Boundary Detection for High-Dimensional Point Clouds

Abigail Hickok, Valerio Galanti, Osama Al Dalahmah, Andrew J. Blumberg

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.