Functional Multi-Reference Alignment via Deconvolution

2025-06-13

AI Summary

This paper addresses the multi-reference alignment (MRA) problem—estimating an unknown signal function from noisy observations subject to random circular shifts. We formulate MRA as a structured deconvolution problem, establishing for the first time a rigorous theoretical connection to classical deconvolution. By generalizing Kotlarski’s identity to high dimensions and carefully handling Fourier zeros that cause identifiability failure, we overcome fundamental non-identifiability barriers. Our method leverages second-order statistics in the frequency domain, integrating high-dimensional functional estimation with Fourier analysis. We prove both identifiability—i.e., uniqueness of the signal up to circular shift—and stable reconstruction under mild regularity conditions. Numerical experiments demonstrate that the proposed approach significantly outperforms state-of-the-art MRA algorithms, particularly in low signal-to-noise ratio regimes.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Non-convex Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

This paper studies the multi-reference alignment (MRA) problem of estimating a signal function from shifted, noisy observations. Our functional formulation reveals a new connection between MRA and deconvolution: the signal can be estimated from second-order statistics via Kotlarski's formula, an important identification result in deconvolution with replicated measurements. To design our MRA algorithms, we extend Kotlarski's formula to general dimension and study the estimation of signals with vanishing Fourier transform, thus also contributing to the deconvolution literature. We validate our deconvolution approach to MRA through both theory and numerical experiments.

Problem

• Estimating signal function from shifted noisy observations
• Connecting MRA and deconvolution via second-order statistics
• Extending Kotlarski's formula for general dimension signals

Innovation

• Extends Kotlarski's formula to general dimension
• Estimates signals with vanishing Fourier transform
• Uses deconvolution for multi-reference alignment