# Homotopy Parametric Simplex Method for Sparse Learning

## TL;DR

This paper introduces a homotopy parametric simplex method (PSM) for sparse learning that efficiently computes solution paths, provides high-precision solutions, and outperforms existing methods in various high-dimensional sparse data analysis tasks.

## Contribution

The paper presents a novel parametric simplex method that efficiently solves regularized linear programs for sparse learning, offering complete solution paths and high accuracy.

## Key findings

- PSM computes solution paths efficiently for various sparse learning models.
- PSM achieves high sparsity and reduces computational cost per iteration.
- Numerical experiments show PSM outperforms existing sparse learning methods.

## Abstract

High dimensional sparse learning has imposed a great computational challenge to large scale data analysis. In this paper, we are interested in a broad class of sparse learning approaches formulated as linear programs parametrized by a {\em regularization factor}, and solve them by the parametric simplex method (PSM). Our parametric simplex method offers significant advantages over other competing methods: (1) PSM naturally obtains the complete solution path for all values of the regularization parameter; (2) PSM provides a high precision dual certificate stopping criterion; (3) PSM yields sparse solutions through very few iterations, and the solution sparsity significantly reduces the computational cost per iteration. Particularly, we demonstrate the superiority of PSM over various sparse learning approaches, including Dantzig selector for sparse linear regression, LAD-Lasso for sparse robust linear regression, CLIME for sparse precision matrix estimation, sparse differential network estimation, and sparse Linear Programming Discriminant (LPD) analysis. We then provide sufficient conditions under which PSM always outputs sparse solutions such that its computational performance can be significantly boosted. Thorough numerical experiments are provided to demonstrate the outstanding performance of the PSM method.