Column Bound for Orthogonal Matrix Factorization (2405.12858v1)

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.