Theoretically and Computationally Convenient Geometries on Full-Rank Correlation Matrices

In contrast to SPD matrices, few tools exist perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as quotient affine-invariant by congruence action positive diagonal space matrices had always been thought a homogeneous space. contrast, we view in this work Lie group and left-invariant metric. This unexpected new viewpoint allows us generalize construction show that main operations can be computed numerically. However, uniqueness logarithm or Fréchet mean are not ensured, which is bad for computing elliptope. Hence, define three families metrics provide Hadamard structures, including two flat. Thus unique. (flat) vector structures particularly appealing because they reduce manifold We also nilpotent structure affine Riemannian/group these four closed form.