TY - GEN

T1 - A new metric on the manifold of kernel matrices with application to matrix geometric means

AU - Sra, Suvrit

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84877770427&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84877770427

SN - 9781627480031

T3 - Advances in Neural Information Processing Systems

SP - 144

EP - 152

BT - Advances in Neural Information Processing Systems 25

T2 - 26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012

Y2 - 3 December 2012 through 6 December 2012

ER -