Optimization over geodesics for exact principal geodesic analysis

Stefan Horst Sommer; Francois Bernard Lauze; Mads Nielsen

doi:10.1007/s10444-013-9308-1

Optimization over geodesics for exact principal geodesic analysis

Stefan Horst Sommer, Francois Bernard Lauze, Mads Nielsen

Department of Computer Science

19 Citations (Scopus)

Abstract

In fields ranging from computer vision to signal processing and statistics, increasing computational power allows a move from classical linear models to models that incorporate non-linear phenomena. This shift has created interest in computational aspects of differential geometry, and solving optimization problems that incorporate non-linear geometry constitutes an important computational task. In this paper, we develop methods for numerically solving optimization problems over spaces of geodesics using numerical integration of Jacobi fields and second order derivatives of geodesic families. As an important application of this optimization strategy, we compute exact Principal Geodesic Analysis (PGA), a non-linear version of the PCA dimensionality reduction procedure. By applying the exact PGA algorithmto synthetic data, we exemplify the differences between the linearized and exact algorithms caused by the non-linear geometry. In addition, we use the numerically integrated Jacobi fields to determine sectional curvatures and provide upper bounds for injectivity radii.

Original language	English
Journal	Advances in Computational Mathematics
Volume	40
Issue number	2
Pages (from-to)	283-313
Number of pages	31
ISSN	1019-7168
DOIs	https://doi.org/10.1007/s10444-013-9308-1
Publication status	Published - Apr 2014

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title = "Optimization over geodesics for exact principal geodesic analysis",

abstract = "In fields ranging from computer vision to signal processing and statistics, increasing computational power allows a move from classical linear models to models that incorporate non-linear phenomena. This shift has created interest in computational aspects of differential geometry, and solving optimization problems that incorporate non-linear geometry constitutes an important computational task. In this paper, we develop methods for numerically solving optimization problems over spaces of geodesics using numerical integration of Jacobi fields and second order derivatives of geodesic families. As an important application of this optimization strategy, we compute exact Principal Geodesic Analysis (PGA), a non-linear version of the PCA dimensionality reduction procedure. By applying the exact PGA algorithmto synthetic data, we exemplify the differences between the linearized and exact algorithms caused by the non-linear geometry. In addition, we use the numerically integrated Jacobi fields to determine sectional curvatures and provide upper bounds for injectivity radii.",

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T1 - Optimization over geodesics for exact principal geodesic analysis

AU - Sommer, Stefan Horst

AU - Lauze, Francois Bernard

AU - Nielsen, Mads

PY - 2014/4

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N2 - In fields ranging from computer vision to signal processing and statistics, increasing computational power allows a move from classical linear models to models that incorporate non-linear phenomena. This shift has created interest in computational aspects of differential geometry, and solving optimization problems that incorporate non-linear geometry constitutes an important computational task. In this paper, we develop methods for numerically solving optimization problems over spaces of geodesics using numerical integration of Jacobi fields and second order derivatives of geodesic families. As an important application of this optimization strategy, we compute exact Principal Geodesic Analysis (PGA), a non-linear version of the PCA dimensionality reduction procedure. By applying the exact PGA algorithmto synthetic data, we exemplify the differences between the linearized and exact algorithms caused by the non-linear geometry. In addition, we use the numerically integrated Jacobi fields to determine sectional curvatures and provide upper bounds for injectivity radii.

AB - In fields ranging from computer vision to signal processing and statistics, increasing computational power allows a move from classical linear models to models that incorporate non-linear phenomena. This shift has created interest in computational aspects of differential geometry, and solving optimization problems that incorporate non-linear geometry constitutes an important computational task. In this paper, we develop methods for numerically solving optimization problems over spaces of geodesics using numerical integration of Jacobi fields and second order derivatives of geodesic families. As an important application of this optimization strategy, we compute exact Principal Geodesic Analysis (PGA), a non-linear version of the PCA dimensionality reduction procedure. By applying the exact PGA algorithmto synthetic data, we exemplify the differences between the linearized and exact algorithms caused by the non-linear geometry. In addition, we use the numerically integrated Jacobi fields to determine sectional curvatures and provide upper bounds for injectivity radii.

KW - Geometric optimization

KW - Principal geodesic analysis

KW - Manifold statistics

KW - Differential geometry

KW - Riemannian metrics

KW - 65K10

KW - 57R99

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DO - 10.1007/s10444-013-9308-1

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JO - Advances in Computational Mathematics

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Optimization over geodesics for exact principal geodesic analysis

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