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Barycentric Subspace Analysis on Manifolds

Xavier Pennec 1, 2
1 ASCLEPIOS - Analysis and Simulation of Biomedical Images
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of $k+1$ reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces. Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).
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Submitted on : Thursday, September 28, 2017 - 11:33:29 AM
Last modification on : Thursday, January 20, 2022 - 5:29:35 PM
Long-term archiving on: : Friday, December 29, 2017 - 2:34:56 PM


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Xavier Pennec. Barycentric Subspace Analysis on Manifolds. Annals of Statistics, Institute of Mathematical Statistics, 2018, 46 (6A), pp.2711-2746. ⟨10.1214/17-AOS1636⟩. ⟨hal-01343881v2⟩



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