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Geodesic Convexity and Covariance Estimation

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1 Author(s)
Wiesel, A. ; Selim & Rachel Benin Sch. of Comput. Sci. & Eng., Hebrew Univ. of Jerusalem, Jerusalem, Israel

Geodesic convexity is a generalization of classical convexity which guarantees that all local minima of g-convex functions are globally optimal. We consider g-convex functions with positive definite matrix variables, and prove that Kronecker products, and logarithms of determinants are g-convex. We apply these results to two modern covariance estimation problems: robust estimation in scaled Gaussian distributions, and Kronecker structured models. Maximum likelihood estimation in these settings involves non-convex minimizations. We show that these problems are in fact g-convex. This leads to straight forward analysis, allows the use of standard optimization methods and paves the road to various extensions via additional g-convex regularization.

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Signal Processing, IEEE Transactions on  (Volume:60 ,  Issue: 12 )