Sylvester-Gallai type theorems for approximate collinearity

TL;DR

This paper extends Sylvester-Gallai theorems to approximate collinearity in noisy settings, showing that points with many nearly collinear triples lie near a low-dimensional space, and introduces the concept of stable locally correctable codes.

Contribution

It provides a stable variant of Sylvester-Gallai theorems for noisy data and introduces stable locally correctable codes, proving their non-existence for constant query complexity.

Findings

Approximately collinear triples imply low-dimensional structure

Stable LCCs with constant queries do not exist

Extension of Sylvester-Gallai theorems to noisy settings

Abstract

We study questions in incidence geometry where the precise position of points is `blurry' (e.g. due to noise, inaccuracy or error). Thus lines are replaced by narrow tubes, and more generally affine subspaces are replaced by their small neighborhood. We show that the presence of a sufficiently large number of approximately collinear triples in a set of points in d dimensional complex space implies that the points are close to a low dimensional affine subspace. This can be viewed as a stable variant of the Sylvester-Gallai theorem and its extensions. Building on the recently found connection between Sylvester-Gallai type theorems and complex Locally Correctable Codes (LCCs), we define the new notion of stable LCCs, in which the (local) correction procedure can also handle small perturbations in the euclidean metric. We prove that such stable codes with constant query complexity do not exist. No impossibility results were known in any such local setting for more than 2 queries.