1. Omer Bobrowski Mathematics Department Duke University Joint work with Sayan Mukherjee 6/19/13

2. The Topology Of Probability Distributions on Manifolds 6/19/13 2 2 Contents Distributions on Compact Manifolds Introduction: Noise and Topology Current & Future Work

3. The Topology Of Probability Distributions on Manifolds 6/19/13 3 3 Contents Distributions on Compact Manifolds Introduction: Noise and Topology Current & Future Work

4. The Topology Of Probability Distributions on Manifolds 6/19/13 4 4 Introduction: Noise and Topology Motivation - Manifold Learning Motivation - Manifold Learning Example: Question: Question: How to choose ? Objective: Objective: Inferring topological features of an unknown space from random samples Objective: Objective: Inferring topological features of an unknown space from random samples More samplesSmallerFiner resolution

5. The Topology Of Probability Distributions on Manifolds 6/19/13 5 5 Introduction: Noise and Topology Homology (Betti numbers) The Topology of Noise The Topology of Noise Noise = a random point cloud Objects of study: Critical points 1. Union of Balls - bicycle 2. Distance Function - cat

6. The Topology Of Probability Distributions on Manifolds 6/19/13 6 6 Introduction: Noise and Topology Betti Numbers Betti Numbers - a topological space - the number of connected components - the number of k-dimensional “holes” or “cycles” ( )Examples:

7. The Topology Of Probability Distributions on Manifolds 6/19/13 7 7 Introduction: Noise and Topology The Distance Function The Distance Function For a finite set The Distance Function: Example: Goal: Study critical points of, for a random is non-differentiable Note:

8. The Topology Of Probability Distributions on Manifolds 6/19/13 8 8 Introduction: Noise and Topology The Morse Index The Morse Index - a smooth (Morse) function - a critical point of f ( ) = # negative eigenvalues of the Hessian at c = # “independent decreasing directions” increasing decreasing

9. The Topology Of Probability Distributions on Manifolds 6/19/13 9 9 Introduction: Noise and Topology Critical Points of the Distance Function Critical Points of the Distance Function Index k critical points are “generated” by subsets of k+1 points increasing decreasing Generated by 1 point Generated by 2 points Generated by 3 points

10. The Topology Of Probability Distributions on Manifolds 6/19/13 10 Introduction: Noise and Topology Morse Theory in One Slide Morse Theory in One Slide Consider the sublevel sets, as Homology changes only at critical levels Critical point with index (generate a hole) (kill a hole)

11. The Topology Of Probability Distributions on Manifolds 6/19/13 11 Contents Distributions on Compact Manifolds Introduction: Noise and Topology Current & Future Work

12. The Topology Of Probability Distributions on Manifolds 6/19/13 12 Distributions on Compact Manifolds - a compact smooth m-dimensional manifold, embedded in - a probability density on - a set of iid points, with density Setup Setup = volume form on - union of d-dimensional balls - distance function in

13. The Topology Of Probability Distributions on Manifolds 6/19/13 13 Distributions on Compact Manifolds Setup & Goals Setup & Goals - a set of random points in - the k -th Betti number of - the number of index- k critical points p of with Limiting behavior of as Link and compare between and Setup: Goal:

14. The Topology Of Probability Distributions on Manifolds 6/19/13 14 Distributions on Compact Manifolds Example Example Union of Balls - bicycle Distance Function - cat Morse Theory

15. The Topology Of Probability Distributions on Manifolds 6/19/13 15 Distributions on Compact Manifolds Previous Work Previous Work M. Kahle – Random geometric complexes M.Kahle & E. Meckes - Limit theorems for Betti numbers of random simplicial complexes O. B. & R.J. Adler - Distance Functions, Critical Points, and Topology for Some Random Complexes P. Niyogi, S. Smale & S. Weinberger – Finding the Homology of Submanifolds with High Confidence from Random Samples The Euclidean Setting The Manifold Setting

16. The Topology Of Probability Distributions on Manifolds 6/19/13 16 Distributions on Compact Manifolds Three Main Regimes Three Main Regimes SubcriticalCriticalSupercritical average number of points in a geodesic -ball

17. The Topology Of Probability Distributions on Manifolds 6/19/13 17 Distributions on Compact Manifolds Subcritical Phase Subcritical Phase

18. The Topology Of Probability Distributions on Manifolds 6/19/13 18 Distributions on Compact Manifolds Subcritical Phase Subcritical Phase Summary: Mostly small disconnected particles (aka “dust”) Very few holes Critical points mostly kill holes Also available: limit variance and distribution

19. The Topology Of Probability Distributions on Manifolds 6/19/13 19 Distributions on Compact Manifolds Critical Phase Critical Phase

20. The Topology Of Probability Distributions on Manifolds 6/19/13 20 Distributions on Compact Manifolds Critical Phase Critical Phase Summary: (qualitative behavior only) (& limits for variance and distribution) Many components and holes The Euler Characteristic:

21. The Topology Of Probability Distributions on Manifolds 6/19/13 21 Distributions on Compact Manifolds Supercritical Phase Supercritical Phase Summary: Highly connected, almost covered Requires (& limits for variance and distribution) No general results for the Betti numbers

22. The Topology Of Probability Distributions on Manifolds 6/19/13 22 Distributions on Compact Manifolds Supercritical Phase Supercritical Phase The connected regime:

23. The Topology Of Probability Distributions on Manifolds 6/19/13 23 Contents Distributions on Compact Manifolds Introduction: Noise and Topology Current & Future Work

24. The Topology Of Probability Distributions on Manifolds 6/19/13 24 Current & Future Work Limits for the Betti number in the critical phase and connection to persistent homology ( ) Recovering the topology of a manifold from noisy samples [Niyogi, Smale, Weinberger - A Topological View of Unsupervised Learning from Noisy Data] [Adler, B, Weinberger - Crackle: The Persistent Homology of Noise]

26. Introduction: Noise and Topology The Čech Complex The Čech Complex Take a set of vertices P (0-simplexes) Draw balls with radius e Intersection of 2 balls  an edge (1-simplex) Intersection of 3 balls  a triangle (2-simplex) Intersection of n balls  a (n-1)-simplex

27. Distributions on Compact Manifolds Supercritical Phase Supercritical Phase = # critical points p with The connected regime: No “small” critical points outside = # critical points p with = volume of a unit ball in

28. Distributions on Compact Manifolds Supercritical Phase Supercritical Phase We can show that if, with then covers with probability (or almost surely) Then: (NSW) (no critical points)