AbstractDifferential geometry is playing an increasing role in manifold learning, and generative modeling in particular. Geometry provides us with well-defined and well-behaved tools for interpolation and statistical analysis on the learned manifolds, and provides a principled solution to the identifiability problem that plagues many generative models. While the geometric approach is elegant it comes with a steep learning curve as the literature is developed from a mathematical rather than applied perspective. In this tutorial we first develop the classic differential geometry needed to understand deterministic manifolds, and then show how this applies to the stochastic setting. We show how to turn the mathematical concepts into simple algorithms that allow for principled data analysis over learned manifolds. Importantly, we require little more mathematical background from the audience than knowledge of Taylor expansions.
I am a professor of Geometry in Machine Learning at the Technical University of Denmark. I received his PhD in computer science from the University of Copenhagen in 2011. During my PhD I spend 6 months as a visiting scholar at UC Berkeley working with Ruzena Bajcsy. Prior to pursuing a PhD I worked as a ”digital lumberjack” in the startup Dralle A/S. I was a postdoc for two years at Perceiving Systems at the Max Planck Institute for Intelligent Systems working with Michael Black. In 2013, I was the sole computer science recipient of the Sapere Aude Research Talent award from the Danish Council for Independent Research, and in 2016 I was the sole computer science Villum Young Investigator. In 2017 I was further awarded a starting grant from the European Research Council. In 2018, I joined the Young Scientists community under the World Economic Forum, and was in the process named one of ‘10 of the most exciting young scientists working in the world today.’ My research interest lie in the span of geometry and statistics. I develop machine learning techniques using geometric constructions, and work on the related numerical challenges. I am particularly interested in random geometries as they naturally appear in learning.