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An Ultra-Fast MLE for Low SNR Multi-Reference Alignment

Shay Kreymer, Amnon Balanov, Tamir Bendory

TL;DR

The paper tackles the challenge of recovering a signal from multiple noisy observations with unknown rotations in the low-SNR regime, a scenario common in single-particle cryo-EM. It develops a fast maximum-likelihood estimator for MRA on $SO(2)$ by applying a low-SNR Taylor expansion to the marginalized log-likelihood, yielding a single-pass frequency-marching algorithm that reconstructs Fourier coefficients sequentially with closed-form phase updates. The key contributions are an analytic low-SNR MLE for $SO(2)$ MRA, a data-driven, non-iterative initialization that rivals EM in accuracy while being orders of magnitude faster, and a roadmap to generalize the approach to 3D cryo-EM via a Procrustes/SVD formulation. This work offers a practical ab initio estimator and a robust initialization for downstream cryo-EM pipelines, potentially enabling faster and scalable high-resolution structure determination.

Abstract

Motivated by single-particle cryo-electron microscopy, multi-reference alignment (MRA) models the task of recovering an unknown signal from multiple noisy observations corrupted by random rotations. The standard approach, expectation-maximization (EM), often becomes computationally prohibitive, particularly in low signal-to-noise ratio (SNR) settings. We introduce an alternative, ultra-fast algorithm for MRA over the special orthogonal group $\mathrm{SO}(2)$. By performing a Taylor expansion of the log-likelihood in the low-SNR regime, we estimate the signal by sequentially computing data-driven averages of observations. Our method requires only one pass over the data, dramatically reducing computational cost compared to EM. Numerical experiments show that the proposed approach achieves high accuracy in low-SNR environments and provides an excellent initialization for subsequent EM refinement.

An Ultra-Fast MLE for Low SNR Multi-Reference Alignment

TL;DR

The paper tackles the challenge of recovering a signal from multiple noisy observations with unknown rotations in the low-SNR regime, a scenario common in single-particle cryo-EM. It develops a fast maximum-likelihood estimator for MRA on by applying a low-SNR Taylor expansion to the marginalized log-likelihood, yielding a single-pass frequency-marching algorithm that reconstructs Fourier coefficients sequentially with closed-form phase updates. The key contributions are an analytic low-SNR MLE for MRA, a data-driven, non-iterative initialization that rivals EM in accuracy while being orders of magnitude faster, and a roadmap to generalize the approach to 3D cryo-EM via a Procrustes/SVD formulation. This work offers a practical ab initio estimator and a robust initialization for downstream cryo-EM pipelines, potentially enabling faster and scalable high-resolution structure determination.

Abstract

Motivated by single-particle cryo-electron microscopy, multi-reference alignment (MRA) models the task of recovering an unknown signal from multiple noisy observations corrupted by random rotations. The standard approach, expectation-maximization (EM), often becomes computationally prohibitive, particularly in low signal-to-noise ratio (SNR) settings. We introduce an alternative, ultra-fast algorithm for MRA over the special orthogonal group . By performing a Taylor expansion of the log-likelihood in the low-SNR regime, we estimate the signal by sequentially computing data-driven averages of observations. Our method requires only one pass over the data, dramatically reducing computational cost compared to EM. Numerical experiments show that the proposed approach achieves high accuracy in low-SNR environments and provides an excellent initialization for subsequent EM refinement.
Paper Structure (10 sections, 1 theorem, 17 equations, 1 figure, 1 algorithm)

This paper contains 10 sections, 1 theorem, 17 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. MLE algorithm for low SNR MRA
  3. Statistical model
  4. Preliminary estimates
  5. Conditional likelihood
  6. The algorithm
  7. Expectation-Maximization (EM)
  8. Computational Complexity
  9. Numerical Experiments
  10. A roadmap to fast maximum-likelihood estimation for cryo-EM

Key Result

Lemma 2.1

For $\sigma \to \infty$, the term inside the $\log$ in eq:log-likleihood is strictly positive, i.e., $\hat{C}_{j,k}[0] + \frac{\Re \bar{\hat{y}}_j[k] \hat{f}[k] \hat{C}_{j,k}[k]}{\sigma^2} > 0$.

Figures (1)

  • Figure 1: Performance comparison of the proposed fast MLE algorithm and standard EM, initialized from a random guess or from the result of the fast MLE algorithm. Panels a, b: Comparison of reconstruction error (a) and runtime (b) versus noise level $\sigma$, with fixed number of observations $n = 100{,}000$. The proposed fast MLE algorithm maintains a constant, low runtime regardless of noise intensity. While the EM algorithm achieves lower reconstruction error, it is orders of magnitude slower. Notably, EM initialized with our method outperforms random initialization in low-noise regimes. Panels c, d: Comparison of accuracy (c) and runtime (d) for varying number of observations $n$, with fixed noise level $\sigma=12$. Both algorithms exhibit the expected $1/n$ error decay. The results highlight a trade-off: our algorithm offers a dramatic reduction in computational cost with only a small difference in accuracy relative to EM.

Theorems & Definitions (3)

  • Lemma 2.1
  • proof
  • Remark 2.2