Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorization

TL;DR

This paper investigates semi-nonnegative matrix factorization (semi-NMF), providing theoretical insights, an SVD-based initialization, an exact algorithm for specific cases, and analyzing its computational complexity and ill-posedness.

Contribution

It introduces a new SVD-based initialization, an exact algorithm for certain matrices, and proves semi-NMF's NP-hardness and potential ill-posedness.

Findings

Semi-NMF error is bounded by the best rank r-1 approximation.

An SVD-based initialization guarantees approximation quality.

The exact algorithm performs well on specific matrix classes.

Abstract

Given a matrix $M$ (not necessarily nonnegative) and a factorization rank $r$, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix $U$ with $r$ columns and a nonnegative matrix $V$ with $r$ rows such that $UV$ is the best possible approximation of $M$ according to some metric. In this paper, we study the properties of semi-NMF from which we develop exact and heuristic algorithms. Our contribution is threefold. First, we prove that the error of a semi-NMF of rank $r$ has to be smaller than the best unconstrained approximation of rank $r-1$. This leads us to a new initialization procedure based on the singular value decomposition (SVD) with a guarantee on the quality of the approximation. Second, we propose an exact algorithm (that is, an algorithm that finds an optimal solution), also based on the SVD, for a certain class of matrices (including nonnegative irreducible matrices) from which we derive an initialization for matrices not belonging to that class. Numerical experiments illustrate that this second approach performs extremely well, and allows us to compute optimal semi-NMF decompositions in many situations. Finally, we analyze the computational complexity of semi-NMF proving its NP-hardness, already in the rank-one case (that is, for $r = 1$), and we show that semi-NMF is sometimes ill-posed (that is, an optimal solution does not exist).