SPECTOR: Spectral Tree Operator Realization for Time-Series Priors

Time series foundation models increasingly rely on synthetic data to supplement limited real-world corpora and to improve the scale and diversity of pretraining. Yet existing generators face a trade-off: compositional kernels offer rich diversity but require cubic-time covariance factorization, while fast parametric simulators scale linearly at the cost of narrower structure and weaker transfer. We introduce SPECTOR (SPECtral Tree Operator Realization), a realization compiler for compositional time-series priors that resolves this trade-off. SPECTOR treats a sum-product kernel expression as a stochastic-process program and recursively compiles its subtrees into efficient realization spaces, without materializing covariance matrices while enabling structural diversity and linear-time sampling. We formalize the semantics of this compiler through a covariance-preserving result for general realization trees, with spectral and feature-space semiring constructions providing stronger law-preserving guarantees for important kernel classes. With 18 kernel primitives, SPECTOR spans stationary, periodic, trending, transient, and stochastic-drift behaviors. Empirically, SPECTOR scales to long contexts and is hundreds of times faster than covariance-based kernel generators, enabling on-the-fly pretraining. Across six architectures and three benchmarks, SPECTOR achieves 74–91% pairwise win rates against competing synthetic priors and narrows the gap to published models pretrained on hybrid real-synthetic corpora.