Geometric Means in a Novel Vector Space Structure on Symmetric Positive‐Definite Matrices

TLDR

The proposed mean coincides with the usual Euclidean mean under the novel vector space structure. The paper introduces the Log‑Euclidean mean, a generalization of the geometric mean for symmetric positive‑definite matrices. The authors construct a Lie‑group structure and a compatible scalar multiplication on SPD matrices, then use bi‑invariant metrics to define the Log‑Euclidean mean as a Riemannian mean, and analyze its invariance properties and relation to the affine‑invariant mean. The Log‑Euclidean mean equals the arithmetic mean of matrix logarithms, shares the determinant with the affine‑invariant mean but generally has a larger trace, and is computationally simpler.

Abstract

In this work we present a new generalization of the geometric mean of positive numbers on symmetric positive‐definite matrices, called Log‐Euclidean. The approach is based on two novel algebraic structures on symmetric positive‐definite matrices: first, a lie group structure which is compatible with the usual algebraic properties of this matrix space; second, a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure. From bi‐invariant metrics on the Lie group structure, we define the Log‐Euclidean mean from a Riemannian point of view. This notion coincides with the usual Euclidean mean associated with the novel vector space structure. Furthermore, this means corresponds to an arithmetic mean in the domain of matrix logarithms. We detail the invariance properties of this novel geometric mean and compare it to the recently introduced affine‐invariant mean. The two means have the same determinant and are equal in a number of cases, yet they are not identical in general. Indeed, the Log‐Euclidean mean has a larger trace whenever they are not equal. Last but not least, the Log‐Euclidean mean is much easier to compute.