发表机构
UC Berkeley; Department of Mathematics, UC Davis（加州大学伯克利分校; 加州大学戴维斯分校数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究在尖峰Wigner模型中，基于强检测需指数时间的猜想，精确刻画了多项式时间弱检测的最优错误权衡，由线性谱统计量实现，并建立了计算Neyman-Pearson引理，将弱检测问题归结为强检测问题。
AI 中文摘要
我们研究了尖峰Wigner模型中假设检验的计算复杂度，该模型是检测大型随机矩阵中低秩结构的典型模型。在“BBP”特征值跃迁之下，预期强检测——即第一类和第二类错误均趋于零——需要指数时间。基于此猜想，我们确定了多项式时间弱检测的极限，精确刻画了第一类和第二类错误之间可能的权衡。具体而言，最优权衡由特定的线性谱统计量实现。因此，弱检测问题完全归结为强检测问题。证明基于Nagda-Raghavendra (2025)和Moitra-Wein (2025)的思想。低度似然比（LDLR）发挥关键作用：任何略微优于LDLR的检验都可以被提升以获得更高的成功概率。这使我们为加性高斯模型的一个子类建立了Neyman-Pearson引理的计算类比：对于给定的超多项式运行时间，第一类和第二类错误之间的最佳权衡要么是通过阈值化LDLR实现的权衡，要么是由强检测产生的平凡权衡。
英文摘要
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.
Comments49 pages
