Author

Sebastian Neumayer

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Preprint Sep 2026

Greedy sampling designs via reduced basis methods: optimal recovery in the uniform norm

We study optimal sampling recovery in reproducing kernel Hilbert spaces (RKHS) in the uniform norm. For every RKHS with bounded kernel, we establish new comparisons between linear sampling widths and Gelfand widths that overcome the known square-root gap, without requiring a measure or a Christoffel-type condition. Our bounds rely on nested sampling designs obtained by kernel interpolation at (weak) P-greedy points. Under additional (polynomial) decay assumptions the decay rate of the Gelfand widths directly transfers to the sampling widths. With either a logarithmic oversampling or passing to the square root of the Gelfand widths we obtain a direct comparison (requiring no decay assumption) between them. This is particularly effective for super-polynomial decay, such as in Paley-Wiener spaces. Our results follow from representations of both widths in terms of kernel translates and yield, in the opposite direction, a new existence result for a sharp reduced basis selection. Numerical experiments for Legendre, mixed-Sobolev, and Paley-Wiener kernels illustrate our findings.

Sebastian Neumayer, Kateryna Pozharska, T. Ullrich · 0 citations

Aug 2026

Data-Driven Regularization with Weak Convexity for Robust Image Reconstruction

Abstract. We describe a practical framework for data-driven regularization in image reconstruction. It combines model expressivity with the guarantees of variational methods. The approach is based on a weakly convex ridge regularizer, defined as the composition of a convolutional filter bank and pointwise potentials constrained to be weakly convex. These potentials are implemented as learnable splines with a strict control of their weak-convexity modulus. The resulting denoisers outperform classic convex regularization techniques as well as competitive benchmarks such as BM3D, while they still correspond to the minimization of a convex energy. The learned regularizers further extend to general inverse problems with provable convergence to critical points. Overall, this framework shows that a controlled relaxation of convexity enables the design of learnable priors that achieve strong empirical performance while preserving mathematical guarantees.

Alexis Goujon, Sebastian Neumayer, Stanislas Ducotterd et al. · SIAM Review · 0 citations