Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A two-phase rank-based algorithm for low-rank matrix completion

Tacildo de Souza Araújo, Douglas S. Gonçalves, Cristiano Torezzan

TL;DR

This work addresses low-rank matrix completion when the target rank is known. It revisits Fixed Rank Soft-Impute (FRSI) and shows convergence to a rank-constrained, observed-entry-consistent set under a strong condition on singular-value behavior, while recognizing non-guaranteed general convergence. The authors then introduce a two-phase algorithm: a Phase One accelerated warm-start that estimates the nuclear-norm parameter λ from rank-aware updates, and a Phase Two accelerated Soft-Impute with adaptive rank tracking to solve the nuclear-norm regularized problem. Empirical results on synthetic and real data (e.g., MovieLens) demonstrate faster recovery and competitive accuracy compared to established methods, validating the practical value of leveraging known rank in a two-phase scheme.

Abstract

Matrix completion aims to recover an unknown low-rank matrix from a small subset of its entries. In many applications, the rank of the unknown target matrix is known in advance. In this paper, first we revisit a recently proposed rank-based heuristic for "known-rank" matrix completion and establish a condition under which the generated sequence is quasi-Fejér convergent to the solution set. Then, by including an acceleration mechanism similar to Nesterov's acceleration, we obtain a new heuristic. Even though the convergence of such heuristic cannot be granted in general, it turns out that it can be very useful as a warm-start phase, providing a suitable estimate for the regularization parameter and a good starting-point, to an accelerated Soft-Impute algorithm. Numerical experiments with both synthetic and real data show that the resulting two-phase rank-based algorithm can recover low-rank matrices, with relatively high precision, faster than other well-established matrix completion algorithms.

A two-phase rank-based algorithm for low-rank matrix completion

TL;DR

This work addresses low-rank matrix completion when the target rank is known. It revisits Fixed Rank Soft-Impute (FRSI) and shows convergence to a rank-constrained, observed-entry-consistent set under a strong condition on singular-value behavior, while recognizing non-guaranteed general convergence. The authors then introduce a two-phase algorithm: a Phase One accelerated warm-start that estimates the nuclear-norm parameter λ from rank-aware updates, and a Phase Two accelerated Soft-Impute with adaptive rank tracking to solve the nuclear-norm regularized problem. Empirical results on synthetic and real data (e.g., MovieLens) demonstrate faster recovery and competitive accuracy compared to established methods, validating the practical value of leveraging known rank in a two-phase scheme.

Abstract

Matrix completion aims to recover an unknown low-rank matrix from a small subset of its entries. In many applications, the rank of the unknown target matrix is known in advance. In this paper, first we revisit a recently proposed rank-based heuristic for "known-rank" matrix completion and establish a condition under which the generated sequence is quasi-Fejér convergent to the solution set. Then, by including an acceleration mechanism similar to Nesterov's acceleration, we obtain a new heuristic. Even though the convergence of such heuristic cannot be granted in general, it turns out that it can be very useful as a warm-start phase, providing a suitable estimate for the regularization parameter and a good starting-point, to an accelerated Soft-Impute algorithm. Numerical experiments with both synthetic and real data show that the resulting two-phase rank-based algorithm can recover low-rank matrices, with relatively high precision, faster than other well-established matrix completion algorithms.
Paper Structure (9 sections, 4 theorems, 32 equations, 2 figures, 4 tables, 3 algorithms)

This paper contains 9 sections, 4 theorems, 32 equations, 2 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Revisiting fixed-rank Soft-Impute (FRSI) and its convergence
  3. A two-phase rank-based algorithm
  4. Numerical results
  5. Tuning the parameter $\beta$
  6. Experiments with synthetic data
  7. Experiments on MovieLens data set
  8. Conclusion
  9. Soft-Impute as a proximal gradient method

Key Result

Proposition 2.1

Let $T: \mathbb{R}^{m \times n} \rightarrow \mathbb{R}^{m \times n}$ be the operator defined above and assume that $\text{rank}(A) = r$. Then, the following properties hold.

Figures (2)

  • Figure 1: Number of iterations vs $\beta$: (a) $n = 1000$, $r=5$, $\epsilon_{\rho} = 10^{-8}$ and $p \in \left\lbrace 92\%, 85\%, 72\%, 50\% \right\rbrace$; (b) $n \in \left\lbrace 500, 1000, 2000, 4000 \right\rbrace$, $r = 5$, $\epsilon_{\rho} = 10^{-8}$, and $p = 40\%$.
  • Figure 2: Optimal value for $\beta$ with $n = 1000$, $r \in \left\lbrace 3, 5, 10, 30, 50, 80, 100 \right\rbrace$, $\epsilon_{\rho} = 10^{-5}$ and $p = 50\%$.

Theorems & Definitions (10)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 3.1