Matrix product constraints by projection methods

TL;DR

This paper introduces projection-based methods for decomposing matrices into factors with specific properties, minimizing their distance from initial guesses, and applies these techniques to challenging exact factorization problems.

Contribution

It develops a general framework for matrix decomposition via projections that enforce structural and property constraints on factors.

Findings

Effective in solving complex exact factorization problems

Provides a flexible approach for matrix decomposition with property constraints

Demonstrates the applicability of projection methods in numerical analysis

Abstract

The decomposition of a matrix, as a product of factors with particular properties, is a much used tool in numerical analysis. Here we develop methods for decomposing a matrix $C$ into a product $X Y$, where the factors $X$ and $Y$ are required to minimize their distance from an arbitrary pair $X_0$ and $Y_0$. This type of decomposition, a projection to a matrix product constraint, in combination with projections that impose structural properties on $X$ and $Y$, forms the basis of a general method of decomposing a matrix into factors with specified properties. Results are presented for the application of these methods to a number of hard problems in exact factorization.