Symmetric Tensor Canonical Polyadic Decomposition Via Probabilistic Inference

Abstract

This paper addresses symmetric tensor canonical polyadic decomposition (CPD) problem from a probabilistic inference perspective. Symmetric tensor CPD aims at decomposing a symmetric tensor into a succinct sum of rank-1 components and has found wide-spread applications in signal processing and machine learning. By exploiting its inherent symmetric structure, a variety of algorithms that leverage advances in algebras and optimizations have been proposed. However, the implementation of these algorithms needs the exact value of tensor rank, which however is unknown in practice and difficult to estimate. Fortunately, recent developments in probabilistic tensor CPD have successfully achieved automatic rank determination by recasting the rank estimation problem into a hyper-parameter inference problem. Nevertheless, the conjugacy property in exponential distribution family, which is essential to permit closed-form derivations of inference algorithms, does not hold for the probabilistic symmetric tensor CPD model. This calls for a novel design of inference algorithm that can achieve accurate and fast CPD with automatic tensor rank determination, and this paper provides such a solution. Numerical results are presented to validate the excellent performance of the proposed algorithm.