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Exploring Data Representation through Mapper: A Geometric Approach

This presentation focuses on the Mapper algorithm, which represents data through graphs, prioritizing geometry while reducing sensitivity to it. It showcases applications, including breast cancer analysis and RNA folding. The talk delves into parametric mappings to simpler spaces, highlighting methods like Principal Components and Hodge theory. Additionally, it discusses the Heat Kernel Methods for shape analysis, offering insights into shape matching and the invariance of features under isometry. Overall, it emphasizes the development of functorial methods for data relationships and fusion.

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Exploring Data Representation through Mapper: A Geometric Approach

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  1. Fodava Review Presentation Stanford Group G. Carlsson, L. Guibas

  2. Mapper • Represents data by graphs rather than scatterplots or clusters • Retains geometry, but is not too sensitive to it • Geometric features yield interesting information about the data

  3. Mapper: Breast Cancer Example (Carlsson, Nicolau) t

  4. Mapper: RNA folding (Carlsson,Guibas, et al)

  5. Parametrization • How to get a tangible feel for features • Get maps to simpler spaces • Principal Components maps to linear spaces • Clustering maps to discrete spaces • Mapper maps to simplicial complexes • Could try to map to circles, trees, etc.

  6. Circular Coordinates • Homotopy classes of maps to circle are correspond to 1d cohomology • Hodge theory analogue gives a map (De Silva, Morozov, Vejdemo-Johansson)

  7. Circular Coordinates - Rotating Cube Example

  8. Heat Kernel Methods for Shape Analysis (Guibas et al) • Heat kernel attached to metric produces for each point of a shape a real value function on the real line. • Produces a faithful signature for the shape • Can be used to match shapes

  9. Heat Kernel Methods for Shape Analysis • Invariant under isometry • Finds features

  10. New Directions • Critical problem: compare shapes of data, find mappings • Related to Data Fusion • Need to develop “functorial” methods for study of data • Will make clear relationships among subsets of data, different data sets.

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