A new metric on the manifold of kernel matrices with application to matrix geometric means

Symmetric positive definite (spd) matrices pervade numerous scientific disciplines, including machine learning and optimization. We consider the key task of measuring distances between two spd matrices; a task that is often nontrivial whenever the distance function must respect the non-Euclidean geometry of spd matrices. Typical non-Euclidean distance measures such as the Riemannian metric δR(X, Y) = ||log(Y-1/2XY-1/2)||F, are computationally demanding and also complicated to use. To allay some of these difficulties, we introduce a new metric on spd matrices, which not only respects non-Euclidean geometry but also offers faster computation than δR while being less complicated to use. We support our claims theoretically by listing a set of theorems that relate our metric to δR(X, Y), and experimentally by studying the nonconvex problem of computing matrix geometric means based on squared distances.

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