Discrete Aware Tensor Completion via Convexized $\ell_0$-Norm Approximation

TL;DR

This paper introduces a novel discrete-aware tensor completion algorithm that incorporates an $ ext{l}_0$-norm regularizer approximated via fractional programming, improving accuracy and convergence for low-rank tensor recovery with discrete entries.

Contribution

It proposes a new low-rank tensor completion method combining nuclear norm minimization with a continuous approximation of the $ ext{l}_0$-norm regularizer, tailored for discrete-valued tensor data.

Findings

Outperforms state-of-the-art tensor completion methods in NMSE

Demonstrates faster convergence in simulations

Effective for image processing with discrete data

Abstract

We consider a novel algorithm, for the completion of partially observed low-rank tensors, where each entry of the tensor can be chosen from a discrete finite alphabet set, such as in common image processing problems, where the entries represent the RGB values. The proposed low-rank tensor completion (TC) method builds on the conventional nuclear norm (NN) minimization-based low-rank TC paradigm, through the addition of a discrete-aware regularizer, which enforces discreteness in the objective of the problem, by an $\ell_0$-norm regularizer that is approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework, in order to solve the proposed problem. Simulation results demonstrate the superior performance of the new method both in terms of normalized mean square error (NMSE) and convergence, compared to the conventional state of-the-art (SotA) techniques, including NN minimization approaches, as well as a mixture of the latter with a matrix factorization approach.