Topology, Big Data and Optimization

Abstract

The idea of using geometry in learning and inference has a long history going back to canonical ideas such as Fisher information, Discriminant analysis, and Principal component analysis. The related area of Topological Data Analysis (TDA) has been developing in the last decade. The idea is to extract robust topological features from data and use these summaries for modeling the data. A topological summary generates a coordinate-free, deformation invariant and highly compressed description of the geometry of an arbitrary data set. Topological techniques are well-suited to extend our understanding of Big Data. These tools do not supplant existing techniques, but rather provide a complementary viewpoint to existing techniques. The qualitative nature of topological features do not give particular importance to individual samples, and the coordinate-free nature of topology generates algorithms and viewpoints well suited to highly complex datasets. With the introduction of persistence and other geometric-topological ideas we can find and quantify local-to-global properties as well as quantifying qualitative changes in data.