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Exact Minimax Optimality of Spectral Methods in Phase Synchronization and Orthogonal Group Synchronization

Anderson Ye Zhang

TL;DR

The paper proves that the spectral method for phase synchronization with missing data and Gaussian noise is minimax optimal for the squared $\ell_2$ loss, achieving the exact leading constant $\tfrac{1}{2}$ in the regime where consistent estimation is possible. It introduces a novel population eigenvector $u^* = z^* \circ \widecheck u$ and a first-order perturbation toolkit that yields sharp $\ell_2$ control of the leading eigenvector and extends to orthogonal group synchronization via eigenspace perturbation. Under conditions $\tfrac{np}{\sigma^2}\to\infty$ and $\tfrac{np}{\log n}\to\infty$ (with $p$ and $\sigma^2$ allowed to vary with $n$), the spectral estimator achieves the minimax lower bound and, in the noiseless case, exact recovery as long as $\tfrac{np}{\log n}\to\infty$. The results demonstrate that spectral methods, not just more complex procedures like MLE, GPM, or SDP, attain optimal statistical performance in these synchronization problems, and the techniques extend to $\mathcal{O}(d)$ synchronization as well as broader low-rank settings.

Abstract

We study the performance of the spectral method for the phase synchronization problem with additive Gaussian noises and incomplete data. The spectral method utilizes the leading eigenvector of the data matrix followed by a normalization step. We prove that it achieves the minimax lower bound of the problem with a matching leading constant under a squared $\ell_2$ loss. This shows that the spectral method has the same performance as more sophisticated procedures including maximum likelihood estimation, generalized power method, and semidefinite programming, as long as consistent parameter estimation is possible. To establish our result, we first have a novel choice of the population eigenvector, which enables us to establish the exact recovery of the spectral method when there is no additive noise. We then develop a new perturbation analysis toolkit for the leading eigenvector and show it can be well-approximated by its first-order approximation with a small $\ell_2$ error. We further extend our analysis to establish the exact minimax optimality of the spectral method for the orthogonal group synchronization.

Exact Minimax Optimality of Spectral Methods in Phase Synchronization and Orthogonal Group Synchronization

TL;DR

The paper proves that the spectral method for phase synchronization with missing data and Gaussian noise is minimax optimal for the squared loss, achieving the exact leading constant in the regime where consistent estimation is possible. It introduces a novel population eigenvector and a first-order perturbation toolkit that yields sharp control of the leading eigenvector and extends to orthogonal group synchronization via eigenspace perturbation. Under conditions and (with and allowed to vary with ), the spectral estimator achieves the minimax lower bound and, in the noiseless case, exact recovery as long as . The results demonstrate that spectral methods, not just more complex procedures like MLE, GPM, or SDP, attain optimal statistical performance in these synchronization problems, and the techniques extend to synchronization as well as broader low-rank settings.

Abstract

We study the performance of the spectral method for the phase synchronization problem with additive Gaussian noises and incomplete data. The spectral method utilizes the leading eigenvector of the data matrix followed by a normalization step. We prove that it achieves the minimax lower bound of the problem with a matching leading constant under a squared loss. This shows that the spectral method has the same performance as more sophisticated procedures including maximum likelihood estimation, generalized power method, and semidefinite programming, as long as consistent parameter estimation is possible. To establish our result, we first have a novel choice of the population eigenvector, which enables us to establish the exact recovery of the spectral method when there is no additive noise. We then develop a new perturbation analysis toolkit for the leading eigenvector and show it can be well-approximated by its first-order approximation with a small error. We further extend our analysis to establish the exact minimax optimality of the spectral method for the orthogonal group synchronization.
Paper Structure (19 sections, 24 theorems, 165 equations, 1 figure)

This paper contains 19 sections, 24 theorems, 165 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Phase Synchronization
  3. No-additive-noise Case
  4. First-order Approximation of The Leading Eigenvector
  5. Sharp $\ell_2$ Analysis of The Spectral Estimator
  6. Orthogonal Group Synchronization
  7. Discussions
  8. Comparison of Spectral Method and Other Methods
  9. Condition on $p$
  10. Other Low-rank Problems
  11. Proofs for Phase Synchronization
  12. Proof of Lemma \ref{['lem:eigenvector_perturbation']}
  13. Proofs of Lemma \ref{['lem:no_additive_noise']}, Proposition \ref{['prop:no_additive_noise']}, and Proposition \ref{['prop:eigenvector_perturbation']}
  14. Proof of Theorem \ref{['thm:phase_main']}
  15. Proofs of Auxiliary Lemmas
  16. ...and 4 more sections

Key Result

Theorem 1

Assume $\frac{np}{\sigma^2}\rightarrow\infty$ and $\frac{np}{\log n}\rightarrow\infty$. There exists some $\delta=o(1)$ such that with high probability, As a consequence, when $\sigma=0$ (i.e., there is no additive noise), the spectral method recovers $z^*$ exactly (up to a phase) with high probability as long as $\frac{np}{\log n}\rightarrow\infty$.

Figures (1)

  • Figure 1: Numerical results for the spectral method, GPM, and SDP in phase synchronization, with $n=100$, $p=0.5$ and $\sigma^2$ varying within $[0,20]$. Left: Error comparison measured by the normalized squared $\ell_2$ loss. Right: Comparison of the high-order term in their errors.

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Proposition 2
  • Theorem 3
  • Lemma 3
  • Proposition 3
  • Lemma 4
  • ...and 31 more