Variational Kernel Design for Internal Noise: Gaussian Chaos Noise, Representation Compatibility, and Reliable Deep Learning

Claims about the benefit of depth depend on the complexity assigned to a representation. We introduce the Variation Brownian Kernel Ladder (VBKL), a path-atomic function-space framework that separates nonlinear recursive dictionary construction from linear variation superposition. Starting from linear projections, each atom recursively composes unit-ball profiles from the Brownian reproducing kernel Hilbert space; the full VBKL space is then the signed-measure variation hull of the completed dictionary. We identify each recursive dictionary as a union of Brownian pullback RKHS balls and establish variation-controlled Hölder regularity, compactness and attainment, and strict growth with depth under a local non-degeneracy condition whose trace lies in the support of the input measure. For associated finite lower-support architectures, we derive Rademacher and generalization bounds through Brownian quadratic chaos, signed threshold traces, and VC entropy. We also construct two-stage approximants by discretizing the outer measure and the selected outer Brownian profiles, obtaining an $M^{-1/2}+m^{-1/2}$ error bound, a sharp interpolation constant $\sqrt{A/2}$, and at most $2M$ active outer-profile basis contributions per evaluation. Controlled experiments illustrate the approximation mechanisms and indicate a favorable limited-data accuracy--complexity trade-off.

We introduce a novel framework for constructing scalable and flexible covariance kernels for Gaussian processes (GPs) by directly learning the covariance structure under a regression-type parameterization induced by Vecchia approximations, using deep neural architectures. Specifically, we model kriging coefficients and conditional standard deviations, deterministic quantities that uniquely characterize the covariance, providing stable and informative learning targets. Exploiting the permutation-equivariant structure of conditioning sets in the Vecchia factorization, we derive a universal representation for permutation-preserving functions and design neural architectures that respect this symmetry, leading to improved training stability and data efficiency. The proposed approach enables expressive, non-stationary kernel learning while maintaining computational scalability, thereby bridging classical GP methodology with modern deep learning.

We study Gaussian regression over the explicit vector-valued Parhi--Nowak deep-RBV^2 architecture with depth L, width w, layer-sum variation budget A, and output bound B. For this O(L w^2)-parameterized architecture, the known lower and upper bounds differ by one factor of depth. We construct a local packing showing that the quadratic depth dependence is intrinsic under an explicit sample-size-dependent radius condition. The packing has log-cardinality Omega(L^2 w^2 log w); its codewords lie in an O(lambda) L^2 ball and are pairwise Omega(lambda)-separated. The main ingredients are a bias-corrected bounded-coefficient approximation theorem and balanced amplification: multiplying a depth-D ReLU network by q can be implemented using one constant channel so that every coefficient grows by only q^(1/D). Translation to vector-valued RBV^2 blocks then has layer-sum cost O(D w^2 q^(1/D)). Gaussian Fano yields a radius-explicit lower bound governed by the output, testing, and representation scales. Under A=B=R, sigma proportional to R, and the stated radius condition, this gives minimax risk at least of order L^2 w^2 log(w) R^2/n. A pseudodimension-based finite-net upper bound gives O-tilde(L^2 w^2 R^2/n) for unbounded Gaussian responses. Thus the minimax risk has quadratic polynomial dependence on depth, up to logarithmic factors, and exhibits a transition to representation-limited behavior at smaller radius.