A theory of meta-factorization

TL;DR

This paper introduces meta-factorization, a unifying theory that describes matrix decompositions as solutions to linear matrix equations, providing insights into existing factorizations and guiding the development of new ones.

Contribution

It presents a novel framework called meta-factorization that reconstructs known factorizations, reveals their internal structures, and suggests ways to create new factorizations.

Findings

Reveals relations between pseudoinverses and matrix decompositions

Provides insights into randomized linear algebra algorithms

Offers a unified view of matrix factorizations

Abstract

We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\"{o}m method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them.