TY - GEN

T1 - On the convexity in Kronecker structured covariance estimation

AU - Wiesel, Ami

PY - 2012

Y1 - 2012

N2 - A classical model for the covariance of a random matrix is the Kronecker product of two smaller covariance matrices associated with the rows and columns. Maximum likelihood estimation in such structures involves a non-convex optimization problem and is traditionally handled via an alternating maximization Flip-Flop technique. We prove that the problem is actually geodesically convex on the manifold of positive definite matrices. This allows for better analysis, efficient numerical solutions, and simple extensions through the use of additional geodesically convex regularizations. As an example, we propose to shrink the solution towards the identity when the number of samples is insufficient. We demonstrate the advantages of this approach using computer simulations.

AB - A classical model for the covariance of a random matrix is the Kronecker product of two smaller covariance matrices associated with the rows and columns. Maximum likelihood estimation in such structures involves a non-convex optimization problem and is traditionally handled via an alternating maximization Flip-Flop technique. We prove that the problem is actually geodesically convex on the manifold of positive definite matrices. This allows for better analysis, efficient numerical solutions, and simple extensions through the use of additional geodesically convex regularizations. As an example, we propose to shrink the solution towards the identity when the number of samples is insufficient. We demonstrate the advantages of this approach using computer simulations.

KW - Geodesic convexity

KW - Kronecker

KW - covariance estimation

KW - log-sum-exp

UR - http://www.scopus.com/inward/record.url?scp=84868254293&partnerID=8YFLogxK

U2 - 10.1109/SSP.2012.6319848

DO - 10.1109/SSP.2012.6319848

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AN - SCOPUS:84868254293

SN - 9781467301831

T3 - 2012 IEEE Statistical Signal Processing Workshop, SSP 2012

SP - 880

EP - 883

BT - 2012 IEEE Statistical Signal Processing Workshop, SSP 2012

T2 - 2012 IEEE Statistical Signal Processing Workshop, SSP 2012

Y2 - 5 August 2012 through 8 August 2012

ER -