Inserted: 1 sep 2009
Last Updated: 30 oct 2009
Journal: Journal of Nonparametric Statistics
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