We prove expression rate bounds of finite-parametric, spectral and neural surrogates for holomorphic maps between separable Hilbert spaces. The surrogates have an encoder-approximator-decoder architecture, with Karhunen-Lo\'{e}ve encoders and frame decoders. We prove expression rate bounds for two classes of finite-parametric surrogates: i) spectral surrogates obtained by N-term truncations of Wiener polynomial chaos expansions and ii) neural surrogates obtained by approximation of parametric maps with deep feedforward neural networks, ReLU and RePU activation functions and uniformly bounded weights. We work under an algebraic decay assumption on the eigenvalues of the covariance of the Gaussian measure on the input space. We obtain convergence rates for mean-square errors, and additionally in first-order Gaussian Sobolev spaces, to account for errors in the approximation of gradients.
A Banach space-valued approximation framework based on symmetrized neural network (SNN) operators generated by a deformation-dependent sigmoidal activation function, yielding a positive, even, normalized, and localized density kernel satisfying the partition of unity.
G. Anastassiou, Seda Karateke, M. Zontul· Axioms· 0 citations
Operator-theoretic generalization bounds for deep multi-output function classes are developed by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces and derive Rademacher complexity bounds for invertible and width-expanding injective architectures.
Mahdi Mohammadigohari, Thomas Borsani, G. Di Fatta· 1 citation
This work introduces the general class of so-called Chernoff-neural operators and shows in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well, and introduces the more specialized class of envelope-neural operators for envelope semigroups which allows for quantit...
Jonas Blessing, Philipp Schmocker, Alessandro Sgarabottolo· 0 citations
A pointwise deterministic bound is derived on the discrepancy between the kernel ridge regression estimator and the ground truth in the corresponding Hilbert space norm for a vector-valued operator under the framework of vector-valued kernel ridge regression.
We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by deep ReLU neural networks. Writing the functional input as $X(t)=\sum_{d\geq1}\xi_d\nu_d(t)$, we quantify the importance of coordinate $d$ through $w_ds_d$, where $s_d$ bounds the magnitude of th...
Shu-Hao Jiao· 0 citations
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