Higher-order low-rank tensors with missing values naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data reconstruction, and so on. These problems can be formulated as low-rank tensor completion (LRTC). Existing methods for LRTC employ matrix nuclear-norm minimization and use the singular value decomposition (SVD) in their algorithms, which become very slow or even not applicable for large-scale problems. To tackle this difficulty, we apply low-rank matrix factorization to each mode unfolding of the tensor in order to enforce low-rankness and update the matrix factors alternatively, which is computationally much cheaper than SVD.

We aim at recovering a low-rank tensor from partial observations , where is the index set of observed entries, and keeps the entries in and zeros out others. We apply low-rank matrix factorization to each mode unfolding of by finding matrices such that for , where is the estimated rank, either fixed or adaptively updated. Introducing one common variable to relate these matrix factorizations, we solve the following model

where and . In the model, , , are weights and satisfy . We use alternating least squares method to solve the model.

Our model is non-convex jointly with respect to and . Although a global solution is not guaranteed, we demonstrate by numerical experiments that our algorithm can reliably recover a wide variety of low-rank tensors, such as the following phase transition plots. In the picture, each target tensor , where and have Gaussian random entries. (a) FaLRTC: the tensor completion method in . (b) MatComp: first reshape the tensor as a matrix and then use the matrix completion solver LMaFit in . (c) TMac-fix: our method with and fixed to . (d) TMac-inc: our method with and using rank-increasing strategy starting from . (e) TMac-dec: our method with and using rank-decreasing strategy starting from .

The results show that our method performs much better than the other two methods.

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