Abstract-Tensor factorization of incomplete data is a powerful technique for imputation of missing entries (also known as tensor completion) by explicitly capturing the latent multilinear structure. One of the most well-known model is CANDECOMP/PARAFAC (CP) style tensor factorization that can be achieved by various optimization algorithms. However, as either the missing ratio or the noise level increases, most of existing CP factorizations are prone to overfitting since the tensor rank, as a tuning parameter, is required to be manually specified. Unfortunately, the determination of tensor rank is a challenging problem especially in the presence of both missing data and noisy measurements. In addition, the existing approaches can only provide the point estimation of latent factors as well as missing entries, which do not take into account the uncertainty information. To address these issues, we formulate CP factorization by a hierarchal 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 an automatic model selection (i.e., rank determination) and noise detection. To learn the model, we develop an elegant deterministic algorithm under the variational Bayesian inference framework as well as the corresponding solution for efficient computation. Therefore, as a parameter-free approach, our model enables us to effectively infer the underlying multilinear factors from an incomplete and noisy tensor data with a low-rank constraint, while also providing the predictive distributions over latent factors and estimations of missing entries. The extensive simulations on synthetic data illustrate the intrinsic capability of recovering the ground truth of tensor rank and preventing the overfitting problem even with extremely large amount of missing entries. Moreover, the results on several real-world applications, including image inpainting and facial image synthesis, demonstrate that our method significantly outperforms state-of-the-art approaches of both tensor factorization and tensor completion in terms of predictive performance.
REF implements a deterministic Bayesian inference algorithm, which formulates CP factorization with a hierarchical probabilistic model and employs Bayesian treatment by incorporating a sparsityinducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination.
Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination
ArXiv
ArXiv
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