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Symmetric Positive Definite (SPD) matrices have been widely used in medicaldata analysis and a number of different Riemannian met-rics were proposed tocompute with them. However, there are very few methodological principlesguiding the choice of one particular metric for a given application. Invarianceunder the action of the affine transformations was suggested as a principle.Another concept is based on symmetries. However, the affine-invariant metricand 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 investigateother principles to restrict the family size.

Is affine invariance well defined on SPD matrices? A principled continuum of metrics