Fundamental limits of detection in the spiked Wigner model
This work addresses a foundational problem in random matrix theory and statistical inference, providing rigorous limits for detection and estimation in spiked models, which is incremental but with broad implications for signal processing and machine learning.
The paper establishes the fundamental limits of detecting a rank-one spike in a Wigner matrix, proving that the log-likelihood ratio is asymptotically normal below a reconstruction threshold and degenerate above, with this threshold marking the same phase transition for both detection and estimation tasks.
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.