Henry Adams

DSCI 475: Topological Data Analysis

                        

Colorado State University, Spring 2021

Instructor: Henry Adams
Email: henry dot adams at colostate dot edu

Lectures: TR 2:00-3:15pm Mountain Time on Zoom
Textbook: None required

Overview: Topological techniques for analyzing high-dimensional or complex data. The shape of data may reflect patterns within; e.g. connected components may correspond to groupings, or a circular shape may correspond to periodic behavior. Topics include clustering, dendrograms, a visual introduction to topology, data modeling and visualization, and selected topics from nonlinear dimensionality reduction, graph-based models of data, Reeb graphs, multi-scale approaches to data, and persistent homology.

Syllabus: Here is the course syllabus.

Videos



Applied Topology 1: Datasets have shape


Applied Topology 2: Topology and homotopy equivalences


Applied Topology 3: A punctured torus is homotopy equivalent to a figure eight


Applied Topology 4: An introduction to the torus and Klein bottle


Applied Topology 5: Spheres in all dimensions


Applied Topology 6: Homology


Applied Topology 7: How do you recover the shape of a dataset?


Applied Topology 8: An introduction to persistent homology


Applied Topology 9: Spaces of 3x3 natural image patches


Applied topology 10: Unsupervised vs supervised learning


Applied topology 11: Clustering and K-means clustering


Applied topology 12: Hierarchical clustering and single-linkage clustering


Applied topology 13: The problem of chaining in single-linkage clustering


Applied topology 14: Čech and Vietoris-Rips simplicial complexes


Applied topology 15: Introduction to a software tutorial for persistent homology and Ripser


Applied topology 16: Sublevelset persistent homology


Applied topology 17: Persistence and local geometry, Part A


Applied topology 18: Persistence and local geometry, Part B


Applied topology 19: Linear dimensionality reduction - Principal Component Analysis (PCA), Part I


Applied topology 20: Linear dimensionality reduction - Principal Component Analysis (PCA), Part II


Applied topology 21: Nonlinear dimensionality reduction - Isomap, Part I


Applied topology 22: Nonlinear dimensionality reduction - Isomap, Part II


Applied topology 23: Paper Introduction: Coordinate-free coverage in sensor networks


Applied topology 24: Evasion paths in mobile sensor networks, Part I


Applied topology 25: Evasion paths in mobile sensor networks, Part II


Applied topology 26: Evasion paths in mobile sensor networks, Part III


Applied topology 27: Evasion paths in mobile sensor networks, Part IV

Schedule

Date Class Topic Remark

Jan 19 Course overview [Logistics]
Jan 21 Topology and data [Slides, Video]

Jan 26 A visual introduction to topology and homotopy equivalences
Jan 28 A visual introduction to homology

Feb 2 A visual introduction to persistent homology
Feb 4 Clustering, k-means clustering

Feb 9 Hierarchical clustering and dendrograms
Feb 11 Point cloud persistent homology [Video]

Feb 16 Case studies: Point cloud persistent homology
Feb 18 Sublevelset persistent homology [Video]

Feb 23 Case studies: Sublevelset persistent homology
Feb 25 Dimensionality reduction: Principal Component Analysis (PCA) [Tutorial]

Mar 2 Nonlinear dimensionality reduction
Mar 4 Reeb graphs and the mapper algorithm [Video]

Mar 9 Coverage problems in sensor networks [Slides]
Mar 11 Coverage problems in sensor networks

Software resources