4.3 Article

Low-rank multi-parametric covariance identification

Journal

BIT NUMERICAL MATHEMATICS
Volume 62, Issue 1, Pages 221-249

Publisher

SPRINGER
DOI: 10.1007/s10543-021-00867-y

Keywords

Covariance approximation; Interpolation on manifolds; Positive-semidefinite matrices; Riemannian metric; Geodesic; Low-rank covariance function; Maximum likelihood

Funding

  1. la Caixa Banking Foundation [100010434, LCF/BQ/AN13/10280009]
  2. Fonds de la Recherche Scientifique (FNRS) under EOS [30468160]
  3. Fonds Wetenschappelijk Onderzoek (FWO)-Vlaanderen under EOS [30468160]
  4. Communaute francaise de Belgique-Actions de Recherche Concertees [ARC 14/19-060]
  5. WBI-World Excellence Fellowship
  6. United States Department of Energy, Office of Advanced Scientific Computing Research (ASCR)
  7. National Physical Laboratory

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The paper presents a differential geometric approach for constructing families of low-rank covariance matrices through interpolation on low-rank matrix manifolds, demonstrating its utility in practical applications such as wind field covariance approximation for unmanned aerial vehicle navigation.
We propose a differential geometric approach for building families of low-rank covariance matrices, via interpolation on low-rank matrix manifolds. In contrast with standard parametric covariance classes, these families offer significant flexibility for problem-specific tailoring via the choice of anchor matrices for interpolation, for instance over a grid of relevant conditions describing the underlying stochastic process. The interpolation is computationally tractable in high dimensions, as it only involves manipulations of low-rank matrix factors. We also consider the problem of covariance identification, i.e., selecting the most representative member of the covariance family given a data set. In this setting, standard procedures such as maximum likelihood estimation are nontrivial because the covariance family is rank-deficient; we resolve this issue by casting the identification problem as distance minimization. We demonstrate the utility of these differential geometric families for interpolation and identification in a practical application: wind field covariance approximation for unmanned aerial vehicle navigation.

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