Nonnegative Low Rank Matrix Completion by Riemannian Optimalization Methods

Abstract

In this paper, we study Riemannian optimization methods for the problem of nonnegative matrix completion that is to recover a nonnegative low rank matrix from its partial observed entries. With the underlying matrix incoherence conditions, we show that when the number $m$ of observed entries are sampled independently and uniformly without replacement, the inexact Riemannian gradient descent method can recover the underlying $n_{1}$-by-$n_{2}$ nonnegative matrix of rank $r$ provided that $m$ is of $\mathcal{O}(r^{2} s \log^2s )$, where $s = \max \{n_{1},n_{2} \}$. Numerical examples are given to illustrate that the nonnegativity property would be useful in the matrix recovery. In particular, we demonstrate the number of samples required to recover the underlying low rank matrix with using the nonnegativity property is smaller than that without using the property.