arXiv research

This paper concerns dictionary learning, i.e., sparse coding, a fundamental representation learning problem. We show that a subgradient descent algorithm, with random initialization, can provably recover orthogonal dictionaries on a natural nonsmooth, nonconvex $\ell_1$ minimization formulation of the problem, under mi…

This paper considers the fundamental problem of learning a complete (orthogonal) dictionary from samples of sparsely generated signals. Most existing methods solve the dictionary (and sparse representations) based on heuristic algorithms, usually without theoretical guarantees for either optimality or complexity. The r…

Anomaly detection in networks often boils down to identifying an underlying graph structure on which the abnormal occurrence rests on. Financial fraud schemes are one such example, where more or less intricate schemes are employed in order to elude transaction security protocols. We investigate the problem of learning …

This paper tackles spatio-temporal information preservation in machine learning.

We propose a new outline for adaptive dictionary learning methods for sparse encoding based on a hierarchical clustering of the training data. Through recursive application of a clustering method, the data is organized into a binary partition tree representing a multiscale structure. The dictionary atoms are defined ad…

We investigate the use of sparse coding and dictionary learning in the context of multitask and transfer learning. The central assumption of our learning method is that the tasks parameters are well approximated by sparse linear combinations of the atoms of a dictionary on a high or infinite dimensional space. This ass…

New algorithms tackle machine learning problems using manifold proximal point methods.

A novel dictionary-based approach for predicting functions.

In this paper, we present new results on using orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries for complex cases (i.e., complex measurement vector, complex dictionary and complex additive white Gaussian noise (CAWGN)). A sufficient condition that OMP can recover …

In signal analysis and synthesis, linear approximation theory considers a linear decomposition of any given signal in a set of atoms, collected into a so-called dictionary. Relevant sparse representations are obtained by relaxing the orthogonality condition of the atoms, yielding overcomplete dictionaries with an exten…

New algorithm recovers sparse signals from linearly sparse dictionaries efficiently.

A large set of signals can sometimes be described sparsely using a dictionary, that is, every element can be represented as a linear combination of few elements from the dictionary. Algorithms for various signal processing applications, including classification, denoising and signal separation, learn a dictionary from …

Dictionary learning is the task of determining a data-dependent transform that yields a sparse representation of some observed data. The dictionary learning problem is non-convex, and usually solved via computationally complex iterative algorithms. Furthermore, the resulting transforms obtained generally lack structure…

New theory shows EDMD works well in chaotic systems.

This paper addresses the problem of identifying a lower dimensional space where observed data can be sparsely represented. This under-complete dictionary learning task can be formulated as a blind separation problem of sparse sources linearly mixed with an unknown orthogonal mixing matrix. This issue is formulated in a…

In this note, we focus on smooth nonconvex optimization problems that obey: (1) all local minimizers are also global; and (2) around any saddle point or local maximizer, the objective has a negative directional curvature. Concrete applications such as dictionary learning, generalized phase retrieval, and orthogonal ten…

Chebyshev Greedy Algorithm is a generalization of the well known Orthogonal Matching Pursuit defined in a Hilbert space to the case of Banach spaces. We apply this algorithm for constructing sparse approximate solutions (with respect to a given dictionary) to convex optimization problems. Rate of convergence results in…

This paper deals with the design of a sensing matrix along with a sparse recovery algorithm by utilizing the probability-based prior information for compressed sensing system. With the knowledge of the probability for each atom of the dictionary being used, a diagonal weighted matrix is obtained and then the sensing ma…

In this paper, we introduce a Deep Convolutional Analysis Dictionary Model (DeepCAM) by learning convolutional dictionaries instead of unstructured dictionaries as in the case of deep analysis dictionary model introduced in the companion paper. Convolutional dictionaries are more suitable for processing high-dimensiona…

Many techniques in computer vision, machine learning, and statistics rely on the fact that a signal of interest admits a sparse representation over some dictionary. Dictionaries are either available analytically, or can be learned from a suitable training set. While analytic dictionaries permit to capture the global st…

Bayesian method improves dictionary learning for complex problems.

This work addresses the problem of learning sparse representations of tensor data using structured dictionary learning. It proposes learning a mixture of separable dictionaries to better capture the structure of tensor data by generalizing the separable dictionary learning model. Two different approaches for learning m…

Sparse representations using learned dictionaries are being increasingly used with success in several data processing and machine learning applications. The availability of abundant training data necessitates the development of efficient, robust and provably good dictionary learning algorithms. Algorithmic stability an…

We consider the dictionary learning problem, where the aim is to model the given data as a linear combination of a few columns of a matrix known as a dictionary, where the sparse weights forming the linear combination are known as coefficients. Since the dictionary and coefficients, parameterizing the linear model are …

A parallel algorithm learns efficient Kronecker product dictionaries.

This paper studies the convergence behaviour of dictionary learning via the Iterative Thresholding and K-residual Means (ITKrM) algorithm. On one hand it is proved that ITKrM is a contraction under much more relaxed conditions than previously necessary. On the other hand it is shown that there seem to exist stable fixe…

Additive models form a widely popular class of regression models which represent the relation between covariates and response variables as the sum of low-dimensional transfer functions. Besides flexibility and accuracy, a key benefit of these models is their interpretability: the transfer functions provide visual means…

In sparse signal representation, the choice of a dictionary often involves a tradeoff between two desirable properties -- the ability to adapt to specific signal data and a fast implementation of the dictionary. To sparsely represent signals residing on weighted graphs, an additional design challenge is to incorporate …

We study the Dictionary Learning (aka Sparse Coding) problem of obtaining a sparse representation of data points, by learning \emph{dictionary vectors} upon which the data points can be written as sparse linear combinations. We view this problem from a geometry perspective as the spanning set of a subspace arrangement,…

Sparse coding in learned dictionaries has been established as a successful approach for signal denoising, source separation and solving inverse problems in general. A dictionary learning method adapts an initial dictionary to a particular signal class by iteratively computing an approximate factorization of a training …

We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our algorithms can decompose a 4-tensor with $n$-dimensional orthonormal components in the…

This paper considers a recently emerged hyperspectral unmixing formulation based on sparse regression of a self-dictionary multiple measurement vector (SD-MMV) model, wherein the measured hyperspectral pixels are used as the dictionary. Operating under the pure pixel assumption, this SD-MMV formalism is special in that…

We present a two-stage approach for learning dictionaries for object classification tasks based on the principle of information maximization. The proposed method seeks a dictionary that is compact, discriminative, and generative. In the first stage, dictionary atoms are selected from an initial dictionary by maximizing…

Two algorithms converge to dictionary learning with geometric rate for non-uniform data.

In this work we propose a new deep learning tool called deep dictionary learning. Multi-level dictionaries are learnt in a greedy fashion, one layer at a time. This requires solving a simple (shallow) dictionary learning problem, the solution to this is well known. We apply the proposed technique on some benchmark deep…

Demixing problems in many areas such as hyperspectral imaging and differential optical absorption spectroscopy (DOAS) often require finding sparse nonnegative linear combinations of dictionary elements that match observed data. We show how aspects of these problems, such as misalignment of DOAS references and uncertain…

Dictionary learning is a cutting-edge area in imaging processing, that has recently led to state-of-the-art results in many signal processing tasks. The idea is to conduct a linear decomposition of a signal using a few atoms of a learned and usually over-completed dictionary instead of a pre-defined basis. Determining …

Traditional dictionary learning methods are based on quadratic convex loss function and thus are sensitive to outliers. In this paper, we propose a generic framework for robust dictionary learning based on concave losses. We provide results on composition of concave functions, notably regarding super-gradient computati…

We consider the problem of sparse coding, where each sample consists of a sparse linear combination of a set of dictionary atoms, and the task is to learn both the dictionary elements and the mixing coefficients. Alternating minimization is a popular heuristic for sparse coding, where the dictionary and the coefficient…

A-DLISTA and VLISTA learn dictionaries and sparse representations under varying sensing matrices.

This article addresses the issue of representing electroencephalographic (EEG) signals in an efficient way. While classical approaches use a fixed Gabor dictionary to analyze EEG signals, this article proposes a data-driven method to obtain an adapted dictionary. To reach an efficient dictionary learning, appropriate s…

We consider the problem of distributed dictionary learning, where a set of nodes is required to collectively learn a common dictionary from noisy measurements. This approach may be useful in several contexts including sensor networks. Diffusion cooperation schemes have been proposed to solve the distributed linear regr…

A dictionary is a database of standard vectors, so that other vectors / signals are expressed as linear combinations of dictionary vectors, and the task of learning a dictionary for a given data is to find a good dictionary so that the representation of data points has desirable features. Dictionary learning and the re…

PUDLE method analyzes and improves unrolled sparse coding networks for dictionary learning.

We present theoretical guarantees for an alternating minimization algorithm for the dictionary learning/sparse coding problem. The dictionary learning problem is to factorize vector samples $y^{1},y^{2},\ldots, y^{n}$ into an appropriate basis (dictionary) $A^*$ and sparse vectors $x^{1*},\ldots,x^{n*}$. Our algorithm …

Structured sparse coding and the related structured dictionary learning problems are novel research areas in machine learning. In this paper we present a new application of structured dictionary learning for collaborative filtering based recommender systems. Our extensive numerical experiments demonstrate that the pres…

This work combines deep learning and sparse coding for CT image reconstruction.

Novel algorithm learns sparse signal representations over topological spaces.