Theoretically and computationally convenient geometries on full-rank correlation matrices. (English) Zbl 1511.53003
Summary: In contrast to SPD matrices, few tools exist to perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as the quotient of the affine-invariant metric by the congruence action of positive diagonal matrices. The space of SPD matrices had always been thought of as a Riemannian homogeneous space. In contrast, we view in this work SPD matrices as a Lie group and the affine-invariant metric as a left-invariant metric. This unexpected new viewpoint allows us to generalize the construction of the quotient-affine metric and to show that the main Riemannian operations can be computed numerically. However, the uniqueness of the Riemannian logarithm or the Fréchet mean are not ensured, which is bad for computing on the elliptope. Hence, we define three new families of Riemannian metrics on full-rank correlation matrices which provide Hadamard structures, including two flat. Thus the Riemannian logarithm and the Fréchet mean are unique. The two (flat) vector space structures are particularly appealing because they reduce the manifold of full-rank correlation matrices to a vector space. We also define a nilpotent group structure for which the affine logarithm and the group mean are unique. We provide the main Riemannian/group operations of these four structures in closed form.
MSC:
| 53-08 | Computational methods for problems pertaining to differential geometry |
| 53B21 | Methods of local Riemannian geometry |
| 15A63 | Quadratic and bilinear forms, inner products |
| 53C22 | Geodesics in global differential geometry |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 58D17 | Manifolds of metrics (especially Riemannian) |
Keywords:
SPD matrices; correlation matrices; Lie group; Lie group actions; quotient-affine metric; Lie-Cholesky metrics; poly-hyperbolic-Cholesky metrics; Euclidean-Cholesky metrics; log-Euclidean-Cholesky metricsReferences:
| [1] | Archakov, I. and Hansen, P. R., A new parametrization of correlation matrices, Econometrica, 89 (2021), pp. 1699-1715. · Zbl 1497.62193 |
| [2] | Arsigny, V., Fillard, P., Pennec, X., and Ayache, N., Log-Euclidean metrics for fast and simple calculus on diffusion tensors, Magn. Reson. Med., 56 (2006), pp. 411-421. |
| [3] | Barachant, A., Bonnet, S., Congedo, M., and Jutten, C., Multiclass brain-computer interface classification by Riemannian geometry, IEEE. Trans. Biomed. Eng., 59 (2012), pp. 920-928, doi:10.1109/TBME.2011.2172210. |
| [4] | Barachant, A., Bonnet, S., Congedo, M., and Jutten, C., Classification of covariance matrices using a Riemannian-based kernel for BCI applications, Neurocomputing, 112 (2013), pp. 172-178. |
| [5] | Besse, A. L., Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, Heidelberg, 1987. · Zbl 0613.53001 |
| [6] | Buser, P. and Karcher, H., Gromov’s almost flat manifolds, Soc. Math. France, Paris, 1981 · Zbl 0459.53031 |
| [7] | David, P., A Riemannian Quotient Structure for Correlation Matrices with Applications to Data Science, Ph.D. thesis, Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, 2019. |
| [8] | David, P. and Gu, W., A Riemannian structure for correlation matrices, Oper. Matrices, 13 (2019), pp. 607-627. · Zbl 1433.62149 |
| [9] | Fletcher, P. T. and Joshi, S., Riemannian geometry for the statistical analysis of diffusion tensor data, Signal Process., 87 (2007), pp. 250-262, doi:10.1016/j.sigpro.2005.12.018. · Zbl 1186.94126 |
| [10] | Gallier, J., Logarithms and Square Roots of Real Matrices, preprint, https://arxiv.org/abs/0805.0245, 2008. |
| [11] | Garba, M. K., Nye, T. M. W., Lueg, J., and Huckemann, S. F., Information geometry for phylogenetic trees, J. Math. Biol., 82 (2021), 19. · Zbl 1460.92141 |
| [12] | Guigui, N. and Pennec, X., A reduced parallel transport equation on Lie groups with a left-invariant metric, in Geometric Science of Information, , Nielsen, F. and Barbaresco, F., eds., Springer, Cham, 2021, pp. 119-126. · Zbl 1490.53072 |
| [13] | Kercheval, A. N., On Rebonato and Jäckel’s parametrization method for finding nearest correlation matrices, Int. J. Pure Appl. Math., 45 (2008), pp. 383-390. · Zbl 1153.91523 |
| [14] | Le Brigant, A., Preston, S. C., and Puechmorel, S., Fisher-Rao geometry of Dirichlet distributions, Differ. Geom. Appl., 74 (2021), 101702. · Zbl 1460.62018 |
| [15] | Lee, J. M., Introduction to Smooth Manifolds, 2nd ed., Springer, New York, 2012. |
| [16] | Li, P., Wang, Q., Zeng, H., and Zhang, L., Local log-Euclidean multivariate Gaussian descriptor and its application to image classification, IEEE Trans. Pattern Anal., 39 (2017), pp. 803-817. |
| [17] | Lin, Z., Riemannian geometry of symmetric positive definite matrices via Cholesky decomposition, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1353-1370, doi:10.1137/18M1221084. · Zbl 07141455 |
| [18] | Marti, G., Goubet, V., and Nielsen, F., cCorrGAN: Conditional correlation GAN for learning empirical conditional distributions in the elliptope, in Geometric Science of Information, , Nielsen, F. and Barbaresco, F., eds., Springer, Cham, 2021, pp. 613-620. · Zbl 07495261 |
| [19] | Miolane, N., Guigui, N., Le Brigant, A., Mathe, J., Hou, B., Thanwerdas, Y., Heyder, S., Peltre, O., Koep, N., Zaatiti, H., Hajri, H., Cabanes, Y., Gerald, T., Chauchat, P., Shewmake, C., Brooks, D., Kainz, B., Donnat, C., Holmes, S., and Pennec, X., Geomstats: A python package for Riemannian geometry in machine learning, J. Mach. Learn. Res., 21 (2020), pp. 1-9, https://hal.inria.fr/hal-02536154. |
| [20] | Nielsen, F. and Sun, K., Clustering in Hilbert’s projective geometry: The case studies of the probability simplex and the elliptope of correlation matrices, in Geometric Structures of Information, Nielsen, F., ed., Springer, Cham, 2019, pp. 297-331. · Zbl 1418.53022 |
| [21] | O’Neill, B., The fundamental equations of a submersion, Michigan Math. J., 13 (1966), pp. 459-469. · Zbl 0145.18602 |
| [22] | Pennec, X., Statistical computing on manifolds: From Riemannian geometry to computational anatomy, in Emerging Trends in Visual Computing: LIX Fall Colloquium, ETVC 2008, Palaiseau, France, 2008. Revised Invited Papers, Lecture Notes in Comput. Sci. 5416, Springer, Berlin, Heidelberg, 2009, pp. 347-386, doi:10.1007/978-3-642-00826-9_16. |
| [23] | Pinheiro, J. C. and Bates, D. M., Unconstrained parametrizations for variance-covariance matrices, Statist. Comput., 6 (1996), pp. 289-296. |
| [24] | Rebonato, R. and Jaeckel, P., The most general methodology to create a valid correlation matrix for risk management and option pricing purposes, J. Risk, 2 (2001), pp. 17-27. |
| [25] | Skovgaard, L. T., A Riemannian geometry of the multivariate normal model, Scand. J. Statist., 11 (1984), pp. 211-223. · Zbl 0579.62033 |
| [26] | Thanwerdas, Y., Riemannian and Stratified Geometries of Covariance and Correlation Matrices, Ph.D. thesis, Université Côte d’Azur and Inria, Sophia Antipolis, France, 2022. |
| [27] | Thanwerdas, Y. and Pennec, X., Geodesics and curvature of the quotient-affine metrics on full-rank correlation matrices, in Proceedings of GSI, 2021 - 5th Conference on Geometric Science of Information, , , Springer, 2021, pp. 93-102. · Zbl 07495206 |
| [28] | Thanwerdas, Y. and Pennec, X., O(n)-invariant Riemannian Metrics on SPD Matrices, preprint, https://arxiv.org/abs/2109.05768, 2021. |
| [29] | Varoquaux, G., Baronnet, F., Kleinschmidt, A., Fillard, P., and Thirion, B., Detection of brain functional-connectivity difference in post-stroke patients using group-level covariance modeling, in Medical Image Computing and Computer Added Intervention, Shen, D., Frangi, A., and Szekely, G., eds., , Springer, Berlin, Heidelberg, 2010, pp. 200-208. |
| [30] | Wang, Z., Vemuri, B. C., Chen, Y., and Mareci, T. H., A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI, IEEE Trans. Med. Imaging, 23 (2004), pp. 930-939. |
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