Related papers: Several Approximation Algorithms for Sparse Best R…
Two approximation algorithms are proposed for $\ell_1$-regularized sparse rank-1 approximation to higher-order tensors. The algorithms are based on multilinear relaxation and sparsification, which are easily implemented and well scalable.…
The goal of this paper is to find a low-rank approximation for a given tensor. Specifically, we give a computable strategy on calculating the rank of a given tensor, based on approximating the solution to an NP-hard problem. In this paper,…
The problem of partitioning a large and sparse tensor is considered, where the tensor consists of a sequence of adjacency matrices. Theory is developed that is a generalization of spectral graph partitioning. A best rank-$(2,2,\lambda)$…
Sparse coding consists in representing signals as sparse linear combinations of atoms selected from a dictionary. We consider an extension of this framework where the atoms are further assumed to be embedded in a tree. This is achieved…
Information is extracted from large and sparse data sets organized as 3-mode tensors. Two methods are described, based on best rank-(2,2,2) and rank-(2,2,1) approximation of the tensor. The first method can be considered as a generalization…
The sparsity constrained rank-one matrix approximation problem is a difficult mathematical optimization problem which arises in a wide array of useful applications in engineering, machine learning and statistics, and the design of…
The approximation of tensors has important applications in various disciplines, but it remains an extremely challenging task. It is well known that tensors of higher order can fail to have best low-rank approximations, but with an important…
The problem of sparse approximation and the closely related compressed sensing have received tremendous attention in the past decade. Primarily studied from the viewpoint of applied harmonic analysis and signal processing, there have been…
Tensor methods are among the most prominent tools for the numerical solution of high-dimensional problems where functions of multiple variables have to be approximated. These methods exploit the tensor structure of function spaces and apply…
We propose a block coordinate descent type algorithm for estimating the rank of a given tensor. In addition, the algorithm provides the canonical polyadic decomposition of a tensor. In order to estimate the tensor rank we use sparse…
In this short paper we bridge two seemingly unrelated sparse approximation topics: continuous sparse coding and low-rank approximations. We show that for a specific choice of continuous dictionary, linear systems with nuclear-norm…
Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in…
We consider the NP-hard problem of finding the closest rank-one binary tensor to a given binary tensor, which we refer to as the rank-one Boolean tensor factorization (BTF) problem. This optimization problem can be used to recover a planted…
Tensor network contractions are widely used in statistical physics, quantum computing, and computer science. We introduce a method to efficiently approximate tensor network contractions using low-rank approximations, where each intermediate…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
Sparse approximations using highly over-complete dictionaries is a state-of-the-art tool for many imaging applications including denoising, super-resolution, compressive sensing, light-field analysis, and object recognition. Unfortunately,…
We study a generalized framework for structured sparsity. It extends the well-known methods of Lasso and Group Lasso by incorporating additional constraints on the variables as part of a convex optimization problem. This framework provides…
In this work, we propose an optimization framework for estimating a sparse robust one-dimensional subspace. Our objective is to minimize both the representation error and the penalty, in terms of the l1-norm criterion. Given that the…
In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9,18], and tensor…
We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse…