发表机构
FPT University, Vietnam（越南富国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明反极布尔函数的黑林格猜想，通过格拉姆矩阵核范数等方法将重叠度最小界扩展到任意独立二元输入信道，为信息论与测试提供相关界。
AI 中文摘要
平衡黑林格猜想比较二元对称噪声下布尔决策诱导的两个输出律的重叠度。我们证明，在整个噪声区间内，单输入坐标使反极决策（其值在所有输入符号反转时也反转）的该重叠度最小。证明使用了平方根信道密度的格拉姆矩阵跨类块的归一化核范数，反极决策的乘法会交换格拉姆矩阵保留的偶和奇傅里叶空间，坐标交换可达到匹配其谱的最小代价，矩阵收缩将该尖锐界扩展到任意独立二元输入信道，固定坐标翻转与随机决策由同一论证及凹性得出，这些扩展为信息论与测试提供了界，包括线性高斯观测及加权边边界不等式。
英文摘要
The balanced Hellinger conjecture compares the overlap of two output laws induced by a Boolean decision under binary symmetric noise. We prove that a single input coordinate minimizes this overlap among antipodal decisions, whose values reverse when all input signs are reversed, throughout the noise interval. The proof uses a normalized nuclear norm of the cross-class block of the Gram matrix of square-root channel densities. Multiplication by an antipodal decision exchanges the even and odd Fourier spaces preserved by the Gram matrix. A coordinate exchange attains the minimum cost of matching their spectra. A matrix contraction extends the sharp bound to arbitrary independent binary-input channels. Fixed coordinate flips and randomized decisions follow from the same argument and concavity. These extensions give bounds for information and testing, including linear Gaussian observations, and a weighted edge-boundary inequality.
Comments15 pages
