The Kähler Mean of Block-Toeplitz Matrices with Toeplitz Structured Blocks

  title={The K{\"a}hler Mean of Block-Toeplitz Matrices with Toeplitz Structured Blocks},
  author={Ben Jeuris and Raf Vandebril},
  journal={SIAM J. Matrix Anal. Appl.},
When one computes an average of positive definite (PD) matrices, the preservation of additional matrix structure is desirable for interpretations in applications. An interesting and widely present structure is that of PD Toeplitz matrices, which we endow with a geometry originating in signal processing theory. As an averaging operation, we consider the barycenter, or minimizer of the sum of squared intrinsic distances. The resulting barycenter, the Kahler mean, is discussed along with its… 
A geometric mean for Toeplitz and Toeplitz-block block-Toeplitz matrices
Abstract Using the symbol functions and their associated Fourier series, we introduce a new definition of geometric mean for all positive semi-definite Toeplitz matrices and positive semi-definite
On the structure of the inverse to Toeplitz-block Toeplitz matrices and of the corresponding polynomial reflection coefficients
  • A. Sakhnovich
  • Mathematics
    Transactions of the American Mathematical Society
  • 2019
The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the 1-D (one-dimensional) case are classical and have numerous applications. We consider the
Inversion of the Toeplitz-block Toeplitz matrices and the structure of the corresponding inverse matrices
The results on the inversion of convolution operators and Toeplitz matrices in the 1-D (one dimensional) case are classical and have numerous applications. We consider a 2-D case of Toeplitz-block
On the inversion of the block double-structured and of the triple-structured Toeplitz matrices and on the corresponding reflection coefficients
Abstract The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the 1-D (one-dimensional) case are classical and have numerous applications. Last
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Gaussian Distributions on Riemannian Symmetric Spaces: Statistical Learning With Structured Covariance Matrices
An original theory of Gaussian distributions on Riemannian symmetric spaces is developed, of their statistical inference, and of their relationship to the concept of Riemansian barycentre, which describes algorithms for density estimation and classification of structured covariance matrices, based on Gaussian distribution mixture models.
Riemannian Laplace Distribution on the Space of Symmetric Positive Definite Matrices
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Mean and Median of PSD Matrices on a Riemannian Manifold: Application to Detection of Narrow-Band Sonar Signals
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On the Riemannian barycentre of a Markov chain
The Riemannian barycentre is one of the most widely used statistical descriptors for probability distributions on Riemannian manifolds. At present, existing algorithms are able to compute the
The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain
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The geometric mean of positive definite matrices is usually identified with the Karcher mean, which possesses all properties—generalized from the scalar case—a geometric mean is expected to satisfy.
Information Geometry of Covariance Matrix: Cartan-Siegel Homogeneous Bounded Domains, Mostow/Berger Fibration and Fréchet Median
Information Geometry has been introduced by Rao, and axiomatized by Chentsov, to define a distance between statistical distributions that is invariant to non-singular parameterization
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Interactions between Symmetric Cone and Information Geometries: Bruhat-Tits and Siegel Spaces Models for High Resolution Autoregressive Doppler Imagery
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Algorithm comparison for Karcher mean computation of rotation matrices and diffusion tensors
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Computationally efficient two-dimensional Capon spectrum analysis
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Positive Definite Matrices
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