A Framework for Exploring High-Dimensional Geometry

To extract useful information from high-dimensional geometric or structural data, we must find low-dimensional projections that are informative and interesting to look at. The conventional, manual-interaction methods used for this purpose are ineffective when the dimensionality of the data is high, or when the geometric models are complex. Standard methods for determining useful low-dimensional views are either limited to discrete data, or to geometric information embedded in at most three dimensions. Since geometric data embedded in dimensions above three have distinct characteristics and visualization requirements, finding directly applicable techniques is a challenge. We present a comprehensive framework for exploring high-dimensional geometric data motivated by projection pursuit techniques. Our approach augments manual exploration by generating sets of salient views that optimize a customizable family of geometry-sensitive measures. These views serve to reveal novel facets of complex manifolds and equations.