Multi-Reference Alignment in High Dimensions: Sample Complexity and Phase Transition

Elad Romanov, Tamir Bendory, Or Ordentlich

Research output: Contribution to journalArticlepeer-review

Abstract

Multi-reference alignment entails estimating a signal in $\mathbb{R}^L$ from its circularly shifted and noisy copies. This problem has been studied thoroughly in recent years, focusing on the finite-dimensional setting (fixed $L$). Motivated by single-particle cryo-electron microscopy, we analyze the sample complexity of the problem in the high-dimensional regime $L\to\infty$. Our analysis uncovers a phase transition phenomenon governed by the parameter $\alpha = L/(\sigma^2\log L)$, where $\sigma^2$ is the variance of the noise. When $\alpha>2$, the impact of the unknown circular shifts on the sample complexity is minor. Namely, the number of measurements required to achieve a desired accuracy $\varepsilon$ approaches $\sigma^2/\varepsilon$ for small $\varepsilon$; this is the sample complexity of estimating a signal in additive white Gaussian noise, which does not involve shifts. In sharp contrast, when $\alpha\leq 2$, the problem is significantly harder, and the sample complexity grows substantially more quickly with $\sigma^2$.
Original languageEnglish
Pages (from-to)494-523
JournalSIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE
Volume3
Issue number2
DOIs
StatePublished - 20 Apr 2021

Keywords

  • multi-reference alignment
  • estimation in high dimension
  • information-theoretic lower bounds
  • mathematics of cryo-EM imaging

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