The sparse dictionary coding framework represents signals as a linear combination of a few predefined dictionary atoms. It has been employed for images, time series, graph signals and recently for 2-way (or 2D) spatio-temporal data employing jointly temporal and spatial dictionaries. Large and over-complete dictionaries enable high-quality models, but also pose scalability challenges which are exacerbated in multi-dictionary settings. Hence, an important problem that we address in this paper is: How to scale multi-dictionary coding for large dictionaries and datasets?
We propose a multi-dictionary atom selection technique for low-rank sparse coding named LRMDS. To enable scalability to large dictionaries and datasets, it progressively selects groups of row-column atom pairs based on their alignment with the data and performs convex relaxation coding via the corresponding sub-dictionaries. We demonstrate both theoretically and experimentally that when the data has a low-rank encoding with a sparse subset of the atoms, LRMDS is able to select them with strong guarantees under mild assumptions. Furthermore, we demonstrate the scalability and quality of LRMDS in both synthetic and real-world datasets and for a range of coding dictionaries. It achieves 3X to 10X speed-up compared to baselines, while obtaining up to two orders of magnitude improvement in representation quality on some of the real world datasets given a fixed target number of atoms.
University of Tehran（德黑兰大学）；Amirkabir University of Technology（阿米尔卡比尔理工大学）
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 le…
Graduate School of Information Sciences and Technology（信息科学技术研究生院）；The University of Tokyo（东京大学）
In dictionary selection, several atoms are selected from finite candidates that successfully approximate given data points in the sparse representation. We propose a novel efficient greedy algorithm for dictionary selection. Not only does our algorithm work much faster than the known methods, but it can also handle more complex sparsity constraints, such as average sparsity. Using numerical experiments, we show that our algorithm outperforms the known methods for dictionary selection, achieving competitive performances with dictionary learning algorithms in a smaller running time.
In recent years, a class of dictionaries have been proposed for multidimensional (tensor) data representation that exploit the structure of tensor data by imposing a Kronecker structure on the dictionary underlying the data. In this work, a novel algorithm called "STARK" is provided to learn Kronecker structured dictionaries that can represent tensors of any order. By establishing that the Kronecker product of any number of matrices can be rearranged to form a rank-1 tensor, we show that Kronecker structure can be enforced on the dictionary by solving a rank-1 tensor recovery problem. Because …
Dictionary learning aims at seeking a dictionary under which the training data can be sparsely represented. Methods in the literature typically formulate the dictionary learning problem as an optimization w.r.t. two variables, i.e., dictionary and sparse coefficients, and solve it by alternating between two stages: sparse coding and dictionary update. The key contribution of this work is a Rank-One Atomic Decomposition (ROAD) formulation where dictionary learning is cast as an optimization w.r.t. a single variable which is a set of rank one matrices. The resulting algorithm is hence single-sta…
In unsupervised learning, an unbiased uniform sampling strategy is typically used, in order that the learned features faithfully encode the statistical structure of the training data. In this work, we explore whether active example selection strategies - algorithms that select which examples to use, based on the current estimate of the features - can accelerate learning. Specifically, we investigate effects of heuristic and saliency-inspired selection algorithms on the dictionary learning task with sparse activations. We show that some selection algorithms do improve the speed of learning, and…
Constrained tensor and matrix factorization models allow to extract interpretable patterns from multiway data. Therefore identifiability properties and efficient algorithms for constrained low-rank approximations are nowadays important research topics. This work deals with columns of factor matrices of a low-rank approximation being sparse in a known and possibly overcomplete basis, a model coined as Dictionary-based Low-Rank Approximation (DLRA). While earlier contributions focused on finding factor columns inside a dictionary of candidate columns, i.e. one-sparse approximations, this work is…
Sparsity-based approaches have been popular in many applications in image processing and imaging. Compressed sensing exploits the sparsity of images in a transform domain or dictionary to improve image recovery from undersampled measurements. In the context of inverse problems in dynamic imaging, recent research has demonstrated the promise of sparsity and low-rank techniques. For example, the patches of the underlying data are modeled as sparse in an adaptive dictionary domain, and the resulting image and dictionary estimation from undersampled measurements is called dictionary-blind compress…
EISLAB, Luleå University of Technology, Sweden（埃斯实验室，吕勒奥理工大学，瑞典）；SKF–LTU University Technology Center and EISLAB, Luleå University of Technology, Sweden（斯凯孚-吕勒奥理工大学技术中心及埃斯实验室，吕勒奥理工大学，瑞典）；Luleå University of Technology, Sweden（吕勒奥理工大学，瑞典）；EISLAB, Luleå University of Technology（埃斯实验室，吕勒奥理工大学）；SKF–LTU University Technology Center and EISLAB, Luleå University of Technology（斯凯孚-吕勒奥理工大学技术中心及埃斯实验室，吕勒奥理工大学）；Luleå University of Technology（吕勒奥理工大学）
Sparse signal representations based on linear combinations of learned atoms have been used to obtain state-of-the-art results in several practical signal processing applications. Approximation methods are needed to process high-dimensional signals in this way because the problem to calculate optimal atoms for sparse coding is NP-hard. Here we study greedy algorithms for unsupervised learning of dictionaries of shift-invariant atoms and propose a new method where each atom is selected with the same probability on average, which corresponds to the homeostatic regulation of a recurrent convolutio…
Dictionary learning aims to find a dictionary under which the training data can be sparsely represented, and it is usually achieved by iteratively applying two stages: sparse coding and dictionary update. Typical methods for dictionary update focuses on refining both dictionary atoms and their corresponding sparse coefficients by using the sparsity patterns obtained from sparse coding stage, and hence it is a non-convex bilinear inverse problem. In this paper, we propose a Rank-One Matrix Decomposition (ROMD) algorithm to recast this challenge into a convex problem by resolving these two varia…
Sparsity and low-rank models have been popular for reconstructing images and videos from limited or corrupted measurements. Dictionary or transform learning methods are useful in applications such as denoising, inpainting, and medical image reconstruction. This paper proposes a framework for online (or time-sequential) adaptive reconstruction of dynamic image sequences from linear (typically undersampled) measurements. We model the spatiotemporal patches of the underlying dynamic image sequence as sparse in a dictionary, and we simultaneously estimate the dictionary and the images sequentially…
Sun Yat-sen University（中山大学）；Huizhou University（惠州学院）
This work investigates how the traditional image classification pipelines can be extended into a deep architecture, inspired by recent successes of deep neural networks. We propose a deep boosting framework based on layer-by-layer joint feature boosting and dictionary learning. In each layer, we construct a dictionary of filters by combining the filters from the lower layer, and iteratively optimize the image representation with a joint discriminative-generative formulation, i.e. minimization of empirical classification error plus regularization of analysis image generation over training image…
Dictionary Subselection Using an Overcomplete Joint Sparsity Model
Institute for Digital Communications (IDCom), The University of Edinburgh（爱丁堡大学数字通信研究所（IDCom））；Paris Diderot University（巴黎第七大学）；Institut Langevin（朗之万研究所）
We analyze the decomposition of a data matrix, assumed to be a superposition of a low-rank component and a component which is sparse in a known dictionary, using a convex demixing method. We provide a unified analysis, encompassing both undercomplete and overcomplete dictionary cases, and show that the constituent components can be successfully recovered under some relatively mild assumptions up to a certain $\textit{global}$ sparsity level. Further, we corroborate our theoretical results by presenting empirical evaluations in terms of phase transitions in rank and sparsity for various diction…
Simula Research Laboratory AS（西穆拉研究实验室有限公司）；University of Innsbruck（因斯布鲁克大学）
This paper extends the recently proposed and theoretically justified iterative thresholding and $K$ residual means algorithm ITKrM to learning dicionaries from incomplete/masked training data (ITKrMM). It further adapts the algorithm to the presence of a low rank component in the data and provides a strategy for recovering this low rank component again from incomplete data. Several synthetic experiments show the advantages of incorporating information about the corruption into the algorithm. Finally, image inpainting is considered as application example, which demonstrates the superior perform…
From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identification
AI Institute in Dynamic Systems, Department of Mechanical Engineering, University of Washington（动态系统人工智能研究所，机械工程系，华盛顿大学）；Department of Mechanical Engineering, University of Hawai‘i at Mānoa（机械工程系，夏威夷大学马诺阿分校）；Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, Universidade Estadual Paulista “Júlio de Mesquita Filho”（数学与计算系，科技学院，保罗斯州立大学“朱利奥·梅斯奎塔·弗洛雷斯”分校）
In the last decade, traditional dictionary learning methods have been successfully applied to various pattern classification tasks. Although these methods produce sparse representations of signals which are robust against distortions and missing data, such representations quite often turn out to be unsuitable if the final objective is signal classification. In order to overcome or at least to attenuate such a weakness, several new methods which incorporate discriminative information into sparse-inducing models have emerged in recent years. In particular, methods for discriminative dictionary l…
The sparsity of signals in a transform domain or dictionary has been exploited in applications such as compression, denoising and inverse problems. More recently, data-driven adaptation of synthesis dictionaries has shown promise compared to analytical dictionary models. However, dictionary learning problems are typically non-convex and NP-hard, and the usual alternating minimization approaches for these problems are often computationally expensive, with the computations dominated by the NP-hard synthesis sparse coding step. This paper exploits the ideas that drive algorithms such as K-SVD, an…
The sparsity of natural signals and images in a transform domain or dictionary has been extensively exploited in several applications such as compression, denoising and inverse problems. More recently, data-driven adaptation of synthesis dictionaries has shown promise in many applications compared to fixed or analytical dictionary models. However, dictionary learning problems are typically non-convex and NP-hard, and the usual alternating minimization approaches for these problems are often computationally expensive, with the computations dominated by the NP-hard synthesis sparse coding step. …
Dimensional Collapse in Transformer Attention Outputs: A Challenge for Sparse Dictionary Learning
Shanghai Innovation Institute, Shanghai, China（上海创新研究院）；OpenMOSS Team, School of Computer Science, Fudan University, Shanghai, China（开放MOSS团队，计算机科学学院，复旦大学）
We propose `Dracula', a new framework for unsupervised feature selection from sequential data such as text. Dracula learns a dictionary of $n$-grams that efficiently compresses a given corpus and recursively compresses its own dictionary; in effect, Dracula is a `deep' extension of Compressive Feature Learning. It requires solving a binary linear program that may be relaxed to a linear program. Both problems exhibit considerable structure, their solution paths are well behaved, and we identify parameters which control the depth and diversity of the dictionary. We also discuss how to derive fea…
University of Minnesota（明尼苏达大学）；Princeton University（普林斯顿大学）
We consider the decomposition of a data matrix assumed to be a superposition of a low-rank matrix and a component which is sparse in a known dictionary, using a convex demixing method. We consider two sparsity structures for the sparse factor of the dictionary sparse component, namely entry-wise and column-wise sparsity, and provide a unified analysis, encompassing both undercomplete and the overcomplete dictionary cases, to show that the constituent matrices can be successfully recovered under some relatively mild conditions on incoherence, sparsity, and rank. We leverage these results to loc…
Ocean University of China（中国海洋大学）；The University of Iowa（爱荷华大学）
In this paper, we propose a novel method for fast face recognition called L1/2 Regularized Sparse Representation using Hierarchical Feature Selection (HSR). By employing hierarchical feature selection, we can compress the scale and dimension of global dictionary, which directly contributes to the decrease of computational cost in sparse representation that our approach is strongly rooted in. It consists of Gabor wavelets and Extreme Learning Machine Auto-Encoder (ELM-AE) hierarchically. For Gabor wavelets part, local features can be extracted at multiple scales and orientations to form Gabor-f…
We propose a Bayesian approach to learn discriminative dictionaries for sparse representation of data. The proposed approach infers probability distributions over the atoms of a discriminative dictionary using a Beta Process. It also computes sets of Bernoulli distributions that associate class labels to the learned dictionary atoms. This association signifies the selection probabilities of the dictionary atoms in the expansion of class-specific data. Furthermore, the non-parametric character of the proposed approach allows it to infer the correct size of the dictionary. We exploit the aforeme…
Department of Computer Science, University of Georgia（佐治亚大学计算机科学系）
In this paper, we investigate the robust dictionary learning (DL) to discover the hybrid salient low-rank and sparse representation in a factorized compressed space. A Joint Robust Factorization and Projective Dictionary Learning (J-RFDL) model is presented. The setting of J-RFDL aims at improving the data representations by enhancing the robustness to outliers and noise in data, encoding the reconstruction error more accurately and obtaining hybrid salient coefficients with accurate reconstruction ability. Specifically, J-RFDL performs the robust representation by DL in a factorized compresse…
A multi-way factor analysis model is introduced for tensor-variate data of any order. Each data item is represented as a (sparse) sum of Kruskal decompositions, a Kruskal-factor analysis (KFA). KFA is nonparametric and can infer both the tensor-rank of each dictionary atom and the number of dictionary atoms. The model is adapted for online learning, which allows dictionary learning on large data sets. After KFA is introduced, the model is extended to a deep convolutional tensor-factor analysis, supervised by a Bayesian SVM. The experiments section demonstrates the improvement of KFA over vecto…
Middle Tennessee State University（中田纳西州立大学）；St. Ambrose University（圣安布罗斯大学）
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 I…
International Center for Theoretical Physics (ICTP)（国际理论物理中心）；Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
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 …
The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）；Cornell University（康奈尔大学）
Low dimensional nonlinear structure abounds in datasets across computer vision and machine learning. Kernelized matrix factorization techniques have recently been proposed to learn these nonlinear structures for denoising, classification, dictionary learning, and missing data imputation, by observing that the image of the matrix in a sufficiently large feature space is low-rank. However, these nonlinear methods fail in the presence of sparse noise or outliers. In this work, we propose a new robust nonlinear factorization method called Robust Non-Linear Matrix Factorization (RNLMF). RNLMF const…
Iterative Atom Refinement: A Monotonicity Principle for Dictionary Learning
Stanford University（斯坦福大学）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Pennsylvania State University（宾夕法尼亚州立大学）；University of California, Merced（加州大学默塞德分校）
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Chinese Academy of Sciences（中国科学院）
Sparse coding, which represents a data point as a sparse reconstruction code with regard to a dictionary, has been a popular data representation method. Meanwhile, in database retrieval problems, learning the ranking scores from data points plays an important role. Up to now, these two problems have always been considered separately, assuming that data coding and ranking are two independent and irrelevant problems. However, is there any internal relationship between sparse coding and ranking score learning? If yes, how to explore and make use of this internal relationship? In this paper, we tr…
Systems and Control Engineering, IIT Bombay（印度理工学院孟买分校系统与控制工程）
Dictionaries are collections of vectors used for representations of random vectors in Euclidean spaces. Recent research on optimal dictionaries is focused on constructing dictionaries that offer sparse representations, i.e., $\ell_0$-optimal representations. Here we consider the problem of finding optimal dictionaries with which representations of samples of a random vector are optimal in an $\ell_2$-sense: optimality of representation is defined as attaining the minimal average $\ell_2$-norm of the coefficients used to represent the random vector. With the help of recent results on rank-$1$ d…
Imperial College London（帝国理工学院）；Middlesex University London（伦敦米德尔塞克斯大学）；University of Oulu（奥卢大学）
Dictionary learning and component analysis models are fundamental for learning compact representations that are relevant to a given task (feature extraction, dimensionality reduction, denoising, etc.). The model complexity is encoded by means of specific structure, such as sparsity, low-rankness, or nonnegativity. Unfortunately, approaches like K-SVD - that learn dictionaries for sparse coding via Singular Value Decomposition (SVD) - are hard to scale to high-volume and high-dimensional visual data, and fragile in the presence of outliers. Conversely, robust component analysis methods such as …
Dictionary based classifiers are a family of algorithms for time series classification (TSC), that focus on capturing the frequency of pattern occurrences in a time series. The ensemble based Bag of Symbolic Fourier Approximation Symbols (BOSS) was found to be a top performing TSC algorithm in a recent evaluation, as well as the best performing dictionary based classifier. A recent addition to the category, the Word Extraction for Time Series Classification (WEASEL), claims an improvement on this performance. Both of these algorithms however have non-trivial scalability issues, taking a consid…
École Centrale de Nantes（南特中央理工学院）；CNRS（法国国家科学研究中心）
In this paper, we propose a new greedy algorithm for sparse approximation, called SLS for Single L_1 Selection. SLS essentially consists of a greedy forward strategy, where the selection rule of a new component at each iteration is based on solving a least-squares optimization problem, penalized by the L_1 norm of the remaining variables. Then, the component with maximum amplitude is selected. Simulation results on difficult sparse deconvolution problems involving a highly correlated dictionary reveal the efficiency of the method, which outperforms popular greedy algorithms and Basis Pursuit D…
Oak Ridge National Laboratory（橡树岭国家实验室）；University of Tennessee at Knoxville（田纳西大学诺克斯维尔分校）
We present a greedy-based approach to construct an efficient single hidden layer neural network with the ReLU activation that approximates a target function. In our approach we obtain a shallow network by utilizing a greedy algorithm with the prescribed dictionary provided by the available training data and a set of possible inner weights. To facilitate the greedy selection process we employ an integral representation of the network, based on the ridgelet transform, that significantly reduces the cardinality of the dictionary and hence promotes feasibility of the greedy selection. Our approach…
Naively trained neural networks tend to experience catastrophic forgetting in sequential task settings, where data from previous tasks are unavailable. A number of methods, using various model expansion strategies, have been proposed recently as possible solutions. However, determining how much to expand the model is left to the practitioner, and often a constant schedule is chosen for simplicity, regardless of how complex the incoming task is. Instead, we propose a principled Bayesian nonparametric approach based on the Indian Buffet Process (IBP) prior, letting the data determine how much to…
Departamento de Procesamiento de Señales, Instituto de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad de la República（共和国大学工程学院电气工程研究所信号处理系（Departamento de Procesamiento de Señales, Instituto de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad de la República））
Matrix factorization is a key tool in data analysis; its applications include recommender systems, correlation analysis, signal processing, among others. Binary matrices are a particular case which has received significant attention for over thirty years, especially within the field of data mining. Dictionary learning refers to a family of methods for learning overcomplete basis (also called frames) in order to efficiently encode samples of a given type; this area, now also about twenty years old, was mostly developed within the signal processing field. In this work we propose two binary matri…
