Methods detect boundaries in noisy high dimensional data clouds
Boundary Detection for High-Dimensional Point Clouds
Machine LearningComputational Geometry
Summary
Detecting edges or boundaries in high-dimensional data is hard, especially when the data is noisy and unevenly spread out. The authors offer two new ways to find these boundaries: one that simplifies the data locally to find a dividing line, and another that checks how evenly data points are spread in different directions. They tested these methods on fake data and real-life examples, including spotting tumor edges in biological data and analyzing AI model behaviors.
What this means in practice
- •For biomedical data analysts: Identify tumor boundaries in spatial gene expression data using robust boundary detection methods.
- •For machine learning engineers: Analyze activation patterns in large language models by detecting geometric boundaries in high-dimensional activation spaces.
Authors
Abigail Hickok, Valerio Galanti, Osama Al Dalahmah, Andrew J. Blumberg
Abstract
We introduce two new methods for boundary detection in high-dimensional point cloud data, intended to work in situations with non-uniform density, high ambient dimension, and substantial noise. Our first method fits a separating hyperplane on local neighborhoods after doing local dimension reduction---this method works best when one can reliably estimate the intrinsic dimension of the data. Our second method measures the directional uniformity of local neighborhoods, and has the best overall performance on synthetic data. In addition to pointwise numerical boundary scores, our algorithms yield an estimated normal vector at each boundary point. We validate our methodology on synthetic data as well as two real examples: we analyze spatial transcriptomics data, where we show that we can detect the boundary of a tumor, and we also study LLM activation geometry.