用于鲁棒有限块长信息与色散分析的卡方几何
Chi-Squared Geometry for Robust Finite-Blocklength Information and Dispersion Analysis
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中文总结 AI 辅助
该研究为离散无记忆信道构建列向卡方几何，提出以η为核心参数的三阶比率展开、双侧色散等价性、经认证的鲁棒设计速率三大结果，无需对数运算即可得到紧的信息与编码速率界。
中文摘要 AI 辅助
我们为离散无记忆信道（DMCs）构建了列向卡方几何，无需计算信道矩阵的对数，就能得到互信息、信道色散和有限块长编码速率的紧的、无对数的界。核心参数是η——转移概率与其输出边缘分布的最坏情况相对偏差，当信道接近全噪声信道$t_{ij}=s_j$时，η恰好很小。我们证明了三个主要结果：（1）三阶比率展开，表明当η→0时，$I(X;Y)/χ^2(X;Y)→1/2$，并带有$O(η)$的偏度修正；（2）双侧色散等价性，用$χ^2(X;Y)$上下界估计$V(X;Y)$，显式常数$c_{\u00b1}(η)→1$；（3）经认证的鲁棒设计速率$R_{\u200bcert}(n,ε)$，其总认证间隙为$O(η)+O(η/\sqrt{n})+O(\log n/n)$。关于I和V的认证界仅需加法、乘法、除法和平方根运算；最终速率还用到$Q^{-1}(ε)$。
英文摘要
We develop a column-wise chi-squared geometry for discrete memoryless channels (DMCs) yielding tight, logarithm-free bounds on mutual information, channel dispersion, and finite-blocklength coding rates without evaluating logarithms of the channel matrix. The key parameter is~\(η\)---the worst-case relative deviation of a transition probability from its output marginal, which is small precisely when the channel is close to the fully noisy channel $t_{ij}=s_j$. We prove three main results: (1) a third-order ratio expansion showing \(I(X;Y)/χ^2(X;Y)\to 1/2\) as \(η\to 0\) with an \(O(η)\) skewness correction; (2) a two-sided dispersion equivalence bounding \(V(X;Y)\) above and below by \(χ^2(X;Y)\) with explicit constants \(c_{\pm}(η)\to 1\); and (3) a certified robust design rate \(R_{\mathrm{cert}}(n,\varepsilon)\) with total certification gap \(O(η)+O(η/\sqrt{n})+O(\log n/n)\). The certified bounds on \(I\) and \(V\) require only addition, multiplication, division, and square roots; the final rate also uses \(Q^{-1}(\varepsilon)\).
