Rényi 可分辨性、噪声稳定性与反收缩性
Rényi Resolvability, Noise Stability, and Anti-contractivity
AI总结：
本文完整解决了任意Rényi参数下的Rényi可分辨性问题，并借此建立其与噪声稳定性的联系，进而推导出与维度无关的q-稳定性界和Markov算子反收缩性不等式。
摘要：
本文研究三个密切相关的主题——Rényi 可分辨性、噪声稳定性和反收缩性。Rényi 可分辨性问题是指在给定信道下，当输入被设定为给定均匀随机变量的函数时，用 Rényi 散度逼近目标输出分布的问题。该问题在 Rényi 参数属于 $(0,2]\cup\{\infty\}$ 范围内时，由本文作者与 Tan 于 2019 年首次研究。在本文中，我们针对 Rényi 参数在整个范围 $\mathbb{R}\cup\{\pm\infty\}$ 内给出了该问题的完整解答。随后，我们将 Rényi 可分辨性问题与噪声稳定性问题联系起来，观察到最大化或最小化一个集合的 $q$-稳定性等价于 Rényi 可分辨性问题的一个变体。通过这一联系，我们给出了关于 $q$-稳定性的与维度无关的精确界。最后，我们将噪声稳定性问题与 Markov 算子（即条件期望算子）的反收缩性联系起来，其中“反收缩性”这一术语由我们引入，指的是众所周知的收缩性/超收缩性的相反性质。我们推导出了与维度无关的精确反收缩性不等式。本文的所有结果都在二值分布上进行了评估。本文的证明主要基于类型方法，特别是加强版的填充-覆盖引理。
英文摘要：
This paper investigates three closely related topics -- Rényi resolvability, noise stability, and anti-contractivity. The Rényi resolvability problem refers to approximating a target output distribution of a given channel in the Rényi divergence when the input is set to a function of a given uniform random variable. This problem for the Rényi parameter in $(0,2]\cup\{\infty\}$ was first studied by the present author and Tan in 2019. In the present paper, we provide a complete solution to this problem for the Rényi parameter in the entire range $\mathbb{R}\cup\{\pm\infty\}$. We then connect the Rényi resolvability problem to the noise stability problem, by observing that maximizing or minimizing the $q$-stability of a set is equivalent to a variant of the Rényi resolvability problem. By such a connection, we provide sharp dimension-free bounds on the $q$-stability. We lastly relate the noise stability problem to the anti-contractivity of a Markov operator (i.e., conditional expectation operator), where the terminology ``anti-contractivity'' introduced by us refers to as the opposite property of the well-known contractivity/hyercontractivity. We derive sharp dimension-free anti-contractivity inequalities. All of the results in this paper are evaluated for binary distributions. Our proofs in this paper are mainly based on the method of types, especially strengthened versions of packing-covering lemmas.
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