Advanced Topics in Topological Data Analysis: Research Highlights
Explore cutting-edge research in Topological Data Analysis (TDA) through topics such as Mapper Stability Measures, Deep Learning advancements, Vectorization techniques, Generalized Persistence modeling, Stochastic Aspects analysis, Formal Properties of Persistence, and Complement Problems in TDA applications.
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Presentation Transcript
Lecture 10: The Future May 17, 2019 Gunnar Carlsson Stanford University
Mapper Stability Measures Use more flexible coverings, Voronoi, Alpha Complex, Witness Different clustering strategies Persistent homology to select useful topological models Measures of how well Mapper represents a metric space
Deep Learning Better features for images Better understanding of text, especially word embeddings Explore the effect of architectures on various things, especially generalization Find more methods for incorporating topology of feature space
Vectorization Complete analysis of tropical functions on one dimensional persistent diagrams Extend tropical functions to multidimensional case
Generalized Persistence Find better ways to represent multidimensional persistence Find descriptions of multidimensional persistence that include both ordinary persistence directions as well as zig-zag ones Use 2D persistence to develop a tool that can be used easily which uses superlevel set persistence for density and distance scale
Stochastic Aspects Develop persistence profiles for metric measure spaces Prove stability theorems for them in presence of Gaussian noise
Formal Properties of Persistence K nneth formula Improve Mayer-Vietoris
Complement Problems Parametrized embedding calculus Unstable Adams spectral sequence for parametrized situation Adapt to create initial guesses for optimization problems
