Brownian Kernel Ladders

2026-06-14

AI Summary

Constructing function spaces that simultaneously exhibit hierarchical compositional structure and mathematical tractability remains a central challenge in statistical learning theory. This work proposes the Brownian Kernel Ladder (BKL) framework, which explicitly encodes network depth into the hierarchical construction of a reproducing kernel Hilbert space by recursively integrating Brownian kernels over function subsets from preceding layers. The BKL framework establishes, for the first time, a direct analytical link between depth and function space structure, yielding a quasi-Banach space endowed with depth-dependent Hölder regularity and strict monotonicity. Leveraging kernel integration, recursive decomposition, and Gaussian complexity analysis, we provide existence guarantees for regularized empirical risk minimization over BKL and derive near-parametric excess risk bounds that are uniformly controlled with respect to both input dimension and network depth.

Abstract

Constructing mathematically tractable function spaces that capture hierarchical compositional representations remains a central challenge in statistical learning theory. We introduce Brownian kernel ladders (BKLs), a recursively defined hierarchy of integral reproducing kernel Hilbert spaces generated through Brownian-kernel integral constructions. Starting from linear functionals, each layer is obtained by integrating Brownian kernels over probability measures supported on subsets of the previous layer, yielding a recursive function-space model in which depth is encoded directly through the hierarchy. Based on this framework, we define canonical BKL spaces together with an associated complexity functional. We establish several analytical and statistical properties of these spaces. In particular, we show that BKL spaces form quasi-Banach spaces, satisfy depth-dependent Hölder regularity estimates, and exhibit strict monotonicity with respect to depth. We further prove existence results for regularized empirical risk minimization and derive Gaussian complexity bounds that remain uniformly controlled with respect to both the ambient dimension and the hierarchy depth. A key ingredient of the analysis is a combinatorial proof technique based on recursive subset decompositions and Brownian-kernel threshold representations. These estimates yield excess-risk guarantees of near-parametric order for regularized empirical risk minimization over BKL spaces. Our results provide a mathematically tractable hierarchical function-space framework for studying compositional representations in deep learning.

Problem

• hierarchical compositional representations
• function spaces
• statistical learning theory
• deep learning
• reproducing kernel Hilbert spaces

Innovation

• Brownian kernel ladders
• hierarchical function spaces
• reproducing kernel Hilbert spaces
• compositional representations
• empirical risk minimization