Moment Constraints and Phase Recovery for Multireference Alignment

2024-09-07

arXiv.org

AI Summary

Multi-reference alignment (MRA) aims to recover an unknown signal from noisy, randomly cyclically shifted observations. This paper proposes a novel framework that unifies the computational efficiency of method-of-moments approaches with the statistical accuracy of maximum-likelihood estimation. Specifically, it models the signal’s power spectrum as residing on a low-dimensional constrained manifold and performs gradient ascent directly on this manifold. We introduce an alternating “iterative alignment–manifold projection” paradigm that jointly optimizes template estimation and manifold constraints. Compared to the expectation-maximization algorithm, our method achieves significantly faster convergence. At low signal-to-noise ratios, it attains higher reconstruction accuracy than classical bispectrum-based methods and demonstrates superior robustness to noise and model mismatch. The core contributions are: (i) the formulation of the power-spectrum manifold for MRA, and (ii) the co-design of alignment and manifold projection within a unified optimization framework—yielding a theoretically rigorous yet computationally efficient solution to MRA.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Recovering signals from noisy shifted samples efficiently
• Combining statistical modeling with moment constraints for accuracy
• Improving reconstruction speed and precision over existing methods

Innovation

• Combines power spectrum manifold constraints with likelihood maximization
• Uses iterative template alignment and manifold projection
• Provides faster reconstruction than EM with higher accuracy