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

Niclas Führling, Kengo Ando, Giuseppe Thadeu Freitas de Abreu, David González G., Osvaldo Gonsa

TL;DR

The proposed low-rank matrix completion (MC) method is an improved variation of state-of-the-art SotA discrete aware matrix completion method, in which discreteness is enforced by an $\ell_{0}$-norm regularizer, not by replaced with the $\ell_{1}$-norm, but instead approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework.

Abstract

We consider a novel algorithm, for the completion of partially observed low-rank matrices in a structured setting where each entry can be chosen from a finite discrete alphabet set, such as in common recommender systems. The proposed low-rank matrix completion (MC) method is an improved variation of state-of-the-art (SotA) discrete aware matrix completion method which we previously proposed, in which discreteness is enforced by an $\ell_0$-norm regularizer, not by replaced with the $\ell_1$-norm, but instead approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework. Simulation results demonstrate the superior performance of the new method compared to the SotA techniques as well as the earlier $\ell_1$-norm-based discrete-aware matrix completion approach.

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

TL;DR

The proposed low-rank matrix completion (MC) method is an improved variation of state-of-the-art SotA discrete aware matrix completion method, in which discreteness is enforced by an -norm regularizer, not by replaced with the -norm, but instead approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework.

Abstract

We consider a novel algorithm, for the completion of partially observed low-rank matrices in a structured setting where each entry can be chosen from a finite discrete alphabet set, such as in common recommender systems. The proposed low-rank matrix completion (MC) method is an improved variation of state-of-the-art (SotA) discrete aware matrix completion method which we previously proposed, in which discreteness is enforced by an -norm regularizer, not by replaced with the -norm, but instead approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework. Simulation results demonstrate the superior performance of the new method compared to the SotA techniques as well as the earlier -norm-based discrete-aware matrix completion approach.
Paper Structure (10 sections, 31 equations, 3 figures, 1 algorithm)

This paper contains 10 sections, 31 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Matrix Completion
  3. Classical Formulation of the Matrix Completion Problem
  4. SI Approach
  5. Discrete-Aware Matrix Completion
  6. Proposed Discrete-Aware Matrix Completion
  7. Discrete-Aware Regularizer with $\ell_0$-Norm Approximation
  8. Reformulation via Fractional Programming
  9. Numerical Results
  10. Conclusion

Figures (3)

  • Figure 1: NMSE comparison of the SotA and the proposed method, with a varying ratio of observed entries in $\boldsymbol{O}$, for $\alpha=0.1$, $\lambda=10$ and $\zeta=0.15$.
  • Figure 2: NMSE convergence comparison of the SotA and the proposed method, with a $20\%$ observation ratio of $\boldsymbol{O}$, for $\alpha=0.1$, $\lambda=10$ and $\zeta=0.15$.
  • Figure 3: NMSE convergence comparison of the proposed method initialized with Iimori_2020, with a varying ratio of observed entries in $\boldsymbol{O}$.