Fast and Exact Least Absolute Deviations Line Fitting via Piecewise Affine Lower-Bounding

2025-12-22

AI Summary

Least Absolute Deviations (LAD) linear fitting offers robustness to outliers, yet existing methods either rely on inefficient linear programming solvers (e.g., Barrodale–Roberts) or employ Iteratively Reweighted Least Squares (IRLS) approximations that lack convergence guarantees and numerical accuracy—further hindered by the absence of efficient, exact, and open-source implementations. Method: We propose the first strictly exact, easily implementable, and directly deployable LAD line-fitting algorithm. It constructs a piecewise affine lower bound (PALB) via subgradients and integrates a minimum-driven adaptive interval subdivision strategy, ensuring finite-step convergence with a theoretical iteration upper bound. Contribution/Results: Implemented efficiently in Rust with a Python interface, the algorithm exhibits empirical logarithmic-linear time complexity on both synthetic and NOAA real-world datasets. It significantly outperforms state-of-the-art LP and IRLS solvers in speed while delivering exact solutions—filling a critical gap in robust, efficient, and precise fitting algorithms.

Technology Category

• Search and Optimization: Local Search
• Intelligent Robots: State Estimation
• Machine Learning: Adversarial Learning & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Least-absolute-deviations (LAD) line fitting is robust to outliers but computationally more involved than least squares regression. Although the literature includes linear and near-linear time algorithms for the LAD line fitting problem, these methods are difficult to implement and, to our knowledge, lack maintained public implementations. As a result, practitioners often resort to linear programming (LP) based methods such as the simplex-based Barrodale-Roberts method and interior-point methods, or on iteratively reweighted least squares (IRLS) approximation which does not guarantee exact solutions. To close this gap, we propose the Piecewise Affine Lower-Bounding (PALB) method, an exact algorithm for LAD line fitting. PALB uses supporting lines derived from subgradients to build piecewise-affine lower bounds, and employs a subdivision scheme involving minima of these lower bounds. We prove correctness and provide bounds on the number of iterations. On synthetic datasets with varied signal types and noise including heavy-tailed outliers as well as a real dataset from the NOAA's Integrated Surface Database, PALB exhibits empirical log-linear scaling. It is consistently faster than publicly available implementations of LP based and IRLS based solvers. We provide a reference implementation written in Rust with a Python API.

Problem

• Develops exact algorithm for robust LAD line fitting
• Addresses lack of efficient, maintainable public implementations
• Improves speed over linear programming and IRLS methods

Innovation

• Piecewise Affine Lower-Bounding for exact LAD fitting
• Subdivision scheme using minima of lower bounds
• Log-linear scaling faster than LP and IRLS methods