Pursuits in Structured Non-Convex Matrix Factorizations

TL;DR

This paper introduces a generalized greedy pursuit framework for structured non-convex matrix factorizations, enabling efficient approximation of matrices with factors from various structured sets, supported by theoretical convergence guarantees and practical experiments.

Contribution

It develops a versatile pursuit algorithm for structured matrix factorization, incorporating a general atomic power method for non-convex subproblems and proving linear convergence in Hilbert spaces.

Findings

Efficient approximation of structured matrices demonstrated on real datasets.

The framework generalizes existing methods like OMP to broader settings.

The atomic power method effectively solves non-convex subproblems.

Abstract

Efficiently representing real world data in a succinct and parsimonious manner is of central importance in many fields. We present a generalized greedy pursuit framework, allowing us to efficiently solve structured matrix factorization problems, where the factors are allowed to be from arbitrary sets of structured vectors. Such structure may include sparsity, non-negativeness, order, or a combination thereof. The algorithm approximates a given matrix by a linear combination of few rank-1 matrices, each factorized into an outer product of two vector atoms of the desired structure. For the non-convex subproblems of obtaining good rank-1 structured matrix atoms, we employ and analyze a general atomic power method. In addition to the above applications, we prove linear convergence for generalized pursuit variants in Hilbert spaces - for the task of approximation over the linear span of arbitrary dictionaries - which generalizes OMP and is useful beyond matrix problems. Our experiments on real datasets confirm both the efficiency and also the broad applicability of our framework in practice.