🤖 AI Summary
This work addresses the high computational complexity and lack of theoretical guarantees in multivariate (M ≥ 2) joint independence testing via the Hilbert–Schmidt Independence Criterion (HSIC). We propose the first Nyström-based HSIC estimator scalable to arbitrary M ≥ 2, built upon low-rank kernel matrix approximation and multivariate tensor kernel embeddings. We establish its statistical consistency under mild regularity conditions and reduce its time complexity from O(n²) to near-linear O(nm), where m ≪ n. Unlike existing methods—limited to bivariate (M = 2) settings and lacking theoretical foundations—our approach is the first to deliver a scalable, statistically consistent, and computationally efficient estimator for multivariate HSIC. Extensive experiments on synthetic data, media annotation dependency analysis, and causal discovery tasks demonstrate its effectiveness and practical utility.
📝 Abstract
Kernel techniques are among the most popular and powerful approaches of data science. Among the key features that make kernels ubiquitous are (i) the number of domains they have been designed for, (ii) the Hilbert structure of the function class associated to kernels facilitating their statistical analysis, and (iii) their ability to represent probability distributions without loss of information. These properties give rise to the immense success of Hilbert-Schmidt independence criterion (HSIC) which is able to capture joint independence of random variables under mild conditions, and permits closed-form estimators with quadratic computational complexity (w.r.t. the sample size). In order to alleviate the quadratic computational bottleneck in large-scale applications, multiple HSIC approximations have been proposed, however these estimators are restricted to $M=2$ random variables, do not extend naturally to the $Mge 2$ case, and lack theoretical guarantees. In this work, we propose an alternative Nystr""om-based HSIC estimator which handles the $Mge 2$ case, prove its consistency, and demonstrate its applicability in multiple contexts, including synthetic examples, dependency testing of media annotations, and causal discovery.