发表机构
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究在球面线性模型中，在谱正则性条件下证明了TAP近似的定量精度，并刻画了后验几何，包括自由能、后验均值距离及带外质量的指数阶。核心贡献是建立了低于涨落尺度的TAP精度和普适后验几何。
AI 中文摘要
我们研究贝叶斯最优球面线性模型，在环境维度和样本量按比例增长的情况下，并假设设计矩阵满足定量的Marchenko–Pastur谱正则性条件。该条件由具有标准化条目和有限四阶矩的归一化独立同分布设计满足，但不要求条目独立，也不对奇异向量施加条件。在此条件下，我们证明了定量的全温度TAP近似，并刻画了后验几何。对于自然的有限纵横比TAP泛函，归一化球面自由能与TAP最优值相差$O_P(p^{-1})$。每个量与其显式确定性等价物之间的差距为$O_P(p^{-1/2})$，且该涨落尺度是尖锐的。在所有全局TAP最大化器上一致地，到球面后验均值的归一化平方欧氏距离为$O_P(p^{-1})$。我们还证明了由岭估计器确定的数据相关带之外的后验质量具有尖锐的指数阶。更精确地说，对所有足够小的带宽$\varepsilon$一致地，该质量的对数至多为$-cp\varepsilon^2+O_P(1)$。对于每个固定的几何允许宽度，球冠构造给出了该质量的匹配指数阶下界。对于每个确定性宽度序列$\varepsilon_p\gg p^{-1/2}$，相应的带捕获渐近上所有的后验质量。
英文摘要
We study the Bayes-optimal spherical linear model as the ambient dimension and sample size grow proportionally, under a quantitative Marchenko--Pastur spectral-regularity condition on the design. This condition is satisfied by normalized i.i.d. designs with standardized entries of finite fourth moment, but does not require entrywise independence or impose conditions on the singular vectors. Under this condition, we prove a quantitative all-temperature TAP approximation and characterize the posterior geometry. For the natural finite-aspect-ratio TAP functional, the normalized spherical free energy and the TAP optimum differ by $O_P(p^{-1})$. Each is within $O_P(p^{-1/2})$ of its explicit deterministic equivalent, and this fluctuation scale is sharp. Uniformly over all global TAP maximizers, the normalized squared Euclidean distance to the spherical posterior mean is $O_P(p^{-1})$. We also prove that the posterior mass outside a data-dependent band determined by the ridge estimator has sharp exponential order. More precisely, uniformly over sufficiently small band widths $\varepsilon$, the logarithm of this mass is at most $-cp\varepsilon^2+O_P(1)$. For every fixed geometrically admissible width, a spherical-cap construction gives a matching exponential-order lower bound on this mass. For every deterministic sequence of widths $\varepsilon_p\gg p^{-1/2}$, the corresponding bands capture asymptotically all posterior mass.
