Sample Complexity Analysis of Multi-Target Detection via Markovian and Hard-Core Multi-Reference Alignment

2025-10-20

AI Summary

This paper investigates the sample complexity of multi-target detection (MTD) under long noisy observations—i.e., detecting and aligning multiple occurrences of an unknown signal at unknown locations. We propose a block-wise strategy that models MTD as a non-i.i.d. multi-reference alignment (MRA) problem and, for the first time, characterize its latent structural constraints via Markov chains and hard-core point processes. Theoretically, we show that the convergence rate matches that of i.i.d. MRA. Leveraging empirical means and moment estimation, combined with exponential-mixture random field theory, we establish that signal recovery in the low-SNR regime requires a sample size scaling as $sigma^{2n_{min}}$, where $n_{min}$ is the minimal order moment of the noise variance—matching the performance of classical i.i.d. MRA. Our core contribution is establishing a rigorous theoretical connection between MTD and structured MRA, yielding a tight characterization of sample complexity.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Topic discovery and tracking

Abstract

Motivated by single-particle cryo-electron microscopy, we study the sample complexity of the multi-target detection (MTD) problem, in which an unknown signal appears multiple times at unknown locations within a long, noisy observation. We propose a patching scheme that reduces MTD to a non-i.i.d. multi-reference alignment (MRA) model. In the one-dimensional setting, the latent group elements form a Markov chain, and we show that the convergence rate of any estimator matches that of the corresponding i.i.d. MRA model, up to a logarithmic factor in the number of patches. Moreover, for estimators based on empirical averaging, such as the method of moments, the convergence rates are identical in both settings. We further establish an analogous result in two dimensions, where the latent structure arises from an exponentially mixing random field generated by a hard-core placement model. As a consequence, if the signal in the corresponding i.i.d. MRA model is determined by moments up to order $n_{min}$, then in the low-SNR regime the number of patches required to estimate the signal in the MTD model scales as $σ^{2n_{min}}$, where $σ^2$ denotes the noise variance.

Problem

• Analyzing sample complexity for multi-target detection in noisy observations
• Developing patching schemes for non-i.i.d. multi-reference alignment models
• Establishing convergence rates for signal estimation in low-SNR regimes

Innovation

• Patching scheme reduces MTD to non-i.i.d. MRA model
• Convergence rates match i.i.d. MRA model with logarithmic factor
• Signal estimation scales with noise variance in low-SNR regime