A stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels and vector-valued interpolation spaces is developed, thereby enabling rich structural modeling with rigorous theoretical guarantees.
This work considers 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 vector-valued reproducing kernel Hilbert space induced by an operator-valued kernel.
This study investigates the use of stochastic gradient descent to learn operators between general Hilbert spaces and establishes upper bounds for convergence rates of the SGD algorithm and derives a minimax lower bound analysis, illustrating that convergence analysis and regularity conditions quantitatively characteriz...
Operator learning is formulated on function spaces, but training data are typically available only through finite-dimensional representations. In encoder--decoder architectures, a matrix-valued kernel on the encoded space induces an operator-valued kernel on the original function spaces, and the corresponding reproduci...
A variation space for nonlinear operators between Hilbert spaces, defined through vector-valued Borel measures of bounded variation, is formulated and analyzed, which yields algebraic approximation and learning rates in contrast to the complexity barriers for Lipschitz and Fr\'echet differentiable operator classes.
Jia-Qi Yang, Lei Shi· 0 citations
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