Langevin on Manifolds

April 1, 2025
By Aathreya Kadambi

Unexpectedly, I saw the exponential map in the ML for Bio and Chem class I’m taking! I was pretty excited to see it. Not only that, but it made me think about the inverse of this map, the “log”. While Lee doesn’t explicitely mention a log map (at least not yet), I think it’s a nice idea. As long as we are in a neighborhood where the exponential map is invertible (or perhaps geodesically complete), we can have a map:

$\log_p : M \rightarrow T_p M$ such that $\exp_p (\log_p q) = q, \qquad \log_p(\exp_p v) = v.$

The exponential is particularly useful because when we want to move “forward” or in a particular direction on a manifold, the interpretation is that the direction we take comes from the tangent space, but then how we actually move on the manifold is given by the exponential map. And if we want to consider instead this notion of how to move in order to get to a particular point, we might think of the log!

This also ties into normal neighborhoods; in a neighborhood around a point, your world practically looks like Euclidean space in terms of how you can move along geodesics.

To be honest, now that I think about it, maybe this isn’t that deep or interesting; it’s practically by definition. But I was just excited to hear about the exponential map outside of my Riemannian geometry class.