Composite Kernel Decomposition of Gaussian Process for Time-Series Foundation Models

Time-series foundation models (TSFMs) rely heavily on synthetic pretraining data sampled from Gaussian processes (GPs) with composite kernels, as in KernelSynth and CauKer. Exact GP sampling requires an $O(T^3)$ covariance decomposition that becomes unstable for long sequences, and fast circulant-embedding samplers do not extend automatically to composed kernels. We introduce CKD-GP, which expands each composite kernel into product terms and samples each in closed form in $O(T \log T)$ time, with no dense factorization. CKD-GP is up to $2{,}645\times$ faster than dense sampling and generates 40M KernelSynth and 10M CauKer series of length 16,384. A TSFM pretrained on these corpora achieves a zero-shot MASE of 0.789 and CRPS of 0.547 on GIFT-Eval.