Related papers: Dictionary Identification - Sparse Matrix-Factoris…
The idea that many important classes of signals can be well-represented by linear combinations of a small set of atoms selected from a given dictionary has had dramatic impact on the theory and practice of signal processing. For practical…
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…
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…
Dictionary learning is a popular approach for inferring a hidden basis or dictionary in which data has a sparse representation. Data generated from the dictionary A (an n by m matrix, with m > n in the over-complete setting) is given by Y =…
In the dictionary learning (or sparse coding) problem, we are given a collection of signals (vectors in $\mathbb{R}^d$), and the goal is to find a "basis" in which the signals have a sparse (approximate) representation. The problem has…
We present a convex formulation of dictionary learning for sparse signal decomposition. Convexity is obtained by replacing the usual explicit upper bound on the dictionary size by a convex rank-reducing term similar to the trace norm. In…
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…
A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical…
A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical…
In sparse recovery, the unique sparsest solution to an under-determined system of linear equations is of main interest. This scheme is commonly proposed to be applied to signal acquisition. In most cases, the signals are not sparse…
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…
Recently, considerable research efforts have been devoted to the design of methods to learn from data overcomplete dictionaries for sparse coding. However, learned dictionaries require the solution of an optimization problem for coding new…
This paper studies the problem of recovering a non-negative sparse signal $\x \in \Re^n$ from highly corrupted linear measurements $\y = A\x + \e \in \Re^m$, where $\e$ is an unknown error vector whose nonzero entries may be unbounded.…
We study the theoretical properties of learning a dictionary from $N$ signals $\mathbf x_i\in \mathbb R^K$ for $i=1,...,N$ via $l_1$-minimization. We assume that $\mathbf x_i$'s are $i.i.d.$ random linear combinations of the $K$ columns…
Factor Analysis is about finding a low-rank plus sparse additive decomposition from a noisy estimate of the signal covariance matrix. In order to get such a decomposition, we formulate an optimization problem using the nuclear norm for the…
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…
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…
Exact recovery of a sparse solution for an underdetermined system of linear equations implies full search among all possible subsets of the dictionary, which is computationally intractable, while l1 minimization will do the job when a…
In dictionary learning, also known as sparse coding, the algorithm is given samples of the form $y = Ax$ where $x\in \mathbb{R}^m$ is an unknown random sparse vector and $A$ is an unknown dictionary matrix in $\mathbb{R}^{n\times m}$…
We propose novel necessary and sufficient conditions for a sensing matrix to be "$s$-good" - to allow for exact $\ell_1$-recovery of sparse signals with $s$ nonzero entries when no measurement noise is present. Then we express the error…