Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization
This paper introduces a novel random-feature construction for Bernstein-Schur kernels, which are products of a finite-feature kernel and a completely monotone shift-invariant kernel. The proposed method combines sketched modulation with radial randomization, achieving linear feature dimension while providing rigorous theoretical guarantees including unbiasedness and operator-norm bounds. The approach is shown to improve efficiency in kernel ridge regression tasks, with a flagship instance being the biased yat-kernel.
[2606.11255] Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization
[Submitted on 8 Jun 2026 (v1), last revised 11 Jun 2026 (this version, v2)]
Title:Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization
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Abstract: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 $d_{\mathrm{eff}}(\lambda)$ the \emph{exact} intrinsic dimension of the matrix variance, so $O((1+\|P\|_{\mathrm{op}}/\lambda)\log(d_{\mathrm{eff}}/\delta))$ radial draws preserve the kernel-ridge solution; tilting the draw by a closed-form whitened leverage improves this to the effective-dimension count $O((1+d_{\mathrm{eff}})\log(d_{\mathrm{eff}}/\delta))$. 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 $b$.
Subjects:
Machine Learning (cs.LG)
Cite as: arXiv:2606.11255 [cs.LG]
(or arXiv:2606.11255v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.11255
arXiv-issued DOI via DataCite
Submission history
From: Taha Bouhsine [view email] [v1] Mon, 8 Jun 2026 21:59:44 UTC (1,334 KB)
[v2] Thu, 11 Jun 2026 16:01:21 UTC (581 KB)
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