Geometries and Interpolations for Symmetric Positive Definite Matrices

Abstract

In this survey we review classical and recently proposed Riemannian metrics and interpolation schemes on the space of symmetric positive definite (SPD) matrices. We perform simulations that illustrate the problem of tensor fattening not only in the usually avoided Frobenius metric, but also in other classical metrics on SPD matrices such as the Wasserstein metric, the affine invariant / Fisher Rao metric, and the log Euclidean metric. For comparison, we perform the same simulations on several recently proposed frameworks for SPD matrices that decompose tensors into shape and orientation. In light of the simulation results, we discuss the mathematical and qualitative properties of these new metrics in comparison with the classical ones. Finally, we explore the nonlinear variation of properties such as shape and scale throughout principal geodesics in different metrics, which affects the visualization of scale and shape variation in tensorial data. With the paper, we will release a software package with Matlab scripts for computing the interpolations and statistics used for the experiments in the paper. 1

Entropy-regularized 2-Wasserstein distance between Gaussian measures

“…The same can be seen in the three-dimensional case, depicted in Fig. 2, visualized using the code accompanying [29]. Here, the ellipsoids are determined by the eigenvectors and -values of the covariance matrix of the corresponding Gaussian, and the colors visualize the level sets of the ellipsoids.…”

“…are multivariate Gaussian distributions. We are interested in obtain explicity formulas for the optimal coupling γ ε solving (17), the Entropy-Kantorovich maximizers (ϕ , ψ ) in ( 22) and the entropic displacement interpolation μ ε t in (29). We start by showing that we can assume, without loss of generality, that μ 0 and μ 1 are centered Gaussian distributions.…”

“…where we used the fact that C Part c. As we have solved for α and β for the optimal plan, the entropic interpolant μ t between μ 0 and μ 1 is given by (29), which we rewrite here…”

“…Other popular divergences for Gaussians include the affine-invariant Riemannian metric [68], corresponding to the Fisher-Rao distance between centered Gaussians, the Alpha Log-Determinant divergences [14], corresponding to Rényi divergences between centered Gaussians, and the log-Euclidean metric [8]. A survey of some of the most common divergences and their resulting geometry on Gaussians can be found in [29]. More recently, applications have driven research into allowing determining optimal divergences for the task at hand, which has raised interest in studying interpolations between different divergences [4,17,78].…”

Continuous Histograms for Anisotropy of 2D Symmetric Piece-Wise Linear Tensor Fields

“…There have been many methods proposed which explicitly try to preserve certain tensor characteristics [3,17]. Recently there appeared a survey on interpolation methods for positive definite tensors [8].…”