Topological data analysis (TDA) is a framework for machine learning that synthesizes and combines machine learning algorithms to identify the shape of data. The technique is responsible for several major breakthroughs in our understanding of science and business. Gunnar Carlsson offers an overview of TDA’s mathematical underpinnings and its practical application through software.
TDA is based on a branch of pure mathematics called topology, which studies the notion of shape. Topology takes on two main tasks: the measurement of shape and the representation of shape. Both tasks are meaningful in the context of large, complex, and high-dimensional datasets. They permit you to measure shape-related properties within the data, such as the presence of loops, and they provide methods for creating compressed representations of datasets that retain features and reflect the relationships among points in the dataset in the form of a topological network or combinatorial graph—a very simple and intuitive object to work with using graph layout algorithms, which makes machine learning algorithms dramatically more effective.
Gunnar Carlsson is a professor of mathematics (emeritus) at Stanford University and is cofounder and president at Ayasdi, which is commercializing products based on machine intelligence and topological data analysis. Gunnar has spent his career devoted to the study of topology, the mathematical study of shape. Originally, his work focused on the pure aspects of the field, but in 2000 he began work on the applications of topology to the analysis of large and complex datasets, which led to a number of projects, notably a multi-university initiative funded by the Defense Advanced Research Projects Agency. He has taught at the University of Chicago, the University of California, San Diego, Princeton University, and, since 1991, Stanford University, where he has served as the chair of the Mathematics Department. He is also a founder of the ATMCS series of conferences focusing on the applications of topology, and is a founding editor of the Journal for Applied and Computational Topology. Gunnar is the author of over 100 academic papers and has given numerous addresses to scholarly meetings. He holds a BA in mathematics from Harvard and a PhD in mathematics from Stanford. He is married with three grown children.
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