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The Derivative of the Matrix Geometric Mean with an Application to the Nonnegative Decomposition of Tensor Grids

  • Bruno IannazzoEmail author
  • Ben Jeuris
  • Filippo Pompili
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 30)

Abstract

We provide an expression for the derivative of the weighted matrix geometric mean, with respect to both the matrix arguments and the weights, that can be easily translated to an algorithm for its computation. As an application, we consider the problem of the approximate decomposition of a tensor grid M, a matrix whose entries are positive definite matrices. For different geometries on the set of positive definite matrices, we derive an approximate decomposition such that any column of M is a barycentric combination of the columns of a smaller tensor grid. This extends the Euclidean case, already considered in the literature, to the geometry in which the barycenter is the matrix geometric mean and the log-Euclidean geometry.

Keywords

Matrix geometric mean Karcher mean Tensor grid Positive definite matrix Nonnegative factorization 

Notes

Acknowledgements

The authors would like to thank the referees for carefully reading the manuscript, providing many insightful comments which improved the presentation of the chapter.

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Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di PerugiaPerugiaItaly
  2. 2.Department of Computer ScienceKU LeuvenLeuvenBelgium
  3. 3.Thomson ReutersTorontoCanada

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