- Pablo Groisman, Universidad de Buenos Aires
- Learning distances to learn topologies, to learn dynamical systems, to learn from chaos.
- 05/12/2022
- 2:30 PM - 3:30 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Olga Turanova (turanova@msu.edu)
Consider a finite sample of points on a manifold embedded in Euclidean space. We'll address the following issues.
1. How we can infer, from the sample, intrinsic distances that are meaningful to understand the data.
2. How we can use these estimated distances to infer the geometry and topology of the manifold (manifold learning).
3. How we can use this knowledge to validate dynamical systems models for chaotic phenomena.
4. Time permitting, we will show applications of this machinery to understand data from the production of songs in canaries.
We will prove that if the sample is given by iid points with density f supported on the manifold, the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges in the sense of Gromov–Hausdorff. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. Then we'll apply this result to estimate the topology of the manifold by constructing intrinsic persistence diagrams (an estimator of the homology of the manifold), which are less sensitive to the particular embedding of the manifold in the Euclidean space and to outliers. We'll also discuss how to use these tools to validate (or to refute) models for chaotic dynamical systems, with applications to the study of birdsongs.
