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Greedy algorithms are widely used for problems in machine learning such as feature selection and set function optimization. Unfortunately, for large datasets, the running time of even greedy algorithms can be quite high. This is because for…

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal…

Greedy Sampling Methods (GSMs) are widely used to construct approximate solutions of Configuration Optimization Problems (COPs), where a loss functional is minimized over finite configurations of points in a compact domain. While effective…

This paper investigates a new learning formulation called structured sparsity, which is a natural extension of the standard sparsity concept in statistical learning and compressive sensing. By allowing arbitrary structures on the feature…

We find the exact order estimates of the approximations of the classes ${\cal F}_{q,r}^{\psi}$ of functions of several variables by greedy approximants in the integral metric. We also obtain the exact order estimates of the best $n$-term…

In this paper we develop the Greedy Recombination Interpolation Method (GRIM) for finding sparse approximations of functions initially given as linear combinations of some (large) number of simpler functions. In a similar spirit to the…

In this paper, we propose a new and simple approach to the approximation algorithms that are modified and improved from our published results. The computational and graphical examples are presented with the aid of Maple procedures.

Several convex formulation methods have been proposed previously for statistical estimation with structured sparsity as the prior. These methods often require a carefully tuned regularization parameter, often a cumbersome or heuristic…

The frame algorithm uses a simple recursive formula to approximate an unknown vector from its frame coefficients. This note introduces an adaptive version of the frame algorithm that maximizes the error reduction between steps in terms of…

Sparse representation of astronomical images is discussed. It is shown that a significant gain in sparsity is achieved when particular mixed dictionaries are used for approximating these types of images with greedy selection strategies.…

A deterministic approximation algorithm is presented for the maximization of non-monotone submodular functions over a ground set of size $n$ subject to cardinality constraint $k$; the algorithm is based upon the idea of interlacing two…

Motivated by the question of optimal functional approximation via compressed sensing, we propose generalizations of the Iterative Hard Thresholding and the Compressive Sampling Matching Pursuit algorithms able to promote sparse in levels…

We present a novel stagewise strategy for improving greedy algorithms for sparse recovery. We demonstrate its efficiency both for synthesis and analysis sparse priors, where in both cases we demonstrate its computational efficiency and…

In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the…

A dedicated algorithm for sparse spectral representation of music sound is presented. The goal is to enable the representation of a piece of music signal, as a linear superposition of as few spectral components as possible. A representation…

In this paper, we propose an algorithm for the construction of low-rank approximations of the inverse of an operator given in low-rank tensor format. The construction relies on an updated greedy algorithm for the minimization of a suitable…

This paper describes a simple greedy D-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards)…

Sparsity-constrained optimization has wide applicability in machine learning, statistics, and signal processing problems such as feature selection and compressive Sensing. A vast body of work has studied the sparsity-constrained…

This paper is devoted to theoretical aspects on optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of nonlinear $m$-term approximation in Banach…

Sparsity learning with known grouping structure has received considerable attention due to wide modern applications in high-dimensional data analysis. Although advantages of using group information have been well-studied by shrinkage-based…