Matrix and Tensor Completion with Noise via Low-rank Deconvolution

David Reixach, Ramon Morros

Keywords: Tensors, Completion, Denoising, Low-rank, Deconvolution

TL;DR: Matrix and Tensor Completion with Noise via Low-rank Deconvolution

Abstract:

Low-rank Deconvolution (LRD) has been recently introduced as a new representation model for multi-dimensional data. In this work we consider its use for tackling the problem of matrix and tensor completion. This model is designed to encode information in a very efficient manner using a limited number of parameters and to be flexible by providing a simple framework that allows for easy inclusion of priors such different types of regularizers while, at the same time, letting for easy algorithms to solve the (generally non-convex) learning process. We suspect that these properties will facilitate the resolution of tensor completion problems, i.e the reconstruction of a tensor from incomplete and randomly corrupted entries. Then, our contribution is twofold: first we show that this model acts as a relaxation of the classical low-rank approach allowing for a greater number of solutions than the imposed by the low-rank constraint while using a similar number
of parameters. And second, we present an algorithm based on a block multi-convex optimization method with nuclear norm minimization and squared total variation regularization that solves the tensor completion problem. Theoretical and empirical results are presented that support our claims.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Submission Number: 171