Is Affine-Invariance Well Defined on SPD Matrices? A Principled Continuum of Metrics

Abstract

Symmetric Positive Definite (SPD) matrices have been widely used in medical data analysis and a number of different Riemannian metrics were proposed to compute with them. However, there are very few methodological principles guiding the choice of one particular metric for a given application. Invariance under the action of the affine transformations was suggested as a principle. Another concept is based on symmetries. However, the affine-invariant metric and the recently proposed polar-affine metric are both invariant and symmetric. Comparing these two cousin metrics leads us to introduce much wider families: power-affine and deformed-affine metrics. Within this continuum, we investigate other principles to restrict the family size.

Exploration of Balanced Metrics on Symmetric Positive Definite Matrices

“…The family of power-affine metrics g A,θ [11] indexed by the power θ = 0 are defined by pullback of the affine-invariant metric by the power function pow θ : (M, θ 2 g A,θ ) −→ (M, g A ):…”

“…In particular, pullbacks of the Euclidean metric and the affine-invariant metric under power diffeomorphisms are detailed in the original paper on kernel metrics [17]. They were later called power-Euclidean [19] and power-affine metrics, or more generally deformed-Euclidean and deformed-affine metrics for an arbitrary diffeomorphism [21]. Moreover, power-Euclidean metrics are mean kernel metrics for any power and poweraffine metrics are mean kernel metrics if and only if the power belongs to [−2, 2] [17].…”

“…In papers where the power diffeomorphisms are used to define power-Euclidean, power-affine and alpha-Procrustes metrics [17,19,18,21,20], it is often noticed that the limit when the power tends to 0 is the log-Euclidean metric. This is actually a general fact.…”

“…In previous works, we introduced two principles for building families of Riemannian metrics that share interesting properties: the principle of deformed metrics [21] and the principle of balanced bilinear forms [22]. Deforming metrics (or datasets of SPD matrices) via a diffeomorphism is a very common procedure to define families of metrics.…”

“…The convex cone of Symmetric Positive Definite (SPD) matrices is a manifold on which several Riemannian metrics were defined: Euclidean, Fisher-Rao/affine-invariant [1,2,3,4,5,6], log-Euclidean [7], Bures-Wasserstein [8,9,10,11,12,13], Bogoliubov-Kubo-Mori [14,15], log-Cholesky [16]... Several families of metrics encompassing them were defined to understand their common properties, their differences and the level of generality of each property: kernel metrics and mean kernel metrics [17,18], power-Euclidean [19], alpha-Procrustes [20], deformed-affine [21], mixed-power-Euclidean [22], extended kernel metrics, bivariate separable metrics [23]... In particular, kernel metrics form a very general family of O(n)-invariant metrics indexed by kernel maps φ : (0, ∞) 2 −→ (0, ∞) acting on the eigenvalues of SPD matrices.…”