Brownian Kernel Ladders

arXiv:2606.15812v2 Announce Type: replace Abstract: We introduce Brownian kernel ladders (BKLs), a recursive hierarchy of integral reproducing kernel Hilbert spaces built from linear functionals by repeatedly integrating Brownian pullback kernels indexed by functions from the preceding layer. The nonnegative 1-homogeneity of the Brownian kernel yields a kernel-preserving canonical spherical normalization and propagates square-root regularity through the hierarchy. Allowing all canonical ladder measures to vary produces a full adaptive BKL envelope with an infimal complexity. For this envelope, we prove depth-dependent Holder and pointwise estimates, quasi-Banach structure, nestedness, and, under a geometric trace condition, strict growth with ballwise separation. We also establish existence of regularized empirical-risk minimizers for continuous losses uniformly bounded below, with almost-everywhere uniqueness of population predictions under strict convexity and pointwise uniqueness under full support. For statistical estimation, we study one realized ladder and finite dictionaries fixed independently of the estimation sample. For a dictionary of M ladders, the Gaussian complexity of the union of radius-r top-layer RKHS balls has n^{-1/2} dependence, no explicit ambient-dimension factor, and model-selection factor 1+sqrt{2ln M}. Corresponding high-probability oracle and excess-risk bounds follow; polynomial-size dictionaries retain a near-parametric rate. The theory separates adaptive representational richness from the statistical cost of ladder selection.

Source: arXiv cs.LG | 2026-08-11