arXiv subjects

Anderson Ye Zhang

Publications and source records attributed to Anderson Ye Zhang.

5 recordsLinked to original sources

arXiv2025

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

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.

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arXiv2025

Misspecified Maximum Likelihood Estimation for Non-Uniform Group Orbit Recovery

We study maximum likelihood estimation (MLE) in the generalized group orbit recovery model, where each observation is generated by applying a random group action and a known, fixed linear operator to an unknown signal, followed by additive noise. This model is motivated by single-particle cryo-electron microscopy (cryo-EM) and can be viewed primarily as a structured continuous Gaussian mixture model. In practice, signal estimation is often performed by marginalizing over the group using a uniform distribution--an assumption that typically does not hold and renders the MLE misspecified. This raises a fundamental question: how does the misspecified MLE perform? We address this question from several angles. First, we show that in the absence of projection, the misspecified population log-likelihood has desired optimization landscape that leads to correct signal recovery. In contrast, when projections are present, the global optimizers of the misspecified likelihood deviate from the true signal, with the magnitude of the bias depending on the noise level. To address this issue, we propose a joint estimation approach tailored to the cryo-EM setting, which parameterizes the unknown distribution of the group elements and estimates both the signal and distribution parameters simultaneously.

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arXiv2024

A Novel and Optimal Spectral Method for Permutation Synchronization

Permutation synchronization is an important problem in computer science that constitutes the key step of many computer vision tasks. The goal is to recover $n$ latent permutations from their noisy and incomplete pairwise measurements. In recent years, spectral methods have gained increasing popularity thanks to their simplicity and computational efficiency. Spectral methods utilize the leading eigenspace $U$ of the data matrix and its block submatrices $U_1,U_2,\ldots, U_n$ to recover the permutations. In this paper, we propose a novel and statistically optimal spectral algorithm. Unlike the existing methods which use $\{U_jU_1^\top\}_{j\geq 2}$, ours constructs an anchor matrix $M$ by aggregating useful information from all of the block submatrices and estimates the latent permutations through $\{U_jM^\top\}_{j\geq 1}$. This modification overcomes a crucial limitation of the existing methods caused by the repetitive use of $U_1$ and leads to an improved numerical performance. To establish the optimality of the proposed method, we carry out a fine-grained spectral analysis and obtain a sharp exponential error bound that matches the minimax rate.

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arXiv2024

Fundamental Limits of Spectral Clustering in Stochastic Block Models

Spectral clustering has been widely used for community detection in network sciences. While its empirical successes are well-documented, a clear theoretical understanding, particularly for sparse networks where degrees are much smaller than $\log n$, remains unclear. In this paper, we address this significant gap by demonstrating that spectral clustering offers exponentially small error rates when applied to sparse networks under Stochastic Block Models. Our analysis provides sharp characterizations of its performance, backed by matching upper and lower bounds possessing an identical exponent with the same leading constant. The key to our results is a novel truncated $\ell_2$ perturbation analysis for eigenvectors, coupled with a new analysis idea of eigenvectors truncation.

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arXiv2024

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

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.

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