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.

翻译：构建能够捕捉层次化组合表示的数学上易处理函数空间，仍是统计学习理论的核心挑战。我们引入布朗核梯子（BKLs），一种通过布朗核积分构造递归定义的积分再生核希尔伯特空间层级结构。从线性泛函出发，每一层通过对前一层子集上概率测度支持的布朗核进行积分获得，从而形成深度直接编码为层级的递归函数空间模型。基于该框架，我们定义了规范BKL空间及其关联复杂度泛函。我们建立了这些空间的若干分析与统计性质，特别是证明了BKL空间构成准巴拿赫空间，满足与深度相关的赫尔德正则性估计，并表现出关于深度的严格单调性。我们进一步证明了正则化经验风险最小化存在性结果，导出了相对于环境维度和层级深度一致可控的高斯复杂度界。分析的关键在于基于递归子集分解与布朗核阈值表示的组合证明技术。这些估计给出了BKL空间上正则化经验风险最小化的近参数阶超额风险保证。我们的结果为深度学习中组合表示研究提供了数学上易处理的层次化函数空间框架。