Bayesian Nonnegative Tensor Completion With Automatic Rank Determination

Abstract

Nonnegative CANDECOMP/PARAFAC (CP) factorization of incomplete tensors is a powerful technique for finding meaningful and physically interpretable latent factor matrices to achieve nonnegative tensor completion. However, most existing nonnegative CP models rely on manually predefined tensor ranks, which introduces uncertainty and leads the models to overfit or underfit. Although the presence of CP models within the probabilistic framework can estimate rank better, they lack the ability to learn nonnegative factors from incomplete data. In addition, existing approaches tend to focus on point estimation and ignore estimating uncertainty. To address these issues within a unified framework, we propose a fully Bayesian treatment of nonnegative tensor completion with automatic rank determination. Benefitting from the Bayesian framework and the hierarchical sparsity-inducing priors, the model can provide uncertainty estimates of nonnegative latent factors and effectively obtain low-rank structures from incomplete tensors. Additionally, the proposed model can mitigate problems of parameter selection and overfitting. For model learning, we develop two fully Bayesian inference methods for posterior estimation and propose a hybrid computing strategy that reduces the time overhead for large-scale data significantly. Extensive simulations on synthetic data demonstrate that our model can recover missing data with high precision and automatically estimate CP rank from incomplete tensors. Moreover, results from real-world applications demonstrate that our model is superior to state-of-the-art methods in image and video inpainting. The code is available at https://github.com/zecanyang/BNTC.