# Maximum vanishing subspace problem, CAT(0)-space relaxation, and block-triangularization of partitioned matrix

## TL;DR

This paper introduces a new approach combining submodular and convex optimization techniques to solve the maximum vanishing subspace problem efficiently, with applications to block-triangularization of partitioned matrices.

## Contribution

It develops a pseudo-polynomial time algorithm for the weighted maximum vanishing subspace problem using CAT(0)-space relaxation, a novel combination of optimization methods.

## Key findings

- The weighted MVSP can be solved efficiently under certain conditions.
- The approach leverages CAT(0)-space convex optimization techniques.
- Implications for canonical block-triangular form of matrices are demonstrated.

## Abstract

In this paper, we address the following algebraic generalization of the bipartite stable set problem. We are given a block-structured matrix (partitioned matrix) $A = (A_{\alpha \beta})$, where $A_{\alpha \beta}$ is an $m_{\alpha}$ by $n_{\beta}$ matrix over field ${\bf F}$ for $\alpha=1,2,\ldots,\mu$ and $\beta = 1,2,\ldots,\nu$. The maximum vanishing subspace problem (MVSP) is to maximize $\sum_{\alpha} \dim X_{\alpha} + \sum_{\beta} \dim Y_{\beta}$ over vector subspaces $X_{\alpha} \subseteq {\bf F}^{m_{\alpha}}$ for $\alpha=1,2,\ldots,\mu$ and $Y_{\beta} \subseteq {\bf F}^{n_{\beta}}$ for $\beta = 1,2,\ldots,\nu$ such that each $A_{\alpha \beta}$ vanishes on $X_{\alpha} \times Y_{\beta}$ when $A_{\alpha \beta}$ is viewed as a bilinear form ${\bf F}^{m_{\alpha}} \times {\bf F}^{n_{\beta}} \to {\bf F}$. This problem arises from a study of a canonical block-triangular form of $A$ by Ito, Iwata, and Murota~(1994), and is closely related to the noncommutative rank of a matrix with indeterminates. We prove that a weighted version (WMVP) of MVSP can be solved in psuedo polynomial time, provided arithmetic operations on ${\bf F}$ can be done in constant time. Our proof is a novel combination of submodular optimization on modular lattice and convex optimization on CAT(0)-space. We present implications of this result on block-triangularization of partitioned matrix.