  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.
