Three problems in kernel methods: proofs and finite-size measurements

Three questions about kernel methods are answered here, and a fourth is measured. For the Gaussian kernel with a fixed bandwidth on the sphere, the minimum-norm interpolant of noisy labels is benign along the sequence d_ℓ = 2^(2^ℓ), n_ℓ = 2^(2^(2ℓ)): its excess risk tends to zero in probability, and almost surely along the sequence, for every fixed-degree spherical-polynomial target (Theorem 5.1). At the exponential sample size n = 2ᵈ the same interpolant is obstructed: the expected excess risk on pure noise is at least (5/36)σ² in the limit, for every fixed bandwidth (Theorem 4.1). For ridge regression on two augmented views of each sample through a random feature map with an odd activation, the optimised objective and the clean prediction risk have the same limits in law as those of a correlated Gaussian surrogate that keeps the shared latent variable of the two views (Theorem 6.1); the proof rests on an exact Gaussianisation of each block conditional on the feature matrix. The hard edge of the degree-two Veronese matrix remains open; the two lemmas that stand between its reduction and the hard edge are measured, and the measurements point to a sharper statement, that the leave-one-out normal has a semicircular spectrum. The proofs are accompanied by finite-size measurements of the quantities they turn on: the leverage certificate of the lower bound, the harmonic dimension gap that separates the benign regime from the obstructed one, and the leverage concentration behind the equivalence. Two further results are evaluated on data: an exact same-sample risk identity for a kernel correction fitted on top of a learned mean, on the EMIT and OCO-2 radiative-transfer emulation corpora, where its sign depends on the component being emulated and the correction fitted to out-of-fold residuals gains nothing; and the compressed-resolvent hypothesis behind growing-rank spiked singular vectors, whose compression error along rays of fixed rank-to-size ratio approaches √(r/n) plus a term of order r/n.