A note on the sample complexity of multi-target detection

2025-01-21

AI Summary

This paper studies the sample complexity of multi-target detection (MTD) under high noise, motivated by cryo-electron microscopy (cryo-EM), where the goal is to recover the 3D structure of biomolecules from noisy 2D images containing multiple copies of a target object, each randomly transformed by elements of a group. We establish the first tight information-theoretic bounds on the sample complexity of MTD under general group actions: a lower bound is derived via reduction to the multi-reference alignment problem and information-theoretic analysis; an upper bound is achieved by constructing an explicit, representation-theoretic recovery algorithm leveraging autocorrelation analysis. Our results precisely characterize how the required sample size depends on the group structure, the distribution over the group, and the geometry of the target space. They reveal fundamental limits of signal estimation in high-noise regimes and provide the first theoretically complete benchmark for sample efficiency in practical multi-target inverse problems such as cryo-EM.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Multi-modal Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• Multi-object Detection
• Cryo-electron Microscopy
• Sample Data Size

Innovation

• Multi-object Detection
• High-noise Environment
• Cryo-electron Microscopy Application