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Interpolation and Extrapolation of Toeplitz Matrices via Optimal Mass Transport

Elvander, Filip LU ; Jakobsson, Andreas LU and Karlsson, Johan (2018) In IEEE Transactions on Signal Processing 66(20). p.5285-5298
Abstract
In this work, we propose a novel method for quantifying distances between Toeplitz structured covariance matrices. By exploiting the spectral representation of Toeplitz matrices, the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain. This may then be interpreted in the covariance domain, suggesting a natural way of interpolating and extrapolating Toeplitz matrices, such that the positive semi-definiteness and the Toeplitz structure of these matrices are preserved. The proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise, and thereby allows for a quantification of the decreased distance between signals when these are corrupted by... (More)
In this work, we propose a novel method for quantifying distances between Toeplitz structured covariance matrices. By exploiting the spectral representation of Toeplitz matrices, the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain. This may then be interpreted in the covariance domain, suggesting a natural way of interpolating and extrapolating Toeplitz matrices, such that the positive semi-definiteness and the Toeplitz structure of these matrices are preserved. The proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise, and thereby allows for a quantification of the decreased distance between signals when these are corrupted by noise. Finally, we illustrate how this approach can be used for several applications in signal processing. In particular, we consider interpolation and extrapolation of Toeplitz matrices, as well as clustering problems and tracking of slowly varying stochastic processes. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Covariance matrices, covariance interpolation, estimation, spectral analysis, optimal mass transport
in
IEEE Transactions on Signal Processing
volume
66
issue
20
pages
14 pages
publisher
IEEE - Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • scopus:85052704347
ISSN
1053-587X
DOI
10.1109/TSP.2018.2866432
language
English
LU publication?
yes
id
db3c322d-6beb-401f-9d84-fd071b73e91f
date added to LUP
2018-09-12 13:41:52
date last changed
2020-01-16 03:30:24
@article{db3c322d-6beb-401f-9d84-fd071b73e91f,
  abstract     = {In this work, we propose a novel method for quantifying distances between Toeplitz structured covariance matrices. By exploiting the spectral representation of Toeplitz matrices, the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain. This may then be interpreted in the covariance domain, suggesting a natural way of interpolating and extrapolating Toeplitz matrices, such that the positive semi-definiteness and the Toeplitz structure of these matrices are preserved. The proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise, and thereby allows for a quantification of the decreased distance between signals when these are corrupted by noise. Finally, we illustrate how this approach can be used for several applications in signal processing. In particular, we consider interpolation and extrapolation of Toeplitz matrices, as well as clustering problems and tracking of slowly varying stochastic processes.},
  author       = {Elvander, Filip and Jakobsson, Andreas and Karlsson, Johan},
  issn         = {1053-587X},
  language     = {eng},
  number       = {20},
  pages        = {5285--5298},
  publisher    = {IEEE - Institute of Electrical and Electronics Engineers Inc.},
  series       = {IEEE Transactions on Signal Processing},
  title        = {Interpolation and Extrapolation of Toeplitz Matrices via Optimal Mass Transport},
  url          = {http://dx.doi.org/10.1109/TSP.2018.2866432},
  doi          = {10.1109/TSP.2018.2866432},
  volume       = {66},
  year         = {2018},
}