4 papers · 1 filter
Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalization
Jia-Qi Yang, Lei Shi
Inspired by the function-space theory of neural networks, we formulate and analyze a variation space for nonlinear operators between Hilbert spaces, defined through vector-valued B…
A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning
Jia-Qi Yang, Lei Shi
We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework…
Learning Operators by Regularized Stochastic Gradient Descent with Operator-valued Kernels
Jia-Qi Yang, Lei Shi
We consider a class of statistical inverse problems involving the estimation of a regression operator from a Polish space to a separable Hilbert space, where the target lies in a v…
Stochastic Gradient Descent for Operator Learning in Hilbert Spaces: Convergence Rates and Minimax Lower Bounds
Lei Shi, Jia-Qi Yang
This study investigates the use of stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We study weak and strong regularity conditions for the targe…