Variation Brownian Kernel Ladders

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.

We study Gaussian regression over the explicit vector-valued Parhi--Nowak deep-RBV^2 architecture with depth L, width w, layer-sum variation budget A, and output bound B. For this O(L w^2)-parameterized architecture, the known lower and upper bounds differ by one factor of depth. We construct a local packing showing that the quadratic depth dependence is intrinsic under an explicit sample-size-dependent radius condition. The packing has log-cardinality Omega(L^2 w^2 log w); its codewords lie in an O(lambda) L^2 ball and are pairwise Omega(lambda)-separated. The main ingredients are a bias-corrected bounded-coefficient approximation theorem and balanced amplification: multiplying a depth-D ReLU network by q can be implemented using one constant channel so that every coefficient grows by only q^(1/D). Translation to vector-valued RBV^2 blocks then has layer-sum cost O(D w^2 q^(1/D)). Gaussian Fano yields a radius-explicit lower bound governed by the output, testing, and representation scales. Under A=B=R, sigma proportional to R, and the stated radius condition, this gives minimax risk at least of order L^2 w^2 log(w) R^2/n. A pseudodimension-based finite-net upper bound gives O-tilde(L^2 w^2 R^2/n) for unbounded Gaussian responses. Thus the minimax risk has quadratic polynomial dependence on depth, up to logarithmic factors, and exhibits a transition to representation-limited behavior at smaller radius.

Deep representation learning often selects hidden features and fits the final predictor on the same sample, so fixed-feature analysis performed after selection can omit selection cost. We study the conditional empirical Rademacher complexity of deep ReLU representations followed by bounded-norm predictors in additive or Lévy-Brownian RKHSs, termed Brownian heads. For a fixed representation, we derive an exact dual identity and sharp bounds in terms of activation mass, the average norm of the observed hidden vectors. Under same-sample selection, the representation supremum induces a quadratic Rademacher process. Brownian layer-cake and Gaussian-projection identities reduce it to coordinatewise or signed projected threshold traces, separating realized scale from selection complexity. For samples with pairwise-distinct inputs, explicit scalar ReLU families match the finite-trace and VC rates up to universal constants at the realized trace-and-envelope level. Induced-norm contraction also yields architecture-level bounds for rectangular, rank-deficient ReLU networks. Experiments verify the sharp bounds and rates, exhibit a selection gap at fixed activation mass, and assess the predictive feasibility of Brownian heads.