Bernstein-Schur Kernels: A Random Features Method for Nonstationary Kernel Learning
Researchers have developed a new random-feature construction for Bernstein-Schur kernels, a class of nonstationary kernels that combine finite-feature and completely monotone shift-invariant kernels. The method works by sketching the finite modulation factor and randomizing the radial factor using Gaussian random Fourier features, achieving feature dimension Dm rather than O(d²). This approach is significant because it extends random feature approximation techniques to kernel classes where standard methods like Bochner sampling do not directly apply.
The paper introduces a dual-randomization strategy for approximating Bernstein-Schur kernels, which occupy a middle ground between shift-invariant and dot-product kernels. The proposed method sketches the modulation component while separately randomizing the completely monotone radial factor by sampling its Bernstein-Widder scale and applying Gaussian random Fourier features. The authors provide theoretical analysis showing unbiasedness, exact variance characterization, operator-norm bounds controlled by intrinsic dimension rather than crude entrywise bounds, and deterministic kernel-ridge stability guarantees. The approach is demonstrated on the biased yat-kernel, where the radial mixture corresponds to inverse-multiquadric spectral sampling, with theoretical results indicating variance-optimality at fixed radial-feature budgets.
What's missing
The paper does not provide empirical validation or computational experiments comparing the proposed method against existing kernel approximation techniques on benchmark datasets. Additionally, practical guidance on selecting the sketch size m and radial-draw count D for real-world applications is not discussed.
What different sources said
- arXiv cs.LGCenter
Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization
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