发表机构
Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究字母表增长时Potts树上鲁棒重构，证明在dλ>1条件下，即使边界噪声极强，根后验渐近不变，无噪声与有噪声重构精度相同。
AI 中文摘要
我们研究了在字母表增长机制 \\(q\to\infty\\) 下，Galton-Watson 树上 \\(q\\) 态 Potts 广播过程的鲁棒重构问题，其中边界深度相对于 \\(q\\) 增长得足够快。我们证明，在 \\(d\lambda > 1\\) 的区域内（其中 \\(d\\) 是期望后代数，\\(\lambda\\) 是 Potts 转移矩阵的非平凡特征值），根后验分布在一大类独立应用于边界标签的噪声信道下渐近不变。即使当 \\(q\to\infty\\) 时单个边界顶点保留真实标签的概率趋于零，这一结论仍然成立。更精确地说，在适当的矩条件和噪声假设下，无噪声和有噪声的根后验分布在期望全变差距离下相互收敛，因此它们具有相同的渐近贝叶斯最优重构精度。
英文摘要
We study robust reconstruction for the \(q\)-state Potts broadcast process on a Galton-Watson tree in the growing-alphabet regime \(q\to\infty\), with boundary depth growing sufficiently quickly in relation to \(q\). We show that, in the regime \(dλ>1\), where \(d\) is the expected number of offspring and \(λ\) is the nontrivial eigenvalue of the Potts transition matrix, the root posterior is asymptotically unchanged by a broad class of noise channels applied independently to the boundary labels. This remains true even when the probability of retaining the true label at an individual boundary vertex tends to zero as \(q\to\infty\). More precisely, under suitable moment and noise assumptions, the noiseless and noisy root posteriors converge to one another in expected total variation, and hence have the same asymptotic Bayes-optimal reconstruction accuracy.
