Abstract
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.
| Original language | English |
|---|---|
| Article number | 6298979 |
| Pages (from-to) | 6182-6189 |
| Number of pages | 8 |
| Journal | IEEE Transactions on Signal Processing |
| Volume | 60 |
| Issue number | 12 |
| DOIs | |
| State | Published - 2012 |
Bibliographical note
Funding Information:Manuscript received February 13, 2012; revised June 06, 2012; accepted August 15, 2012. Date of publication September 11, 2012; date of current version November 20, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Antonio De Maio. This work was partially supported by the Israel Science Foundation Grant 786/11. Parts of this material in this paper was presented in at IEEE Statistical Signal Processing Conference, Ann Arbor, MI, August 2012.
Keywords
- Elliptical distributions
- Kronecker models
- geodesic convexity
- log-sum-exp
- martix variate models
- robust covariance estimation