Column Bound for Orthogonal Matrix Factorization

TL;DR

This paper establishes bounds on the number of columns needed for orthogonal matrix factorization by linking it to the Coupon Collector's Problem, revealing how sparsity influences tractability.

Contribution

It introduces a novel theoretical bound on the column count for OMF based on sparsity and probabilistic analysis, connecting combinatorial and matrix factorization concepts.

Findings

Derived a lower bound on columns p for OMF based on sparsity

Connected the problem to the Coupon Collector's Problem

Provided explicit formula involving parameters n and θ

Abstract

This article explores the intersection of the Coupon Collector's Problem and the Orthogonal Matrix Factorization (OMF) problem. Specifically, we derive bounds on the minimum number of columns $p$ (in $\mathbf{X}$) required for the OMF problem to be tractable, using insights from the Coupon Collector's Problem. Specifically, we establish a theorem outlining the relationship between the sparsity of the matrix $\mathbf{X}$ and the number of columns $p$ required to recover the matrices $\mathbf{V}$ and $\mathbf{X}$ in the OMF problem. We show that the minimum number of columns $p$ required is given by $p = \Omega \left(\max \left\{ \frac{n}{1 - (1 - \theta)^n}, \frac{1}{\theta} \log n \right\} \right)$, where $\theta$ is the i.i.d Bernoulli parameter from which the sparsity model of the matrix $\mathbf{X}$ is derived.