Sparse multi-reference alignment: sample complexity and computational hardness

TL;DR

This paper investigates the sample and computational complexities of sparse multi-reference alignment, revealing the trade-offs between different algorithms and their feasibility for cryo-EM applications.

Contribution

It establishes the statistical feasibility of sparse MRA with noise proportional to the square of observations and analyzes the computational hardness of various algorithms.

Findings

Projection-based algorithms achieve optimal estimation but are exponentially complex.

Bispectrum methods offer polynomial-time solutions with suboptimal estimation.

Convex relaxations provide polynomial algorithms for sparse signals at optimal rates.

Abstract

Motivated by the problem of determining the atomic structure of macromolecules using single-particle cryo-electron microscopy (cryo-EM), we study the sample and computational complexities of the sparse multi-reference alignment (MRA) model: the problem of estimating a sparse signal from its noisy, circularly shifted copies. Based on its tight connection to the crystallographic phase retrieval problem, we establish that if the number of observations is proportional to the square of the variance of the noise, then the sparse MRA problem is statistically feasible for sufficiently sparse signals. To investigate its computational hardness, we consider three types of computational frameworks: projection-based algorithms, bispectrum inversion, and convex relaxations. We show that a state-of-the-art projection-based algorithm achieves the optimal estimation rate, but its computational complexity is exponential in the sparsity level. The bispectrum framework provides a statistical-computational trade-off: it requires more observations (so its estimation rate is suboptimal), but its computational load is provably polynomial in the signal's length. The convex relaxation approach provides polynomial time algorithms (with a large exponent) that recover sufficiently sparse signals at the optimal estimation rate. We conclude the paper by discussing potential statistical and algorithmic implications for cryo-EM.