From Weak to Strong Testing in Gaussian Models

2026-09-28

AI Summary

This study investigates the computational complexity of hypothesis testing in the Spiked Wigner model, focusing on the boundaries between weak and strong detection under polynomial-time constraints and their associated error trade-offs. Methodologically, by leveraging the low-degree likelihood ratio (LDLR) and linear spectral statistics, we establish a computational analogue of the Neyman–Pearson lemma, demonstrating that weak detection can be fully reduced to strong detection. The primary contributions include precisely characterizing the optimal trade-off between Type I and Type II errors, determining the polynomial-time weak detection limit, and revealing the intrinsic hardness of weak detection from a computational complexity perspective. Ultimately, this work provides a novel paradigm for detection theory in Gaussian models.

Abstract

We study the computational complexity of hypothesis testing in the spiked Wigner model, a prototypical model for detecting low-rank structure in a large random matrix. Below the "BBP" eigenvalue transition, it is expected that strong detection --- with both type I and II errors vanishing --- requires exponential time. Assuming this as a conjecture, we determine the limits of polynomial-time weak detection, exactly characterizing the possible tradeoffs between type I and II errors. Specifically, the optimal tradeoff is achieved by a particular linear spectral statistic. Thus, the question of weak detection is entirely reduced to that of strong detection. The proof builds on ideas of Nagda-Raghavendra (2025) and Moitra-Wein (2025). The low-degree likelihood ratio (LDLR) plays a key role: any test that slightly beats the LDLR can be boosted to have an even higher success probability. This leads us to establish a computational analogue of the Neyman-Pearson lemma for a subclass of additive Gaussian models: for a given super-polynomial runtime, the best possible tradeoff between type I and II errors is either the one achieved by thresholding the LDLR, or the trivial tradeoff that results from strong detection.

Innovation

• Spiked Wigner Model
• Low-Degree Likelihood Ratio
• Computational Neyman-Pearson Lemma
• Weak Detection
• Linear Spectral Statistic