AI 中文总结
该研究提出统一信息论框架，推导极小极大分位数下界，结合多种信息度量，在不同模型中验证其有效性，揭示信息控制需适配恢复分辨率与似然比尾部。
AI 中文摘要
我们开发了一个用于对极小极大分位数进行下界估计的统一信息论框架。该框架的出发点是适配损失函数的Neyman-Pearson元逆定理，其在每个损失半径和置信水平下对极小极大成功概率进行约束。该约束将先验和损失诱导的小球几何结构与实验的统计可区分性分离开，并在辅助输出分布上进行优化。对这一单一测试约束的不同松弛方式会产生基于f-信息性、Sibson信息I_α、最大泄漏（Maximal Leakage）和Amemiya范数的逆定理，经典的Fano和Le Cam下界作为特殊情况被恢复。该框架还阐明了为何不同信息度量适用于不同的恢复准则：最大泄漏对一类对称精确恢复问题是精确的，我们利用这一恒等式推导了平衡高斯加权随机块模型中完整极小极大精确恢复风险的有限样本界，以及各向同性有界能量噪声下低秩矩阵估计的双侧有限样本极小极大分位数界。对于近似汉明恢复，我们展示了一个异构二元模型，其中优化后的有限Sibson阶会产生强逆定理，而最大泄漏的特例则是平凡的。最后，对于单坐标泊松定位，通过其Amemiya范数使用的Bennett型Young函数恢复了精确的成功概率尺度，而经典Fano和固定功率松弛则严格更弱。这些结果表明，精确的极小极大分位数逆定理需要使信息控制同时适配恢复分辨率和似然比尾部。
英文摘要
We develop a unified information-theoretic framework for lower bounding minimax quantiles. The starting point is a loss-adapted Neyman--Pearson metaconverse that bounds the minimax success probability at every loss threshold and confidence level. The bound separates the small-ball behaviour of the prior under the loss from the statistical distinguishability of the observation model, and is optimised over an auxiliary output distribution. Different relaxations of this Neyman--Pearson bound yield converses based on \(f\)-informativity, Sibson mutual information \(I_α\), Maximal Leakage, and Amemiya norms. Classical Fano and Le Cam lower bounds are recovered as special cases. The framework also clarifies why different information measures are suited to different recovery criteria. Maximal Leakage is exact for a class of symmetric exact-recovery problems. We use this identity to derive finite-sample bounds on the full minimax exact-recovery risk in the balanced Gaussian weighted stochastic block model, as well as two-sided finite-sample minimax-quantile bounds for low-rank matrix estimation under isotropic bounded-energy noise. For approximate Hamming recovery, we exhibit a heterogeneous binary model in which an optimised finite Sibson order yields a strong converse while the Maximal Leakage specialisation is trivial. Finally, for one-coordinate Poisson localisation, a Bennett-type Young function used through its Amemiya norm recovers the exact success-probability scale, whereas classical Fano and fixed-power relaxations are strictly weaker. These results show that sharp converses for minimax quantiles require adapting the information measure to the recovery resolution, whether exact or approximate, and to the tail behaviour of the likelihood ratio.
