Various Performance Bounds on the Estimation of Low-Rank Probability Mass Function Tensors from Partial Observations

Probability mass function (PMF) estimation using a low-rank model for the PMF tensor has gained increased popularity in recent years. However, its performance evaluation relied mostly on empirical testing. In this work, we derive theoretical bounds on the attainable performance under this model assumption. We begin by deriving the constrained Cramér-Rao Bound (CCRB) on the low-rank decomposition parameters, and then extend the CCRB to bounds on the mean square error in the resulting estimates of the PMF tensor's elements, as well as on the mean Kullback-Leibler divergence (KLD) between the estimated and true PMFs. The asymptotic tightness of these bounds is demonstrated by comparing them to the performance of the Maximum Likelihood estimate in a small-scale simulation example.