In this paper, we propose a data-driven leak localization method for water distribution networks (WDNs) which combines two complementary approaches: graph-based interpolation and dictionary classification. The former estimates the complete WDN hydraulic state (i.e., hydraulic heads) from real measurements at certain nodes and the network graph. Then, these actual measurements, together with a subset of valuable estimated states, are used to feed and train the dictionary learning scheme. Thus, the meshing of these two methods is explored, showing that its performance is superior to either approach alone, even deriving different mechanisms to increase its resilience to classical problems (e.g., dimensionality, interpolation errors, etc.). The approach is validated using the L-TOWN benchmark proposed at BattLeDIM2020.
State-of-the-art approaches for clustering high-dimensional data utilize deep auto-encoder architectures. Many of these networks require a large number of parameters and suffer from a lack of interpretability, due to the black-box nature of the auto-encoders. We introduce Mixture Model Auto-Encoders (MixMate), a novel architecture that clusters data by performing inference on a generative model. Derived from the perspective of sparse dictionary learning and mixture models, MixMate comprises several auto-encoders, each tasked with reconstructing data in a distinct cluster, while enforcing sparsity in the latent space. Through experiments on various image datasets, we show that MixMate achieves competitive performance compared to state-of-the-art deep clustering algorithms, while using orders of magnitude fewer parameters.
Aiming at high-dimensional (HD) data acquisition and analysis, snapshot compressive imaging (SCI) obtains the 2D compressed measurement of HD data with optical imaging systems and reconstructs HD data using compressive sensing algorithms. While the Plug-and-Play (PnP) framework offers an emerging solution to SCI reconstruction, its intrinsic denoising process is still a challenging problem. Unfortunately, existing denoisers in the PnP framework either suffer limited performance or require extensive training data. In this paper, we propose an efficient and effective shallow-learning-based algorithm for video SCI reconstruction. Revisiting dictionary learning methods, we empower the PnP framework with a new denoiser, the kernel singular value decomposition (KSVD). Benefited from the advent of KSVD, our algorithm retains a good trade-off among quality, speed, and training difficulty. On a variety of datasets, both quantitative and qualitative evaluations of our simulation results demonstrate the effectiveness of our proposed method. In comparison to a typical baseline using total variation, our method achieves around $2$ dB improvement in PSNR and 0.2 in SSIM. We expect that our proposed PnP-KSVD algorithm can serve as a new baseline for video SCI reconstruction.
Comparing structured objects such as graphs is a fundamental operation involved in many learning tasks. To this end, the Gromov-Wasserstein (GW) distance, based on Optimal Transport (OT), has proven to be successful in handling the specific nature of the associated objects. More specifically, through the nodes connectivity relations, GW operates on graphs, seen as probability measures over specific spaces. At the core of OT is the idea of conservation of mass, which imposes a coupling between all the nodes from the two considered graphs. We argue in this paper that this property can be detrimental for tasks such as graph dictionary or partition learning, and we relax it by proposing a new semi-relaxed Gromov-Wasserstein divergence. Aside from immediate computational benefits, we discuss its properties, and show that it can lead to an efficient graph dictionary learning algorithm. We empirically demonstrate its relevance for complex tasks on graphs such as partitioning, clustering and completion.
Nonlinear phenomena can be analyzed via linear techniques using operator-theoretic approaches. Data-driven method called the extended dynamic mode decomposition (EDMD) and its variants, which approximate the Koopman operator associated with the nonlinear phenomena, have been rapidly developing by incorporating machine learning methods. Neural ordinary differential equations (NODEs), which are a neural network equipped with a continuum of layers, and have high parameter and memory efficiencies, have been proposed. In this paper, we propose an algorithm to perform EDMD using NODEs. NODEs are used to find a parameter-efficient dictionary which provides a good finite-dimensional approximation of the Koopman operator. We show the superiority of the parameter efficiency of the proposed method through numerical experiments.
While open databases are an important resource in the Deep Learning (DL) era, they are sometimes used "off-label": data published for one task are used for training algorithms for a different one. This work aims to highlight that in some cases, this common practice may lead to biased, overly-optimistic results. We demonstrate this phenomenon for inverse problem solvers and show how their biased performance stems from hidden data preprocessing pipelines. We describe two preprocessing pipelines typical of open-access databases and study their effects on three well-established algorithms developed for Magnetic Resonance Imaging (MRI) reconstruction: Compressed Sensing (CS), Dictionary Learning (DictL), and DL. In this large-scale study we performed extensive computations. Our results demonstrate that the CS, DictL and DL algorithms yield systematically biased results when naively trained on seemingly-appropriate data: the Normalized Root Mean Square Error (NRMSE) improves consistently with the preprocessing extent, showing an artificial increase of 25%-48% in some cases. Since this phenomenon is generally unknown, biased results are sometimes published as state-of-the-art; we refer to that as subtle data crimes. This work hence raises a red flag regarding naive off-label usage of Big Data and reveals the vulnerability of modern inverse problem solvers to the resulting bias.
We consider increasingly complex models of matrix denoising and dictionary learning in the Bayes-optimal setting, in the challenging regime where the matrices to infer have a rank growing linearly with the system size. This is in contrast with most existing literature concerned with the low-rank (i.e., constant-rank) regime. We first consider a class of rotationally invariant matrix denoising problems whose mutual information and minimum mean-square error are computable using techniques from random matrix theory. Next, we analyze the more challenging models of dictionary learning. To do so we introduce a novel combination of the replica method from statistical mechanics together with random matrix theory, coined spectral replica method. This allows us to derive variational formulas for the mutual information between hidden representations and the noisy data of the dictionary learning problem, as well as for the overlaps quantifying the optimal reconstruction error. The proposed method reduces the number of degrees of freedom from $\Theta(N^2)$ matrix entries to $\Theta(N)$ eigenvalues (or singular values), and yields Coulomb gas representations of the mutual information which are reminiscent of matrix models in physics. The main ingredients are a combination of large deviation results for random matrices together with a new replica symmetric decoupling ansatz at the level of the probability distributions of eigenvalues (or singular values) of certain overlap matrices and the use of HarishChandra-Itzykson-Zuber spherical integrals.
Phase retrieval aims at reconstructing unknown signals from magnitude measurements of linear mixtures. In this paper, we consider the phase retrieval with dictionary learning problem, which includes an additional prior information that the measured signal admits a sparse representation over an unknown dictionary. The task is to jointly estimate the dictionary and the sparse representation from magnitude-only measurements. To this end, we study two complementary formulations and develop efficient parallel algorithms by extending the successive convex approximation framework using a smooth majorization. The first algorithm is termed compact-SCAphase and is preferable in the case of less diverse mixture models. It employs a compact formulation that avoids the use of auxiliary variables. The proposed algorithm is highly scalable and has reduced parameter tuning cost. The second algorithm, referred to as SCAphase, uses auxiliary variables and is favorable in the case of highly diverse mixture models. It also renders simple incorporation of additional side constraints. The performance of both methods is evaluated when applied to blind sparse channel estimation from subband magnitude measurements in a multi-antenna random access network. Simulation results demonstrate the efficiency of the proposed techniques compared to state-of-the-art methods.
Convolutional sparse coding improves on the standard sparse approximation by incorporating a global shift-invariant model. The most efficient convolutional sparse coding methods are based on the alternating direction method of multipliers and the convolution theorem. The only major difference between these methods is how they approach a convolutional least-squares fitting subproblem. This letter presents a solution to this subproblem, which improves the efficiency of the state-of-the-art algorithms. We also use the same approach for developing an efficient convolutional dictionary learning method. Furthermore, we propose a novel algorithm for convolutional sparse coding with a constraint on the approximation error.
Few-shot classification (FSC) is one of the most concerned hot issues in recent years. The general setting consists of two phases: (1) Pre-train a feature extraction model (FEM) with base data (has large amounts of labeled samples). (2) Use the FEM to extract the features of novel data (with few labeled samples and totally different categories from base data), then classify them with the to-be-designed classifier. The adaptability of pre-trained FEM to novel data determines the accuracy of novel features, thereby affecting the final classification performances. To this end, how to appraise the pre-trained FEM is the most crucial focus in the FSC community. It sounds like traditional Class Activate Mapping (CAM) based methods can achieve this by overlaying weighted feature maps. However, due to the particularity of FSC (e.g., there is no backpropagation when using the pre-trained FEM to extract novel features), we cannot activate the feature map with the novel classes. To address this challenge, we propose a simple, flexible method, dubbed as Class-Irrelevant Mapping (CIM). Specifically, first, we introduce dictionary learning theory and view the channels of the feature map as the bases in a dictionary. Then we utilize the feature map to fit the feature vector of an image to achieve the corresponding channel weights. Finally, we overlap the weighted feature map for visualization to appraise the ability of pre-trained FEM on novel data. For fair use of CIM in evaluating different models, we propose a new measurement index, called Feature Localization Accuracy (FLA). In experiments, we first compare our CIM with CAM in regular tasks and achieve outstanding performances. Next, we use our CIM to appraise several classical FSC frameworks without considering the classification results and discuss them.
This paper propose a novel dictionary learning approach to detect event action using skeletal information extracted from RGBD video. The event action is represented as several latent atoms and composed of latent spatial and temporal attributes. We perform the method at the example of fall event detection. The skeleton frames are clustered by an initial K-means method. Each skeleton frame is assigned with a varying weight parameter and fed into our Gradual Online Dictionary Learning (GODL) algorithm. During the training process, outlier frames will be gradually filtered by reducing the weight that is inversely proportional to a cost. In order to strictly distinguish the event action from similar actions and robustly acquire its action unit, we build a latent unit temporal structure for each sub-action. We evaluate the proposed method on parts of the NTURGB+D dataset, which includes 209 fall videos, 405 ground-lift videos, 420 sit-down videos, and 280 videos of 46 otheractions. We present the experimental validation of the achieved accuracy, recall and precision. Our approach achieves the bestperformance on precision and accuracy of human fall event detection, compared with other existing dictionary learning methods. With increasing noise ratio, our method remains the highest accuracy and the lowest variance.
This work presents an approach for image reconstruction in clinical low-dose tomography that combines principles from sparse signal processing with ideas from deep learning. First, we describe sparse signal representation in terms of dictionaries from a statistical perspective and interpret dictionary learning as a process of aligning distribution that arises from a generative model with empirical distribution of true signals. As a result we can see that sparse coding with learned dictionaries resembles a specific variational autoencoder, where the decoder is a linear function and the encoder is a sparse coding algorithm. Next, we show that dictionary learning can also benefit from computational advancements introduced in the context of deep learning, such as parallelism and as stochastic optimization. Finally, we show that regularization by dictionaries achieves competitive performance in computed tomography (CT) reconstruction comparing to state-of-the-art model based and data driven approaches.
The emerging wide area monitoring systems (WAMS) have brought significant improvements in electric grids' situational awareness. However, the newly introduced system can potentially increase the risk of cyber-attacks, which may be disguised as normal physical disturbances. This paper deals with the event and intrusion detection problem by leveraging a stream data mining classifier (Hoeffding adaptive tree) with semi-supervised learning techniques to distinguish cyber-attacks from regular system perturbations accurately. First, our proposed approach builds a dictionary by learning higher-level features from unlabeled data. Then, the labeled data are represented as sparse linear combinations of learned dictionary atoms. We capitalize on those sparse codes to train the online classifier along with efficient change detectors. We conduct numerical experiments with industrial control systems cyber-attack datasets. We consider five different scenarios: short-circuit faults, line maintenance, remote tripping command injection, relay setting change, as well as false data injection. The data are generated based on a modified IEEE 9-bus system. Simulation results show that our proposed approach outperforms the state-of-the-art method.
With the recent success of representation learning methods, which includes deep learning as a special case, there has been considerable interest in developing representation learning techniques that can incorporate known physical constraints into the learned representation. As one example, in many applications that involve a signal propagating through physical media (e.g., optics, acoustics, fluid dynamics, etc), it is known that the dynamics of the signal must satisfy constraints imposed by the wave equation. Here we propose a matrix factorization technique that decomposes such signals into a sum of components, where each component is regularized to ensure that it satisfies wave equation constraints. Although our proposed formulation is non-convex, we prove that our model can be efficiently solved to global optimality in polynomial time. We demonstrate the benefits of our work by applications in structural health monitoring, where prior work has attempted to solve this problem using sparse dictionary learning approaches that do not come with any theoretical guarantees regarding convergence to global optimality and employ heuristics to capture desired physical constraints.
3D hand-object pose estimation is an important issue to understand the interaction between human and environment. Current hand-object pose estimation methods require detailed 3D labels, which are expensive and labor-intensive. To tackle the problem of data collection, we propose a semi-supervised 3D hand-object pose estimation method with two key techniques: pose dictionary learning and an object-oriented coordinate system. The proposed pose dictionary learning module can distinguish infeasible poses by reconstruction error, enabling unlabeled data to provide supervision signals. The proposed object-oriented coordinate system can make 3D estimations equivariant to the camera perspective. Experiments are conducted on FPHA and HO-3D datasets. Our method reduces estimation error by 19.5% / 24.9% for hands/objects compared to straightforward use of labeled data on FPHA and outperforms several baseline methods. Extensive experiments also validate the robustness of the proposed method.
There exist many high-dimensional data in real-world applications such as biology, computer vision, and social networks. Feature selection approaches are devised to confront with high-dimensional data challenges with the aim of efficient learning technologies as well as reduction of models complexity. Due to the hardship of labeling on these datasets, there are a variety of approaches on feature selection process in an unsupervised setting by considering some important characteristics of data. In this paper, we introduce a novel unsupervised feature selection approach by applying dictionary learning ideas in a low-rank representation. Dictionary learning in a low-rank representation not only enables us to provide a new representation, but it also maintains feature correlation. Then, spectral analysis is employed to preserve sample similarities. Finally, a unified objective function for unsupervised feature selection is proposed in a sparse way by an $\ell_{2,1}$-norm regularization. Furthermore, an efficient numerical algorithm is designed to solve the corresponding optimization problem. We demonstrate the performance of the proposed method based on a variety of standard datasets from different applied domains. Our experimental findings reveal that the proposed method outperforms the state-of-the-art algorithm.
This paper addresses sparse signal reconstruction under various types of structural side constraints with applications in multi-antenna systems. Side constraints may result from prior information on the measurement system and the sparse signal structure. They may involve the structure of the sensing matrix, the structure of the non-zero support values, the temporal structure of the sparse representationvector, and the nonlinear measurement structure. First, we demonstrate how a priori information in form of structural side constraints influence recovery guarantees (null space properties) using L1-minimization. Furthermore, for constant modulus signals, signals with row-, block- and rank-sparsity, as well as non-circular signals, we illustrate how structural prior information can be used to devise efficient algorithms with improved recovery performance and reduced computational complexity. Finally, we address the measurement system design for linear and nonlinear measurements of sparse signals. Moreover, we discuss the linear mixing matrix design based on coherence minimization. Then we extend our focus to nonlinear measurement systems where we design parallel optimization algorithms to efficiently compute stationary points in the sparse phase retrieval problem with and without dictionary learning.
The objective of this study is to derive functional networks for the autism spectrum disorder (ASD) population using the group ICA and dictionary learning model together and to classify ASD and typically developing (TD) participants using the functional connectivity calculated from the derived functional networks. In our experiments, the ASD functional networks were derived from resting-state functional magnetic resonance imaging (rs-fMRI) data. We downloaded a total of 120 training samples, including 58 ASD and 62 TD participants, which were obtained from the public repository: Autism Brain Imaging Data Exchange I (ABIDE I). Our methodology and results have five main parts. First, we utilize a group ICA model to extract functional networks from the ASD group and rank the top 20 regions of interest (ROIs). Second, we utilize a dictionary learning model to extract functional networks from the ASD group and rank the top 20 ROIs. Third, we merged the 40 selected ROIs from the two models together as the ASD functional networks. Fourth, we generate three corresponding masks based on the 20 selected ROIs from group ICA, the 20 ROIs selected from dictionary learning, and the 40 combined ROIs selected from both. Finally, we extract ROIs for all training samples using the above three masks, and the calculated functional connectivity was used as features for ASD and TD classification. The classification results showed that the functional networks derived from ICA and dictionary learning together outperform those derived from a single ICA model or a single dictionary learning model.
Dictionary learning consists of finding a sparse representation from noisy data and is a common way to encode data-driven prior knowledge on signals. Alternating minimization (AM) is standard for the underlying optimization, where gradient descent steps alternate with sparse coding procedures. The major drawback of this method is its prohibitive computational cost, making it unpractical on large real-world data sets. This work studies an approximate formulation of dictionary learning based on unrolling and compares it to alternating minimization to find the best trade-off between speed and precision. We analyze the asymptotic behavior and convergence rate of gradients estimates in both methods. We show that unrolling performs better on the support of the inner problem solution and during the first iterations. Finally, we apply unrolling on pattern learning in magnetoencephalography (MEG) with the help of a stochastic algorithm and compare the performance to a state-of-the-art method.
The dictionary learning problem, representing data as a combination of a few atoms, has long stood as a popular method for learning representations in statistics and signal processing. The most popular dictionary learning algorithm alternates between sparse coding and dictionary update steps, and a rich literature has studied its theoretical convergence. The success of dictionary learning relies on access to a "good" initial estimate of the dictionary and the ability of the sparse coding step to provide an unbiased estimate of the code. The growing popularity of unrolled sparse coding networks has led to the empirical finding that backpropagation through such networks performs dictionary learning. We offer the theoretical analysis of these empirical results through PUDLE, a Provable Unrolled Dictionary LEarning method. We provide conditions on the network initialization and data distribution sufficient to recover and preserve the support of the latent code. Additionally, we address two challenges; first, the vanilla unrolled sparse coding computes a biased code estimate, and second, gradients during backpropagated learning can become unstable. We show approaches to reduce the bias of the code estimate in the forward pass, and that of the dictionary estimate in the backward pass. We propose strategies to resolve the learning instability by tuning network parameters and modifying the loss function. Overall, we highlight the impact of loss, unrolling, and backpropagation on convergence. We complement our findings through synthetic and image denoising experiments. Finally, we demonstrate PUDLE's interpretability, a driving factor in designing deep networks based on iterative optimizations, by building a mathematical relation between network weights, its output, and the training set.