We revisit the fundamental question of simple-versus-simple hypothesis testing with an eye towards computational complexity, as the statistically optimal likelihood ratio test is often computationally intractable in high-dimensional settings. In the classical spiked Wigner model (with a general i.i.d. spike prior) we show that an existing test based on linear spectral statistics achieves the best possible tradeoff curve between type I and type II error rates among all computationally efficient tests, even though there are exponential-time tests that do better. This result is conditional on an appropriate complexity-theoretic conjecture, namely a natural strengthening of the well-established low-degree conjecture. Our result shows that the spectrum is a sufficient statistic for computationally bounded tests (but not for all tests). To our knowledge, our approach gives the first tool for reasoning about the precise asymptotic testing error achievable with efficient computation. The main ingredients required for our hardness result are a sharp bound on the norm of the low-degree likelihood ratio along with (counterintuitively) a positive result on achievability of testing. This strategy appears to be new even in the setting of unbounded computation, in which case it gives an alternate way to analyze the fundamental statistical limits of testing.

翻译：我们重新审视简单对简单假设检验这一基本问题，并着眼于计算复杂性——因为统计最优的似然比检验在高维场景下通常计算上难以实现。在经典尖峰维格纳模型（具有一般独立同分布的尖峰先验）中，我们证明一种基于线性谱统计量的现有检验方法能在所有计算高效检验中实现第一类与第二类错误率的最佳权衡曲线，尽管存在效果更优的指数时间检验。该结论依赖于合理的复杂性理论猜想，即对既定低度猜想的自然强化。我们的结果表明：谱统计量对计算受限检验是充分统计量（但非对所有检验成立）。据我们所知，本研究首次提供了分析高效计算所能达到精确检验错误的工具。我们的困难性结论所需的核心要素包括低度似然比范数的锐利界，以及（反直觉地）检验可达性的正向结果。即使在无界计算条件下，这一策略似乎也具有新颖性——它能作为分析检验基本统计极限的替代方法。