Walks on hyperplane arrangements and optimization of piecewise linear functions

TL;DR

This paper introduces an exact iterative algorithm for minimizing certain convex piecewise linear functions defined on hyperplane arrangements, with applications in robust regression and permutation-based residuals.

Contribution

It presents a novel method that efficiently finds improving directions in complex arrangements using Birkhoff's Theorem, extending applicability beyond rank estimators.

Findings

Algorithm effectively minimizes convex functions on hyperplane arrangements.

Uses Birkhoff's Theorem to handle exponential complexity in arrangements.

Applicable to robust regression and permutation-based residuals.

Abstract

We propose an exact iterative algorithm for minimization of a class of continuous cell-wise linear convex functions on a hyperplane arrangement. Our particular setup is motivated by evaluation of so-called rank estimators used in robust regression, where every cell of the underlying arrangement corresponds to a permutation of residuals (and we also show that the class of function for which the method works is more general). The main obstacle in the construction of the algorithm is how to find an improving direction while standing in a point incident with exponentially many cells of the arrangement. We overcome this difficulty using Birkhoff Theorem which allows us to express the cone of improving directions in the exponential number of cells using a linear system with quadratic number of variables only.