We show that the maximum expected inner product between a random vector and the standard normal vector over all couplings subject to a mutual information constraint or regularization is equivalent to a truncated integral involving the rate-distortion function, up to universal multiplicative constants. The proof is based on a lifting technique, which constructs a Gaussian process indexed by a random subset of the type class of the probability distribution involved in the information-theoretic inequality, and then applying a form of the majorizing measure theorem.

翻译：我们证明，在互信息约束或正则化条件下，所有耦合中随机向量与标准正态向量之间最大期望内积等价于一个涉及率-失真函数的截断积分（仅相差通用乘法常数）。证明基于一种提升技术，该技术构造了一个由信息论不等式所涉概率分布的典型类随机子集索引的高斯过程，随后应用了主导测度定理的一种形式。