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Moment Constraints and Phase Recovery for Multireference Alignment

Vahid Shahverdi, Emanuel Ström, Joakim Andén

TL;DR

This work addresses multireference alignment by constraining reconstructions to the phase manifold determined by the power spectrum, and then maximizing the data likelihood via gradient-based optimization on this manifold. The authors introduce Moment-Constrained Alignment (MCA), an iterative algorithm that alternates between template alignment and projection onto the phase manifold, yielding convergence guarantees and competitive accuracy relative to EM and bispectrum-based methods. Theoretical contributions include the smoothness of the infinite-data loss, a gradient formula, finite-set characterizations of critical points, and probabilistic convergence bounds for the iterative scheme. Empirically, MCA demonstrates improved speed over EM and robustness to noise compared to the third-order moment approach, with strong performance across several signal types and noise regimes. The work also outlines extensions to cryo-EM and discusses open questions on consistency and broader applicability.

Abstract

Multireference alignment (MRA) refers to the problem of recovering a signal from noisy samples subject to random circular shifts. Expectation--maximization (EM) and variational approaches use statistical modeling to achieve high accuracy at the cost of solving computationally expensive optimization problems. The method of moments, instead, achieves fast reconstructions by utilizing the power spectrum and bispectrum to determine the signal up to shift. Our approach combines the two philosophies by viewing the power spectrum as a manifold on which to constrain the signal. We then maximize the data likelihood function on this manifold with a gradient-based approach to estimate the true signal. Algorithmically, our method involves iterating between template alignment and projections onto the manifold. The method offers increased speed compared to EM and demonstrates improved accuracy over bispectrum-based methods.

Moment Constraints and Phase Recovery for Multireference Alignment

TL;DR

This work addresses multireference alignment by constraining reconstructions to the phase manifold determined by the power spectrum, and then maximizing the data likelihood via gradient-based optimization on this manifold. The authors introduce Moment-Constrained Alignment (MCA), an iterative algorithm that alternates between template alignment and projection onto the phase manifold, yielding convergence guarantees and competitive accuracy relative to EM and bispectrum-based methods. Theoretical contributions include the smoothness of the infinite-data loss, a gradient formula, finite-set characterizations of critical points, and probabilistic convergence bounds for the iterative scheme. Empirically, MCA demonstrates improved speed over EM and robustness to noise compared to the third-order moment approach, with strong performance across several signal types and noise regimes. The work also outlines extensions to cryo-EM and discusses open questions on consistency and broader applicability.

Abstract

Multireference alignment (MRA) refers to the problem of recovering a signal from noisy samples subject to random circular shifts. Expectation--maximization (EM) and variational approaches use statistical modeling to achieve high accuracy at the cost of solving computationally expensive optimization problems. The method of moments, instead, achieves fast reconstructions by utilizing the power spectrum and bispectrum to determine the signal up to shift. Our approach combines the two philosophies by viewing the power spectrum as a manifold on which to constrain the signal. We then maximize the data likelihood function on this manifold with a gradient-based approach to estimate the true signal. Algorithmically, our method involves iterating between template alignment and projections onto the manifold. The method offers increased speed compared to EM and demonstrates improved accuracy over bispectrum-based methods.
Paper Structure (19 sections, 22 theorems, 106 equations, 10 figures, 1 algorithm)

This paper contains 19 sections, 22 theorems, 106 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Aligning Methods
  3. Non-aligning Methods
  4. Preliminary Results
  5. Discriminant and Alignment
  6. Convergence to Infinite-Data Limit
  7. Phase Recovery
  8. Properties of the Expected Loss Function
  9. Critical Points
  10. Low-SNR Regime -- Aligning with Noise
  11. Reconstruction algorithm
  12. Efficient Computations
  13. Experiments
  14. Conclusion and Outlook
  15. Proofs from \ref{['sec:phase-recovery']}
  16. ...and 4 more sections

Key Result

Lemma 2.10

\newlabellem:properties_L_sigma0 Let $z$ be a vector in $\mathbb{R}^L \setminus \mathcal{P}_L$ and $x \in \mathcal{M}_{m_2}$. For all constants $\lambda,\tau>0$,

Figures (10)

  • Figure 1: Illustration of the MCA algorithm presented in this paper, applied to a step function signal with $N = 1000$ at $\text{SNR} = 1$. The arrows indicate data flow between stages of the algorithm -- merging of two edges means that the data is combined. The "align" step corresponds to computing $a_i = \sigma(z_k\,;\xi_i)$ for all $i=1,\dots,N$, the "average" step consists of computing $\overline{\sigma}(z_k\,;\mathcal{X}_N)=\tfrac{1}{N}\sum_{i=1}^N a_i$ and the "match moments" step consists of projecting $\overline{\sigma}(z_k\,;\mathcal{X}_N)$ onto the phase manifold $\mathcal{M}_{m_2}$. The converged output is indicated in red.
  • Figure 1: A group element in $\operatorname{CO}(8)$ acting on a 3-dimensional phase manifold. The solid circles on the left rotate under this group element (the dashed circles represent the negative frequencies with conjugate phase). The permutation of the pair of points in the middle is due to the eigenvalue $-1$ of the group element.
  • Figure 1: Samples of the unshifted data (left) correlation between the true underlying signal and samples (middle) and distribution of the estimated shifts over $10^7$ samples (right), at different SNR. The $96\%$ confidence intervals are marked in yellow in the left and middle plots.
  • Figure 1: Normalized root-mean-square error for sawtooth, square and smooth signal, averaged over 40 runs with $N=10^4$ samples and varying $\sigma$. The 100% range is shaded and the 60% range of outcomes is shown as dashed lines.
  • Figure 2: Black and white images: Einstein's iconic photograph (left) and alignment of 80,000 images of pure white noise with that photograph using cross-correlation. Color images: Downsampled Fourier amplitudes of their left-hand side images, with phases indicated as white vectors. The aligned picture was first observed in shatsky2009method, and is related to the phenomenon of confirmation bias from noise, as discussed in balanov2024confirmation.
  • ...and 5 more figures

Theorems & Definitions (54)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Lemma 2.10
  • ...and 44 more