Piecewise Convex Function Estimation and Model Selection
This work provides a method for function estimation in scenarios where the underlying function has piecewise convexity, which is incremental as it builds on existing convex optimization techniques.
The paper tackles the problem of estimating a piecewise convex or concave function from noisy data, addressing both known and unknown region counts, and reduces the known-region case to a finite-dimensional convex optimization in dual space.
Given noisy data, function estimation is considered when the unknown function is known apriori to consist of a small number of regions where the function is either convex or concave. When the regions are known apriori, the estimate is reduced to a finite dimensional convex optimization in the dual space. When the number of regions is unknown, the model selection problem is to determine the number of convexity change points. We use a pilot estimator based on the expected number of false inflection points.