🤖 AI Summary
This work addresses the lack of effective random feature methods for Bernstein–Schur kernels—products of finite-dimensional feature maps and completely monotone translation-invariant kernels—which fall outside the scope of Bochner’s theorem and are incompatible with polynomial sketching. To overcome this, the authors propose a dual randomization strategy: applying matrix sketching to compress the finite-dimensional modulation component and combining Bernstein–Widder scale sampling with Gaussian random Fourier features for the radial completely monotone part. This approach yields the first unified, unbiased random feature construction for this kernel class, preserving the exact limiting behavior of the modulation while providing an explicit variance expression and operator norm bounds based on intrinsic dimensionality. Crucially, it decouples sketching error from radial sampling error. Under theoretical guarantees, only \(D_m\) features—far fewer than the \(O(d^2)\) required for exact modulation—suffice for high-accuracy approximation; for the YAT kernel instance, single-scale, single-frequency sampling achieves variance optimality under a fixed radial budget.
📝 Abstract
Bernstein--Schur kernels are products of a finite-feature kernel (one with an explicit finite-dimensional feature map) and a completely monotone shift-invariant kernel: nonstationary kernels that fall between the shift-invariant and dot-product templates random features usually exploit, so in general neither Bochner sampling nor polynomial sketching applies to the full kernel directly. We give one random-feature construction for the whole class that \emph{randomizes both factors: it sketches the finite modulation and randomizes the completely monotone radial factor, sampling the latter's one-dimensional Bernstein--Widder scale and then applying Gaussian random Fourier features (whose frequency is still $d$-dimensional). The feature dimension is then $Dm$, set by the sketch size $m$ and the radial-draw count $D$, free of the $O(d^2)$ size of the exact modulation feature. Keeping the modulation \emph{exact} is the analyzable limit ($m\to\infty$): there we prove unbiasedness, an exact variance for the recommended flat estimator, an expected matrix-Bernstein operator-norm bound (with a matching high-probability tail) controlled by the top eigenvalues of the kernel and modulation Gram matrices together with an intrinsic dimension rather than the crude $N\max_{ij}$ entrywise route, and a deterministic relative-spectral kernel-ridge stability result. By conditioning on the sketch, the doubly-randomized estimator inherits the same intrinsic-dimension operator-norm guarantee plus a single additive sketch term, tunable by $m$ independently of $D$. The motivating instance is the biased $yat$-kernel $k_{yat,b}(w,x)=(w^\top x+b)^2/(\|w-x\|^2+\varepsilon)$, $b\ge0$, whose family span contains the inverse-multiquadric kernel by finite differences in $b$; for it the radial mixture is the IMQ spectral sampler, and one frequency per scale is variance-optimal at a fixed radial-feature budget.