Fisher Information and the Minimum Mean-Square Error for Locating Closely Spaced Point Sources
Abstract
We determine the optimal average error of locating closely spaced sources from noisy superpositions of a known smooth response with unknown complex coefficients. Relative phases control how changes in one location can be offset by the other unknowns, so local Fisher information alone does not determine the attainable error. For every fixed number K≥2 of sources with independent uniform phases, we prove matching Bayes risk bounds jointly in separation and noise, for all sufficiently small separations and all noise levels. Odd source counts have two risk regimes; even counts have an intermediate regime in which mean squared error is linear in the noise standard deviation. A projected least-squares estimator attains these orders without knowing the phases or the prior density. Bounded nonuniform phase laws preserve the orders under stated support conditions, whereas priors concentrated near poorly identified configurations can change the exponents. A common finite comparison yields both a global inverse bound and paths controlled by the original prior. These give sufficient conditions under which the risk has the order of the pointwise capped Cramér--Rao bound, and a lower-tail calculation explains the power and logarithmic regimes. We also give conditional guarantees when the cluster center, separation, and source count are unknown, with an additional low-noise requirement for count selection. A calibrated covariance factor model illustrates the same criterion beyond source localization.
Related
- Locating a Seymour vertex in oriented graphs having minimum outdegree at most 7
- On the discrete Fisher information
- A Sharp Fisher-Information Inequality for Boolean Pairs: Proof of the Amari--Kobayashi Conjecture
- Conditional Fisher-Information Central Limit Theorems under Log-Concavity with Information-Theoretic Consequences
- A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow
BibTeX
@misc{wen2026fisher,
title = {{Fisher Information and the Minimum Mean-Square Error for Locating Closely Spaced Point Sources}},
author = {Chuling Wen and Jian Lu and Qingtang Jiang and Charles K. Chui},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32018},
eprint = {2610.32018},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32018},
}
RIS
TY - UNPB
TI - Fisher Information and the Minimum Mean-Square Error for Locating Closely Spaced Point Sources
AU - Chuling Wen
AU - Jian Lu
AU - Qingtang Jiang
AU - Charles K. Chui
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32018
UR - https://arxiv.org/abs/2610.32018
AB - We determine the optimal average error of locating closely spaced sources from noisy superpositions of a known smooth response with unknown complex coefficients. Relative phases control how changes in one location can be offset by the other unknowns, so local Fisher information alone does not determine the attainable error. For every fixed number K≥2 of sources with independent uniform phases, we prove matching Bayes risk bounds jointly in separation and noise, for all sufficiently small separations and all noise levels. Odd source counts have two risk regimes; even counts have an intermediate regime in which mean squared error is linear in the noise standard deviation. A projected least-squares estimator attains these orders without knowing the phases or the prior density. Bounded nonuniform phase laws preserve the orders under stated support conditions, whereas priors concentrated near poorly identified configurations can change the exponents. A common finite comparison yields both a global inverse bound and paths controlled by the original prior. These give sufficient conditions under which the risk has the order of the pointwise capped Cramér--Rao bound, and a lower-tail calculation explains the power and logarithmic regimes. We also give conditional guarantees when the cluster center, separation, and source count are unknown, with an additional low-noise requirement for count selection. A calibrated covariance factor model illustrates the same criterion beyond source localization.
ER -