# Operator-valued Kernels for Learning from Functional Response Data

arXiv: https://arxiv.org/abs/1510.08231 · HTML: https://mlsanity.com/paper/1510.08231/

## Key facts

- Cited by 60 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2015.

## Built on by (most cited first, 30 of 60)

- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Conditional Distributional Treatment Effect with Kernel Conditional Mean Embeddings and U-Statistic Regression](https://mlsanity.com/paper/2102.08208/) (2021)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Transformer Meets Boundary Value Inverse Problems](https://mlsanity.com/paper/2209.14977/) (2022)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm](https://mlsanity.com/paper/2312.07186/) (2023)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [ProKeR: A Kernel Perspective on Few-Shot Adaptation of Large Vision-Language Models](https://mlsanity.com/paper/2501.11175/) (2025)
- [Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms](https://mlsanity.com/paper/2405.14778/) (2024)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [A foundational neural operator that continuously learns without forgetting](https://mlsanity.com/paper/2310.18885/) (2023)
- [Reproducing kernel Hilbert C*-module and kernel mean embeddings](https://mlsanity.com/paper/2101.11410/) (2021)
- [Nonparametric learning of kernels in nonlocal operators](https://mlsanity.com/paper/2205.11006/) (2022)
- [Learning "best" kernels from data in Gaussian process regression. With application to aerodynamics](https://mlsanity.com/paper/2206.02563/) (2022)
- [Data adaptive RKHS Tikhonov regularization for learning kernels in operators](https://mlsanity.com/paper/2203.03791/) (2022)
- [Entangled Kernels -- Beyond Separability](https://mlsanity.com/paper/2101.05514/) (2021)
- [Learning in RKHM: a C^*-Algebraic Twist for Kernel Machines](https://mlsanity.com/paper/2210.11855/) (2022)
- [Nonlinear functional regression by functional deep neural network with kernel embedding](https://mlsanity.com/paper/2401.02890/) (2024)
- [Hyperparameter Optimization for Randomized Algorithms: A Case Study on Random Features](https://mlsanity.com/paper/2407.00584/) (2024)
- [Operator Learning at Machine Precision](https://mlsanity.com/paper/2511.19980/) (2025)
- [Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis](https://mlsanity.com/paper/2105.12778/) (2021)
- [Koopman Spectrum Nonlinear Regulators and Efficient Online Learning](https://mlsanity.com/paper/2106.15775/) (2021)
- [Vector-Valued Least-Squares Regression under Output Regularity Assumptions](https://mlsanity.com/paper/2211.08958/) (2022)
- [Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator](https://mlsanity.com/paper/2412.16588/) (2024)
- [Handling Hard Affine SDP Shape Constraints in RKHSs](https://mlsanity.com/paper/2101.01519/) (2021)
- [Diffeomorphic Measure Matching with Kernels for Generative Modeling](https://mlsanity.com/paper/2402.08077/) (2024)

## Abstract (arXiv)

In this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions: a data is represented by a set of functions, and the label is also a function. We focus on the use of reproducing kernel Hilbert space theory to learn from such functional data. Basic concepts and properties of kernel-based learning are extended to include the estimation of function-valued functions. In this setting, the representer theorem is restated, a set of rigorously defined infinite-dimensional operator-valued kernels that can be valuably applied when the data are functions is described, and a learning algorithm for nonlinear functional data analysis is introduced. The methodology is illustrated through speech and audio signal processing experiments.