# Optimization for L1-Norm Error Fitting via Data Aggregation

## TL;DR

This paper introduces a data aggregation algorithm that guarantees global optimality for a broad class of L1-norm error fitting problems, including regression and Procrustes, with improved computational efficiency.

## Contribution

It generalizes the AID algorithm to solve any L1-norm error fitting model with a fitting function satisfying certain assumptions, ensuring monotonic convergence to the global optimum.

## Key findings

- The algorithm outperforms existing benchmarks in speed for L1 regression subset selection.
- Performance improves as data size increases.
- Applicable to multi-dimensional problems with arbitrary constraints.

## Abstract

We propose a data aggregation-based algorithm with monotonic convergence to a global optimum for a generalized version of the L1-norm error fitting model with an assumption of the fitting function. The proposed algorithm generalizes the recent algorithm in the literature, aggregate and iterative disaggregate (AID), which selectively solves three specific L1-norm error fitting problems. With the proposed algorithm, any L1-norm error fitting model can be solved optimally if it follows the form of the L1-norm error fitting problem and if the fitting function satisfies the assumption. The proposed algorithm can also solve multi-dimensional fitting problems with arbitrary constraints on the fitting coefficients matrix. The generalized problem includes popular models such as regression and the orthogonal Procrustes problem. The results of the computational experiment show that the proposed algorithms are faster than the state-of-the-art benchmarks for L1-norm regression subset selection and L1-norm regression over a sphere. Further, the relative performance of the proposed algorithm improves as data size increases.