Calculus of Variations and Geometric Measure Theory

F. Balabdaoui - K. Rufibach - F. Santambrogio

Least Squares estimation of two ordered monotone regression curves

created by santambro on 01 Sep 2009
modified on 30 Oct 2009


Accepted Paper

Inserted: 1 sep 2009
Last Updated: 30 oct 2009

Journal: Journal of Nonparametric Statistics
Year: 2009

This is not a true calculus of variations paper. It's an application of convex optimization to statistics. But, curiously enough, the main issue which is addressed (L2 projection on increasing functions) is used in some optimal transport papers as well.


To project (in the $L^2$ metric) a single function $f$ on the set of increasing functions one takes the primitive of $f$, then its convex hull, and then takes the derivative. Here the problem is more complicated: project a pair $(f,g)$ on the set of pairs of increasing functions, the former smaller than the latter. It is a regression problem with applications in statistics. The solution is not explicit but we solve it numerically through a projected subgradient algorithm from convex optimization.

Keywords: regression, monotone curves, subgradient descent