Geometric anomaly detection in data

TL;DR

This paper introduces a local topological approach for detecting interfaces and anomalies in high-dimensional data by stratifying data into submanifolds, effectively identifying singular regions and intersections.

Contribution

The paper presents a novel topological method for anomaly detection that can identify singularities and intersections in high-dimensional datasets, even without direct sampling from those regions.

Findings

Successfully identified surface intersections in 24D space

Detected self-intersections of a minimal surface in 3D

Algorithm is highly parallelizable and scalable

Abstract

This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can optimally stratify a given dataset into clusters, each of which is in turn well-approximable by a suitable submanifold of the ambient space. Since these approximating submanifolds might have different dimensions, we are able to detect non-manifold like singular regions in data even when none of the data points have been sampled from those singularities. We showcase this method by identifying the intersection of two surfaces in the 24-dimensional space of cyclo-octane conformations, and by locating all the self-intersections of a Henneberg minimal surface immersed in 3-dimensional space. Due to the local nature of the required topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors.