Nonparametric low-rank tensor imputation

Completion or imputation of three-way data arrays with missing entries is a basic problem encountered in various areas, including bio-informatics, image processing, and preference analysis. If available, prior information about the data at hand should be incorporated to enhance performance of the imputation method adopted. This is the motivation behind the proposed low-rank tensor estimator which leverages the correlation across slices of the data cube in the form of reproducing kernels. The rank of the tensor estimate is controlled by a novel regularization on the factors of its PARAFAC decomposition. Such a regularization is inspired by a reformulation of the nuclear norm for matrices, which allows to bypass the challenge that rank and singular values of tensors are unrelated quantities. The proposed technique is tested on MRI data of the brain with 30% missing data, resulting in a recovery error of -17dB.