Multireference Alignment using Semidefinite Programming

TL;DR

This paper introduces a semidefinite programming approach for multireference alignment, providing a polynomial-time approximation that outperforms existing methods and offers stability guarantees under a random noise model.

Contribution

It proposes a novel SDP relaxation for the multireference alignment problem, analyzes its stability under noise, and connects it to phase correlation methods, reducing computational complexity.

Findings

SDP approximation performs well in typical instances

Stability guarantees under a random noise model

Dropping positivity constraints links to phase correlation

Abstract

The multireference alignment problem consists of estimating a signal from multiple noisy shifted observations. Inspired by existing Unique-Games approximation algorithms, we provide a semidefinite program (SDP) based relaxation which approximates the maximum likelihood estimator (MLE) for the multireference alignment problem. Although we show that the MLE problem is Unique-Games hard to approximate within any constant, we observe that our poly-time approximation algorithm for the MLE appears to perform quite well in typical instances, outperforming existing methods. In an attempt to explain this behavior we provide stability guarantees for our SDP under a random noise model on the observations. This case is more challenging to analyze than traditional semi-random instances of Unique-Games: the noise model is on vertices of a graph and translates into dependent noise on the edges. Interestingly, we show that if certain positivity constraints in the SDP are dropped, its solution becomes equivalent to performing phase correlation, a popular method used for pairwise alignment in imaging applications. Finally, we show how symmetry reduction techniques from matrix representation theory can simplify the analysis and computation of the SDP, greatly decreasing its computational cost.