Rank regularization and Bayesian inference for tensor completion and extrapolation
This work addresses tensor completion for applications like brain imaging and gene expression, but it is incremental as it builds on existing Bayesian and regularization methods.
The paper tackles the problem of tensor completion with missing entries by proposing a novel rank regularizer for PARAFAC decomposition within a Bayesian framework, achieving recovery errors of -10dB and -15dB on brain imaging and yeast gene expression datasets with 50% and 15% missing entries, respectively.
A novel regularizer of the PARAFAC decomposition factors capturing the tensor's rank is proposed in this paper, as the key enabler for completion of three-way data arrays with missing entries. Set in a Bayesian framework, the tensor completion method incorporates prior information to enhance its smoothing and prediction capabilities. This probabilistic approach can naturally accommodate general models for the data distribution, lending itself to various fitting criteria that yield optimum estimates in the maximum-a-posteriori sense. In particular, two algorithms are devised for Gaussian- and Poisson-distributed data, that minimize the rank-regularized least-squares error and Kullback-Leibler divergence, respectively. The proposed technique is able to recover the "ground-truth'' tensor rank when tested on synthetic data, and to complete brain imaging and yeast gene expression datasets with 50% and 15% of missing entries respectively, resulting in recovery errors at -10dB and -15dB.