Desingularization of Bounded-Rank Matrix Sets
Valentin Khrulkov, Ivan Valer'evich Oseledets · SIAM Journal on Matrix Analysis and Applications · 2018
The conventional ways to solve optimization problems on low-rank matrix sets which appear in a great number of applications tend to ignore its underlying structure of an algebraic variety and existence of singular points. This leads to the 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 algebraic geometry and show how to desingularize these sets. Our main result is an algorithm which uses only bounded functions of singular values and hence does not suffer from the issue described above.