Related papers: On the Minimal Overcompleteness Allowing Universal…
Interpretability benefits the theoretical understanding of representations. Existing word embeddings are generally dense representations. Hence, the meaning of latent dimensions is difficult to interpret. This makes word embeddings like a…
Over the past years, there are increasing interests in recovering the signals from undersampling data where such signals are sparse under some orthogonal dictionary or tight framework, which is referred to be sparse synthetic model. More…
Many applications like audio and image processing show that sparse representations are a powerful and efficient signal modeling technique. Finding an optimal dictionary that generates at the same time the sparsest representations of data…
Finding the sparse representation of a signal in an overcomplete dictionary has attracted a lot of attention over the past years. This paper studies ProSparse, a new polynomial complexity algorithm that solves the sparse representation…
We consider the problem of dictionary learning under the assumption that the observed signals can be represented as sparse linear combinations of the columns of a single large dictionary matrix. In particular, we analyze the minimax risk of…
The minimum average number of bits need to describe a random variable is its entropy, assuming knowledge of the underlying statistics On the other hand, universal compression supposes that the distribution of the random variable, while…
Sparse data models, where data is assumed to be well represented as a linear combination of a few elements from a dictionary, have gained considerable attention in recent years, and their use has led to state-of-the-art results in many…
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 $\mathbb{R}^{n\times m}$…
Distributional models are derived from co-occurrences in a corpus, where only a small proportion of all possible plausible co-occurrences will be observed. This results in a very sparse vector space, requiring a mechanism for inferring…
Given an overcomplete dictionary $A$ and a signal $b$ that is a linear combination of a few linearly independent columns of $A$, classical sparse recovery theory deals with the problem of recovering the unique sparse representation $x$ such…
It is now well established that sparse signal models are well suited to restoration tasks and can effectively be learned from audio, image, and video data. Recent research has been aimed at learning discriminative sparse models instead of…
In the first part of the series papers, we set out to answer the following question: given specific restrictions on a set of samplers, what kind of signal can be uniquely represented by the corresponding samples attained, as the foundation…
This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a…
We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary.…
Sparse coding is a crucial subroutine in algorithms for various signal processing, deep learning, and other machine learning applications. The central goal is to learn an overcomplete dictionary that can sparsely represent a given input…
Sparse representations using overcomplete dictionaries have proved to be a powerful tool in many signal processing applications such as denoising, super-resolution, inpainting, compression or classification. The sparsity of the…
In a previous paper [Adcock & Huybrechs, 2019] we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but…
Sparse representations with learned dictionaries have been successful in several image analysis applications. In this paper, we propose and analyze the framework of ensemble sparse models, and demonstrate their utility in image restoration…
A simple sparse coding mechanism appears in the sensory systems of several organisms: to a coarse approximation, an input $x \in \R^d$ is mapped to much higher dimension $m \gg d$ by a random linear transformation, and is then sparsified by…
Word embeddings have found their way into a wide range of natural language processing tasks including those in the biomedical domain. While these vector representations successfully capture semantic and syntactic word relations, hidden…