An Alternating Bayesian Approach to PARAFAC Decomposition of Tensors

The PARAllel FACtor (PARAFAC) decomposition is known as one of the most commonly used tools in tensor signal/data processing. Unfortunately, its classical algorithms barely take the potential statistical and/or deterministic prior information of the decomposed tensor into consideration, while the modern ones are usually problem oriented, which limits their applications. To fill in this gap, the PARAFAC decomposition of a tensor is brought into the framework of Bayesian inference in this paper. By introducing transition models for the loading/factor matrices of a tensor, the PARAFAC decomposition can be formulated as an alternating Bayesian filter. By means of the flexibility of the Bayesian filter, the proposed filtering decomposition approach illustrated can cover two commonly used priors-parametric and time transition ones. Under the linear Gaussian assumption, the proposed filter can be implemented as an alternating (matrix) Kalman filter. Analyses show that the performance of the proposed filter is similar to the reported ALS-based algorithms when priors are unavailable. The results of numerical simulations show that our Bayesian approach outperforms the reported PARAFAC decomposition algorithms in the literature, especially for the cases where the statistical and/or deterministic priors are offered such as the target tracking application of a bistatic ULA multiple-input multiple-output radar system.