Enhancing the Tensor Normal via Geometrically Parameterized Cholesky Factors
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908320015581184 |
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| author | Simonis, Quinn Wells, Martin T. |
| author_facet | Simonis, Quinn Wells, Martin T. |
| contents | In this article, we explore Bayesian extensions of the tensor normal model through a geometric expansion of the multi-way covariance's Cholesky factor inspired by the Fréchet mean under the log-Cholesky metric. Specifically, within a tensor normal framework, we identify three structural components in the covariance of the vectorized data. By parameterizing vector normal covariances through such a Cholesky factor representation, analogous to a finite average of multiway Cholesky factors, we eliminate one of these structural components without compromising the analytical tractability of the likelihood, in which the multiway covariance is a special case. Furthermore, we demonstrate that a specific class of structured Cholesky factors can be precisely represented under this parameterization, serving as an analogue to the Pitsianis-Van Loan decomposition. We apply this model using Hamiltonian Monte Carlo in a fixed-mean setting for two-way covariance relevancy detection of components, where efficient analytical gradient updates are available, as well as in a seasonally-varying covariance process regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enhancing the Tensor Normal via Geometrically Parameterized Cholesky Factors Simonis, Quinn Wells, Martin T. Methodology Computation In this article, we explore Bayesian extensions of the tensor normal model through a geometric expansion of the multi-way covariance's Cholesky factor inspired by the Fréchet mean under the log-Cholesky metric. Specifically, within a tensor normal framework, we identify three structural components in the covariance of the vectorized data. By parameterizing vector normal covariances through such a Cholesky factor representation, analogous to a finite average of multiway Cholesky factors, we eliminate one of these structural components without compromising the analytical tractability of the likelihood, in which the multiway covariance is a special case. Furthermore, we demonstrate that a specific class of structured Cholesky factors can be precisely represented under this parameterization, serving as an analogue to the Pitsianis-Van Loan decomposition. We apply this model using Hamiltonian Monte Carlo in a fixed-mean setting for two-way covariance relevancy detection of components, where efficient analytical gradient updates are available, as well as in a seasonally-varying covariance process regime. |
| title | Enhancing the Tensor Normal via Geometrically Parameterized Cholesky Factors |
| topic | Methodology Computation |
| url | https://arxiv.org/abs/2504.10645 |