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Gaussian Processes (GPs) define a prior over functions that can be updated to a posterior once we have observed data. I've been working with scalar-valued GPs, i.e. functions $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ such that \begin{equation} f(x_1,...,x_d) \sim GP(m(x), K(x,x')) \end{equation} However, I was wondering if Gaussian Processes can be used for vector-valued response, i.e. when $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{n}$? This could be useful if you are trying to jointly estimate $n$ parameters.

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Yes, Gaussian processes can be used to approximate vector valued functions.

This is done in spatial statistics where it's called cokriging and in machine learning in the field of multi-task learning. There's a relatively recent survey paper by Alvarez et al. which includes Gaussian processes this as part of a wider survey of kernel methods for vector valued functions. There are about ~120 reference spread across the ML and stats literature so the topic has attraced a fair amount of academic interest.

For comparison to what you've discussed above, I think the definition given by Alvarez et al. was instructive. Assume you're interested in learning a function $f$ where $f: \chi = \mathbb{R}^p \rightarrow \mathbb{R}^D$, and you have training data $S = (\mathbf{X}, \mathbf{Y}) = \{(x_1, y_1), . . . ,(x_N , y_N )\}$. Then:

The vector-valued function $f$ is assumed to follow a Gaussian process $$ f ∼ GP(\mathbf{m}, \mathbf{K}), $$ where $\mathbf{m} \in \mathbb{R}^D$ is a vector which components are the mean functions ${m_d(x)}^D_{d=1}$ of each output and $\mathbf{K}$ is a positive matrix valued function [that is, $\mathbf{K} : \chi \times \chi \rightarrow R^{D \times D}$, such that for any $\mathbf{x}, \mathbf{x ′}$ we have $\mathbf{K}(\mathbf{x}, \mathbf{x}′)$ is a positive semi-definite matrix]. The entries $(\mathbf{K}(\mathbf{x}, \mathbf{x}′))_{d,d′}$ in the matrix $\mathbf{K}(\mathbf{x}, \mathbf{x}′)$ correspond to the covariances between the outputs $f_d(\mathbf{x})$ and $f_{d′} (\mathbf{x} ′)$ and express the degree of correlation or similarity between them. For a set of inputs $\mathbf{X}$, the prior distribution over the vector $f(\mathbf{X})$ is given by $$ f(\mathbf{X}) ∼ N (m(\mathbf{X}), \mathbf{K}(\mathbf{X}, \mathbf{X})), $$ where $m(\mathbf{X})$ is a vector that concatenates the mean vectors associated to the outputs and the covariance matrix $\mathbf{K}(\mathbf{X}, \mathbf{X})$ is the block partitioned matrix where $m(\mathbf{X})$ is a vector that concatenates the mean vectors associated to the outputs and the covariance matrix $\mathbf{K}(\mathbf{X}, \mathbf{X})$ is the block partitioned matrix [given by the $ND \times ND$ entries $(\mathbf{K}(\mathbf{x}_i, \mathbf{x}_j ))_{d,d′}$ for $i,j\in \{1 \ldots N \}$ and $d, d' \in \{1 .. D \}$]

Alvarez, Mauricio A., Lorenzo Rosasco, and Neil D. Lawrence. "Kernels for vector-valued functions: A review." Foundations and Trends® in Machine Learning 4.3 (2012): 195-266.

I do recommend the paper, I thought it was very interesting and I hope that helps!

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  • $\begingroup$ If $f$ is vector-valued taking values in R^m, then $f(X)$ is a set of vectors f(x_1),...f(x_n), or equivalently a nxm matrix. Can you then please clarify what it means to write $f(X) \sim N(m(X),K(X,X))$? Are we talking here about a normal distribution for a nxm matrix? If so what you wrote above does not match with the dimensions and input data for a "matrix normal distribution" as described here: en.wikipedia.org/wiki/Matrix_normal_distribution $\endgroup$
    – math_lover
    Sep 13, 2021 at 20:44

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