Efficient Matrix Factorization Via Householder Reflections

TL;DR

This paper introduces a new matrix factorization method using Householder reflections, providing guarantees for exact and approximate recovery of factors with fewer data columns, advancing orthogonal dictionary learning techniques.

Contribution

It presents a novel factorization approach involving Householder matrices, with theoretical guarantees for exact and approximate recovery from limited data, and offers potential for improved algorithms.

Findings

Exact recovery with constant columns in data matrix

Approximate recovery in polynomial time with logarithmic columns

Potential applications in orthogonal dictionary learning

Abstract

Motivated by orthogonal dictionary learning problems, we propose a novel method for matrix factorization, where the data matrix $\mathbf{Y}$ is a product of a Householder matrix $\mathbf{H}$ and a binary matrix $\mathbf{X}$. First, we show that the exact recovery of the factors $\mathbf{H}$ and $\mathbf{X}$ from $\mathbf{Y}$ is guaranteed with $\Omega(1)$ columns in $\mathbf{Y}$ . Next, we show approximate recovery (in the $l\infty$ sense) can be done in polynomial time($O(np)$) with $\Omega(\log n)$ columns in $\mathbf{Y}$ . We hope the techniques in this work help in developing alternate algorithms for orthogonal dictionary learning.