Data Analysis using Computational Topology and Geometric Statistics

Mathematical scientists of diverse backgrounds are being asked to apply the techniques of their specialty to data which is greater in both size and complexity than that which has been studied previously. Large, high-dimensional data sets, for which traditional linear methods are inadequate, pose challenges in representation, visualization, interpretation and analysis. A common finding is that these massive data sets require the development of new theory and that these advances are dependent on increasing technical sophistication. Two such data-analytic techniques that have recently developed independently of each other have come to the fore, namely, Geometric Statistics and Computational Topology. Although the former uses geometric arguments, while the latter uses algebraic-topological arguments, and hence they appear disparate, there is substantial commonality and overlap just as in the more traditional overlap between geometry and topology. Thus the purpose of this workshop is to bring together these two research directions and explore their overlap, particularly in the service of statistical data analysis. A standard paradigm assumes that the data comes from some underlying geometric structure, such as a curved submanifold or a singular algebraic variety. The observed data is obtained as a random sample from this space, and the objective is to statistically recover features of the underlying space and/or the distribution that generated the sample. In Geometric Statistics one uses the underlying Riemannian structure to recover quantitative information concerning the probability distribution and/or functionals thereof. The idea is to extend statistical estimation techniques to functions over Riemannian manifolds, utilizing spectral methods adapted to the Riemannian structure. One then considers the magnitude of the statistical accuracy of these estimators. Considerable progress has been achieved in terms of optimal estimation in the minimax sense. These ideas have far reaching implications in the analysis of high-dimensional data such as, for example, in astronomy, biomechanics, medical imaging, microwave engineering and texture analysis. In Computational Topology, one attempts to recover more qualitative global features of the underlying data instead, such as connectedness, or the number of holes, or the existence of obstructions to certain constructions, based upon the random sample. In other words, one hopes to recover the underlying topology. An advantage of topology is that it is stable under deformations and thus insensitive to errors introduced in the sampling. A combinatorial construction such as the alpha complex or the Čech complex converts the discrete data into an object for which it is possible to compute the topology. However, it is quickly apparent that such a