Quantifying the dependence between high-dimensional random variables is central to statistical learning and inference. Two classical methods are canonical correlation analysis (CCA), which identifies maximally correlated projected versions of the original variables, and Shannon's mutual information, which is a universal dependence measure that also captures high-order dependencies. However, CCA only accounts for linear dependence, which may be insufficient for certain applications, while mutual information is often infeasible to compute/estimate in high dimensions. This work proposes a middle ground in the form of a scalable information-theoretic generalization of CCA, termed max-sliced mutual information (mSMI). mSMI equals the maximal mutual information between low-dimensional projections of the high-dimensional variables, which reduces back to CCA in the Gaussian case. It enjoys the best of both worlds: capturing intricate dependencies in the data while being amenable to fast computation and scalable estimation from samples. We show that mSMI retains favorable structural properties of Shannon's mutual information, like variational forms and identification of independence. We then study statistical estimation of mSMI, propose an efficiently computable neural estimator, and couple it with formal non-asymptotic error bounds. We present experiments that demonstrate the utility of mSMI for several tasks, encompassing independence testing, multi-view representation learning, algorithmic fairness, and generative modeling. We observe that mSMI consistently outperforms competing methods with little-to-no computational overhead.

本文提出了一种信息理论的广义CCA方法，称为最大切片互信息（mSMI），它通过低维投影识别高维变量之间的最大互信息，能捕捉数据中的复杂依赖关系并具有较快的计算和可扩展的估计能力，同时保留了Shannon互信息的优秀结构特性和独立性的识别，提出了高效计算的神经估计器，并给出了其非渐近误差界。实验结果表明，在独立性检测、多视图表示学习、算法公平性和生成建模等任务中，mSMI方法在几乎没有计算开销的情况下始终优于竞争方法。