# Efficient Second-Order Shape-Constrained Function Fitting

## TL;DR

This paper introduces efficient algorithms for fitting one-dimensional shape-constrained functions, including monotonicity and convexity, with near-linear time complexity, advancing the computational methods for such problems.

## Contribution

The paper presents the first near-linear-time algorithms for second-order shape-constrained function fitting, applicable to a broad class of shape constraints and utilizing a novel geometric interpretation.

## Key findings

- Algorithms achieve $O(n ext{ log } rac{U}{ ext{error}})$ time for general shape constraints.
- A simple $O(n)$ greedy algorithm for unweighted convex regression.
- Generalization to DAGs is as hard as linear programming.

## 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 $\varepsilon$ in $O\left(n \log \frac{U}{\varepsilon} \right)$ 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.