ShapeFit: Exact location recovery from corrupted pairwise directions

TL;DR

ShapeFit introduces a convex optimization approach for exact location recovery from pairwise directions, robust to a constant fraction of adversarial corruptions, applicable in high-dimensional and three-dimensional settings.

Contribution

The paper presents a novel convex program for location recovery that guarantees exact recovery under adversarial corruption in high dimensions and for 3D, with rigorous theoretical proofs.

Findings

Exact recovery with high probability in high dimensions under Erdős-Rényi graph models.

Exact recovery in 3D with a constant fraction of corruptions per location.

The convex program is simple and computationally efficient.

Abstract

Let $t_1,\ldots,t_n \in \mathbb{R}^d$ and consider the location recovery problem: given a subset of pairwise direction observations $\{(t_i - t_j) / \|t_i - t_j\|_2\}_{i<j \in [n] \times [n]}$, where a constant fraction of these observations are arbitrarily corrupted, find $\{t_i\}_{i=1}^n$ up to a global translation and scale. We propose a novel algorithm for the location recovery problem, which consists of a simple convex program over $dn$ real variables. We prove that this program recovers a set of $n$ i.i.d. Gaussian locations exactly and with high probability if the observations are given by an \erdosrenyi graph, $d$ is large enough, and provided that at most a constant fraction of observations involving any particular location are adversarially corrupted. We also prove that the program exactly recovers Gaussian locations for $d=3$ if the fraction of corrupted observations at each location is, up to poly-logarithmic factors, at most a constant. Both of these recovery theorems are based on a set of deterministic conditions that we prove are sufficient for exact recovery.