Expectation-maximization for low-SNR multi-reference alignment

TL;DR

This paper analyzes the expectation-maximization algorithm for multi-reference alignment at low signal-to-noise ratios, revealing fundamental convergence bottlenecks, sensitivity to initialization, and the impact of Fourier phase preservation on reconstruction quality.

Contribution

It provides a detailed characterization of EM's convergence dynamics, initialization dependence, and introduces the 'Ghost of Newton' instability in low-SNR MRA.

Findings

EM exhibits a two-phase convergence with exponential decay rates depending on SNR.

Iteration complexity to reach accurate reconstruction scales as SNR^{-2}.

Fourier phases are preserved during EM iterations, while magnitudes contract slowly.

Abstract

We study the multi-reference alignment (MRA) problem of recovering a signal from noisy observations acted on by unknown random circular shifts. While the information-theoretic limits of MRA are well characterized in many settings, the algorithmic behavior at low signal-to-noise ratio (SNR), the regime of practical interest, remains poorly understood. In this paper, we analyze the expectation-maximization (EM) algorithm, a widely used method for MRA, and characterize its convergence dynamics and initialization dependence in the low-SNR limit. On the convergence side, we prove a two-phase phenomenon near the ground truth as $\mathrm{SNR}\to 0$: an initial contraction with error decaying as $\exp(-\, \mathrm{SNR} \cdot t)$ followed by a much slower phase scaling as $\exp(- \,\mathrm{SNR}^2 \cdot t)$, where $t$ is the iteration number. This yields an iteration-complexity lower bound $T \gtrsim \mathrm{SNR}^{-2}$ to reach a small fixed target accuracy, revealing a severe computational bottleneck at low SNR. We also identify a finite-sample instability, which we term \emph{Ghost of Newton}, in which EM initially approaches the ground truth but later diverges, degrading reconstruction quality. On the bias side, we analyze EM in the noise-only setting ($\mathrm{SNR}=0$), a regime referred to as Einstein from Noise, to highlight its pronounced sensitivity to initialization. We prove that the EM map preserves the Fourier phases of the initialization across all iterations, while the corresponding Fourier magnitudes contract toward zero at a slow rate of $(1+T)^{-1/2}$. Consequently, although the amplitudes vanish in the limit of $T \to \infty$ iterations, the reconstructed structure continues to reflect the geometry encoded by the template's Fourier phases. Together, these results expose fundamental computational and initialization-driven limitations of EM for MRA in the low-SNR regime.