Title

Why Curvature in L-Curve: Combining Soft Constraints

Authors

Uram Anibal Sosa Aguirre, University of Texas at El PasoFollow
Martine Ceberio, University of Texas at El PasoFollow
Vladik Kreinovich, University of Texas at El PasoFollow

Publication Date

12-2010

Comments

Technical Report: UTEP-CS-10-58

Published in Proceedings of the Fourth International Workshop on Constraint Programming and Decision Making CoProD'11, El Paso, Texas, March 17, 2011.

Abstract

In solving inverse problems, one of the successful methods of determining the appropriate value of the regularization parameter is the L-curve method of combining the corresponding soft constraints, when we plot the curve describing the dependence of the logarithm $x$ of the mean square difference on the logarithm $y$ of the mean square non-smoothness, and select a point on this curve at which the curvature is the largest. This method is empirically successful, but from the theoretical viewpoint, it is not clear why we should use curvature and not some other criterion. In this paper, we show that reasonable scale-invariance requirements lead to curvature and its generalizations.