Abstract
Matrix data sets are common nowadays like in biomedical imaging where the Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) modality produces data sets of 3D symmetric positive definite matrices anchored at voxel positions capturing the anisotropic diffusion properties of water molecules in biological tissues. The space of symmetric matrices can be partially ordered using the Löwner ordering, and computing extremal matrices dominating a given set of matrices is a basic primitive used in matrix-valued signal processing. In this letter, we design a fast and easy-to-implement iterative algorithm to approximate arbitrarily finely these extremal matrices. Finally, we discuss on extensions to matrix clustering.
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Notes
- 1.
Although addition preserves the symmetric property, beware that the product of two symmetric matrices may be not symmetric.
- 2.
Those definitions extend to Hermitian matrices \(M_d(\mathbb {C})\).
- 3.
Also often written Loewner in the literature, e.g., see Siotani (1967).
- 4.
For example, consider \(P=\mathrm {diag}(1,2)\) and \(Q=\mathrm {diag}(2,1)\) then \(P-Q=\mathrm {diag}(-1,1)\) and \(Q-P=\mathrm {diag}(1,-1)\).
- 5.
For example, \(S=\mathrm {diag}(-1,1)\) is dominated by \(P=\mathrm {diag}(1=|-1|,1)\) (by taking the absolute values of the eigenvalues of S).
- 6.
In 2D, we sample \(v=[\cos \theta , \sin \theta ]^\top \) for \(\theta \in [0,2\pi [\). In 3D, we use spherical coordinates \(v=[\sin \theta \cos \phi , \sin \theta \sin \phi , \cos \theta ]^\top \) for \(\theta \in [0,2\pi [\) and \(\phi \in [0,\pi [\).
- 7.
References
Allamigeon, X., Gaubert, S., Goubault, E., Putot, S., & Stott, N. (1980). A scalable algebraic method to infer quadratic invariants of switched systems. In 2015 International Conference on Embedded Software (EMSOFT) (pp. 75–84), October 2015.
Angulo, J. (2013). Matrix Information Geometry. In F. Nielsen & R. Bhatia (Eds.), Supremum/infimum and nonlinear averaging of positive definite symmetric matrices (pp. 3–33). Heidelberg: Springer.
Bădoiu, M., & Clarkson, K. L. (2003). Smaller core-sets for balls. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’03, Philadelphia, PA, USA, 2003 (pp 801–802). Society for Industrial and Applied Mathematics.
Bădoiu, M., & Clarkson, K. L. (2008). Optimal core-sets for balls. Computational Geometry, 40(1), 14–22.
Bhatia, R. (2009). Positive definite matrices. Princeton: Princeton university press.
Boissonnat, J.-D., Cérézo, A., Devillers, O., Duquesne, J., & Yvinec, M. (1996). An algorithm for constructing the convex hull of a set of spheres in dimension \(d\). Computational Geometry, 6(2), 123–130.
Boissonnat, J.-D., & Karavelas, M. I. (2003). On the combinatorial complexity of euclidean Voronoi cells and convex hulls of \(d\)-dimensional spheres. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 305–312). Society for Industrial and Applied Mathematics.
Burgeth, B., Bruhn, A., Didas, S., Weickert, J., & Welk, M. (2007). Morphology for matrix data: Ordering versus PDE-based approach. Image and Vision Computing, 25(4), 496–511.
Burgeth, B., Bruhn, A., Papenberg, N., Welk, M., & Weickert, J. (2007). Mathematical morphology for matrix fields induced by the Loewner ordering in higher dimensions. Signal Processing, 87, 277–290.
Calvin, J. A., & Dykstra, R. L. (1991). Maximum likelihood estimation of a set of covariance matrices under Löwner order restrictions with applications to balanced multivariate variance components models. The Annals of Statistics, 19, 850–869.
Chen, K. (2009). On coresets for \(k\)-median and \(k\)-means clustering in metric and euclidean spaces and their applications. SIAM Journal on Computing, 39(3), 923–947.
Fischer, K., & Gärtner, B. (2004). The smallest enclosing ball of balls: Combinatorial structure and algorithms. International Journal of Computational Geometry & Applications, 14(04n05), 341–378.
Fischer, K., Gärtner, B., & Kutz, M. (2003). Fast smallest-enclosing-ball computation in high dimensions. In G. Di Battista & U. Zwick (Eds.), Algorithms-ESA 2003 (pp. 630–641). Heidelberg: Springer.
Förstner, W. (1986). A feature based correspondence algorithm for image matching. International Archives of Photogrammetry and Remote Sensing, 26(3), 150–166.
Hill, R. D., & Waters, S. R. (1987). On the cone of positive semidefinite matrices. Linear Algebra and its Applications, 90, 81–88.
Jambawalikar, S., & Kumar, P. (2008). A note on approximate minimum volume enclosing ellipsoid of ellipsoids. In International Conference on Computational Sciences and Its Applications, 2008. ICCSA’08 (pp. 478–487). IEEE.
Kumar, P., Mitchell, J. S. B., & Yildirim, E. A. (2003). Approximate minimum enclosing balls in high dimensions using core-sets. Journal of Experimental Algorithmics (JEA), 8, Article No. 1.1.
Mihelic, J., & Robic, B. (2003). Approximation algorithms for the \(k\)-center problem: An experimental evaluation. In Selected papers of the International Conference on Operations Research (SOR 2002) (p. 371) Heidelberg:Springer.
Nielsen, F., & Bhatia, R. (2013). Matrix Information Geometry. Heidelberg: Springer Publishing Company, Incorporated.
Siotani, M. (1967). Some applications of Loewner’s ordering on symmetric matrices. Annals of the Institute of Statistical Mathematics, 19(1), 245–259.
Tsai, M.-T. (2007). Maximum likelihood estimation of Wishart mean matrices under Löwner order restrictions. Journal of Multivariate Analysis, 98(5), 932–944.
Acknowledgements
This work was carried out during the Matrix Information Geometry (MIG) workshop (Nielsen and Bhatia 2013), organized at École Polytechnique, France in February 2011 (https://www.sonycsl.co.jp/person/nielsen/infogeo/MIG/). Frank Nielsen dedicates this work to the memory of his late father Gudmund Liebach Nielsen who passed away during the last day of the workshop.
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Nielsen, F., Nock, R. (2017). Fast \((1+\epsilon )\)-Approximation of the Löwner Extremal Matrices of High-Dimensional Symmetric Matrices. In: Nielsen, F., Critchley, F., Dodson, C. (eds) Computational Information Geometry. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-47058-0_6
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