Traveling the Robot Learning Manifold: A Tale of Geometries and Inductive Biases

Robot motions are fundamentally governed by non-Euclidean geometries. Robot state spaces are non-linear manifolds, various robotic variables exhibit distinct geometric characteristics, and collected data often resides in curved spaces. Despite that many problems naturally lend themselves to geometric interpretations, these underlying structures are often relegated to the background. In the modern era of data-driven robotics, this omission creates a critical gap, as many contemporary learning algorithms operate on representations that inadvertently ignore or distort the natural geometries of robotics.

In this lecture, I will discuss how differential geometry — arising from data structure, physics, and prior knowledge — provides a rigorous framework to construct representations and learning algorithms that respect and exploit these natural geometries. I will demonstrate that the performance of diverse algorithms is significantly enhanced by considering the intrinsic geometric characteristics of data, show that the complex dynamic properties of robots are more elegantly learned and accurately controlled within physics-based geometric configuration spaces, and illustrate that imposing structured geometry on latent spaces allows for richer representations. Ultimately, I will highlight that explicitly encoding differential-geometric structures lead to improved performance, more data-efficient learning, sound guarantees, and robust generalization, ensuring that the data-driven robots of tomorrow remain mathematically and physically grounded in the real world.