Approximation schemes for functions of positive-definite matrix values

Nir Sharon, Uri Itai

Research output: Contribution to journalArticlepeer-review

Abstract

In recent years, there has been an enormous interest in developing methods for the approximation of manifold-valued functions. In this paper, we focus on the manifold of symmetric positive-definite (SPD) matrices. We investigate the use of SPD-matrix means to adapt linear positive approximation methods to SPD-matrix-valued functions. Specifically, we adapt corner-cutting subdivision schemes and Bernstein operators. We present the concept of admissible matrix means and study the adapted approximation schemes based on them. Two important cases of admissible matrix means are treated in detail: the exp-log and the geometric matrix means. We derive special properties of the approximation schemes based on these means. The geometric mean is found to be superior in the sense of preserving more properties of the data, such as monotonicity and convexity. Furthermore, we give error bounds for the approximation of univariate SPD-matrix-valued functions by the adapted operators.

Original languageEnglish
Pages (from-to)1436-1468
Number of pages33
JournalIMA Journal of Numerical Analysis
Volume33
Issue number4
DOIs
StatePublished - Oct 2013

Keywords

  • Bernstein operators
  • approximation of matrix-valued functions
  • corner-cutting subdivision schemes
  • matrix means for symmetric positive-definite matrices
  • symmetric positive-definite matrices

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