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A Majorization-Minimization Gauss-Newton Method for 1-Bit Matrix Completion

Xiaoqian Liu, Xu Han, Eric C. Chi, Boaz Nadler

TL;DR

This work tackles 1-bit matrix completion by introducing MMGN, a Majorization-Minimization Gauss-Newton method that converts the nonconvex rank-constrained likelihood problem into a sequence of standard low-rank matrix completion subproblems. Each surrogate minimization uses a factorization Theta = UV^T and a single Gauss-Newton update, yielding a per-iteration cost of O(|Omega| r) and practical descent guaranteed by Armijo line search. MMGN achieves comparable or better estimation accuracy than state-of-the-art methods while offering substantial speedups, especially when the observation fraction is small or the matrix is highly spiky. The approach also demonstrates robust performance across logistic and probit link models and scales well to large real datasets, with potential extensions to other quantization schemes and theoretical convergence guarantees left for future work.

Abstract

In 1-bit matrix completion, the aim is to estimate an underlying low-rank matrix from a partial set of binary observations. We propose a novel method for 1-bit matrix completion called Majorization-Minimization Gauss-Newton (MMGN). Our method is based on the majorization-minimization principle, which converts the original optimization problem into a sequence of standard low-rank matrix completion problems. We solve each of these sub-problems by a factorization approach that explicitly enforces the assumed low-rank structure and then apply a Gauss-Newton method. Using simulations and a real data example, we illustrate that in comparison to existing 1-bit matrix completion methods, MMGN outputs comparable if not more accurate estimates. In addition, it is often significantly faster, and less sensitive to the spikiness of the underlying matrix. In comparison with three standard generic optimization approaches that directly minimize the original objective, MMGN also exhibits a clear computational advantage, especially when the fraction of observed entries is small.

A Majorization-Minimization Gauss-Newton Method for 1-Bit Matrix Completion

TL;DR

This work tackles 1-bit matrix completion by introducing MMGN, a Majorization-Minimization Gauss-Newton method that converts the nonconvex rank-constrained likelihood problem into a sequence of standard low-rank matrix completion subproblems. Each surrogate minimization uses a factorization Theta = UV^T and a single Gauss-Newton update, yielding a per-iteration cost of O(|Omega| r) and practical descent guaranteed by Armijo line search. MMGN achieves comparable or better estimation accuracy than state-of-the-art methods while offering substantial speedups, especially when the observation fraction is small or the matrix is highly spiky. The approach also demonstrates robust performance across logistic and probit link models and scales well to large real datasets, with potential extensions to other quantization schemes and theoretical convergence guarantees left for future work.

Abstract

In 1-bit matrix completion, the aim is to estimate an underlying low-rank matrix from a partial set of binary observations. We propose a novel method for 1-bit matrix completion called Majorization-Minimization Gauss-Newton (MMGN). Our method is based on the majorization-minimization principle, which converts the original optimization problem into a sequence of standard low-rank matrix completion problems. We solve each of these sub-problems by a factorization approach that explicitly enforces the assumed low-rank structure and then apply a Gauss-Newton method. Using simulations and a real data example, we illustrate that in comparison to existing 1-bit matrix completion methods, MMGN outputs comparable if not more accurate estimates. In addition, it is often significantly faster, and less sensitive to the spikiness of the underlying matrix. In comparison with three standard generic optimization approaches that directly minimize the original objective, MMGN also exhibits a clear computational advantage, especially when the fraction of observed entries is small.
Paper Structure (22 sections, 3 theorems, 24 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 22 sections, 3 theorems, 24 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prior Work
  3. Our Contributions
  4. Notation and Organization
  5. The MMGN Method
  6. The Majorization-Minimization Principle
  7. Inexact Majorization-Minimization via Gauss-Newton
  8. Related Problems
  9. Logistic PCA.
  10. Binary matrix factorization.
  11. Link prediction in network analysis.
  12. Numerical Studies
  13. Simulation Settings
  14. Non-spiky Matrices
  15. Varying the noise level
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $\Phi(\theta)$ be a CDF that satisfies assumptions A1 and A2. The following is a majorization of $\ell({\bm{\mathbf{\Theta}}})$ at ${\bm{\tilde{\mathbf}{\Theta}}}$ where $c({\bm{\tilde{\mathbf}{\Theta}}})$ depends on ${\bm{\tilde{\mathbf}{\Theta}}}$ but not on ${\bm{\mathbf{\Theta}}}$, and

Figures (7)

  • Figure 1: Probit model: Relative error, Hellinger distance, and runtime versus noise level $\sigma$ for a non-spiky underlying matrix of size $m \times n = 1000 \times 1000$ and rank $r^* = 1$ at observation fraction $\rho = 0.3$.
  • Figure 2: Probit model ($\sigma = 1$): Relative error, Hellinger distance, and runtime versus observation fraction $\rho$ for a non-spiky underlying matrix of size $m \times n = 1000 \times 1000$ and rank $r^* = 1$.
  • Figure 3: Probit model ($\sigma = 0.18$): Relative error, Hellinger distance, and runtime versus matrix dimension $n$ for a square non-spiky underlying matrix of rank $r^* = 5$ with observation fraction $\rho = 0.8$.
  • Figure 4: Probit model ($\sigma = 0.18$): Relative error, Hellinger distance, and runtime versus true rank $r^*$ for a non-spiky underlying matrix of size $m \times n = 1000 \times 1000$ and observation fraction $\rho = 0.8$.
  • Figure 5: Probit model ($\sigma = 0.18$): Relative error (using the training set), log-likelihood (on the testing set), and runtime versus estimated rank $\hat{r}$ for a non-spiky underlying matrix of size $m \times n = 1000 \times 1000$, true rank $r^*=5$, and observation fraction $\rho = 0.8$. LBFGS produced log-likelihood of negative infinity at $\hat{r}\in \{11, 12\}$, so the corresponding box plots were missing.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Corollary 2.1
  • Corollary 2.2