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Tightness of SDP and Burer-Monteiro Factorization for Phase Synchronization in High-Noise Regime

Anderson Ye Zhang

TL;DR

This work advances the understanding of phase synchronization by showing that, in the high-noise regime, the BM factorization and SDP relaxations are exponentially close to the MLE, with an explicit bound $\mathbb E\big(n^{-2}\|\hat Z^{\textsc{BM},m} - \hat z^{\textsc{MLE}}(\hat z^{\textsc{MLE}})^{\mathrm{H}}\|_{\mathrm{F}}^2\big) \le C\exp(-n/(8\sigma^{2})) + n^{-10}$. It develops a deterministic fixed-point framework that unifies the MLE and BM as fixed points of two mappings $F_1$ and $F_m$, derives a contraction-type relation between their distances, and then analyzes the MLE via a Lipschitz surrogate $G$ together with a leave-one-out scheme to obtain exponential bounds. The results extend beyond SDP to BM factorization with general rank parameter $m\ge 2$, showing the BM–MLE deviation decays exponentially with $n/\sigma^{2}$ and that tightness against the MLE holds in the prescribed high-noise regime. Overall, the paper provides a rigorous, entrywise understanding of the SDP/BM relaxations’ behavior under heavy noise and offers a pathway to analyze similar low-rank relaxations in related synchronization problems.

Abstract

We study the difference between the maximum likelihood estimation (MLE) and its semi-definite programming (SDP) relaxation for the phase synchronization problem, where $n$ latent phases are estimated based on pairwise observations corrupted by Gaussian noise at a level $σ$. While previous studies have established that SDP coincides with the MLE when $σ\lesssim \sqrt{n / \log n}$, the behavior in the high-noise regime $σ\gtrsim \sqrt{n / \log n}$ remains unclear. We address this gap by quantifying the deviation between the SDP and the MLE in the high-noise regime as $\exp(-c \frac{n}{σ^2})$, indicating an exponentially small discrepancy. In fact, we establish more general results for the Burer-Monteiro (BM) factorization that covers the SDP as a special case: it has the exponentially small deviation from the MLE in the high-noise regime and coincides with the MLE when $σ$ is small. To obtain our results, we develop a refined entrywise analysis of the MLE that is beyond the existing $\ell_\infty$ analysis in literature.

Tightness of SDP and Burer-Monteiro Factorization for Phase Synchronization in High-Noise Regime

TL;DR

This work advances the understanding of phase synchronization by showing that, in the high-noise regime, the BM factorization and SDP relaxations are exponentially close to the MLE, with an explicit bound . It develops a deterministic fixed-point framework that unifies the MLE and BM as fixed points of two mappings and , derives a contraction-type relation between their distances, and then analyzes the MLE via a Lipschitz surrogate together with a leave-one-out scheme to obtain exponential bounds. The results extend beyond SDP to BM factorization with general rank parameter , showing the BM–MLE deviation decays exponentially with and that tightness against the MLE holds in the prescribed high-noise regime. Overall, the paper provides a rigorous, entrywise understanding of the SDP/BM relaxations’ behavior under heavy noise and offers a pathway to analyze similar low-rank relaxations in related synchronization problems.

Abstract

We study the difference between the maximum likelihood estimation (MLE) and its semi-definite programming (SDP) relaxation for the phase synchronization problem, where latent phases are estimated based on pairwise observations corrupted by Gaussian noise at a level . While previous studies have established that SDP coincides with the MLE when , the behavior in the high-noise regime remains unclear. We address this gap by quantifying the deviation between the SDP and the MLE in the high-noise regime as , indicating an exponentially small discrepancy. In fact, we establish more general results for the Burer-Monteiro (BM) factorization that covers the SDP as a special case: it has the exponentially small deviation from the MLE in the high-noise regime and coincides with the MLE when is small. To obtain our results, we develop a refined entrywise analysis of the MLE that is beyond the existing analysis in literature.

Paper Structure

This paper contains 23 sections, 21 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Deterministic Analysis Through Fixed Points
  3. Introducing an Equivalent Representation of the BM Factorization and a Different Loss Function
  4. Introducing $F_1$ and $F_m$: $\hat{z}^\textsc{MLE}$ and $\hat{V}^{\textsc{BM},m}$ Are Their Fixed Points
  5. A Contraction-type Result for $\ell_m(F_m(V), F_1(z))$ and $\ell_m(V, z)$
  6. Implications on $\ell_m(\hat{V}^{\textsc{BM},m},\hat{z}^\textsc{MLE})$
  7. Analysis on MLE
  8. High-level Idea of Our Analysis
  9. Step 1: Approximate $\hat{z}^\textsc{MLE}$ by Fixed Points of Lipschitz Mappings
  10. Step 2: Leave-One-Out Analysis for Fixed Points of $G(\cdot,s,t)$
  11. Exponential Bounds
  12. Discussions
  13. Computation and Optimization Landscape
  14. Role of $m$
  15. Tightness of the SDP
  16. ...and 8 more sections

Key Result

Theorem 1.1

Suppose $m\in\mathbb{N}\setminus\{1\}$.

Figures (1)

  • Figure 1: Left: A visualization of the geometric relationship among the SDP, the MLE, and the ground truth. Right: Summary of Theorem \ref{['thm:intro2']}: The distance between the BM factorization and the MLE decays exponentially as $\frac{n}{\sigma^2}$ increases. The distance becomes 0, indicating tightness, when $\frac{n}{\sigma^2}\gtrsim \log n$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 28 more