Potts半群的非线性Log-Sobolev不等式及其在重构问题中的应用
Non-linear Log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems
摘要：
考虑完全图上的随机游动半群，我们称之为Potts半群。Diaconis和Saloff-Coste计算了相对熵与Dirichlet形式之比的最大值，得到了$2$ -log-Sobolev不等式（$2$ -LSI）中的常数$α_2$。在本文中，我们得到了最好的可能的非线性不等式有关熵和Dirichlet形式（即，$p$ -NLSI，$p\ge1$）。作为一个例子，我们展示了$α_1 = 1+\frac{1+o(1)}{\log k}$。通过集成$1$ -NLSI，我们得到了新的强数据处理不等式（SDPI），这反过来又使我们能够改进Mossel和Peres的结果的重建阈值上的Potts模型的树。一个特殊的情况是重建颜色的$k$着色树的根的所有叶子的颜色的知识的问题。我们表明，有一个非平凡的重建概率的树的分支数应至少是$$\frac{\log k}{\log k - \log(k-1)} = (1-o(1))k\log k.$$这恢复以前的结果（Sly和Bhatnagar等人）。（稍微）更具一般性，但更重要的是避免了对任何着色专门参数的需要。同样，我们改进了现有技术的弱恢复阈值的随机块模型与$k$平衡组，所有$k\ge 3$。为了进一步展示我们的方法的力量，我们证明了最佳的非重建结果的广播树模型与高斯内核，关闭一个缺口留下开放的Eldan等人。这些改进倡导信息理论的方法作为一个有用的补充，传统的技术起源于统计物理。
英文摘要：
Consider the semigroup of random walk on a complete graph, which we call the Potts semigroup. Diaconis and Saloff-Coste computed the maximum of the ratio of the relative entropy and the Dirichlet form obtaining the constant $α_2$ in the $2$-log-Sobolev inequality ($2$-LSI). In this paper, we obtain the best possible non-linear inequality relating entropy and the Dirichlet form (i.e., $p$-NLSI, $p\ge1$). As an example, we show $α_1 = 1+\frac{1+o(1)}{\log k}$. By integrating the $1$-NLSI we obtain the new strong data processing inequality (SDPI), which in turn allows us to improve results of Mossel and Peres on reconstruction thresholds for Potts models on trees. A special case is the problem of reconstructing color of the root of a $k$-colored tree given knowledge of colors of all the leaves. We show that to have a non-trivial reconstruction probability the branching number of the tree should be at least $$\frac{\log k}{\log k - \log(k-1)} = (1-o(1))k\log k.$$ This recovers previous results (of Sly and Bhatnagar et al.) in (slightly) more generality, but more importantly avoids the need for any coloring-specialized arguments. Similarly, we improve the state-of-the-art on the weak recovery threshold for the stochastic block model with $k$ balanced groups, for all $k\ge 3$. To further show the power of our method, we prove optimal non-reconstruction results for a broadcasting on trees model with Gaussian kernels, closing a gap left open by Eldan et al. These improvements advocate information-theoretic methods as a useful complement to the conventional techniques originating from the statistical physics.
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