Monday, October 19th
Speaker: Ryan Robinett (@robinett)
Realizing Riemannian Manifolds from Empirical Point Clouds
The widely-accepted hypothesis that empirical data distributions can often be modeled as lower-dimensional manifolds embedded in a higher dimensional space is the foundation of virtually all dimensionality reduction techniques; e.g., principal components analysis (PCA) assumes that the distribution can be well-approximated by a linear submanifold, while methods like t-distributed stochastic neighborhood embedding (tSNE) and locally linear embedding (LLE) allow for more varied topology at the cost of metric information. While current dimensionality reduction methods exploit the locally Euclidean structure of manifolds, they all fail to create actual manifold representations: representations in terms of coordinate charts and transition maps, possibly with an additional Riemannian metric. Such representation allows for the rich library of techniques from Differential Geometry to be applied to empirical data distributions, allowing for more meaningful leverage of the Manifold Hypothesis. We present a technique for creating local coordinate charts, transition maps between coordinate charts, and local forms of the Riemannian metric in order to create non-degenerate manifold representations from empirical point clouds.