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

Wed Dec 05 07:00 PM -- 12:00 AM (PST) @ Harrah’s Special Events Center 2nd Floor #None
Symmetric positive definite (spd) matrices are remarkably pervasive in a multitude of scientific disciplines, including machine learning and optimization. We consider the fundamental task of measuring distances between two spd matrices; a task that is often nontrivial whenever an application demands the distance function to respect the non-Euclidean geometry of spd matrices. Unfortunately, typical non-Euclidean distance measures such as the Riemannian metric $\riem(X,Y)=\frob{\log(X\inv{Y})}$, are computationally demanding and also complicated to use. To allay some of these difficulties, we introduce a new metric on spd matrices: this metric not only respects non-Euclidean geometry, it also offers faster computation than $\riem$ while being less complicated to use. We support our claims theoretically via a series of theorems that relate our metric to $\riem(X,Y)$, and experimentally by studying the nonconvex problem of computing matrix geometric means based on squared distances.

Author Information

Suvrit Sra (MIT)

Suvrit Sra is a faculty member within the EECS department at MIT, where he is also a core faculty member of IDSS, LIDS, MIT-ML Group, as well as the statistics and data science center. His research spans topics in optimization, matrix theory, differential geometry, and probability theory, which he connects with machine learning --- a key focus of his research is on the theme "Optimization for Machine Learning” (http://opt-ml.org)

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