[Paper Review] Maximum vanishing subspace problem, CAT(0)-space relaxation, and block-triangularization of partitioned matrix

Masaki Hamada|arXiv (Cornell University)|May 5, 2017

Complexity and Algorithms in Graphs32 references22 citations

TL;DR

This paper introduces a weighted maximum vanishing subspace problem (WMVSP) for partitioned matrices and proves it can be solved in pseudo-polynomial time using a novel combination of submodular optimization on modular lattices and convex optimization on CAT(0)-spaces. The key contribution is a polynomial-time algorithm for block-triangularization of partitioned matrices via proximal point methods on CAT(0)-spaces, resolving an open problem on canonical forms for such matrices.

ABSTRACT

In this paper, we consider computing the degree of the Dieudonné determinant of a polynomial matrix A = A_l + A_{l-1} s + ⋯ + A₀ s^l, where each A_d is a linear symbolic matrix, i.e., entries of A_d are affine functions in symbols x₁, …, x_m over a field K. This problem is a natural "weighted analog" of Edmonds' problem, which is to compute the rank of a linear symbolic matrix. Regarding x₁, …, x_m as commutative or noncommutative, two different versions of weighted and unweighted Edmonds' problems can be considered. Deterministic polynomial-time algorithms are unknown for commutative Edmonds' problem and have been proposed recently for noncommutative Edmonds' problem. The main contribution of this paper is to establish a deterministic polynomial-time reduction from (non)commutative weighted Edmonds' problem to unweighed Edmonds' problem. Our reduction makes use of the discrete Legendre conjugacy between the integer sequences of the maximum degree of minors of A and the rank of linear symbolic matrices obtained from the coefficient matrices of A. Combined with algorithms for noncommutative Edmonds' problem, our reduction yields the first deterministic polynomial-time algorithm for noncommutative weighted Edmonds' problem with polynomial bit-length bounds. We also give a reduction of the degree computation of quasideterminants and its application to the degree computation of noncommutative rational functions.

Motivation & Objective

• To address the open problem of computing a canonical block-triangular form for partitioned matrices under equivalence transformations.
• To generalize the bipartite stable set problem to a linear algebraic setting involving block-structured matrices over a field F.
• To develop a polynomial-time algorithm for the weighted maximum vanishing subspace problem (WMVSP), which generalizes MVSP.
• To establish a novel computational framework combining submodular optimization on modular lattices and convex optimization on CAT(0)-spaces.

Proposed method

• Formulate the weighted maximum vanishing subspace problem (WMVSP) as maximizing a weighted sum of subspace dimensions under bilinear vanishing constraints.
• Model the lattice of subspaces of vector spaces as a modular lattice and extend submodular functions to the orthoscheme complex K(L), a CAT(0)-space.
• Apply a CAT(0)-space version of the proximal point algorithm (PPA) to solve the resulting convex optimization problem.
• Use the unique geodesic structure of CAT(0)-spaces to define and compute proximal mappings iteratively.
• Construct a recursive algorithm (q-DM and q-DMreg) that uses optimal solutions of WMVSP with specially designed weights to extract extremal mv-subspaces and build a quasi-Dulmage-Mendelsohn (q-DM) decomposition.
• Prove that under specific weight schemes, optimal WMVSP solutions yield extremal (minimal or maximal) mv-subspaces, enabling recursive decomposition.

Experimental results

Research questions

• RQ1 Can the weighted maximum vanishing subspace problem (WMVSP) be solved in pseudo-polynomial time for partitioned matrices over a field F?
• RQ2 Does convex optimization on CAT(0)-spaces provide a viable framework for solving discrete optimization problems on infinite modular lattices of subspaces?
• RQ3 Can extremal mv-subspaces (minimal and maximal) be uniquely identified via appropriate weight assignments in WMVSP?
• RQ4 Is there a polynomial-time algorithm to compute a canonical block-triangular form for a partitioned matrix using WMVSP solutions?
• RQ5 Can the quasi-Dulmage-Mendelsohn decomposition be computed algorithmically via repeated solution of WMVSP with tailored weights?

Key findings

• WMVSP can be solved in pseudo-polynomial time, polynomial in m, n, and the maximum weight W, assuming constant-time arithmetic over F.
• The solution relies on a novel combination of submodular optimization on modular lattices and convex optimization on CAT(0)-spaces via a proximal point algorithm.
• Extremal mv-subspaces (minimal and maximal) are uniquely obtained as optimal solutions of WMVSP under specific weight schemes (e.g., Cα = m+2, Dβ = m+1 for maximality).
• A recursive algorithm (q-DM and q-DMreg) computes a quasi-Dulmage-Mendelsohn decomposition by recursively identifying mv-subspaces using WMVSP with tailored weights.
• The algorithm runs in polynomial time because each WMVSP instance has bounded weights, and Theorem 1.1 ensures pseudo-polynomial solvability.
• The method establishes a constructive link between mv-subspaces and canonical block-triangular forms, resolving an open problem posed by Ito, Iwata, and Murota (1994).