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

Niclas Führling, Getuar Rexhepi, Giuseppe Abreu

TL;DR

This work tackles the problem of completing low-rank tensors whose entries come from a discrete alphabet by introducing a discrete-aware tensor completion (DaLRTC) framework. The method augments the conventional nuclear-norm objective with a differentiable $l_0$-norm–based discrete regularizer, which is convexified via fractional programming and optimized with a proximal-gradient algorithm. The approach jointly leverages a data-fidelity term, a multi-linear nuclear-norm prior, and a discrete prior to improve recovery from partial observations, demonstrated to achieve superior NMSE and convergence over state-of-the-art NN-based and matrix-factorization-based methods on color images. This discrete-aware TC is applicable to image processing and broader multidimensional data problems with discrete-valued measurements, offering improved reconstruction quality in scenarios with partial and quantized 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.

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

TL;DR

This work tackles the problem of completing low-rank tensors whose entries come from a discrete alphabet by introducing a discrete-aware tensor completion (DaLRTC) framework. The method augments the conventional nuclear-norm objective with a differentiable -norm–based discrete regularizer, which is convexified via fractional programming and optimized with a proximal-gradient algorithm. The approach jointly leverages a data-fidelity term, a multi-linear nuclear-norm prior, and a discrete prior to improve recovery from partial observations, demonstrated to achieve superior NMSE and convergence over state-of-the-art NN-based and matrix-factorization-based methods on color images. This discrete-aware TC is applicable to image processing and broader multidimensional data problems with discrete-valued measurements, offering improved reconstruction quality in scenarios with partial and quantized 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 -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.
Paper Structure (14 sections, 49 equations, 3 figures, 1 algorithm)

This paper contains 14 sections, 49 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical Formulation of the Matrix and Tensor Completion Problem
  4. Classical Solution to the Matrix and Tensor Completion Problem
  5. SI Approach
  6. Discrete-Aware Matrix Completion
  7. Simple Low-Rank Tensor Completion (SiLRTC) Algorithm
  8. Matrix Factorization Approach
  9. Proposed Method
  10. Discrete Regularization via $l_0$-Norm Approximation
  11. Reformulation via Fractional Programming
  12. Iterative Solution via Proximal Gradient
  13. Numerical Results
  14. Conclusion

Figures (3)

  • Figure 1: Illustration of the original tensor in form of an image, as well as its partially observed, sampled version, with $20\%$ observed entries.
  • Figure 2: Illustration of the $l_0$-norm approximation with $L=1$ and a varying smoothing factor $\alpha$.
  • Figure 3: NMSE comparison of the SotA and the proposed method, with a varying ratio of observed entries in $\boldsymbol{O}$, for $\alpha=0.01$, $\lambda=65$ and $\zeta=0.5$.