# Brownian Kernel Ladders

arXiv: https://arxiv.org/abs/2606.15812 · HTML: https://mlsanity.com/paper/2606.15812/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 30 references that are in the map.
- First posted on arXiv 14 Jun 2026.

## Builds on (most cited first, 2 of 2)

- [Neural reproducing kernel Banach spaces and representer theorems for deep networks](https://mlsanity.com/paper/2403.08750/) (2024)
- [Deep Networks are Reproducing Kernel Chains](https://mlsanity.com/paper/2501.03697/) (2025)

## Built on by (most cited first, 2 of 2)

- [Variation Brownian Kernel Ladders](https://mlsanity.com/paper/2608.13882/) (2026)
- [Common Covariance Geometry and Certification for Brownian Kernel Ladders](https://mlsanity.com/paper/2609.29525/) (2026)

## Abstract (arXiv)

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 H"older 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+√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.