
-we extract face embeddings of real images and their corresponding adversarial images, under the
-obfuscation setting, via the white-box matcher (FaceNet) and a black-box matcher (ArcFace). In total,
-we extract feature vectors from 1,456 face images of 10 subjects in the LFW dataset [52]. In Figure 9,
-we plot the correlation heatmap between face features of real images, their corresponding adversarial
-masks and adversarial images. First, we observe that face embeddings of real images extracted by
-FaceNet and ArcFace are correlated in a similar fashion. This indicates that both matchers extract
-features with related pairwise correlations. Consequently, perturbing salient features for FaceNet
-can lead to high attack success rates for ArcFace as well. The similarity among the correlation
-distributions of both matchers can also be observed when adversarial masks and adversarial images
-are input to the matchers. That is, receptive fields for automatic face recognition systems attend to
-similar regions in the face.
-12
-
-Probe
-Adv. Mask
-Visualization
-Adv. Image
-0.12
-0.26
-Figure 8: State-of-the-art face matchers can be evaded by slightly perturbing salient facial regions, such as
-eyebrows, eyeballs, and nose (cosine similarity obtained via ArcFace [3]).
-Figure 9: Correlation between face features extracted via FaceNet and ArcFace from 1,456 images belonging to
-10 subjects.
-To further illustrate the distributions of the embeddings of real and synthesized images, we plot
-the 2D t-SNE visualization of the face embeddings for the 10 subjects in Figure 10. The identity
-clusterings can be clearly observed from both real and adversarial images. In particular, the adversarial
-counterpart of each subject forms a new cluster that draws closer to the adversarial clusterings of
-other subjects. This shows that AdvBiom perturbs only salient pixels related to face identity while
-maintaining a semantic meaning in the feature space, resulting in a similar manifold of synthesized
-faces to that of real faces.
-4.2.5
-Controllable Perturbation
-The perturbation loss, Lperturbation is bounded by a hyper-parameter, ϵ, i.e., the L2 norm of the
-adversarial mask must be at least ϵ. Without this constraint, the adversarial mask becomes a blank
-image with no changes to the probe. With ϵ, we can observe a trade-off between the attack success
-rate and the structural similarity between the probe and synthesized adversarial face (Fig. 11). A
-higher ϵ leads to less perturbation restriction, resulting in a higher attack success rate at the cost of a
-lower structural similarity. For an impersonation attack, this implies that the adversarial image may
-13
-
-Real Image
-Adversarial Mask
-Adversarial Image
-0.8
-FaceNet
-0.4
-0.0
-ArcFace
--0.4FaceNet
-Real Image
-Adversarial Image (Obfuscation)
-ArcFace
-Figure 10: 2D t-SNE visualization of face representations extracted via FaceNet and ArcFace from 1,456 images
-belonging to 10 subjects.
-contain facial features from both the hacker and the target. In our experiments, we chose ϵ = 8.0 and
-ϵ = 3.0 for impersonation and obfuscation attacks, respectively.
-4
-6
-8
-10
-12
-14
-16
-0.81
-0.92
-0.95
-0.76
-0.69
-5
-13
-21
-39
-52
-60
-66
-Hyper-parameter (ε)
-Success Rate (%)
-Structural Similarity
-ε = 4.0
-ε = 8.0
-ε = 10.0
-ε = 16.0
-Figure 11: Trade-off between attack success rate and structural similarity for impersonation attacks.
-4.2.6
-Attacks via AdvBiom Beyond Faces
-We now show that the AdvBiommethod, coupled with the proposed Minutiae Displacement and
-Distortion Modules, can be extended to effectively generate adversarial fingerprints which are visually
-similar to corresponding probe fingerprints while evading two state-of-the-art COTS fingerprint
-matchers as well as a deep network-based fingerprint matcher.
-14
-
-FS: 0.97
-FS: 0.92
-(a) Enrolled Mate
-VS: 235 | FS: 0.96
-VS: 172 | FS: 0.99
-(b) Input Probe
-VS: 31 | FS: 0.92
-VS: 10 | FS: 0.92
-(c) AdvBiom
-VS: 134 | FS: 0.96
-VS: 104 | FS: 0.96
-(d) DeepFool [16]
-VS: 139 | FS: 0.95
-VS: 104 | FS: 0.96
-(d) PGD [17]
-Figure 12: Example probe and corresponding mate fingerprints along with synthesized adversarial probes. (a)
-Two example mate fingerprints from NIST SD4 [55], and (b) the corresponding mates. Adversarial probe
-fingerprints using different approaches are shown in: (c) proposed synthesis method, AdvBiom; (d-e) state-of-
-the-art methods, DeepFool and PGD respectively. VeriFinger v11.0 match score (probe v. mate) - VS, and the
-fingerprintness score (degree of similarity of a given image to a fingerprint pattern) - FS ∈ [0,1] [56], which
-ranges from 1 (the highest) to 0 (the lowest), are given below each image. A VS of above 48 (at 0.01% FAR)
-indicates a successful match between the probe and the mate. The proposed attack AdvBiom successfully
-evades COTS and deep network-based matchers, while maintaining visual fingerprint perceptibility and high
-fingerprintness scores.
-Grosz et. al [57] showed that random minutiae position displacements and non-linear distortions
-drastically affected the performance of COTS fingerprint matchers. AdvBiom builds upon these two
-perturbations and when given a probe fingerprint, can synthesize an adversarial fingerprint image that
-retains all of the original fingerprint attributes except the identity, i.e. a fingerprint recognition system
-should not match the adversarial fingerprint to the probe fingerprint (obfuscation attack).
-Figure 13 shows the schematic of AdvBiom conditioned for fingerprints. The following subsections
-explain the major components of the approach in detail.
-Minutiae Displacement Module
-While the authors in [57] showed the effectiveness of random
-minutiae position displacements on COTS matchers, they studied the effect of this perturbation by
-directly modifying the minutiae template instead of the fingerprint image (pixel space). However, it
-may be difficult to obtain the minutiae template of a given fingerprint image using COTS minutiae
-extractors rather than the source fingerprint image itself. Thus, we propose a minutiae displacement
-module Gdisp which, given a fingerprint image, displaces its minutiae points in random directions by
-a predefined distance. To extract minutiae points from a fingerprint image, we employ a minutiae map
-extractor (M) from [58]. For a fingerprint image of width w and height h, M outputs a 12 channel
-heat map H ∈ Rh×w×12, where if H(i,j,c), value of the heat map at position (i,j) and channel c, is
-greater than a threshold mt and is the local maximum in its 5 × 5 × 3 neighboring cube, a minutiae is
-marked at (i,j). The minutiae direction θ is calculated by maximising the quadratic interpolation with
-respect to:
-f
-�
-(c − 1) × π
-6
-�
-= H (i, j, (c − 1)%12)
-(9)
-f
-�
-c × π
-6
-�
-= H(i, j, c)
-(10)
-f
-�
-(c + 1) × π
-6
-�
-= H(i, j, (c + 1)%12)
-(11)
-Figure 14 shows a fingerprint image and its corresponding 12 channel minutiae map. Once M
-extracts a minutiae map Hprobe from the input probe fingerprint x, we detect minutiae points by
-applying a threshold of 0.2 on Hprobe and finding closed contours. Each detected contour, at say
-15
-
-𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑀𝑎𝑝
-Extractor (ℳ)
-𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑡𝑜𝑟
-(𝒟)
-GAN Loss
-ℒgan
-𝑀𝑖𝑛. 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
-Module (𝒢𝑑𝑖𝑠𝑝)
-Probe Fingerprint
-Displaced Fingerprint
-Original M.Map
-𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑃𝑖𝑥.
-Displacement
-Target M.Map
-M.Map Sim Loss
-ℒmmap_sim
-Pixel Loss
-ℒpixel
-Predicted M.Map
-M.Map Dis Loss
-ℒmmap_dis
-𝐷𝑖𝑠𝑡𝑜𝑟𝑡𝑖𝑜𝑛
-Module (𝒢𝑑𝑖𝑠𝑡)
-𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑀𝑎𝑝
-Extractor (ℳ)
-Adversarial Fingerprint
-Figure 13: Schematic of AdvBiom for generating adversarial fingerprints. Given a probe fingerprint image, it is
-passed to Gdisp which randomly displaces its minutiae points. The distortion module (Gdist) identifies control
-points on the displaced fingerprint and non-linearly distorts the image to output the adversarial fingerprint. The
-solid black arrows show the forward pass of the network while the dotted black arrows show the propagation of
-the losses.
-location (i, j), is displaced by a predefined L1 distance d = |∆i|+|∆j|, giving us the target minutiae
-map Htarget.
-0
-𝜋/6
-𝜋/3
-𝜋/2
-2𝜋/3
-5𝜋/6
-𝜋
-7𝜋/6
-4𝜋/3
-3𝜋/2
-5𝜋/3
-11𝜋/6
-Figure 14: The 12 channel minutiae map of an example fingerprint image shown on the left. The minutiae points
-(shown in red) are marked by a COTS minutiae extractor. The bright spots in each channel image indicate the
-spatial location of minutiae points while the kth channel (k ∈ [0, 11]) indicate the contributions of minutiae
-points to the kπ/6 orientation.
-The minutiae displacement module Gdisp is essentially an autoencoder conditioned on the probe
-fingerprint x and the target minutiae map Htarget. It learns to generate a displaced fingerprint xdisp
-whose predicted minutiae map Hpred is as close as possible to the target minutiae map Htarget in the
-pixel space. To achieve this, we have three losses that govern Gdisp:
-Lmmap_sim = ||Htarget − Hpred||1
-(12)
-Lmmap_dis =
-1
-||Htarget − Hprobe||1
-(13)
-, where Lmmap_sim is the minutiae map similarity loss which minimises the distance between the
-predicted and target minutiae map, while the minutiae map dissimilarity loss Lmmap_dis maximises
-16
-
-G
-QU
-Q
-Q
-QQ
-Q
-Gthe distance between the predicted and probe minutiae map. In figure 15, we show two example
-probe fingerprints and their corresponding displaced fingerprints after passing through Gdisp.
-Distortion Module
-One of the most noteworthy conclusions from [57] was that non-linear distor-
-tions to minutiae points was one of the most successful perturbations to lower the similarity scores
-between perturbed and corresponding unperturbed fingerprints. Again, the non-linear distortion
-was applied to all the minutiae points in the template and not to the image. Thus, our next step in
-generating adversarial fingerprints consists of a distortion module Gdist which learns to distort salient
-points in a fingerprint image.
-The architecture of Gdist consists of an encoder conditioned on the input probe fingerprint x and the
-target minutiae map Hprobe. The output from the encoder is a predefined number of control points11 c.
-The non-linear distortion model proposed in [59], learned using a thin plate spline (TPS) model [60]
-from 320 already distorted fingerprint videos, was employed to calculate the displacements of the
-predicted control points. The hyper-parameter σ is used to indicate the extent of the distortion. The
-control points and their displacements are then fed to a differentiable warping module [61] to get the
-resultant adversarial fingerprint xadv.
-To limit the magnitude of non-linear distortion and to ensure that xdisp and xadv are close to the
-probe fingerprint x, we introduce pixel loss between the image pairs (x, xdisp) and (x, xadv):
-Lpixel = 1
-n
-�
-i,j
-|xi,j − xdispi,j|+ 1
-n
-�
-i,j
-|xi,j − xadvi,j|
-(14)
-Figure 16 shows two displaced fingerprints and their corresponding output from Gdist.
-Discriminator
-In order to guide the generative modules Gdisp and Gdist to synthesize realistic
-fingerprint images, we introduce a fully convolutional network as a patch-based discriminator D.
-The job of the discriminator is to distinguish between real fingerprint images x and the generated
-adversarial fingerprint images xadv. This is accomplished through the GAN loss:
-Lgan = logD(x) + log(1 − D(xadv))
-(15)
-The proposed approach AdvBiom is trained in an end-to-end manner with respect to the following
-objective function:
-(16)
-L = Lgan + λmmap_simLmmap_sim + λmmap_disLmmap_dis + λpixelLpixel
-where the hyper-parameters λmmap_sim, λmmap_dis, and λpixel denote the relative importance of
-their respective losses. Once trained, AdvBiom can generate an adversarial fingerprint image for
-any input probe fingerprint and can be tested on any fingerprint matcher regardless of the feature
-extraction method (minutiae or deep-features).
-11Control points are points in an image to which non-linear distortion is applied.
-Probe Fingerprint
-Displaced Fingerprint
-Probe Fingerprint
-Displaced Fingerprint
-Figure 15: Example probe fingerprints from NIST SD4 [55] and their corresponding output from the minutiae
-displacement module Gdisp. The minutiae points (shown in red) are marked using a COTS minutiae extractor.
-17
-
-Displaced Fingerprint
-Distorted Fingerprint
-Displaced Fingerprint
-Distorted Fingerprint
-Figure 16: Fingerprints in the left column are example displaced fingerprints from Gdisp. The distortion module
-Gdist predicts control points (marked in blue) and distorts the images based on their displacements (red arrows)
-using the non-linear distortion model from [59]. The resultant distorted fingerprint images are shown in the right
-column.
-Successful
-Attacks
-Failed
-Attacks
-Original 
-Probe
-Adv. Probe
-(AdvFinge)
-Mate
-Original 
-Probe
-Adv. Probe
-(AdvFinge)
-Mate
-VS: 36 | FS: 0.82 
-VS: 47 | FS: 0.88 
-VS: 109 | FS: 0.87 
-VS: 76 | FS: 0.89 
-(AdvBiom)
-(AdvBiom)
-Figure 17: Example successful and failed adversarial fingerprints attack using AdvBiom on NIST SD4 [55]. The
-VeriFinger matching scores (probe v. mate): VS, and fingerprintness [56] scores: FS, of adversarial probes are
-shown below their respective triplet. Note that the VeriFinger matching threshold is 48 at 0.01% FAR.
-Evaluation Metrics:
-The requirement of a good adversarial fingerprints generator is to evade
-fingerprint matchers while preserving fingerprint attributes and being model-agnostic. Thus, in order
-to quantify the performance of adversarial attacks generated by AdvBiom and other state-of-the-art
-baselines, we employ the following evaluation metrics:
-• True Accept Rate (TAR): The extent to which an adversarial attack can evade a fingerprint
-matcher is measured by the drop in TAR at an operational setting, say 0.01% False Accept
-Rate (FAR).
-• Fingerprintness: Soweon and Jain [56] proposed a domain-specific metric called finger-
-printness to measure the degree of similarity of a given image to a fingerprint pattern.
-Fingerprintness ranges from [0,1] where higher the score, higher the probability of the
-pattern in the image corresponding to a fingerprint pattern.
-• NFIQ 2.0: Lastly, we use NFIQ 2.0 [62] quality scores to evaluate the fingerprint quality
-of adversarial fingerprint images. NFIQ scores range from [0,100] where a score of 100
-depicts the highest fingerprint quality.
-Note that since non-linear distortions change the structure of the image, using the structural similarity
-index (SSIM) metric is inappropriate as it essentially measures the local change in structures of the
-image pairs.
-Datasets: We train AdvBiom on an internal dataset of 120,000 rolled fingerprint images. Furthermore,
-we evaluate the performance of the proposed fingerprint adversarial attack and other baselines on:
-• 2,000 fingerprint pairs from NIST SD4 [55]
-• 27,000 fingerprint pairs from NIST SD14 [63]
-18
-
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-0甲Accuracy
-Adversarial Attacks
-Original
-Probes
-FGSM
-I-FGSM
-Deep
-Fool
-PGD
-Adv
-Biom
-TAR
-(%)
-at
-0.01%
-FAR
-NIST
-SD4
-VeriFinger
-99.05
-95.20
-98.30
-95.00
-97.60
-56.25
-Innovatrics
-97.00
-93.00
-95.50
-92.65
-94.75
-41.35
-DeepPrint
-94.55
-36.20
-64.15
-30.40
-68.75
-46.35
-NIST
-SD14
-VeriFinger
-99.42
-95.20
-98.30
-95.00
-97.60
-37.67
-Innovatrics
-98.24
-90.84
-95.68
-91.32
-94.01
-25.69
-DeepPrint
-96.52
-48.70
-84.48
-31.44
-64.28
-69.42
-FVC
-2004
-DB1 A
-VeriFinger
-94.89
-91.60
-91.53
-86.92
-92.69
-22.31
-Innovatrics
-94.15
-87.36
-85.68
-82.32
-88.75
-5.52
-DeepPrint
-75.36
-13.22
-33.31
-6.87
-27.39
-20.62
-Table 2: True Accept Rate (TAR) @ 0.01% FAR of AdvBiom along with state-of-the-art baselines attacks on
-three datasets - NIST SD4 [55], NIST SD14 [63], and FVC 2004 DB1 A [64]. 2 COTS fingerprint matchers -
-VeriFinger v11.0 [65] and Innovatrics v7.6.0.627 [66], and a deep network-based matcher DeepPrint [67] were
-employed for the evaluation. It is observed that DeepPrint, a deep network-based matcher, is susceptible to all
-types of adversarial attacks while VeriFinger and Innovatrics are more robust.
-• 558 fingerprints from DB1 A of FVC 2004 [64], consisting of 1,369 genuine pairs.
-Experimental Settings: AdvBiom was trained using the Adam optimizer with β1 as 0.5 and β2 as
-0.9. The hyper-parameters were empirically set to λmmap_sim = 0.05, λmmap_dis = 500000, and
-λpixel = 1000 for convergence. Based on the conclusions drawn in [57], d, c, and λ were set to
-20, 16, and 2.0 respectively for optimal effectiveness against fingerprint matchers while ensuring
-fingerprint realism. AdvBiom was trained for 16,000 steps using Tensorflow r1.14.0 on an Intel Core
-i7-11700F @ 2.50GHz CPU with a RTX 3070 GPU. On the same machine, AdvBiom can synthesize
-an adversarial fingerprint within 0.35 seconds.
-Fingerprint Authentication Systems: Since AdvBiom is a black-box attack, we do not require any
-fingerprint authentication system while training the network. However, we evaluate AdvBiom and
-other baseline attacks on two COTS fingerprint matchers and one deep network-based matcher:
-• VeriFinger v11.0 [65]
-• Innovatrics v7.6.0.627 [66]
-• DeepPrint [67]
-Comparison with Prevailing Fingerprint Adversarial Generators
-We show the performance
-of our method AdvBiom as compared to other state-of-the-art attacks in Table 2. We observe that
-the TAR of two COTS and a deep network-based fingerprint matcher for the aforementioned three
-datasets. It is to note that all the baseline attacks [14, 15, 17, 16] are white-box attacks and were
-trained using DeepPrint [67]. It is evident from Table 2 that AdvBiom is the most successful attack on
-COTS matchers VeriFinger and Innovatrics, and is also able to effectively evade a deep network-based
-fingerprint matcher, namely DeepPrint. It can also be observed that while COTS fingerprint matchers
-are robust to most adversarial attacks, DeepPrint is very susceptible to the same attacks since it
-heavily relies on the texture of the fingerprint which is majorly affected by adversarial attacks.
-A successful adversarial attack should not only evade fingerprint matchers but should also preserve
-fingerprint attributes. In order to observe the effect of adversarial attacks on fingerprint pattern
-in images, we plot the fingerprintness [56] distribution of 2,000 probes from NIST SD4 [55] for
-AdvBiom as well as for other baseline attacks. Since all the state-of-the-art baselines essentially add
-noise to each pixel in the image, they do not change the structure of the fingerprint and thus do not
-19
-
-0.6
-0.7
-0.8
-0.9
-1.0
-Fingerprintedness Scores
-0
-2
-4
-6
-8
-10
-12
-Probability of occurence
-Original Probes (  = 0.91)
-AdvFinge (  = 0.86)
-FGSM (  = 0.89)
-I-FGSM (  = 0.91)
-PGD (  = 0.90)
-DeepFool (  = 0.90)
-AdvBiom
-Figure 18: Fingerprintness [56] distribution of 2,000
-probes from NIST SD4 with respect to AdvBiom and
-other state-of-the-art baselines attacks.
-0
-20
-40
-60
-80
-100
-NFIQ Score
-0.000
-0.005
-0.010
-0.015
-0.020
-0.025
-Probability of occurence
-Original Probes (  = 41.70)
-AdvFinge (  = 31.46)
-FGSM (  = 43.30)
-I-FGSM (  = 44.23)
-PGD (  = 41.28)
-DeepFool (  = 43.42)
-AdvBiom
-Figure 19: NFIQ 2.0 [62] quality scores distribution
-of 2,000 probes from NIST SD4 [55] with respect to
-AdvBiom and other baselines attacks.
-affect fingerprintness scores. AdvBiom , on the other hand, displaces minutiae points and non-linearly
-distorts the image, and still maintains a high mean fingerprintness score of µ = 0.86.
-Furthermore, we also compute the NFIQ 2.0 [62] quality scores distribution (figure 19) of the original
-and adversarial probes from NIST SD4 [55]. As shown in figure 12, baseline attacks tend to minutely
-perturb image pixels to generate adversarial fingerprints and as a result do not have much of an effect
-on the quality scores. AdvBiom , on the other hand, provides an optimal solution by successfully
-attacking fingerprint matchers while maintaining high fingerprintness and NFIQ scores.
-Genuine and Imposter Scores Distribution
-To determine the effect of adversarial fingerprint
-on both genuine and imposter pairs, we plot the genuine and imposter scores distribution of NIST
-SD4 [55] in figure 20 before and after applying AdvBiom . We computed a total of 2,000 genuine
-and 20,000 imposter scores for the evaluation. It can be observed that the genuine scores drastically
-decrease and shift to the left of the axis as their mean drops from 183.87 to 55.55 after the attack.
-However, the imposter scores remain unaffected with the mean imposter score changing by only 0.53.
-0
-100
-200
-300
-400
-Matching Scores
-0.00
-0.02
-0.04
-0.06
-0.08
-0.10
-0.12
-0.14
-Probability of Occurence
-Genuine Scores | Before Attack (  = 183.88)
-Genuine Scores | After Attack (  = 55.55)
-Imposter Scores | Before Attack (  = 6.00)
-Imposter Scores | After Attack (  = 6.52)
-48: Matching Threshold at 0.01% FAR
-(a) Using VeriFinger SDK [65]
-(b) Using Innovatrics SDK [66]
-1.00
-0.75
-0.50
-0.25
-0.00
-0.25
-0.50
-0.75
-1.00
-Matching Scores
-0.00
-0.02
-0.04
-0.06
-0.08
-0.10
-0.12
-Probability of Occurence
-Genuine Scores | Before Attack (  = 0.94)
-Genuine Scores | After Attack (  = 0.78)
-Imposter Scores | Before Attack (  = 0.10)
-Imposter Scores | After Attack (  = 0.09)
-0.837: Matching Threshold at 0.01% FAR
-(c) using DeepPrint [67]
-Figure 20: Genuine and imposter scores distribution of NIST SD4 [55] before and after the adversarial attack
-AdvBiom using three state-of-the-art fingerprint matchers - VeriFinger v11.0 [65], Innovatrics v7.6.0.627 [66],
-and DeepPrint [67]. Here, µ refers to the mean of the scores distribution. In all the three cases, the genuine
-scores shift towards the left while the imposter scores do not get affected by the attack.
-Is AdvBiom Biased Towards Certain Fingerprint Types?
-The generated adversarial fingerprint
-from AdvBiom is conditioned on the input probe fingerprint. Thus, it is essential to check if there is a
-relation between the amount of perturbation applied and the fingerprint type. The confusion matrix
-for the five fingerprint types (left loop, right loop, whorl, arch, tented arch) before and after applying
-AdvBiom on the 2,000 probes of NIST SD4 [55] is shown in table 3. Note that we use NIST SD4 for
-this evaluation since it has a uniform number of fingerprint images per each type (400 fingerprints
-per type). It is evident from the table that all five fingerprint types are almost equally susceptible to
-the attack, and thus the attack crafted AdvBiom is not biased towards a particular fingerprint type.
-20
-
-Genuine Scores LBefore Attack (u = 590.00)
-Genuine Scores 1After Attack (μu = 48.72)
-0.6
-Imposter Scores /Before Attack (μu = 1.34)
-Imposter Scores l After Attack (μ = 1.4o)
-0.5
-Probability of Occurence
-40:Matching Threshold at 0.01% FAR
-0.4
-0.3
-0.2
-0.1
-0.0
-200
-600
-0
-400
-800
-1000
-Matching Scores0.012
-0.010
-0.008
-0.006
-0.004
-0.002
-0.000
-0
-200
-400
-600
-800
-1000Before Attack
-After Attack
-TAR
-L: 99.75%
-R: 99.25%
-W: 99.50%
-T: 99.25%
-A: 97.50%
-FAR
-L: 0%
-R: 0%
-W: 0%
-T: 0%
-A: 0%
-FRR
-L: 0.25%
-R: 0.75%
-W: 0.50%
-T: 0.75%
-A: 2.50%
-TRR
-L: 100%
-R: 100%
-W: 100%
-T: 100%
-A: 100%
-TAR
-L: 59.00%
-R: 56.50%
-W: 58.75%
-T: 56.00%
-A: 57.00%
-FAR
-L: 0%
-R: 0%
-W: 0%
-T: 0%
-A: 0%
-FRR
-L: 41.00%
-R: 43.50%
-W: 41.25%
-T: 44.00%
-A: 43.00%
-TRR
-L: 100%
-R: 100%
-W: 100%
-T: 100%
-A: 100%
-Table 3: Confusion matrix for five fingerprint types (left loop: L, right loop: R, whorl: W, tented arch: T, arch:
-A) from NIST SD4 [55] before and after the adversarial attack using AdvBiom. Here, TAR = True Accept Rate,
-FAR = False Accept Rate, FRR = False Reject Rate, and TRR = True Reject Rate. Note that the matching
-threshold was 48 at 0.01% FAR using the COTS fingerprint matcher VeriFinger. AdvBiom is not biased towards
-any fingerprint type.
-5
-Conclusions
-We show that a new method of adversarial synthesis, namely AdvBiom, that automatically generates
-adversarial face images with imperceptible perturbations evading state-of-the-art biometric matchers.
-With the help of a GAN, and the proposed perturbation and identity losses, AdvBiom learns the set
-of pixel locations required by face matchers for identification and only perturbs those salient facial
-regions (such as eyebrows and nose). Once trained, AdvBiom generates high quality and perceptually
-realistic adversarial examples that are benign to the human eye but can evade state-of-the-art black-
-box face matchers, while outperforming other state-of-the-art adversarial face methods. Beyond
-faces, we show for the first time that such a method with the proposed Minutiae Displacement and
-Distortion Modules can also evade state-of-the-art automated fingerprint recognition systems.
-References
-[1] D. E. Rumelhart and J. L. McClelland, Learning Internal Representations by Error Propagation,
-pp. 318–362. 1987.
-[2] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional
-neural networks,” Commun. ACM, vol. 60, p. 84–90, may 2017.
-[3] J. Deng, J. Guo, N. Xue, and S. Zafeiriou, “Arcface: Additive angular margin loss for deep
-face recognition,” in Proceedings of the IEEE/CVF conference on computer vision and pattern
-recognition, pp. 4690–4699, 2019.
-[4] W. Liu, Y. Wen, Z. Yu, M. Li, B. Raj, and L. Song, “Sphereface: Deep hypersphere embedding
-for face recognition,” in Proceedings of the IEEE conference on computer vision and pattern
-recognition, pp. 212–220, 2017.
-[5] F. Schroff, D. Kalenichenko, and J. Philbin, “Facenet: A unified embedding for face recogni-
-tion and clustering,” in Proceedings of the IEEE conference on computer vision and pattern
-recognition, pp. 815–823, 2015.
-[6] P. Grother, G. Quinn, and P. Phillips, “Report on the evaluation of 2d still-image face recognition
-algorithms,” 2010-06-17 2010.
-[7] N. Carlini and D. Wagner, “Towards evaluating the robustness of neural networks,” in 2017 ieee
-symposium on security and privacy (sp), pp. 39–57, Ieee, 2017.
-[8] Y. Dong, F. Liao, T. Pang, H. Su, J. Zhu, X. Hu, and J. Li, “Boosting adversarial attacks with
-momentum,” in Proceedings of the IEEE conference on computer vision and pattern recognition,
-pp. 9185–9193, 2018.
-[9] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing adversarial examples,”
-arXiv preprint arXiv:1412.6572, 2014.
-[10] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards deep learning models
-resistant to adversarial attacks,” arXiv preprint arXiv:1706.06083, 2017.
-21
-
-[11] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, and R. Fergus,
-“Intriguing properties of neural networks,” arXiv preprint arXiv:1312.6199, 2013.
-[12] S.-M. Moosavi-Dezfooli, A. Fawzi, O. Fawzi, and P. Frossard, “Universal adversarial pertur-
-bations,” in Proceedings of the IEEE conference on computer vision and pattern recognition,
-pp. 1765–1773, 2017.
-[13] UIDAI, “Unique Identification Authority of India.” https://uidai.gov.in, 2022.
-[14] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing adversarial examples,”
-in 3rd International Conference on Learning Representations, ICLR, 2015.
-[15] A. Kurakin, I. J. Goodfellow, and S. Bengio, “Adversarial examples in the physical world,” in
-5th International Conference on Learning Representations, ICLR, 2017.
-[16] S. Moosavi-Dezfooli, A. Fawzi, and P. Frossard, “Deepfool: A simple and accurate method to
-fool deep neural networks,” in IEEE Conference on Computer Vision and Pattern Recognition,
-CVPR, pp. 2574–2582, IEEE Computer Society, 2016.
-[17] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards deep learning models
-resistant to adversarial attacks,” in 6th International Conference on Learning Representations,
-ICLR, 2018.
-[18] N. Carlini and D. A. Wagner, “Towards evaluating the robustness of neural networks,” in IEEE
-Symposium on Security and Privacy, SP, pp. 39–57, IEEE Computer Society, 2017.
-[19] C. Xiao, J. Zhu, B. Li, W. He, M. Liu, and D. Song, “Spatially transformed adversarial examples,”
-in 6th International Conference on Learning Representations, ICLR, 2018.
-[20] K. Eykholt, I. Evtimov, E. Fernandes, B. Li, A. Rahmati, C. Xiao, A. Prakash, T. Kohno, and
-D. Song, “Robust physical-world attacks on deep learning visual classification,” in IEEE/CVF
-Conference on Computer Vision and Pattern Recognition, pp. 1625–1634, 2018.
-[21] N. Papernot, P. D. McDaniel, S. Jha, M. Fredrikson, Z. B. Celik, and A. Swami, “The limitations
-of deep learning in adversarial settings,” in IEEE European Symposium on Security and Privacy,
-EuroS&P, pp. 372–387, IEEE, 2016.
-[22] A. Kurakin, I. J. Goodfellow, and S. Bengio, “Adversarial machine learning at scale,” in 5th
-International Conference on Learning Representations, ICLR, 2017.
-[23] Y. Dong, F. Liao, T. Pang, H. Su, J. Zhu, X. Hu, and J. Li, “Boosting adversarial attacks
-with momentum,” in IEEE Conference on Computer Vision and Pattern Recognition, CVPR,
-pp. 9185–9193, IEEE Computer Society, 2018.
-[24] C. Xie, Z. Zhang, Y. Zhou, S. Bai, J. Wang, Z. Ren, and A. L. Yuille, “Improving transferability
-of adversarial examples with input diversity,” in Proceedings of the IEEE/CVF Conference on
-Computer Vision and Pattern Recognition, pp. 2730–2739, 2019.
-[25] D. Deb, J. Zhang, and A. K. Jain, “Advfaces: Adversarial face synthesis,” in 2020 IEEE
-International Joint Conference on Biometrics (IJCB), pp. 1–10, IEEE, 2020.
-[26] Y. Liu, X. Chen, C. Liu, and D. Song, “Delving into transferable adversarial examples and
-black-box attacks,” in 5th International Conference on Learning Representations, ICLR, 2017.
-[27] W. Xiang, H. Su, C. Liu, Y. Guo, and S. Zheng, “Improving the robustness of adversarial attacks
-using an affine-invariant gradient estimator,” Available at SSRN 4095198, 2021.
-[28] Y. Dong, T. Pang, H. Su, and J. Zhu, “Evading defenses to transferable adversarial examples by
-translation-invariant attacks,” in Proceedings of the IEEE/CVF Conference on Computer Vision
-and Pattern Recognition, pp. 4312–4321, 2019.
-[29] S. Cheng, Y. Dong, T. Pang, H. Su, and J. Zhu, “Improving black-box adversarial attacks with a
-transfer-based prior,” Advances in neural information processing systems, vol. 32, 2019.
-22
-
-[30] A. Ilyas, L. Engstrom, A. Athalye, and J. Lin, “Black-box adversarial attacks with limited
-queries and information,” in International Conference on Machine Learning, pp. 2137–2146,
-PMLR, 2018.
-[31] Y. Li, L. Li, L. Wang, T. Zhang, and B. Gong, “Nattack: Learning the distributions of adver-
-sarial examples for an improved black-box attack on deep neural networks,” in International
-Conference on Machine Learning, pp. 3866–3876, PMLR, 2019.
-[32] Y. Dong, H. Su, B. Wu, Z. Li, W. Liu, T. Zhang, and J. Zhu, “Efficient decision-based black-
-box adversarial attacks on face recognition,” in Proceedings of the IEEE/CVF Conference on
-Computer Vision and Pattern Recognition, pp. 7714–7722, 2019.
-[33] W. Brendel, J. Rauber, and M. Bethge, “Decision-based adversarial attacks: Reliable attacks
-against black-box machine learning models,” arXiv preprint arXiv:1712.04248, 2017.
-[34] M. Sharif, S. Bhagavatula, L. Bauer, and M. K. Reiter, “Accessorize to a crime: Real and stealthy
-attacks on state-of-the-art face recognition,” CCS ’16, (New York, NY, USA), p. 1528–1540,
-Association for Computing Machinery, 2016.
-[35] M. Sharif, S. Bhagavatula, L. Bauer, and M. K. Reiter, “A general framework for adversarial
-examples with objectives,” ACM Transactions on Privacy and Security (TOPS), vol. 22, no. 3,
-pp. 1–30, 2019.
-[36] S. Komkov and A. Petiushko, “Advhat: Real-world adversarial attack on arcface face id system,”
-in 2020 25th International Conference on Pattern Recognition (ICPR), pp. 819–826, IEEE,
-2021.
-[37] D.-L. Nguyen, S. S. Arora, Y. Wu, and H. Yang, “Adversarial light projection attacks on face
-recognition systems: A feasibility study,” in Proceedings of the IEEE/CVF conference on
-computer vision and pattern recognition workshops, pp. 814–815, 2020.
-[38] B. Yin, W. Wang, T. Yao, J. Guo, Z. Kong, S. Ding, J. Li, and C. Liu, “Adv-makeup: A new
-imperceptible and transferable attack on face recognition,” arXiv preprint arXiv:2105.03162,
-2021.
-[39] X. Liu, F. Shen, J. Zhao, and C. Nie, “Rstam: An effective black-box impersonation attack on
-face recognition using a mobile and compact printer,” arXiv preprint arXiv:2206.12590, 2022.
-[40] A. J. Bose and P. Aarabi, “Adversarial attacks on face detectors using neural net based con-
-strained optimization,” in 2018 IEEE 20th International Workshop on Multimedia Signal
-Processing (MMSP), pp. 1–6, IEEE, 2018.
-[41] N. Cauli, A. Ortis, and S. Battiato, “Fooling a face recognition system with a marker-free
-label-consistent backdoor attack,” in Image Analysis and Processing – ICIAP 2022: 21st Inter-
-national Conference, Lecce, Italy, May 23–27, 2022, Proceedings, Part II, (Berlin, Heidelberg),
-p. 176–185, Springer-Verlag, 2022.
-[42] L. Yang, Q. Song, and Y. Wu, “Attacks on state-of-the-art face recognition using attentional
-adversarial attack generative network,” Multimedia tools and applications, vol. 80, no. 1,
-pp. 855–875, 2021.
-[43] Y. Zhong and W. Deng, “Towards transferable adversarial attack against deep face recognition,”
-IEEE Transactions on Information Forensics and Security, vol. 16, pp. 1452–1466, 2020.
-[44] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: A
-simple way to prevent neural networks from overfitting,” Journal of Machine Learning Research,
-vol. 15, no. 56, pp. 1929–1958, 2014.
-[45] S. Jia, B. Yin, T. Yao, S. Ding, C. Shen, X. Yang, and C. Ma, “Adv-attribute: Inconspicuous and
-transferable adversarial attack on face recognition,” arXiv preprint arXiv:2210.06871, 2022.
-[46] H. Qiu, C. Xiao, L. Yang, X. Yan, H. Lee, and B. Li, “Semanticadv: Generating adversarial
-examples via attribute-conditioned image editing,” in European Conference on Computer Vision,
-pp. 19–37, Springer, 2020.
-23
-
-[47] S. Shan, E. Wenger, J. Zhang, H. Li, H. Zheng, and B. Y. Zhao, “Fawkes: protecting privacy
-against unauthorized deep learning models,” in USENIX, pp. 1589–1604, 2020.
-[48] A. Dabouei, S. Soleymani, J. Dawson, and N. Nasrabadi, “Fast geometrically-perturbed adver-
-sarial faces,” in WACV, 2019.
-[49] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and
-Y. Bengio, “Generative adversarial networks,” Communications of the ACM, vol. 63, no. 11,
-pp. 139–144, 2020.
-[50] E. L. Denton, S. Chintala, R. Fergus, et al., “Deep generative image models using a laplacian
-pyramid of adversarial networks,” Advances in neural information processing systems, vol. 28,
-2015.
-[51] D. Yi, Z. Lei, S. Liao, and S. Z. Li, “Learning face representation from scratch,” arXiv preprint
-arXiv:1411.7923, 2014.
-[52] G. B. Huang, M. Ramesh, T. Berg, and E. Learned-Miller, “Labeled faces in the wild: A database
-for studying face recognition in unconstrained environments,” Tech. Rep. 07-49, University of
-Massachusetts, Amherst, October 2007.
-[53] K. Zhang, Z. Zhang, Z. Li, and Y. Qiao, “Joint face detection and alignment using multitask
-cascaded convolutional networks,” IEEE SPL, vol. 23, no. 10, pp. 1499–1503, 2016.
-[54] P. J. Grother, M. Ngan, and K. Hanaoka, “Ongoing Face Recognition Vendor Test (FRVT), Part
-2: Identification,” NIST Interagency Report, 2018.
-[55] NIST,
-“NIST
-Special
-Database
-4.”
-https://www.nist.gov/srd/
-nist-special-database-4, 2022.
-[56] S. Yoon and A. K. Jain, “Is there a fingerprint pattern in the image?,” in 2013 International
-Conference on Biometrics (ICB), pp. 1–8, 2013.
-[57] S. A. Grosz, J. J. Engelsma, N. G. P. Jr., and A. K. Jain, “White-box evaluation of fingerprint
-matchers,” CoRR, vol. abs/1909.00799, 2019.
-[58] K. Cao, D. Nguyen, C. Tymoszek, and A. K. Jain, “End-to-end latent fingerprint search,” IEEE
-Trans. Inf. Forensics Secur., vol. 15, pp. 880–894, 2020.
-[59] X. Si, J. Feng, J. Zhou, and Y. Luo, “Detection and rectification of distorted fingerprints,” IEEE
-Trans. Pattern Anal. Mach. Intell., vol. 37, no. 3, pp. 555–568, 2015.
-[60] F. L. Bookstein, “Principal warps: Thin-plate splines and the decomposition of deformations,”
-IEEE Trans. Pattern Anal. Mach. Intell., vol. 11, no. 6, pp. 567–585, 1989.
-[61] F. Cole, D. Belanger, D. Krishnan, A. Sarna, I. Mosseri, and W. T. Freeman, “Synthesizing
-normalized faces from facial identity features,” in Proceedings of the IEEE Conference on
-Computer Vision and Pattern Recognition (CVPR), July 2017.
-[62] E. Tabassi, “NFIQ 2.0: NIST Fingerprint image quality,” NISTIR 8034, 2016.
-[63] NIST,
-“NIST
-Special
-Database
-14.”
-https://www.nist.gov/srd/
-nist-special-database-14, 2022.
-[64] D. Maio, D. Maltoni, R. Cappelli, J. L. Wayman, and A. K. Jain, “Fvc2004: Third fingerprint
-verification competition,” in International conference on biometric authentication, pp. 1–7,
-Springer, 2004.
-[65] Neurotechnology, “VeriFinger SDK.” https://www.neurotechnology.com, 2022.
-[66] Innovatrics, “Innovatrics SDK.” https://www.innovatrics.com/innovatrics-abis/,
-2022.
-[67] J. J. Engelsma, K. Cao, and A. K. Jain, “Learning a fixed-length fingerprint representation,”
-IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 1–1, 2019.
-24
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-arXiv:2301.03521v1  [math.SP]  9 Jan 2023
-GREEN’S FUNCTIONS FOR FIRST-ORDER SYSTEMS OF
-ORDINARY DIFFERENTIAL EQUATIONS WITHOUT THE
-UNIQUE CONTINUATION PROPERTY
-STEVEN REDOLFI AND RUDI WEIKARD
-Abstract. This paper is a contribution to the spectral theory associated with
-the differential equation Ju′ + qu = wf on the real interval (a, b) when J is
-a constant, invertible skew-Hermitian matrix and q and w are matrices whose
-entries are distributions of order zero with q Hermitian and w non-negative.
-Under these hypotheses it may not be possible to uniquely continue a solution
-from one point to another, thus blunting the standard tools of spectral theory.
-Despite this fact we are able to describe symmetric restrictions of the maximal
-relation associated with Ju′ + qu = wf and show the existence of Green’s
-functions for self-adjoint relations even if unique continuation of solutions fails.
-1. Introduction
-This paper is a contribution to the spectral theory for the differential equation
-Ju′ + qu = wf
-posed on the real interval (a, b) when J is a constant, invertible, and skew-Hermitian
-n × n-matrix while the entries of the matrices q and w are distributions of order
-zero1 with q Hermitian and w non-negative. Ghatasheh and Weikard [7] studied this
-equation under the additional hypothesis that initial value problems have unique
-balanced2 solutions in the space of functions of locally bounded variation.
-The equation Ju′ + qu = wf has, of course, been investigated by many people
-when the coefficients q and w are locally integrable. In that situation initial value
-problems always have unique solutions. This is not necessarily the case when the
-measures induced by q or w have discrete components. It appears that an equation
-with measure coefficients was first considered in 1952, when Krein [8] modelled a
-vibrating string. In 1964 Atkinson [2] suggested to unify the treatment of differen-
-tial and difference equations by writing them as systems of integral equation where
-integrals were to be viewed as matrix-valued Riemann-Stieltjes integrals. Atkinson
-explained that the presence of point masses may prevent the continuation of so-
-lutions across such points and posed a condition avoiding that problem but more
-Date: 11. May 2022.
-This is a preprint of an article published in Integral Equations and Operator Theory which is
-available online at https://doi.org/10.1007/s00020-022-02703-6.
-©2022.
-This manuscript version is made available under the CC-BY-NC-ND 4.0 license
-http://creativecommons.org/licenses/by-nc-nd/4.0/.
-1Recall that distributions of order 0 are distributional derivatives of functions of locally
-bounded variation and hence may be thought of, on compact subintervals of (a, b), as measures.
-For simplicity we might use the word measure instead of distribution of order 0 below.
-2A function of locally bounded variation is called balanced, if its values at any given point are
-averages of its left- and right-hand limits at that point.
-1
-
-2
-STEVEN REDOLFI AND RUDI WEIKARD
-restrictive than the one posed in [7]. In 1999 Savchuk and Shkalikov [10] treated
-Schr¨odinger equations with potentials in the Sobolev space W −1,2
-loc
-. Their paper was
-very influential and spurred many further developments. Nevertheless, Eckhardt
-et al. [5] showed in 2013, with the help of quasi-derivatives or, equivalently, by
-writing the equation as a system, that a treatment without leaving the realm of
-locally integrable coefficients is possible. In the same year Eckhardt and Teschl [6]
-investigated 2×2-systems with diagonal measure-valued matrices q and w requiring
-essentially Atkinson’s condition.
-A more thorough account of the subject’s history is given in [7]. The papers
-[5] and [6], mentioned above, may also serve as excellent sources, with perhaps
-different emphases, of this history.
-One feature of systems of first-order equations is that, generally, they are repre-
-sented by linear relations rather than linear operators. There is a well-developed
-spectral theory for linear relations initiated by Arens [1], see also Orcutt [9], and
-Bennewitz [3]. The most important results (for our purposes) are also surveyed in
-Appendix B of [7].
-Existence or uniqueness of solutions of an initial value problem for Ju′+qu = wf
-fails when, for some x ∈ (a, b), the matrices
-B±(x, 0) = J ± 1
-2∆q(x)
-are not invertible. Here ∆q(x) = Q+(x)−Q−(x) when Q denotes an anti-derivative
-of q. Equivalently, ∆q(x) = dQ({x}) where dQ is the measure (locally) generated
-by q. Assuming the unique continuation property for solutions of Ju′ + qu = wf
-Ghatasheh and Weikard defined maximal and minimal relations Tmax and Tmin
-associated with the differential equation Ju′ + qu = wf and showed that Tmax
-is the adjoint of Tmin. They characterized the self-adjoint restrictions of Tmax, if
-any, with the aid of boundary conditions and proved that resolvents are given as
-integral operators, i.e., the existence of a Green’s function for any such self-adjoint
-relation T . Under even more restrictive conditions they also showed the existence
-of a Fourier transform diagonalizing T .
-Campbell, Nguyen, and Weikard [4] defined maximal and minimal relations and
-showed that Tmax = T ∗
-min without the hypothesis of unique continuation of so-
-lutions. Our goal here is to advance their ideas. In particular, even though the
-equation Ju′ + qu = w(λu + f) may have infinitely many linearly independent
-solutions the deficiency indices, i.e., the number of linearly independent solutions
-of Ju′ + qu = ±iwu of finite positive norm, is still bounded by n, the size of the
-system. We show that symmetric restrictions of Tmax, in particular the self-adjoint
-ones, are still given by posing boundary conditions and we show that the resolvents
-of self-adjoint restrictions are integral operators by proving the existence of Green’s
-functions.
-We will not approach the problem of Fourier transforms and eigenfunction ex-
-pansions but hope to return to it in future work.
-The material in this paper is arranged as follows. In Section 2 we recall the
-circumstances under which existence and uniqueness of solutions to initial value
-problems does hold and investigate the sets of those x ∈ (a, b) and λ ∈ C giving
-rise to trouble.
-Then, in Section 3 we discuss the manifold of solutions of our
-differential equation in the special case when a and b are regular endpoints. These
-results are instrumental in Section 4 where we investigate the deficiency indices
-
-GREEN’S FUNCTIONS
-3
-of the minimal relation and its symmetric extensions but without the assumption
-that a and b are regular. Before we prove the existence of Green’s functions for
-self-adjoint restrictions of the maximal relation in Section 6 we discuss the role
-played by non-trivial solutions of zero norm in Section 5.
-Let us add a few words about notation. D′0((a, b)) is the space of distributions of
-order 0, i.e., the space of distributional derivatives of functions of locally bounded
-variation. Any function u of locally bounded variation has left- and right-hand
-limits denoted by u− and u+, respectively. Also, u is called balanced if u = u# =
-(u+ + u−)/2. The space of balanced functions of bounded variation defined on
-(a, b) is denoted by BV#((a, b)) while BV#
-loc((a, b)) stands for the space of balanced
-functions of locally bounded variation.
-We use
-1 to denote an identity matrix
-of appropriate size and superscripts ⊤ and ∗ indicate transposition and adjoint,
-respectively. The sum of two closed only trivially intersecting subspaces S and T of
-some Hilbert space (i.e., their direct sum) is denoted by S ⊎ T ; if S and T are even
-orthogonal we may use ⊕ instead of ⊎. The orthogonal complement of a subspace
-S of a Hilbert space H is denoted by H ⊖ S or by S⊥. For c1, ..., cN ∈ Cn we
-abbreviate the column vector (c⊤
-1 , ..., c⊤
-N)⊤ ∈ CnN by (c1, ..., cN)⋄.
-2. Preliminaries
-Throughout this paper we assume the following hypothesis to be in force.
-Hypothesis 2.1. J is a constant, invertible and skew-Hermitian n × n-matrix.
-Both q and w are in D′0((a, b))n×n, w is non-negative and q Hermitian.
-Given that w is non-negative it gives rise to a positive measure on (a, b) and we
-denote the space of functions f which satisfy
-�
-f ∗wf < ∞ by L2(w). This space
-permits the semi-inner product ⟨f, g⟩ =
-�
-f ∗wg (note that ⟨f, f⟩ may be 0 without
-f being 0).
-Consider the differential equation
-Ju′ + (q − λw)u = wf
-(2.1)
-where λ is a complex parameter and f an element of L2(w). The latter condition
-guarantees that wf is in D′0((a, b))n. We will search for solutions in BV#
-loc((a, b))n.
-In this case each term in (2.1) is a distribution of order 0 so that it makes sense to
-pose the equation.
-The point a is called a regular endpoint for Ju′ + qu = wf, if there is a point
-c ∈ (a, b) such that the left-continuous anti-derivatives Q and W of q and w are
-of bounded variation on (a, c). In this case q and w may be thought of as finite
-measures on (a, c). Similarly, b is called regular, if Q and W are of bounded variation
-on (c, b). If an endpoint is not regular, it is called singular. Not surprisingly, the
-study of our problem is less complicated when the endpoints are regular and we
-will use this fact to our advantage.
-Despite our earlier denigration of the existence and uniqueness theorem of so-
-lutions of initial value problems it continues to play a crucial role. The following
-theorem was proved in [7].
-Theorem 2.2. Suppose r ∈ D′0((a, b))n×n, g ∈ D′0((a, b))n and that the matrices
-1 ± ∆r(x)/2 are invertible for all x ∈ (a, b). Let x0 be a point in (a, b). Then the
-initial value problem u′ = ru + g, u(x0) = u0 ∈ Cn has a unique balanced solution
-u ∈ BV#
-loc((a, b))n.
-
-4
-STEVEN REDOLFI AND RUDI WEIKARD
-If a is a regular endpoint we may pose an initial condition (for u+) at a. Simi-
-larly, if b is regular we may prescribe u−(b) as the initial condition.
-Suppose now that u is a solution of (2.1). Treating either side of this equation
-as a measure (restricted to a compact subset of (a, b)) evaluation at a singleton {x}
-shows that
-J(u+(x) − u−(x)) + ∆q−λw(x)u#(x) = ∆w(x)f(x)
-or, equivalently,
-B+(x, λ)u+(x) − B−(x, λ)u−(x) = ∆w(x)f(x)
-(2.2)
-when we define
-B±(x, λ) = J ± 1
-2
-�
-∆q(x) − λ∆w(x)
-�
-.
-Note that, if B+(x, λ) is not invertible, we could be in one of the following two
-situations: (i) a solution given on (a, x) may fail to exist on (x, b) or (ii) there
-are infinitely many ways to continue a solution on (a, x) to (x, b). An analogous
-statement holds, of course, if B−(x, λ) is not invertible.
-Let us now investigate the circumstances when a pair (x, λ) gives such trouble.
-Define the sets Λx = {λ ∈ C : det(B+(x, λ)) det(B−(x, λ)) = 0} and Ξλ = {x ∈
-(a, b) : det(B+(x, λ)) det(B−(x, λ)) = 0}. First note, since B−(x, λ) = −B+(x, λ)∗,
-we have that Ξλ = Ξλ and that each Λx is symmetric with respect to the real axis.
-Also, Λx is empty unless at least one of ∆q(x) and ∆w(x) is different from 0 and
-hence for all but countably many x. Next, we claim that Λx is finite as soon as it
-misses one point. To see this suppose that B+(x, λ0) is invertible and that λ ̸= λ0.
-Since
-B+(x, λ) = (λ0 − λ)B+(x, λ0)
-�1
-2B+(x, λ0)−1∆w(x) − 1/(λ − λ0)
-�
-we see that B+(x, λ) fails to be invertible only if 1/(λ−λ0) is an eigenvalue of some
-n × n-matrix. A similar statement holds, of course, for B− proving our claim.
-The really bad points x, namely those where Λx = C, are thus contained in Ξ0.
-Here we wish to remove the hypothesis Ξ0 = ∅ posed in [7]. On any subinterval of
-(a, b) on which q gives rise to a finite measure we find that �∞
-k=1 ∥∆q(xk)∥ must be
-finite, when k �→ xk is a sequence of distinct points in that interval. It follows now
-that Ξ0 is a discrete set. One shows similarly that, for any fixed complex number
-λ the set Ξλ is discrete.
-Lemma 2.3. Suppose [s, t] ⊂ (a, b) and (s, t)∩Ξ0 = ∅. Then we have that Λ(s,t) =
-�
-x∈(s,t) Λx is a discrete subset of C.
-Proof. There are only finitely many points x in (s, t) where ∥J−1∆q(x)∥ > 1. Using
-a Neumann series one sees that only at such points the norm of B+(x, 0)−1 can be
-larger than 2∥J−1∥. Thus there is a positive number C such that ∥B+(x, 0)−1∥ ≤ C
-for all x ∈ (s, t). Now suppose that B+(x, λ) is not invertible and that |λ| ≤ R.
-Then 1/λ is an eigenvalue of 1
-2B+(x, 0)−1∆w(x). This requires that ∥∆w(x)∥ ≥
-2/(RC) and thus can happen only for finitely many x ∈ (s, t). Since similar argu-
-ments work for B− the number of points in �
-x∈(s,t) Λx which lie in a disk of radius
-R centered at 0 must be finite.
-□
-We remark that, when one of the anti-derivatives of q and w is only locally of
-bounded variation, the set �
-x∈(a,b) Λx need not be discrete even if every Λx is finite.
-
-GREEN’S FUNCTIONS
-5
-Theorem 2.4. Suppose [s, t] ⊂ (a, b) and (s, t) ∩ Ξ0 = ∅. If u0 ∈ Cn and λ ∈
-C \ Λ(s,t), then the initial value problem Ju′ + qu = λwu, u+(s) = u0 has a unique
-balanced solution in (s, t). Moreover, u(x, ·) for x ∈ (s, t) as well as u−(t, ·) are
-analytic in C \ Λ(s,t) and meromorphic on C. An analogous statement holds when
-the initial condition is posed at t.
-Proof. The first claim is simply a consequence of Theorem 2.2. When x ∈ (s, t)
-the analyticity of u(x, ·) in C \ Λ(s,t), which is an open set, was proved in Section
-2.3 of [7]. If we modify q and w by setting them 0 on [t, b) we do not change the
-solution on (s, t). The solution for the modified problem evaluated at t is analytic
-and coincides with u−(t, ·) proving its analyticity. It remains to show that a point
-λ0 ∈ Λ(s,t) can merely give rise to poles.
-We know already that there are only finitely many points x in (s, t) where one of
-B±(x, λ0) fails to be invertible. Suppose x′ and x′′ are two consecutive such points.
-If we know the solution on (s, x′) and that u−(x′, ·) has, at worst, a pole at λ0,
-then the solution in (x′, x′′) is determined by the initial value
-u+(x′, λ) = B+(x′, λ)−1B−(x′, λ)u−(x′, λ)
-which also has, at worst, a pole at λ0 since this is true for B+(x′, λ)−1. For x ∈ (s, t)
-the claim follows now by induction. To prove that u−(t, ·) is also meromorphic we
-proceed as before and modify q and w on [t, b).
-□
-3. Solving the differential equation
-Our goal in this section is to investigate the set of solutions of the differential
-equation Ju′ + (q − λw)u = wf on (a, b) under a strengthened hypothesis.
-Hypothesis 3.1. In addition to Hypothesis 2.1 we ask that a and b are regular
-endpoints for Ju′ + qu = wf.
-Moreover, given the partition
-a = x0 < x1 < x2 < ... < xN < xN+1 = b
-(3.1)
-of (a, b) we require that Ξ0 ⊂ {x1, ..., xN}. We then consider only λ for which both
-B+(x, λ) and B−(x, λ) are invertible unless x is in {x1, ..., xN}.
-This hypothesis is in force throughout this section but later only if explicitly
-mentioned. We emphasize that Ξ0 is finite when a and b are regular. Also, the set
-of permissible λ, which we call Ω0, is symmetric with respect to the real axis and
-avoids only a discrete set.
-On each interval (xj, xj+1) we let Uj(·, λ) be a fundamental matrix of balanced
-solutions of the homogeneous differential equation Ju′ + (q − λw)u = 0 such that
-limx↓xj Uj(x, λ) =
-1. The existence of these fundamental matrices is guaranteed by
-Theorem 2.2. The general balanced solution u of the non-homogeneous equation
-Ju′ + (q − λw)u = wf on (xj, xj+1) satisfies, according to Lemma 3.3 in [7],
-u−(x) = U −
-j (x, λ)
-�
-cj + J−1
-�
-(xj,x)
-Uj(·, λ)∗wf
-�
-for any cj ∈ Cn. Define
-Uj(xj+1, λ) =
-lim
-x↑xj+1 Uj(x, λ)
-and
-Ij(f, λ) =
-�
-(xj,xj+1)
-Uj(·, λ)∗wf.
-
-6
-STEVEN REDOLFI AND RUDI WEIKARD
-Using u+(xj) = cj and u−(xj) = Uj−1(xj, λ)(cj−1 + J−1Ij−1(f, λ)) in equation
-(2.2) gives
-(−B−(xj, λ)Uj−1(xj, λ), B+(xj, λ))
-�cj−1
-cj
-�
-= ∆w(xj)f(xj) + B−(xj, λ)Uj−1(xj, λ)J−1Ij−1(f, λ).
-We need to consider these equations for j = 1, ..., N simultaneously. This gives rise
-to the system
-B(λ)˜u = F0(f, λ)
-(3.2)
-where ˜u = (c0, ..., cN)⋄, B(λ), to be specified presently, is in CnN×n(N+1), and
-F0(f, λ) is in CnN.
-The two-diagonal block-matrix structure of B suggests the
-introduction of matrices E⊤ and E⊥, which, respectively, strip the first and last
-n components off a vector in their domain Cn(N+1). If we also define the block-
-matrices
-B(λ) = diag(B+(x1, λ), ..., B+(xN, λ)),
-U(λ) = diag(U0(x1, λ), ..., UN−1(xN, λ)),
-and J = diag(J, ..., J) and when we note that
-B(λ)∗ = diag(−B−(x1, λ), ..., −B−(xN, λ)),
-we obtain
-B(λ) = B(λ)∗U(λ)E⊥ + B(λ)E⊤.
-(3.3)
-The vector F0(f, λ) is given by
-F0(f, λ) = R(f) − B(λ)∗U(λ)J −1I(f, λ)
-with R(f) = ((∆wf)(x1), ..., (∆wf)(xN))⋄ and I(f, λ) = (I0(f, λ), ..., IN−1(f, λ))⋄.
-We now have the following theorem.
-Theorem 3.2. The differential equation Ju′ + (q − λw)u = wf has a solution u
-on (a, b) if and only if ˜u = (u+(x0), ..., u+(xN))⋄ is a solution of equation (3.2).
-In particular, in the homogeneous case, where f = 0, the space of solutions has
-dimension n(N + 1) − rk B(λ) ≥ n.
-We note that rk B(λ) = n when N = 1 so that the space of solutions of Ju′ +
-(q − λw)u = 0 is then exactly n-dimensional. For N = 2, however, consider the
-example (a, b) = R, J =
-� 0 −1
-1
-0
-�
-, q =
-� 0 2
-2 0
-�
-(δ1 − δ2), w =
-� 2 0
-0 0
-�
-(δ1 + δ2), where the
-δk are Dirac point measures concentrated on {k}. It shows that the dimension of
-the space of solutions of Ju′ + (q − λw)u = 0 may be strictly larger than n.
-Next we investigate the connection between the right-hand limits of a solution u
-of the homogeneous equation Ju′ + (q − λw)u = 0 at the points x0, ..., xN (given
-by the vector ˜u) and the vector ˆu = (u(x1), ..., u(xN))⋄. We have ˆu = D(λ)˜u where
-D(λ) = 1
-2(U(λ)E⊥ + E⊤)
-(3.4)
-is again a two-diagonal block-matrix. If N ≥ 2 we will also introduce the matrices
-Bm(λ) and Dm(λ) which are obtained by deleting the first and last n columns from
-B(λ) and D(λ), respectively. If N = 1 we should think of Bm(λ) and Dm(λ) as
-maps from the trivial vector space to Cn. Their adjoints are the map from Cn to
-{0}. With this understanding the following results hold also for N = 1 even though
-they then involve “matrices” with no rows or columns.
-
-GREEN’S FUNCTIONS
-7
-Lemma 3.3. D(λ)∗B(λ) − B(λ)∗D(λ) = diag(−J, 0, ..., 0, J) and Dm(λ)∗B(λ) −
-Bm(λ)∗D(λ) = 0.
-Proof. This follows since U(λ)∗J U(λ) = J which, in turn, follows from Lemma 3.2
-in [7].
-□
-Lemma 3.4. The map v �→ B(λ)v, restricted to ker D(λ), is a bijection onto
-ker Dm(λ)∗. Similarly, the map v �→ D(λ)v, restricted to ker B(λ), is a bijection
-onto ker Bm(λ)∗. In particular, dim ker D(λ) = dim ker Dm(λ)∗ and dim ker B(λ) =
-dim ker Bm(λ)∗.
-Proof. The identity Dm(λ)∗B(λ)−Bm(λ)∗D(λ) = 0 shows that B(λ) maps ker D(λ)
-to ker Dm(λ)∗ as well as that D(λ) maps ker B(λ) to ker Bm(λ)∗.
-If v ∈ ker B(λ) ∩ ker D(λ) one shows that E⊥v = E⊤v = 0 using the definitions
-(3.3) and (3.4) of B and D and the fact that B(λ) − B(λ)∗ = 2J . This, of course,
-implies that v = 0 and hence the injectivity of both B(λ)|ker D(λ) and D(λ)|ker B(λ).
-Clearly, both D(λ) and Dm(λ)∗, having invertible matrices along their main
-diagonal, are of full rank.
-The rank-nullity theorem shows therefore that their
-kernels both have dimension n. This proves surjectivity of B(λ)|ker D(λ).
-Finally, assume that v ∈ ker Bm(λ)∗. Then v = D(λ)x for some x ∈ Cn(N+1)
-which implies that 0 = Bm(λ)∗D(λ)x = Dm(λ)∗B(λ)x. The first part of the proof
-shows that there is a y ∈ ker D(λ) such that B(λ)y = B(λ)x. Hence v = D(λ)(x���y)
-where x − y ∈ ker B(λ).
-□
-The following theorem establishes a connection between solutions of the differ-
-ential equation Ju′ + (q − λw)u = 0 and elements of ker Bm(λ)∗.
-Theorem 3.5. If u is a solution of Ju′ + (q − λw)u = 0 on (a, b), then ˆu =
-(u(x1), ..., u(xN))⋄ is in ker Bm(λ)∗.
-If, in addition, u+(a) = u−(b) = 0, then
-ˆu ∈ ker B(λ)∗ (a subspace of ker Bm(λ)∗).
-Conversely, if ˆu ∈ ker Bm(λ)∗, then Ju′ + (q − λw)u = 0 has a unique solution
-u on (a, b) such that (u(x1), ..., u(xN))⋄ = ˆu. If, indeed, ˆu ∈ ker B(λ)∗, we further
-have u+(a) = u−(b) = 0.
-Let us emphasize that supp u ⊂ [x1, xN] when u+(a) = u−(b) = 0.
-Proof. If u solves Ju′ + (q − λw)u = 0, then, by Theorem 3.2, ˜u ∈ ker B(λ).
-Lemma 3.4 shows then that ˆu = D(λ)˜u is in ker Bm(λ)∗. If u+(a) = u−(b) = 0,
-then Lemma 3.3 gives 0 = B(λ)∗D(λ)˜u = B(λ)∗ˆu.
-Conversely, assume that ˆu ∈ ker Bm(λ)∗ = D(λ)(ker B(λ)).
-Then there is a
-unique vector ˜u ∈ ker B(λ) such that ˆu = D(λ)˜u, which, in turn, defines a unique
-solution u of Ju′+(q−λw)u = 0 such that (u(x1), ..., u(xN))⋄ = ˆu. If ˆu ∈ ker B(λ)∗,
-then, according to Lemma 3.3, diag(−J, 0, ..., 0, J)˜u = 0 which shows that u+(a) =
-u−(b) = 0.
-□
-Given an algebraic system Ax = b we know that there exist solutions only if
-b ∈ ran A = (ker A∗)⊥. For the differential equation Ju′ + (q − λw)u = wf with
-integrable coefficients q and w the unique continuation property for the solutions
-gives rise to the variation of constants formula, which then guarantees the existence
-of solutions for any non-homogeneity f (within reason). In the present situation,
-
-8
-STEVEN REDOLFI AND RUDI WEIKARD
-however, the problem of existence raises its head and we now set out to give neces-
-sary and sufficient conditions for f guaranteeing the existence of a solution in the
-spirit of Linear Algebra.
-Lemma 3.6. If ˜v ∈ ker B(λ) and ˆv = D(λ)˜v, then
-˜v∗E∗
-⊥ = −ˆv∗B(λ)∗U(λ)J −1
-and
-˜v∗E∗
-⊤ = ˆv∗B(λ)J −1.
-Moreover, if f ∈ L2(w) and Jv′ + (q − λw)v = 0, then
-�
-v∗wf = ˆv∗F0(f, λ) + ˆv∗B(λ)J −1˜I(f, λ) = ˆv∗F0(f, λ) + ˜v∗E∗
-⊤˜I(f, λ)
-where (v(x1), ..., v(xN))⋄ = ˆv = D(λ)˜v and ˜I(f, λ) = (0, ..., 0, IN(f, λ))⋄ ∈ CnN.
-Proof. Using the definitions (3.3) and (3.4) of B and D and the identities B(λ) −
-B(λ)∗ = 2J and U(λ)∗J U(λ) = J we obtain that B(λ)˜v = 0 implies
-B(λ)D(λ)˜v = U(λ)∗−1J E⊥˜v
-and
-B(λ)∗D(λ)˜v = −J E⊤˜v.
-Taking adjoints gives the first claim since ˆv = D(λ)˜v.
-The second claim is an immediate consequence of this, since
-�
-v∗wf = ˆv∗R(f) + ˜v∗(I0(f, λ), ..., IN (f, λ))⋄
-= ˆv∗R(f) + ˜v∗E∗
-⊥I(f, λ) + ˜v∗E∗
-⊤˜I(f, λ)
-= ˆv∗R(f) − ˆv∗B(λ)∗U(λ)J −1I(f, λ) + ˆv∗B(λ)J −1˜I(f, λ)
-= ˆv∗F0(f, λ) + ˆv∗B(λ)J −1˜I(f, λ).
-□
-Theorem 3.7. The differential equation Ju′ + (q − λw)u = wf has a solution on
-(a, b) if and only if
-�
-v∗wf = 0 for every solution v of Jv′ + (q − λw)v = 0 which
-vanishes at a and b.
-Proof. By Theorem 3.2 the solution u exists if and only if the system (3.2) has a
-solution ˜u = (u+(x0), ..., u+(xN))⋄. This, in turn, happens if and only if F0(f, λ) ∈
-ran B(λ) = (ker B(λ)∗)⊥.
-By Theorem 3.5 the solutions of Jv′ +(q −λw)v = 0 which vanish at a and b are
-in one-to-one correspondence with elements of ker B(λ)∗. Since v+(xN) = 0 we have
-˜v∗E∗
-⊤˜I(f, λ) = 0 and then, from Lemma 3.6, we obtain ˆv∗F0(f, λ) =
-�
-v∗wf.
-□
-In the case of unique continuation of solutions the condition that v vanishes at a
-or b implies, of course, that v = 0. Consequently, Ju′ + (q − λw)u = wf has then a
-solution for any f ∈ L2(w). The set of all solutions is thus obtained by adding the
-general solution of Ju′ +(q −λw)u = 0 whose dimension is n(N + 1)−rkB(λ) ≥ n.
-Theorem 3.8. The differential equation Ju′ + (q − λw)u = wf has a solution on
-(a, b) which vanishes at a and b if and only if
-�
-v∗wf = 0 for every solution v of
-Jv′ + (q − λw)v = 0.
-Proof. For u to vanish at a and b it is required that u+(x0) = 0 and u+(xN) =
-−J−1IN(f, λ). The system (3.2) is therefore equivalent to
-Bm(λ)(c1, ..., cN−1)⋄ = F0(f, λ) + B(λ)J −1˜I(f, λ).
-The proof is now analogous to the one for Theorem 3.7.
-□
-
-GREEN’S FUNCTIONS
-9
-We conclude this section by “counting” the solutions of Ju′ + qu = λwu which
-are not compactly supported.
-More precisely, we will determine the dimension
-of the quotient space of all solutions of Ju′ + qu = λwu modulo the space of
-compactly supported solutions. Theorem 3.5 shows that the space of all solutions of
-Ju′+qu = λwu is in one-to-one correspondence with ker Bm(λ)∗ and that the space
-of compactly supported solutions of Ju′+qu = λwu is in one-to-one correspondence
-with ker B(λ)∗. We therefore define
-˜n(λ) = dim(ker Bm(λ)∗/ ker B(λ)∗) = dim ker Bm(λ)∗ − dim ker B(λ)∗.
-Lemma 3.9. ˜n(λ) + ˜n(λ) = 2n.
-Proof. Since rk B(λ) = rk B(λ)∗, the rank-nullity theorem implies
-dim ker B(λ) = n(N + 1) − rk B(λ)∗ = n + dim ker B(λ)∗.
-Hence, using also the analogous equation for λ,
-dim ker B(λ) − dim ker B(λ)∗ + dim ker B(λ) − dim ker B(λ)∗ = 2n.
-Lemma 3.4 gives that dim ker B(λ) = dim ker Bm(λ)∗ yielding the claim.
-□
-From Theorem 2.4 we know that the matrices Uj(xj+1, ·) are meromorphic on C
-with poles at most at points in the complement of Ω0. It follows that the entries of
-B are also meromorphic. Since the meromorphic functions on C form a field there
-is a row-echelon matrix ˜B with meromorphic entries such that B˜u = 0 has the same
-solutions as ˜B˜u = 0. Now define a set Ω as Ω0 without the set of all poles of ˜B as
-well as their complex conjugates, and the set of zeros and their conjugates of any
-of the pivots of ˜B.
-Theorem 3.10. If λ ∈ Ω, then dim ker B(λ) = dim ker B(λ) and ˜n(λ) = n.
-Proof. The construction of Ω entails that rk B(λ) = rk ˜B(λ) = rk B(λ) if λ ∈ Ω.
-Since ˜n(λ) = dim ker B(λ) − dim ker B(λ)∗ = dim ker B(λ) + n − dim ker B(λ) we
-obtain ˜n(λ) = n.
-□
-4. Symmetric restrictions of Tmax
-Given a differential equation Ju′ + qu = wf we now define associated minimal
-and maximal relations. Recall that L2(w) is the space of functions f such that
-�
-f ∗wf < ∞. First we define
-Tmax = {(u, f) ∈ L2(w) × L2(w) : u ∈ BV#
-loc((a, b))n, Ju′ + qu = wf}.
-Subsequently we will always tacitly assume that u ∈ BV#
-loc((a, b))n, when we use
-u′. Next, let
-Tmin = {(u, f) ∈ Tmax : supp u is compact in (a, b)}.
-Note that these are spaces of pairs of functions. To employ the power of functional
-analysis we need to realize these relations in Hilbert spaces. Therefore we introduce,
-as usual, the space L2(w) as the quotient of L2(w) modulo the subspace of all u ∈
-L2(w) for which ∥u∥2 =
-�
-u∗wu = 0. Denoting the equivalence class corresponding
-to u by [u] we now set
-Tmax = {([u], [f]) ∈ L2(w) × L2(w) : (u, f) ∈ Tmax}
-
-10
-STEVEN REDOLFI AND RUDI WEIKARD
-and
-Tmin = {([u], [f]) ∈ Tmax : (u, f) ∈ Tmin}.
-Here (and elsewhere) we choose brevity over precision: whenever we have a pair
-([u], [f]) in Tmax we choose u and f such that (u, f) ∈ Tmax.
-Define the vector space
-L0 = {u ∈ BV#
-loc((a, b))n : Ju′ + qu = 0 and ∥u∥ = 0}.
-In many cases this space is trivial and some authors restrict their attention to the
-case where it is; this is then called the definiteness condition. However, we will
-not do so here. Note that ∥u∥ = 0 if and only if wu is the zero distribution. The
-significance of L0 stems from the following fact. Suppose ([u], [f]) ∈ Tmax and that
-there are u, v ∈ [u] and f, g ∈ [f] such that Ju′ + qu = wf and Jv′ + qv = wg.
-Then J(u − v)′ + q(u − v) = w(f − g) = 0 as well as w(u − v) = 0, i.e., u − v ∈ L0.
-In other words, in the presence of a non-trivial space L0, the class [u] has many
-representatives of locally bounded variation satisfying the differential equation for a
-given class [f] (the choice of a representative of [f], on the other hand, is irrelevant).
-In Section 5 we will describe a procedure to choose a representative of [u] in a
-distinctive way.
-In [4] it was proved that Tmin is symmetric, indeed that T ∗
-min = Tmax. In this case
-it is well-known that von Neumann’s theorem holds. Setting Dλ = {([u], λ[u]) ∈
-Tmax} it states that
-Tmax = Tmin ⊎ Dλ ⊎ Dλ
-when Im λ ̸= 0. Moreover, when λ = ±i, these direct sums are even orthogonal. It
-is also known that the dimension of Dλ does not change as λ varies in either the
-upper or the lower half plane. The numbers n± = dim D±i are called deficiency
-indices of Tmin and we are now setting out to investigate these.
-If u is a solution of Ju′ +qu = λwu which is compactly supported then (u, λu) ∈
-Tmin and ([u], λ[u]) ∈ Tmin ∩ Dλ. If λ is not real, then Tmin ∩ Dλ is trivial and it
-follows that compactly supported solutions of Ju′ + qu = λwu do not contribute to
-the corresponding deficiency index. We now have, as a corollary of Theorem 3.10,
-that the deficiency indices of Tmin cannot be more than n if a and b are regular
-endpoints. We do not state this result separately since it is included in the next
-theorem about the general case.
-Thus, to emphasize, we allow in the following a and b to be either regular or
-singular endpoints. Let τk, k ∈ Z, be a strictly increasing sequence in (a, b) having
-a and b as its only limit points and such that all points in Ξ0 are among the
-τk. Considering now only the interval Ik = (τ−k, τk) we set xj = τ−k+j for j =
-0, ..., N + 1 = 2k. We can then introduce the objects from Section 3. To emphasize
-their dependence on k we will add a superscript (k) to those objects. We have then,
-in particular, the matrices B(k), B(k)
-m and the sets Ω(k) of permissible values of λ.
-We now define Ω = �∞
-k=1 Ω(k) and note that Ω is symmetric with respect to the
-real axis and misses only countably many values from C.
-Now fix a non-real λ ∈ Ω. If u is a solution of Ju′+qu = λwu on (a, b) we denote
-its restriction to the interval Ik by u(k). We are interested in the quotient space Xk
-of all solutions of Ju′ +qu = λwu on Ik modulo the compactly supported solutions.
-If u is a solution of Ju′+qu = λwu on Ik we denote the associated equivalence class
-in Xk by ⌊u⌋k. A compactly supported solution u of Ju′ + qu = λwu on Ik can be
-extended by 0 to all of (a, b) yielding an element in Tmin ∩ Dλ. This implies, since
-
-GREEN’S FUNCTIONS
-11
-Im λ ̸= 0, that ∥u∥2 =
-�
-Ik u∗wu = 0 and shows that Xk is a normed space with the
-norm given by ∥u∥2
-k =
-�
-Ik u∗wu. According to Theorem 3.5 the quotient space Xk
-is isomorphic to ker B(k)
-m (λ)∗/ ker B(k)(λ)∗ and, by Theorem 3.10, its dimension is
-equal to n since λ ∈ Ω ⊂ Ω(k).
-Theorem 4.1. The deficiency indices of Tmin are less than or equal to n.
-Proof. Fix a non-real λ ∈ Ω. Suppose u1, ..., um are solutions of Ju′ + qu = λwu
-such that [u1], ..., [um] are linearly independent elements of Dλ. We will show below
-that there is an interval Ip = (τ−p, τp) such that ⌊u(p)
-1 ⌋p, ..., ⌊u(p)
-m ⌋p are linearly
-independent elements of Xp. Hence m ≤ n, the dimension of Xp. Since deficiency
-indices are constant in either half-plane they cannot be larger than n.
-We will now prove the existence of Ip by induction. That is we prove that, for
-every k ∈ {1, ..., m}, there is an interval Iℓk such that the restrictions of u1, ..., uk
-to Iℓk generate linearly independent elements ⌊u(ℓk)
-1
-⌋ℓk, ..., ⌊u(ℓk)
-k
-⌋ℓk of Xℓk. Once
-this is achieved we set p = ℓm.
-Suppose k = 1 and let Iℓ1 be an interval such that ∥u(ℓ1)
-1
-∥ > 0. By what we
-argued above we know that u(ℓ1)
-1
-is not compactly supported in Iℓ1 and thus gives
-rise to a non-zero (and hence linearly independent) element of Xℓ1.
-Now suppose we had already shown our claim for some k < m. If ⌊u(ℓk)
-1
-⌋ℓk, ...,
-⌊u(ℓk)
-k+1⌋ℓk are already linearly independent as elements of Xℓk we choose ℓk+1 = ℓk
-and our induction step is complete. Otherwise, there are unique complex numbers
-α1, ..., αk such that
-∥(α1u1 + ... + αkuk + uk+1)(ℓk)∥ℓk = 0.
-However, there must be an interval Iℓk+1 ⊃ Iℓk where
-∥(α1u1 + ... + αkuk + uk+1)(ℓk+1)∥ℓk+1 > 0
-on account that [u1], ..., [uk+1] are linearly independent. It follows now that, as ele-
-ments of Xℓk+1 the vectors ⌊u(ℓk+1)
-1
-⌋ℓk+1, ..., ⌊u(ℓk+1)
-k+1
-⌋ℓk+1 are linearly independent.
-This completes our induction step also in this case.
-□
-Corollary 4.2. If a and b are regular, then n+ = n−.
-Proof. Fix a non-real λ in Ω. Since a and b are regular, the set Ξλ = Ξλ is finite.
-Thus we may assume that it is contained in Ik = (τ−k, τk) for some appropriate k.
-Then dim ker B(k)(λ) is the number of linearly independent solutions of Ju′ + qu =
-λwu. Theorem 3.10 shows that Ju′ + qu = λwu has the same number of linearly
-independent solutions. Any of these solutions has finite norm but some may have
-norm 0. Now note, that if u is a solution of Ju′+qu = λwu of norm 0, then we have
-wu = 0, so that u is also a solution of Ju′ + qu = λwu. Therefore n+ = n−.
-□
-As mentioned above, it is well-known, even in the case of relations, that von
-Neumann’s theorem E∗ = E⊕Di⊕D−i holds when E is a closed symmetric relation
-in H × H when H is a Hilbert space. In our case, when d = dim Di ⊕ D−i is finite,
-as we just showed, we can use Theorem B.5 in [7] to characterize the symmetric
-restriction of Tmax in terms of boundary conditions. We state that theorem here
-for easy reference. The operator J appearing there is defined by J (u, f) = (f, −u)
-for u, f ∈ H.
-
-12
-STEVEN REDOLFI AND RUDI WEIKARD
-Theorem 4.3. Suppose E is a closed symmetric relation in H × H with d =
-dim Di ⊕ D−i < ∞ and that m ≤ d/2 is a natural number or 0. If A : E∗ → Cd−m
-is a surjective linear operator such that E ⊂ ker A and AJ A∗ has rank d−2m then
-ker A is a closed symmetric restriction of E∗ for which the dimension of (ker A)⊖E
-is m. Conversely, every closed symmetric restriction of E∗ is the kernel of such a
-linear operator A. Finally, ker A is self-adjoint if and only if AJ A∗ = 0 (entailing
-m = d/2).
-A second ingredient for our next considerations is Lagrange’s identity (or Green’s
-formula). If (u, f) and (v, g) are in Tmax, then v∗wf and g∗wu are finite measures.
-Therefore v∗Ju′ + v′∗Ju = v∗wf − g∗wu is also a finite measure. Its antiderivative
-v∗Ju is of bounded variation and thus has limits at a and b. Integration now gives
-Lagrange’s identity
-(v∗Ju)−(b) − (v∗Ju)+(a) = ⟨v, f⟩ − ⟨g, u⟩.
-(4.1)
-Note the right-hand side, and hence the left-hand side, does not change upon choos-
-ing different representatives in place of u, f, v, or g.
-Now, if (v, g) is an element of Di⊕D−i, then (u, f) �→ ⟨(v, g), (u, f)⟩ is a bounded
-linear functional on Tmax. Conversely, since Tmax is a Hilbert space, a bounded
-linear functional on Tmax is given by (u, f) �→ ⟨(v, g), (u, f)⟩ for some (v, g) ∈ Tmax.
-When it is also known that Tmin is in the kernel of this functional, (v, g) may be
-chosen in Di ⊕ D−i. Hence, in our situation, the operator A from Theorem 4.3
-is given by d − m linearly independent elements in Di ⊕ D−i. Lagrange’s identity
-implies that the entries of the matrix AJ A∗ are then given by
-(AJ A∗)k,ℓ = ⟨(vk, gk), (gℓ, −vℓ)⟩ = (g∗
-kJgℓ)−(b) − (g∗
-kJgℓ)+(a).
-(4.2)
-Therefore we arrive at the following theorem.
-Theorem 4.4. Let d = n+ + n− and suppose that m ≤ min{n+, n−}. If (v1, g1),
-..., (vd−m, gd−m) are linearly independent elements of Di⊕D−i such that the matrix
-defined in (4.2) has rank d − 2m, then
-T = {(u, f) ∈ Tmax : (g∗
-j Ju)−(b) − (g∗
-j Ju)+(a) = 0 for j = 1, ..., d − m}
-(4.3)
-is a closed symmetric restriction of Tmax.
-Conversely, if T is a closed symmetric restriction of Tmax and m is the dimen-
-sion of T ⊖ Tmin, then T is given by (4.3) for appropriate elements (v1, g1), ...,
-(vd−m, gd−m) of Di ⊕ D−i for which the matrix defined in (4.2) has rank d − 2m.
-For self-adjoint restrictions of Tmax it is hence necessary and sufficient that n+ =
-n− = m = d−m and that (g∗
-kJgℓ)−(b)−(g∗
-kJgℓ)+(a) = 0 for all 1 ≤ k, ℓ ≤ m = d/2.
-5. The space L0
-We mentioned earlier that the class [u] does not have a unique balanced repre-
-sentative when ([u], [f]) ∈ Tmax, if the space L0 has non-trivial elements. In this
-section we describe a procedure to choose a representative in a distinctive way.
-To this end we assume, without loss of generality, that B+(τ0, 0) = B−(τ0, 0) = J
-so that solutions of our differential equations are continuous at τ0. Define N0 =
-{h(τ0) : h ∈ L0} and for each k ∈ N both Nk = {h+(τk) : h ∈ L0, supp h ⊂ [τk, b)}
-and N−k = {h−(τ−k) : h ∈ L0, supp h ⊂ (a, τ−k]}. Then, for k ∈ N0, we say that a
-function u ∈ BV#
-loc((a, b))n satisfies condition (±k), if u±(τ±k) is perpendicular to
-N±k (using always the upper sign or always the lower sign).
-
-GREEN’S FUNCTIONS
-13
-Lemma 5.1. Suppose ([u], [f]) ∈ Tmax. Then there is a unique balanced v ∈ [u]
-such that (v, f) ∈ Tmax and v satisfies condition (k) for every k ∈ Z.
-Proof. First consider uniqueness. Suppose u and v are two functions satisfying the
-given conditions. Then u − v ∈ L0 and hence (u − v)(τ0)∗t(τ0) = 0 for t = u and
-t = v. Subtract these equations to find (u−v)(τ0) = 0, and thus u = v on (τ−1, τ1).
-Moreover, h1 = (u − v)χ[τ1,b) and h−1 = (u − v)χ(a,τ−1] are in L0. Conditions (1)
-and (−1) show therefore that (u−v)+(τ1) and (u−v)−(τ−1) are also 0 which proves
-that u = v on (τ−2, τ2). Induction informs us now that u = v everywhere.
-We now turn to existence. Pick a balanced representative u ∈ [u] such that
-(u, f) ∈ Tmax. There is an element h0 ∈ L0 such that the orthogonal projection of
-u(τ0) onto N0 equals h0(τ0). Thus v0 = u − h0 satisfies (v0, f) ∈ Tmax, v0 ∈ [u],
-and condition (0).
-Next, there is an element h1 ∈ L0 with support in [τ1, b) such that the orthogonal
-projection of v+
-0 (τ1) onto N1 equals h+
-1 (τ1). We now define v1 = v0 − h1. Then
-(v1, f) ∈ Tmax, v1 ∈ [u], and v1 satisfies condition (1). Notice that v1 = v0 on
-(a, τ1) implying that v1 also satisfies condition (0).
-Proceeding recursively, we may define, for each k ∈ N, functions hk ∈ L0 sup-
-ported in [τk, b) such that vk = u−�k
-j=0 hj satisfies conditions (0), ..., (k), vk ∈ [u],
-and (vk, f) ∈ Tmax.
-Since, for a fixed x ∈ [τ0, b), only finitely many of the numbers hk(x) are different
-from 0, we find that the sequence k �→ vk converges pointwise to a function ˜v ∈ [u]
-satisfying conditions (k) for all k ∈ N0 and (˜v, f) ∈ Tmax. We can now repeat
-this process for negative integers starting from the function ˜v instead of u arriving
-eventually at a function v ∈ [u] satisfying conditions (k) for all k ∈ Z and (v, f) ∈
-Tmax.
-□
-We denote the operator which assigns the function v just constructed to a given
-element ([u], [f]) ∈ Tmax by E. If Im = (τ−m, τm) we also define Em : Tmax →
-BV#(Im)n by composing E with the restriction to the interval Im.
-Note that
-BV#(Im)n is a Banach space with the norm |||u|||m defined as the sum of the
-variation of u over Im and the norm of u(τ0).
-Theorem 5.2. The operator Em : Tmax → BV#(Im)n is bounded.
-Proof. Due to the closed graph theorem we merely have to show that Em is a
-closed operator. Thus assume that the sequence ([uj], [fj]) converges to ([u], [f]) in
-Tmax and that Em([uj], [fj]) converges to v in BV#(Im)n and hence pointwise. To
-simplify notation we assume that Em([uj], [fj])) and Em([u], [f]) are the restrictions
-of uj and u, respectively, to the interval Im. We need to show that u = v on Im.
-First note that u±
-j (τ±k) ∈ N ⊥
-±k and
-��u±
-j (τ±k) − v±(τ±k)
-�� → 0 imply that v
-satisfies conditions (±k) for each k ∈ {0, ..., m − 1}. For ℓ ∈ {−m, m − 1} and
-x ∈ (τℓ, τℓ+1) we have
-u−
-j (x) = U −
-ℓ (x)
-�
-u+
-j (τℓ) + J−1
-�
-(τℓ,x)
-U ∗
-ℓ wfj
-�
-when Uℓ denotes the fundamental matrix of Ju′ + qu = 0 on the interval (τℓ, τℓ+1)
-satisfying U +
-ℓ (τℓ) =
-1. Taking the limit as j → ∞ gives
-v−(x) = U −
-ℓ (x)
-�
-v+(τℓ) + J−1
-�
-(τℓ,x)
-U ∗
-ℓ wf
-�
-
-14
-STEVEN REDOLFI AND RUDI WEIKARD
-since the integral may be considered as a vector of scalar products which are, of
-course, continuous. The variation of constants formula shows that v is a balanced
-solution for Jv′ + qv = wf on (τℓ, τℓ+1). We also have
-J(u+
-j (τℓ) − u−
-j (τℓ)) + ∆q(τℓ)uj(τℓ) = ∆w(τℓ)fj(τℓ).
-(5.1)
-The fact that [fj] converges to [f] in L2(w) implies, on account of the positivity
-of w, that ∆w(τℓ)fj(τℓ) converges to ∆w(τℓ)f(τℓ).
-Therefore taking a limit in
-(5.1) shows, in conjunction with the previous observations, that Jv′ + qv = wf on
-the interval Im. Since u satisfies the same equation we have that u − v satisfies
-J(u − v)′ + q(u − v) = 0 on Im.
-Next we show w(u − v) = 0 on Im. Fatou’s lemma implies
-0 ≤
-�
-Im
-(u − v)∗w(u − v) ≤ lim inf
-j→∞
-�
-Im
-(u − uj)∗w(u − uj) = 0.
-It follows that w(u − v) = 0 on Im.
-Finally, a variant of Lemma 5.1 shows now that u = v.
-□
-6. Green’s function
-Now suppose that we have a self-adjoint restriction T of Tmax. The resolvent set
-of T is the set of those λ for which T − λ : dom(T ) → L2(w) is bijective, i.e.,
-̺(T ) = {λ ∈ C : ker(T − λ) = {0}, ran(T − λ) = L2(w)}
-which is an open set. We denote its complement, the spectrum of T , by σ(T ).
-Since T is self-adjoint, σ(T ) is a subset of R.
-If λ ∈ ̺(T ), then the resolvent
-Rλ = (T − λ)−1 is a bounded linear operator from L2(w) to dom(T ). We now
-define Rλ : L2(w) → BV#
-loc((a, b))n by
-Rλ[f] = E((Rλ[f], λRλ[f] + [f])).
-Thus Rλ[f] is the unique solution of Ju′ + qu = w(λu + f) in L2(w) satisfying
-condition (k) for every k ∈ Z.
-We will now show that Rλ is an integral operator. Its kernel G is called a Green’s
-function for T .
-Theorem 6.1. If T is a self-adjoint restriction of Tmax, then there exists, for given
-x ∈ (a, b) and λ ∈ ̺(T ), a matrix G(x, ·, λ) such that the columns of G(x, ·, λ)∗ are
-in L2(w) and
-(Rλ[f])(x) =
-�
-G(x, ·, λ)wf.
-(6.1)
-Proof. Fix x ∈ Im and λ ∈ ̺(T ). Consider the restriction of Rλ[f] to the interval
-Im. Since Em and Rλ are bounded operators the map [f] �→ (Rλ[f])(x) is a bounded
-linear map from L2(w) to Cn. Hence there are elements [g1], ..., [gn] ∈ L2(w) such
-that the k-th component of (Rλ[f])(x) equals ⟨[gk], [f]⟩. Let these be the columns
-of the matrix-valued function G(x, ·, λ)∗. Then we obtain (6.1).
-□
-One wishes to complement this fairly abstract existence result by a more concrete
-one where Green’s function is given in terms of solutions of the differential equation
-as is done in the classical case, see, for instance, Zettl [11]. This was also achieved
-in [7] under the assumption that Ξ0 is empty and minor generalizations of this
-are certainly possible. Such an explicit construction of Green’s function, where
-possible, is the cornerstone of many other results in spectral theory, in particular
-
-GREEN’S FUNCTIONS
-15
-the development of a spectral transformation and more detailed information about
-the resolvent, e.g., the compactness of the resolvent in the regular case. Due to
-the difficulties posed by the absence of an existence and uniqueness theorem for
-initial value problems we have, so far, not been able to obtain such a construction
-in general. However, we hope to return to this issue in the future.
-7. Example
-In this section we treat an example where the matrices B±(x, λ) fail to be invert-
-ible for infinitely many x and all λ, in other words where Ξ0 is infinite and Λx = C
-for all x ∈ Ξ0 (recall that in [7] the hypothesis Ξ0 = ∅ was made causing each Λx
-to be finite). The example is Ju′ + qu = wf on (a, b) = R where
-J =
-�
-0
-−1
-1
-0
-�
-, q =
-�
-0
-2
-2
-0
-� �
-k∈Z
-(δ2k − δ2k+1), and, w =
-�
-2
-0
-0
-0
-� �
-k∈Z
-δk
-with δk denoting the Dirac point measure concentrated on {k}. Since we are seeking
-balanced solutions we need the matrices
-B−(2k − 1, λ) =
-�λ
-0
-2
-0
-�
-and
-B+(2k − 1, λ) =
-�−λ
-−2
-0
-0
-�
-as well as
-B−(2k, λ) =
-�
-λ
-−2
-0
-0
-�
-and
-B+(2k, λ) =
-�
-−λ
-0
-2
-0
-�
-.
-If x is not an integer we have B±(x, λ) = J. Note that f ∈ L2(w) if and only if
-k �→ f1(k) is in ℓ2(Z) and any element in L2(w) is uniquely determined by these
-values (here f1 denotes the first component of f).
-In any interval (k, k + 1) solutions of Ju′ + qu = w(λu + f) are constant, say
-(αk, βk)⊤. At x = 2k − 1 the equation
-B+(2k − 1, λ)u+(2k − 1) − B−(2k − 1, λ)u−(2k − 1) = (2f1(2k − 1), 0)⊤
-implies α2k−2 = 0 and
-− λα2k−1 − 2β2k−1 = 2f1(2k − 1).
-(7.1)
-Similarly, at x = 2k we get α2k = 0 and
-− λα2k−1 + 2β2k−1 = 2f1(2k).
-(7.2)
-We can now describe the space Tmax. A pair (u, f) is in Tmax if and only if the
-sequences k �→ f1(k) and k �→ u1(k) are in ℓ2(Z), f1(2k) = −f1(2k − 1), u1(2k) =
-u1(2k − 1), and
-u =
-�
-k∈Z
-� �2u1(2k)
-f1(2k)
-�
-χ#
-(2k−1,2k) +
-� 0
-β2k
-�
-χ#
-(2k,2k+1)
-�
-with arbitrary numbers β2k. Note that ∥u∥2 = 4 �
-k∈Z |u1(2k)|2.
-Choosing here f = 0 shows that 0 is an eigenvalue of Tmax with infinite multi-
-plicity. Choosing f = 0 and requiring ∥u∥ = 0 determines the space L0. Indeed,
-L0 =
-� �
-k∈Z
-�
-0
-β2k
-�
-χ#
-(2k,2k+1) : β2k ∈ C
-�
-
-16
-STEVEN REDOLFI AND RUDI WEIKARD
-which is infinite-dimensional. We now define the sequence τ setting τ0 = 1/2 and,
-for k ∈ N, τk = k and τ−k = 1 − k. A solution u of Ju′ + qu = w(λu + f) always
-satisfies condition (2k + 1) and it satisfies condition (2k) exactly when β2k = 0.
-For f = 0 equations (7.1) and (7.2) show that no non-zero λ can be an eigenvalue
-of Tmax. In particular, the deficiency indices n± are 0, i.e., Tmax is self-adjoint. Now
-choose λ ̸= 0 and f arbitrary in L2(w). Then
-(Rλf)(x) = − 1
-2λ
-�
-k∈Z
-�2f1(2k − 1) + 2f1(2k)
-λf1(2k − 1) − λf1(2k)
-�
-χ#
-(2k−1,2k)(x)
-(7.3)
-is the unique solution of Ju′ + qu = w(λu + f) satisfying condition (k) for any
-k ∈ Z. Since
-∥Rλf∥2 =
-�
-k∈Z
-2|(Rλf)1(k)|2 =
-1
-|λ|2
-�
-k∈Z
-|f1(2k − 1) + f1(2k)|2
-(7.4)
-is finite we have that C \ {0} is the resolvent set of Tmax.
-We now define H = {u ∈ L2(w) : u1(2k − 1) = u1(2k)} and H∞ = {f ∈ L2(w) :
-f1(2k − 1) = −f1(2k)}. These spaces are orthogonal to each other and their direct
-sum is L2(w). Equation (7.4) shows that ker Rλ = H∞. Moreover, we have
-Tmax = (H × {0}) ⊕ ({0} × H∞).
-This is an instance of a general feature for a self-adjoint linear relation T : if H is
-the closure of the domain of T , H∞ the orthogonal complement of H, and T0 =
-T ∩ (H × H), then T = T0 ⊕ ({0} × H∞). The former summand is then a linear
-operator densely defined in H called the operator part of T . The latter summand
-is called the multi-valued part of T .
-We end this example by identifying Green’s function for our example. It may
-be guessed by looking at equation (7.3). In any case one can check directly that
-(Rλf)(x) =
-�
-G(x, ·, λ)wf. Note that the second column of G is irrelevant since
-the second row of w is 0. When x is not integer G(x, y, λ) is given by
-�
-k∈Z
-�
-− 1
-λ
-�1
-0
-0
-0
-�
-+ 1
-2
-� 0
-1
-−1
-0
-�
-sgn(x − y)
-�
-χ#
-(2k−1,2k)(x)χ#
-(2k−1,2k)(y).
-If x is an integer we have instead
-G(2k − 1, y, λ) = 1
-2
-lim
-x↓2k−1 G(x, y, λ)
-and
-G(2k, y, λ) = 1
-2 lim
-x↑2k G(x, y, λ).
-References
-[1] Richard Arens. Operational calculus of linear relations. Pacific J. Math., 11:9–23, 1961.
-[2] F. V. Atkinson. Discrete and continuous boundary problems. Mathematics in Science and
-Engineering, Vol. 8. Academic Press, New York-London, 1964.
-[3] Christer Bennewitz. Symmetric relations on a Hilbert space. Pages 212–218. Lecture Notes
-in Math., Vol. 280, 1972.
-[4] Kevin Campbell, Minh Nguyen, and Rudi Weikard. On the spectral theory for first-order
-systems without the unique continuation property. Linear Multilinear Algebra, 69(12):2315–
-2323, 2021. Published online: 04 Oct 2019.
-[5] Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, and Gerald Teschl. Weyl-Titchmarsh the-
-ory for Sturm-Liouville operators with distributional potentials. Opuscula Math., 33(3):467–
-563, 2013.
-[6] Jonathan Eckhardt and Gerald Teschl. Sturm-Liouville operators with measure-valued coef-
-ficients. J. Anal. Math., 120:151–224, 2013.
-
-GREEN’S FUNCTIONS
-17
-[7] Ahmed Ghatasheh and Rudi Weikard. Spectral theory for systems of ordinary differential
-equations with distributional coefficients. J. Differential Equations, 268(6):2752–2801, 2020.
-[8] M. G. Kre˘ın. On a generalization of investigations of Stieltjes. Doklady Akad. Nauk SSSR
-(N.S.), 87:881–884, 1952.
-[9] Bruce Call Orcutt. Canonical differential equations. PhD thesis, University of Virginia, 1969.
-[10] A. M. Savchuk and A. A. Shkalikov. Sturm-Liouville operators with singular potentials. Math-
-ematical Notes, 66(6):741–753, 1999. Translated from Mat. Zametki, Vol. 66, pp. 897–912
-(1999).
-[11] Anton Zettl. Sturm-Liouville theory, volume 121 of Mathematical Surveys and Monographs.
-American Mathematical Society, Providence, RI, 2005.
-Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL
-35226-1170, USA
-Email address: [email protected], [email protected]
-

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-Towards Dependable Autonomous Systems
-Based on Bayesian Deep Learning Components
-Fabio Arnez∗, Huascar Espinoza†, Ansgar Radermacher∗ and Franc¸ois Terrier∗
-∗Universit´e Paris-Saclay, CEA, List, F-91120, Palaiseau, France
-{name.lastname}@cea.fr
-†KDT JU, TO 56 05/16, B-1049 Brussels, Belgium
-[email protected]
-Abstract—As
-autonomous
-systems
-increasingly
-rely
-on
-Deep Neural Networks (DNN) to implement the navigation
-pipeline functions, uncertainty estimation methods have be-
-come paramount for estimating confidence in DNN predictions.
-Bayesian Deep Learning (BDL) offers a principled approach to
-model uncertainties in DNNs. However, in DNN-based systems,
-not all the components use uncertainty estimation methods
-and typically ignore the uncertainty propagation between them.
-This paper provides a method that considers the uncertainty
-and the interaction between BDL components to capture the
-overall system uncertainty. We study the effect of uncertainty
-propagation in a BDL-based system for autonomous aerial
-navigation. Experiments show that our approach allows us to
-capture useful uncertainty estimates while slightly improving the
-system’s performance in its final task. In addition, we discuss the
-benefits, challenges, and implications of adopting BDL to build
-dependable autonomous systems.
-Index Terms—Bayesian Deep Learning, Uncertainty Propaga-
-tion, Unmanned Aerial Vehicle, Navigation, Dynamic Depend-
-ability
-I. INTRODUCTION
-Navigation in complex environments still represents a big
-challenge for autonomous systems (AS). Particular instances
-of this problem are autonomous driving and autonomous aerial
-navigation in the context of self-driving cars and Unmanned
-Aerial Vehicles (UAVs), respectively. In both cases, the naviga-
-tion task is addressed by first acquiring rich and complex raw
-sensory information (e.g., from camera, radar, LiDAR, etc.),
-which is then processed to drive the autonomous agent towards
-its goal. Usually, this process is done in sequence, where tasks
-and specific software components are linked together in the so-
-called perception-planning-control software pipeline [1], [2].
-Over the last decade, Deep Neural Networks (DNNs) have
-become a popular choice to implement navigation pipeline
-components thanks to their effectiveness in processing com-
-plex sensory inputs, and their powerful representation learning
-that surpasses the performance of traditional methods. Cur-
-rently, three main paradigms exist to develop and train navi-
-gation components based on DNNs: Modular (isolated), End-
-to-End (E2E) learning, and mixed or hybrid approaches [2].
-Preprint version. Accepted and presented at the 18th European Depend-
-able Computing Conference (EDCC), Zaragoza, Spain, 2022. Digital Object
-Identifier (DOI) is available in the preprint description.
-Fig. 1. UAV BDL-based Aerial Navigation Pipeline: The downstream control
-component gets predictions of the previous perception component as input and
-must take their uncertainty into account.
-Despite the remarkable progress in representation learning,
-DNNs should also represent the confidence in their predictions
-to deploy them in safety-critical systems. McAllister et al. [2]
-proposed using Bayesian Deep Learning (BDL) to implement
-the components from navigation pipelines or stacks. Bayesian
-methods offer a principled framework to model and capture
-system uncertainty. However, if the Bayesian approach is
-followed, all the components in the system pipeline should
-use BDL to enable uncertainty propagation in the pipeline.
-Hence, BDL components should admit uncertainty information
-as an input to account for the uncertainty from the outputs of
-preceding BDL components (See Fig. 1).
-In recent years, a large body of literature has employed
-uncertainty estimation methods in robotic tasks thanks to
-its potential to improve the safety of automated functions
-[3], and the capacity to increase the task performance [4],
-[5]. However, uncertainty is captured partially in navigation
-pipelines that utilize DNNs. BDL methods are used mainly in
-perception tasks, and downstream components (e.g., planning
-and control) usually ignore the uncertainty from the preceding
-components or do not capture uncertainty in their predictions.
-Although some works propagate downstream perceptual
-uncertainty from intermediate representations [6]–[8], the
-overall system output does not take into account all the
-uncertainty sources from DNN components in the pipeline.
-Moreover, proposed frameworks for dynamic dependability
-management that use uncertainty information focus only on
-DNN-based perception tasks [9], [10], ignoring uncertainty
-propagation through the system pipeline, the interactions be-
-arXiv:2301.05297v1  [cs.RO]  12 Jan 2023
-
-Perception
-Controltween uncertainty-aware components, and the potential impact
-on system performance and safety.
-Quantifying uncertainty in a BDL-based system (i.e., a
-pipeline of BDL components) still remains a challenging task.
-Uncertainties from BDL components must be assembled in
-a principled way to provide a reliable measure of overall
-system uncertainty, based on which safe decisions can be
-made [2], [11]. In this paper, we propose to capture the
-uncertainty along a pipeline of BDL components and study
-the impact of uncertainty propagation on the aerial navigation
-task in a UAV. In addition, we propose an uncertainty-centric
-dynamic dependability management framework to cope with
-the challenges that arise from propagating uncertainty through
-BDL-based systems.
-II. RELATED WORK
-A. Neural Network Uncertainty Estimation
-Bayesian neural networks (BNN) have been widely used
-to represent the confidence in the predictions. A proper con-
-fidence representation in DNN predictions can be achieved
-by modeling two sources of uncertainty: aleatoric (data)
-and epistemic (model) uncertainty. For epistemic uncertainty,
-Bayesian inference is used to estimate the posterior predic-
-tive distribution. In practice, approximate Bayesian inference
-methods are often used [12]–[15] since the posterior on the
-model parameters p(θ | D) is intractable in DNNs.
-To model data uncertainty, [14], [16] propose to incorporate
-additional outputs to represent the parameters (mean and vari-
-ance) of a Gaussian distribution. Loquercio et al. [17] forward
-propagate sensor noise through the DNN. This approach does
-not require retraining, however, it assumes a fixed uncertainty
-value for the sensor noise at the input. Another family of
-methods aim to capture complex stochastic patterns such
-as multimodality or heteroscedasticity (aleatoric uncertainty)
-using latent variables (LV) as input. When BNNs are used with
-LV (BNN+LV), both types of uncertainty can be captured [18],
-[19]. In this approach, a BNN receives an input combined with
-a random disturbance coming from an LV (i.e., features are
-partially stochastic). In contrast, this paper considers that a
-BNN can receive a complete stochastic features at the input.
-B. Uncertainty in DNN-Based Navigation
-In an autonomous driving context, perception uncertainty is
-captured from implicit [8] and explicit representations [7] and
-used downstream for scene motion forecasting and trajectory
-planning respectively. In reinforcement learning, input uncer-
-tainty has been employed for model-based [20] and model-free
-control policies [21]. In the former case, a collision predictor
-uncertainty is passed to a model predictive controller. In the
-latter, perception uncertainty is mapped to the control policy
-uncertainty using heuristics. In the context of aerial navigation,
-a few works have considered uncertainty. [17] uses a fixed
-uncertainty value for sensors as an input to a control policy.
-[6] extends the work from [22] to use the uncertainty from
-perception noisy representations downstream in a BNN control
-policy. Although these approaches use perception uncertainty
-in downstream components, not all the DNN components in
-the pipeline employ uncertainty estimation methods.
-C. Uncertainty-based Dependability Frameworks
-For the deployment of dependable autonomous systems that
-use machine learning (ML) components, Trapp et al. [23] and
-Henne et al. [9] conceptualized the use and runtime monitoring
-of perception uncertainty to ensure safe behavior on AS. To
-model system behavior, probabilistic graphical models (PGMs)
-and, in particular, Bayesian Networks (BNs) have been used in
-dependability research for safety and reliability analyses and
-risk assessment applications [24]. BNs allow incorporating ex-
-pert domain knowledge, model complex relationships between
-components, and enable decision-making under uncertainty.
-In the context of autonomous aviation systems, [10] proposes
-a method for quantifying system assurance using perception
-component uncertainty and dynamic BNs. For autonomous
-vehicles, [25] offers a framework for dynamic risk assessment,
-using BNs to predict the behavior intents of other traffic par-
-ticipants. Unlike these works, this paper considers uncertainty
-from Bayesian deep learning components beyond perception.
-III. SYSTEM TASK FORMULATION
-In this paper, we address the problem of autonomous aerial
-navigation. The goal of the autonomous agent (i.e., UAV) is
-to navigate through a set of gates with unknown locations
-disposed in a circular track. Following prior work from [6],
-[22], the navigation architecture consists of two DNN-based
-components: one for perception and the other for control
-(see Fig. 2). Both DNNs are trained following the hybrid
-paradigm. To achieve the agent goal, the navigation task is
-formulated as a sequential-decision making problem, where
-a sequence of control actions are produced given environ-
-ment observations. In this regard, the simulation environment
-provides at each time step an observation comprised of an
-RGB image x acquired from a front-facing camera on the
-UAV. The perception component defines an encoder function
-qφ : X → Z that maps the input image x to a rich low
-dimensional representation z ��� R10. Next, a control policy
-πw : Z → Y maps the compact representation z to control
-commands y = [ ˙x, ˙y, ˙z, ˙ψ] ∈ R4, corresponding to linear and
-angular (yaw) velocities in the UAV body frame.
-In the perception component, a cross-modal variational
-autoencoder (CMVAE) [22], [26] is used to learn a rich and
-robust compact representation. A CMVAE is a variant of the
-traditional variational autoencoder (VAE) [27] that learns a
-single latent representation for multiple data modalities. In
-this case, the perception dataset Dp has two data modalities:
-the RGB images and the pose of the gate relative to the UAV
-body-frame. During training, the CMVAE encoder qφ maps an
-input image x to a noisy representation with mean µφ(x) and
-variance σ2
-φ(x) in the latent space, from where latent vectors
-z are sampled, z ∼ N(µφ, σ2
-φ). Next, a latent vector z is used
-to reconstruct the input image and estimate the gate pose (i.e.,
-recover the two data modalities) using two DNNs, a decoder
-and a feed-forward network. The CMVAE encoder qφ is based
-
-Fig. 2. System architecture for aerial navigation
-on the Dronet architecture [28], and additional constraints
-on the latent space are imposed through the loss function to
-promote the learning of robust disentangled representations.
-Once the perception component is trained, the downstream
-control task (control policy π) uses a feed-forward network to
-operate on the latent vectors z at the output of the CMVAE
-encoder qφ to predict UAV velocities. To this end, the control
-policy network is added at the output of the perception
-encoder qφ, forming the navigation pipeline DNN. The control
-component π uses a control imitation learning dataset (Dc).
-During training, we freeze the perception encoder qφ to update
-only the control policy network. For more information about
-the general architecture for aerial navigation, datasets, and
-training procedures, we refer the reader to [6], [22].
-IV. METHODOLOGY
-A. Uncertainty from Perception Representations
-Although the CMVAE encoder qφ employs Bayesian in-
-ference to obtain latent vectors z, CMVAE does not capture
-epistemic uncertainty since the encoder lacks a distribution
-over parameters φ. To capture uncertainty in the perception
-encoder we follow prior work from [29], [30] that attempts to
-capture epistemic uncertainty in VAEs. We adapt the CMVAE
-to capture the posterior qΦ(z | x, Dp) as shown in (1).
-qΦ(z | x, Dp) =
-�
-q(z | x, φ)p(φ | Dp)dφ
-(1)
-To approximate (1), we take a set Φ = {φm}M
-m of encoder
-parameters samples φm ∼ p(φ | Dp), to obtain a set of
-latent samples {zm}M
-m=1 ∼ qΦ(z | x, Dp) at the output
-of the encoder. In practice, we modify CMVAE by adding
-a dropout layer in the encoder. Then, we use Monte Carlo
-Dropout (MCD) [12] to approximate the posterior on the
-encoder weights p(φ | Dp). Finally, for a given input image x
-we perform M stochastic forward passes (with dropout “turned
-on”) to compute a set of M latent vector samples z at runtime.
-B. Input Uncertainty for Control
-In BDL, downstream uncertainty propagation assumes that
-a neural network component is able to handle or admit uncer-
-tainty at the input. In our navigation case, this implies that the
-DNN-based controller is able to handle the uncertainty coming
-from the perception encoder qΦ. To capture the navigation
-model uncertainty (overall system uncertainty at the output
-of the controller), we use the Bayesian approach to compute
-the posterior predictive distribution for target variable y∗
-associated with a new input image x∗, as shown in (2).
-p(y∗ | x∗, Dc, Dp) =
-��
-π(y | z, w)p(w | Dc)qΦ(z | x∗, Dp)dwdz
-(2)
-The integrals from (2) are intractable, and we rely on
-approximations to obtain an estimation of the predictive dis-
-tribution. The posterior p(w | Dc) is difficult to evaluate,
-thus we can approximate the inner integral using an ensemble
-of neural networks [15]. In practice, we train an ensemble
-of N probabilistic control policies {πn(y | z, wn)}N
-n=1,
-with weights {wn}N
-n=1 ∼ p(w|D). Each control policy πn
-in the ensemble predicts the mean µ and variance σ2 for
-each velocity command, i.e., yµ
-=
-[µ ˙x, µ ˙y, µ ˙z, µ ˙ψ] and
-yσ2 = [σ2
-˙x, σ2
-˙y, σ2
-˙z, σ2
-˙ψ]. For training the control policy we
-use imitation learning and the heteroscedastic loss function,
-as suggested by [14], [16].
-The outer integral is approximated by taking a set of
-samples from the perception component latent space. In
-[6] latent samples are drawn using the encoder mean and
-variance z ∼ N(µφ, σ2
-φ). For the sake of simplicity, we
-directly use the samples obtained in the perception component
-{zm}M
-m ∼ qΦ(z | x, Dp) to take into account the epistemic
-uncertainty from the previous stage. Finally, the predictions
-that we get from passing each latent vector z through each
-ensemble member are used to estimate the posterior predictive
-distribution in (2). From the control policy perspective, using
-multiple latent samples z can be seen as taking a better
-“picture” of the latent space (perception representation) to
-gather more information about the environment.
-V. EXPERIMENTS & DISCUSSION
-For our experiments, we seek to study the impact of
-uncertainty propagation in the navigation pipeline. In par-
-ticular, we seek to answer the following research questions:
-RQ1. How does uncertainty from perception representations
-affect downstream component uncertainty estimation quality?
-RQ2. Can uncertainty propagation improve system perfor-
-mance? RQ3. Could uncertainty-aware components in the
-pipeline help detect challenging scenes that can threaten the
-system mission? To answer these questions we perform a
-quantitative and qualitative comparison between uncertainty-
-aware aerial navigation models.
-A. Experimental setup
-1) Navigation Model Baselines: All the navigation archi-
-tectures are based on [22] and are implemented using PyTorch.
-Table I shows the uncertainty-aware navigation architectures
-used in our experiments, detailing the type of perception
-component, the number of latent variable samples (LVS), the
-type of control policy, and the number of control prediction
-
-TrainingOnly
-Ensemble
-Probabilistic NeuralNetworks
-2
-CMVAE
-Yμl
-2
-元1
-Yol
-Z1
-Yμ3
-Z2
-:
-元3
-Yo3
-p(y* I x*, Dc, Dp)
-2
-Yμ5
-Yo5
-元5
-q(z / x, Dp)
-[Tn(y I z, Wn))N
-Perception
-ControlTABLE I
-UNCERTAINTY-AWARE NAVIGATION MODELS IN THE EXPERIMENTS
-Model
-Perception (qφ)
-LVS
-Control Policy (π)
-CPS
-M0
-MCD-CMVAE
-32
-Ensemble (N = 5) Prob.
-160
-M1
-CMVAE
-32
-Ensemble (N = 5) Prob.
-160
-M2
-CMVAE
-1
-Ensemble (N = 5) Prob.
-5
-M3
-CMVAE
-32
-Deterministic
-32
-M4
-CMVAE
-1
-Prob.
-1
-samples (CPS) at the output of the system. For instance, M0
-represents our Bayesian navigation pipeline. M0 perception
-component captures epistemic uncertainty using MCD with
-32 forward passes for each input to get 32 latent variable pre-
-dictions. For the sake of simplicity, perception predictions are
-directly used as latent variable samples in downstream control.
-The control component uses an ensemble of 5 probabilistic
-control policies obtaining 160 control prediction samples. M1
-to M4 partially capture uncertainty in the pipeline since
-they use a deterministic perception component (CMVAE). For
-the control component, M1 and M2 take 32 and 1 latent
-variable samples (LVS) respectively, and use the samples later
-with an ensemble of 5 probabilistic control policies capturing
-epistemic and aleatoric uncertainty; M3 uses 32 LVS, and
-the control component is completely deterministic; M4 uses
-1 LVS with a probabilistic control policy to capture aleatoric
-uncertainty. For UAV control, we use the expected value of
-the predicted velocities means at the output of the control
-component [14], i.e., ˆyµ = E([µ ˙x, µ ˙y, µ ˙z, µ ˙ψ]).
-2) Datasets: We use two independent datasets for each
-component in the navigation pipeline. The perception CMVAE
-uses a dataset (Dp) of 300k images where a gate is visible and
-gate-pose annotations area available. The control component
-uses a dataset (Dc) of 17k images with UAV velocity anno-
-tations. Dc is collected by flying the UAV in a circular track
-with gates, using traditional methods for trajectory planning
-and control (see [22] for more details). The perception dataset
-is divided into 80% for training, and the remaining 20% for
-validation and testing. The control dataset uses a split of 90%
-for training and the remaining for validation and testing. In
-both cases the image size is 64x64 pixels. In addition, using the
-validation data from Dp and Dc, we generate refined validation
-sub-datasets with images that have: exactly one visible gate
-(ideal situation), no visible gate in front, and multiple gates
-visible. The last two types of images represent situations that
-can pose a risk to the system task. Each sub-dataset contains
-200 images.
-B. Experiments
-In the context of RQ2, we use the validation dataset from the
-control component to measure the regression Expected Cali-
-bration Error (ECE) [31] to compare the quality of uncertainty
-estimates from navigation models at the output of the system,
-(i.e., the control component output).
-In order to answer RQ2, we evaluate our navigation archi-
-tecture under controlled simulations using the AirSim simu-
-(a) Circular track view without noise (left) and with noise (right).
-(b) UAV mission scenes
-Fig. 3. UAV Mission: Navigation tracks and scenes from birds-eye view, and
-view from UAV perspective
-lation environment. The UAV mission resembles the scenario
-and the conditions observed in the training dataset. Therefore,
-we use a circular track with eight equally spaced gates posi-
-tioned initially in a radius of 8m and constant height. To assess
-the system performance to perturbations in the environment,
-we generate new tracks adding random noise to each gate
-radius and height.
-In the context of the AirSim [32] simulation environment, a
-track is entirely defined by a set of gates, their poses in three-
-dimensional space, and the expected navigation direction of
-the agent. For perception-based navigation, the complexity of
-a track resides in the “gate-visibility” difficulty [33], [34], i.e.,
-how well the camera Field-of-View (FoV) captures the gate. A
-natural way to increase track complexity is by adding a random
-displacement to the position of each gate. A track without
-random displacement in the gates has a circular fashion. Gate
-position randomness alters the shape of the track, affecting the
-gate visibility, i.e., gates are: not visible, partially visible, or
-multiple gates can be captured in the UAV FoV as presented
-in Fig. 3. The images from these scenarios are challenging
-given its potential impact on system performance.
-To measure the system performance we take the average
-number of gates passed in all generated tracks. For track
-generation we use a random seed to produce circular tracks
-with two levels of noise in the gates offset, i.e., each random
-seed generates two (reproducible) noisy tracks. In total, we use
-6 random seeds to produce 12 tracks, 6 tracks per noise level.
-The two noise levels are a combination of Gate Radius Noise
-(GRN) and Gate Height Noise (GHN). Finally, all navigation
-models are tested in the same generated tracks for a fair
-comparison, and each model has 3 trials per track.
-To address RQ3, we perform a qualitative comparison of
-the component predicted densities using scenes (images) from
-challenging situations during the UAV mission. To this end,
-we first use the images from the generated sub-datasets. Next,
-we use the scenes from Fig. 3b as an input to the Bayesian
-navigation model M0 to analyze the effect uncertainty prop-
-agation under specific situations.
-
-口口
-口
-口
-口TABLE II
-UNCERTAINTY-AWARE NAVIGATION MODELS:
-ECE & AVG. NUMBER OF GATES PASSED
-Model
-ECE (↓)
-Performance with Track Gate Noise (↑)
-GRN ∼ U[−1.0, 1.0)
-GHN ∼ U[0, 2.0)
-GRN ∼ U[−1.5, 1.5)
-GHN ∼ U[0, 3.0)
-M0
-0.00700
-19.77
-9.22
-M1
-0.00129
-17.67
-6.0
-M2
-0.00136
-17.33
-4.0
-M3
-0.05709
-8.33
-5.0
-M4
-0.00050
-15.16
-4.38
-C. Results
-Table II summarizes the ECE for all the navigation models
-using the validation dataset from the control component. M4
-has the best uncertainty quality since the model learned to
-predict the noise from the data using the heteroscedastic
-loss function. On the contrary, M2 has the worst calibration
-results caused by the deterministic control choice and its
-inability to learn the data uncertainty. M1 and M2 have
-similar values since both receive the one noisy encoding from
-perception. However, M1 takes multiple samples from the
-noisy perception encoding which causes a reduction of the
-ECE value. Finally, M0 shows a higher ECE value compared
-to the previous models. This is caused by applying MCD in
-the perception CMVAE and the dispersion of the latent codes
-at the output of the perception encoder qΦ. The uncertainty
-quality of the downstream control is slightly affected because
-the control component did not see the same perception encod-
-ing dispersion (uncertainty) during training.
-For RQ2, Table II presents the navigation performance
-results for all the navigation models. In general, learning to
-predict uncertainty in the control component can boost the
-performance significantly. However, for M3, sampling from a
-noisy perception representation adds sufficient diversity to the
-downstream control predictions, resulting in better decisions
-than M2 in tracks with higher noise levels. In M4, the good
-performance suggests that the track noise observed at test time
-(lower noise level), resembles the data noise observed during
-the training of the single probabilistic model.
-In case of M0, the diversity from perception prediction
-samples improves the performance. Interestingly, the perfor-
-mance difference with other models is not significant. This
-situation can make us wonder if an uncertainty estimation is
-needed along the whole pipeline. Nonetheless, we believe that
-performance similarity is rooted in how we use our model
-predictions and uncertainties. The control output is computed
-by taking the mean and variance of the policy ensemble
-mixture, and only the mean values are passed to the UAV
-control. However, the multimodal predictions in Fig. 5 show
-that admitting perception uncertainty (samples) at the input of
-the control component permits the representation of ambiguity
-in the predictions. Hence, a proper use of predictions and
-associated uncertainties is needed. For example, in a bi-modal
-predictive distribution at the output, we can use the modes
-(a) Visible gate sub-dataset
-(b) No visible gate sub-dataset
-(c) Multiple gates visible sub-dataset
-Fig. 4. Navigation model standard deviation (ˆσ) prediction comparison
-(i.e., distribution peaks) instead of the expected value to avoid
-sub-optimal control decisions (e.g., near distribution valleys).
-In the context of RQ3, Fig. 4 shows the estimated uncer-
-tainty densities (ˆσ) for each velocity command at the output
-of the system, using the images from the generated datasets.
-In this case, M0 allows higher uncertainty estimates while
-reducing the dispersion in the sub-datasets from each situation.
-Fig. 5 shows M0 predictions at the output of the perception
-(z) and control (ˆµ, ˆσ) components. Predictions are made using
-the three sample images from Fig. 3b, using the LVS and CPS
-to estimate the densities.
-M0 perception and control outputs show high uncertainty
-(dispersion) values when a gate is not visible (mid-right). The
-ˆµ ˙y density suggests that the UAV control predictions will
-follow the training dataset (Dc) bias, rotating clockwise and
-moving to the right when no gate is in-front. Interestingly,
-the predicted densities in the bottom plots show that M0 is
-able to represent the ambiguity in the input, i.e. sample image
-
-DoubleorMultipleVisibleGatesSubdataset:PredictedStandardDeviationDensities
-1.2
-Navigation Model
-Velocity (m/s) or (deg/s)
-Mo
-1.0
-Mi
-M2
-0.8
-0.2
-0.0
-ox
-ModelPredictionVisibleGate Subdataset:Predicted Standard DeviationDensities
-1.2
-Navigation Model
-Mo
-1.0
-Mi
-M2
-0.8
-0.2
-0.0
-ox
-Mode/PredictionNoVisibleGate Subdataset:Predicted Standard DeviationDensities
-1.2
-Navigation Model
-Velocity (m/s) or (deg/s)
-Mo
-1.0
-M1
-M2
-0.8
-0.6
-0.4
-0.2
-0.0
-x
-ModelPrediction(a) Single gate prediction densities
-(b) No visible gate prediction densities
-(c) Double gate prediction densities
-Fig. 5. Bayesian navigation model M0: Perception qΦ predictions z density (left); Predicted velocity ˆµ density (mid); Predicted velocity ˆσ (right)
-.
-with two gates. The predicted densities have a multimodal
-distribution (two peaks) for ˆµ ˙y and ˆσ ˙y commands. Further,
-the predicted densities for the latent vector z show that
-the uncertainty from perception outputs is different for each
-type of sample, which is suitable for the early detection
-of anomalies based on uncertainty information. In addition,
-detecting multi-modality in prediction distributions can help
-expressing situations where decisions must be made.
-D. Dynamic Dependability Management using Uncertainty
-from DNN-Based Systems
-Based on the results and observations in the previous
-sub-sections, uncertainty propagation through a DNN-based
-can impact downstream component predictions and their per-
-formance. Thus, using uncertainty information to improve
-system dependability or safety can be a challenging task. For
-example, building monitoring functions based on uncertainty
-information is no simple task. The uncertainty intervals we ob-
-served for different situations present overlaps that can lead to
-false-positive or false-negative verdicts. Moreover, the multi-
-modal nature of some predictions under specific conditions or
-scenes demands knowledge of multiple intervals around the
-monitored uncertainty value. Therefore dependable and safe
-automated systems require more than a simple composition of
-predicates around some confidence measures.
-Towards building dependable autonomous systems, we pro-
-pose to align with previous frameworks that leverage percep-
-tion uncertainty (cf. subsection II-C). However existing frame-
-works for system dependability do not consider the impact
-of uncertainty propagation in uncertainty-aware systems. To
-overcome these new challenges, we propose to capture and
-use uncertainty beyond perception and consider as well the
-uncertainty from downstream components along the navigation
-pipeline, as presented in Fig.6 1 . Our approach for dynamic
-dependability management takes inspiration from [35] and
-focuses on safety. Therefore, we propose an architecture for
-dynamic risk assessment and management where we devise
-three functional blocks, as shown in Fig. 6 2 : Monitoring
-functions, risk estimation and behavior arbitration modules.
-1) Monitoring Functions: Monitoring is a widely-known
-dependability technique for runtime verification intended to
-track system variables (e.g. component inputs and outputs).
-In the automotive domain, SOTIF and ISO26262 suggest the
-use of monitoring functions as a solution for error detection in
-hardware and software components [36]. Monitoring functions
-are designed using a set of rules, based on a model of the
-system and its environment, and the properties they should
-guarantee. Hence, monitors basically perform a binary classi-
-fication task to check if a property holds or not.
-Designing monitoring functions for ML components is
-different given the probabilistic nature of the outputs and
-the difficulty in specifying the component behavior at design
-time. For ML-based components in general, typical monitoring
-
-PerceptionCMVAEEncodergoPredictionDensities
-value
-iable
-vari
-.atent
-Zo
-Z1
-Z2
-Z3
-Z4
-Z5
-Z6
-Z7
-Z8
-Z9
-LatentvectorzvariablesControl EnsembleMixtureVelocity μPredictionDensities
-Mo Prediction
-1.75
-μx
-1.50
-py
-1.25
-1.00
-0.75
-0.50
-0.25 
-0.00
--1.0
-0.5
-0.0 
-0.5 
-1.0 
-1.5 
-2.0 
-2.5
-3.0
-3.5 
-Predictedvelocityμ(m/s)or(deg/s)Control Ensemble Mixture Velocity  Prediction Densities
-3.5 
-Mo Prediction
-3.0 
-ox
-oy
-2.5 
-Density
-02
-2.0
-1.5 
-1.0
-0.5 
-0.0
-0.0 
-0.2 
-0.4 
-0.6
-0.8 
-1.0
-Predicted velocity (m/s)or (deg/s)Perception CMVAE Encoder go Prediction Densities
-value
-variable
-.atent
--3
-Zo
-Z1
-Z2
-Z3
-Z4
-Z5
-Z6
-Z7
-Z8
-Zg
-LatentvectorzvariablesControl Ensemble MixtureVelocity μPredictionDensities
-2.00
-Mo Prediction
-μx
-1.75
-ily
-1.50
-1.25
-1.00
-0.75 
-0.50 
-0.25 
-0.00
-1.0
-0.5
- 0.0 
-0.5 
-1.0
-1.5
-2.0
-2.5 
-3.0 
-3.5 
-Predictedvelocityμ (m/s)or(deg/s)ControlEnsembleMixtureVelocityoPredictionDensities
-Mo Prediction
-3.0 
-0x
-2.5
-oy
-02
-1.5
-1.0
-0.5 
-0.0 
-0.0 
-0.2
-0.4
-0.6
-0.8 
-1.0
-Predicted velocity  (m/s)or (deg/s)Perception CMVAE Encoder go Prediction Densities
-value
-variable
-.atent
-Zo
-Z1
-Z2
-Z3
-Z4
-Z5
-Z6
-Z7
-Z8
-Zg
-LatentvectorzvariablesControl Ensemble Mixture Velocityμ Prediction Densities
-3.5 
-Mo Prediction
-3.0 
-px
-2.5 
-Density
-2.0
-1.5
-1.0
-0.5 
-0.0 
-0.5
-0.0 
-0.5 
-1.0 
-1.5
-2.0
-2.5 
-3.0
-Predicted velocity μ(m/s)or(deg/s)Control Ensemble Mixture Velocity  Prediction Densities
-4.0
-Mo Prediction
-ox
-3.5 
-<
-3.0 
-02
-2.0
-1.5
-1.0 
- 0.5 
-0.0 
-0.0 
-0.2 
-0.4 
-0.6 
-0.8 
-1.0 
-Predicted velocity (m/s)or (deg/s)Fig. 6. Runtime risk assessment & management framework
-function tasks include Out-of-Distribution (OoD) detection or
-Out-of-Boundary (OoB) detection and can be implemented
-with rules, data-driven methods or a mix of both.
-2) Probabilistic Inference for Risk Assessment: To enable
-dynamic uncertainty-aware reasoning and provide context to
-risk estimates, we propose to use Bayesian networks. Fol-
-lowing the methodology described in [24], BNs for risk
-assessment and safety can be constructed using a combination
-of expert domain knowledge and data. The experts provide a
-model of causal relations and can have support from traditional
-dependability analysis (e.g., fault tree analysis) to build the BN
-structure while system data is used to provide the conditional
-probabilities between random variables.
-In our framework, the BN of the system can receive
-the predictions from components in the pipeline (probability
-distributions) and the verdicts from monitoring functions ap-
-plied to system sensors, component predictions, and relevant
-environmental variables. The output of the BN is represented
-by all the critical events identified by experts. Hence, during
-inference, the BN estimates the probability of a critical event,
-which is used along with its severity to compute the system’s
-risk at runtime [37]. Though we focus on risk assessment, in a
-general way the output of BNs can be any assurance measure
-variables linked to dependability attributes [10]. Further, the
-BN should handle uncertain evidence [38] to preserve the
-probabilistic nature of component and monitor predictions.
-3) Behavior Arbitration: The last building block in our
-framework aims at keeping the system in a safe state by
-taking or discarding navigation pipeline predictions. Safe
-decisions must be made in the presence of high-risk values in
-a given context caused by erroneous component predictions or
-associated uncertainties and external environmental variables.
-To this end, we propose using Behavior Trees (BTs) to adopt
-different system behaviors while facing high-risk situations.
-BTs are sophisticated modular decision-making engines for
-reactive and fault-tolerant task execution [39]. Compositions
-of BTs can preserve safety and robustness properties [40] and
-are widely adopted tools in robotics. In the context of our
-system, we can have a dedicated behavior to search for a gate
-when we detect that there are no gates in the UAV FoV. This
-behavior will put the system back into a state where the levels
-of uncertainty do not represent a risk.
-VI. CONCLUSION
-We presented a method to capture and propagate uncertainty
-along a navigation pipeline implemented with Bayesian deep
-learning components for UAV aerial navigation. We analyzed
-the effect of uncertainty propagation regarding system com-
-ponent predictions and performance. Our experiments show
-that our approach to capturing and propagating uncertainty
-along the system can provide valuable predictions for decision-
-making and identifying situations that are critical for the
-system. However, proper use and management of component
-predictions and uncertainty estimates are needed to create
-dependable and highly-performant systems. In this sense and
-based on our observations, we also proposed a framework for
-system dependability management using system uncertainty
-and focused on safety. In future work, we aim to implement
-our proposed dependability framework and explore sampling-
-free methods [41] for uncertainty estimation to reduce the
-computational budget and memory footprint in our approach.
-ACKNOWLEDGMENT
-This work has received funding from the COMP4DRONES
-project, under ECSEL Joint Undertaking (JU) grant agreement
-N°826610. The ECSEL JU receives support from the European
-Union’s Horizon 2020 research and innovation programme and
-from Spain, Austria, Belgium, Czech Republic, France, Italy,
-Latvia, Netherlands.
-REFERENCES
-[1] S. Grigorescu, B. Trasnea, T. Cocias, and G. Macesanu, “A survey
-of deep learning techniques for autonomous driving,” Journal of Field
-Robotics, vol. 37, no. 3, pp. 362–386, 2020.
-[2] R. McAllister, Y. Gal, A. Kendall, M. Van Der Wilk, A. Shah, R. Cipolla,
-and A. Weller, “Concrete problems for autonomous vehicle safety: Ad-
-vantages of bayesian deep learning,” in Proceedings of the Twenty-Sixth
-International Joint Conference on Artificial Intelligence.
-International
-Joint Conferences on Artificial Intelligence, Inc., 2017.
-[3] R. Michelmore, M. Kwiatkowska, and Y. Gal, “Evaluating uncertainty
-quantification in end-to-end autonomous driving control,” arXiv preprint
-arXiv:1811.06817, 2018.
-[4] F. Nozarian, C. M¨uller, and P. Slusallek, “Uncertainty quantification
-and calibration of imitation learning policy in autonomous driving.” in
-TAILOR, 2020, pp. 146–162.
-[5] E. Ohn-Bar, A. Prakash, A. Behl, K. Chitta, and A. Geiger, “Learning
-situational driving,” in Proceedings of the IEEE/CVF Conference on
-Computer Vision and Pattern Recognition, 2020, pp. 11 296–11 305.
-[6] F. Arnez, H. Espinoza, A. Radermacher, and F. Terrier, “Improving
-robustness of deep neural networks for aerial navigation by incorporating
-input uncertainty,” in International Conference on Computer Safety,
-Reliability, and Security.
-Springer, 2021, pp. 219–225.
-[7] B. Ivanovic, K.-H. Lee, P. Tokmakov, B. Wulfe, R. McAllister,
-A. Gaidon, and M. Pavone, “Heterogeneous-agent trajectory forecasting
-incorporating class uncertainty,” arXiv preprint arXiv:2104.12446, 2021.
-[8] S. Casas, C. Gulino, S. Suo, K. Luo, R. Liao, and R. Urtasun,
-“Implicit latent variable model for scene-consistent motion forecasting,”
-in Computer Vision–ECCV 2020: 16th European Conference, Glasgow,
-UK, August 23–28, 2020, Proceedings, Part XXIII 16.
-Springer, 2020,
-pp. 624–641.
-[9] M. Henne, A. Schwaiger, and G. Weiss, “Managing uncertainty of ai-
-based perception for autonomous systems.” in AISafety@ IJCAI, 2019.
-
-Bayesian Deep Learning-Based Navigation Pipeline
-(Uncertainty Propagation Between Components)
-Sensors
-Perception
-Planning
-Control
-Monitoring Functions
-(Data-Driven / STL / Mixed)
-Environment
-Graphical Model
-Probabilistic
-Behavior
-Arbitration
-个
-个
-Expertknowledge&Data
-Risk Estimation
-2
-Risk Assessment & Management[10] E. Asaadi, E. Denney, and G. Pai, “Quantifying assurance in learning-
-enabled systems,” in International Conference on Computer Safety,
-Reliability, and Security.
-Springer, 2020, pp. 270–286.
-[11] A. Lavin, C. M. Gilligan-Lee, A. Visnjic, S. Ganju, D. Newman,
-S. Ganguly, D. Lange, A. G. Baydin, A. Sharma, A. Gibson et al.,
-“Technology readiness levels for machine learning systems,” arXiv
-preprint arXiv:2101.03989, 2021.
-[12] Y. Gal and Z. Ghahramani, “Dropout as a bayesian approximation:
-Representing model uncertainty in deep learning,” in international
-conference on machine learning, 2016, pp. 1050–1059.
-[13] Y. Gal, J. Hron, and A. Kendall, “Concrete dropout,” in Advances in
-neural information processing systems, 2017, pp. 3581–3590.
-[14] B. Lakshminarayanan, A. Pritzel, and C. Blundell, “Simple and scalable
-predictive uncertainty estimation using deep ensembles,” in Advances in
-neural information processing systems, 2017, pp. 6402–6413.
-[15] F. K. Gustafsson, M. Danelljan, and T. B. Sch¨on, “Evaluating scalable
-bayesian deep learning methods for robust computer vision,” arXiv
-preprint arXiv:1906.01620, 2019.
-[16] A. Kendall and Y. Gal, “What uncertainties do we need in bayesian
-deep learning for computer vision?” in Advances in neural information
-processing systems, 2017, pp. 5574–5584.
-[17] A. Loquercio, M. Segu, and D. Scaramuzza, “A general framework for
-uncertainty estimation in deep learning,” IEEE Robotics and Automation
-Letters, vol. 5, no. 2, pp. 3153–3160, 2020.
-[18] S. Depeweg, J. Hern´andez-Lobato, F. Doshi-Velez, and S. Udluft,
-“Learning and policy search in stochastic dynamical systems with
-bayesian neural networks,” in 5th International Conference on Learning
-Representations, ICLR 2017-Conference Track Proceedings, 2017.
-[19] S. Depeweg, J.-M. Hernandez-Lobato, F. Doshi-Velez, and S. Udluft,
-“Decomposition of uncertainty in bayesian deep learning for efficient
-and risk-sensitive learning,” in International Conference on Machine
-Learning.
-PMLR, 2018, pp. 1184–1193.
-[20] B. L¨utjens, M. Everett, and J. P. How, “Safe reinforcement learning
-with model uncertainty estimates,” in 2019 International Conference on
-Robotics and Automation (ICRA).
-IEEE, 2019, pp. 8662–8668.
-[21] T. Fan, P. Long, W. Liu, J. Pan, R. Yang, and D. Manocha, “Learning
-resilient behaviors for navigation under uncertainty,” in 2020 IEEE
-International Conference on Robotics and Automation (ICRA).
-IEEE,
-2020, pp. 5299–5305.
-[22] R. Bonatti, R. Madaan, V. Vineet, S. Scherer, and A. Kapoor, “Learning
-visuomotor policies for aerial navigation using cross-modal representa-
-tions,” arXiv preprint arXiv:1909.06993, 2019.
-[23] M. Trapp, D. Schneider, and G. Weiss, “Towards safety-awareness
-and dynamic safety management,” in 2018 14th European Dependable
-Computing Conference (EDCC).
-IEEE, 2018, pp. 107–111.
-[24] S. Kabir and Y. Papadopoulos, “Applications of bayesian networks and
-petri nets in safety, reliability, and risk assessments: A review,” Safety
-science, vol. 115, pp. 154–175, 2019.
-[25] J. Reich, M. Wellstein, I. Sorokos, F. Oboril, and K.-U. Scholl, “Towards
-a software component to perform situation-aware dynamic risk assess-
-ment for autonomous vehicles,” in Dependable Computing–EDCC 2021
-Workshops: DREAMS, DSOGRI, SERENE 2021, Munich, Germany,
-September 13, 2021, Proceedings.
-Springer Nature, 2021, p. 3.
-[26] A. Spurr, J. Song, S. Park, and O. Hilliges, “Cross-modal deep varia-
-tional hand pose estimation,” in Proceedings of the IEEE Conference on
-Computer Vision and Pattern Recognition, 2018, pp. 89–98.
-[27] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” arXiv
-preprint arXiv:1312.6114, 2013.
-[28] A. Loquercio, A. I. Maqueda, C. R. Del-Blanco, and D. Scaramuzza,
-“Dronet: Learning to fly by driving,” IEEE Robotics and Automation
-Letters, vol. 3, no. 2, pp. 1088–1095, 2018.
-[29] E. Daxberger and J. M. Hern´andez-Lobato, “Bayesian variational
-autoencoders for unsupervised out-of-distribution detection,” arXiv
-preprint arXiv:1912.05651, 2019.
-[30] A. Jesson, S. Mindermann, U. Shalit, and Y. Gal, “Identifying causal-
-effect inference failure with uncertainty-aware models,” Advances in
-Neural Information Processing Systems, vol. 33, pp. 11 637–11 649,
-2020.
-[31] V. Kuleshov, N. Fenner, and S. Ermon, “Accurate uncertainties for deep
-learning using calibrated regression,” arXiv preprint arXiv:1807.00263,
-2018.
-[32] S. Shah, D. Dey, C. Lovett, and A. Kapoor, “Airsim: High-fidelity visual
-and physical simulation for autonomous vehicles,” in Field and service
-robotics.
-Springer, 2018, pp. 621–635.
-[33] R. Madaan, N. Gyde, S. Vemprala, M. Brown, K. Nagami, T. Taubner,
-E. Cristofalo, D. Scaramuzza, M. Schwager, and A. Kapoor, “Airsim
-drone racing lab,” arXiv preprint arXiv:2003.05654, 2020.
-[34] Y. Song, M. Steinweg, E. Kaufmann, and D. Scaramuzza, “Autonomous
-drone racing with deep reinforcement learning,” in 2021 IEEE/RSJ
-International Conference on Intelligent Robots and Systems (IROS).
-IEEE, 2021, pp. 1205–1212.
-[35] P. Moosbrugger, K. Y. Rozier, and J. Schumann, “R2u2: monitoring
-and diagnosis of security threats for unmanned aerial systems,” Formal
-Methods in System Design, vol. 51, no. 1, pp. 31–61, 2017.
-[36] S. Mohseni, M. Pitale, V. Singh, and Z. Wang, “Practical solutions
-for machine learning safety in autonomous vehicles,” arXiv preprint
-arXiv:1912.09630, 2019.
-[37] J. Eggert, “Risk estimation for driving support and behavior planning
-in intelligent vehicles,” at-Automatisierungstechnik, vol. 66, no. 2, pp.
-119–131, 2018.
-[38] A. B. Mrad, V. Delcroix, S. Piechowiak, P. Leicester, and M. Abid,
-“An explication of uncertain evidence in bayesian networks: likelihood
-evidence and probabilistic evidence,” Applied Intelligence, vol. 43, no. 4,
-pp. 802–824, 2015.
-[39] M. Colledanchise and L. Natale, “On the implementation of behavior
-trees in robotics,” IEEE Robotics and Automation Letters, vol. 6, no. 3,
-pp. 5929–5936, 2021.
-[40] M. Colledanchise and P. ¨Ogren, “How behavior trees modularize ro-
-bustness and safety in hybrid systems,” in 2014 IEEE/RSJ International
-Conference on Intelligent Robots and Systems.
-IEEE, 2014, pp. 1482–
-1488.
-[41] B. Charpentier, O. Borchert, D. Z¨ugner, S. Geisler, and S. G¨unnemann,
-“Natural posterior network: Deep bayesian predictive uncertainty for ex-
-ponential family distributions,” arXiv preprint arXiv:2105.04471, 2021.
-

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-arXiv:2301.11598v1  [math.NA]  27 Jan 2023
-Practical Sketching Algorithms for Low-Rank
-Tucker Approximation of Large Tensors
-Wandi Dong1, Gaohang Yu1*, Liqun Qi2,1,3 and Xiaohao Cai4
-1Department of Mathematics, Hangzhou Dianzi University,
-Hangzhou, 310018, China.
-2Huawei Theory Research Lab, Hong Kong, China.
-3Department of Applied Mathematics, Hongkong Polytechnic
-University, Hong Kong, China.
-4School of Electronics and Computer Science, University of
-Southampton, Southampton, SO17 1BJ, UK.
-*Corresponding author(s). E-mail(s): [email protected];
-Contributing authors: [email protected];
-[email protected]; [email protected];
-Abstract
-Low-rank approximation of tensors has been widely used in high-
-dimensional data analysis. It usually involves singular value decom-
-position (SVD) of large-scale matrices with high computational com-
-plexity. Sketching is an effective data compression and dimension-
-ality reduction technique applied to the low-rank approximation of
-large matrices. This paper presents two practical randomized algo-
-rithms for low-rank Tucker approximation of large tensors based
-on sketching and power scheme, with a rigorous error-bound analy-
-sis. Numerical experiments on synthetic and real-world tensor data
-demonstrate the competitive performance of the proposed algorithms.
-Keywords: tensor sketching, randomized algorithm, Tucker decomposition,
-subspace power iteration, high-dimensional data
-MSC Classification: 68W20 , 15A18 , 15A69
-1
-
-2
-Sketching Algorithms for Low-Rank Tucker Approximation
-1 Introduction
-In practical applications, high-dimensional data, such as color images, hyper-
-spectral images and videos, often exhibit a low-rank structure. Low-rank
-approximation of tensors has become a general tool for compressing and
-approximating high-dimensional data and has been widely used in scientific
-computing, machine learning, signal/image processing, data mining, and many
-other fields [1]. The classical low-rank tensor factorization models include,
-e.g., Canonical Polyadic decomposition (CP) [2, 3], Tucker decomposition [4–
-6], Hierarchical Tucker (HT) [7, 8], and Tensor Train decomposition (TT)
-[9]. This paper focuses on low-rank Tucker decomposition, also known as
-the low multilinear rank approximation of tensors. When the target rank
-of Tucker decomposition is much smaller than the original dimensions, it
-will have good compression performance. For a given Nth-order tensor X ∈
-RI1×I2×...×IN , the low-rank Tucker decomposition can be formulated as the
-following optimization problem, i.e.,
-min
-Y ∥X − Y∥2
-F ,
-(1)
-where Y ∈ RI1×I2×...×IN , with rank(Y(n)) ≤ rn for n = 1, 2, . . ., N, Y(n) is the
-mode-n unfolding matrix of Y, and rn is the rank of the mode-n unfolding
-matrix of X.
-For the Tucker approximation of higher-order tensors, the most fre-
-quently used non-iterative algorithms are the improved algorithms for the
-higher-order singular value decomposition (HOSVD) [5], the truncated higher-
-order SVD (THOSVD) [10] and the sequentially truncated higher-order SVD
-(STHOSVD) [11]. Although the results of THOSVD and STHOSVD are usu-
-ally sub-optimal, they can use as reasonable initial solutions for iterative
-methods such as higher-order orthogonal iteration (HOOI) [10]. However, both
-algorithms rely directly on SVD when computing the singular vectors of inter-
-mediate matrices, requiring large memory and high computational complexity
-when the size of tensors is large.
-Strikingly, randomized algorithms can reduce the communication among
-different levels of memories and are parallelizable. In recent years, many schol-
-ars have become increasingly interested in randomized algorithms for finding
-approximation Tucker decomposition of large-scale data tensors [12–17, 19, 20].
-For example, Zhou et al. [12] proposed a randomized version of the HOOI
-algorithm for Tucker decomposition. Che and Wei [13] proposed an adaptive
-randomized algorithm to solve the multilinear rank of tensors. Minster et al.
-[14] designed randomized versions of the THOSVD and STHOSVD algorithms,
-i.e., R-STHOSVD. Sun et al. [17] presented a single-pass randomized algorithm
-to compute the low-rank Tucker approximation of tensors based on a practical
-matrix sketching algorithm for streaming data, see also [18] for more details.
-Regarding more randomized algorithms proposed for Tucker decomposition,
-please refer to [15, 16, 19, 20] for a detailed review of randomized algorithms
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-3
-for solving Tucker decomposition of tensors in recent years involving, e.g., ran-
-dom projection, sampling, count-sketch, random least-squares, single-pass, and
-multi-pass algorithms.
-This paper presents two efficient randomized algorithms for finding the
-low-rank Tucker approximation of tensors, i.e., Sketch-STHOSVD and sub-
-Sketch-STHOSVD summarized in Algorithms 6 and 8, respectively. The main
-contributions of this paper are threefold. Firstly, we propose a new one-pass
-sketching algorithm (i.e., Algorithm 6) for low-rank Tucker approximation,
-which can significantly improve the computational efficiency of STHOSVD.
-Secondly, we present a new matrix sketching algorithm (i.e., Algorithm 7) by
-combining the two-sided sketching algorithm proposed by Tropp et al. [18]
-with subspace power iteration. Algorithm 7 can accurately and efficiently com-
-pute the low-rank approximation of large-scale matrices. Thirdly, the proposed
-Algorithm 8 can deliver a more accurate Tucker approximation than sim-
-pler randomized algorithms by combining the subspace power iteration. More
-importantly, sub-Sketch-STHOSVD can converge quickly for any data tensors
-and independently of singular value gaps.
-The rest of this paper is organized as follows. Section 2 briefly introduces
-some basic notations, definitions, and tensor-matrix operations used in this
-paper and recalls some classical algorithms, including THOSVD, STHOSVD,
-and R-STHOSVD, for low-rank Tucker approximation. Our proposed two-
-sided sketching algorithm for STHOSVD is given in Section 3. In Section 4,
-we present an improved algorithm with subspace power iteration. The effec-
-tiveness of the proposed algorithms is validated thoroughly in Section 5 by
-numerical experiments on synthetic and real-world data tensors. We conclude
-in Section 6.
-2 Preliminary
-2.1 Notations and basic operations
-Some common symbols used in this paper are shown in the following Table 1.
-Table 1 Common symbols used in this paper.
-Symbols
-Notations
-a
-scalar
-A
-matrix
-X
-tensor
-X(n)
-mode-n unfolding matrix of X
-×n
-mode-n product of tensor and matrix
-In
-identity matrix with size n × n
-σi(A)
-the ith largest singular value of A
-A⊤
-transpose of A
-A†
-pseudo-inverse of A
-
-4
-Sketching Algorithms for Low-Rank Tucker Approximation
-We denote an Nth-order tensor X ∈ RI1×I2×...×IN with entries given by
-xi1,i2,...,iN, 1 ≤ in ≤ In, n = 1, 2, ..., N. The Frobenius norm of X is defined as
-∥X∥F =
-�
-�
-�
-�
-I1,I2,...,IN
-�
-i1,i2,...,iN
-x2
-i1,i2,...,iN .
-The mode-n tensor-matrix multiplication is a frequently encountered operation
-in tensor computation. The mode-n product of a tensor X ∈ RI1×I2×...×IN
-by a matrix A ∈ RK×In (with entries ak,in) is denoted as Y = X ×n A ∈
-RI1×...×In−1×K×In+1×...×IN, with entries
-yi1,...,in−1,k,in+1,...,iN =
-In
-�
-in=1
-xi1,...,in−1,in,in+1,...,iNak,in.
-The mode-n matricization of higher-order tensors is the reordering of ten-
-sor elements into a matrix. The columns of mode-n unfolding matrix X(n) ∈
-RIn×(�
-N̸=n IN ) are the mode-n fibers of X. More specifically, a element
-(i1, i2, ..., iN) of X is maps on a element (in, j) of X(n), where
-j = 1 +
-N
-�
-k=1,k̸=n
-[(ik − 1)
-k−1
-�
-m=1,m̸=n
-Im].
-Let the rank of mode-n unfolding matrix X(n) is rn, the integer array
-(r1, r2, ..., rN) is Tucker-rank of Nth-order tensor X, also known as the mul-
-tilinear rank. The Tucker decomposition of X with rank (r1, r2, ..., rN) is
-expressed as
-X = G ×1 U (1) ×2 U (2) . . . ×N U (N),
-(2)
-where G ∈ Rr1×r2×...×rN is the core tensor, and {U (n)}N
-n=1 with U (n) ∈ RIn×rn
-is the mode-n factor matrices. The graphical illustration of Tucker decom-
-position for a third-order tensor shows in Figure 1. We denote an optimal
-rank-(r1, r2, ..., rN) approximation of a tensor X as ˆ
-Xopt, which is the optimal
-Tucker approximation by solving the minimization problem in (1). Below we
-Fig. 1 Tucker decomposition of a third-order tensor.
-present the definitions of some concepts used in this paper.
-
-B
-9
-3
-2
-ASketching Algorithms for Low-Rank Tucker Approximation
-5
-Definition 1 (Kronecker products) The Kronecker product of matrices A ∈ Rm×n
-and B ∈ Rk×l is defined as
-A ⊗ B =
-
-
-a11B
-a12B
-... a1nB
-a21B
-a22B
-... a2nB
-:
-:
-...
-:
-am1B am2B ... amnB
-
- ∈ Rmk×nl.
-The Kronecker product helps express Tucker decomposition. The Tucker
-decomposition in (2) implies
-X(n) = U (n)G(n)(U (N) ⊗ ... ⊗ U (n+1) ⊗ U (n−1) ⊗ ... ⊗ U (1))⊤.
-Definition 2 (Standard normal matrix) The elements of a standard normal matrix
-follow the real standard normal distribution (i.e., Gaussian with mean zero and
-variance one) form an independent family of standard normal random variables.
-Definition 3 (Standard Gaussian tensor) The elements of a standard Gaussian
-tensor follow the standard Gaussian distribution.
-Definition 4 (Tail energy) The jth tail energy of a matrix X is defined as
-τ 2
-j (X) :=
-min
-rank(Y )<j ∥X − Y ∥2
-F =
-�
-i≥j
-σ2
-i (X).
-2.2 Truncated higher-order SVD
-Since the actual Tucker rank of large-scale higher-order tensor is hard to com-
-pute, the truncated Tucker decomposition with a pre-determined truncation
-(r1, r2, ..., rN) is widely used in practice. THOSVD is a popular approach to
-computing the truncated Tucker approximation, also known as the best low
-multilinear rank-(r1, r2, ..., rN) approximation, which reads
-min
-G; U(1),U(2),··· ,U(N) ∥X − G ×1 U (1) ×2 U (2) · · · ×N U (N)∥2
-F
-s.t.
-U (n)⊤U (n) = Irn, n ∈ {1, 2, ..., N}.
-Algorithm 1 THOSVD
-Require: tensor X ∈ RI1×I2×...×IN and target rank (r1, r2, . . . , rN)
-Ensure: Tucker approximation ˆ
-X = G ×1 U (1) ×1 U (2) · · · ×N U (N)
-1: for n = 1, 2, . . ., N do
-2:
-(U (n), ∼, ∼) ← truncatedSVD(X(n), rn)
-3: end for
-4: G ← X×1U (1)⊤ ×2 U (2)⊤ · · · ×N U (N)⊤
-
-6
-Sketching Algorithms for Low-Rank Tucker Approximation
-Algorithm 1 summarizes the THOSVD approach. Each mode is processed
-individually in Algorithm 1. The low-rank factor matrices of mode-n unfolding
-matrix X(n) are computed through the truncated SVD, i.e.,
-X(n) =
-�
-U (n)
-˜
-U (n)
-� �S(n)
-˜
-S(n)
-� �V (n)⊤
-˜
-V (n)⊤
-�
-∼= U (n)S(n)V (n)⊤,
-where U (n)S(n)V (n)⊤ is a rank-rn approximation of X(n), the orthogonal
-matrix U (n) ∈ RIn×rn is the mode-n factor matrix of X in Tucker decomposi-
-tion, S(n) ∈ Rrn×rn and V (n) ∈ RI1...In−1In+1...IN×rn. Once all factor matrices
-have been computed, the core tensor G can be computed as
-G = X×1U (1)⊤ ×2 U (2)⊤ · · · ×N U (N)⊤ ∈ Rr1×r2×...×rN.
-Then, the Tucker approximation ˆ
-X of X can be computed as
-ˆ
-X = G ×1 U (1) ×2 U (2) · · · ×N U (N)
-= X ×1 (U (1)U (1)⊤) ×2 (U (2)U (2)⊤) · · · ×N (U (N)U (N)⊤).
-With the notation ∆2
-n(X) ≜ �In
-i=rn+1 σ2
-i (X(n)) and ∆2
-n(X) ≤ ∥X − ˆ
-Xopt∥2
-F
-[14], the error-bound for Algorithm 1 can be stated in the following Theorem 1.
-Theorem 1 ([11], Theorem 5.1) Let ˆ
-X = G ×1 U(1) ×2 U(2) · · · ×N U(N) be the
-low multilinear rank-(r1, r2, ..., rN) approximation of a tensor X ∈ RI1×I2×...×IN by
-THOSVD. Then
-∥X − ˆ
-X ∥2
-F ≤
-N
-�
-n=1
-∥X ×n (IIn − U(n)U(n)⊤)∥2
-F =
-N
-�
-n=1
-In
-�
-i=rn+1
-σ2
-i (X(n))
-=
-N
-�
-n=1
-∆2
-n(X ) ≤ N∥X − ˆ
-Xopt∥2
-F .
-2.3 Sequentially truncated higher-order SVD
-Vannieuwenhoven et al.[11] proposed one more efficient and less computation-
-ally complex approach for computing approximate Tucker decomposition of
-tensors, called STHOSVD. Unlike THOSVD algorithm, STHOSVD updates
-the core tensor simultaneously whenever a factor matrix has computed.
-Given the target rank (r1, r2, . . . , rN) and a processing order sp
-:
-{1, 2, ..., N}, the minimization problem (1) can be formulated as the following
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-7
-optimization problem
-min
-U(1),··· ,U(N) ∥X − X ×1 (U (1)U (1)⊤) ×2 (U (2)U (2)⊤) · · · ×N (U (N)U (N)⊤)∥2
-F
-=
-min
-U(1),··· ,U(N)(∥X ×1 (I1 − U (1)U (1)⊤)∥2
-F + ∥ ˆ
-X (1) ×2 (I2 − U (2)U (2)⊤)∥2
-F +
-· · · + ∥ ˆ
-X (N−1) ×N (IN − U (N)U (N)⊤)∥2
-F )
-= min
-U(1)(∥X ×1 (I1 − U (1)U (1)⊤)∥2
-F + min
-U(2)(∥ ˆ
-X (1) ×2 (I2 − U (2)U (2)⊤)∥2
-F +
-min
-U(3)(· · · + min
-U(N) ∥ ˆ
-X (N−1) ×N (IN − U (N)U (N)⊤)∥2
-F ))),
-(3)
-where
-ˆ
-X (n) = X ×1 (U (1)U (1)⊤) ×2 (U (2)U (2)⊤) · · · ×n (U (n)U (n)⊤), n =
-1, 2, ..., N − 1, denote the intermediate approximation tensors.
-Algorithm 2 STHOSVD
-Require: tensor X ∈ RI1×I2×...×IN , target rank (r1, r2, . . . , rN), and process-
-ing order sp : {i1, i2, . . . , iN}
-Ensure: Tucker approximation ˆ
-X = G ×1 U (1) ×2 U (2) . . . ×N U (N)
-1: G ← X
-2: for n = i1, i2, . . . , iN do
-3:
-(U (n), S(n), V (n)⊤) ← truncatedSVD(G(n), rn)
-4:
-G ← foldn(S(n)V (n)⊤) (% forming the updated tensor from its mode-n
-unfolding)
-5: end for
-In Algorithm 2, the solution U (n) of problem (3) can be obtained via
-truncatedSVD(G(n), rn), where G(n) is mode-n unfolding matrix of the (n−1)-
-th intermediate core tensor G = X ×n−1
-i=1 U (i)⊤ ∈ Rr1×r2×...×rn−1×In×...×IN,
-i.e.,
-G(n) =
-�
-U (n)
-˜
-U (n)
-� �S(n)
-˜
-S(n)
-� �V (n)⊤
-˜
-V (n)⊤
-�
-∼= U (n)S(n)V (n)⊤,
-where the orthogonal matrix U (n)
-is the mode-n factor matrix, and
-S(n)V (n)⊤ ∈ Rrn×r1...rn−1In+1...IN is used to update the n-th intermediate core
-tensor G. Function foldn(S(n)V (n)⊤) tensorizes matrix S(n)V (n)⊤ into ten-
-sor G ∈ Rr1×r2×...×rn×In+1×...×IN. When the target rank rn is much smaller
-than In, the size of the updated intermediate core tensor G is much smaller
-than original tensor. This method can significantly improve computational
-performance. STHOSVD algorithm possesses the following error-bound.
-Theorem 2 ([11], Theorem 6.5) Let ˆ
-X = G ×1 U(1) ×2 U(2) . . . ×N U(N) be the
-low multilinear rank-(r1, r2, ..., rN) approximation of a tensor X ∈ RI1×I2×...×IN by
-
-8
-Sketching Algorithms for Low-Rank Tucker Approximation
-STHOSVD with processsing order sp : {1, 2, . . . , N}. Then
-∥X − ˆ
-X ∥2
-F =
-N
-�
-n=1
-∥ ˆ
-X (n−1) − ˆ
-X (n)∥2
-F ≤
-N
-�
-n=1
-∥X ×n (IIn − U(n)U(n)⊤)∥2
-F
-=
-N
-�
-n=1
-∆2
-n(X ) ≤ N∥X − ˆ
-Xopt∥2
-F .
-Although STHOSVD has the same error-bound as THOSVD, it is less com-
-putationally complex and requires less storage. As shown in Section 5 for the
-numerical experiment, the running (CPU) time of the STHOSVD algorithm
-is significantly reduced, and the approximation error has slightly better than
-that of THOSVD in some cases.
-2.4 Randomized STHOSVD
-When the dimensions of data tensors are enormous, the computational cost
-of the classical deterministic algorithm TSVD for finding a low-rank approx-
-imation of mode-n unfolding matrix can be expensive. Randomized low-rank
-matrix algorithms replace original large-scale matrix with a new one through
-a preprocessing step. The new matrix contains as much information as possi-
-ble about the rows or columns of original data matrix. Its size is smaller than
-original matrix, allowing the data matrix to be processed efficiently and thus
-reducing the memory requirements for solving low-rank approximation of large
-matrix.
-Algorithm 3 R-SVD
-Require: matrix A ∈ Rm×n, target rank r, and oversampling parameter p ≥ 0
-Ensure: low-rank approximation matrix ˆA = ˆU ˆS ˆV ⊤ of A
-1: Ω ← randn(n, r + p)
-2: Y ← AΩ
-3: (Q, ∼) ← thinQR(Y )
-4: B ← Q⊤A
-5: (U, S, V ⊤) ← thinSVD(B)
-6: ˆU ← QU(:, 1 : r)
-7: ˆS ← S(1 : r, 1 : r), ˆV ← V (:, 1 : r)
-N. Halko et al. [21] proposed a randomized SVD (R-SVD) for matrices. The
-preprocessing stage of the algorithm is performed by right multiplying original
-data matrix A ∈ Rm×n with a random Gaussian matrix Ω ∈ Rn×r. Each
-column of the resulting new matrix Y = AΩ ∈ Rm×r is a linear combination
-of the columns of original data matrix. When r < n, the size of matrix Y
-is smaller than A. The oversampling technique can improve the accuracy of
-solutions. Subsequent computations are summarised in Algorithm 3, where
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-9
-randn generates a Gaussian random matrix, thinQR produces an economy-size
-of the QR decomposition, and thinSVD is the thin SVD decomposition. When
-A is dense, the arithmetic cost of Algorithm 3 is O(2(r + p)mn + r2(m + n))
-flops, where p > 0 is the oversampling parameter satisfying r+p ≤ min{m, n}.
-Algorithm 3 is an efficient randomized algorithm for computing rank-r
-approximations to matrices. Minster et al. [14] applied Algorithm 3 directly
-to the STHOSVD algorithm and then presented a randomized version of
-STHOSVD (i.e., R-STHOSVD), see Algorithm 4.
-Algorithm 4 R-STHOSVD
-Require: tensor X ∈ RI1×I2×...×IN , targer rank (r1, r2, . . . , rN), processing
-order sp : {i1, i2, . . . , iN}, and oversampling parameter p ≥ 0
-Ensure: Tucker approximation ˆ
-X = G ×1 U (1) ×2 U (2) . . . ×N U (N)
-1: G ← X
-2: for n = i1, i2, . . . , iN do
-3:
-( ˆU, ˆS, ˆV ⊤) ← R-SVD(G(n), rn, p) (cf. Algorithm 3)
-4:
-U (n) ← ˆU
-5:
-G ← foldn( ˆS ˆV ⊤)
-6: end for
-3 Sketching algorithm for STHOSVD
-A drawback of R-SVD algorithm is that when both dimensions of the inter-
-mediate matrices are enormous, the computational cost can still be high. To
-resolve this problem, we could resort to the two-sided sketching algorithm for
-low-rank matrix approximation proposed by Joel A. Tropp et al. [22]. The
-preprocessing of sketching algorithm needs two sketch matrices to contain
-information regarding the rows and columns of input matrix A ∈ Rm×n. Thus
-we should choose two sketch size parameters k and l, s.t. , r ≤ k ≤ min{l, n},
-0 < l ≤ m. The random matrices Ω ∈ Rn×k and Ψ ∈ Rl×m are fixed indepen-
-dent standard normal matrices. Then we can multiply matrix A left and right
-respectively to obtain random sketch matrices Y ∈
-Rm×k and W ∈ Rl×n,
-which collect sufficient data about the input matrix to compute the low-rank
-approximation. The dimensionality and distribution of the random sketch
-matrices determine the approximation’s potential accuracy, with larger values
-of k and l resulting in better approximations but also requiring more storage
-and computational cost.
-The sketching algorithm for low-rank approximation is given in Algorithm
-5. Function orth(A) in Step 2 produces an orthonormal basis of A. Using
-orthogonalization matrices will achieve smaller errors and better numerical
-stability than directly using the randomly generated Gaussian matrices. In
-particular, when A is dense, the arithmetic cost of Algorithm 5 is O((k +
-l)mn + kl(m + n)) flops. Algorithm 5 is simple, practical, and possesses the
-sub-optimal error-bound as stated in the following Theorem 3. In Theorem 3,
-
-10
-Sketching Algorithms for Low-Rank Tucker Approximation
-Algorithm 5 Sketch for low-rank approximation
-Require: matrix A ∈ Rm×n, and sketch size parameters k, l
-Ensure: rank-k approximation ˆA = QX of A
-1: Ω ← randn(n, k), Ψ ← randn(l, m)
-2: Ω ← orth(Ω), Ψ⊤ ← orth(Ψ⊤)
-3: Y ← AΩ
-4: W ← ΨA
-5: (Q, ∼) ← thinQR(Y )
-6: X ← (ΨQ)†W
-function f(s, t) := s/(t − s − 1)(t > s + 1 > 1). The minimum in Theorem
-3 reveals that the low rank approximation of given matrix A automatically
-exploits the decay of tail energy.
-Theorem 3 ([22], Theorem 4.3) Assume that the sketch size parameters satisfy
-l > k + 1, and draw random test matrices Ω ∈ Rn×k and Ψ∈ Rl×m independently
-forming the standard normal distribution. Then the rank-k approximation ˆA obtained
-from Algorithm 5 satisfies
-E ∥ A − ˆA ∥2
-F ≤ (1 + f(k, l)) · min
-̺<k−1(1 + f(̺, k)) · τ 2
-̺+1(A)
-=
-k
-l − k − 1 · min
-̺<k−1
-k
-k − ̺ − 1 · τ 2
-̺+1(A).
-Using the two-sided sketching algorithm to leverage STHOSVD algorithm,
-we propose a practical sketching algorithm for STHOSVD named Sketch-
-STHOSVD. We summarize the procedures of Sketch-STHOSVD algorithm in
-Algorithm 6, with its error analysis stated in Theorem 4.
-Algorithm 6 Sketch-STHOSVD
-Require: tensor X ∈ RI1×I2×...×IN , targer rank (r1, r2, . . . , rN), processing
-order sp : {i1, i2, . . . , iN}, and sketch size parameters {l1, l2, ..., lN}
-Ensure: Tucker approximation ˆ
-X = G ×1 U (1) ×2 U (2) . . . ×N U (N)
-1: G ← X
-2: for n = i1, i2, . . . , iN do
-3:
-(Q, X) ← Sketch(G(n), rn, ln) (cf. Algorithm 5)
-4:
-U (n) ← Q
-5:
-G ← foldn(X)
-6: end for
-Theorem 4 Let ˆ
-X = G ×1 U(1) ×2 U(2) . . . ×N U(N) be the Tucker approximation
-of a tensor X ∈ RI1×I2×...×IN by the Sketch-STHOSVD algorithm (i.e., Algorithm
-6) with target rank rn < In, n = 1, 2, ..., N, sketch size parameters {l1, l2, ..., lN} and
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-11
-processing order sp : {1, 2, . . . , N}. Then
-E{Ωj}N
-j=1∥X − �
-X ∥2
-F ≤
-N
-�
-n=1
-rn
-ln − rn − 1
-min
-̺n<rn−1
-rn
-rn − ̺n − 1∆2
-n(X )
-≤
-N
-�
-n=1
-rn
-ln − rn − 1
-min
-̺n<rn−1
-rn
-rn − ̺n − 1∥X − ˆ
-Xopt∥2
-F .
-Proof Combining Theorem 2 and Theorem 3, we have
-E{Ωj}N
-j=1∥X − �
-X ∥2
-F
-=
-N
-�
-n=1
-E{Ωj}N
-j=1∥ ˆ
-X (n−1) − ˆ
-X (n)∥2
-F
-=
-N
-�
-n=1
-E{Ωj}n−1
-j=1
-�
-EΩn∥ ˆ
-X (n−1) − ˆ
-X (n)∥2
-F
-�
-=
-N
-�
-n=1
-E{Ωj}n−1
-j=1
-�
-EΩn∥G(n−1) ×n−1
-i=1 U(i)×n(I − U(n)U(n)⊤)∥2
-F
-�
-≤
-N
-�
-n=1
-E{Ωj}n−1
-j=1
-�
-EΩn∥(I − U(n)U(n)⊤)Gn−1
-n
-)∥2
-F
-�
-≤
-N
-�
-n=1
-E{Ωj}n−1
-j=1
-rn
-ln − rn − 1
-min
-̺n<rn−1
-rn
-rn − ̺n − 1
-In
-�
-i=rn+1
-σ2
-i (G(n−1)
-(n)
-)
-≤
-N
-�
-n=1
-E{Ωj}n−1
-j=1
-rn
-ln − rn − 1
-min
-̺n<rn−1
-rn
-rn − ̺n − 1∆2
-n(X )
-=
-N
-�
-n=1
-rn
-ln − rn − 1
-min
-̺n<rn−1
-rn
-rn − ̺n − 1∆2
-n(X )
-≤
-N
-�
-n=1
-rn
-ln − rn − 1
-min
-̺n<rn−1
-rn
-rn − ̺n − 1∥X − ˆ
-Xopt∥2
-F .
-□
-We assume the processing order for STHOSVD, R-STHOSVD, and Sketch-
-STHOSVD algorithms is sp : {1, 2, ..., N}. Table 2 summarises the arithmetic
-cost of different algorithms for the cases related to the general higher-order
-tensor X ∈ RI1×I2×...×IN with target rank (r1, r2, . . . , rN) and the special
-cubic tensor X ∈ RI×I×...×I with target rank (r, r, ..., r). Here the tensors are
-dense and the target ranks rj ≪ Ij, j = 1, 2, . . ., N.
-
-12
-Sketching Algorithms for Low-Rank Tucker Approximation
-Table 2 Arithmetic cost for the algorithms THOSVD, STHOSVD, R-STHOSVD, and
-the proposed Sketch-STHOSVD.
-Algorithm
-X ∈ RI1×I2×...×IN
-X ∈ RI×I×...×I
-THOSVD
-O(
-N
-�
-j=1
-Ij I1:N + �N
-j=1 r1:j Ij:N )
-O(NIN+1 +
-N
-�
-j=1
-rj IN−j+1)
-STHOSVD
-O(
-N
-�
-j=1
-Ij r1:j−1Ij:N +
-N
-�
-j=1
-r1:j Ij+1:N )
-O(
-N
-�
-j=1
-rj−1IN−j+2 + rj IN−j)
-R-STHOSVD
-O(
-N
-�
-j=1
-r1:jIj:N +
-N
-�
-j=1
-r1:j Ij+1:N )
-O(
-N
-�
-j=1
-rj IN−j+1 + rj IN−j )
-Sketch-STHOSVD
-O(
-N
-�
-j=1
-rj lj(Ij + r1:j−1Ij+1:N ) +
-N
-�
-j=1
-r1:j Ij+1:N )
-O(
-N
-�
-j=1
-rl(I + rj−1IN−j ) + rj IN−j )
-4 Sketching algorithm with subspace power
-iteration
-When the size of original matrix is very large or the singular spectrum of
-original matrix decays slowly, Algorithm 5 may produce a poor basis in many
-applications. Inspired by [23], we suggest using the power iteration technique
-to enhance the sketching algorithm by replacing A with (AA⊤)qA, where q
-is a positive integer. According to the SVD decomposition of matrix A, i.e.,
-A = USV ⊤, we know that (AA⊤)qA = US2q+1V ⊤. It can see that A and
-(AA⊤)qA have the same left and right singular vectors, but the latter has a
-faster decay rate of singular values, making its tail energy much smaller.
-Algorithm 7 Sketching algorithm with subspace power iteration (sub-
-Sketch)
-Require: matrix A ∈ Rm×n, sketch size parameters k, l, and integer q > 0
-Ensure: rank-k approximation ˆA = QX of A
-1: Ω ← randn(n, k), Ψ ← randn(l, m)
-2: Ω ← orth(Ω), Ψ⊤ ← orth(Ψ⊤)
-3: Y = AΩ, W = ΨA
-4: Q0 ← thinQR(Y )
-5: for j = 1, . . . , q do
-6:
-ˆYj = A⊤Qj−1
-7:
-( ˆQj, ∼) ← thinQR( ˆYj)
-8:
-Yj = A ˆQj
-9:
-(Qj, ∼) ← thinQR(Yj)
-10: end for
-11: Q = Qq
-12: X ← (ΨQ)†W
-Although power iteration can improve the accuracy of Algorithm 5 to some
-extent, it still suffers from a problem, i.e., during the execution with power
-iteration, the rounding errors will eliminate all information about the singular
-modes associated with the singular values. To address this issue, we propose an
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-13
-improved sketching algorithm by orthonormalizing the columns of the sample
-matrix between each application of A and A⊤, see Algorithm 7. When A is
-dense, the arithmetic cost of Algorithm 7 is O((q + 1)(k + l)mn + kl(m + n))
-flops. Numerical experiments show that a good approximation can achieve
-with a choice of 1 or 2 for subspace power iteration parameter [21].
-Algorithm 8 sub-Sketch-STHOSVD
-Require: tensor X ∈ RI1×I2×...×IN , targer rank (r1, r2, . . . , rN), processing
-order sp : {i1, i2, . . . , iN}, sketch size parameters {l1, l2, ..., lN}, and integer
-q > 0
-Ensure: Tucker approximation ˆ
-X = G ×1 U (1) ×2 U (2) . . . ×N U (N)
-1: G ← X
-2: for n = i1, i2, . . . , iN do
-3:
-(Q, X) ← sub-Sketch(G(n), rn, ln, q) (cf. Algorithm 7)
-4:
-U (n) ← Q
-5:
-G ← foldn(X)
-6: end for
-Using Algorithm 7 to compute the low-rank approximations of intermedi-
-ate matrices, we can obtain an improved sketching algorithm for STHOSVD,
-called sub-Sketch-STHOSVD, see Algorithm 8. The error-bound for Algorithm
-8 states in the following Theorem 5. Its proof is deferred in Appendix.
-Theorem 5 Let ˆ
-X = G ×1 U(1) ×2 U(2) . . . ×N U(N) be the Tucker approximation
-of a tensor X ∈ RI1×I2×...×IN obtained by the sub-Sketch-STHOSVD algorithm
-(i.e., Algorithm 8) with target rank rn < In, n = 1, 2, ..., N, sketch size parameters
-{l1, l2, ..., lN} and processing order p : {1, 2, . . . , N}. Let ̟k ≡
-σk+1
-σk
-denote the
-singular value gap, then
-E{Ωj}N
-j=1∥X − �
-X ∥2
-F ≤
-N
-�
-n=1
-(1 + f(rn, ln)) ·
-min
-̺n<rn−1(1 + f(̺n, rn)̟r4q) · τ 2
-̺+1(X(n))
-≤
-N
-�
-n=1
-(1 + f(rn, ln)) ·
-min
-̺n<rn−1(1 + f(̺n, rn)̟r4q)∥X − ˆ
-Xopt∥2
-F .
-Proof See Appendix.
-□
-5 Numerical experiments
-This section conducts numerical experiments with synthetic data and
-real-world data, including comparisons between the traditional THOSVD,
-STHOSVD algorithms, the R-STHOSVD algorithm proposed in [14], and our
-
-14
-Sketching Algorithms for Low-Rank Tucker Approximation
-proposed algorithms Sketch-STHOSVD and sub-Sketch-STHOSVD. Regard-
-ing the numerical settings, the oversampling parameter p = 5 is used in
-Algorithm 3, the sketch parameters ln = rn + 2, n = 1, 2, . . ., N, are used
-in Algorithms 6 and 8, and the power iteration parameter q = 1 is used in
-Algorithm 8.
-5.1 Hilbert tensor
-Hilbert tensor is a synthetic and supersymmetric tensor, with each entry
-defined as
-Xi1i2...in =
-1
-i1 + i2 + ... + in
-, 1 ≤ in ≤ In, n = 1, 2, ..., N.
-In the first experiment, we set N = 5 and In = 25, n = 1, 2, . . . , N. The target
-rank is chosen as (r, r, r, r, r), where r ∈ [1, 25]. Due to the supersymmetry of
-the Hilbert tensor, the processing order in the algorithms does not affect the
-final experimental results, and thus the processing order can be directly chosen
-as sp : {1, 2, 3, 4, 5}.
-0
-5
-10
-15
-20
-25
-Target rank
-10-15
-10-10
-10-5
-100
-Relative Error
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-0
-5
-10
-15
-20
-25
-Target rank
-10-1
-100
-101
-102
-Running Time
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-Fig. 2 Results comparison on the Hilbert tensor with a size of 25 × 25 × 25 × 25 × 25 in
-terms of numerical error (left) and CPU time (right).
-The results of different algorithms are given in Figure 2. It shows that our
-proposed algorithms (i.e., Sketch-STHOSVD and sub-Sketch-STHOSVD) and
-algorithm R-STHOSVD outperform the algorithms THOSVD and STHOSVD.
-In particular, the error of the proposed algorithms Sketch-STHOSVD and sub-
-Sketch-STHOSVD is comparable to R-STHOSVD (see the left plot in Figure
-2), while they both use less CPU time than R-STHOSVD (see the right plot in
-Figure 2). This result demonstrates the excellent performance of the proposed
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-15
-algorithms and indicates that the two-sided sketching method and the subspace
-power iteration used in our algorithms can indeed improve the performance of
-STHOSVD algorithm.
-For a large-scale test, we use a Hilbert tensor with a size of 500×500×500
-and conduct experiments using ten different approximate multilinear ranks. We
-perform the tests ten times and report the algorithms’ average running time
-and relative error in Table 3 and Table 4, respectively. The results show that
-the randomized algorithms can achieve higher accuracy than the deterministic
-algorithms. The proposed Sketch-STHOSVD algorithm is the fastest, and the
-sub-Sketch-STHOSVD algorithm achieves the highest accuracy efficiently.
-Table 3 Results comparison in terms of the CPU time (in second) on the Hilbert tensor
-with a size of 500 × 500 × 500 as the target rank increases.
-Target rank
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-(10,10,10)
-17.18
-7.49
-0.92
-0.86
-0.98
-(20,20,20)
-23.13
-8.87
-1.25
-1.05
-1.48
-(30,30,30)
-24.91
-9.35
-1.66
-1.53
-2.16
-(40,40,40)
-28.05
-10.41
-1.94
-1.44
-2.11
-(50,50,50)
-29.44
-11.39
-2.07
-1.67
-2.43
-(60,60,60)
-30.14
-11.07
-2.37
-1.90
-2.77
-(70,70,70)
-29.44
-11.18
-2.57
-2.10
-3.02
-(80,80,80)
-29.65
-12.30
-3.05
-2.54
-3.75
-(90,90,90)
-31.11
-12.80
-3.80
-2.80
-4.33
-(100,100,100)
-32.22
-13.51
-4.04
-3.07
-4.61
-Table 4 Results comparison in terms of the relative error on the Hilbert tensor with a
-size of 500 × 500 × 500 as the target rank increases.
-Target rank
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-(10,10,10)
-2.7354e-06
-2.7347e-06
-2.7347e-06
-1.1178e-05
-2.7568e-06
-(20,20,20)
-1.1794e-12
-1.1793e-12
-1.1794e-12
-7.1408e-12
-1.2677e-12
-(30,30,30)
-4.6574e-15
-3.2739e-15
-3.2201e-15
-4.0641e-15
-2.0182e-15
-(40,40,40)
-4.4282e-15
-3.4249e-15
-2.8212e-15
-2.1562e-15
-1.7860e-15
-(50,50,50)
-4.1628e-15
-3.2342e-15
-2.6823e-15
-2.3205e-15
-1.8625e-15
-(60,60,60)
-4.1214e-15
-3.1271e-15
-2.3652e-15
-2.2920e-15
-1.7472e-15
-(70,70,70)
-4.1085e-15
-3.0000e-15
-2.1761e-15
-2.0499e-15
-1.6370e-15
-(80,80,80)
-4.0956e-15
-3.1350e-15
-1.8382e-15
-1.8209e-15
-1.6424e-15
-(90,90,90)
-4.0792e-15
-3.3742e-15
-1.8102e-15
-1.7193e-15
-1.5264e-15
-(100,100,100)
-4.0390e-15
-3.0571e-15
-1.7323e-15
-1.6304e-15
-1.4957e-15
-5.2 Sparse tensor
-In this experiment, we test the performance of different algorithms on a sparse
-tensor X ∈ R200×200×200, i.e.,
-X =
-10
-�
-i=1
-γ
-i2 xi ◦ yi ◦ zi +
-200
-�
-i=11
-1
-i2 xi ◦ yi ◦ zi.
-Where xi, yi, zi ∈ Rn are sparse vectors all generated using the sprand com-
-mand in MATLAB with 5% nonzeros each, and γ is a user-defined parameter
-
-16
-Sketching Algorithms for Low-Rank Tucker Approximation
-20
-40
-60
-80
-100
-Target rank
-10-3
-10-2
-Relative Error
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-10-4
-10-3
-Relative Error
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-10-6
-10-5
-Relative Error
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-10-1
-100
-Running Time
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-10-1
-100
-Running Time
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-10-1
-100
-Running Time
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-Fig. 3 Results comparison on a sparse tensor with a size of 200 × 200 × 200 in terms of
-numerical error (first row) and CPU time (second row).
-which determines the strength of the gap between the first ten terms and the
-rest terms. The target rank is chosen as (r, r, r), where r ∈ [20, 100]. The exper-
-imental results show in Figure 3, in which three different values γ = 2, 10, 200
-are tested. The increase of gap means that the tail energy will be reduced, and
-the accuracy of the algorithms will be improved. Our numerical experiments
-also verified this result.
-Figure 3 demonstrates the superiority of the proposed sketching algo-
-rithms. In particular, we see that the proposed Sketch-STHOSVD is the fastest
-algorithm, with a comparable error against R-STHOSVD; the proposed sub-
-Sketch-STHOSVD can reach the same accuracy as the STHOSVD algorithm
-but in much less CPU time; and the proposed sub-Sketch-STHOSVD achieves
-much better low-rank approximation than R-STHOSVD with similar CPU
-time.
-Now we consider the influence of noise on algorithms’ performance. Specif-
-ically, the sparse tensor X with noise is designed in the same manner as in
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-17
-20
-40
-60
-80
-100
-Target rank
-0.19
-0.195
-0.2
-0.205
-0.21
-0.215
-0.22
-0.225
-Relative Error
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-0.045
-0.05
-0.055
-0.06
-Relative Error
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-1.45
-1.5
-1.55
-1.6
-1.65
-1.7
-1.75
-1.8
-1.85
-1.9
-1.95
-Relative Error
-10-3
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-10-1
-100
-Running Time
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-10-1
-100
-Running Time
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-20
-40
-60
-80
-100
-Target rank
-10-1
-100
-Running Time
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-Fig. 4 Results comparison on a 200×200×200 sparse tensor with noise in terms of numerical
-error (first row) and CPU time (second row).
-[24], i.e.,
-ˆ
-X = X + δK,
-where K is a standard Gaussian tensor and δ is used to control the noise
-level. Let δ = 10−3 and keep the rest parameters the same as the settings
-in the previous experiment. The relative error and running time of different
-algorithms are shown in Figure 4. In Figure 4, we see that noise indeed affects
-the accuracy of the low-rank approximation, especially when the gap is small.
-However, the influence of noise does not change the conclusion obtained on
-the case without noise. The accuracy of our sub-Sketch-STHOSVD algorithm
-is the highest among the randomized algorithms. As γ increases, sub-Sketch-
-STHOSVD can achieve almost the same accuracy as that of THOSVD and
-STHOSVD in a comparable CPU time against R-STHOSVD.
-
-18
-Sketching Algorithms for Low-Rank Tucker Approximation
-5.3 Real-world data tensor
-In this experiment, we test the performance of different algorithms on a colour
-image, called HDU picture1, with a size of 1200 × 1800 × 3. We also evaluate
-the proposed sketching algorithms on the widely used YUV Video Sequences2.
-Taking the ‘hall monitor’ video as an example and using the first 30 frames, a
-three order tensor with a size of 144 × 176 × 30 is then formed for this test.
-Firstly, we conduct an experiment on the HDU picture with target rank
-(500, 500, 3), and compare the PSNR and CPU time of different algorithms.
-The experimental result is shown in Figure 5, which shows that the PSNR
-of sub-Sketch-STHOSVD, THOSVD and STHOSVD is very similar (i.e.,
-∼ 40) and that sub-Sketch-STHOSVD is more efficient in terms of CPU
-time. R-STHOSVD and Sketch-STHOSVD are also very efficient compared to
-sub-Sketch-STHOSVD; however, the PSNR they achieve is 5 dB less than sub-
-Sketch-STHOSVD. Then we conduct separate numerical experiments on the
-HDU picture and the ‘hall monitor’ video clip as the target rank increases, and
-compare these algorithms in terms of the relative error, CPU time and PSNR,
-see Figure 6 and Figure 7. These experimental results again demonstrate
-the superiority (i.e., low error and good approximation with high efficiency)
-of the proposed sub-Sketch-STHOSVD algorithm in computing the Tucker
-decomposition approximation.
-Original
-THOSVD (2.62; 40.61)
-STHOSVD (1.89; 40.65)
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-(0.61; 34.72)
-(0.55; 34.63)
-(0.84; 39.97)
-Fig. 5 Results comparison on a HDU picture with a size of 1200 × 1800 × 3 in terms of
-PSNR (i.e., peak signal-to-noise ratio) and CPU time. The target rank is (500,500,3). The
-two values in e.g. (2.62; 40.61) represent the CPU time and the PSNR, respectively.
-In the last experiment, a larger-scale real-world tensor data is used. We
-choose a color image (called the LONDON picture) with a size of 4775×7155×3
-as the test image and consider the influence of noise. The LONDON picture
-1https://www.hdu.edu.cn/landscape
-2http://trace.eas.asu.edu/yuv/index.html
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-19
-0
-200
-400
-600
-800
-1000
-Target rank
--11
--10
--9
--8
--7
--6
--5
--4
-Relative Error
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-0
-200
-400
-600
-800
-1000
-Target rank
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-Running Time
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-0
-200
-400
-600
-800
-1000
-Target rank
-20
-25
-30
-35
-40
-45
-50
-55
-PSNR
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-Fig. 6 Results comparison on a HDU picture with size of 1200 × 1800 × 3 in terms of
-numerical error (left), CPU time (middle) and PSNR (right). The HDU picture is with target
-rank (r, r, 3), r ∈ [50, 1000].
-0
-20
-40
-60
-80
-100
-Target rank
--9
--8
--7
--6
--5
--4
--3
-Relative Error
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-0
-20
-40
-60
-80
-100
-Target rank
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.045
-0.05
-0.055
-Running Time
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-0
-20
-40
-60
-80
-100
-Target rank
-10
-15
-20
-25
-30
-35
-PSNR
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-Fig. 7 Results comparison on the ‘hall monitor’ grey video with size of 144 × 176 × 30 in
-terms of numerical error (left), CPU time (middle) and PSNR (right). The ‘hall monitor’
-grey video is with target rank (r, r, 10), r ∈ [5, 100].
-with white Gaussian noise is generated using the awgn(X,SNR) built-in function
-in MATLAB. We set the target rank as (50,50,3) and SNR to 20. The results
-comparisons without and with white Gaussian noise are respectively shown in
-Figure 8 and Figure 9 in terms of the CPU time and PSNR. Moreover, we also
-test the algorithms on the LONDON picture as the target rank increases. The
-results regarding the relative error, the CPU time and the PSNR are reported
-in Tables 5, 6 and 7, respectively. On the whole, the results again show the
-consistent performance of the proposed methods.
-
-20
-Sketching Algorithms for Low-Rank Tucker Approximation
-Original
-THOSVD (154.95; 24.07)
-STHOSVD (49.34; 24.09)
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-(1.29; 21.27)
-(1.17; 21.09)
-(1.29; 23.65)
-Fig. 8 Results comparison on LONDON picture with a size of 4775 × 7155 × 3 in terms of
-CPU time and PSNR. The target rank is (50,50,3).
-Noisy picture(PSNR=16.92)
-THOSVD (160.59; 20.54)
-STHOSVD (50.16; 20.54)
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-(1,25; 19.37)
-(1.15; 19.25)
-(1.45; 20.45)
-Fig. 9 Results comparison on LONDON picture with a size of 4775 × 7155 × 3 and white
-Gaussian noise in terms of CPU time and PSNR. The target rank is (50,50,3).
-In summary, the numerical results show the superiority of the sub-sketch
-STHOSVD algorithm for large-scale tensors with or without noise. We can see
-that sub-Sketch-STHOSVD could achieve close approximations to that of the
-deterministic algorithms in a time similar to other randomized algorithms.
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-21
-Table 5 Results comparison in terms of the relative error on the LONDON picture with a
-size of 4775 × 7155 × 3 as the target rank increases.
-Target rank
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-(10,10,10)
-0.019037
-0.019025
-0.031000
-0.040006
-0.020756
-(20,20,20)
-0.012669
-0.012644
-0.023467
-0.027398
-0.013703
-(30,30,30)
-0.010168
-0.010124
-0.018354
-0.020451
-0.010965
-(40,40,40)
-0.008630
-0.008599
-0.015792
-0.017029
-0.009443
-(50,50,50)
-0.007576
-0.007532
-0.013917
-0.015333
-0.008286
-(60,60,60)
-0.006778
-0.006710
-0.012967
-0.013589
-0.007359
-(70,70,70)
-0.006119
-0.006049
-0.011813
-0.011886
-0.006687
-(80,80,80)
-0.005532
-0.005491
-0.010658
-0.011148
-0.006123
-(90,90,90)
-0.005076
-0.005023
-0.010018
-0.010378
-0.005602
-(100,100,100)
-0.004669
-0.004619
-0.009249
-0.009578
-0.005172
-Table 6 Results comparison in terms of the CPU time (in second) on the LONDON
-picture with a size of 4775 × 7155 × 3 as the target rank increases.
-Target rank
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-(10,10,10)
-156.13
-49.22
-0.94
-0.99
-1.12
-(20,20,20)
-165.22
-77.64
-1.24
-1.48
-1.56
-(30,30,30)
-241.11
-76.57
-1.69
-1.39
-1.69
-(40,40,40)
-242.08
-74.25
-1.57
-1.45
-1.68
-(50,50,50)
-268.71
-72.85
-1.51
-1.45
-1.80
-(60,60,60)
-265.52
-77.80
-1.75
-1.51
-2.26
-(70,70,70)
-241.95
-77.82
-1.93
-1.78
-2.24
-(80,80,80)
-264.86
-73.53
-1.86
-1.74
-2.31
-(90,90,90)
-274.73
-72.67
-1.93
-1.83
-2.16
-(100,100,100)
-283.88
-86.42
-2.24
-2.20
-2.46
-Table 7 Results comparison in terms of the PSNR on the LONDON picture with a size
-of 4775 × 7155 × 3 as the target rank increases.
-Target rank
-THOSVD
-STHOSVD
-R-STHOSVD
-Sketch-STHOSVD
-sub-Sketch-STHOSVD
-(10,10,10)
-20.06
-20.07
-17.96
-16.86
-19.70
-(20,20,20)
-21.84
-21.84
-19.18
-18.51
-21.50
-(30,30,30)
-22.79
-22.81
-20.25
-19.78
-22.46
-(40,40,40)
-23.50
-23.52
-20.90
-20.57
-23.11
-(50,50,50)
-24.07
-24.09
-21.45
-21.03
-23.68
-(60,60,60)
-24.55
-24.60
-21.76
-21.55
-24.20
-(70,70,70)
-25.00
-25.05
-22.16
-22.13
-24.61
-(80,80,80)
-25.43
-25.47
-22.61
-22.41
-25.00
-(90,90,90)
-25.81
-25.85
-22.87
-22.72
-25.38
-(100,100,100)
-26.17
-26.22
-23.22
-23.07
-25.73
-6 Conclusion
-In this paper we proposed efficient sketching algorithms, i.e., Sketch-
-STHOSVD and sub-Sketch-STHOSVD, to calculate the low-rank Tucker
-approximation of tensors by combining the two-sided sketching technique with
-the STHOSVD algorithm and using the subspace power iteration. Detailed
-error analysis is also conducted. Numerical results on both synthetic and real-
-world data tensors demonstrate the competitive performance of the proposed
-algorithms in comparison to the state-of-the-art algorithms.
-Acknowledgements
-We would like to thank the anonymous referees for their comments and sug-
-gestions on our paper, which lead to great improvements of the presentation.
-
-22
-Sketching Algorithms for Low-Rank Tucker Approximation
-G. Yu’s work was supported in part by National Natural Science Foundation
-of China (No. 12071104) and Natural Science Foundation of Zhejiang Province
-(No. LD19A010002).
-Appendix
-Lemma 1 [[25], Theorem 2] Let ̺ < k − 1 be a positive natural number and Ω ∈
-Rk×n be a Gaussian random matrix. Suppose Q is obtained from Algorithm 7. Then
-∀A ∈ Rm×n, we have
-EΩ∥A − QQ⊤A∥2
-F ≤ (1 + f(̺, k)̟4q
-k ) · τ 2
-̺+1(A).
-(4)
-Lemma 2 [[22], Lemma A.3] Let A ∈ Rm×n be an input matrix and ˆA = QX
-be the approximation obtained from Algorithm 7. The approximation error can be
-decomposed as
-∥A − ˆA∥2
-F = ∥A − QQ⊤A∥2
-F + ∥X − Q⊤A∥2
-F .
-(5)
-Lemma 3 [[22], Lemma A.5] Assume Ψ ∈ Rl×n is a standard normal matrix
-independent from Ω. Then
-EΨ∥X − Q⊤A∥2
-F = f(k, l) · ∥A − QQ⊤A∥2
-F .
-(6)
-The error-bound for Algorithm 7 can be shown in Lemma 4 below.
-Lemma 4 Assume the sketch size parameter satisfies l > k + 1. Draw random
-test matrices Ω ∈ Rn×k and Ψ∈ Rl×m independently from the standard normal
-distribution. Then the rank-k approximation ˆA obtained from Algorithm 7 satisfies
-E ∥ A − ˆA ∥2
-F ≤ (1 + f(k, l)) · min
-̺<k−1(1 + f(̺, k)̟k
-4q) · τ 2
-̺+1(A).
-Proof Using equations (4), (5) and (6), we have
-E ∥ A − ˆA ∥2
-F = EΩ∥A − QQ⊤A∥2
-F + EΩEΨ∥X − Q⊤A∥2
-F
-= (1 + f(k, l)) · EΩ∥A − QQ⊤A∥2
-F
-≤ (1 + f(k, l)) · (1 + f(̺, k)̟k
-4q) · τ 2
-̺+1(A).
-After minimizing over eligible index ̺ < k − 1, the proof is completed.
-□
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-23
-We are now in the position to prove Theorem 5. Combining Theorem 2
-and Lemma 4, we have
-E{Ωj}N
-j=1∥X − �
-X ∥2
-F
-=
-N
-�
-n=1
-E{Ωj}N
-j=1∥ ˆ
-X (n−1) − ˆ
-X (n)∥2
-F
-=
-N
-�
-n=1
-E{Ωj}n−1
-j=1
-�
-EΩn∥ ˆ
-X (n−1) − ˆ
-X (n)∥2
-F
-�
-=
-N
-�
-n=1
-E{Ωj}n−1
-j=1
-�
-EΩn∥G(n−1) ×n−1
-i=1 U (i)×n(I − U (n)U (n)⊤)∥2
-F
-�
-≤
-N
-�
-n=1
-E{Ωj}n−1
-j=1
-�
-EΩn∥(I − U (n)U (n)⊤)G(n−1)
-(n)
-)∥2
-F
-�
-≤
-N
-�
-n=1
-E{Ωj}n−1
-j=1 (1 + f(rn, ln)) ·
-min
-̺n<rn−1(1 + f(̺n, rn)̟r
-4q)
-In
-�
-i=rn+1
-σ2
-i (G(n−1)
-(n)
-)
-≤
-N
-�
-n=1
-E{Ωj}n−1
-j=1 (1 + f(rn, ln)) ·
-min
-̺n<rn−1(1 + f(̺n, rn)̟r4q)∆2
-n(X)
-=
-N
-�
-n=1
-(1 + f(rn, ln)) ·
-min
-̺n<rn−1(1 + f(̺n, rn)̟r
-4q)∆2
-n(X)
-≤
-N
-�
-n=1
-(1 + f(rn, ln)) ·
-min
-̺n<rn−1(1 + f(̺n, rn)̟r
-4q)∥X − ˆ
-Xopt∥2
-F ,
-which completes the proof of Theorem 5.
-References
-[1] Comon, P.: Tensors: A brief introduction. IEEE Signal Processing Maga-
-zine. 31(3), 44-53(2014)
-[2] Hitchcock, F. L.: Multiple Invariants and Generalized Rank of a P-
-Way Matrix or Tensor. Journal of Mathematics and Physics. 7(1-4),
-39-79(1928)
-[3] Kiers, H. A. L.: Towards a standardized notation and terminology in
-multiway analysis. Journal of Chemometrics Society. 14(3), 105-122(2000)
-[4] Tucker, L. R.: Implications of factor analysis of three-way matrices for
-measurement of change. Problems in measuring change. 15, 122-137(1963)
-
-24
-Sketching Algorithms for Low-Rank Tucker Approximation
-[5] Tucker, L. R.: Some mathematical notes on three-mode factor analysis.
-Psychometrika. 31(3), 279-311(1966)
-[6] De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singu-
-lar value decomposition. SIAM journal on Matrix Analysis Applications.
-21(4), 1253-1278(2000)
-[7] Hackbusch, W., K¨uhn, S.: A new scheme for the tensor representation.
-Journal of Fourier analysis applications. 15(5), 706-722(2009)
-[8] Grasedyck, L.: Hierarchical Singular Value Decomposition of Tensors.
-SIAM journal on Matrix Analysis Applications. 31(4), 2029-2054 (2010)
-[9] Oseledets, I. V.: Tensor-train decomposition. SIAM Journal on Scientific
-Computing. 33(5), 2295-2317(2011)
-[10] De Lathauwer, L., De Moor, B., Vandewalle, J.: On the best rank-1 and
-rank-(r1, r2,...,rn) approximation of higher-order tensors. SIAM journal
-on Matrix Analysis Applications. 21(4), 1324-1342(2000)
-[11] Vannieuwenhoven, N., Vandebril, R., Meerbergen, K.: A new truncation
-strategy for the higher-order singular value decomposition. SIAM Journal
-on Scientific Computing. 34(2), A1027-A1052(2012)
-[12] Zhou, G., Cichocki, A., Xie, S.: Decomposition of big tensors with low
-multilinear rank. arXiv preprint, arXiv:1412.1885(2014)
-[13] Che, M., Wei, Y.: Randomized algorithms for the approximations of
-Tucker and the tensor train decompositions. Advances in Computational
-Mathematics. 45(1), 395-428(2019)
-[14] Minster, R., Saibaba, A. K., Kilmer, M. E.: Randomized algorithms for
-low-rank tensor decompositions in the Tucker format. SIAM Journal on
-Mathematics of Data Science. 2(1), 189-215 (2020)
-[15] Che, M., Wei, Y., Yan, H.: The computation of low multilinear rank
-approximations of tensors via power scheme and random projection.
-SIAM Journal on Matrix Analysis Applications. 41(2), 605-636 (2020)
-[16] Che, M., Wei, Y., Yan, H.: Randomized algorithms for the low multilin-
-ear rank approximations of tensors. Journal of Computational Applied
-Mathematics. 390(2), 113380(2021)
-[17] Sun, Y., Guo, Y., Luo, C., Tropp, J., Udell, M.: Low-rank tucker approx-
-imation of a tensor from streaming data. SIAM Journal on Mathematics
-of Data Science. 2(4), 1123-1150(2020)
-
-Sketching Algorithms for Low-Rank Tucker Approximation
-25
-[18] Tropp, J. A., Yurtsever, A., Udell, M., Cevher, V.: Streaming low-rank
-matrix approximation with an application to scientific simulation. SIAM
-Journal on Scientific Computing. 41(4), A2430-A2463(2019)
-[19] Malik, O. A., Becker, S.: Low-rank tucker decomposition of large tensors
-using tensorsketch. Advances in neural information processing systems.
-31, 10116-10126 (2018)
-[20] Ahmadi-Asl, S., Abukhovich, S., Asante-Mensah, M. G., Cichocki, A.,
-Phan, A. H., Tanaka, T.: Randomized algorithms for computation of
-Tucker decomposition and higher order SVD (HOSVD). IEEE Access. 9,
-28684-28706(2021)
-[21] Halko, N., Martinsson, P.-G., Tropp, J. A.: Finding structure with ran-
-domness: Probabilistic algorithms for constructing approximate matrix
-decompositions. SIAM review. 53(2), 217-288 (2011)
-[22] Tropp, J. A., Yurtsever, A., Udell, M., Cevher, V.: Practical sketching
-algorithms for low-rank matrix approximation. SIAM Journal on Matrix
-Analysis Applications. 38(4), 1454-1485(2017)
-[23] Rokhlin, V., Szlam, A., Tygert, M.: A randomized algorithm for princi-
-pal component analysis. SIAM Journal on Matrix Analysis Applications,
-31(3), 1100-1124(2009)
-[24] Xiao, C., Yang, C., Li, M.: Efficient Alternating Least Squares Algorithms
-for Low Multilinear Rank Approximation of Tensors. Journal of Scientific
-Computing. 87(3), 1-25(2021)
-[25] Zhang, J., Saibaba, A. K., Kilmer, M. E., Aeron, S.: A randomized tensor
-singular value decomposition based on the t-product. Numerical Linear
-Algebra with Applications. 25(5), e2179(2018)
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-1 
- 
-Describing NMR chemical exchange by effective phase diffusion approach 
-Guoxing Lin* 
-Carlson School of Chemistry and Biochemistry, Clark University, Worcester, MA 01610, USA 
- 
-*Email: [email protected] 
- 
-Abstract 
-        This paper proposes an effective phase diffusion method to analyze chemical exchange in nuclear 
-magnetic resonance (NMR). The chemical exchange involves spin jumps around different sites where the 
-spin angular frequencies vary, which leads to a random phase walk viewed from the rotating frame 
-reference. Therefore, the random walk in phase space can be treated by the effective phase diffusion 
-method. Both the coupled and uncoupled phase diffusions are considered; additionally, it includes normal 
-diffusion as well as fractional diffusion. Based on these phase diffusion equations, the line shape of NMR 
-exchange spectrum can be analyzed. By comparing these theoretical results with the conventional theory, 
-this phase diffusion approach works for fast exchange, ranging from slightly faster than intermediate 
-exchange to very fast exchange. For normal diffusion models, the theoretically predicted curves agree 
-with those predicted from traditional models in the literature, and the characteristic exchange time 
-obtained from phase diffusion with a fixed jump time is the same as that obtained from the conventional 
-model. However, the phase diffusion with a monoexponential time distribution gives a characteristic 
-exchange time constant which is half of that obtained from the traditional model. Additionally, the 
-fractional diffusion obtains a significantly different line shape than that predicted based on normal 
-diffusion.  
-Keywords: NMR, chemical exchange, Mittag-Leffler function, phase diffusion 
- 
-1. 
-Introduction 
-       Chemical exchange is a powerful nuclear magnetic resonance (NMR) technique to detect dynamics 
-behavior in biological and polymer systems at the atomic level [1,2,3,4]. Chemical exchange NMR 
-monitors spin jumping around different environmental sites due to changes in conformational or chemical 
-states. In chemical exchange, the angular frequency of the spin precession changes, which results in 
-observable line shape changes in the NMR spectrum. Although chemical exchange has been a established 
-tool, theoretical developments are still needed to better understand the chemical exchange NMR, 
-particularly for complex systems.  
-      The characteristic exchange time could follow a complicated distribution. Many theoretical models 
-have been developed to analyze chemical exchanges in NMR [3, 5, 6]. The two-site exchange model based 
-on the modified-Bloch equation successfully interprets many NMR lines hape [5], where the jump time in 
-the exchange equation is a fixed constant. However, in a real system, the exchange time could follow a 
-distribution such as the exponential function in the Gaussian exchange model reported in [7]. 
-Additionally, a complex exchange distribution could exist in complicated conformational change or 
-diffusion-induced exchange, such as Xenon diffusing in the heterogeneous system [8].   In a complex 
-system, a monoexponential time distribution may not be sufficient to explain the dynamics behavior.  For 
-complicated system  the time distribution function could be the Mittag-Leffler function (MLF) 𝐸𝛼 (− (
-𝑡
-𝜏)
-𝛼
-)  
-[9,10], or a stretched exponential function (SEF) exp (− (
-𝑡
-𝜏)
-𝛼
-) , where α is the time-fractional derivative 
-order, and 𝜏 is the characteristic time. The MLF 𝐸𝛼(−𝑡𝛼) = ∑
-(−𝑡𝛼)𝑛
-Γ(𝑛𝛼+1)
-∞
-𝑛=0
-, can be reduced to a SEF 
-
-2 
- 
-exp (−
-𝑡𝛼
-Γ(1+𝛼)) when 𝑡 is small. The SEF exp (− (
-𝑡
-𝜏)
-𝛼
-) is the same as the Kohlrausch-Williams-Watts (KWW) 
-function [11,12,13,14]], a well-known time correlation function in macromolecular systems. The Mittag-
-Leffler function-based distribution is heavy-tailed. Mittag Leffler function has been employed to analyze 
-anomalous NMR dynamics processes such as PFG anomalous diffusion [15,16], and anomalous NMR 
-relaxation [17,18,19,20]. Currently chemical theories of NMR are still difficult to handle these complex 
-distributions.    
-       A phase diffusion method is proposed in this paper to explain the NMR chemical exchange. Most 
-current methods are real space approaches, such as the modified Bloch exchange equations [3,4,6], while 
-the Gaussian exchange model is a phase space method based on evaluating the accumulated phase 
-variance for the random phase process during the exchange process. As the spin phase in the chemical 
-exchange undergoes a random walk in the rotating frame reference, an effective phase diffusion method 
-will be employed in this paper to analyze the exchange. Effective phase diffusion has been applied to 
-analyze PFG diffusion and NMR relaxation. It has advantages over the traditional methods: It can provide 
-the exact phase distribution that cannot be obtained by conventional real space theoretical method; 
-additionally, the NMR signal can be directly obtained from vector sum by Fourier transform in phase 
-space, which makes the analysis intuitive and often simplifies the solving process; furthermore, the phase 
-diffusion method could be straightforwardly applied to anomalous dynamics process based on fractional 
-diffusion [18, 21, 22, 23,24,25].   Both the normal and fractional phase diffusion are considered, where the 
-exchange time can be a simple constant or a certain type of distribution. 
-      Additionally, each the individual phase jump length is proportional to the jump time and the angular 
-frequency. Because both the jump time and angular frequency fluctuate and obey certain types of 
-distributions, the distribution of phase jump length could be either strongly or weakly correlated to the 
-jump time distribution. The phase walk with weak phase time correlation can be treated by the uncoupled 
-diffusion, while the strong correlation may require coupled diffusion model [26]. The traditional 
-uncoupled diffusion has been successful in explaining many transport phenomena. However, it is 
-insufficient to account for the divergence of the second moment of Levy flight processes [26], where a 
-coupled diffusion is needed. 
-    The rest of the paper is organized as follows. Section 2.1 treats the simplest normal diffusion with a 
-fixed jump time: The obtained exchange time agrees well with the traditional two-site exchange. While 
-the phase diffusion with jump time distribution is presented in Section 2.2: Firstly, the general expressions 
-for phase evolution in chemical exchange are derived in Section 2.2.1; secondly, the normal diffusion is 
-presented in Section 2.2.2, with both uncoupled and coupled diffusion, and it is found that the exchange 
-time constant is two-time faster than that of the traditional model and the fixed jump time diffusion result; 
-thirdly, the fractional diffusion with MLF based jump time distribution is derived in Section 2.2.3, where 
-the uncoupled diffusion is handled by time-fractional diffusion equation, and the coupled fractional 
-diffusion is handled by coupled random walk model [27]. The results here give additional insights into 
-the NMR chemical exchange, which could improve the analysis of NMR and magnetic resonance imaging 
-(MRI) experiments, particularly in complicated systems.  
-2. Theory 
- The chemical exchange occurs when the spin jumps among different sites where the spin precession 
-frequencies are different [1,2,5]. The precession frequencies of the spin moment are proportional to the 
-intensity of the local magnetic field, which is affected by the surrounding electron cloud and nearby spin 
-moments [2]. For simplicity, we consider only the basic exchange between two sites with equal 
-populations [3,5] in this paper, neglecting the relaxation effect. The average precession angular frequencies 
-for these two sites are arbitrarily set as  𝜔1 and 𝜔2 respectively, with 𝜔1  < 𝜔2 and 
-
-3 
- 
- ∆𝜔 = 𝜔2 − 𝜔1.  
- If the angular frequency of the rotating frame reference is set as  
-𝜔1+𝜔2
-2
-;  the two sites have relative 
-angular frequencies -𝜔0 and  𝜔0, respectively, 𝜔0 =
-𝜔2−𝜔1
-2
-.   
- 
- The phase of spin undergoes chemical exchange during a time interval 𝜏 changes either by 𝜔0𝜏 or 
--𝜔0𝜏  depending on the site. From the traditional exchange equations for chemical exchange, the spin 
-always jumps to a site with a different angular frequency after a time interval 𝜏.   Here, the choice for 
-the next sites is assumed to be random; the next site's angular frequency could be either the same or 
-different. This assumption may be more realistic; for instance, after a time interval 𝜏, a spin may 
-successfully jump to another site or return to the original site; or in a heterogeneous system, the spin 
-moves to a similar environment with the same frequency or a different environment with a different 
-frequency.   The random phase jump can be viewed as a random walk process in phase space and 
-analyzed by phase diffusion [16,17].   Both the normal and fractional phase diffusion will be 
-considered in the following. 
-2.1 Simple normal diffusion with a fixed jump time 
-     If the jump time interval 𝜏 is a constant, the random phase jumps with average jump length ∆𝜙 equaling  
-−𝜔0𝜏 or 𝜔0𝜏. The effective phase diffusion constant 𝐷𝜙,𝑠 for such a simple normal diffusion can be obtained 
-by [16] 
- 
- 
- 
-𝐷𝜙,𝑠 =
-〈(∆𝜙)2〉
-2𝜏
-=
-(𝜔0𝜏)2
-2𝜏
-=
-𝜔02
-2 τ, 
- 
- 
- 
-(1) 
-and the normal phase diffusion equation can be described by [16,17] 
-𝑑𝑃(𝜙,𝑡)
-𝑑𝑡
-= 𝐷𝜙,𝑠Δ𝑃(𝜙, 𝑡), 
- 
- 
- 
- 
-( 2) 
-where 𝜙 is the phase and 𝑃(𝜙, 𝑡) is the probability density function of spin at time t with 𝜙. The solution 
-of Eq. (3) is [16] 
-𝑃(𝜙, 𝑡) =
-1
-√4𝜋𝐷𝜙,𝑠𝑡 𝑒𝑥𝑝 [−
-𝜙2
-4𝐷𝜙,𝑠𝑡].   
- 
- 
-(3) 
-The total magnetization 𝑀(𝑡) is obtained by  
-𝑀(𝑡) = ∫
-𝑑𝜙
-∞
-−∞
-𝑒𝑖𝜙𝑃(𝜙, 𝑡) = 𝑒𝑥𝑝(−𝐷𝜙,𝑠𝑡),  
- 
- 
-(4) 
-which is a time domain signal. By Fourier transform, we have the frequency domain signal  
-𝑆(𝜔) = 
-𝐷𝜙,𝑠
-𝐷𝜙,𝑠2+𝜔2 =
-𝜔02
-2 τ
-(
-𝜔02
-2 τ)
-2
-+𝜔2
-.   
- 
- 
- 
-(5) 
-Note that both 𝜔 and 𝜔0 are the angular frequencies in the rotating frame reference.  
-2.2 Diffusion with waiting time distribution  
-2.2.1 General expressions for phase evolution in chemical exchange 
-     A more realistic exchange time should follow a certain type of time distribution function 𝜑(𝑡), which is 
-often related to the time correlation function 𝐺(𝑡) by 𝜑(𝑡) = −
-𝑑𝐺(𝑡)
-𝑑𝑡 . A commonly used simple time 
-correlation function is the mono exponential distribution; in contrast, in a complicated system, it can be a 
-Mittage Leffler function [20]  or stretched exponential function such as the KWW function [11-14].  
-    For a spin starting jumps at a time 𝑡′ from the site with frequency  𝜔𝑖,0, the probability of acquiring 
-phase  𝜔𝑖,0𝑡′+ 𝜙 at time t is 
-
-4 
- 
-𝑃𝜔𝑖,0(𝜙, 𝑡) = 𝜑(𝑡′)𝑃(𝜙, 𝑡 − 𝑡′),     
- 
- 
-(6) 
-where the phase change 𝜔𝑖,0𝑡′ is obtained from time 0 to time 𝑡′  when the spin stays immobile at the site, 
-and 𝑃(𝜙, 𝑡 − 𝑡′) is the phase PDF resulting from the diffusion, or random walk in the phase space during  
-𝑡 − 𝑡′. Summing all possible magnetization vectors with different phase 𝜔𝑖,0𝑡′ + 𝜙, at time t, the net 
-magnetization 𝑀𝜔𝑖,0(𝑡′, 𝑡) contributed from these spins beginning to jump randomly from time 𝑡′ is 
-𝑀𝜔𝑖,0(𝑡′, 𝑡) = ∫
-𝑑𝜙
-∞
-−∞
-𝑒𝑖𝜔𝑖,0𝑡′+𝜙𝑃𝜔𝑖,0(𝜙, 𝑡 − 𝑡′) = 𝑒𝑖𝜔𝑖,0𝑡′𝜑(𝑡′)𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1   , 
-(7) 
-where 
-𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1 = ∫
-𝑑𝜙
-∞
-−∞
-𝑒𝑖𝑘𝜙𝑃(𝜙, 𝑡 − 𝑡′).  
- 
- 
-(8) 
-The total magnetization from all spins in the systems at time t is   
-𝑀(𝑡) = ∫ 𝑑𝑡′
-𝑡
-0
- ∑ 𝑝𝑖𝑀𝜔𝑖,0(𝑡′, 𝑡)
-𝑖
-,   
- 
- 
-(9) 
-where 𝑝𝑖 is the population of spins at site i. For simplicity, only exchange with two equal population sites 
-will be considered here;  let 𝜔1,0 = −𝜔0, 𝜔2,0 = 𝜔0 and all the subindexes i will be dropped out throughout 
-the rest of the paper.   For a two-site system with equal populations 𝑝1 = 𝑝2 = 
-1
-2, 
-𝑀(𝑡) = ∫ 𝑑𝑡′
-𝑡
-0
- 1
-2 [𝑀−𝜔0(𝑡′, 𝑡) + 𝑀𝜔0(𝑡′, 𝑡)] 
-               
-        = ∫ 𝑑𝑡′
-𝑡
-0
- 
-1
-2 [𝑒−𝑖𝜔0𝑡′ + 𝑒𝑖𝜔0𝑡′] 𝜑(𝑡′)𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1 
-         = ∫ 𝑑𝑡′
-𝑡
-0
-  𝐵(𝑡′)𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1 ,                                                  (10a) 
-where  
-𝐵(𝑡′) =
-1
-2 [𝑒−𝑖𝜔0𝑡′ + 𝑒𝑖𝜔0𝑡′]𝜑(𝑡′).                                           (10b) 
-Eq. (10a) involves the convolution of 𝐵(𝑡′) and 𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1. In Laplace representation [25],  
-𝑀(𝑠) = 𝐵(𝑠)𝑝(𝑘, 𝑠)|𝑘=1;     
- 
- 
- 
-(11) 
-in many cases (the coupled diffusion in the paper), the frequency domain signal can be obtained by the 
-Fourier transform of  𝑀(𝑡): 
-𝑆(𝜔) = ∫
-𝑒𝑖𝜔𝑡𝑀(𝑡)𝑑𝑡
-∞
-0
-= 𝐵(𝜔)𝑝(𝑘, 𝜔)|𝑘=1 
-   
-(12) 
-2.2.2 Normal diffusion with monoexponential distribution function 
-Now, let us consider if the jump time follows a monoexponential distribution  𝜑(𝑡) described by 
-[25] 
-𝜑(𝑡) =
-1
-𝜏 exp (−
-𝑡′
-𝜏 ) ,   
- 
- 
- 
- 
-(13) 
-whose Laplace representation is  
-𝜑(𝑠) =
-1
-𝑠𝜏+1   
- 
- 
- 
- 
- 
- (14) 
-According to Eq. (10), 
- 𝐵(𝑡) =
-1
-2 [𝑒−𝑖𝜔0𝑡′ + 𝑒𝑖𝜔0𝑡′]
-1
-𝜏 exp (−
-𝑡′
-𝜏 ),    
- 
-(15a) 
-whose Laplace representation is [25] 
-
-5 
- 
-𝐵(𝑠) = 
-1
-2 [
-1
-𝜏(𝑠−𝑖𝜔0)+1 +
-1
-𝜏(𝑠+𝑖𝜔0)+1] ≈
-1
-1+𝜔02𝜏2+𝜏𝑠(1−𝜔02𝜏2) =
-1
-1+𝜔02𝜏2
-1+𝑠
-𝜏(1−𝜔02𝜏2)
-1+𝜔02𝜏2
-.   
-(15b) 
-I. 
-Uncoupled normal diffusion 
-The spin angular frequency is often affected by a random fluctuating magnetic field, which is produced 
-by surrounding spins undergoing the thermal motion [**]; additionally, the angular frequency could be 
-affected by the electron cloud change during the exchange process;  further, the chemical exchange may 
-take place because the spin moves among different domains in a heterogeneous system where the 
-frequency fluctuating around positive or negative 𝜔0 . This angular frequency can be denoted as 𝜔, and 
-the  average of its absolute value is 〈|𝜔|〉 = 𝜔0. Because 𝜔 is randomly fluctuating, the individual random 
-phase jump ∆𝜙 = 𝜔��𝑗𝑢𝑚𝑝  randomly fluctuates for each jump time 𝜏𝑗𝑢𝑚𝑝; the space and time uncoupled 
-phase diffusion could be applied to treat the phase random walk, a more complicated coupled diffusion 
-will be considered in Section in subsequence.   For an uncoupled diffusion,   
-〈𝜏𝑗𝑢𝑚𝑝〉 = ∫
-𝑡
-𝜏 exp (−
-𝑡
-𝜏) 𝑑𝑡 = 𝜏
-∞
-0
-,  
- 
- 
- 
-(16) 
-〈𝜏𝑗𝑢𝑚𝑝
-2
-〉 = ∫
-𝑡2
-𝜏 exp (−
-𝑡
-𝜏) 𝑑𝑡 = 2𝜏2
-∞
-0
-.   
- 
- 
- 
-(17) 
-The average phase jump length square 〈(∆𝜙)2〉 is 
- 
-〈(∆𝜙)2〉 = 〈(|𝜔|𝜏𝑗𝑢𝑚𝑝)2〉 = 〈𝜔2〉〈𝜏𝑗𝑢𝑚𝑝
-2
-〉 = 𝜔0
-22𝜏2 = 2𝜔0
-2𝜏2.  
- 
-(18) 
-Such an uncoupled random walk has an effective phase diffusion constant 
- 
-𝐷𝜙 =
-〈(∆𝜙)2〉
-2〈𝜏𝑗𝑢𝑚𝑝〉 =
-〈(𝜔𝜏𝑗𝑢𝑚𝑝)2〉
-2𝜏
-=
-〈𝜔2〉〈𝜏𝑗𝑢𝑚𝑝
-2
-〉
-2𝜏
-=
-𝜔022𝜏2
-2𝜏
-= 𝜔0
-2𝜏.   
- 
- 
-(19) 
-With 𝐷𝜙 , the normal phase diffusion equation can be described by [18] 
-𝑑𝑃(𝜙,𝑡)
-𝑑𝑡
-= 𝐷𝜙Δ𝑃(𝜙, 𝑡). 
- 
- 
- 
- 
-(20) 
-From Eq (20), the probability density function is  
-𝑃(𝜙, 𝑡) =
-1
-√4𝜋𝐷𝜙𝑡 𝑒𝑥𝑝 [−
-𝜙2
-4𝐷𝜙𝑡].     
- 
- 
- (21) 
-Substituting Eq. (21) into Eq. (8), we have  
-𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1 = ∫
-𝑑𝜙
-∞
-−∞
-𝑃(𝜙, 𝑡 − 𝑡′) = 𝑒𝑥𝑝[−𝐷𝜙(𝑡 − 𝑡′)],   
- 
- (22) 
-whose Laplace transform representation is 
-𝑝(𝑘, 𝑠)|𝑘=1 =
-1
-𝑠+𝐷𝜙.   
- 
- 
- 
- 
- 
-(23) 
-Substituting Eqs. (15b) and (23) into Eq. (11) yields 
-𝑀(𝑠) = 𝐵(𝑠)𝑝(𝑘, 𝑠)|𝑘=1=
-1
-1+𝜔02𝜏2
-1+𝑠
-𝜏(1−𝜔02𝜏2)
-1+𝜔02𝜏2
-1
-𝑠+𝐷𝜙.   
- 
- 
-(24) 
-From 𝑀(𝑠), the inverse Laplace transform gives 
-𝑀(𝑡) = ∫
-𝑑𝑡 
-∞
-0
-1
-𝜏(1−𝜔02𝜏2) exp (−
-𝑡′
-𝜏(1−𝜔02𝜏2)
-1+𝜔02𝜏2
-) 𝑒𝑥𝑝[−𝐷𝜙(𝑡 − 𝑡′)].  
-  (25) 
-
-6 
- 
-Eq. (25) includes the convolution of two parts, the 
-1
-𝜏(1−𝜔02𝜏2) exp (−
-𝑡′
-𝜏(1−𝜔02𝜏2)
-1+𝜔02𝜏2
-) comes from the Fourier 
-transform of the 𝜑(𝑡), while 𝑒𝑥𝑝[−𝐷𝜙(𝑡 − 𝑡′)] results from the phase diffusion;  the Frequency domain 
-signal can be obtained from the Fourier Transform of expression (25) as  
-𝑆(𝜔) =
-1
-1+𝜔02𝜏2
-1+[
-𝜏(1−𝜔02𝜏2)
-1+𝜔02𝜏2 ]
-2
-𝜔2
- 
-𝐷𝜙
-𝐷𝜙2+𝜔2.    
- 
- 
-(26) 
-II. 
- Coupled normal diffusion with monoexponential distribution function 
-The coupled random phase walk has a joint probability function 𝜓(𝜙, 𝑡)  expressed by {25} 
-𝜓(𝜙, 𝑡) = 𝜑(𝑡)Φ(𝜙|𝑡), 
- 
- 
- 
- 
- (27) 
-where 𝜑(𝑡) is the waiting time function, and  Φ(𝜙|𝑡) is the conditional probability that a phase jump length 
-𝜙 requiring time t. In Fourier-Laplace representation, the probability density function of a coupled random 
-walk has been derived in Ref. [25] as  
-𝑃(𝑘, 𝑠) =
-Ψ(𝑘,𝑠)
-1−𝜓(𝑘,𝑠)  ,  
- 
- 
- 
- 
- (28) 
-where 𝜓(𝑘, 𝑠) is the joint probability, and Ψ(𝜙, 𝑡) is the PDF for the phase displacement of the last, 
-incomplete walk, which is  [25] 
-Ψ(𝜙, 𝑡) = 𝛿( 𝜙) ∫
-𝜑(𝑡′)𝑑𝑡′
-∞
-𝑡
-,  
- 
- 
- 
- (29a) 
-and 
- Ψ(𝑘, 𝑠) = 
-1−𝜑(𝑠)
-𝑠
-.   
- 
- 
- 
- 
-(29b) 
-By substituting Eq. (29b) into Eq. (28), it arrives [25] 
-𝑃(𝑘, 𝑠) =
-Ψ(𝑘,𝑠)
-1−𝜓(𝑘,𝑠)  = 
-1−𝜑(𝑠)
-𝑠
-1
-1−𝜓(𝑘,𝑠).   
- 
- 
- 
-(30) 
-In the chemical exchange, the joint probability could be described by  
-Φ(𝜙|𝑡) =
-1
-2 𝛿(|𝜙| − 𝜔0𝑡),  
- 
- 
- 
-(31a) 
-𝜓(𝜙, 𝑡) =
-1
-2 𝜑(𝑡)𝛿(|𝜙| − 𝜔0𝑡),   
- 
- 
- 
-(31b) 
-and 
-𝜓(𝑘, 𝑠)=∫ 𝑒𝑖𝑘𝜙−𝑠𝑡𝜓(𝜙, 𝑡)𝑑𝜙𝑑𝑡= 
-1
-2 [
-1
-𝜏(𝑠−𝑖𝑘𝜔0)+1 +
-1
-𝜏(𝑠+𝑖𝑘𝜔0)+1] ≈
-1+𝜏𝑠
-1+𝑘2𝜔02𝜏2+2𝜏𝑠. 
- 
-(31c) 
-Substitute Eq. (31c) into Eq. (30), and we get 
-𝑃(𝑘, 𝑠) =
-Ψ(𝑘,𝑠)
-1−𝜓(𝑘,𝑠)  = 
-1−𝜑(𝑠)
-𝑠
-1
-1−𝜓(𝑘,𝑠) =
-𝜏
-1−
-1+𝜏𝑠
-1+𝑘2𝜔02𝜏2+2𝜏𝑠
-.   
- 
-(32) 
-and 
-  
- 
- 
- 
- 
-𝑝(𝑘, 𝑠)|𝑘=1 =
-𝜏
-1−
-1+𝜏𝑠
-1+𝜔02𝜏2+2𝜏𝑠
-.   
- 
- 
-  
-(33) 
- 
-Substituting Eqs. (15b) and (33) into Eq. (11) yields 
-
-7 
- 
- 
-𝑀(𝑠) =
-1+𝜏𝑠
-1+𝜔02𝜏2+2𝜏𝑠
-𝜏
-1−
-1+𝜏𝑠
-1+12𝜔02𝜏2+2𝜏𝑠
-=
-𝜏(1+𝜏𝑠)
-𝜔02𝜏2+𝜏𝑠 ≈
-𝜏
-𝜏𝑠(1−𝜔02𝜏2)+𝜔02𝜏2 =
-𝜏/𝜔02𝜏2
-𝜏𝑠(1−𝜔02𝜏2)/𝜔02𝜏2+1   
- 
-(34) 
-From 𝑀(𝑠), the inverse Laplace transform gives 
-𝑀(𝑡) =
-1
-1−𝜔02𝜏2 exp (−
-𝑡
-𝜏(1−𝜔02𝜏2)
-𝜔02𝜏2
-).   
- 
- 
- 
-  (35) 
-The frequency domain NMR signal can be obtained from the Fourier Transform of ���(𝑡) in Eq. (35) as  
-𝑆(𝜔) =
-1
-1−𝜔02𝜏2 
-𝜏(1−𝜔02𝜏2)
-𝜔02𝜏2
-(
-𝜏(1−𝜔02𝜏2)
-𝜔02𝜏2
-)
-2
-𝜔2+1
-= 
-𝜏
-𝜏2(1−𝜔02𝜏2)2𝜔2+𝜔02𝜏2.   
- 
-  (36) 
-2.2.3  Fractional phase diffusion 
-For a complicated system, the time correlation function may not be a simple monoexponential function, 
-such as the Kohlrausch-Williams-Watts (KWW) function, or Mittag-Leffler function, and the 
-corresponding phase diffusion could be an anomalous diffusion [21-24]. The time fractional phase 
-diffusion will be investigated here, and the time correlation function is assumed as a MLF, 
-G(t) = 𝐸𝛼 (− (
-𝑡
-𝜏)
-𝛼
-),  
- 
- 
- 
- 
-(37) 
- and its waiting time distribution function will be a heavy-tailed time distribution [27] 
-𝜑𝑓(𝑡) = −
-𝑑
-𝑑𝑡 𝐸𝛼 (− (
-𝑡
-𝜏)
-𝛼
-) ,   
- 
- 
- 
- (38) 
-whose Laplace transform is [25,27] 
-𝜑𝑓(𝑠) =
-1
-𝑠𝛼𝜏𝛼+1.  
- 
- 
- 
- 
-  (39) 
-Based on Eqs. (10b), (38) and (39), we have  
-𝐵(𝑠) =  𝑅𝑒𝜑𝑓(𝑠 + 𝑖𝜔) = 𝑅𝑒 1
-2 [
-1
-𝜏𝛼
-𝜔0
-𝛼 [cos (𝜋
-2 𝛼 − 𝑠𝛼
-𝜔0) + 𝑖 sin (𝜋
-2 𝛼 − 𝑠𝛼
-𝜔0)] + 1
-𝜏𝛼
-] 
-≈
-𝜔0𝛼𝜏𝛼(cos𝜋
-2𝛼+
-1
-𝜔0𝛼𝜏𝛼)
-1+𝜔02𝛼𝜏2𝛼+2𝜔0𝛼𝜏𝛼cos𝜋
-2𝛼+𝑠𝛼𝜔0𝛼−1𝜏𝛼sin𝜋
-2𝛼
-1−𝜔02𝛼𝜏2𝛼
-𝜔0𝛼𝜏𝛼cos𝜋
-2𝛼+1 
-=
-𝑐
-1+𝑠𝜏′ ,      
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
-(40a) 
-where 
- 𝑐 =
-𝜔0𝛼𝜏𝛼(cos𝜋
-2𝛼+
-1
-𝜔0𝛼𝜏𝛼)
-1+𝜔02𝛼𝜏2𝛼+2𝜔0𝛼𝜏𝛼cos𝜋
-2𝛼 ,        
- 
- 
- 
- 
- 
- 
-(40b) 
-and   
-   
- 
- 
-𝜏′ =
-𝛼𝜔0𝛼−1𝜏𝛼sin𝜋
-2𝛼
-1−𝜔02𝛼𝜏2𝛼
-𝜔0𝛼𝜏𝛼cos𝜋
-2𝛼+1
-1+𝜔02𝛼𝜏2𝛼+2𝜔0𝛼𝜏𝛼cos𝜋
-2𝛼 .        
- 
- 
- 
-(40c) 
- 
-I. 
-Uncoupled fractional diffusion  
- 
- 
-For such a time distribution function, the phase diffusion constant can be calculated according to Ref. [16, 
-
-8 
- 
-22,23] as 
- 
- 
- 
- 
-𝐷𝜙𝑓 =
-〈(∆𝜙)2〉
-2Γ(1+𝛼)𝜏𝛼.  
- 
- 
- 
-(41a) 
-The average phase jump may be assumed as  〈(∆𝜙)2〉 = 𝜔0
-22𝜏2, then 
- 
- 
- 
- 
- 
- 
- 𝐷𝜙𝑓 =
-𝜔02𝜏2−𝛼
-Γ(1+𝛼)𝜏𝛼  .    
- 
- 
- 
- (41b) 
-With 𝐷𝜙𝑓 , the fractional phase diffusion equation can be described by [16,17,21,23,24] 
- 𝑡𝐷∗
-𝛼𝑃𝑓 = 𝐷𝜙𝑓Δ𝑃𝑓(𝜙, 𝑡).  
- 
- 
- 
-  (42) 
-where 0 < 𝛼, 𝛽 ≤ 2, 𝐷𝑓𝑟 is the rotational diffusion coefficient, a is the spherical radius, 𝑡𝐷∗
-𝛼  is the Caputo 
-fractional derivative defined by [22,23] 
- 𝑡𝐷∗
-𝛼𝑓(𝑡): = {
-1
-𝛤(𝑚−𝛼) ∫
-𝑓(𝑚)(𝜏)𝑑𝜏
-(𝑡−𝜏)𝛼+1−𝑚 , 𝑚 − 1 < 𝛼 < 𝑚,
-𝑡
-0
-𝑑𝑚
-𝑑𝑡𝑚 𝑓(𝑡), 𝛼 = 𝑚,
-   
- 
-   
-Fourier transform of Eq. (42) give [ 
-                                                                𝑡𝐷∗
-𝛼𝑝(𝑘, 𝑡) = −𝐷𝜙𝑓𝑘2𝑝(𝑘, 𝑡).  
-  
- 
- 
- 
- (43) 
-The solution of Eq. (43) is  𝑝(𝑘, 𝑡) = 𝐸𝛼[−𝐷𝜙𝑓𝑘2𝑡𝛼] [16,17], whose Laplace representation is [22,23,25] 
- 
- 
- 
- 
- 
-𝑝(𝑘, 𝑠) =
-𝑆𝛼−1
-𝑆𝛼+𝐷𝜙𝑓𝑘2,  
- 
- 
- 
- 
-(44) 
-then 
- 
- 
- 
- 
- 
-𝑝(𝑘, 𝑠)|𝑘=1 =
-𝑆𝛼−1
-𝑆𝛼+𝐷𝜙𝑓.   
- 
- 
- 
- 
-(45) 
-Substituting Eqs. (40) and (45) into Eq. (11), we get 
- 
-𝑀(𝑠) =
-𝑐
-1+𝑠𝜏′
-𝑆𝛼−1
-𝑆𝛼+𝐷𝜙𝑓,     
- 
- 
- 
- 
-(46) 
-whose inverse Laplace transform gives  
-M(𝑡) = ∫ 𝑑𝑡′ 
-𝑡
-0
-𝑐
-𝜏′ exp (−
-𝑡′
-𝜏′)𝐸𝛼[−𝐷𝜙𝑓(𝑡 − 𝑡′)𝛼].    
- 
- 
-(47) 
-  The NMR signal can be obtained from the Fourier transform of M(𝑡), which is  
-𝑆(𝜔) = 𝐵(𝜔) ∙ 𝐸(𝜔) =
-𝑐
-1+𝜏′2𝜔2 ∙
-𝜔𝛼−1(
-1
-𝐷𝜙𝑓
-) sin(𝜋
-2𝛼)
-𝜔2𝛼(
-1
-𝐷𝜙𝑓
-)
-2
-+2𝜔𝛼(
-1
-𝐷𝜙𝑓
-) cos(𝜋
-2𝛼)+1
-.                            (48) 
-II. Coupled fractional diffusion 
-Similarly,  for the coupled normal diffusion, the joint probability could be described by [25] 
-Φ(𝜙|𝑡) =
-1
-2 𝛿(|𝜙| − 𝜔0𝑡),   
- 
- 
- 
- 
-(49a) 
-𝜓(𝜙, 𝑡) =
-1
-2 𝜑(𝑡)𝛿(|𝜙| − 𝜔0𝑡),      
- 
- 
- 
-(49b) 
-𝜓(𝑘, 𝑠)=∫ 𝑒𝑖𝑘𝜙−𝑠𝑡𝜓(𝜙, 𝑡)𝑑𝜙𝑑𝑡= 
-1
-2 [
-1
-𝜏𝛼(𝑠−𝑖𝑘𝜔0)𝛼+1 +
-1
-𝜏𝛼(𝑠+𝑖𝑘𝜔0)𝛼+1].                        
- (49c) 
-and 
-
-9 
- 
-𝜓(1, 𝑠) ≈
-1
-2 [
-1
-𝜏𝛼(𝑠−𝑖𝜔0)𝛼+1 +
-1
-𝜏𝛼(𝑠+𝑖𝜔0)𝛼+1].  
- 
- 
- 
- (50)  
-Compared to Eq. (40a), it is obvious that 
-𝜓(1, 𝑠) = 𝐵(𝑠) =
-𝑐
-1+𝑠𝜏′ .   
- 
- 
- 
-  (51) 
-Substituted Eq. (51) into Eq. (30), we get 
-𝑝(𝑘, 𝑠)|𝑘=1 =
-Ψ(𝑘,𝑠)
-1−𝜓(1,𝑠)  = 
-1−𝜑(𝑠)
-𝑠
-1
-1−𝜓(1,𝑠)=
-𝜏𝛼𝑠𝛼−1
-1���
-𝑐
-1+𝑠𝜏′
-.    
- 
- 
-(52) 
-Eq. (52) can be substituted into Eq. (11) to give 
-𝑀(𝑠) = 𝐵(𝑠)𝑝(𝑘, 𝑠)|𝑘=1 =
-𝑐
-1+𝑠𝜏′
-𝜏𝛼𝑠𝛼−1
-1−
-𝑐
-1+𝑠𝜏′
-=
-𝑐𝜏𝛼𝑠𝛼−1
-1+𝑠𝜏′−𝑐 =
-1
-𝑆𝛼−1
-𝑐𝜏𝛼
-1+𝑠𝜏′−𝑐=
-1
-𝑆𝛼−1
-𝑐𝜏𝛼
-1−𝑐
-1+𝑠 𝜏′
-1−𝑐
-,      
- 
-(53) 
-whose inverse Laplace transform yields 
- 𝑀(𝑡) = ∫ 𝑑𝑡′
-1
-Γ(1−𝛼) 𝑡′−𝛼 𝑐
-𝑡
-0
-𝜏𝛼 1
-𝜏′ exp (−
-𝑡−𝑡′
-𝜏′
-1−𝑐
-) . 
- 
- 
-(54) 
-The Fourier transform of 𝑀(𝑡) gives NMR frequency domain signal 
-𝑆(𝜔) = sin (
-𝜋
-2 𝛼) |𝜔|𝛼−1 𝑐𝜏𝛼
-𝜏′
-𝜏′
-1−𝑐
-1+( 𝜏′
-1−𝑐)
-2
-𝜔2 = sin (
-𝜋
-2 𝛼) |𝜔|𝛼−1
-𝑐𝜏𝛼
-1−𝑐
-1+( 𝜏′
-1−𝑐)
-2
-𝜔2.  
-(55) 
-3. Results  
- 
-     A phase diffusion equation method is proposed to describe the effect of chemical exchange on NMR 
-spectrum, based on uncoupled and coupled normal and fractional diffusions. The exchange between two 
-sites with equal populations is considered, and the theoretical expressions are organized in Table 1.    
-Table 1  
- Comparison of theoretical NMR line shape expressions from phase diffusion method to traditional results for 
-chemical exchange between two sites with equal populations. 
-Frequency domain signal expression from phase diffusion results: 
-Simple phase diffusion with a constant jump time 
-𝑆(𝜔) = 
-𝐷𝜙,𝑠
-𝐷𝜙,𝑠2+𝜔2, 𝐷𝜙,𝑠 =
-𝜔02
-2 τ. 
-Normal phase diffusion with monoexponential function 
-Uncoupled diffusion 
-𝑆(𝜔) =
-1
-1+𝜔02𝜏2
-1+[
-𝜏(1−𝜔02𝜏2)
-1+𝜔02𝜏2 ]
-2
-𝜔2  
-𝐷𝜙
-𝐷𝜙2+𝜔2,  𝐷𝜙 = 𝜔0
-2𝜏. 
-Coupled diffusion 
-𝑆(𝜔) = 
-𝜏
-𝜏2(1−𝜔0
-2𝜏2)2𝜔2+𝜔0
-2𝜏2. 
-Fractional phase diffusion with heavy-tailed time distribution 
- 𝑐 =
-𝜔0𝛼𝜏𝛼(cos𝜋
-2𝛼+
-1
-𝜔0𝛼𝜏𝛼)
-1+𝜔0
-2𝛼𝜏2𝛼+2𝜔0
-𝛼𝜏𝛼cos𝜋
-2𝛼 , 𝜏′ =
-𝛼𝜔0𝛼−1𝜏𝛼sin𝜋
-2𝛼
-1−𝜔02𝛼𝜏2𝛼
-𝜔0𝛼𝜏𝛼cos𝜋
-2𝛼+1
-1+𝜔0
-2𝛼𝜏2𝛼+2𝜔0
-𝛼𝜏𝛼cos𝜋
-2𝛼  
-Uncoupled diffusion 
-𝑆(𝜔) =
-𝑐
-1+𝜏′2𝜔2 ∙
-𝜔𝛼−1(
-1
-𝐷𝜙𝑓) sin(𝜋
-2𝛼)
-𝜔2𝛼(
-1
-𝐷𝜙𝑓)
-2
-+2𝜔𝛼(
-1
-𝐷𝜙𝑓)cos(𝜋
-2𝛼)+1
-,  𝐷𝜙 =
-𝜔02𝜏2
-Γ(1+𝛼)𝜏𝛼 
-Coupled diffusion 
-𝑆(𝜔) = sin (𝜋
-2 𝛼) |𝜔|𝛼−1
-𝑐𝜏𝛼
-1 − 𝑐
-1 + ( 𝜏′
-1 − 𝑐)
-2
-𝜔2
- 
-Frequency domain signal expression from traditional method: 
- 
-𝑆(𝜔) =
-𝜔02𝜏
-2
-[𝜏
-2(𝜔0
-2−𝜔2)]
-2
-+𝜔2  [3,4,5] 
-
-10 
- 
- 
- 
-phasediff_tau_2overkex_kex30dw_beta_0.75_122722 - Copy
-Traditional
-Fixed_jump_time_diffusion
-Uncoupled_normal_diffusion
-Coupled_Normal_diffusion
-Uncoupled_fractional_diffusion
-Coupled_fractional_diffusion
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-0.01
-0.02
-0.03
-0.04
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-/2  (Hz)
-S()
-a
-  = 2'  
-  traditional model, fixed time diffusion
-'  coupled and uncoupled 
-     norrmal and fractional diffusion
- = 2'  =(15)
-phasediff_tau_2overkex_kex5dw_beta_0.75_122722 - Copy
-Traditional
-Fixed_jump_time_diffusion
-Uncoupled_normal_diffusion
-Coupled_Normal_diffusion
-Uncoupled_fractional_diffusion
-Coupled_fractional_diffusion
-0
-0.01
-0.02
-0.03
-0.04
-0.05
--150
--100
--50
-0
-50
-100
-150
-S()
-/2  (Hz)
- = 2'  =(2.5)
-b
-  traditional model, fixed time diffusion
-'  coupled and uncoupled 
-     norrmal and fractional diffusion
- 
-0
-0.005
-0.01
-0.015
-0.02
--150
--100
--50
-0
-50
-100
-150
-phasediff_tau_2overkex_kex2dw_beta_0.75_122722 - Copy
-Traditional
-Fixed_jump_time_diffusion
-Uncoupled_normal_diffusion
-Coupled_Normal_diffusion
-Uncoupled_fractional_diffusion
-Coupled_fractional_diffusion
-S()
-/2  (Hz)
-c
- = 2' = 1 
-0
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
-0.007
--150
--100
--50
-0
-50
-100
-150
-phasediff_tau_2overkex_kex1dw_beta_0.75_122722 - Copy
-Traditional
-Fixed_jump_time_diffusion
-Uncoupled_normal_diffusion
-Coupled_Normal_diffusion
-Uncoupled_fractional_diffusion
-Coupled_fractional_diffusion
-/2  (Hz)
-S()
-d
- = 2'  =1(0.5)
- 
-0
-0.001
-0.002
-0.003
-0.004
-0.005
--150
--100
--50
-0
-50
-100
-150
-phasediff_tau_2overkex_kex0.6dw_beta_0.75_122722 - Copy
-Traditional
-Fixed_jump_time_diffusion
-Uncoupled_normal_diffusion
-Coupled_Normal_diffusion
-Uncoupled_fractional_diffusion
-Coupled_fractional_diffusion
-S()
-/2  (Hz)
-e
- = 2'  = 1 (0.3)
-tau = 2/kex  = 2/0.6 dw
-0
-0.002
-0.004
-0.006
-0.008
-0.01
--150
--100
--50
-0
-50
-100
-150
-phasediff_tau_2overkex_kex0.1dw_beta_0.75_122722 - Copy
-Traditional
-Fixed_jump_time_diffusion
-Uncoupled_normal_diffusion
-Coupled_Normal_diffusion
-Uncoupled_fractional_diffusion
-Coupled_fractional_diffusion
-S()
-/2  (Hz)
-f
- = 2'  =(0.05 )
- 
-Fig. 1  The comparison among the various theoretical results obtained from the phase diffusion models and those 
-obtained by the traditional two-site exchange model, all equations listed in Table 1, with ∆𝜔/2𝜋 = 100 Hz, and 𝛼 = 0.75 
-for fractional phase diffusion. 
- 
- 
- 
- 
- 
- 
-
-11 
- 
-4. Discussion 
- 
-      In rotating frame reference, the spin phase in chemical exchange undergoes random phase jumps, 
-which can be intrinsically describe by either uncoupled effective phase diffusion equation or coupled 
-random walk.  
-       Figure 1 shows the comparison among the various theoretical results obtained from the phase diffusion 
-models and those obtained by the traditional two-site exchange model, all equations listed in Table 1. From 
-Figure 1, when the exchange is sufficiently fast, 𝜏 = 2𝜏′ ≤ 1/∆𝜔, the theoretical curves from diffusion with 
-a fixed jump time, uncoupled and coupled normal diffusion almost overlap with that predicted from the 
-traditional model.  However, the exchange time constant 𝜏 for the traditional model and the fixed time 
-diffusion is two times as 𝜏′ for the coupled and uncoupled normal diffusion with the monoexponential 
-distribution. The difference in exchange time could be explained by the following: the effective phase 
-diffusion constant is 
-𝜔02
-2 τ  for diffusion with a fixed jump time τ; in contrast, it is 𝜔0
-2τ for the uncoupled 
-diffusion with a monoexponential time distribution. The two-time difference in diffusion coefficients 
-resulted from the 〈𝜏𝑗𝑢𝑚𝑝
-2
-〉 = ∫
-𝑡2
-𝜏 exp (−
-𝑡
-𝜏) 𝑑𝑡 = 2𝜏2
-∞
-0
-, while in is the fixed time jump 〈𝜏𝑗𝑢𝑚𝑝
-2
-〉 = 𝜏2. The same 
-phase diffusion coefficient 𝜔0
-2τ has been used in Ref. [17] to obtain NMR relaxation expressions, which 
-replicate the traditional NMR relaxation theories;  these NMR relaxation expressions have been verified by 
-numerous experimental results; although this theoretical and experimental confirm is from relaxation 
-NMR, it still provide a strong support to select 𝜔0
-2τ rather than 
-𝜔02
-2 τ  as a phase diffusion coefficient, 
-considering both the exchange and relaxation are random phase walk processes. Therefore, in the analysis 
-of NMR chemical exchange line shape, the exchange time constant could be a two-time difference 
-depending on the employed models.  
-  Additionally, in Figure 1, the exchange line shapes in normal diffusion and fractional diffusion are 
-significantly different. The spectrum line from coupled fractional diffusion is broader than that of 
-uncoupled fractional diffusion, which may be reasonable because there is a more direct effect of heavy-
-tailed time distribution on the phase length in the coupled fractional diffusion than that of uncoupled 
-fractional diffusion.    Meanwhile,   the effect of coupled and uncoupled diffusion on the NMR line shape 
-is different in normal and fractional diffusions; the difference between the coupled and uncoupled 
-diffusion is negligible in normal diffusion but significant in fractional diffusion.  
-      Both the theoretical curves from the coupled and uncoupled fractional phase diffusion become 
-narrower when the fractional derivative parameter 𝛼 decreases.  The overlapped curves in Figure 2 imply 
-the fractional diffusion results reduce to the normal diffusion results when 𝛼 = 1. While, Figure 3 shows 
-the changes in coupled and uncoupled fractional diffusion among different fractional derivative orders, 𝛼 
-= 1, 0.9, 0.75 and 0.5. The smaller the 𝛼  is, the broader the NMR peak is.  Additionally, the middle part and 
-the end part of the fractional diffusion curves have different features: the middle part is a narrow peak, 
-while the end parts are broad shoulders. The narrower peak could come from fast exchange time, while 
-broader end shoulders come from slow exchange. In the view of the traditional model, this could be 
-interpreted as a bimodal exchange. However, both the fast and slow exchange times come from the same 
-heavy-tailed time distribution.  
-     The diffusion method proposed here shows excellent results in the fast exchange range, but it 
-encounters challenges in slow exchange.  This difficulty in slow exchange results from that the diffusion 
-limit is not met because the experimental time window in NMR is not infinite. It requires further effort to 
-overcome the hurdle.   The current method can be combined with other anomalous diffusion models, such 
-as the fractal derivative [28,29,30]. Further research is needed to understand and apply the models, 
-
-12 
- 
-particularly the fractional diffusion model, and to extend the current method for multiple sites and unequal 
-population exchange.   
- 
- 
- 
-0
-0.002
-0.004
-0.006
-0.008
-0.01
-0.012
--150
--100
--50
-0
-50
-100
-150
-phasediff_tau_2overkex_kex2dw_beta_1_122722 3:20:13 PM 12/29/2022
-Uncoupled_normal_diffusion
-Uncoupled_fractional_diffusion
-' =  1/  =
-S()
-/2  (Hz)
-a
- 
-0
-0.002
-0.004
-0.006
-0.008
-0.01
-0.012
--150
--100
--50
-0
-50
-100
-150
-phasediff_tau_2overkex_kex2dw_beta_1_122722
-Coupled_Normal_diffusion
-Coupled_fractional_diffusion
-S()
-/2  (Hz)
-b
-' =  1/  =
- 
- 
-Fig. 2  The fractional diffusion results reduce to the normal diffusion results when 𝛼 = 1, and ∆𝜔/2𝜋 = 100 Hz. 
- 
-0
-0.005
-0.01
-0.015
-0.02
--100
--50
-0
-50
-100
-Uncoupled Fractional Diffusion
- = 
- = 
- = 
- = 
-S()
-a
-/2  (Hz)
-' =  1   =
-' =  1/
-0
-0.005
-0.01
-0.015
-0.02
--100
--50
-0
-50
-100
-Coupled Fractional Diffusion
- = 
- = 
- = 
- = 
-S()
-b
-/2  (Hz)
-' =  1/
- 
- 
-Fig. 3  The changes in coupled and uncoupled fractional diffusion among different fractional derivative orders, 𝛼 = 1, 
-0.9, 0.75 and 0.5, and ∆𝜔/2𝜋 = 100 Hz. 
- 
- 
- 
- 
- 
-
-13 
- 
-5. Conclusion 
-This paper proposes a phase diffusion method to describe the chemical exchange NMR spectrum. The 
-major conclusions are summarized in the following: 
-1. This method directly analyzes the spin system evolution in phase space rather than real space used 
-by most other traditional models.   
-2. The line shape difference between coupled and uncoupled phase diffusion is not obvious in 
-normal diffusion but significant in fractional diffusion.  
-3. There is a significant difference in the line shape between the normal and fractional diffusions. 
-4. Unlike the traditional method, the exchange time constant can follow certain types of distributions. 
-Additionally, the exchange time constant is two times faster based on the monoexponential time 
-distribution than that obtained by the traditional model.  
-5. The method could be extended to multiple sites and unequal population chemical exchange. 
-Furthermore, this phase diffusion method could be combined with other phase diffusion equations 
-in relaxation and PFG diffusion to deal with more complicated scenarios. 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
-
-14 
- 
-References 
- 
-1. A. Abragam, Principles of Nuclear Magnetism, Clarendon Press, Oxford, 1961. 
-2. C. P. Slichter. Principles of magnetic resonance, Springer series in Solid‐State Sciences, Vol. 
-1, Ed by M. Cardoua, P. Fulde and H. J. Queisser, Springer‐Verlag, Berlin (1978). 
-3. A. G. Palmer, H. Koss, Chapter Six - Chemical Exchange, Editor(s): A. J. Wand, Methods in 
-Enzymology, Academic Press, Volume 615, 2019, 177-236. 
-4. J. I. Kaplan, G. Fraenkel, NMR of Chemically Exchanging Systems, Academic Press, New 
-York, 1980. 
-5. C. S. Johnson,  Chemical rate processes and magnetic resonance, Adv. Magn. Reson. 1, 33 
-(1965). 
-6. N. Daffern, C. Nordyke, M. Zhang, A. G. Palmer, J. E. Straub. Dynamical Models of Chemical 
-Exchange in Nuclear Magnetic Resonance Spectroscopy, The Biophysicist 3(1) (2022): 13-34 
-7. J. M. Schurr, B. S. Fujimoto, R. Diaz, B. H. Robinson. 1999. Manifestations of slow site exchange 
-processes in solution NMR: a continuous Gaussian exchange model, J. Magn. Reson. 140 
-(1999) 404–431. 
-8. G. Lin, A. A. Jones. A lattice model for the simulation of one and two dimensional 129Xe 
-exchange spectra produced by translational diffusion, Solid State Nuclear Magnetic 
-Resonance. 26(2) (2004) 87-98. 
-9. R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin. Mittag-Leffler Functions, Related Topics 
-and Applications; Springer: Berlin, 2014. 
-10. T. Sandev, Z. Tomovsky, Fractional equations and models. Theory and applications, Springer 
-Nature Switzerland AG, Cham, Switzerland, 2019. 
-11. R. Kohlrausch, Theorie des elektrischen Rückstandes in der Leidner Flasche, Annalen der 
-Physik und Chemie. 91 (1854) 179–213. 
-12. G. Williams, D. C. Watts, Non-Symmetrical Dielectric Relaxation Behavior Arising from a 
-Simple Empirical Decay Function, Transactions of the Faraday Society 66 (1970) 80–85. 
-13. T. R. Lutz, Y. He, M. D. Ediger, H. Cao, G. Lin, A. A. Jones, Macromolecules 36 (2003) 1724. 
-14. E. Krygier, G. Lin, J. Mendes, G. Mukandela, D. Azar, A.A. Jones, J. A.Pathak, R. H. Colby, S. 
-K. Kumar, G. Floudas, R. Krishnamoorti, R. Faust, Macromolecules 38 (2005), 7721. 
-15. G. Lin, General pulsed-field gradient signal attenuation expression based on a fractional 
-integral modified-Bloch equation, Commu Nonlinear Sci. Numer. Simul. 63 (2018) 404-420.. 
-16. G. Lin, An effective phase shift diffusion equation method for analysis of PFG normal and 
-fractional diffusions, J. Magn. Reson. 259 (2015) 232–240. 
-17. G. Lin, Describing NMR relaxation by effective phase diffusion equation, Communications in 
-Nonlinear Science and Numerical Simulation. 99 (2021) 105825. 
-18. R. L. Magin, Weiguo Li, M. P. Velasco, J. Trujillo, D. A. Reiter, A. Morgenstern, R. G. Spencer, 
-Anomalous NMR relaxation in cartilage matrix components and native cartilage: Fractional-
-order models, J. Magn. Reson. 210 (2011)184-191. 
-
-15 
- 
- 
-19. T. Zavada, N. Südland, R. Kimmich, T.F. Nonnenmacher, Propagator representation of 
-anomalous diffusion: The orientational structure factor formalism in NMR, Phys. Rev. 60 
-(1999) 1292-1298. 
-20. G. Lin, Describe NMR relaxation by anomalous rotational or translational diffusion, Commu 
-Nonlinear Sci. Numer. Simul. 72 (2019) 232. 
-21. W. Wyss, J. Math. Phys. (1986) 2782-2785. 
-22. A. I. Saichev, G.M. Zaslavsky, Fractional kinetic equations: Solutions and applications. Chaos 
-7 (1997) 753–764. 
-23. R. Gorenflo, F. Mainardi, Fractional Diffusion Processes: Probability Distributions and 
-Continuous Time Random Walk, in: Lecture Notes in Physics, No 621, Springer-Verlag, Berlin, 
-2003, pp. 148–166. 
-24. F. Mainardi, Yu. Luchko, G. Pagnini, The fundamental solution of the space-time fractional 
-diffusion equation, Fract. Calc. Appl. Anal. 4 (2001) 153–192. 
-25. Y. Povstenko. Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, 
-Birkhäuser, New York, 2015. 
-26. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics 
-approach, Phys. Rep. 339 (2000) 1–77. 
-27. G. Germano, M. Politi, E. Scalas, R. L. Schilling. Phys Rev E 2009;79:066102. 
-28. W. Chen, Time space fabric underlying anomalous diffusion, Chaos Solitons Fractals 28 (2006) 
-923. 
-29. W. Chen, H. Sun, X. Zhang, D. Korošak, Anomalous diffusion modeling by fractal and 
-fractional derivatives, Comput. Math. Appl. 5 (5) (2010) 1754–1758. 
-30. H. Sun, M.M. Meerschaert, Y. Zhang, J. Zhu, W. Chen, A fractal Richards’ equation to capture 
-the non-Boltzmann scaling of water transport in unsaturated media, Adv. Water Resour. 52 
-(2013) 292–295. 
-

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-Multiple phenotypes in HL60 leukemia cell population
-Yue Wang1,2, Joseph X. Zhou3, Edoardo Pedrini3, Irit Rubin3, May Khalil3, Hong
-Qian2, and Sui Huang3
-1Department of Computational Medicine, University of California, Los Angeles,
-California, United States of America
-2Department of Applied Mathematics, University of Washington, Seattle,
-Washington, United States of America
-3Institute for Systems Biology, Seattle, Washington, United States of America
-Abstract
-Recent studies at individual cell resolution have revealed phenotypic heterogene-
-ity in nominally clonal tumor cell populations. The heterogeneity affects cell growth
-behaviors, which can result in departure from the idealized exponential growth. Here
-we measured the stochastic time courses of growth of an ensemble of populations of
-HL60 leukemia cells in cultures, starting with distinct initial cell numbers to capture
-the departure from the exponential growth model in the initial growth phase. De-
-spite being derived from the same cell clone, we observed significant variations in the
-early growth patterns of individual cultures with statistically significant differences in
-growth kinetics and the presence of subpopulations with different growth rates that
-endured for many generations. Based on the hypothesis of existence of multiple inter-
-converting subpopulations, we developed a branching process model that captures the
-experimental observations.
-1
-Introduction
-Cancer has long been considered a genetic disease caused by oncogenic mutations in so-
-matic cells that confer a proliferation advantage. According to the clonal evolution theory,
-accumulation of random genetic mutations produces cell clones with cancerous cell phe-
-notype. Specifically, cells with the novel genotype(s) may display increased proliferative
-fitness and gradually out-grow the normal cells, break down tissue homeostasis and gain
-other cancer hallmarks [15]. In this view, a genetically distinct clone of cells dominates the
-cancer cell population and is presumed to be uniform in terms of the phenotype of indi-
-vidual cells within an isogenic clone. In this traditional paradigm, non-genetic phenotypic
-variation within one clone is not taken into account.
-1
-arXiv:2301.03782v1  [q-bio.PE]  10 Jan 2023
-
-With the advent of systematic single-cell resolution analysis, however, non-genetic cell
-heterogeneity within clonal (cancer) cell populations is found to be universal [33]. This
-feature led to the consideration of the possibility of biologically (qualitatively) distinct
-(meta)stable cell subpopulations due to gene expression noise, representing intra-clonal
-variability of features beyond the rapid random micro-fluctuations.
-Hence, transitions
-between the subpopulations, as well as heterotypic interactions among them may influence
-cell growth, migration, drug resistance, etc. [39, 13, 9]. Thus, an emerging view is that
-cancer is more akin to an evolving ecosystem [11] in which cells form distinct subpopulations
-with persistent characteristic features that determine their mode of interaction, directly
-or indirectly via competition for resources [10, 36]. However, once non-genetic dynamics
-is considered, cell “ecology” differs fundamentally from the classic ecological system in
-macroscopic biology: the subpopulations can reversibly switch between each other whereas
-species in an ecological population do not convert between each other [7]. This affords
-cancer cell populations a remarkable heterogeneity, plasticity and evolvability, which may
-play important roles in their growth and in the development of resistance to treatment
-[30].
-Many new questions arise following the hypothesis that phenotypic heterogeneity and
-transitions between phenotypes within one genetic clone are important factors in cancer.
-Can tumors arise, as theoretical considerations indicate, because of a state conversion
-(within one clone) to a phenotype capable of faster, more autonomous growth as opposed
-to acquisition of a new genetic mutation that confers such a selectable phenotype [55,
-1, 18, 34, 33, 56, 23, 41]? Is the macroscopic, apparently sudden outgrowth of a tumor
-driven by a new fastest-growing clone (or subpopulation) taking off exponentially, or due
-to the cell population reaching a critical mass that permits positive feedback between its
-subpopulations that stimulates outgrowth, akin to a collectively autocatalytic set [17]?
-Should therapy target the fastest growing subpopulations, or target the interactions and
-interconversions of cancer cells?
-At the core of these deliberations is the fundamental question on the mode of tumor
-cell population growth that now must consider the influence of inherent phenotypic hetero-
-geneity of cells and the non-genetic (hence potentially reversible) inter-conversion of cells
-between the phenotypes that manifest various growth behaviors and the interplay between
-these two modalities.
-Traditionally tumor growth has been described as following an exponential growth law,
-motivated by the notion of uniform cell division rate for each cell, i.e. a first order growth
-kinetics [29]. But departure from the exponential model has long been noted. To better fit
-experimental data, two major modifications have been developed, namely the Gompertz
-model and the West law model [53]. While no one specific model can adequately describe
-any one tumor, each model highlights certain aspects of macroscopic tumor kinetics, mainly
-the maximum size and the change in growth rate at different stages. These models however
-are not specifically motivated by cellular heterogeneity. Assuming non-genetic heterogene-
-ity with transitions between the cell states, the population behavior is influenced by many
-2
-
-intrinsic and extrinsic factors that are both variable and unpredictable at the single-cell
-level. Thus, unlike macroscopic population dynamics [43], tumor growth cannot be ad-
-equately captured by a deterministic model, but a stochastic cell and population level
-kinetic model is more realistic.
-Using stochastic processes in modeling cell growth via clonal expansion has a long
-history [54]. An early work is the Luria-Delbr¨uck model, which assumes cells grow deter-
-ministically, with wildtype cells mutating and becoming (due to rare and quasi-irreversible
-mutations) cells with a different phenotype randomly [28]. Since then, there have been
-many further developments that incorporate stochastic elements into the model, such as
-those proposed by Lea and Coulson [25], Koch [22], Moolgavkar and Luebeck [27], and
-Dewanji et al. [8]. We can find various stochastic processes: Poisson processes [2], Markov
-chains [14], and branching processes [19], or even random sums of birth-death processes [8],
-all playing key roles in the mathematical theories of cellular clonal growth and evolution.
-These models have been applied to clinical data on lung cancer [31], breast cancer [37],
-and treatment of cancer [38].
-At single-cell resolution, another cause for departure from exponential growth is the
-presence of positive (growth promoting) cell-cell interactions (Allee effect) in the early
-phase of population growth, such that cell density plays a role in stimulating division,
-giving rise to the critical mass dynamics [20, 24].
-To understand the intrinsic tumor growth behavior (change of tumor volume over time)
-it is therefore essential to study tumor cell populations in culture which affords detailed
-quantitative analysis of cell numbers over time, unaffected by the tumor microenvironment,
-and to measure departure from exponential growth.
-This paper focuses on stochastic
-growth of clonal but phenotypically heterogeneous HL60 leukemia cells with near single-cell
-sensitivities in the early phase of growth, that is, in sparse cultures. We and others have in
-the past years noted that at the level of single cells, each cell behaves akin to an individual,
-differently from another, which can be explained by the slow correlated transcriptome-wide
-fluctuations of gene expression [4, 26]. Given the phenotypic heterogeneity and anticipated
-functional consequences, grouping of cells is necessary. Such classification would require
-molecular cell markers for said functional implication, but such markers are often difficult
-to determine a priori.
-Here, since most pertinent to cancer biology, we directly use a
-functional marker that is of central relevance for cancer: cell division, which maps into cell
-population growth potential — in brief “cell growth”.
-Therefore, we monitored longitudinally the growth of cancer cell populations seeded at
-very small numbers of cells (1, 4, or 10 cells) in statistical ensembles of microcultures (wells
-on a plate of wells). We found evidence that clonal HL60 leukemia cell populations contain
-subpopulations that exhibit diverse growth patterns.
-Based on statistical analysis, we
-propose the existence of three distinctive cell phenotypic states with respect to cell growth.
-We show that a branching process model captures the population growth kinetics of a
-population with distinct cell subpopulations. Our results suggest that the initial phase cell
-growth (“take-off” of a cell culture) in the HL60 leukemic cells is predominantly driven by
-3
-
-the fast-growing cell subpopulation. Reseeding experiments revealed that the fast-growing
-subpopulation could maintain its growth rate over several cell generations, even after the
-placement in a new environment. Our observations underscore the need to not only target
-the fast-growing cells but also the transition to them from the other cell subpopulations.
-2
-Results
-2.1
-Experiment of the cell population growth from distinct initial cell
-numbers.
-To expose the variability of growth kinetics as a function of initial cell density N0 (“initial
-seed number”), HL60 cells were sorted into wells of a 384-well plate (0.084 cm2 area)
-to obtain “statistical ensembles” of replicate microcultures (wells) of the same condition,
-distinct only by N0. Based on prior titration experiments to determine ranges of interest
-for N0 and statistical power, for this experiment we plated 80 wells with N0 = 10 cells
-(N0 = 10-cell group), 80 wells with N0 = 4 cells (N0 = 4-cell group), and 80 wells with
-N0 = 1 cell (N0 = 1-cell group). Cells were grown in the same conditions for 23 days (for
-details of cell culture and sorting, see the Methods section). Digital images were taken
-every 24 hours for each well from Day 4 on, and the area occupied by cells in each well
-was determined using computational image analysis. We had previously determined that
-one area unit equals approximately 500 cells. This is consistent and readily measurable
-because the relatively rigid and uniformly spherical HL60 cells grow as a non-adherent
-“packed” monolayer at the bottom of the well. Note that we are interested in the initial
-exponential growth (and departure from it) and not in the latter phases when the culture
-becomes saturated as has been the historical focus of analysis (see Introduction).
-Wells that have reached at least 5 area units were considered for the characterization
-of early phase (before plateau) growth kinetics by plotting the areas in logarithmic scale as
-a function of time (Fig. 1). All the N0 = 10-cell wells required 3.6-4.6 days to grow from
-5 area units to 50 area units (mean=4.05, standard deviation=0.23). For the N0 = 1-cell
-wells, we observed a diversity of behaviors. While some of the cultures only took 3.5-5
-days to grow from 5 area units to 50 area units, others needed 6-7.2 days (mean=5.02,
-standard deviation=0.75). The N0 = 4-cell wells had a mean=4.50 days and standard
-deviation=0.44 to reach that same population size.
-To examine the exponential growth model, in Fig. 2 (left panel), we plotted the per
-capita growth rate versus cell population size, where each point represents a well (popu-
-lation) at a time point. As expected, as the population became crowded, the growth rate
-decreased toward zero. But in the earlier phase, many populations in the N0 = 1-cell group
-had a lower per capita growth rate than those in the N0 = 10-cell group, even at the same
-population size – thus departing from the expected behavior of exponential growth. The
-weighted Welch’s t-test showed that the difference in these growth rates was significant
-(see the Methods section).
-4
-
-While qualitative differences in the behaviors of cultures with different initial seeding
-cell numbers N0 can be expected for biological reasons (see below), in the elementary
-exponential growth model, the difference of growth rate should disappear when populations
-with distinct seeding numbers are aligned for the same population size that they have
-reached as in Fig. 2. A simple possibility is that the deviations of expected growth rates
-emanate from difference in cell-intrinsic properties.
-Some cells grew faster, with a per
-capita growth rate of 0.6 ∼ 0.9 (all N0 = 10-cell wells and some N0 = 1-cell wells), while
-some cells grew slower, with a per capita growth rate of 0.3 ∼ 0.5 (some of the N0 = 1-
-cell wells). In other words, there is intrinsic heterogeneity in the cell population that is
-not “averaged out” in the culture with low N0, and the sampling process exposes these
-differences between the cells that appear to be relatively stable.
-To illustrate the inherent diversity of initial growth rates, in Fig. 3 (left panel), we
-display the daily cell-occupied areas plotted on a linear scale starting from Day 4. All wells
-with seed of N0 = 10 or N0 = 4 cells grew exponentially. Among the N0 = 1-cell wells,
-14 populations died out. Four wells in the N0 = 1-cell group had more than 10 cells on
-Day 8 but never grew exponentially, and had fewer than 1000 cells after 15 days (on Day
-23). For these non-growing or slow-growing N0 = 1-cell wells, the per capita growth rate
-was 0 ∼ 0.2. In comparison, all the N0 = 10-cell wells needed at most 15 days to reach
-the carrying capacity (around 80 area units, or 40000 cells). See Table 1 for a summary of
-the N0 = 1-cell group’s growth patterns. This behavior is not idiosyncratic to the culture
-system because they recapitulate a pilot experiment performed in the larger scale format
-of 96-well plates (not shown).
-From the above experimental observations, we asserted that there might be at least
-three stable cell growth phenotypes in a population: a fast type, whose growth rate was
-0.6 ∼ 0.9/day for non-crowded conditions; a moderate type, whose growth rate was 0.3 ∼
-0.5/day for non-crowded conditions; and a slow type, whose growth rate was 0 ∼ 0.2/day
-for the non-crowded population.
-The graphs of Fig. 3 also revealed other phenomena of growth kinetics: (1) Most
-N0 = 4-cell wells plateaued by Day 14 to Day 17, but some lagged significantly behind.
-(2) Similarly, four wells in the N0 = 1-cell group exhibited longer lag-times before the
-exponential growth phase, and never reached half-maximal cell numbers by Day 23. These
-outliers reveal intrinsic variability and were taken into account in the parameter scanning
-(see the Methods section).
-2.2
-Reseeding experiments revealing the enduring intrinsic growth pat-
-terns.
-When a well in the N0 = 1-cell group had grown to 10 cells, population behavior was
-still different from those in the N0 = 10-cell group at the outset. In view of the spate of
-recent results revealing phenotypic heterogeneity, we hypothesized that the difference was
-cell-intrinsic as opposed to being a consequence of the environment (e.g., culture medium
-5
-
-Growth pattern
-Well label
-Day 1
-Day 8
-Day 14
-Day 19
-Day 23
-No growth,
-extinction
-162,167,170,176,
-177,179,182,183,
-186,201,234,236,
-239,240
-1
-<10
-<10
-∼0
-Empty
-Slow growth,
-no exponential
-growth
-165
-1
-89
-∼300
-∼350
-∼500
-166
-1
-36
-∼110
-∼120
-∼150
-178
-1
-43
-∼140
-∼170
-∼200
-211
-1
-16
-∼90
-∼200
-∼400
-Delayed
-exponential
-growth
-163
-1
-12
-∼130
-∼300
-∼5000
-181
-1
-44
-∼270
-∼550
-∼5500
-193
-1
-25
-∼200
-∼800
-∼9000
-204
-1
-21
-∼100
-∼600
-∼6000
-Normal
-exponential
-growth
-200 and
-many others
-1
-∼130
-∼20000
-∼40000
-(full)
-∼40000
-(full)
-Table 1: The population of some wells in the N0 = 1-cell group in the growth experiment
-with different initial cell numbers, where ∼ meant approximate cell number. These wells
-illustrated different growth patterns from those wells starting with N0 = 10 or N0 = 4
-cells. Such differences implied that cells from wells with different initial cell numbers were
-essentially different.
-6
-
-Time (days) to
-reach one half area
-11
-12
-13
-14
-15
-16–20
->20
-Faster wells
-26
-2
-1
-2
-1
-0
-0
-Slower wells
-0
-0
-0
-1
-1
-25
-5
-Table 2: The distribution of time needed for each well to reach the “half area” population
-size in the reseeding experiment. We reseeded equal numbers of cells that grew faster (from
-a full well) and cells that grew slower (from a half-full well), and cultivated them under the
-same new fresh medium environment to compare their intrinsic growth rates. The results
-showed that faster growing cells, even reseeded, still grew faster.
-in N0 = 1 vs N0 = 10 -cell wells).
-To test our hypothesis and exclude differences in the culture environment as determi-
-nants of growth behavior, we reseeded the cells that exhibited the different growth rates
-in fresh cultures. We started with a number of N0 = 1-cell wells. After a period of almost
-3 weeks, again some wells showed rapid proliferation, with cells covering the well, while
-others were half full and yet others wells were almost empty. We collected cells from the full
-and half-full wells and reseeded them into 32 wells each (at about N0 = 78 cells per well).
-These 64 wells were monitored for another 20 days. We found that most wells reseeded
-from the full well took around 11 days to reach the population size of a half-full well, while
-most wells reseeded from the half-full well required around 16 ∼ 20 days to reach the same
-half full well population size. Five wells reseeded from the half-full wells were far from even
-reaching half full well population size by Day 20 (see Table 2). Permutation test showed
-that this difference in growth rate was significant (see the Methods section).
-This reseeding experiment shows that the difference in growth rate was maintained
-over multiple generations, even after slowing down in the plateau phase (full well) and
-was maintained when restarting a microculture at low density in fresh medium devoid of
-secreted cell products. Therefore, it is plausible that there exists endogenous heterogeneity
-of growth phenotypes in the clonal HL60 cell line and that these distinct growth phenotypes
-are stable for at least 15 ∼ 20 cell generations.
-2.3
-Quantitative analysis of experimental results.
-In the experiments with different initial cell numbers N0, we observed at least three patterns
-with different growth rates, and the reseeding showed that these growth patterns were
-endogenous to the cells. Therefore, we propose that each growth pattern discussed above
-corresponded to a cell phenotype that dominated the population: fast, moderate, and slow.
-In the initial seeding of cells that varies N0, the cells were randomly chosen (by FACS);
-thus, their intrinsic growth phenotypes were randomly distributed. During growth, the
-population of a well would be dominated by the fastest type that existed in the seeding
-cells, thus qualitatively, we have following scenarios: (1) A well in the N0 = 10-cell group
-7
-
-almost certainly had at least one initial cell of fast type, and the population would be
-dominated by fast type cells. Different wells had almost the same growth rate, reaching
-saturation at almost the same time. (2) For an N0 = 1-cell well, if the only initial cell is of
-the fast type, then the population has only the fast type, and the growth pattern will be
-close to that of N0 = 10-cell wells. If the only initial cell is of the moderate type, then the
-population could still grow exponentially, but with a slower growth rate. This explains why
-after reaching 5 area units, many but not all N0 = 1-cell wells were slower than N0 = 10-
-cell wells. (3) Moreover, in such an N0 = 1-cell well with a moderate type initial cell, the
-cell might not divide quite often during the first few days due to randomness of entering
-the cell cycle. This would lead to a considerable delay in entering the exponential growth
-phase. (4) By contrast, for an N0 = 1-cell well with a slow type initial cell, the growth rate
-could be too small, and the population might die out or survive without ever entering the
-exponential growth phase in duration of the experiment. (5) Most N0 = 4-cell wells had at
-least one fast type initial cell, and the growth pattern was the same as N0 = 10-cell wells.
-A few N0 = 4-cell wells only had moderate and slow cells, and thus had slower growth
-patterns.
-The above verbal argument is shown in Fig. 4 and entails mathematical modeling with
-the appropriate parameters that relate the relative frequency of these cell types in the
-original population, their associated growth and transition rates to examine whether it
-explains the data.
-2.4
-Branching process model.
-To construct a quantitative dynamical model to recapitulate the growth dynamics differ-
-ences from cell populations with distinct initial seed cell numbers N0, and three intrinsic
-types of proliferation behaviors, we used a multi-type discrete-time branching process.
-The traditional method of population dynamics based on ordinary differential equation
-(ODE), which is deterministic and has continuous variables, is not suited when the cell
-population is small as is the case for the earliest stage of proliferation from a few cells
-being studied in our experiments. Deterministic models are also unfit because with such
-small populations and measurements at single-cell resolution, stochasticity in cell activity
-does not average out. The nuanced differences between individual cells cannot be captured
-by a different deterministic mechanism of each individual cell, and the only information
-available is the initial cell number. Thus, the unobservable nuances between cells are taken
-care of by a stochastic model.
-Given the small populations, our model should be purely stochastic, without determin-
-istic growth. The focus is the concrete population size of a finite number (three) of types,
-thus Poisson processes are not suitable. Markov chains can partially describe the propor-
-tions under some conditions [47], but population sizes are known, not just their ratios,
-therefore Markov chains are not necessary. Even the lifted Markov chains [48] and random
-dynamical systems [52] are not applicable in this situation, since the population should be
-8
-
-non-negative. Branching processes can describe the population size of multiple types with
-symmetric and asymmetric division, transition, and death [19]. Also, the parameters can
-be temporally and spatially inhomogeneous, which is convenient. Therefore, we utilized
-branching processes in our model.
-In the branching process, each cell during each time interval independently and ran-
-domly chooses a behavior: division, death, or stagnation in the quiescent state, whose rates
-depend on the cell growth type. Denoting the growth rate and death rate of the fast type
-by gF and dF respectively, and the population size of fast type cells on Day n by F(n), the
-population at Day n + 1 is:
-F(n + 1) =
-F(n)
-�
-i=1
-Ai,
-where Ai for different i are independent. Ai represents the descendants of a fast type cell i
-after one day. It equals 2 with probability gF, 0 with probability dF, and 1 with probability
-1 − gF − dF. Therefore, given F(n), the distribution of F(n + 1) is:
-P[F(n + 1) = N] =
-�
-2a+b=N
-F(n)!
-a!b![F(n) − a − b]!ga
-Fd[F(n)−a−b]
-F
-(1 − gF − dF)b,
-where the summation is taken for all non-negative integer pairs (a, b) with 2a + b = N.
-Moderate and slow types evolve similarly, with their corresponding growth rates gM, gS,
-and death rates dM, dS.
-As shown in Fig. 2, the growth rates gF, gM, and gS should be decreasing functions of
-the total population. In our model, we adopted a quadratic function.
-We performed a parameter scan to show that our model could reproduce experimental
-phenomena for a wide range of model parameters (see details in Table 3).
-The simulation results are shown on the right panels of Figs. 1–3, in comparison with
-the experimental data in the left. Our model qualitatively captured the growth patterns
-of groups with different initial seeding cell numbers. For example, in Fig. 2, when wells
-were less than half full (cell number < 20000), most wells in the N0 = 10-cell group grew
-faster than the N0 = 1-cell group even when they had the same cell number. In Fig. 3,
-all wells in the N0 = 10-cell group in our model grew quickly until saturation. Similar to
-the experiment, some wells in the N0 = 1-cell group in our model never grew, while some
-began to take off very late.
-In our model, the high extinction rate in the N0 = 1-cell group (14/80) was explained
-as “bad luck” at the early stage, since birth rate and death rate were close, and a cell could
-easily die without division. Another possible explanation for such a difference in growth
-rates was that the population would be 10 small colonies when starting from 10 initial cells,
-while starting from 1 initial cell, the population would be 1 large colony. With the same
-area, 10 small colonies should have a larger total perimeter, thus larger growth space and
-larger growth rate than that of 1 large colony. However, we carefully checked the photos,
-9
-
-Parameters
-Appearance of experimental phenomena
-pF
-pM
-pS
-d
-g0
-r
-Feature 1
-Feature 2
-Feature 3
-Feature 4
-0.4
-0.4
-0.2
-0.01
-0.5
-0.1
-Yes
-Yes
-Yes
-Yes
-0.4
-0.4
-0.2
-0
-0.5
-0.1
-Yes
-Yes
-Yes
-Yes
-0.4
-0.4
-0.2
-0.05
-0.5
-0.1
-Yes
-Yes
-Yes
-Yes
-0.4
-0.4
-0.2
-0.1
-0.5
-0.1
-No
-Yes
-Yes
-No
-0.4
-0.4
-0.2
-0.01
-0.45
-0.1
-Yes
-Yes
-Yes
-Yes
-0.4
-0.4
-0.2
-0.01
-0.6
-0.1
-Yes
-Yes
-Yes
-Yes
-0.4
-0.4
-0.2
-0.01
-0.4
-0.1
-Yes
-Yes
-Yes
-No
-0.4
-0.4
-0.2
-0.01
-0.5
-0.05
-Yes
-Yes
-Yes
-Yes
-0.4
-0.4
-0.2
-0.01
-0.5
-0
-Yes
-Yes
-Yes
-Yes
-0.4
-0.4
-0.2
-0.01
-0.5
-0.15
-Yes
-Yes
-Yes
-No
-0.4
-0.4
-0.2
-0.01
-0.5
-0.2
-No
-Yes
-Yes
-No
-0.3
-0.5
-0.2
-0.01
-0.5
-0.1
-Yes
-Yes
-Yes
-Yes
-0.5
-0.3
-0.2
-0.01
-0.5
-0.1
-Yes
-Yes
-Yes
-Yes
-0.4
-0.5
-0.1
-0.01
-0.5
-0.1
-Yes
-Yes
-Yes
-Yes
-0.4
-0.3
-0.3
-0.01
-0.5
-0.1
-Yes
-Yes
-Yes
-Yes
-0.5
-0.4
-0.1
-0.01
-0.5
-0.1
-Yes
-Yes
-Yes
-Yes
-0.3
-0.4
-0.3
-0.01
-0.5
-0.1
-Yes
-Yes
-Yes
-Yes
-0.1
-0.1
-0.8
-0.01
-0.5
-0.1
-No
-Yes
-Yes
-No
-0.5
-0.5
-0
-0.01
-0.5
-0.1
-Yes
-Yes
-No
-Yes
-0
-0.5
-0.5
-0.01
-0.5
-0.1
-No
-Yes
-Yes
-Yes
-0.5
-0
-0.5
-0.01
-0.5
-0.1
-Yes
-No
-Yes
-No
-1
-0
-0
-0.01
-0.5
-0.1
-Yes
-No
-No
-No
-Table 3: Performance of our model with different parameters. Here we adjusted the param-
-eters of our model in a wide range and observed whether the model could still reproduce
-four important “features” in the experiment. This parameter scan showed that our model
-is robust under perturbations on parameters. Here pF, pM, pS are the probabilities that an
-initial cell is of fast, moderate, or slow type; d is the death rate; g0 is the growth factor;
-r is the range of the random modifier. See the Methods section for explanations of these
-parameters.
-Feature 1, all wells in the N0 = 10-cell group were saturated; Feature 2,
-presence of late-growing wells in the N0 = 1-cell group; Feature 3, presence of non-growing
-wells in the N0 = 1-cell group; Feature 4, different growth rates at the same population
-size between the N0 = 10-cell group and the N0 = 1-cell group.
-10
-
-and found that almost all wells produced 1 large colony with nearly the same shape, and
-there was no significant relationship between colony perimeter and growth rate.
-3
-Discussion
-As many recent single-cell level data have shown, a tumor can contain multiple distinct
-subpopulations engaging in interconversions and interactions among them that can in-
-fluence cancer cell proliferation, death, migration, and other features that contribute to
-malignancy [33, 55, 1, 18, 34, 56, 20, 24, 5, 32, 6]. Presence of these two intra-population
-behaviors can be manifest as departure from the elementary model of exponential growth
-[35] (in the early phase of population growth, far away from carrying capacity of the culture
-environment which is trivially non-exponential). The exponential growth model assumes
-uniformity of cell division rates across all cells (hence a population doubling rate that is
-proportional to a given population size N(t)) and the absence of cell-cell interactions that
-affect cell division and death rates. Investigating the “non-genetic heterogeneity” hypoth-
-esis of cancer cells quantitatively is therefore paramount for understanding cancer biology
-but also for elementary principles of cell population growth.
-As an example, here we showed that clonal cell populations of the leukemia HL60
-cell line are heterogeneous with regard to growth behaviors of individual cells that can
-be summarized in subpopulations characterized by a distinct intrinsic growth rates which
-were revealed by analysis of the early population growth starting with microcultures seeded
-with varying (low) cell number N0.
-Since we have noted only very weak effect of cell-cell interactions on cell growth be-
-haviors (Allee effect) in this cell line (as opposed to another cell tumor cell line in which
-we found that departure from exponential growth could be explained by the Allee effect
-[20]), we focused on the very presence among HL60 cells of subpopulations with distinct
-proliferative capacity as a mechanism for the departure of the early population growth
-curve from exponential growth.
-The reseeding experiment demonstrated that the characteristic growth behaviors of
-subpopulations could be inherited across cell generations and after moving to a new envi-
-ronment (fresh culture), consistent with long-enduring endogenous properties of the cells.
-This result might be explained by cells occupying distinct stable cell states (in a multi-
-stable system). Thus, we introduced multiple cell types with different growth rates in our
-stochastic model. Specifically, in a branching process model, we assumed the existence
-of three types: fast, moderate, and slow cells. The model we built could replicate the
-key features in the experimental data, such as different growth rates at the same popula-
-tion size between the N0 = 10-cell group and the N0 = 1-cell group, and the presence of
-late-growing and non-growing wells in the N0 = 1-cell group.
-While we were able to fit the observed behaviors in which the growth rate depended not
-only on N(t) but also on N0, the existence of the three or even more cell types still needs
-11
-
-to be verified experimentally. For instance, statistical cluster analysis of transcriptomes of
-individual cells by single-cell RNA-seq [3] over the population may identify the presence
-of transcriptomically distinct subpopulations that could be isolated (e.g., after association
-with cell surface markers) and evaluated separately for their growth behaviors. We might
-apply inference methods on such sequencing data to determine the gene regulatory relations
-that lead to multiple phenotypes [50, 44], although the causal relationship might not always
-be determined [49]. Besides, since the existence of transposons might affect the growth
-rates, corresponding analysis should be conducted [21, 40].
-The central assumption of coexistence of multiple subpopulations in the cell line stock
-must be accompanied by the second assumption that there are transitions between these
-distinct cell populations. For otherwise, in the stock population the fastest growing cell
-would eventually outgrow the slow growing cells. Furthermore, one has to assume a steady-
-state in which the population of slow growing cells are continuously replenished from the
-population of fast-growing cells. Finally, we must assume that the steady-state proportions
-of the subpopulations are such that at low seeding wells with N0 = 1 cells, there is a sizable
-probability that a microculture receives cells from each of the (three) presumed subtypes of
-cells. The number of wells in the ensemble of replicate microcultures for each N0- condition
-has been sufficiently large for us to make the observations and inform the model, but a
-larger ensemble would be required to determine with satisfactory accuracy the relative
-proportions of the cell types in the parental stock population.
-Transitions might also have been happening during our experiment. For example, those
-late growing wells in the N0 = 1-cell group could be explained by such a transition: Initially,
-only slow type cells were present, but once one of these slow growing cells switched to the
-moderate type, an exponential growth ensued at the same rate that is intrinsic to that of
-moderate cells.
-If there are transitions, what is the transition rate? Our reseeding experiments are
-compatible with a relatively slow rate for interconversion of growth behaviors in that the
-same growth type was maintained across 30 generations. An alternative to the principle
-of transition at a constant intrinsic to each of the types of cells may be that transition
-is extrinsically determined. Specifically, the seeding in the “lone” condition of N0 = 1
-may induce a dormant state, that is a transition to a slower growth mode that is then
-maintained, on average over 30+ generations, with occasional return to the faster types
-that account for the delayed exponential growth. The lack of experimental data might be
-partially made up by inference methods [51].
-This model however would bring back the notion of “environment awareness”, or the
-principle of a “critical density” for growth implemented by cell-cell interaction (Allee effect)
-which we had deliberately not considered (see above) since it was not necessary. We do not
-exclude this possibility which could be experimentally tested as follows: Cultivate N0 = 1-
-cell wells for 20 days when the delayed exponential growth has happened in some wells,
-but then use the cells of those wells with fast-growing population (which should contain of
-the fast type) to restart the experiment, seeded at N0 = 10, 4, 1 cells. If wells with different
-12
-
-seeding numbers exhibit the same growth rates, then the growth difference in the original
-experiment is solely due to preexisting (slow interconverting) cell phenotypes. If now the
-N0 = 1-cell wells resumes the typical slow growth, this would indicate a density induced
-transition to the slow growth type. If cell-cell interaction needs to be taken into account,
-certain results in developmental biology might help, since they study the emergence of
-patterns through strong cell-cell interactions [46, 45, 42].
-In the spirit of Occam’s razor, and given the technical difficulty in separate experiments
-to demonstrate cell-cell interactions in HL60 cells, we were able to model the observed
-behaviors with the simplest assumption of cell-autonomous properties, including existence
-of multiple states (growth behaviors) and slow transitions between them but without cell
-density dependence or interactions.
-Taken together, we showed that one manifestation of the burgeoning awareness of ubiq-
-uitous cell phenotype heterogeneity in an isogenic cell population is the presence of distinct
-intrinsic types of cells that slowly interconvert among them, resulting in a stationary popu-
-lation composition. The differing growth rates of the subtypes and their stable proportions
-may be an elementary characteristic of a given population that by itself can account for the
-departure of early population growth kinetics from the basic exponential growth model.
-4
-Methods
-4.1
-Setup of growth experiment with different initial cell numbers.
-HL60 cells were maintained in IMDM wGln, 20% FBS(heat inactivated), 1% P/S at a
-cell density between 3 × 105 and 2.5 × 106 cells/ml (GIBCO). Cells were always handled
-and maintained under sterile conditions (tissue culture hood; 37◦C, 5% CO2, humidified
-incubator). At the beginning of the experiment, cells were collected, washed two times in
-PBS, and stained for vitality (Trypan blue GIBCO). The population of cells was first gated
-for morphology and then for vitality staining. Only Trypan negative cells were sorted (BD
-FACSAria II). The cells were sorted in a 384 well plate with IMDM wGln, 20% FBS(heat
-inactivated), and 1% P/S (GIBCO).
-Cell population growth was monitored using a Leica microscope (heated environmental
-chamber and CO2 levels control) with a motorized tray. Starting from Day 4, the 384
-well plate was placed inside the environmental chamber every 24 hours. The images were
-acquired in a 3 × 3 grid for each well; after acquisition, the 9 fields were stitched into a
-single image. Software ImageJ was applied to identify and estimate the area occupied by
-“entities” in each image. The area (proportional to cell number) was used to follow the
-cell growth.
-13
-
-4.2
-Setup of reseeding experiment for growth pattern inheritance.
-HL60 cells were cultivated for 3 weeks, and then we chose one full well and one half full
-well. We supposed the full well was dominated by fast type cells, and the half-full well
-was dominated by moderate type cells, which had lower growth rates. We reseeded cells
-from these two wells and cultivated them in two 96-well (rows A-H, columns 1-12) plates.
-In each plate, B2-B11, D2-D11, and F2-F11 wells started with 78 fast cells, while C2-C11,
-E2-E11, and G2-G11 wells started with 78 moderate cells. Rows A, H, columns 1, 12 had
-no cells and no media, and we found that wells in rows B, G, columns 2, 11, which were
-the outmost non-empty wells, evaporated much faster than inner wells. Therefore, the
-growth of cells in those wells was much slower than inner wells. Hence we only considered
-inner wells, where D3-D10 and F3-F10 started with fast cells, C3-C10 and E3-E10 started
-with moderate cells, namely 32 fast wells and 32 moderate wells in total.
-During the
-experiment, no media was added. Each day, we observed those wells to check whether
-their areas exceeded one-half of the whole well. The experiment was terminated after 20
-days.
-4.3
-Weighted Welch’s t-test.
-The weighted Welch’s t-test is used to test the hypothesis that two populations have equal
-mean, while sample values have different weights [12].
-Assume for group i (i = 1, 2),
-the sample size is Ni and the jth sample is the average of cj
-i independent and identically
-distributed variables. Let Xj
-i be the observed average for the jth sample. Set ν1 = N1 − 1,
-ν2 = N2 − 1. Define
-¯
-Xi
-W = (
-Ni
-�
-j=1
-Xj
-i cj)/(
-Ni
-�
-j=1
-)cj,
-s2
-i,W =
-Ni[�Ni
-j=1(Xj
-i )2cj]/(�Ni
-j=1 cj
-i) − Ni( ¯
-Xi
-W )2
-Ni − 1
-,
-t =
-¯
-X1
-W − ¯
-X2
-W
-�
-s2
-1,W
-N1 +
-s2
-2,W
-N2
-,
-ν =
-(
-s2
-1,W
-N1 +
-s2
-2,W
-N2 )2
-s4
-1,W
-N2
-1 ν1 +
-s4
-2,W
-N2
-2 ν2
-.
-If two populations have equal mean, then t satisfies the t-distribution with degree of freedom
-ν.
-The weighted Welch’s t-test was applied to the growth experiment with different initial
-cell numbers, in order to determine whether the growth rates during exponential phase
-14
-
-(5–50 area units) were different between groups. Here Xj
-i corresponded to growth rate,
-and cj
-i corresponded to cell area. The p-value for N0 = 10-cell group vs. N0 = 4-cell group
-was 2.12 × 10−8; the p-value for N0 = 10-cell group vs. N0 = 1-cell group was smaller than
-10−12; the p-value for N0 = 4-cell group vs. N0 = 1-cell group was 5.35 × 10−5. Therefore,
-the growth rate difference between any two groups was statistically significant.
-4.4
-Permutation Test.
-The permutation test is a non-parametric method to test whether two samples are signifi-
-cantly different with respect to a statistic (e.g., sample mean) [16]. It is easy to calculate
-and fits our situation, thus we adopt this test rather than other more complicated tests,
-such as the Mann-Whitney test.
-For two samples {x1, · · · , xm}, {y1, · · · , yn}, consider
-the null hypothesis: the mean of x and y are the same. For these samples, calculate the
-mean of the first sample: µ0 =
-1
-m
-� xi. Then we randomly divide these m + n samples
-into two groups with size m and n: {x′
-1, · · · , x′
-m}, {y′
-1, · · · , y′
-n}, such that each permuta-
-tion has equal probability. For these new samples, calculate the mean of the first sample:
-µ′
-0 = 1
-m
-� x′
-i. Then the two-sided p-value is defined as
-p = 2 min{P(µ0 ≤ µ′
-0), 1 − P(µ0 ≤ µ′
-0)}.
-If µ0 is an extreme value in the distribution of µ′
-0, then the two sample means are different.
-In the reseeding experiment, the mean time of exceeding half well for the fast group
-was 11.4375 days. For all
-�64
-32
-�
-possible result combinations, only 7 combinations had equal
-or less mean time. Thus the p-value was 2 × 7/
-�64
-32
-�
-= 7.6 × 10−18. This indicated that the
-growth rate difference between fast group and moderate group was significant.
-4.5
-Model Details.
-The simulation time interval was half day, but we only utilized the results in full days. For
-each initial cell, the probabilities of being fast, moderate or slow type, pF, pM, pS, were 0.4,
-0.4, 0.2.
-Each half day, a fast type cell had probability d to die, and probability gF to divide.
-The division produced two fast cells, capturing the intrinsic growth behavior that is to
-some extent inheritable. Denote the total cell number of previous day as N, then
-gF = g0(1 − N2/C2) + δ,
-where δ is a random variable that satisfies the uniform distribution on [−r, r], and it is a
-constant for all cells in the same well. If gF < 0, set gF = 0. If gF > 1 − d, set gF = 1 − d.
-In the simulation displayed, death rate d = 0.01, carrying capacity C = 40000, growth
-factor g0 = 0.5, and the range of random modifier r = 0.1.
-Each half day, a moderate type cell had probability d to die, and probability gM to
-divide. The division produced two moderate cells. gM = gF/1.5.
-15
-
-Similarly, each half day, a slow type cell had probability d to die, and probability gS to
-divide. The division produced two slow-growing cells. gS = gF/3.
-4.6
-Parameter scan.
-Since growth is measured by the area covered by cells, we could not experimentally verify
-most assumptions of our model, or determine the values of parameters.
-Therefore, we
-performed a parameter scan by evaluating the performance of our model for different sets
-of parameters.
-We adjusted 6 parameters: initial type probabilities pF, pM, pS, death
-rate d, growth factor g0, and random modifier r. We checked whether these 4 features
-observable in the experiment could be reproduced: growth of all wells in the N0 = 10-cell
-group to saturation; existence of late-growing wells in the N0 = 1-cell group; existence of
-non-growing wells in the N0 = 1-cell group; difference in growth rates in the N0 = 10-cell
-group and the N0 = 1-cell group at the same population size. Table 3 shows the results
-of the performance of simulations with the various parameter sets. Within a wide range
-of parameters, our model is able to replicate the experimental results shown in Figs. 1–3,
-indicating that our model is robust under perturbations.
-Acknowledgements
-We would like to thank Ivana Bozic, Yifei Liu, Georg Luebeck, Weili Wang, Yuting Wei
-and Lingxue Zhu for helpful advice and discussions.
-References
-[1] Angelini, E., Wang, Y., Zhou, J. X., Qian, H., and Huang, S. A model for
-the intrinsic limit of cancer therapy: Duality of treatment-induced cell death and
-treatment-induced stemness. PLOS Comput. Biol. 18, 7 (2022), e1010319.
-[2] Bartoszynski, R., Brown, B. W., McBride, C. M., and Thompson, J. R.
-Some nonparametric techniques for estimating the intensity function of a cancer re-
-lated nonstationary Poisson process. Ann. Stat. (1981), 1050–1060.
-[3] Bhartiya, D., Kausik, A., Singh, P., and Sharma, D. Will single-cell rnaseq
-decipher stem cells biology in normal and cancerous tissues?
-Hum. Reprod. Update
-27, 2 (2021), 421–421.
-[4] Chang, H. H., Hemberg, M., Barahona, M., Ingber, D. E., and Huang,
-S. Transcriptome-wide noise controls lineage choice in mammalian progenitor cells.
-Nature 453, 7194 (2008), 544–547.
-16
-
-[5] Chapman, A., del Ama, L. F., Ferguson, J., Kamarashev, J., Wellbrock,
-C., and Hurlstone, A. Heterogeneous tumor subpopulations cooperate to drive
-invasion. Cell Rep. 8, 3 (2014), 688–695.
-[6] Chen, X., Wang, Y., Feng, T., Yi, M., Zhang, X., and Zhou, D. The overshoot
-and phenotypic equilibrium in characterizing cancer dynamics of reversible phenotypic
-plasticity. J. Theor. Biol. 390 (2016), 40–49.
-[7] Clark, W. H. Tumour progression and the nature of cancer. Br. J. Cancer 64, 4
-(1991), 631.
-[8] Dewanji, A., Luebeck, E. G., and Moolgavkar, S. H. A generalized Luria–
-Delbr¨uck model. Math. Biosci. 197, 2 (2005), 140–152.
-[9] Durrett, R., Foo, J., Leder, K., Mayberry, J., and Michor, F. Intratumor
-heterogeneity in evolutionary models of tumor progression. Genetics 188, 2 (2011),
-461–477.
-[10] Egeblad, M., Nakasone, E. S., and Werb, Z.
-Tumors as organs: Complex
-tissues that interface with the entire organism. Dev. Cell 18, 6 (2010), 884–901.
-[11] Gatenby, R. A., Cunningham, J. J., and Brown, J. S. Evolutionary triage gov-
-erns fitness in driver and passenger mutations and suggests targeting never mutations.
-Nat. Commun. 5 (2014), 5499.
-[12] Goldberg, L. R., Kercheval, A. N., and Lee, K.
-T-statistics for weighted
-means in credit risk modeling. J. Risk Finance 6, 4 (2005), 349–365.
-[13] Gunnarsson, E. B., De, S., Leder, K., and Foo, J. Understanding the role of
-phenotypic switching in cancer drug resistance. J. Theor. Biol 490 (2020), 110162.
-[14] Gupta, P. B., Fillmore, C. M., Jiang, G., Shapira, S. D., Tao, K., Kuper-
-wasser, C., and Lander, E. S. Stochastic state transitions give rise to phenotypic
-equilibrium in populations of cancer cells. Cell 146, 4 (2011), 633–644.
-[15] Hanahan, D., and Weinberg, R. A. Hallmarks of cancer: The next generation.
-Cell 144, 5 (2011), 646–674.
-[16] Hastie, T., Tibshirani, R., and Friedman, J. The Elements of Statistical Learn-
-ing, 2nd ed. Springer, New York, 2016.
-[17] Hordijk, W., Steel, M., and Dittrich, P.
-Autocatalytic sets and chemical
-organizations: modeling self-sustaining reaction networks at the origin of life. New J.
-Phys. 20, 1 (2018), 015011.
-17
-
-[18] Howard, G. R., Johnson, K. E., Rodriguez Ayala, A., Yankeelov, T. E.,
-and Brock, A. A multi-state model of chemoresistance to characterize phenotypic
-dynamics in breast cancer. Sci. Rep. 8, 1 (2018), 1–11.
-[19] Jiang, D.-Q., Wang, Y., and Zhou, D. Phenotypic equilibrium as probabilistic
-convergence in multi-phenotype cell population dynamics. PLOS ONE 12, 2 (2017),
-e0170916.
-[20] Johnson, K. E., Howard, G., Mo, W., Strasser, M. K., Lima, E. A., Huang,
-S., and Brock, A. Cancer cell population growth kinetics at low densities deviate
-from the exponential growth model and suggest an Allee effect. PLOS Biol. 17, 8
-(2019), e3000399.
-[21] Kang, Y., Gu, C., Yuan, L., Wang, Y., Zhu, Y., Li, X., Luo, Q., Xiao, J.,
-Jiang, D., Qian, M., et al. Flexibility and symmetry of prokaryotic genome re-
-arrangement reveal lineage-associated core-gene-defined genome organizational frame-
-works. mBio 5, 6 (2014), e01867–14.
-[22] Koch, A. L. Mutation and growth rates from Luria-Delbr¨uck fluctuation tests. Mutat.
-Res. -Fund. Mol. M. 95, 2-3 (1982), 129–143.
-[23] Kochanowski, K., Sander, T., Link, H., Chang, J., Altschuler, S. J., and
-Wu, L. F. Systematic alteration of in vitro metabolic environments reveals empirical
-growth relationships in cancer cell phenotypes. Cell Rep. 34, 3 (2021), 108647.
-[24] Korolev, K. S., Xavier, J. B., and Gore, J. Turning ecology and evolution
-against cancer. Nat. Rev. Cancer 14, 5 (2014), 371–380.
-[25] Lea, D. E., and Coulson, C. A. The distribution of the numbers of mutants in
-bacterial populations. J. Genet. 49, 3 (1949), 264–285.
-[26] Li, Q., Wennborg, A., Aurell, E., Dekel, E., Zou, J.-Z., Xu, Y., Huang, S.,
-and Ernberg, I. Dynamics inside the cancer cell attractor reveal cell heterogeneity,
-limits of stability, and escape. Proc. Natl. Acad. Sci. U.S.A. 113, 10 (2016), 2672–2677.
-[27] Luebeck, G., and Moolgavkar, S. H. Multistage carcinogenesis and the incidence
-of colorectal cancer. Proc. Natl. Acad. Sci. 99, 23 (2002), 15095–15100.
-[28] Luria, S. E., and Delbr¨uck, M. Mutations of bacteria from virus sensitivity to
-virus resistance. Genetics 28, 6 (1943), 491.
-[29] Mackillop, W. J. The growth kinetics of human tumours. Clin. Phys. Physiol. M.
-11, 4A (1990), 121.
-[30] Meacham, C. E., and Morrison, S. J.
-Tumour heterogeneity and cancer cell
-plasticity. Nature 501, 7467 (2013), 328–337.
-18
-
-[31] Newton, P. K., Mason, J., Bethel, K., Bazhenova, L. A., Nieva, J., and
-Kuhn, P.
-A stochastic markov chain model to describe lung cancer growth and
-metastasis. PLOS ONE 7, 4 (2012), e34637.
-[32] Niu, Y., Wang, Y., and Zhou, D. The phenotypic equilibrium of cancer cells:
-From average-level stability to path-wise convergence. J. Theor. Biol. 386 (2015),
-7–17.
-[33] Pisco, A. O., and Huang, S. Non-genetic cancer cell plasticity and therapy-induced
-stemness in tumour relapse: ‘What does not kill me strengthens me’. Br. J. Cancer
-112, 11 (2015), 1725–1732.
-[34] Sahoo, S., Mishra, A., Kaur, H., Hari, K., Muralidharan, S., Mandal, S.,
-and Jolly, M. K. A mechanistic model captures the emergence and implications of
-non-genetic heterogeneity and reversible drug resistance in ER+ breast cancer cells.
-NAR Cancer 3, 3 (2021), zcab027.
-[35] Skehan, P., and Friedman, S. J. Non-exponential growth by mammalian cells in
-culture. Cell Prolif. 17, 4 (1984), 335–343.
-[36] Sonnenschein, C., and Soto, A. M. Somatic mutation theory of carcinogenesis:
-Why it should be dropped and replaced. Mol. Carcinog. 29, 4 (2000), 205–211.
-[37] Speer, J. F., Petrosky, V. E., Retsky, M. W., and Wardwell, R. H. A
-stochastic numerical model of breast cancer growth that simulates clinical data. Cancer
-Res. 44, 9 (1984), 4124–4130.
-[38] Spina, S., Giorno, V., Rom´an-Rom´an, P., and Torres-Ruiz, F. A stochastic
-model of cancer growth subject to an intermittent treatment with combined effects:
-Reduction in tumor size and rise in growth rate. Bull. Math. Biol. 76, 11 (2014),
-2711–2736.
-[39] Tabassum, D. P., and Polyak, K. Tumorigenesis: It takes a village. Nat. Rev.
-Cancer 15, 8 (2015), 473–483.
-[40] Wang, Y.
-Algorithms for finding transposons in gene sequences.
-arXiv preprint
-arXiv:1506.02424 (2015).
-[41] Wang, Y. Some Problems in Stochastic Dynamics and Statistical Analysis of Single-
-Cell Biology of Cancer. Ph.D. thesis, University of Washington, 2018.
-[42] Wang, Y. Two metrics on rooted unordered trees with labels. Algorithms for Molec-
-ular Biology 17, 1 (2022), 1–17.
-19
-
-[43] Wang, Y., Dessalles, R., and Chou, T. Modelling the impact of birth control
-policies on China’s population and age: effects of delayed births and minimum birth
-age constraints. Royal Society Open Science 9, 6 (2022), 211619.
-[44] Wang, Y., and He, S. Inference on autoregulation in gene expression. arXiv preprint
-arXiv:2201.03164 (2022).
-[45] Wang, Y., Kropp, J., and Morozova, N. Biological notion of positional informa-
-tion/value in morphogenesis theory. International Journal of Developmental Biology
-64, 10-11-12 (2020), 453–463.
-[46] Wang, Y., Minarsky, A., Penner, R., Soul´e, C., and Morozova, N. Model
-of morphogenesis. Journal of Computational Biology 27, 9 (2020), 1373–1383.
-[47] Wang, Y., Mistry, B. A., and Chou, T. Discrete stochastic models of selex:
-Aptamer capture probabilities and protocol optimization. The Journal of Chemical
-Physics 156, 24 (2022), 244103.
-[48] Wang, Y., and Qian, H. Mathematical representation of Clausius’ and Kelvin’s
-statements of the second law and irreversibility. Journal of Statistical Physics 179, 3
-(2020), 808–837.
-[49] Wang, Y., and Wang, L. Causal inference in degenerate systems: An impossibility
-result. In International Conference on Artificial Intelligence and Statistics (2020),
-PMLR, pp. 3383–3392.
-[50] Wang, Y., and Wang, Z. Inference on the structure of gene regulatory networks.
-Journal of Theoretical Biology 539 (2022), 111055.
-[51] Wang, Y., Zhang, B., Kropp, J., and Morozova, N. Inference on tissue trans-
-plantation experiments. Journal of Theoretical Biology 520 (2021), 110645.
-[52] Ye, F. X.-F., Wang, Y., and Qian, H. Stochastic dynamics: Markov chains and
-random transformations. Discrete & Continuous Dynamical Systems-B 21, 7 (2016),
-2337.
-[53] Yorke, E. D., Fuks, Z., Norton, L., Whitmore, W., and Ling, C. C. Model-
-ing the development of metastases from primary and locally recurrent tumors: Com-
-parison with a clinical data base for prostatic cancer.
-Cancer Res. 53, 13 (1993),
-2987–2993.
-[54] Zheng, Q. Progress of a half century in the study of the Luria–Delbr¨uck distribution.
-Math. Biosci. 162, 1 (1999), 1–32.
-[55] Zhou, D., Wang, Y., and Wu, B.
-A multi-phenotypic cancer model with cell
-plasticity. J. Theor. Biol. 357 (2014), 35–45.
-20
-
-[56] Zhou, J. X., Pisco, A. O., Qian, H., and Huang, S. Nonequilibrium population
-dynamics of phenotype conversion of cancer cells. PLOS ONE 9, 12 (2014), e110714.
-21
-
-Figure 1: Growth curves of the experiment (left) and simulation (right), starting from
-the time of reaching 5 area units (experiment) or having 2500 cells (simulation), with a
-logarithm scale for the y-axis. The time required for reaching 5 area units was determined
-by exponential extrapolation, as reliable imaging started at > 5 area units. The x-axis is
-the time from reaching 5 area units (experiment) or 2500 cells (simulation). Red, green,
-or blue curves correspond to 10, 4, or 1 initial cell(s). Although starting from the same
-population level, patterns are different for distinct initial cell numbers. The N0 = 1-cell
-group has higher diversity.
-22
-
-experimental
-80
-cell area
-40
-20
-10-cell group
-4-cell group
-10
-1-cell group
-5
-0
-5
-10
-15
-time (day)
-80
-cell area
-40
-20simulation
-40000
-cell number
-20000
-10000
-5000
-2500
-0
-5
-10
-15
-time (day)
-40000
-ell number
-20000
-100005
-0
-5
-10
-15
-time (day)
-80
-cell area
-40
-20
-10
-5
-0
-5
-10
-15
-time (day)8
-QQQ
-2500
-0
-5
-10
-15
-time (day)
-40000
-cell number
-20000
-10000
-5000
-2500
-0
-5
-10
-15
-time (day)20
-40
-60
-80
-cell area
-0
-0.5
-1
-1.5
-growth rate
-experimental
-10-cell group
-4-cell group
-1-cell group
-0
-20
-40
-60
-80
-cell area
-0
-0.5
-1
-1.5
-growth rate
-1
-2
-3
-4
-cell number
-104
-0
-0.5
-1
-1.5
-growth rate
-simulation
-0
-1
-2
-3
-4
-cell number
-104
-0
-0.5
-1
-1.5
-growth rate
-Figure 2: Per capita growth rate (averaged within one day) vs. cell population for the
-experiment (left) and simulation (right). Each point represents one well in one day. Red,
-green, or blue points correspond to 10, 4, or 1 initial cell(s).
-23
-
-0
-5
-10
-15
-20
-time (day)
-0
-20
-40
-60
-80
-cell area
-experimental
-10-cell group
-4-cell group
-1-cell group
-0
-5
-10
-15
-20
-time (day)
-5
-10
-20
-40
-80
-cell area
-0
-5
-10
-15
-20
-time (day)
-0
-1
-2
-3
-4
-cell number
-104
-simulation
-0
-5
-10
-15
-20
-time (day)
-2500
-5000
-10000
-20000
-40000
-cel number
-Figure 3: Growth curves of the experiments with different initial cell numbers N0 (left)
-and growth curves of corresponding simulation (right). Each curve describes the change in
-the cell population (measured by area or number) over a well along time. Red, green, or
-blue curves correspond to N0 = 10, 4, or 1 initial cell(s).
-24
-
-Figure 4: Schematic illustration of the qualitative argument: Three cell types and growth
-patterns (three colors) with different seeding numbers. One N0 = 10-cell well will have
-at least one fast type cell with high probability, which will dominate the population. One
-N0 = 1-cell well can only have one cell type, thus in the microculture ensemble of replicate
-wells, three possible growth patterns for wells can be observed.
-25
-
-fast
-moderate
-slow
-fast
-fast
-moderate
-slow

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-Springer Nature 2021 LATEX template
-Homeostatic regulation of renewing tissue cell
-populations via crowding control: stability,
-robustness and quasi-dedifferentiation
-Cristina Parigini1,2,3 and Philip Greulich1,2*
-1*School of Mathematical Sciences, University of Southampton,
-Southampton, United Kingdom.
-2Institute for Life Sciences, University of Southampton,
-Southampton, United Kingdom.
-3Te P¯unaha ¯Atea - Space Institute, University of Auckland,
-Auckland, New Zealand.
-*Corresponding author(s). E-mail(s): [email protected];
-Contributing authors: [email protected];
-Abstract
-To maintain renewing epithelial tissues in a healthy, homeostatic state,
-(stem) cell divisions and differentiation need to be tightly regulated.
-Mechanisms of homeostatic control often rely on crowding control: cells
-are able to sense the cell density in their environment (via various
-molecular and mechanosensing pathways) and respond by adjusting
-division, differentiation, and cell state transitions appropriately. Here
-we determine, via a mathematically rigorous framework, which general
-conditions for the crowding feedback regulation (i) must be minimally
-met, and (ii) are sufficient, to allow the maintenance of homeosta-
-sis in renewing tissues. We show that those conditions naturally allow
-for a degree of robustness toward disruption of regulation. Further-
-more, intrinsic to this feedback regulation is that stem cell identity is
-established collectively by the cell population, not by individual cells,
-which implies the possibility of ‘quasi-dedifferentiation’, in which cells
-committed to differentiation may reacquire stem cell properties upon
-depletion of the stem cell pool. These findings can guide future exper-
-imental campaigns to identify specific crowding feedback mechanisms.
-Keywords: keyword1, Keyword2, Keyword3, Keyword4
-1
-arXiv:2301.05321v1  [q-bio.TO]  12 Jan 2023
-
-Springer Nature 2021 LATEX template
-2
-Homeostatic regulation of renewing tissue cell populations via crowding control
-1 Introduction
-Many adult tissues are renewing, that is, terminally differentiated cells are
-steadily removed and replaced by new cells produced by the division of cycling
-cells (stem cells and progenitor cells), which then differentiate. In order to
-maintain those tissues in a healthy, homeostatic state, (stem) cell divisions
-and differentiation must be tightly balanced. Adult stem cells are the key
-players in maintaining and renewing such tissues due to their ability to produce
-cells through cell division and differentiation persistently [1]. However, the
-underlying cell-intrinsic and extrinsic factors that regulate a homeostatic state
-are complex and not always well understood.
-Several experimental studies have identified mechanisms and pathways that
-regulate homeostasis. For example, cell crowding can trigger delamination and
-thus loss of cells in Drosophila back [2], and differentiation in cultured human
-colon, various zebrafish epiderimises, and canine kidney cells [3, 4]. On the
-other hand, cell crowding can affect cell proliferation: overcrowding can inhibit
-proliferation [5], whereas a reduction in the cell density, obtained, for example,
-by stretching a tissue [6] causes an increase in proliferative activity (both
-shown in cultured canine kidney cells). Although the mechanisms to mediate
-this regulation are not always clear, experimental studies on mechanosensing
-showed that cell overcrowding reduces cell motility and consequently produces
-a compression on cells that inhibits cell proliferation [5, 7]. Another mechanism
-utilising crowding feedback is the competition for limited growth signalling
-factors [8]. More specifically, in the mouse germ line, cells in the niche respond
-to a growth factor (FGF5) that promotes proliferation over differentiation,
-which they deplete upon being exposed to it. Therefore, the more cells are
-in the niche, the less FGF5 is available per cell, and the less proliferation (or
-more differentiation) occurs.
-Despite differing in the involved molecular pathways and many other
-details, all these regulatory mechanisms are, in essence, sensing the cell den-
-sity in their environment and responding by adjusting their propensities to
-divide, differentiate, die, or emigrate from the tissue. This class of mechanisms,
-for which cell fate propensities depend on the cell density, can be classified as
-crowding feedback regulation: the cell density determines the cells’ prolifera-
-tion and differentiation, which affects their population dynamics and thus the
-cell density. However, the crowding response to changes in cell density cannot
-be arbitrary in order to maintain homeostasis. It must provide a (negative)
-feedback, in the sense that cells sense the cell density and adjust proliferation,
-differentiation, and cell loss, such that the cell density is decreased if it is too
-high and increased if it is too low. For simple tissues consisting of a single
-cell type with a unique cell state, it is relatively straightforward to give the
-conditions for crowding feedback to maintain homeostasis successfully. In this
-case, when the cell division rate decreases with cell density and differentiation
-and or death rate increase with cell density, a homeostatic state is maintained.
-However, such conclusions are not as simple to make when a tissue consists of
-a complex lineage hierarchy and a multitude of underlying cellular states. In
-
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-3
-the latter, more realistic case, conditions for successful homeostatic regulation
-– in which case we speak of crowding control – may take more complex forms.
-Previous studies based on mathematical modelling have shed some light
-on quantitative mechanisms for homeostatic control [9–13]. In particular, in
-[13], a mathematical assessment of crowding feedback modelling shows that
-a (dynamic) homeostatic state exists under reasonable biological conditions.
-Nevertheless, the case of dynamic homeostasis considered there may not nec-
-essarily be a steady state but could also exhibit oscillations in cell numbers
-(as does realistically happen in the uterus during the menstrual cycle). While
-the criterion presented in [13] provides a valid sufficient condition for dynamic
-homeostasis, it relies on a rather abstract mathematical quantity – the domi-
-nant eigenvalue of the dynamical matrix – that is difficult, if not impossible,
-to measure in reality.
-Here, we wish to generalise previous findings and seek to identify general
-conditions for successful homeostatic control if propensities for cell division,
-differentiation, and loss are responsive to variations in cell density. More
-precisely, we derive conditions that must be minimally fulfilled (necessary
-conditions) and conditions which are sufficient, to ensure that homeostasis pre-
-vails. To identify and formulate those conditions, we note that homeostasis is
-a property of the tissue cell population dynamics, which can be mathemati-
-cally expressed as a dynamical system. Even if a numerically exact formulation
-of the dynamics may not be possible, one can formulate generic yet mathe-
-matically rigorous conditions by referring to the criteria for the existence of
-stable steady states in the cell population dynamics of renewing tissues. We
-will derive those conditions by mathematical, analytical means, augmented by
-a numerical analysis testing the limits of those conditions.
-We will also show that homeostatic control by crowding feedback possesses
-inherent robustness to failures and perturbations of the regulatory pathways,
-which may occur through external influences (e.g. wide-spread biochemical fac-
-tors) and genetic mutations. Finally, we will assess the response of cells when
-the pool of stem cells is depleted. Crucially, we find that inherent to crowd-
-ing feedback control is that formerly committed progenitor cells reacquire
-self-renewal capacity without substantial changes in their internal states. Ded-
-ifferentiation has been widely reported under conditions of tissue regeneration
-[14, 15] or when stem cells are depleted [16–19], which is usually thought to
-involve a substantial reprogramming of the cell-intrinsic states towards a stem
-cell type. On the other hand, our analysis suggests the possibility of “quasi”-
-dedifferentiation, the reversion from a committed cell to a stem cell by a
-mere quantitative adjustment of the pacing of proliferation and differentiation,
-without a substantial qualitative change in its expression profiles.
-
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-Homeostatic regulation of renewing tissue cell populations via crowding control
-2 Modelling of tissue cell dynamics under
-crowding feedback
-We seek to assess the conditions for homeostasis in renewing tissue cell pop-
-ulations, that is, either a steady state of the tissue cell population (strict
-homeostasis) or long-term, bounded oscillations or fluctuations (dynamic
-homeostasis), which represent well-defined constraints on the dynamics of the
-tissue cell population. To this end, we will here derive a formal, mathematical
-representation of the tissue cell dynamics under crowding feedback.
-The cell population is fully defined by (i) the number of cells, (ii) the
-internal (biochemical and mechanical) states of each cell, and (iii) the spatial
-position of cells. We assume that a cell’s behaviour can depend on the cell
-density and the states of cells in its close cellular environment. As we examine
-a situation close to a homeostatic state, we assume that the cell density is
-homogeneous over the range of interaction between cells, which expands over
-a volume V . Hence, the cell density ρ is proportional to the average number
-of cells, ¯n, in that volume, ρ =
-¯n
-V . Similarly, we define the number of cells
-in internal state i as ni, and the cell density of cells in internal state i as
-ρi = ¯ni
-V , where ¯ni is the expected value of ni. As we consider only the crowding
-feedback response of cells, which only accounts for the cell densities ρi but
-not the explicit position of cells, the spatial configuration (iii) is not relevant
-to our considerations. Thus, the configuration of the cell population and its
-time evolution is entirely determined by the average number of cells in each
-state i, as a function of time t, ¯ni(t). The configuration of cell numbers ni
-can change only through three processes: (1) cell division, whereby it must be
-distinguished between the cell state of daughter cells, (2) the transition from
-one cell state to another, (3) loss of a cell, through cell death or emigration
-out of the tissue. Following the lines of Refs. [13, 20] and denoting as Xi,j,k a
-cell in internal states i, j, k, respectively, we can formalise these events as:
-cell division: Xi
-λirjk
-i
-−−−→ Xj + Xk
-(1)
-cell state transition: Xi
-ωij
-−−→ Xj
-(2)
-cell loss: Xi
-γi
-−→ ∅ ,
-(3)
-where the symbols above the arrows denote the dynamical rates of the transi-
-tions, i.e. the average frequency at which such events occur. In particular, γi
-is the rate at which a cell in state i is lost, ωij the rate at which a cell changes
-its state from i to j and λirjk
-i
-denotes the rate at which a cell i divides to pro-
-duce two daughter cells, one in state j and one in state k (i = j, j = k, k = i
-are possible). For later convenience, we distinguish here the overall rate of cell
-division in state i, λi and the probability rjk
-i
-that such a division produces
-daughter cells in states j and k.
-Since we consider a situation where cells can respond to the cell densities
-ρi via crowding feedback, all the rates and probabilities (λi, γi, ωij, rjk
-i ) may
-
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-depend on the cell densities of either state j, ρj. For convenience, we discretise
-the number of states in case the state space is a continuum and only distinguish
-states which have substantially different propensities (λi, γi, ωij, rjk
-i ). Without
-loss of generality, we assume that there are m states, that is, i, j, k = 1, ..., m
-(for a rigorous argument for the discretisation of the state space, see [13]).
-The rates given above denote the average number of events happening per
-time unit. Thus, we can express the total rate of change of the average (i.e.
-expected) number of cells ¯ni(t), that is, the derivative ˙¯ni = d¯ni
-dt , in terms of
-the rates of those events. This defines a set of ordinary differential equations.
-Following the lines of Refs. [13, 20], we can write ˙ni as,
-˙¯ni =
-��
-j
-ωji¯nj + λj
-��
-k
-rik
-j + rki
-j
-�
-¯nj
-�
-− ¯ni
-�
-λi + γi +
-�
-j
-ωij
-�
-,
-(4)
-where for convenience, we did not write the time dependence explicitly, i.e.
-ni = ni(t), and all parameters may depend on the cell densities ρj. Since V is
-constant, we can divide by V to equivalently express this in terms of the cell
-state densities, ρi = ¯ni
-V , and then write Eq. (4) compactly as,
-d
-dtρ(t) = A(ρ(t)) ρ(t)
-(5)
-where ρ = (ρ1, ρ2, ...) is the vector of cell state densities and A(ρ) is the matrix,
-A =
-�
-�
-λ1 − �
-j̸=1 κ1j − γ1
-κ21
-κ31
-· · ·
-κ12
-λ2 − �
-j̸=2 κ2j − γ2 κ32
-· · ·
-κ1m
-κ2m
-· · · λm − �
-j̸=m κmj − γm
-�
-� ,
-(6)
-in which κij = λi2rj
-i + ωij, with rj
-i = �
-k(rjk
-i
-+ rkj
-i )/2, is the total transition
-rate, that combines all transitions from Xi to Xj by cell divisions and direct
-state transitions (again, all parameters may depend on ρ, as therefore also
-does A). We can thus generally write the elements of the matrix A, aij with
-i, j = 1, ..., m as
-aij =
-� λi − γi − �
-k̸=i κik
-for i = j
-κji
-for i ̸= j
-(7)
-We now make the mild assumption that divisions of the form Xi → Xj+Xk
-are effectively three events, namely, cell duplication, Xi → Xi + Xi coupled to
-cell state changes, Xi → Xj and Xi → Xk, if j ̸= i or k ̸= i. In this view, the
-parameters relevant for crowding feedback are the total cell state transition
-propensities κij and the cell division rate λi, as in (6), instead of ωij and rjk
-i .
-These equations describe a dynamical system which, for given initial con-
-ditions, determines the time evolution of the cell densities, ρi(t). Crucially,
-
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-this description allows for a rigorous mathematical definition of what a home-
-ostatic state is, and to apply tools of dynamical systems analysis to determine
-the circumstances under which a homeostatic state prevails. In particular, we
-define a strict homeostatic state as a steady state of the system, (5), when the
-cell numbers – and thus cell densities, given that V is fixed – in each state
-do not change, mathematically expressed as dρ
-dt = 0 (a fixed point of the sys-
-tem). A dynamic homeostatic state is when cell densities may also oscillate
-or fluctuate but remain bounded and thus possess a finite long-term average
-cell population (in which case the system either approaches a steady state or
-limit cycles – that is, oscillations – or chaotic but bounded behaviour). Based
-on these definitions, we can now analyse under which circumstances crowd-
-ing feedback can maintain those states, which in the case of strict homeostasis
-requires, in addition, that the corresponding steady state is stable.
-2.1 Cell types and lineage hierarchies
-According to [13], cell population dynamics of the type (5) can be associated
-with a cell state network, in which each state is a node, and the nodes are
-connected through cell state transition (direct transitions and cell divisions).
-Furthermore, by decomposing this network in strongly connected components
-(SCCs), the cell fate model can be viewed as a directed acyclic network [21],
-generally called the condensed network. Here, we follow the definitions of [13]
-and define a cell type as an SCC of the cell state network, so that any cell states
-connected via cyclic cell state trajectories (sequences of cell state transitions)
-are of the same type, and the condensed network of cell types represents the
-cell lineage hierarchy. This definition ensures that cells of the same type have
-the same lineage potential (outgoing cell state trajectories) and that the stages
-of the cell cycle are associated with the same cell type. In this context, we
-will in the following also speak of differentiation when a cell state transition
-between different cell types occurs.
-Each cell type can be classified as self-renewing, declining or hyper-
-proliferating, depending on the dominant eigenvalue µ (called growth parame-
-ter) of the dynamical matrix A (from Eq. (5) ff.) reduced to that SCC. This
-is µ = 0 for self-renewing cell types, when cell numbers of that type remain
-constant over time, µ < 0 (µ > 0) for the declining (hyperproliferating) types
-when cell numbers decline (increase) in the long term [13]. Importantly, for the
-population dynamics to be strictly homeostatic, which means that a steady
-state of model (5) exists, the cell type network must fulfil strict rules. These
-are: (i) at least one self-renewing cell type (with µ = 0) must exist; (ii) self-
-renewing cell types must stay at an apex of the condensed network; (iii) all
-the other cells must be of declining types. This means that the critical task of
-homeostatic control is to ensure that the kinetic parameters of the cell type at
-the apex of the cell lineage hierarchy are fine-tuned to maintain exactly µ = 0.
-Therefore, we can restrict our analysis to find conditions for the cell type
-at the lineage hierarchy’s apex to be self-renewing, which we will do in the
-following. Other cell types simply need that differentiation (transition towards
-
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-7
-another cell type) or loss is faster than proliferation, so that they become
-declining cell types, µ < 0, but those rates do not require fine-tuning and thus
-trivially regulated. We note that when we consider only cell states of the type
-at the apex of the cell lineage hierarchy, any differentiation event is – according
-to this restricted model – a cell loss event and included as event occurring with
-rates γi. Given that cell loss from a cell type at the lineage apex is rare, we
-will therefore in the following also denote the rates γi simply as differentiation
-rates.
-3 Results
-We will now determine necessary and sufficient conditions for the establish-
-ment of strict and dynamical homeostasis when subject to crowding feedback,
-which we here define through the derivatives of the dynamical parameters
-λi, rjk
-i , ωij, γi as a function of the cell densities. As argued before, we only need
-to consider cell types at an apex of the cell type network, which, for home-
-ostasis to prevail, must have a growth parameter (i.e. dominant eigenvalue of
-matrix A in Eq. (6)) µ = 0. Furthermore, we assume that the apex cell type
-resides in a separate stem cell niche. Therefore, the parameters only depend
-on cell densities ρi of states associated with that cell type, i.e. we can write
-A = A(ρ), where ρ = �
-i∈S ρi comprises only cell states of the apex cell type
-S. Provided that, the matrix elements are functions of ρ, and therefore also µ
-is a function of ρ. Thus, self-renewal corresponds to a non-trivial fixed point,
-ρ∗, of Eq. (5), restricted to cell type S, for which the dominant eigenvalue of
-A is zero, that is µ(ρ∗) = 0 (ρ∗ = �
-i∈S ρ∗
-i ).
-For convenience, we will often generally refer to parameters as αi, i =
-1, ..., 2m + m2, where αi stands for any of the parameters, {λi, γi, κij|i, j =
-1, ..., m}, respectively1. Hence, we study which conditions the functions αi(ρ)
-must meet to maintain homeostasis. In particular, we study how those param-
-eters qualitatively change with the cell density – increase or decrease – that
-is, we study how the sign and magnitude of derivatives α′
-i :=
-dαi
-dρ
-affects
-homeostasis.
-A crucial property of the matrix A(ρ) is that it is always a Metzler matrix,
-since all its off-diagonal elements, κij ≥ 0. Since the cell state network of a
-cell type is strongly connected, we can further state that A(ρ) is irreducible.
-Notably, for irreducible Metzler matrices holds the Perron-Frobenius theorem
-[22], and thus A(ρ) possesses a simple, real dominant eigenvalue µ. Besides,
-it as left and right eigenvectors associated with µ, respectively indicated as v
-and w, which are strictly positive, that is, all their entries are vi > 0, wi > 0.
-From this follows that the partial derivative of the dominant eigenvalue µ by
-1More
-precisely,
-αi|i=1,..,m
-:=
-λi, αi|i=m+1,..,2m
-:=
-γi−m, αi|i=2m+1,..,2m+m2
-:=
-κ⌊(i−2m)/m⌋,i−⌊(i−2m)/m⌋m
-
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-the i, j-th element of A, aij = [A]ij is always positive:
-∂µ
-∂aij
-= viwj
-vw > 0
-(8)
-where the left equality is according to [23] and is generally valid for simple
-eigenvalues. Here, v is assumed to be in row form, and vw thus corresponds
-to a scalar product.
-3.1 Sufficient condition for dynamic homeostasis
-In [13], it was shown that a dynamic homeostatic state, where cell numbers
-may change over time but stay bounded, is assured if, 2
-µ′(ρ) < 0 for all ρ > 0.
-(9)
-This sufficient condition requires that the dominant eigenvalue of A as a func-
-tion of the cell density, µ(ρ), is a strictly decreasing function of cell density.
-Also, the range of this function must be sufficiently large so that it has a root,
-i.e. a value ρ∗ with µ(ρ∗) = 0 must exist for the function µ(ρ).
-Assuming that a non-trivial steady state, ρ∗ > 0, exists, we now translate
-the sufficient condition for a dynamic homeostatic state, Eq. (9), into condi-
-tions on the parameters as a function of the cell density, αi(ρ). In particular,
-we can write,
-µ′(ρ) =
-�
-ij
-∂µ
-∂aij
-∂aij
-∂ρ =
-�
-ij
-viwj
-vw a′
-ij =
-�
-i
-viwi
-vw a′
-ii +
-�
-i,j̸=i
-viwj
-vw a′
-ij
-=
-�
-i
-viwi
-vw
-�
-�λ′
-i − γ′
-i −
-�
-j̸=i
-κ′
-ij
-�
-� +
-�
-i,j̸=i
-vjwi
-vw κ′
-ij ,
-(10)
-where we used Eq. (8) and the explicit forms of aij, the elements of the matrix
-A according to Eq. (7). Provided that all the parameters depend on ρ, condition
-(9) results in:
-0 > µ′ =⇒ 0 >
-�
-i
-viwi (λ′
-i − γ′
-i) + wi
-�
-j̸=i
-(vj − vi)κ′
-ij
-for all ρ > 0 ,
-(11)
-While we cannot give an explicit general expression for the dominant eigen-
-vectors v, w, this condition is sufficiently fulfilled if each term of the sum on
-the right-hand side of Eq. (11) is negative. More restrictively, we have Eq. (11)
-2In [13], this condition, defined through dependency on cell number, can be directly translated
-into a condition on the cell density derivative if the volume is assumed as a constant.
-
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-sufficiently fulfilled if
-�
-�
-�
-�
-�
-λ′
-i ≤ 0, γ′
-i ≥ 0 for all i
-λ′
-i < 0 or γ′
-i > 0 at for least one i
-κ′
-ij = 0 for all i, j
-for ρ > 0
-(12)
-This means that, excluding rates that are zero, which are biologically mean-
-ingless, if no direct state transitions within a cell type are subject to crowding
-feedback (κ′
-ij = 0), while all (non-zero) cell division rates depend negatively
-on ρ (λ′
-i < 0), and differentiation rates depend positively (γ′
-i > 0), for all
-attainable levels of ρ, then dynamical homeostasis is ensured.
-Alternatively, we can rewrite Eq. (11) as
-0 >
-�
-i
-viwi
-vw
-�
-�λ′
-i − γ′
-i −
-�
-j̸=i
-κ′
-ij +
-�
-j̸=i
-vj
-vi
-κ′
-ij
-�
-�
-for all ρ > 0 ,
-(13)
-which, due to
-vj
-vi
-> 0, implies another sufficient condition for dynamic
-homeostasis:
-�
-�
-�
-�
-�
-λ′
-i ≤ 0, γ′
-i ≥ 0 for all i
-λ′
-i < 0 or γ′
-i > 0 at for least one i
-κ′
-ij ≤ 0 with |�
-j κ′
-ij| ≤ γ′
-i − λ′
-i for all i, j
-(14)
-The above condition is less restrictive than Eq. (12), allowing for some non-
-zero crowding feedback dependency of state transition rates κij, as long as the
-crowding feedback strength of the total outgoing transition rate of each state
-does not outweigh the feedback on proliferation and differentiation rate of that
-state (if there is).
-3.2 Necessary condition for strict homeostasis
-We now consider the circumstances under which a strict homeostatic is main-
-tained, that is, when a steady state of the cell population exists and is
-asymptotically stable.
-A necessary condition for the existence of a steady state ρ∗ (irrespective
-of stability) has been given in [13], namely, that the cell type at the apex of
-the lineage hierarchy is self-renewing, i.e. its dynamical matrix A has µ = 0.
-µ depends on the cell density ρ of the apex cell type, since the dynamical
-parameters αi and thus A depend on ρ. As before, it is required that µ(ρ∗)
-has sufficient range so that a value ρ∗ with µ(ρ∗) = 0 exists. This condition is
-fulfilled if the range of the feedback parameters αi(ρ) is sufficiently large. In
-that case there exists an eigenvector ρ∗ with A(ρ∗)ρ∗ = 0, which can be chosen
-by normalisation to fulfil �
-i∈S ρ∗
-i = ρ∗. Thus, ρ∗ is a fixed point (steady state)
-
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-of the cell population system (5). Hence, we need to establish what is required
-for this state to be asymptotically stable.
-To start with, we give the Jacobian matrix of the system (5) at the fixed
-point ρ∗ :
-[J]ij = ∂[A(ρ)ρ]i
-∂ρj
-����
-ρ=ρ∗
-= a∗
-ij + ηi ,
-(15)
-where
-ηi =
-�
-k
-a′
-ikρ∗
-k .
-(16)
-Here and in the following, we assume the derivatives to be taken at the steady
-state, i.e. a′
-ij :=
-daij
-dρ |ρ=ρ∗. The eigenvalues of the Jacobian matrix J at ρ∗
-determine the stability of the steady state ρ∗: it is asymptotically stable if and
-only if the real part of all eigenvalues of J(ρ∗) is negative.
-The Routh-Hurwitz theorem [24] states that for a polynomial to have only
-roots with negative real part, all its coefficients must necessarily be positive.
-Given that the eigenvalues of the Jacobian matrix J are the roots of its char-
-acteristic polynomial, a necessary condition for ρ∗ to be asymptotically stable
-is that the coefficients of the characteristic polynomial of J are all positive.
-Let us start by considering a self-renewing cell type with exactly two cell
-states being at the apex of a lineage hierarchy. This system has a 2 × 2 dynam-
-ical matrix A and Jacobian J, whereby A is irreducible and has dominant
-eigenvalue µA = 0. The characteristic polynomial of a generic 2×2 matrix, M,
-is
-P M(s) = s2 + pM
-1 s + pM
-0 .
-(17)
-with pM
-1 = −tr(M) and pM
-0 = det(M). In particular, since A has an eigenvalue
-zero,
-pA
-0 = det(A) = a11a22 − a12a21 = 0 .
-(18)
-From this follows that the right and left eigenvectors to the matrix A
-associated with the dominant eigenvalue µA = 0, w and v, are:
-w =
-�
-−a22
-a21
-�
-and v =
-�
-−a22 a12
-�
-.
-(19)
-From the Jacobian matrix J, we get equivalently,
-pJ
-0 = det(J) = (a21 − a22)(−a22η1 + a12η2)
-a22
-= vη |w|
-a22
-,
-(20)
-with the L1-norm |w| = w1 + w2 = −a22 + a213. Here we used the form of J
-in Eq. (15) with η = (η1, η2) from (16), as well as the relations (18) and (19),
-and we factorised the determinant.
-3Note that aii is always negative or zero
-
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-From Eq. (10), we can further establish:
-µ′ =
-�
-ij
-viwj
-vw a′
-ij =
-�
-ij
-|w|
-ρ∗
-viρ∗
-j
-vw a′
-ij = |w|
-ρ∗
-vη
-vw
-(21)
-= − a22pJ
-0
-ρ∗pJ
-1 a22
-.
-(22)
-Here, we used that ρ∗ is a dominant right eigenvector, and thus ρ∗ =
-ρ∗
-|w|w,
-and furthermore we used the definition of ηi = �
-j a′
-ijρ∗
-j, we substituted Eq.
-(20), and used that vw = a2
-22 + a12a21 = −pA
-1 a22. Finally, we get:
-pJ
-0 = −µ′ρ∗pA
-1 .
-(23)
-Notably, we can show that this relation also holds for higher dimensions by
-explicitly computing the coefficients of characteristic polynomials pA,J
-i
-, the
-eigenvalues and eigenvectors, and then evaluating both sides of the equation.
-For systems with three states, this can be done analytically. For systems with
-4,5, and 6 states we tested relation (23) numerically by generating N =1000
-random matrices with entries chosen from a uniform distribution4. In each
-case, this relation was fulfilled. Hence we are confident that this relation holds
-up to 6 states, and it is reasonable to expect this to hold also for larger systems.
-Since A has a simple dominant eigenvalue µA = 0, we can factorise one term
-from the characteristic polynomial, P(s) = sQ(s) knowing that all roots of
-Q(s) are negative. Applying the Routh-Hurwitz necessary condition to Q(s), it
-follows that the coefficients of the polynomial Q, pQ
-i > 0, where i = 1, 2, ..., n−
-1. Thus, pA
-1 > 0 and considering that ρ∗ > 0 by definition, then for having
-pJ
-0 > 0 we must require µ′ < 0. Therefore, a necessary condition for a stable,
-strict homeostatic state is
-0 > µ′ =⇒ 0 >
-�
-i
-viwi (λ′
-i − γ′
-i) + wi
-�
-j̸=i
-(vj − vi)κ′
-ij
-������
-ρ=ρ∗
-,
-(24)
-where on the right-hand side, we used Eq. (11). This condition is bound to the
-validity of Eq. (23), that is, we can show it analytically for up to three states
-and numerically up to 6 states. Nonetheless, we also expect this to be true for
-larger systems.
-One way to satisfy this necessary condition is if at ρ = ρ∗
-�
-�
-�
-�
-�
-�
-i ≤ 0, γ′
-i ≥ 0 for all i
-λ′
-i < 0 or γ′
-i > 0 at for least one i
-κ′
-ij = 0
-.
-(25)
-4The diagonal elements of the random matrix are tuned using a local optimiser (fmincon
-function of Matab) so that the matrix has a zero dominant eigenvalue.
-
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-Notably, the necessary conditions (24) and (25) only differ from the suffi-
-cient conditions for dynamic heterogeneity, Eqs. (11) and (12), by needing to
-be fulfilled only at the steady-state cell density ρ∗, whereas to ensure dynamic
-homeostasis, those should be valid for a sufficiently large range of ρ.
-3.3 Sufficient condition for strict homeostasis
-Now we assess under which circumstances a strict homeostatic state is assured
-to prevail.
-First of all, the necessary conditions from above need to be fulfilled. In
-particular, the parameter functions αi(ρ) must have a sufficient range so that
-µ(ρ) has a root, i.e. ρ∗ with µ(ρ∗) = 0 exists, from which the existence of
-a steady state follows. The question now is whether we can find sufficient
-conditions assuring that the fixed point ρ∗ with �
-i ρ∗
-i = ρ∗ is (asymptotically)
-stable.
-Let us define a matrix B(x), x = (x1, ..., xm) with bij(x) = [B]ij(x) =
-a∗
-ij +xi. Hence, B(xi = 0) = A(ρ∗) and B(xi = ηi) = J, where J, the Jacobian
-matrix, and ηi are defined as in (15) and (16), respectively. We consider now the
-dominant eigenvalue as function of the entries of B, µ[B] := µ({bij}|i,j=1,...,m)
-(the square brackets are chosen to denote the difference from the function
-µ(ρ)). For sufficiently small ηi, we can then express the dominant eigenvalue
-of the Jacobian matrix J, µ[J], relative to the dominant eigenvalues of A∗ :=
-A(ρ∗) as,
-µ[J] = µ[A∗] +
-�
-i
-∂µ
-∂xi
-|xi=0 ηi + O(η2) ,
-(26)
-with,
-∂µ
-∂xi
-|xi=0 =
-�
-ij
-∂µ
-∂bij
-∂bij
-∂xi
-|xi=0 =
-�
-ij
-∂µ
-∂aij
-|B=A∗ ,
-(27)
-since for x = 0, bij = aij for all i, j. It follows that for sufficiently small5 ηi,
-and if all ηi < 0, we have
-µJ = µ[A∗](ρ∗) +
-�
-i
-∂µB
-∂xi
-|xi=0ηi + O(η2
-i ) ≈
-�
-i
-∂µA
-∂aij
-ηi < 0
-(28)
-since all ∂µA
-∂aij > 0 (according to Eq. (8)) and µA(ρ∗) = 0. Hence, since µJ < 0,
-the steady state ρ∗ is asymptotically stable if all ηi < 0. Thus, we get a
-sufficient condition for asymptotic stability of the steady state ρ∗:
-0 > ηi = ρ∗
-i (λ′
-i − γ′
-i) +
-�
-k̸=i
-(κ′
-kiρ∗
-k − κ′
-ikρ∗
-i ) > −ϵi for all i
-(29)
-5That is, there exist ϵi > 0 so that this is valid for any |ηi| < ϵi
-
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-where ϵi > 0 is sufficiently small. As this is an asymptotically stable steady
-state, it corresponds to a controlled strict homeostatic state. In this case,
-even if the cell numbers are disturbed (to some degree), the cell population is
-regulated to return to the strict homeostatic state.
-Notably, condition (29) is fulfilled if,
-�
-�
-�
-�
-�
-�
-�
-�
-�
-λ′
-i ≤ 0, γ′
-i ≥ 0 for all i
-λ′
-i < 0 or γ′
-i > 0 at for least one i
-κ′
-ij = 0
-and |λ′
-i|, |γ′
-i|, < ϵi
-(30)
-Furthermore, we may soften the condition on κij to
-κ′
-ij
-κ′
-ji <
-ρ∗
-j
-ρ∗
-i to allow also
-some degree of feedback for the κij.
-The conditions (30) are very similar to the ones for dynamic homeostasis,
-(12), but here these conditions only need to be fulfilled at ρ = ρ∗, whereas
-for dynamic homeostasis they need to be fulfilled for a sufficient range of ρ.
-Moreover, in addition to the qualitative nature of the feedback (related to
-the signs of λ′
-i, γ′
-i), the ‘strength’ of the crowding feedback, i.e. the absolute
-values of λ′
-i, γ′
-i must not be ‘too large’, that is, smaller than ϵi. We cannot, in
-general and for all system sizes, give a definite value for the feedback strength
-bound ϵi below which strict homeostasis is assured. Nevertheless, by using the
-sufficient stability criterion based on the Routh-Hurwitz criterion [24] we can
-identify those bounds for systems with up to three cell states, which guides
-expectations for larger systems. The details of this criterion and the necessary
-derivations are shown in Appendix A. There, we show that for systems with one
-or two cell states, ϵi = ∞, which means that asymptotic stability is ensured for
-arbitrary feedback strengths. For systems with three cell states, we can assure
-that ϵi = ∞ if certain further conditions are met (see Eq. (A13)). Otherwise,
-ϵi can be determined implicitly from the roots of a quadratic form (Eq. (A14)),
-and thus stability may depend on the magnitude of the feedback. In principle,
-such bounds can be found for larger systems too, but the algebraic complexity
-of this process renders it unfeasible to do this in practical terms.
-3.4 Robustness to perturbations and failures
-Now, we wish to assess the robustness of the above crowding control mecha-
-nism, i.e. what occurs if it is disrupted, for example, by the action of toxins,
-other environmental cues, or by cell mutations. More precisely, we will study
-what happens if one or more feedback pathways, here characterised as a param-
-eter αi with α′
-i ̸= 0 fulfilling the conditions for (dynamic or strict) homeostatic
-control, is failing, that is, it becomes α′
-i = 0. We will first address the case
-of tissue-extrinsic factors, i.e. those affecting all the cells in the tissue, and
-then the case of single-cell mutations. In the latter case, only a single cell
-would initially show a dysregulated behaviour, yet, if this confers a proliferative
-advantage, it can lead to hyperplasia and possibly cancer [25–27].
-
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-Homeostatic regulation of renewing tissue cell populations via crowding control
-First, we note that the sufficient condition for strict homeostasis, given
-by Eq. (30), is overly restrictive. In a tissue cell type under crowding feed-
-back control with λ′
-i < 0 and γ′
-i > 0 for more than one i, there is a degree
-of redundancy. That is, if the feedback is removed for one or more of these
-parameters (changing to λ′
-i = 0 and, or γ′
-i = 0), then the sufficient condition
-for a strict homeostatic state remains fulfilled as long as at least one λ′
-i or
-γ′
-i remains non-zero. This possible redundancy confers a degree of robustness,
-meaning that feedback pathways can be removed – setting α′
-i = 0 – without
-losing homeostatic control. Since the necessary conditions, Eqs. (24), are even
-less restrictive, tissue homeostasis may even tolerate more severe disruptions
-that reverse some feedback pathways, e.g. switching from λ′
-i < 0 to λ′
-i > 0,
-as long as other terms in the sum on the right-hand side of (24) compensate
-for this changed sign, ensuring that the sum as a whole is negative. In any
-case, it is important to remind the underlying assumption for which a non-
-trivial steady state exists. In case the variability of the kinetic parameters is
-not enough to assure the condition µ(ρ∗ = 0), then the tissue will degenerate,
-either shrinking and eventually disappearing or indefinitely growing.
-From the above considerations, we conclude that if crowding control applies
-to more than one parameter αi, that is, α′
-i ̸= 0 with appropriate sign and
-magnitude, homeostasis is potentially robust to feedback disruption. This may
-include a simple variation of the feedback function α′
-i but also perturbation in
-the feedback functions shape and complete feedback failure, α′
-i = 0.
-An illustrative example of this situation is shown in Figure 1. Here, the time
-evolution of the cell density is shown for a three-state cell fate model, which has
-been computed by integration of the dynamical system (5) (the details of this
-model are given in Appendix B as Eq. (B15) and illustrated in Figure B1). Four
-kinetic parameters are regulated via crowding control satisfying the sufficient
-condition for strict homeostasis, (30). Then, starting from this homeostatic
-configuration, feedback disruption is introduced at a time equal to zero. In one
-case (“Single failure”), a single kinetic parameter suffers a complete failure of
-the type α′
-i = 0. In this case, the remaining feedback functions compensate
-for this failure, and a new homeostatic condition is achieved. Instead, in the
-second case (“Multiple failures”), failures are applied so that three of the four
-kinetic parameters initially regulated do not adjust with cell density6. Notably,
-the only feedback function left satisfies the condition for asymptotic stability,
-(30). Nevertheless, the variability of this kinetic parameter is not enough to
-assure the existence of a steady state, since in this case, the function µ(ρ) does
-not possess any root. Hence µ > 0 for all ρ, leading to an indefinite growth of
-the cell population. Additional test cases are presented in Appendix B.2.
-So far, we modelled the feedback dysregulation as acting on a global scale,
-thus changing the whole tissue’s dynamics behaviour. This situation represents
-a feedback mechanism affected by cell-extrinsic signals, in which any dysregu-
-lation applies to all the cells in the same way. However, dysregulation can also
-6Only in this example, feedback control fails upon multiple failures, while in general, multiple
-failures may still be compensated to maintain homeostatic control.
-
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-2
-4
-6
-8
-10
-Homeostasis
-Single failure
-Multiple failures
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--0.05
-0
-0.05
-0.1
-0.15
-Homeostasis
-Single failure
-Multiple failures
-Fig. 1
-Cell dynamics in terms of cell density, scaled by the steady state in the homeostatic
-case, as a function of time (left) and the corresponding variation of the dominant eigenvalue µ
-(right). Time is scaled by the inverse of ¯α = mini α∗
-i . The homeostatic model is perturbed at
-a time equal to zero to include feedback failure of the type α′
-i = 0. In the case where only one
-feedback function fails (“Single failure”), the system is able to achieve and maintain a new
-homeostatic state, characterised by a constant cell density and a zero dominant eigenvalue.
-In case more than one feedback fails (“Multiple failures”), the cell dynamics are unstable
-since a steady state does not exist and µ > 0 for all ρ. The cell fate model corresponds to
-model (B15) with parameters given in Table B1 and Table B2.
-act at the single-cell level, for example, when DNA mutations occur. In this
-case, the impact of the dysregulation is slightly different, as explained in the
-following.
-Suppose, upon disruption of crowding control in a single cell, for example,
-by DNA mutations, a sufficient number of crowding feedback pathways remain
-so that there is a steady state and the sufficient condition (30) is still fulfilled.
-In that case, homeostasis is retained, just as when this occurs in a tissue-wide
-disruption. However, if the homeostatic control of that single cell fails such that
-the cell becomes hyperproliferative, µ > 0, or declining, µ < 0, the tissue may
-still remain homeostatic. If µ < 0, the single mutated cell will be lost, upon
-which only a population of crowding controlled cells remain, which remain in
-homeostasis. If µ > 0 in a single cell, hyper-proliferation is not ensured either:
-while the probability for mutated cells to grow in numbers is larger than to
-decline, due to the low numbers, mere randomness can lead to the loss of the
-mutated cell with a non-zero probability, which results in the extinction of
-the dysregulated mutant7. In that case, the mutant cells go extinct and the
-tissue remains homeostatic despite the disruption of homeostatic control in
-the mutated cells; a stark contrast to disruption on the tissue level. Otherwise,
-if the mutant clone (randomly) survives, it will continue to hyper-proliferate
-and eventually dominate the tissue, which is thus rendered non-homeostatic.
-However, the tissue divergence time scale may be much longer than the case
-where the same dysregulation occurs in all cells.
-The deterministic cell population model (5) is suitable for describing the
-average cell numbers. Nevertheless, it fails to describe the stochastic nature of
-7For example, in the case of a single state with cell division rate λ and loss rate γ – a simple
-branching process – the probability for a mutant with µ > 0, that is, λ > γ, to establish is 1−γ/λ,
-which is less than certainty.
-
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-Homeostatic regulation of renewing tissue cell populations via crowding control
-single-cell fate choice. Thus, assessing a single cell’s impact on tissue dynamics
-requires stochastic modelling. To that end, we implemented this situation as
-a Markov process with the same rates as the tissue cell population dynamics
-model8 (see Appendix B.3 for more details).
-In Figure 2, we show numerical simulation results of a stochastic version
-of the model used for previous results in Figure 1, depicted in terms of tissue
-cell density as a function of time. Here, two possible realisations of the same
-stochastic process are presented. We note that the initially homeostatic tissue
-results in stochastic fluctuations of the cell density, which remain, on average,
-constant. At a time equal to zero, a single cell in this tissue switches behaviour,
-presenting multiple failures which, if applied to all the cells, would determine
-the growth of the tissue (corresponding to Multiple failures curve in Figure 1).
-In one instance of the stochastic simulation, however, the mutated clone goes
-extinct after some time, leaving a tissue globally unaffected by the mutation.
-On the other hand, in another instance, the mutated clone prevails, leading to
-the growth of the tissue cell population. The fact that vastly different outcomes
-can occur with the same parameters and starting conditions demonstrates the
-impact of stochasticity in the case of a single-cell mutation.
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-1.2
-1.4
-Homeostasis
-Multiple Failure (instance #1)
-Multiple Failure (instance #2)
-Fig. 2
-Numerical simulation results of a stochastic version of the model used in Figure 1
-upon disruption of crowding control in a single cell, mimicking a DNA mutation. At a
-time equal to 0, the initially homeostatic model is disrupted with a single cell presenting
-multiple failures in the feedback control, as in Figure 1. Two instances of simulations run
-with identical parameters are presented. The rescaled cell density ρ/ρ∗ is shown as a function
-of the time, scaled by the inverse of ¯α = mini α∗
-i . Whilst the mutated cell and its progeny go
-extinct in one instance (#1), in the other (#2), mutated cells prevail and hyper-proliferate
-so that tissue homeostasis is lost. The simulation stops when the clone goes extinct or when
-instability is detected. Full details of the simulation are provided in Appendix B.3.
-3.5 Quasi-dedifferentiation
-In the previous section, we addressed the case where external or cell-intrinsic
-factors disrupt homeostatic control in self-renewing cells of a tissue. However,
-8While a Markov process is an approximation which not necessarily reflects the probability
-distribution of subsequent event times realistically, it is often sufficient to assess the qualitative
-behaviour of a system with low numbers, subject to random influences from the environment.
-
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-situations such as injury, poisoning, or cell radiation might also affect home-
-ostasis in other ways. An example is when stem cells are completely depleted
-from the tissue. In this context, many studies about tissue regeneration after
-injury report evidence of cell plasticity [17, 18], when committed cells regain
-the potential of the previously depleted stem cells. Cell dedifferentiation is just
-an example where differentiated cells return to an undifferentiated state as a
-response to tissue damage. Lineage tracing experiments confirmed this feature
-in vivo in several cases [16, 28–30].
-In the following, we assess how committed progenitor cells respond to the
-depletion of the stem cell pool if they are under crowding feedback control.
-Without loss of generality, let us consider an initially homeostatic scenario
-where there is a self-renewing (i.e. stem) cell type (S) – with growth param-
-eter µ = 0 – at the apex of a lineage hierarchy, and a committed progenitor
-cell type (C) – with µ < 0, but with at least one state that has a non-zero
-cell division rate – below type S in the hierarchy, as depicted in Figure 3.
-Based on this cell fate model, S-cells proliferate and differentiate into C-cells
-while maintaining the S-cell population. The C-cells also proliferate and dif-
-ferentiate into other downstream cell types which we do not explicitly consider
-here. C-cells do not maintain their own population; only the steady influx of
-new cells of that type via differentiation of S-cells into C-cells maintains the
-latter population (see [13]). We further assume that both S- and C-cells are
-under appropriate crowding control, fulfilling both the sufficient conditions for
-dynamic homeostasis, (12), and for stable, strict homeostasis, (30).
-Based on the above modelling, we can write the dynamics of the cell
-densities belonging to the committed progenitor type as,
-d
-dtρc = Ac(ρc)ρc + u ,
-(31)
-where ρc = (ρms+1, ρms+2, .., ρms+mc) are the cell densities in the committed
-C-type, with ms being the number of states of the self-renewing S-type. Ac is
-the dynamical matrix restricted to states in the C-type and ui = �ms
-j=1 κjiρj
-is a constant vector quantifying the influx of cells into the C-type.
-First, we note that the Jacobian matrix of a committed cell type, described
-by (31), J =
-�
-∂A(ρc)ρc
-∂ρj
-�
-j=ms+1,...,ms+mc
-, has the same form as a cell type at the
-apex of the hierarchy, since u does not depend on the densities ρms+1,...,ms+mc.
-From this follows that if C-cells are regulated by crowding control, fulfilling the
-conditions (30), then also the population of C-cells is stable around a steady
-state ρ∗
-c, albeit with a growth parameter µc(ρ∗
-c) < 09.
-We now consider the scenario where all stem cells are depleted at some
-point, as was experimentally done in [16, 18]. This would stop any replen-
-ishment of C-cells through differentiation of S-cells, corresponding to setting
-9This can be seen when multiplying the steady state condition for (31), Ac(ρs, ρc)ρc + u = 0
-with a positive left dominant eigenvector v, giving, µcvρ∗
-c + vu = 0. Since ρ∗ and v have all
-positive entries and u is non-negative, this equation can only be fulfilled for µc < 0.
-
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-Homeostatic regulation of renewing tissue cell populations via crowding control
-Fig. 3 Sketch representative of the quasi-dedifferentiation scenario. A homeostatic system
-enclosed in the black box comprises two cell types: a stem cell type, S, (blue) and a com-
-mitted cell type, C, (green). In the unperturbed homeostatic scenario, S is self-renewing,
-characterised by a growth parameter at the steady state µ∗ = 0, and C is transient, with
-a growth parameter at the steady state µ∗ < 0. Both cell types are subject to crowding
-control, fulfilling both conditions (12), and (30). By removing the stem cell type XS, the
-committed cell type acquires self-renewing property through crowding control, effectively
-becoming a stem cell type (see Figure 4).
-u = 0 in (31). Hence we end up with the dynamics ˙ρc = A(ρc)ρc. Now, assum-
-ing that the function µ(ρ) has sufficient range, so that µ(ρ∗∗
-c ) = 0 for some
-ρ∗∗
-c , and provided that A(ρc) is under crowding control fulfilling the sufficient
-conditions for asymptotic stability of a steady state, then, following our argu-
-ments from section 3.3, the population of C-cells will attain a stable steady
-state. In other words, those previously committed cells become self-renewing
-cells. Also, since they now reside at the apex of the lineage hierarchy (given
-that S-cells are absent), they effectively become stem cells.
-Hence, under crowding control, previously committed progenitor cells
-(committed cells that can divide) will automatically become stem cells if the
-original stem cells are depleted. Commonly, such a reversion of a committed cell
-type to a stem cell type would be called ‘dedifferentiation’ or ‘reprogramming’.
-However, in this case, no genuine reversion of cell states occurs; previously
-committed cells do not transition back to states associated with the stem cell
-type. Instead, they respond by crowding feedback and adjust their dynamical
-rates so that µ becomes zero, hence attaining a self-renewing cell type. Cru-
-cially, this new stem cell type is fundamentally different to the original one
-and still most similar to the original committed type. We call this process
-quasi-dedifferentiation. The quasi-dedifferentiation follows the same reversion
-of proliferative potential as in ‘genuine’ dedifferentiation but without explicit
-reversion in the cell state trajectories.
-The following numerical example illustrates this situation. We focus on the
-cell dynamics of a single C-type regulated via crowding feedback (detail of
-the model are provided in Appendix B.4). The cell density as a function of
-the time, shown in Figure 4, is obtained by integrating the corresponding cell
-population model according to Eq. (5). The system is initially in a homeostatic
-condition, meaning that there is a constant influx of cells from some upstream
-self-renewing types. Such upstream types are assumed to be properly regulated
-such that this cell influx is constant over time. At a time equal to zero, the cell
-influx becomes suddenly zero, representing an instantaneous removal of all the
-
-Xo-
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--10
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-Fig. 4 Cell dynamics of an initially committed cell type C (µ < 0) upon removal of all stem
-cells. (Left) Cell density scaled by the steady-state density as a function of time. (Right)
-Corresponding variation of the dominant eigenvalue µc. Time is scaled by the inverse of
-¯α = mini α∗
-i . It is assumed that a stem cell type, S, initially resides in the lineage hierarchy
-above the committed cell type (as in Figure 3). S cells differentiate into C cells, which is
-modelled as a constant cell influx of C-cells (S is not explicitly simulated). At a time equal
-to zero, a sudden depletion of S cells is modelled by stopping the cell influx. After some
-transitory phase, the cell population stabilises around a new steady state and becomes self-
-renewing with µc = 0. The full description of the dynamical model, which corresponds to
-model (B15) with parameters given in Table B1, is reported in Appendix B.4.
-self-renewing cells from the tissue. A new homeostatic condition is achieved
-after a transitory phase thanks to the crowding feedback acting on the C-
-type. This example demonstrates how an initially committed cell type, i.e.
-with µc < 0, regulated via crowding feedback, might be able to switch, upon
-disruption, to a self-renewing behaviour µc = 0.
-4 Discussion
-For maintaining healthy adult tissue, the tissue cell population must be
-maintained in a homeostatic state. Here, we assessed one of the most com-
-mon generalised regulation mechanisms of homeostasis, which we refer to as
-crowding feedback. Based on this, progenitor cells (stem cells and committed
-progenitors) adjust their propensities to divide, differentiate, and die, accord-
-ing to the surrounding density of cells, which they sense via biochemical or
-mechanical signals. For this purpose, we used a generic mathematical model
-introduced before in Refs. [13, 20], which describes tissue cell population
-dynamics in the most generic way, including cell divisions, cell state transi-
-tions, and cell loss / differentiation. Based on this model, we rigorously define
-what is meant when speaking of a ‘homeostatic state’, introducing two notions:
-a strict homeostasis is a steady state of the tissue cell population dynamics,
-while dynamical homeostasis allows, in addition to strict homeostasis, for oscil-
-lations and fluctuations, as long as a finite long-term average cell population
-is maintained (such as the endometrium during the menstrual cycle).
-By analysing this dynamical system, we find several sufficient and necessary
-conditions for homeostasis. These conditions are formulated in terms of how
-the propensities of cell division, differentiation, and cell state changes, of cells
-
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-Homeostatic regulation of renewing tissue cell populations via crowding control
-whose type is at the apex of an adult cell lineage hierarchy, may depend on
-their cell density. We find that when, for a wide range of cell density values,
-the cell division propensity of at least one state decreases with cell density or
-the differentiation propensity increases with it, while other propensities (e.g.
-of cell state transitions) are not affected by the cell density, then dynamic
-homeostasis prevails (12). For strict homeostasis to prevail, this only needs
-to be fulfilled at the steady state itself, but in addition, the magnitude of
-the feedback strength may not be too large (30). We can derive explicit and
-implicit expressions for the bound on feedback strength for systems of two
-and three-cell states but cannot do so for arbitrary systems. Furthermore, we
-find that a necessary condition for strict homeostasis is that the conditions for
-dynamic homeostasis are met at least at the steady state cell density.
-A direct consequence of the conditions we found is that they allow for a
-considerable degree of redundancy when more than one propensity depends
-appropriately on the cell density. Hence feedback pathways, that is, cell dynam-
-ics parameters depending on the cell density, may serve as ‘back-ups’ to each
-other if one fails. We demonstrate that this confers robustness to the home-
-ostatic system in that one or more crowding feedback pathways may fail, yet
-the tissue remains in homeostasis.
-Finally, we assess how crowding feedback regulation affects the response of
-committed progenitor cells to a complete depletion of all stem cells. We showed
-that committed cells which can divide and are under appropriate crowding
-feedback control (that is, meeting the sufficient conditions (12) and (30)), will
-necessarily, without additional mechanisms or assumptions, reacquire stem cell
-identity, that is, become self-renewing and are at the apex of the lineage hierar-
-chy. Notably, while this process resembles that of dedifferentiation, it does not
-involve explicit reprogramming, in that the cell state transitions are reversed.
-Instead, only the cell fate propensities adjust to the changing environment by
-balancing proliferation and differentiation as is required for self-renewal. While
-these are purely theoretical considerations, and such a process has not yet
-been experimentally found, we predict that it must necessarily occur under the
-appropriate conditions. This can be measured by assessing the gene expression
-profiles (e.g. via single-cell RNA sequencing) of cells that ‘dedifferentiate’, i.e.
-reacquire stemness after depletion of stem cells. Moreover, those considerations
-yield further, more general insights:
-• Stem cell identity is neither the property of individual cells nor is it strictly
-associated with particular cell types or states. Any cell that can divide and
-differentiate, committed or not, may become a stem cell under appropriate
-environmental control.
-• From the latter follows that stemness is a property determined by the
-environment, not the cell itself.
-• ‘Cell plasticity’ might need to be seen in a wider context. Usually, cell
-plasticity is associated with a change of a cell’s type when subjected to
-environmental cues, which involves a substantial remodelling of the cell’s
-morphology and biochemical state. However, we see that a committed cell
-
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-may turn into a stem cell simply by adjusting the pace of the cell cycle
-and differentiation processes to the environment. This may not require
-substantial changes in the cell’s state.
-This exemplifies that homeostatic control through crowding feedback is not
-only a way to render homeostasis stable and robust, but also to create stem
-cell identities as a collective property of the tissue cell population.
-Acknowledgments.
-We thank Ben MacArthur and Ruben Sanchez-Garcia
-for valuable discussions.
-Declarations
-PG is supported by an MRC New Investigator Award, Grant number
-MR/R026610/1. The code generated for numerical computations in the cur-
-rent study is available on Github, https://github.com/cp4u17/Feedback. No
-other data was generated for this work.
-Contributions are as follows: C.P. and P.G. conceptualised the paper, C.P.
-and P.G. did the mathematical analysis, C.P. did the numerical analysis, P.G.
-supervised the work.
-The authors have no competing interests to declare that are relevant to the
-content of this article.
-Appendix A
-Asymptotic stability assessment
-based on Routh-Hurwitz
-A.1
-Background
-In control system theory, a commonly used method for assessing the stability
-of a linear system is the Routh-Hurtwiz (RH) criterion [24]. It is an algebraic
-criterion providing a necessary and sufficient condition on the parameters of a
-dynamic system of arbitrary order to ensure the dynamics are asymptotically
-stable. In particular, the criterion defines a set of conditions on the coefficients,
-pi, of the characteristic polynomial, P(s), written as
-P(s) = sn +
-n
-�
-i=1
-pisn−i ,
-(A1)
-in which n corresponds to the dimension of the system. Note that the notation
-used in this section, based on that from [24], is different from that of the main
-text, where pi is the polynomial coefficient of ith order.
-A first result of the RH criterion is that a necessary condition for the
-dynamical system to be asymptotically stable is that all the coefficients must
-be positive, that is,
-pi > 0, for all i .
-(A2)
-
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-Additional conditions on the polynomial coefficients are added for a necessary
-and sufficient condition. These conditions are based on Routh’s array, written
-as
-�
-�����
-1 p2 p4 ... 0
-p1 p3 ...
-b1 b2 ...
-c1
-...
-�
-�����
-,
-(A3)
-in which the first two rows contain all the coefficients of the characteristic
-polynomial, and the following ones are recursively computed as
-bi = −
-det
-�
-1
-p2i
-p1 p2i+1
-�
-p1
-,
-(A4)
-ci = −
-det
-�
-p1 p2i+1
-b1
-bi
-�
-b1
-,
-(A5)
-and so on until a zero is encountered. The RH criterion states that the system is
-asymptotically stable if and only if the elements in the first column of Routh’s
-array are positive.
-Based on that, it can be easily shown that for a second-order polynomial,
-the necessary condition (A2) is also sufficient for asymptotic stability (a.s.)
-since b1 = p1p2, which means that
-The system is a. s.
-⇐⇒ pi > 0, for i = 1, 2 .
-(A6)
-Instead, the necessary and sufficient condition for a polynomial of order three
-results in
-The system is a. s.
-⇐⇒ pi > 0, for i = 1, 2, 3 and p1p2 − p3 > 0 .
-(A7)
-The same reasoning can be applied to higher-order dynamics to derive
-additional conditions on the coefficients pi.
-A.2
-Verification of the necessary condition for
-asymptotic stability
-The Matlab code for verifying (23) is provided in https://github.com/
-cp4u17/Feedback.git.
-The strategy used is to evaluate each term in Eq. (23) and simply compare
-the left and right-hand sides of the equation. We followed a symbolic approach
-(based on the Matlab symbolic toolbox) for an arbitrary three-state model. A
-numerical approach was used instead for higher-order dynamics, specifically
-4, 5 and 6 state cell fate models. To do so, we randomly defined the cell
-dynamical matrix at the steady state, A(ρ∗), and its derivative with respect
-
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-to ρ. Entries were chosen from a uniform distribution and, for assuring a zero
-dominant eigenvalue for A(ρ∗), a local optimiser (fmincon function of Matlab)
-was used to find appropriate diagonal elements. For each dimension of the cell
-fate model, we tested up to 1000 random cases.
-A.3
-Sufficient condition for asymptotic stability
-In this section, we will indicate with the superscripts A and J the coefficients of
-the characteristic polynomial expressed as Eq. (A1) respectively of the matrix
-of the dynamical system, Eq. (6), and those of the Jacobian matrix, Eq. (15).
-For a two and three-state system, the following relations can be alge-
-braically derived
-pJ
-1 = pA
-1 −
-�
-i
-ηi .
-(A8)
-where ηi is according to Eq. (16). Again, considering that pA
-1 > 0, if all ηi ≤ 0
-then pJ
-1 > 0.
-Hence, the above relation implies that in a two-state system, the RH cri-
-terion given by Eq. (A6) is fulfilled when η ≤ 0, with at least one negative
-component (otherwise J = A) and therefore the system is asymptotically sta-
-ble. We recall that asking ηi ≤ 0 without further constraints is equivalent to
-the previously derived condition (30) with ϵi = ∞.
-For applying the RH criterion to a three-state cell dynamic system, given
-by Eq. (A7), we need to evaluate the sign of pJ
-2 and then that of pJ
-1 pJ
-2 − pJ
-3 .
-To do so, we first write
-pJ
-2 = pA
-2 −
-�
-i
-fiηi ,
-(A9)
-in which fi = �
-j aji − Tr(A) for i = 1, 2, 3. Since the off-diagonal elements
-are non-negative, and the trace of A is negative, then fi > 0 for i = 1, 2, 3.
-That means that if all ηi ≤ 0 then pJ
-2 > 0. Concerning the term pJ
-1 pJ
-2 − pJ
-3 ,
-this can be written as a quadratic form in η =
-�
-η1, η2, η3
-�
-as
-pJ
-1 pJ
-2 − pJ
-3 = Q(η) = ηT AQη + bT
-Qη + cQ ,
-(A10)
-in which
-AQ =
-�
-�
-f1 f1 f1
-f2 f2 f2
-f3 f3 f3
-�
-� ,
-(A11)
-bQ = −pA
-1
-�
-�
-f1
-f2
-f3
-�
-� − pA
-2
-vw
-�
-�
-v3(w3 − w1) + v2(w2 − w1)
-v3(w3 − w2) + v1(w1 − w2)
-v2(w2 − w3) + v1(w1 − w3)
-�
-� ,
-(A12)
-and cQ = pA
-1 pA
-2 . Here, v = (v1, v2, v3) is a left dominant eigenvector and
-w = (w1, w2, w3) a right dominant eigenvector.
-We now note that the matrix AQ is semidefinite positive since two eigen-
-values are zero (the rows are two-fold degenerate) and one is positive, equal
-to Tr(AQ) = �
-i fi, and cQ > 0. We now distinguish two cases, depending on
-
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-Homeostatic regulation of renewing tissue cell populations via crowding control
-the sign of bQ elements. First, if bQ ≤ 0, then Q(η) > 0 for any η ≤ 0. Since
-fi, pA
-1 , pA
-2 , vw > 0, we get a sufficient condition for bQ ≤ 0, namely,
-0 ≤ v3(w3 − w1) + v2(w2 − w1)
-(A13)
-0 ≤ v3(w3 − w2) + v1(w1 − w2)
-0 ≤ v2(w2 − w3) + v1(w1 − w3)
-In that case, asymptotic stability and thus crowding feedback control is assured
-for any η < 0, and thus the bound for feedback strength is ϵi = ∞ for i =
-1, 2, 3.
-Otherwise, if there is at least one positive element in bQ, then Q(η) > 0
-only if |ηi| < ϵi, where ϵ = (ϵ1, ϵ2, ϵ3) are the absolute values of the solutions
-to the equation Q(η) = 0, that is – given that ηi are negative – the solution to,
-0 = ϵT AQϵ − bT
-Qϵ + cQ .
-(A14)
-Importantly, we note that the elements of bQ depend uniquely on the proper-
-ties of the dynamical system and therefore, they can be determined without
-requiring the knowledge of the parameter derivatives, i.e. the specific crowding
-feedback dependencies.
-The Matlab code for verifying (A8), (A9) and (A10) is provided in
-https://github.com/cp4u17/Feedback.git.
-Appendix B
-Test case
-B.1
-Asymptotic stability
-This section reports the details of the model used for numerical examples
-presented in the main text. The cell dynamics correspond to the following
-three-state cell fate model
-X1
-λ1
-−→ X1 + X1,
-X1
-ω13
-−−→ X3,
-X1
-γ1
-−→ ∅
-X2
-ω21
-−−→ X1,
-X2
-ω23
-−−→ X3,
-X2
-γ2
-−→ ∅
-X3
-λ3
-−→ X3 + X3,
-X3
-ω31
-−−→ X1,
-X3
-ω32
-−−→ X2,
-(B15)
-whose network is shown in Figure B1. In such a model, for simplicity, we only
-consider symmetric self-renewing divisions so that κij = ωij. Also, we apply
-the crowding feedback to division rates, λi, and differentiation rates γi. In this
-way, it is straightforward to apply the sufficient condition (30) for asymptotic
-stability since κ′
-ij = 0 for all i, j.
-Hence, each kinetic parameter of the type αi ∈ {λj, γj}j=1,...,3 is expressed
-as a function of ρ, whilst those of the type αi ∈ {κjk}j,k=1,...,3 are constant. In
-particular, we chose a Hill function [31] where αi(ρ) = ci + kiρni/(Kni
-i
-+ ρni)
-in case αi is a differentiation rate, so that α′
-i = ∂αi/∂ρ > 0, and αi(ρ) =
-
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-ci +ki/(Kni
-i +ρ/ni) in case it is a proliferation rate, so that α′
-i < 0. According
-to (30) this choice assures that, if there is a value ρ = ρ∗ for which µ(ρ∗) = 0,
-this corresponds to an asymptotically stable steady state.
-The parameter values used in our example are reported in Table B1, and
-the profiles of the proliferation and differentiation rates as a function of ρ are
-shown in Figure B2. Based on these values, the steady state corresponds to
-ρ∗ = 1 (arbitrary unit). As expected, the dominant eigenvalue of the Jacobian
-at the steady state is negative (µJ = −1.21).
-To test the dynamical behaviour of the tissue cell population, we numer-
-ically solved the system of ODEs (5) for different initial conditions based on
-the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 function).
-The results are shown in Figure B3 as the time evolution of ρ, normalised
-by the steady-state ρ∗, (left panels), and of the dominant eigenvalue, µ (right
-panels). The label H indicates an initial condition corresponding to the self-
-renewing state ρ∗, that is, the system is initially in homeostasis. In the
-simulations labelled as P− and P+, we applied perturbation in the initial
-state ρ∗ = (ρ∗
-1, ρ∗
-2, ρ∗
-3), which are, respectively,
-�
-0.8ρ∗
-1, 0.75ρ∗
-2, 0.85ρ∗
-3
-�
-and
-�
-1.5ρ∗
-1 1.1ρ∗
-2 1.2ρ∗
-3
-�
-. As expected, in all these cases, the feedback’s effect is sta-
-bilising the system so that it returns to the steady state upon perturbation,
-ρ → ρ∗, (asymptotic stability) and thus regains self-renewal property, µ → 0,
-over time.
-Fig. B1
-Cell state network representing a cell type composed of three states. The links
-represent direct transitions, ωij; symmetric divisions occur with rates λi and differentiation
-with rate γi, where subscripts i, j = 1, 2, 3 indicate the corresponding cell state, as per model
-(B15).
-B.2
-Failure of feedback function
-Based on the cell fate model regulated via crowding feedback described in
-the previous section, we assess the impact of failure in one or more feedback
-functions. In particular, the failure of the crowding regulation is modelled,
-assuming one or more kinetic parameters as a constant. Five different failure
-test cases are assessed. For doing so, we chose αi = (1 + C)α∗
-i being constant
-instead of depending on ρ, in which α∗ is the value at the steady state when
-
-M
-Y1
-XI
-23
-1
-013
-021
-1
-031
--
-X
-X2
-023
-032Springer Nature 2021 LATEX template
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-Homeostatic regulation of renewing tissue cell populations via crowding control
-k
-K
-n
-α∗
-α′
-λ1
-0.74
-0.57
-2.00
-0.61
--0.84
-λ3
-7.79
-2.07
-2.00
-1.53
--0.56
-γ1
-3.07
-1.22
-2.00
-1.28
-1.48
-γ2
-2.28
-0.43
-2.00
-1.97
-0.61
-κ13
-–
-0.95
-0.00
-κ21
-–
-1.44
-0.00
-κ23
-–
-1.71
-0.00
-κ31
-–
-2.03
-0.00
-κ32
-–
-1.35
-0.00
-Table B1
-Values of the Hill function parameters describing the kinetic parameters in
-case of homeostasis regulation via crowding feedback for the cell fate model (B15). The
-generic kinetic parameters (represented as αi in the right columns of the table) are a
-function of the total cell density, ρ, and are given by γi(ρ) = c + kρn/(Kn + ρn) and
-λi(ρ) = c + k/(Kn + ρn) with i = 1, 2, 3. A common value c = 0.05 is assumed. State
-transition rates ωij, are constant and equal to κij. For such a cell fate dynamics, the steady
-state is ρ∗ = 1. The unit of the kinetic parameter is arbitrary and therefore omitted. Unless
-specified otherwise, these values apply to all the numerical examples presented in this work.
-0
-0.5
-1
-1.5
-2
-0
-0.5
-1
-1.5
-2
-2.5
-0
-0.5
-1
-1.5
-2
--4
--2
-0
-2
-4
-Fig. B2
-Proliferation and differentiation rates (left panels, with α as a generic placeholder
-for parameters), and their derivative with respect to ρ (right panels) as functions of cell
-density normalised by the steady-state ρ∗ for the cell fate model (B15) schematised in
-Figure B1. The profiles in the left panel correspond to Hill functions defined in Table B1.
-there are no failures (reported in Table B1) and C is a constant (reported in
-Table B2). Five test cases, indicated as F1−5, are assessed.
-In test case F1, only one feedback fails. Three of the four kinetic parameters
-fail in cases F2−4. Finally, F5 represents a case where all the feedback functions
-fail. The corresponding variability of the dominant eigenvalue, µ, as a function
-of the cell density is shown in Figure B4. It is clear that whilst F1−4 cases
-satisfy the sufficient condition for strict homeostasis, (30), in test cases F5,
-the dominant eigenvalue being constant means that there is no homeostatic
-regulation. Importantly, there is no steady state in test cases F2,4 since the
-dominant eigenvalue is always positive in one case or negative in the other.
-Based on these assumptions, we numerically solved the system of ODEs
-(5) using the explicit Runge-Kutta Dormand-Prince method (Matlab ode45
-function). The failure test cases start at time 0 from an initially homeostatic
-condition, H. The results are shown in Figure B5 as the time evolution of
-ρ, normalised by the homeostatic steady-state, ρ∗, (left panels), and of the
-
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-0
-5
-10
-15
-0.8
-1
-1.2
-1.4
-H
-P-
-P+
-0
-5
-10
-15
--0.6
--0.4
--0.2
-0
-0.2
-0.4
-H
-P-
-P+
-Fig. B3
-Effect of perturbation of homeostasis under crowding control, when feedback
-parameters are according to Table B1. (Left) Cell density ρ, scaled by the steady-state ρ∗,
-as a function of time. (Right) Corresponding variation of the dominant eigenvalue µ. Time
-is scaled by the inverse of ¯α = mini α∗
-i . Three different initial condition are tested: H,
-corresponds to the steady state ρ∗ = (ρ∗
-1, ρ∗
-2, ρ∗
-3), P− to
-�0.8ρ∗
-1, 0.75ρ∗
-2, 0.85ρ∗
-3
-�
-and P+
-to
-�1.5ρ∗
-1, 1.1ρ∗
-1, 1.2ρ∗
-1
-�
-. Since the steady state is asymptotically stable, thanks to crowding
-control, the cell population remain in, or return to, a homeostatic state characterised by
-µ = 0.
-Parameter
-F1
-F2
-F3
-F4
-F5
-λ1
-+5%
-+5%
-+5%
--20%
--5%
-λ3
--
-+5%
-+5%
--20%
--5%
-γ1
--
--5%
--
-+20%
--5%
-γ2
--
--
--5%
--
--5%
-Table B2
-Value of the constant C in the feedback failure test cases. Whenever a failure
-in the feedback of one kinetic parameter α occurs, that parameter is modelled as a
-constant, α = (1 + C)α∗, in which the steady-state value, α∗, is reported in Table B1. Test
-cases F1 and F2 correspond to those presented in the main text (Figure 1).
-dominant eigenvalue, µ, (right panels). Note that the cases F1,2 correspond
-respectively to the Single failure and Multiple failures reported in the
-main text (Figure 1).
-In two cases, F1,3, despite a single or multiple feedback functions failing, a
-new homeostatic condition is reached after some time, where µ = 0. However,
-suppose a different set of feedback fails, like in F2,4, such that the dominant
-eigenvalue is respectively positive or negative for any ρ. In that case, no steady
-state can be attained, and the tissue cell population will hyper-proliferate or
-decline in the long term. Hence, even if the condition for asymptotic stability
-is met, there is no steady state. Finally, if homeostasis is not regulated at
-all, as in F5, then the population dynamics only depend on the value of the
-dominant eigenvalue (the cell dynamical model (5) turns linear). In the case
-shown, µ > 0 and therefore, the cell population diverges.
-B.3
-Single cell mutation scenario
-To assess the tissue dynamics with a single-cell mutation, as presented in the
-main text, we modelled the clonal dynamics, namely, the dynamics of single
-cells and their progeny. For doing so, we considered the model (B15) as a
-
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-0
-0.5
-1
-1.5
-2
--2
-0
-2
-4
-H
-F1
-F2
-F3
-F4
-F5
-Fig. B4
-Variation of the dominant eigenvalue µ as a function of the cell density, ρ,
-normalised by the reference homeostatic state value, ρ∗. The curve H corresponds to the
-reference homeostatic model presented in Appendix B.1. The other curves, F1−5, represent
-different sets of feedback failure, as reported in Table B2.
--10
-0
-10
-20
-30
-40
-0
-0.5
-1
-1.5
-2
-2.5
-H
-F1
-F2
-F3
-F4
-F5
--10
-0
-10
-20
-30
-40
--0.6
--0.4
--0.2
-0
-0.2
-H
-F1
-F2
-F3
-F4
-F5
-Fig. B5
-Failure of feedback control. (Left) Cell density, scaled by the steady state in the
-homeostatic case, as a function of time. (Right) Corresponding variation of the dominant
-eigenvalue µ. Time is scaled by the inverse of ¯α = mini α∗
-i . The homeostatic model, H, is
-perturbed at a time equal to zero to include the feedback failure reported in Table B2. Whilst
-in F1,3, the regulation is able to achieve and maintain a new homeostatic state (µ = 0),
-the remaining case fails to regulate the cell population, leading to an indefinite growth or
-shrinking of the tissue.
-Markov process with the same numerical rates as before, but now events are
-treated as stochastic. Then, we run numerical simulations using the Gillespie
-algorithm [32] to evaluate this model. In particular, the results presented in
-this work are based on 100 independent instances, where each instance is a
-possible realisation of the stochastic process. We chose a total cell number
-N0 = 5000 as the initial condition (cell density is based on unitary volume).
-In real tissues, the number of cells could be a few orders of magnitude larger.
-However, this number is sufficiently large to avoid the extinction of the process
-in the time scale analysed, so once rescaled, these dynamics are representative
-of those in the tissue. All the simulations are stopped when the mutated clone
-goes extinct or divergence of the dynamics is detected, defined as reaching
-N = 5N0.
-
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-From an implementation point of view, we consider a cell fate model
-represented by two disconnected cell state networks to model the tissue dynam-
-ics, including the mutated cell. One network corresponds to the unperturbed
-test case H, and the other to the dysregulated one, F2 (both described in
-Appendix B.2). The simulation starts with N0 cells in the H network, dis-
-tributed in each state proportionally to the expected steady-state distribution
-in the tissue, and no cells in the F2 network. Thus, since the two networks
-are disconnected, F2 remains empty, and the simulation represents the tissue
-dynamics before the dysregulation. At a time equal to zero, we moved one
-cell from a random state in the H network to the corresponding state in the
-F2 one. This simulation represents the tissue dynamics, including the single
-mutated cell.
-In Figure B6 (left), all the trajectories where the mutated clones go extinct
-are shown. In these cases, the tissue dynamics remain globally unaffected by
-the mutation. Due to the stochastic nature of the process, mutant clones can
-go extinct even if the growth parameter is positive. That is, even in cases where
-divergence would be observed for the tissue-wide disruption. However, this does
-not occur in all the instances. The right panel of the same figure shows those
-instances where the mutated clone does not go extinct and eventually prevails,
-resulting in diverging cell population dynamics. For the chosen parameters,
-this divergence of the mutated clone is detected in 6% of all cases. Surprisingly,
-only a few clones survive despite a proliferative advantage, but this is plausible
-for a small fitness advantage (For example, in the case of a single state with
-cell division rate λ and loss rate γ – a simple branching process [33] – the
-probability for the a mutant with µ > 0, that is, λ > γ, to establish is 1−γ/λ,
-which can be very low for λ ≈ γ).
-In the main text (Figure 2), only one profile for each scenario is shown,
-respectively. They correspond to instance #24 for the homeostatic case and
-instance #43 for the diverging case.
-B.4
-Quasi-dedifferentiation
-The numerical example presented in the main text is based on the same cell
-fate model described in Appendix B.1. To model the dynamics of a committed
-cell type, we choose a constant non-negative u =
-�
-0.02 0.07 0.06
-�T to model
-for the cell influx. For such a model, the steady state, ρ∗
-c, is asymptotically
-stable.
-The figures presented in the main text are based on the numerical integra-
-tion of the system of ordinary differential equation (31). In particular, we used
-the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 function).
-References
-[1] National Institute of Health: Stem Cell Basics (2016). https://stemcells.
-nih.gov/info/basics
-
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-0
-5
-10
-15
-20
-0.8
-0.9
-1
-1.1
-1.2
-H
-F2
-0
-20
-40
-60
-80
-0.8
-0.9
-1
-1.1
-1.2
-H
-F2
-Fig. B6
-Results of numerical simulations of the stochastic process representing the cell
-dynamics, according to section B.3. The cell density, scaled by the steady state in the
-homeostatic case, as a function of the time is shown for 100 random instances. Each shown
-trajectory is the result of a different instance of the stochastic process. At a time equal to
-zero, the cell mutation is modelled as a switch of a single random cell from the homeostatic
-H cell dynamics to the F2 model assessed in Appendix B.2. On the left panel, only the
-trajectories for which the mutated clone goes extinct are shown. The right panel shows the
-trajectories in which the mutated clone prevails. Dynamics are scaled by ¯α = mini{α∗
-i }.
-[2] Marinari, E., Mehonic, A., Curran, S., Gale, J., Duke, T., Baum, B.:
-Live-cell delamination counterbalances epithelial growth to limit tis-
-sue overcrowding. Nature 484(7395), 542–545 (2012). https://doi.org/10.
-1038/nature10984
-[3] Eisenhoffer, G.T., Loftus, P.D., Yoshigi, M., Otsuna, H., Chien, C.-B.,
-Morcos, P.A., Rosenblatt, J.: Crowding induces live cell extrusion to main-
-tain homeostatic cell numbers in epithelia. Nature 484(7395), 546–549
-(2012). https://doi.org/10.1038/nature10999
-[4] Eisenhoffer, G.T., Rosenblatt, J.: Bringing balance by force: Live cell
-extrusion controls epithelial cell numbers. Trends in Cell Biology 23(4),
-185–192 (2013). https://doi.org/10.1016/j.tcb.2012.11.006
-[5] Puliafito, A., Hufnagel, L., Neveu, P., Streichan, S., Sigal, A., Fygenson,
-D.K., Shraiman, B.I.: Collective and single cell behavior in epithelial con-
-tact inhibition. Proceedings of the National Academy of Sciences 109(3),
-739–744 (2012). https://doi.org/10.1016/j.juro.2012.06.073
-[6] Gudipaty, S.A., Lindblom, J., Loftus, P.D., Redd, M.J., Edes, K., Davey,
-C.F., Krishnegowda, V., Rosenblatt, J.: Mechanical stretch triggers rapid
-epithelial cell division through Piezo1. Nature 543(7643), 118–121 (2017).
-https://doi.org/10.1038/nature21407
-[7] Shraiman, B.I.: Mechanical feedback as a possible regulator of tis-
-sue growth. Proceedings of the National Academy of Sciences 102(9),
-3318–3323 (2005). https://doi.org/10.1073/pnas.0404782102
-[8] Kitadate, Y., J¨org, D.J., Tokue, M., Maruyama, A., Ichikawa, R.,
-
-Springer Nature 2021 LATEX template
-Homeostatic regulation of renewing tissue cell populations via crowding control
-31
-Tsuchiya, S., Segi-Nishida, E., Nakagawa, T., Uchida, A., Kimura-
-Yoshida, C., Mizuno, S., Sugiyama, F., Azami, T., Ema, M., Noda, C.,
-Kobayashi, S., Matsuo, I., Kanai, Y., Nagasawa, T., Sugimoto, Y., Taka-
-hashi, S., Simons, B.D., Yoshida, S.: Competition for Mitogens Regulates
-Spermatogenic Stem Cell Homeostasis in an Open Niche. Cell Stem Cell
-24(1), 79–92 (2019). https://doi.org/10.1016/j.stem.2018.11.013
-[9] Johnston, M.D., Edwards, C.M., Bodmer, W.F., Maini, P.K., Chapman,
-S.J.: Mathematical modeling of cell population dynamics in the colonic
-crypt and in colorectal cancer. Proceedings of the National Academy
-of Sciences 104(10), 4008–4013 (2007). https://doi.org/10.1073/pnas.
-0611179104
-[10] Sun, Z., Komarova, N.L.: Stochastic modeling of stem-cell dynamics with
-control Zheng. Math Bioscience 240(2), 231–240 (2012). https://doi.org/
-10.1016/j.mbs.2012.08.004
-[11] Bocharov, G., Quiel, J., Luzyanina, T., Alon, H., Chiglintsev, E., Cheresh-
-nev, V., Meier-Schellersheim, M., Paul, W.E., Grossman, Z.: Feedback
-regulation of proliferation vs. differentiation rates explains the depen-
-dence of CD4 T-cell expansion on precursor number. Proceedings of the
-National Academy of Sciences of the United States of America 108(8),
-3318–3323 (2011). https://doi.org/10.1073/pnas.1019706108
-[12] Greulich, P., Simons, B.D.: Dynamic heterogeneity as a strategy of
-stem cell self-renewal. Proceedings of the National Academy of Sciences
-113(27), 7509–7514 (2016). https://doi.org/10.1073/pnas.1602779113
-[13] Greulich, P., MacArthur, B.D., Parigini, C., S´anchez-Garc´ıa, R.J.: Uni-
-versal principles of lineage architecture and stem cell identity in renewing
-tissues. Development 148, 194399 (2021)
-[14] Donati, G., Rognoni, E., Hiratsuka, T., Liakath-Ali, K., Hoste, E., Kar,
-G., Kayikci, M., Russell, R., Kretzschmar, K., Mulder, K.W., Teich-
-mann, S.A., Watt, F.M.: Wounding induces dedifferentiation of epidermal
-Gata6+ cells and acquisition of stem cell properties. Nature Cell Biology
-19, 603–613 (2017). https://doi.org/10.1038/ncb3532
-[15] Jopling, C., Boue, S., Belmonte, J.C.I.: Dedifferentiation, transdifferenti-
-ation and reprogramming: three routes to regeneration. Nature Reviews
-Molecular Cell Biology 12, 79 (2011)
-[16] Tata, P.R., Mou, H., Pardo-Saganta, A., Zhao, R., Prabhu, M., Law, B.M.,
-Vinarsky, V., Cho, J.L., Breton, S., Sahay, A., Medoff, B.D., Rajagopal,
-J.: Dedifferentiation of committed epithelial cells into stem cells in vivo.
-Nature 503(7475), 218–223 (2013). https://doi.org/10.1038/nature12777
-
-Springer Nature 2021 LATEX template
-32
-Homeostatic regulation of renewing tissue cell populations via crowding control
-[17] Tetteh, P.W., Farin, H.F., Clevers, H.: Plasticity within stem cell hier-
-archies in mammalian epithelia. Trends in Cell Biology 25(2), 100–108
-(2015). https://doi.org/10.1016/j.tcb.2014.09.003
-[18] Tetteh, P.W., Basak, O., Farin, H.F., Wiebrands, K., Kretzschmar, K.,
-Begthel, H., van den Born, M., Korving, J., De Sauvage, F.J., van Es, J.H.,
-Van Oudenaarden, A., Clevers, H.: Replacement of Lost Lgr5-Positive
-Stem Cells through Plasticity of Their Enterocyte-Lineage Daughters.
-Cell Stem Cell 18(2), 203–213 (2016). https://doi.org/10.1016/j.stem.
-2016.01.001
-[19] Murata, K., Jadhav, U., Madha, S., van Es, J., Dean, J., Cavazza, A.,
-Wucherpfennig, K., Michor, F., Clevers, H., Shivdasani, R.A.: Ascl2-
-Dependent Cell Dedifferentiation Drives Regeneration of Ablated Intesti-
-nal Stem Cells. Cell Stem Cell 26(3), 377–3906 (2020). https://doi.org/
-10.1016/j.stem.2019.12.011
-[20] Parigini, C., Greulich, P.: Universality of clonal dynamics poses funda-
-mental limits to identify stem cell self-renewal strategies. eLife 9, 1–44
-(2020). https://doi.org/10.7554/eLife.56532
-[21] Bollob´as, B.: Modern Graph Theory, 1st edn. Graduate Texts in Mathe-
-matics 184. Springer, New York (1998)
-[22] MacCluer, C.R.: The Many Proofs and Applications of Perron’s Theorem.
-SIAM Review 42(3), 487–498 (2000)
-[23] Horn,
-R.A.,
-Johnson,
-C.R.:
-Matrix
-Analysis,
-2nd
-edn.
-Cam-
-bridge University Press, Cambridge (1985). https://doi.org/10.1017/
-cbo9780511810817
-[24] Franklin, G.F., Powell, J.D., Emami-Naeini, A.: Feedback Control of
-Dynamic Systems, 7th edn. Prentice Hall Press, USA (2014)
-[25] Tomasetti, C., Vogelstein, B., Parmigiani, G.: Half or more of the somatic
-mutations in cancers of self-renewing tissues originate prior to tumor
-initiation. Proceedings of the National Academy of Sciences 110(6),
-1999–2004 (2013). https://doi.org/10.1073/pnas.1221068110
-[26] Colom, B., Jones, P.H.: Clonal analysis of stem cells in differentiation
-and disease. Current Opinion in Cell Biology 43, 14–21 (2016). https:
-//doi.org/10.1016/j.ceb.2016.07.002
-[27] Rodilla, V., Fre, S.: Cellular plasticity of mammary epithelial cells under-
-lies heterogeneity of breast cancer. Biomedicines 6(4), 9–12 (2018). https:
-//doi.org/10.3390/biomedicines6040103
-
-Springer Nature 2021 LATEX template
-Homeostatic regulation of renewing tissue cell populations via crowding control
-33
-[28] Tata, P.R., Rajagopal, J.: Cellular plasticity: 1712 to the present day.
-Current Opinion in Cell Biology 43, 46–54 (2016). https://doi.org/10.
-1016/j.ceb.2016.07.005
-[29] Merrell,
-A.J.,
-Stanger,
-B.Z.:
-Adult
-cell
-plasticity
-in
-vivo:
-De-
-differentiation and transdifferentiation are back in style. Nature Reviews
-Molecular Cell Biology 17(7), 413–425 (2016). https://doi.org/10.1038/
-nrm.2016.24
-[30] Puri, S., Folias, A.E., Hebrok, M.: Plasticity and dedifferentiation within
-the pancreas: Development, homeostasis, and disease. Cell Stem Cell
-16(1), 18–31 (2015). https://doi.org/10.1016/j.stem.2014.11.001
-[31] Lei, J., Levin, S.A., Nie, Q.: Mathematical model of adult stem cell
-regeneration with cross-talk between genetic and epigenetic regulation.
-Proceedings of the National Academy of Sciences 111(10) (2014). https:
-//doi.org/10.1073/pnas.1324267111
-[32] Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions.
-The Journal of Physical Chemistry 81(25), 2340–2361 (1977). https://
-doi.org/10.1021/j100540a008
-[33] Haccou, P., Jagers, P., Vatutin, V.A.: Branching Processes: Varia-
-tion, Growth, and Extinction of Populations. Cambridge University
-Press, Cambridge (2005). https://doi.org/10.2277/0521832209. http://
-pure.iiasa.ac.at/7598/
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