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

@article{Jeuris2016TheKM, 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.}, year={2016}, volume={37}, pages={1151-1175} }

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…

## 28 Citations

A geometric mean for Toeplitz and Toeplitz-block block-Toeplitz matrices

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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…

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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…

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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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- Computer Science, MathematicsArXiv
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It is proved that if a1, . . . , ap are continuous and positive functions, or are in the Wiener algebra with some further conditions, then means of geometric type, such as the ALM, the NBMP and the Karcher mean of quasi-Toeplitz positive definite matrices associated with a 1, .

Gaussian Distributions on Riemannian Symmetric Spaces: Statistical Learning With Structured Covariance Matrices

- Mathematics, Computer ScienceIEEE Transactions on Information Theory
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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.

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- Mathematics, Computer ScienceEntropy
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New probability distributions, called Riemannian Laplace distributions on the space P m, are introduced and it is shown that these distributions provide a statistical foundation for the concept of the RiemANNian median, which offers improved robustness in dealing with outliers.

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- Mathematics, Computer ScienceIEEE Transactions on Signal Processing
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It is demonstrated that geometric computing in the Siegel–Klein disk allows one to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in theSiegel–Poincaré disk model, and to approximate fast and numerically the S Spiegel-Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.

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