On Weighted Low-Rank Approximation

TL;DR

This paper studies the problem of low-rank matrix approximation under a weighted Frobenius norm, exploring the number of possible approximations and their dependence on weight values.

Contribution

It introduces conjectures on the maximum number of approximations and analyzes how these depend on the weights, advancing understanding of weighted low-rank approximation.

Findings

Conjecture that the number of approximations is at most min(m, n).

Analysis of how approximation solutions vary with different weight configurations.

Provides insights into the structure of weighted low-rank approximation solutions.

Abstract

Our main interest is the low-rank approximation of a matrix in R^m.n under a weighted Frobenius norm. This norm associates a weight to each of the (m x n) matrix entries. We conjecture that the number of approximations is at most min(m, n). We also investigate how the approximations depend on the weight-values.