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

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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 $\sigma^{2n_{\min}}$, where $\sigma^2$ denotes the noise variance.

Subjects:

Signal Processing (eess.SP); Information Theory (cs.IT)

Cite as: arXiv:2510.17775 [eess.SP]

(or arXiv:2510.17775v3 [eess.SP] for this version)

https://doi.org/10.48550/arXiv.2510.17775

arXiv-issued DOI via DataCite

Submission history

From: Carlos Esteve-Yagüe [view email] [v1] Mon, 20 Oct 2025 17:35:19 UTC (2,063 KB) [v2] Mon, 15 Dec 2025 14:20:36 UTC (557 KB) [v3] Mon, 30 Mar 2026 09:12:14 UTC (1,039 KB)