Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices: Multifidelity covariance estimation via regression on the manifold of symmetric positive definite matrices
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| Titel: | Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices: Multifidelity covariance estimation via regression on the manifold of symmetric positive definite matrices |
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| Autoren: | Aimee Maurais, Terrence Alsup, Benjamin Peherstorfer, Youssef M. Marzouk |
| Quelle: | SIAM Journal on Mathematics of Data Science. 7:189-223 |
| Publication Status: | Preprint |
| Verlagsinformationen: | Society for Industrial & Applied Mathematics (SIAM), 2025. |
| Publikationsjahr: | 2025 |
| Schlagwörter: | FOS: Computer and information sciences, covariance estimation, Computer Science - Machine Learning, Numerical solutions of ill-posed problems in abstract spaces, regularization, Numerical solutions to equations with nonlinear operators, multifidelity methods, Numerical Analysis (math.NA), estimation on manifolds, Statistics - Computation, Machine Learning (cs.LG), Positive matrices and their generalizations, cones of matrices, Numerical solutions to equations with linear operators, Special polytopes (linear programming, centrally symmetric, etc.), General nonlinear regression, FOS: Mathematics, Mathematics - Numerical Analysis, Riemannian geometry, Mahalanobis distance, Statistics on manifolds, Computation (stat.CO), statistical coupling |
| Beschreibung: | We introduce a multifidelity estimator of covariance matrices formulated as the solution to a regression problem on the manifold of symmetric positive definite matrices. The estimator is positive definite by construction, and the Mahalanobis distance minimized to obtain it possesses properties enabling practical computation. We show that our manifold regression multifidelity (MRMF) covariance estimator is a maximum likelihood estimator under a certain error model on manifold tangent space. More broadly, we show that our Riemannian regression framework encompasses existing multifidelity covariance estimators constructed from control variates. We demonstrate via numerical examples that the MRMF estimator can provide significant decreases, up to one order of magnitude, in squared estimation error relative to both single-fidelity and other multifidelity covariance estimators. Furthermore, preservation of positive definiteness ensures that our estimator is compatible with downstream tasks, such as data assimilation and metric learning, in which this property is essential. To appear in the SIAM Journal on Mathematics of Data Science (SIMODS) |
| Publikationsart: | Article |
| Dateibeschreibung: | application/xml |
| Sprache: | English |
| ISSN: | 2577-0187 |
| DOI: | 10.1137/23m159247x |
| DOI: | 10.48550/arxiv.2307.12438 |
| Zugangs-URL: | http://arxiv.org/abs/2307.12438 |
| Rights: | arXiv Non-Exclusive Distribution |
| Dokumentencode: | edsair.doi.dedup.....e48c1d93b9af81a553e823fcef02109c |
| Datenbank: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstract: | We introduce a multifidelity estimator of covariance matrices formulated as the solution to a regression problem on the manifold of symmetric positive definite matrices. The estimator is positive definite by construction, and the Mahalanobis distance minimized to obtain it possesses properties enabling practical computation. We show that our manifold regression multifidelity (MRMF) covariance estimator is a maximum likelihood estimator under a certain error model on manifold tangent space. More broadly, we show that our Riemannian regression framework encompasses existing multifidelity covariance estimators constructed from control variates. We demonstrate via numerical examples that the MRMF estimator can provide significant decreases, up to one order of magnitude, in squared estimation error relative to both single-fidelity and other multifidelity covariance estimators. Furthermore, preservation of positive definiteness ensures that our estimator is compatible with downstream tasks, such as data assimilation and metric learning, in which this property is essential.<br />To appear in the SIAM Journal on Mathematics of Data Science (SIMODS) |
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| ISSN: | 25770187 |
| DOI: | 10.1137/23m159247x |