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December 2018 Barycentric subspace analysis on manifolds
Xavier Pennec
Ann. Statist. 46(6A): 2711-2746 (December 2018). DOI: 10.1214/17-AOS1636

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 Subspace Analysis (BSA).

Citation

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Xavier Pennec. "Barycentric subspace analysis on manifolds." Ann. Statist. 46 (6A) 2711 - 2746, December 2018. https://doi.org/10.1214/17-AOS1636

Information

Received: 1 June 2016; Revised: 1 July 2017; Published: December 2018
First available in Project Euclid: 7 September 2018

zbMATH: 06968597
MathSciNet: MR3851753
Digital Object Identifier: 10.1214/17-AOS1636

Subjects:
Primary: 60D05 , 62H25
Secondary: 58C06 , 62H11

Keywords: Barycenter , flag of subspaces , Fréchet mean , Manifold , PCA

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6A • December 2018
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