Variation Brownian Kernel Ladders

• URL: http://arxiv.org/abs/2608.13882v1
• Date: Fri, 14 Aug 2026 02:15:08 GMT
• Title: Variation Brownian Kernel Ladders
• Abstract summary: We introduce the emphVariation Brownian Kernel Ladder (VBKL), a path-atomic function-space framework.<n>Starting from linear projections, each atom composes unit-ball profiles from the Brownian reproducing kernel Hilbert space.<n>The full VBKL space is then the signed-measure variation hull of the completed dictionary.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Claims about the benefit of depth depend on the complexity assigned to a representation. We introduce the \emph{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.

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