Desingularization of bounded-rank matrix sets

TL;DR

This paper introduces a desingularization method for low-rank matrix sets using algebraic geometry, resulting in algorithms that avoid issues caused by near-zero singular values.

Contribution

It proposes a novel desingularization approach for low-rank matrix sets that leads to algorithms free from singular value inversion problems.

Findings

Developed an algebraic geometry-based desingularization algorithm.

The algorithm uses only bounded functions of singular values.

Results show improved stability in low-rank matrix optimization.

Abstract

Conventional ways to solve optimization problems on low-rank matrix sets which appear in great number of applications ignore its underlying structure of an algebraic variety and existence of singular points. This leads to appearance of inverses of singular values in algorithms and since they could be close to $0$ it causes certain problems. We tackle this problem by utilizing ideas from the algebraic geometry and show how to desingularize these sets. Our main result is algorithm which uses only bounded functions of singular values and hence does not suffer from the issue described above.