AI 中文总结
本文针对相关源压缩的Kullback-Leibler散度最大化问题，证明k级函数为最优，部分解决Amari-Kobayashi猜想，并在贝叶斯分布式1比特假设检验中验证其最优性。
AI 中文摘要
受Amari和Kobayashi提出的一个猜想的启发，我们研究了识别一对布尔函数的问题，这对函数能最大化通过分别压缩两个相关源得到的两个分布之间的Kullback-Leibler散度。当参考分布对应于独立源时，该问题可简化为最大化互信息的问题，Pichler、Piantanida和Matz已证明独裁者函数在该问题中是最优的。对于最大化Fisher信息的问题（可视为本文所研究问题的局部版本），Amari和Kobayashi猜想奇偶函数是最优的。对于无偏的布尔函数对，以及非负相关区域中的相同函数对，我们证明散度和Fisher信息均由k级函数最大化，k级函数即傅里叶系数仅支撑在k级上的函数。由于k级函数包含奇偶函数，这为Amari和Kobayashi的猜想提供了部分解决。此外，在贝叶斯分布式1比特假设检验框架中，我们证明k级函数在所有函数对中是最优的。最后，我们还讨论了本文所研究问题的单函数版本，该版本可视为Courtade和Kumar猜想的散度类似问题。
英文摘要
Motivated by a conjecture of Amari and Kobayashi, we study the problem of identifying pairs of Boolean functions that maximize the Kullback-Leibler divergence between two distributions obtained by separately compressing two correlated sources. When the reference distribution corresponds to independent sources, this problem reduces to the problem of maximizing mutual information, for which the optimality of dictator functions has been proved by Pichler, Piantanida, and Matz. For the problem of maximizing Fisher information, which can be viewed as a local version of the problem studied in this paper, Amari and Kobayashi conjectured that parity functions are optimal. For unbiased pairs of Boolean functions, and for identical pairs in the nonnegative correlation regime, we prove that both the divergence and the Fisher information are maximized by level-$k$ functions, namely, functions whose Fourier coefficients are supported only on level $k$. Since level-$k$ functions include parity functions, this gives a partial resolution of the conjecture of Amari and Kobayashi. Furthermore, in the framework of Bayesian distributed one-bit hypothesis testing, we prove that level-$k$ functions are optimal among all pairs of functions. Finally, we also discuss the one function version of the problem studied in this paper, which can be regarded as the divergence analogue of the Courtade and Kumar conjecture.
Comments24 pages, 2 figures
