A note on the sample complexity of multi-target detection

TL;DR

The paper addresses the sample complexity of multi-target detection (MTD) in the high-noise regime, motivated by cryo-EM, by deriving upper bounds via autocorrelation analysis and lower bounds via reductions to multi-reference alignment (MRA). It analyzes three MTD variants: (i) 1D signals with circular translations, (ii) 2D images under SO(2) rotations, and (iii) 1D signals without group action, showing that key cases yield $\omega(\sigma^6)$ scaling under suitable conditions, with some bounds proven and others conjectured. The core methodological contributions are the autocorrelation framework and the reduction to MRA to transfer known lower bounds, establishing fundamental limits on orbit recovery from noisy mixtures. The work provides a principled basis for understanding estimation limits in cryo-EM-like settings and guides future development of autocorrelation-based algorithms for structure determination under severe noise.

Abstract

This work studies the sample complexity of the multi-target detection (MTD) problem, which involves recovering a signal from a noisy measurement containing multiple instances of a target signal in unknown locations, each transformed by a random group element. This problem is primarily motivated by single-particle cryo-electron microscopy (cryo-EM), a groundbreaking technology for determining the structures of biological molecules. We establish upper and lower bounds for various MTD models in the high-noise regime as a function of the group, the distribution over the group, and the arrangement of signal occurrences within the measurement. The lower bounds are established through a reduction to the related multi-reference alignment problem, while the upper bounds are derived from explicit recovery algorithms utilizing autocorrelation analysis. These findings provide fundamental insights into estimation limits in noisy environments and lay the groundwork for extending this analysis to more complex applications, such as cryo-EM.

Figures (1)

• Figure 1: (a) Single-particle electron microscopy reconstructs 3D structures from 2D projections bendory2020single. Particles in vitrified ice form a micrograph, modeled using the MTD framework \ref{['eqn:MTDmodelGeneral']}, where $x_i = \Pi(g_i \cdot x)$, with $g_i$ a 3D rotation and $\Pi$ as a tomographic projection. In high noise levels, direct particle detection is infeasible, but 3D reconstruction might be feasible by directly processing the micrographs bendory2023toward. (b) The estimation error (RMSE) versus observation length for MTD with 2D rotations based on autocorrelation analysis (taken from kreymer2022two). As seen empirically, the image can be estimated using the third-order autocorrelation. (c) An MTD observation of 1D signals in which identical signals $x_i = x$ are located at unknown positions in the observation $y$. In the high noise regime, their locations cannot be estimated, but the signal $x$ can be estimated accurately. The estimation error (RMSE) scales as $\sigma^3$ in high noise levels, consistent across autocorrelation analysis and expectation-maximization (taken from lan2020multi).

Theorems & Definitions (28)

• Definition 2.1: Separation
• Remark 2.2
• Definition 2.3: Sample complexity
• Definition 3.1: Empirical autocorrelations
• Definition 3.2: Autocorrelation ensemble mean
• Proposition 3.3
• Proposition 3.4
• Proposition 4.1
• Proposition 5.1
• Proposition 5.2