When Kernel Ridge Regression Meets the Hölder-Zygmund Class: Minimax Optimality and Failure of Properness

We study kernel ridge regression for nonparametric regression over the Hölder-Zygmund class. Using an RKHS equivalent to a Sobolev space of smoothness s+d/2, we prove that misspecified KRR attains the minimax L2 rate n^{-2s/(2s+d)}. We also show that properness fails in the Hölder-Zygmund norm: even for the zero regression function with Gaussian noise, the expected squared Hölder-Zygmund norm of the KRR noise component grows as log n.

Optimal Multiscale Learning of Linear Operators

We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data. In wavelet coordinates, the problem is recast as an infinite-dimensional matrix regression problem with a heterogeneous two-sided multiscale structure. We establish minimax rates under Sobolev operator-norm loss and construct a finite-resolution blockwise least-squares estimator attaining these rates. The analysis reveals a nonuniform local estimation difficulty across scales, which can be exploited algorithmically: by assigning scale-adaptive sample sizes, the estimator achieves the optimal computational cost among dense least-squares implementations.