Resolution-Independent Analysis of Encoder--Decoder Operator Learning via Limiting Kernels
Abstract
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 reproducing kernel Hilbert spaces are isometrically isomorphic. As the input and output resolutions increase, the induced kernels converge to a limiting kernel, in the sense of operator-norm convergence of their associated integral operators, allowing regularity assumptions to be stated independently of the encoding resolution. For regularized stochastic gradient descent, we establish upper bounds for decreasing and fixed step sizes, separating the encoding and regularization terms from optimization terms of order t−θ and T−θ', respectively, for any θ,θ'∈(0,1). We further prove lower bounds showing that these encoding-induced terms are generally unavoidable. The analysis is further extended to encoder--decoder neural networks through the limiting neural tangent kernel (NTK), yielding error bounds with an additional finite-width term and polynomial parameter and sample complexity guarantees when the encoding errors decay algebraically. The framework covers matrix-valued kernels constructed from radial and dot product kernels, NTKs arising from wide encoder--decoder neural networks, and encoder--decoder pairs based on Fourier, Legendre polynomial, wavelet, PCA, or pointwise sampling representations.
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BibTeX
@misc{shi2026resolutionindependent,
title = {{Resolution-Independent Analysis of Encoder--Decoder Operator Learning via Limiting Kernels}},
author = {Lei Shi and Jia-Qi Yang and Ding-Xuan Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13798},
eprint = {2609.13798},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13798},
}
RIS
TY - UNPB
TI - Resolution-Independent Analysis of Encoder--Decoder Operator Learning via Limiting Kernels
AU - Lei Shi
AU - Jia-Qi Yang
AU - Ding-Xuan Zhou
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13798
UR - https://arxiv.org/abs/2609.13798
AB - 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 reproducing kernel Hilbert spaces are isometrically isomorphic. As the input and output resolutions increase, the induced kernels converge to a limiting kernel, in the sense of operator-norm convergence of their associated integral operators, allowing regularity assumptions to be stated independently of the encoding resolution. For regularized stochastic gradient descent, we establish upper bounds for decreasing and fixed step sizes, separating the encoding and regularization terms from optimization terms of order t^−θ and T^−θ', respectively, for any θ,θ'∈(0,1). We further prove lower bounds showing that these encoding-induced terms are generally unavoidable. The analysis is further extended to encoder--decoder neural networks through the limiting neural tangent kernel (NTK), yielding error bounds with an additional finite-width term and polynomial parameter and sample complexity guarantees when the encoding errors decay algebraically. The framework covers matrix-valued kernels constructed from radial and dot product kernels, NTKs arising from wide encoder--decoder neural networks, and encoder--decoder pairs based on Fourier, Legendre polynomial, wavelet, PCA, or pointwise sampling representations.
ER -