Deep Operator-Valued Spectral Kernel Networks

Deep Spectral Kernel Networks (DSKNs) construct nonstationary representations via paired-frequency maps, but independent phase sampling collapses cross-frequency interactions in expectation at initialization to a stationary covariance. We show that sharing a single phase per frequency pair restores the exact harmonizable integrand. Building on this, we introduce Deep Operator-Valued Spectral Kernel Networks (DOSKNs), a primal framework that extends deep spectral learning to vector-valued outputs while decoupling spectral resolution, layer width, and subspace rank. Grouped output coupling enforces block-rank constraints, regularizing cross-dimension interactions and lowers parameter count. Furthermore, we introduce a trace-constrained squared group regularizer with an exact $\mathcal{O}(M\log M)$ proximal map. Empirical evaluations confirm that DOSKN addresses stationary collapse, cuts coupling parameters by up to 76%, and consistently matches or outperforms unconstrained models.