Nyström $M$-Hilbert-Schmidt Independence Criterion
This work addresses scalability issues in kernel-based independence testing for large-scale applications involving multiple variables, representing an incremental improvement over existing approximations limited to two variables.
The authors tackled the computational bottleneck of the Hilbert-Schmidt independence criterion (HSIC) for multiple random variables (M≥2) by proposing a Nyström-based estimator, proving its consistency and demonstrating applicability in synthetic examples, dependency testing, and causal discovery.
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 $M\ge 2$ case, and lack theoretical guarantees. In this work, we propose an alternative Nyström-based HSIC estimator which handles the $M\ge 2$ case, prove its consistency, and demonstrate its applicability in multiple contexts, including synthetic examples, dependency testing of media annotations, and causal discovery.