SATA: Stochastic Algebraic Topology and Applications
Abstract
The Stochastic Algebraic Topology and Applications (SATA) project aims to exploit recent advances in the complementary areas of topology and stochastic processes to tackle a wide range of data analytic problems of broad importance. Treating data topologically is crucial in scenarios in which it is important to detect, localize, and perhaps perform an initial classification of objects without attempting to completely characterize them. Adding a stochastic element allows for the almost pervasive situation in which the data itself is imperfectly observed due to the presence of background noise. As current probabilistic and statistical methodology is ill suited to detect such qualitative structures, the project aims to develop generic stochastic models whose topological structures are amenable to mathematical analysis, as a first step towards implementation of a broader, more quantitative program. Core topics include random functions on manifolds, random manifolds created by random embeddings, and random manifolds arising in machine learning, along with their theoretical and practical interplay. Secondary topics include the analysis of associated algorithms, and the topological understanding of random spaces that arise in particular stochastic models. We have also studied implementation and application of these ideas on some problems coming from engineering and physics. Initial results were obtained regarding the statistics of random functions, the application and analysis of Morse theory in random settings, and on the complexity of the basic topological inference problems in data analysis. Significant progress has been made in the areas of the Statistics of Random Functions; Morse Theory, Critical Points, Betti Numbers and Random Complexes; and Random Manifolds and Random Embeddings.
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 01, 2012
- Accession Number
- ADA627871
Entities
People
- Jonathan E. Taylor
- Robert Adler
- Shmuel Weinberger
- Yuliy Baryshnikov