Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Scattered Manifold-Valued Data Approximation

by P. Grohs and M. Sprecher and T. Yu

(Report number 2014-08)

Abstract
We consider the problem of approximating a smooth function \(f\) from an Euclidean domain to a manifold \(M\) by scattered samples \((f(\xi_i))_{i\in \mathcal{I}}\), where the data sites \((\xi_i)_{i\in \mathcal{I}}\) are assumed to be locally close but can otherwise be far apart points scattered throughout the domain. We introduce a natural approximant based on combining the moving least square method and the Karcher mean. We prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart. The analysis also tells us that the use of Karcher's mean (dependent on a Riemannian metric and the associated exponential map) is inessential and one can replace it by a more general notion of `center of mass' based on a general retraction on the manifold. Consequently, we can substitute the Karcher mean by a more computationally efficient mean. We illustrate our work with numerical results which confirm our theoretical findings.

Keywords: Riemannian data, manifold-valued function, approximation, scattered data, model reduction

@Techreport{GSY14_558,
  author = {P. Grohs and M. Sprecher and T. Yu},
  title = {Scattered Manifold-Valued Data Approximation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-08},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-08.pdf },
  year = {2014}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).