Efficient Second-Order Shape-Constrained Function Fitting

David Durfee, Yu Gao, Anup B. Rao, Sebastian Wild

Abstract: We give an algorithm to compute a one-dimensional shape-constrained function that best fits given data in weighted- $$L_{\infty }$$ norm. We give a single algorithm that works for a variety of commonly studied shape constraints including monotonicity, Lipschitz-continuity and convexity, and more generally, any shape constraint expressible by bounds on first- and/or second-order differences. Our algorithm computes an approximation with additive error $$\epsilon $$ in $$O\left( n \log \frac{U}{\epsilon } \right) $$ time, where U captures the range of input values. We also give a simple greedy algorithm that runs in O(n) time for the special case of unweighted $$L_{\infty }$$ convex regression. These are the first (near-)linear-time algorithms for second-order-constrained function fitting. To achieve these results, we use a novel geometric interpretation of the underlying dynamic programming problem. We further show that a generalization of the corresponding problems to directed acyclic graphs (DAGs) is as difficult as linear programming.