Geometric anomaly detection in data.

Area of Science:

• Data science
• Topology
• Computational geometry

Background:

• High-dimensional data analysis often assumes data lies on a single manifold.
• Standard methods struggle with datasets that deviate from this manifold hypothesis.
• Detecting anomalies and interfaces in complex data remains a significant challenge.

Purpose of the Study:

• To develop a systematic framework for detecting interfaces and anomalies in datasets that may not conform to the manifold hypothesis.
• To provide a method for partitioning complex data into regions that can be approximated by individual manifolds.
• To identify singular regions and intersections within high-dimensional data.

Main Methods:

• Computing the local topology of small regions around each data point.
• Partitioning datasets into disjoint classes based on local topological features.
• Approximating each data partition with a single manifold of potentially different intrinsic dimensions.

Main Results:

• Successfully identified the intersection of two surfaces in a 24-dimensional cyclo-octane conformation space.
• Located all self-intersections of a Henneberg minimal surface in 3D space.
• Demonstrated the ability to discover singular regions even when data points are not sampled precisely from singularities.

Conclusions:

• The proposed local topology framework effectively detects interfaces and anomalies in data that violate the manifold hypothesis.
• The method allows for the stratification of complex datasets into simpler, manifold-approximated components.
• The computational approach is locally based, enabling efficient parallel processing for large datasets.