# Kernels for Vector-Valued Functions: a Review

@article{lvarez2012KernelsFV, title={Kernels for Vector-Valued Functions: a Review}, author={Mauricio A {\'A}lvarez and Lorenzo Rosasco and Neil D. Lawrence}, journal={ArXiv}, year={2012}, volume={abs/1106.6251} }

Kernel methods are among the most popular techniques in machine learning. From a regularization perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the regularization functional through the notion of reproducing kernel Hilbert spaces. From a probabilistic perspective they are the key in the context of Gaussian processes, where the kernel function is known as the covariance function. Traditionally, kernel methods have been…

## 618 Citations

### Multi-task Learning in vector-valued reproducing kernel Banach spaces with the ℓ1 norm

- Computer Science, MathematicsJ. Complex.
- 2021

### Online Learning with Multiple Operator-valued Kernels

- Computer ScienceArXiv
- 2013

Two online algorithms for learning a vector-valued function f while taking into account the output structure are described, one of which extends the standard kernel-based online learning algorithm NORMA from scalar-valued to operator-valued setting and the other addresses the limitation of pre-defining theoutput structure in ONORMA by learning sequentially a linear combination of operator- valued kernels.

### Random Fourier Features For Operator-Valued Kernels

- Computer Science, MathematicsACML
- 2016

A general principle for Operator-valued Random Fourier Feature construction relies on a generalization of Bochner's theorem for translation-invariant operator-valued Mercer kernels and proves the uniform convergence of the kernel approximation for bounded and unbounded operator random Fourier features.

### Multi-task Learning in Vector-valued Reproducing Kernel Banach Spaces with the $\ell^1$ Norm.

- Computer Science, Mathematics
- 2019

A class of vector-valued reproducing kernel Banach spaces with the $\ell^1$ norm is constructed so that the constructed spaces could have desirable properties including the crucial linear representer theorem.

### Kernel Mean Embedding of Distributions: A Review and Beyonds

- Computer ScienceFound. Trends Mach. Learn.
- 2017

A comprehensive review of existing work and recent advances in the Hilbert space embedding of distributions, and to discuss the most challenging issues and open problems that could lead to new research directions.

### Operator-valued Kernels for Learning from Functional Response Data

- Computer ScienceJ. Mach. Learn. Res.
- 2016

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…

### Learning with Operator-valued Kernels in Reproducing Kernel Krein Spaces

- Mathematics, Computer ScienceNeurIPS
- 2020

This work considers operator-valued kernels which might not be necessarily positive definite, and an iterative Operator based Minimum Residual (OpMINRES) algorithm is proposed for solving the loss stabilization problem.

### Online Learning with Operator-valued Kernels

- Computer ScienceESANN
- 2015

An online algorithm, OLOK, is described that extends the standard kernel-based online learning algorithm NORMA from scalar-valued to operator-valued setting and reports a cumulative error bound that holds both for classification and regression.

### On the Dualization of Operator-Valued Kernel Machines

- Computer ScienceArXiv
- 2019

This work investigates how to use the duality principle to handle different families of loss functions, yet unexplored within vv-RKHSs.

### Nonlinear Functional Output Regression: a Dictionary Approach

- Computer ScienceAISTATS
- 2021

In many fields, each data instance consists in a high number of measurements of the same underlying phenomenon. Such high dimensional data generally enjoys strong smoothness across features which can…

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