Geometric anomaly detection in data

The quest for low-dimensional models which approximate high -dimensional data is pervasive across the physical, natural, and social sciences. The dominant paradigm underlying most standard modeling techniques assumes that the data are concentrated near a single unknown manifold of relatively small intrinsic dimen-sion. Here, we present a systematic framework for detecting interfaces and related anomalies in data which may fail to sat-isfy the manifold hypothesis. By computing the local topology of small regions around each data point, we are able to par-tition a given dataset into disjoint classes, each of which can be individually approximated by a single manifold. Since these manifolds may have different intrinsic dimensions, local topol-ogy discovers singular regions in data even when none of the points have been sampled precisely from the singularities. We showcase this method by identifying the intersection of two sur-faces in the 24-dimensional space of cyclo-octane conformations and by locating all of the self-intersections of a Henneberg min-imal surface immersed in 3-dimensional space. Due to the local nature of the topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors.