Bayesian Tensor Completion and Decomposition with Automatic CP Rank Determination Using MGP Shrinkage Prior

Abstract

Tensor completion, which completes high-dimensional data with missing entries, has many applications, such as recommender systems and image inpainting. Low-rank CP decomposition is one of the popular methods in tensor completion and is an extension of matrix decomposition to higher order tensors. However, unlike matrix factorization, it is NP-hard to obtain the rank of CP decomposition directly. In this paper, our objective is simultaneously achieving tensor completion and rank determination in CP decomposition. This can be achieved using Bayesian CP decomposition with Multiplicative Gamma Process (MGP) as the prior distribution. MGP is a distribution that decays the components. Using MGP, the proposed method avoids duplication of components and enables highly accurate rank estimation in Bayesian tensor modeling. In addition, MGP helps to reduce noise sensitivity and estimation time. Numerical experiments using artificial data and image data demonstrate the effectiveness of the proposed method.