Vector-Valued Gaussian Processes and their Kernels on a Class of Metric Graphs
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2025
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| author | Filosi, Tobia Porcu, Emilio Emery, Xavier Agostinelli, Claudio Alegrìa, Alfredo |
| author_facet | Filosi, Tobia Porcu, Emilio Emery, Xavier Agostinelli, Claudio Alegrìa, Alfredo |
| contents | Despite the increasing importance of stochastic processes on linear networks and graphs, current literature on multivariate (vector-valued) Gaussian random fields on metric graphs is elusive. This paper challenges several aspects related to the construction of proper matrix-valued kernels structures. We start by considering matrix-valued metrics that can be composed with scalar- or matrix-valued functions to implement valid kernels associated with vector-valued Gaussian fields. We then provide conditions for certain classes of matrix-valued functions to be composed with the univariate resistance metric and ensure positive semidefiniteness. Special attention is then devoted to Euclidean trees, where a substantial effort is required given the absence of literature related to multivariate kernels depending on the $\ell_1$ metric. Hence, we provide a foundational contribution to certain classes of matrix-valued positive semidefinite functions depending on the $\ell_1$ metric. This fact is then used to characterise kernels on Euclidean trees with a finite number of leaves. Amongst those, we provide classes of matrix-valued covariance functions that are compactly supported. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10208 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vector-Valued Gaussian Processes and their Kernels on a Class of Metric Graphs Filosi, Tobia Porcu, Emilio Emery, Xavier Agostinelli, Claudio Alegrìa, Alfredo Statistics Theory Despite the increasing importance of stochastic processes on linear networks and graphs, current literature on multivariate (vector-valued) Gaussian random fields on metric graphs is elusive. This paper challenges several aspects related to the construction of proper matrix-valued kernels structures. We start by considering matrix-valued metrics that can be composed with scalar- or matrix-valued functions to implement valid kernels associated with vector-valued Gaussian fields. We then provide conditions for certain classes of matrix-valued functions to be composed with the univariate resistance metric and ensure positive semidefiniteness. Special attention is then devoted to Euclidean trees, where a substantial effort is required given the absence of literature related to multivariate kernels depending on the $\ell_1$ metric. Hence, we provide a foundational contribution to certain classes of matrix-valued positive semidefinite functions depending on the $\ell_1$ metric. This fact is then used to characterise kernels on Euclidean trees with a finite number of leaves. Amongst those, we provide classes of matrix-valued covariance functions that are compactly supported. |
| title | Vector-Valued Gaussian Processes and their Kernels on a Class of Metric Graphs |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2501.10208 |