最具信息量的比特及其超越：Courtade--Kumar 猜想的证明与多比特扩展
The Most Informative Bit and Beyond: A Proof of the Courtade--Kumar Conjecture and Multibit Extensions
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- Northeastern University（东北大学）
- Fidian
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中文总结 AI 辅助
本文证明了 Courtade--Kumar 猜想（任意布尔函数通过二元对称信道的信息量不超过单坐标），并构造缩短汉明码量化器证明 k≥8 时坐标投影非最优，提出 2≤k≤7 时坐标投影最优的猜想。
中文摘要 AI 辅助
设 \\(X=(X_1,\ldots,X_n)\\) 在布尔立方体上均匀分布，并设 \\(Y\\) 是通过将其各坐标独立地经过交叉概率为 \\(p\\) 的二元对称信道而得到的。对于一个 \\(k\\)-比特量化器 \\(Q(X)\\)，报告 \\(k\\) 个输入坐标可保留 \\(k(1-h_2(p))\\) 比特的信息，这使得坐标投影成为一种自然的量化基准。对于 \\(k=1\\)，这就是 Courtade--Kumar 猜想：每个布尔函数 \\(f\\) 都满足 \\(I(f(X);Y)\leq 1-h_2(p)\\)，且坐标函数达到等号。我们在所有维度和所有交叉概率下证明了该猜想，且无需平衡假设。证明过程追踪了随着信道相关性变化时相对于坐标的信息间隙。在单调重排之后，一个熵流恒等式将该间隙的导数与相关布尔决策集的边边界联系起来。一个尖锐的局部熵比较、傅里叶分析以及对关键集合的对数 Sobolev 估计表明，任何正间隙都必须在噪声降低时增大，这与它在无噪声端点处的非正值相矛盾。对于 \\(k\geq8\\)，出现了一种编码理论现象。我们构造了一个缩短汉明码量化器，它在 \\(k=8\\) 时超越了坐标基准，并且一个直和论证将该失效扩展到所有 \\(k\geq8\\)。因此，结构化编码可以比坐标选择保留更多信息。在几个结构化码族中的穷举搜索以及针对 \\(2\leq k\leq7\\) 的随机码本上的蒙特卡洛搜索发现，结构化码本优于随机码本，但仍低于坐标基准。这促使我们提出猜想：对于 \\(2\leq k\leq7\\)，坐标投影是最优的。
英文摘要
Let \(X=(X_1,\ldots,X_n)\) be uniform on the Boolean cube, and let \(Y\) be obtained by passing its coordinates independently through a binary symmetric channel with crossover probability \(p\). For a \(k\)-bit quantizer \(Q(X)\), reporting \(k\) input coordinates retains \(k(1-h_2(p))\) bits of information, making coordinate projections a natural quantization benchmark. For \(k=1\), this is the Courtade--Kumar conjecture: every Boolean function \(f\) satisfies \(I(f(X);Y)\leq 1-h_2(p)\), with equality for coordinate functions. We prove the conjecture in every dimension and for every crossover probability, without a balance assumption. The proof tracks the information gap relative to a coordinate as the channel correlation varies. After a monotone rearrangement, an entropy-flow identity relates the derivative of this gap to the edge boundary of the associated Boolean decision set. A sharp local entropy comparison, Fourier analysis, and logarithmic Sobolev estimates for pivotal sets imply that any positive gap must grow as the noise decreases, contradicting its nonpositive value at the noiseless endpoint. For \(k\geq8\), a coding-theoretic phenomenon emerges. We construct a shortened-Hamming-code quantizer that beats the coordinate benchmark for \(k=8\), and a direct-sum argument extends the failure to every \(k\geq8\). Thus structured coding can retain more information than coordinate selection. Exhaustive searches within several structured code families and Monte Carlo searches over random codebooks for \(2\leq k\leq7\) find that structured codebooks outperform random ones but remain below the coordinate benchmark. This motivates our conjecture that coordinate projections are optimal for \(2\leq k\leq7\).
