B0993
Title: Cramer-Rao bounds for CANDECOMP/PARAFAC non-negative tensor decomposition
Authors: Carlos Llosa - Sandia National Laboratories (United States) [presenting]
Daniel M Dunlavy - Sandia National Laboratories (United States)
Richard B Lehoucq - Sandia National Laboratories (United States)
Arvind Prasadan - Sandia National Laboratories (United States)
Oscar Lopez - Florida Atlantic University (United States)
Abstract: A Cramer-Rao lower bound (CRLB) is presented on the variance of estimates of factor matrices in CANDECOMP/PARAFAC (CP) non-negative tensor decomposition when these are obtained from minimizing the Kullback-Leibler loss. While the tensor decomposition is the maximum likelihood (ML) estimator of a Poisson model, differentiating this Poisson log-likelihood is challenging. To overcome this challenge, it is first demonstrated that the traditional algorithm used for fitting the tensor decomposition is an instance of an expectation-maximization (EM) algorithm. The associated complete log-likelihood is easier to differentiate, and the observed Fisher information matrix (FIM) is expressed in terms of conditional expectations of its gradient and Hessian. By expressing the condition number of the FIM in terms of tensor rank, order and size gauging the stability of a given tensor decomposition (whether the CRLB is finite or not) is possible. The novel FIM expression can also be used to formulate faster Newton-Raphson and Fisher scoring algorithms for ML estimation. The bounds are studied in detailed experiments where we vary the signal-to-noise (SNR) ratio, and in real-world datasets with varied SNR levels.
