Piecewise Convex Function Estimation: Representations, Duality and Model Selection

TL;DR

This paper introduces a method for estimating piecewise convex functions using spline estimates that adaptively preserve convexity properties, with a dual formulation simplifying the optimization process.

Contribution

It presents a novel robust penalized likelihood approach, duality results, and adaptive smoothing techniques for piecewise convex function estimation.

Findings

Adaptive halfwidth kernel smoothing decreases in regions of rapid change.

Representation results and smoothness properties are derived for convexity-prescribed estimates.

Dual formulation reduces the estimation to finite-dimensional convex optimization.

Abstract

We consider spline estimates which preserve prescribed piecewise convex properties of the unknown function. A robust version of the penalized likelihood is given and shown to correspond to a variable halfwidth kernel smoother where the halfwidth adaptively decreases in regions of rapid change of the unknown function. When the convexity change points are prescribed, we derive representation results and smoothness properties of the estimates. A dual formulation is given which reduces the estimate is reduced to a finite dimensional convex optimization in the dual space.