Diffusion Variational Autoencoders

A standard Variational Autoencoder, with a Euclidean latent space, is structurally incapable of capturing topological properties of certain datasets. To remove topological obstructions, we introduce Diffusion Variational Autoencoders with arbitrary manifolds as a latent space. A Diffusion Variational Autoencoder uses transition kernels of Brownian motion on the manifold. In particular, it uses properties of the Brownian motion to implement the reparametrization trick and fast approximations to the KL divergence. We show that the Diffusion Variational Autoencoder is capable of capturing topological properties of synthetic datasets. Additionally, we train MNIST on spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a natural dataset like MNIST does not have latent variables with a clear-cut topological structure, training it on a manifold can still highlight topological and geometrical properties.Comment: 10 pages, 8 figures Added an appendix with derivation of asymptotic expansion of KL divergence for heat kernel on arbitrary Riemannian manifolds, and an appendix with new experiments on binarized MNIST. Added a previously missing factor in the asymptotic expansion of the heat kernel and corrected a coefficient in asymptotic expansion KL divergence; further minor edit

Optimal rates of convergence for persistence diagrams in Topological Data Analysis

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results

The Topology of Probability Distributions on Manifolds

Let $P$ be a set of $n$ random points in $R^d$, generated from a probability measure on a $m$-dimensional manifold $M \subset R^d$. In this paper we study the homology of $U(P,r)$ -- the union of $d$-dimensional balls of radius $r$ around $P$, as $n \to \infty$, and $r \to 0$. In addition we study the critical points of $d_P$ -- the distance function from the set $P$. These two objects are known to be related via Morse theory. We present limit theorems for the Betti numbers of $U(P,r)$, as well as for number of critical points of index $k$ for $d_P$. Depending on how fast $r$ decays to zero as $n$ grows, these two objects exhibit different types of limiting behavior. In one particular case ($n r^m > C \log n$), we show that the Betti numbers of $U(P,r)$ perfectly recover the Betti numbers of the original manifold $M$, a result which is of significant interest in topological manifold learning

Statistical Inference using the Morse-Smale Complex

The Morse-Smale complex of a function $f$ decomposes the sample space into cells where $f$ is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize, and compare multivariate functions. In this paper, we present some statistical results on estimating Morse-Smale complexes. This allows us to derive new results for two existing methods: mode clustering and Morse-Smale regression. We also develop two new methods based on the Morse-Smale complex: a visualization technique for multivariate functions and a two-sample, multivariate hypothesis test.Comment: 45 pages, 13 figures. Accepted to Electronic Journal of Statistic

Towards Stratification Learning through Homology Inference

A topological approach to stratification learning is developed for point cloud data drawn from a stratified space. Given such data, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving a stratification for each radius level. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata, and we prove a result which guarantees the correctness of our clustering, given certain topological conditions; some geometric intuition for these topological conditions is also provided. Our correctness result is then given a probabilistic flavor: we give bounds on the minimum number of sample points required to infer, with probability, which points belong to the same strata. Finally, we give an explicit algorithm for the clustering, prove its correctness, and apply it to some simulated data.Comment: 48 page

Quantum Gravity as Topological Quantum Field Theory

The physics of quantum gravity is discussed within the framework of topological quantum field theory. Some of the principles are illustrated with examples taken from theories in which space-time is three dimensional.Comment: 23 pages, amstex, JMP special issue (deadline permitting). (Text not changed