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Geodesics and Curvature of the Quotient-Affine Metrics on Full-Rank Correlation Matrices

Abstract : Correlation matrices are used in many domains of neurosciences such as fMRI, EEG, MEG. However, statistical analyses often rely on embeddings into a Euclidean space or into Symmetric Positive Definite matrices which do not provide intrinsic tools. The quotient-affine metric was recently introduced as the quotient of the affine-invariant metric on SPD matrices by the action of diagonal matrices. In this work, we provide most of the fundamental Riemannian operations of the quotient-affine metric: the expression of the metric itself, the geodesics with initial tangent vector, the Levi-Civita connection and the curvature.
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https://hal.archives-ouvertes.fr/hal-03157992
Contributor : Yann Thanwerdas Connect in order to contact the contributor
Submitted on : Tuesday, May 11, 2021 - 12:34:53 PM
Last modification on : Friday, July 9, 2021 - 2:01:54 PM

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Yann Thanwerdas, Xavier Pennec. Geodesics and Curvature of the Quotient-Affine Metrics on Full-Rank Correlation Matrices. GSI 2021 - 5th conference on Geometric Science of Information, Jul 2021, Paris, France. pp.93-102, ⟨10.1007/978-3-030-80209-7_11⟩. ⟨hal-03157992v3⟩

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