Fodava Review Presentation

Stanford GroupG. Carlsson, L. Guibas

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

Mapper: Breast Cancer Example (Carlsson, Nicolau)

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

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.

Circular Coordinates

• Homotopy classes of maps to circle are correspond to 1d cohomology

• Hodge theory analogue gives a map (De Silva, Morozov, Vejdemo-Johansson)

Circular Coordinates - Rotating Cube Example

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

Heat Kernel Methods for Shape Analysis

• Invariant under isometry• Finds features

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.