Hermite Interpolation and data processing errors on Riemannian Matrix Manifolds

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Abstract

The main contribution of this paper is twofold: On the one hand, a general framework for performing Hermite interpolation on Riemannian manifolds is presented. The method is applicable if algorithms for the associated Riemannian exponential and logarithm mappings are available. This includes many of the matrix manifolds that arise in practical Riemannian computing applications such as data analysis and signal processing, computer vision and image processing, structured matrix optimization problems, and model reduction. On the other hand, we expose a natural relation between data processing errors and the sectional curvature of the manifold in question. This provides general error bounds for manifold data processing methods that rely on Riemannian normal coordinates. Numerical experiments are conducted for the compact Stiefel manifold of rectangular column-orthogonal matrices. As use cases, we compute Hermite interpolation curves for orthogonal matrix factorizations such as the singular value decomposition and the QR-decomposition.

Original languageEnglish
JournalSIAM Journal on Scientific Computing
Volume42
Issue number5
Pages (from-to)A2593–A2619
ISSN1064-8275
DOIs
Publication statusPublished - 2020

Keywords

  • Hermite interpolation
  • matrix manifold
  • SVD
  • QR decomposition
  • Riemannian Computing
  • Riemannian exponential
  • QR-decomposition
  • Matrix manifold
  • Riemannian logarithm

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