# Fundamental limits of detection in the spiked Wigner model

arXiv: https://arxiv.org/abs/1806.09588 · HTML: https://mlsanity.com/paper/1806.09588/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2018.

## Built on by (most cited first, 18 of 18)

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- [Signal-Plus-Noise Decomposition of Nonlinear Spiked Random Matrix Models](https://mlsanity.com/paper/2405.18274/) (2024)
- [Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices](https://mlsanity.com/paper/2203.00821/) (2022)
- [Subspace clustering in high-dimensions: Phase transitions & Statistical-to-Computational gap](https://mlsanity.com/paper/2205.13527/) (2022)
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- [Learning with Restricted Boltzmann Machines: Asymptotics of AMP and GD in High Dimensions](https://mlsanity.com/paper/2505.18046/) (2025)
- [Computational and statistical lower bounds for low-rank estimation under general inhomogeneous noise](https://mlsanity.com/paper/2510.08541/) (2025)

## Abstract (arXiv)

We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the non-spiked one is asymptotically normal below a certain reconstruction threshold which is not necessarily of a "spectral" nature, and that it is degenerate above. This establishes the maximal region of contiguity between the planted and null models. It is known that this threshold also marks a phase transition for estimating the spike: the latter task is possible above the threshold and impossible below. Therefore, both estimation and detection undergo the same transition in this random matrix model. We also provide further information about the performance of the optimal test. Our proofs are based on Gaussian interpolation methods and a rigorous incarnation of the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington--Kirkpatrick spin-glass model.