# Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learning

arXiv: https://arxiv.org/abs/1608.05639 · HTML: https://mlsanity.com/paper/1608.05639/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Aug 2016.

## Built on by (most cited first, 15 of 15)

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- [Entangled Kernels -- Beyond Separability](https://mlsanity.com/paper/2101.05514/) (2021)
- [FAVOR#: Sharp Attention Kernel Approximations via New Classes of Positive Random Features](https://mlsanity.com/paper/2302.00787/) (2023)
- [Gaussian Process Regression under Computational and Epistemic Misspecification](https://mlsanity.com/paper/2312.09225/) (2023)
- [Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operators](https://mlsanity.com/paper/2603.00971/) (2026)
- [Learning dissipative Hamiltonian dynamics with reproducing kernel Hilbert spaces and random Fourier features](https://mlsanity.com/paper/2410.18656/) (2024)
- [From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systems](https://mlsanity.com/paper/2506.15906/) (2025)
- [Geometric Dictionary Learning of Dynamical Systems with Optimal Transport](https://mlsanity.com/paper/2605.18276/) (2026)
- [Self-Adaptive Learning and Model Predictive Control for Tracking Unknown Dynamics with No Regret](https://mlsanity.com/paper/2607.26370/) (2026)

## Abstract (arXiv)

This paper presents a framework for computing random operator-valued feature maps for operator-valued positive definite kernels. This is a generalization of the random Fourier features for scalar-valued kernels to the operator-valued case. Our general setting is that of operator-valued kernels corresponding to RKHS of functions with values in a Hilbert space. We show that in general, for a given kernel, there are potentially infinitely many random feature maps, which can be bounded or unbounded. Most importantly, given a kernel, we present a general, closed form formula for computing a corresponding probability measure, which is required for the construction of the Fourier features, and which, unlike the scalar case, is not uniquely and automatically determined by the kernel. We also show that, under appropriate conditions, random bounded feature maps can always be computed. Furthermore, we show the uniform convergence, under the Hilbert-Schmidt norm, of the resulting approximate kernel to the exact kernel on any compact subset of Euclidean space. Our convergence requires differentiable kernels, an improvement over the twice-differentiability requirement in previous work in the scalar setting. We then show how operator-valued feature maps and their approximations can be employed in a general vector-valued learning framework. The mathematical formulation is illustrated by numerical examples on matrix-valued kernels.