Abstract:
Consider a finite sample of points on a manifold embedded in Euclidean space. We'll address the following issues.
1. How to infer, from the sample, intrinsic distances that are meaningful to understand the cloud of points.
2. How to use these estimated distances to infer the geometry and topology of the manifold (manifold learning).
3. How to 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 i.i.d. points with density f supported on the manifold, the metric space defined by the sample endowed with a computable metric given by an Euclidean FPP model, converges in the sense of Gromov–Hausdorff to an intrinsic object (i.e. independent of the embedding in Euclidean space).
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.
Biography:
Pablo Groisman is Visiting Associate Professor of Mathematics at NYU Shanghai. He is also a Principal Researcher at CONICET-Argentina and Professor at University of Buenos Aires. He holds a PhD from University of Buenos Aires, Argentina.
Professor Groisman's research interests include stochastic processes, interacting particle systems, conditioned evolution and probabilistic aspects of machine learning. His work has appeared in journals such as Communications in Pure and Applied Mathematics, Stochastic Processes and their Applications, Annales de l'Institut Henri Poincaré, Bernoulli, Communications in Partial Differential Equations and Electronic Journal of Probability.
Professor Groisman received the “Manuel Sadovsky” award in Mathematics from the Argentinian National Academy of Sciences. For more about Professor Groisman visit his page here.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai