Learning polytopes with fixed facet directions

TL;DR

This paper presents a method for reconstructing polytopes with fixed facet directions using support function evaluations, involving convex quadratic programming, geometric analysis, and an algorithm with convergence guarantees.

Contribution

It introduces a quadratic programming approach for polytope reconstruction with fixed facet directions and analyzes conditions for uniqueness and convergence.

Findings

Least-squares estimate is a convex quadratic program.

Provides a combinatorial characterization for uniqueness.

Algorithm converges to the true shape with increasing data.

Abstract

We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction in this case. We provide an algorithm that, under mild assumptions, converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed.