Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination

--CANDECOMP/P ARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. T o address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. T o learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance. For instance, a video sequence can be represented by a third-order tensor with dimensionality of height width time; an image ensemble measured under multiple conditions can be represented by a higher order tensor with dimensionality ofpixel person pose illumination . T ensor factorization enables us to explicitly take into account the structure information by effectively capturing the multilinear interactions among multiple latent factors. Therefore, its theory and algorithms have been an active area of study during the past decade (see e.g., [1], [2]), and have been successfully applied to various application fields, such as face recognition, social network analysis, image and video completion, and brain signal processing. This issue has attracted a great deal of research interest in tensor completion in recent years. The objective of tensor factorization of incomplete data is to capture the underlying multilinear factors from only partially observed entries, which can in turn predict the missing entries.