Kernel Granger Component Analysis for Nonlinear Directed Component Discovery

Granger Component Analysis (GCA) discovers latent components of multivariate time series that exhibit directed temporal dependence, but is limited to linear representations. We introduce Kernel Granger Component Analysis (KGCA), a nonlinear extension that learns directed latent components in a reproducing kernel Hilbert space. KGCA optimizes a ridge-regularized Granger predictive objective using explicit envelope-theorem gradients, and incorporates a time-reversal criterion to resolve directional identifiability. We also describe a scalable random Fourier feature (RFF) variant that approximates the kernel map explicitly and avoids forming the full Gram matrix. To avoid spurious directionality from post hoc component selection, we use a restricted evaluation protocol that measures directed structure intrinsic to the learned representation. Empirically, KGCA and RFF-KGCA recover reliable nonlinear directed components, exhibit positive directionality gaps, and avoid spurious directionality under independent-process controls. These results show that kernelizing GCA enables nonlinear directed component discovery while preserving the interpretability and explicit optimization structure of the original framework.