Putting nonnegative matrix factorization to the test: A tutorial derivation of pertinent Cramer? Rao bounds and performance benchmarking

Nonnegative matrix factorization (NMF) is a useful tool in a broad range of applications, from signal separation to computer vision and machine learning. NMF is a hard (NP-hard) computational problem for which various approximate solutions have been developed over the years. Given the widespread interest in NMF and its applications, it is perhaps surprising that the pertinent Cram?r?Rao lower bound (CRLB) on the accuracy of the nonnegative latent factor estimates has not been worked out in the literature. In hindsight, one reason may be that the required computations are more subtle than usual: the problem involves constraints and ambiguities that must be dealt with, and the Fisher information matrix is always singular. We provide a concise tutorial derivation of the CRLB for both symmetric NMF and asymmetric NMF, using the latest CRLB tools, which should be of broad interest for analogous derivations in related factor analysis problems. We illustrate the behavior of these bounds with respect to model parameters and put some of the best NMF algorithms to the test against one another and the CRLB. The results help illuminate what can be expected from the current state of art in NMF algorithms, and they are reassuring in that the gap to optimality is small in relatively sparse and low rank scenarios.