A Universal Reproducing Kernel Hilbert Space from Polynomial Alignment and IMQ Distance

An approach to construct explicit integral representations for two-layer ReLU networks is presented, which provides relatively simple representations for any multivariate polynomial. Quantitative bounds are provided for a particular, sharpened ReLU integral representation, which involves a harmonic extension and a projection. The bounds demonstrate that functions can be approximated with $L^{2}(\mathcal{D})$ errors that do not depend explicitly on dimension or degree, but rather the coefficients of their monomial expansions and the distribution $\mathcal{D}$. We also present a connection to the RKHS of the exponential kernel $K(x,y)=\exp\left(\left\langle x,y\right\rangle \right)$, and a very simple integral representation involving additionally multiplication via a fixed function which has better quantitative bounds.

Bernstein--Schur kernels are products of a finite-feature kernel and a completely monotone shift-invariant kernel: nonstationary kernels falling between the shift-invariant and dot-product templates random features exploit, so neither Bochner sampling nor polynomial sketching applies to the full kernel directly. We give one random-feature construction for the whole class that randomizes both factors: it sketches the finite modulation and samples the radial factor's one-dimensional Bernstein--Widder scale before applying Gaussian random Fourier features, giving feature dimension $Dm$, free of the $O(d^2)$ size of the exact modulation feature. With the modulation kept exact (the $m\to\infty$ limit), we prove unbiasedness, an exact variance, and a matrix-Bernstein operator-norm bound controlled by the top kernel and modulation eigenvalues and an intrinsic dimension rather than the crude $N\max_{ij}$ route. Whitening this argument at the ridge makes the effective dimension $d_{\mathrm{eff}}(λ)$ the \emph{exact} intrinsic dimension of the matrix variance, so $O((1+\|P\|_{\mathrm{op}}/λ)\log(d_{\mathrm{eff}}/δ))$ radial draws preserve the kernel-ridge solution; tilting the draw by a closed-form whitened leverage improves this to the effective-dimension count $O((1+d_{\mathrm{eff}})\log(d_{\mathrm{eff}}/δ))$. Conditioning on the sketch carries every guarantee to the deployed doubly-randomized estimator up to one additive sketch term, and all hold for the whole class with the modulation Gram in place of the polynomial one. The flagship instance is the biased $yat$-kernel $k_{yat,b}(w,x)=(w^\top x+b)^2/(\|w-x\|^2+\varepsilon)$, whose family span contains the inverse-multiquadric kernel by finite differences in $b$.

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.