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Nonconvex Deterministic Matrix Completion by Projected Gradient Descent Methods

Hang Xu, Song Li, Junhong Lin

TL;DR

The paper addresses deterministic low‑rank matrix completion by sampling entries on Ramanujan graphs and analyzes nonconvex PGD approaches based on Burer–Monteiro factorization. It proves that PGD converges linearly to the incoherent ground‑truth under a benign initialization and sufficient samples, with a rate tied to the ground‑truth condition number $\kappa$, and introduces a scaled PGD variant that achieves a linear rate independent of $\kappa$ at higher sample complexity. The work integrates Ramanujan‑graph spectral properties and incoherence to establish restricted strong convexity/smoothness in the nonconvex lift, provides initialization guarantees, and demonstrates the methods’ practical efficiency via numerical experiments. Overall, the results show that deterministic sampling via Ramanujan graphs enables efficient, provably convergent nonconvex matrix completion with computational advantages over nuclear‑norm minimization in large‑scale settings, with potential applicability to related low‑rank recovery problems.

Abstract

We study deterministic matrix completion problem, i.e., recovering a low-rank matrix from a few observed entries where the sampling set is chosen as the edge set of a Ramanujan graph. We first investigate projected gradient descent (PGD) applied to a Burer-Monteiro least-squares problem and show that it converges linearly to the incoherent ground-truth with respect to the condition number \k{appa} of ground-truth under a benign initialization and large samples. We next apply the scaled variant of PGD to deal with the ill-conditioned case when \k{appa} is large, and we show the algorithm converges at a linear rate independent of the condition number \k{appa} under similar conditions. Finally, we provide numerical experiments to corroborate our results.

Nonconvex Deterministic Matrix Completion by Projected Gradient Descent Methods

TL;DR

The paper addresses deterministic low‑rank matrix completion by sampling entries on Ramanujan graphs and analyzes nonconvex PGD approaches based on Burer–Monteiro factorization. It proves that PGD converges linearly to the incoherent ground‑truth under a benign initialization and sufficient samples, with a rate tied to the ground‑truth condition number , and introduces a scaled PGD variant that achieves a linear rate independent of at higher sample complexity. The work integrates Ramanujan‑graph spectral properties and incoherence to establish restricted strong convexity/smoothness in the nonconvex lift, provides initialization guarantees, and demonstrates the methods’ practical efficiency via numerical experiments. Overall, the results show that deterministic sampling via Ramanujan graphs enables efficient, provably convergent nonconvex matrix completion with computational advantages over nuclear‑norm minimization in large‑scale settings, with potential applicability to related low‑rank recovery problems.

Abstract

We study deterministic matrix completion problem, i.e., recovering a low-rank matrix from a few observed entries where the sampling set is chosen as the edge set of a Ramanujan graph. We first investigate projected gradient descent (PGD) applied to a Burer-Monteiro least-squares problem and show that it converges linearly to the incoherent ground-truth with respect to the condition number \k{appa} of ground-truth under a benign initialization and large samples. We next apply the scaled variant of PGD to deal with the ill-conditioned case when \k{appa} is large, and we show the algorithm converges at a linear rate independent of the condition number \k{appa} under similar conditions. Finally, we provide numerical experiments to corroborate our results.
Paper Structure (19 sections, 20 theorems, 151 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 19 sections, 20 theorems, 151 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Matrix Completion
  3. Our Contributions
  4. Related Work
  5. Notations and Organization
  6. Preliminaries
  7. Ramanujan Graph
  8. Incoherence Conditions
  9. Problem Setup and Main Results
  10. Nonconvex Model via Matrix Decomposition
  11. Projected Gradient Descent
  12. Scaled Projected Gradient Descent
  13. Convergent Analysis of Scaled PGD
  14. Proof Sketch
  15. Useful Lemmas
  16. ...and 4 more sections

Key Result

Theorem 1

Assume the matrix $\pmb{M}^* \in \mathbb{R}^{n_1\times n_2}$ satisfies the assumptions ${\rm \pmb{\mathscr{A}}_1}$ and ${\rm \pmb{\mathscr{A}}_2}$ with $\delta_d \leq \frac{1}{64}$, the adjacency matrix $\pmb{G}$ of the graph satisfies the assumptions ${\rm \pmb{\mathscr{G}}_1}$, ${\rm \pmb{\mathscr Meanwhile,

Figures (2)

  • Figure 1: Frequency of success for probabilistic and deterministic matrix completion.
  • Figure 2: The relative errors of PGD and Scaled PGD with $r=3, 5$ under different condition numbers $\kappa = 1,5,10$ for deterministic matrix completion.

Theorems & Definitions (40)

  • Definition 1: Asymmetric Ramanujan graph, BV2020
  • Remark 1
  • Definition 2
  • Theorem 1
  • Remark 2
  • Definition 3
  • Lemma 1
  • Theorem 2
  • Remark 3
  • Proposition 1: Local Curvature Condition
  • ...and 30 more