How Many Directions Determine a Shape: A VC Theory for Modulii Spaces of Shapes
Talkat Robust and High-Dimensional Statistics (Simons Institute workshop, 2018) · 2018-10-29..11-02·
summary
Proves that turning meshes into direction-wise topological summaries is information-preserving: the Euler characteristic transform and persistent homology transform are injective when using all directions, while empirically one needs about 162 directions in 2D and 700 directions in 3D. Gives a VC-style sampling bound that scales with a lower curvature Δ and a cap on critical points, smooths ECC into L2 for functional modeling, builds Gaussian-process/likelihood models on the transforms, and demonstrates gains on 106 primate heel bones and 92 glioblastoma MRIs where topology helps explain survival and localize discriminative regions.
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chapters (29)
- 0:00Introduction
- 2:10Evolutionary Trees as Broccoli
- 3:05Models of Shapes
- 3:42Shape Spaces
- 6:29Divyamorphism Paste
- 7:43Goals
- 8:19What is a shape
- 9:05Definition of a shape
- 9:54Constructible sets and functions
- 10:24Euler characteristic
- 11:14Euler characteristic curve
- 11:42Persistent homology
- 12:10Persistence diagram
- 14:13Euler characteristic transformed
- 16:16Robustness
- 17:12Distances between shapes
- 17:59Is this transform invertible
- 18:44Shapiros Inversion Theorem
- 19:19How Many Directions Determine a Shape
- 21:26Two Constraints
- 21:56Resolution Parameter
- 24:56Probability Models
- 26:48Gaussian Process
- 27:01Real Data
- 27:50Doug
- 33:15Data Set
- 35:37Diffusion of Shapes
- 37:12Practical Applications
- 38:17Random Mixture Model
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