Riemannian and stratified geometries on covariance and correlation matrices
Following the proposal of the affine-invariant and log-Euclidean families of metrics on SPD matrices, a zoo of other Riemannian and non-Riemannian metrics were proposed in the literature. With the PhD thesis of Yann Thanwerdas (ATSI PhD prize of EDSTIC 2022), we investigated families of Riemannian and stratified geometries on covariance and correlation matrices.
Families of metrics on SPD matrices
For SPD matrices, Yann Thanwerdas characterized in the first paper all the continuous metrics invariant by O(n) by means of three multivariate continuous functions, that encompass the classical metrics (log-Euclidean, BKM, power-Euclidean) for which we detail all the basic Riemannian operations. In the second paper, we related the specific family of mixed-Euclidean metrics to different types of divergence used in information geometry.
Yann Thanwerdas, Xavier Pennec.
The geometry of mixed-Euclidean metrics on symmetric positive definite matrices.
Differential Geometry and its Applications, 2022, 81 (101867).
DOI : 10.1016/j.difgeo.2022.101867.
[PDF].
Metrics on correlations matrices
Correlation matrices are covariance matrices that are right and left renormalized by the diagonal matrix of inverse standard deviations. These objects live in an edged stratified compact space in which it is difficult to handle with the classical Euclidean metric. A first idea to change the metric of the interior of this space, the open elliptope, is to take the quotient of the affine-invariant metric by the renormalization group. Unfortunately, we showed in [Thanwerdas & Pennec, GSI 2021] that the curvature of this quotient-affine metric is of non-constant sign and unbounded from above, which makes this geometry practically very complex to work with in practice.
We shows in this paper that the open elliptope can be provided with other non-positive curvature or even flat metrics that allow working in Hadamard or Euclidean spaces, very similarly to the famous affine-invariant metrics or log-Euclidean on SPD matrices. These metrics could generate a very strong interest in neuroscience because the brain networks extracted from functional magnetic imaging are in fact encoded by series of correlation matrices.
Yann Thanwerdas, Xavier Pennec.
Theoretically and computationally convenient geometries on full-rank correlation matrices
SIAM Journal on Matrix Analysis and Applications, 2022, 43 (4), pp.1851-1872.
DOI : 10.1137/22M1471729.
[PDF].