The Shape of Things: Topological Data Analysis

Abstract

An interesting feature of much modern Big Data is that the data we collect, or the data we want to analyze, are not necessarily in the traditional matrix or array form familiar from our textbooks. They may be coerced to such a format for relative ease of analysis, but this is not a strong justification. Past columns have explored new methods that exploit the natural structure of such data sets more directly. Topological data analysis (TDA) is one such method. Much daunting mathematics lies behind the methods of TDA, but it is possible to gain an idea and understanding of the approach and its potential usefulness even without a deep dive into the intricacies of topology, homology classes, and the like. In fact, the basic idea is quite simple: to study data through their low-dimension topological features, which translate into connected components (dimension 0), loops (dimension 1), and voids (dimension 2). Higher dimensions do exist, but often do not contain much useful information. For threedimensional data, up to the second dimension topological features can be considered at most. A good analogy to make the meaning of these features concrete is a piece of Swiss cheese. The piece of cheese itself is one connected component. The holes that are apparent on the The Shape of Things: Topological Data Analysis