Tag: differential geometry

Machine Learning, Math

The geometry of data: the missing metric tensor and the Stein score [Part II]

Credit: ESA/Webb, NASA & CSA, J. Rigby. / The James Webb Space Telescope captures gravitational lensing, a phenomenon that can be modeled using differential geometry.

Note: This is a continuation of the previous post: Thoughts on Riemannian metrics and its connection with diffusion/score matching [Part I], so if you haven’t read it yet, please consider reading as I won’t be re-introducing in depth the concepts (e.g., the two scores) that I described there already. This article became a bit long, so if you are familiar already with metric tensors and differential geometry you can just skip the first part.

I was planning to write a paper about this topic, but my spare time is not that great so I decided it would be much more fun and educative to write this article in form of a tutorial. If you liked it, please consider citing it:

Cite this article as: Christian S. Perone, "The geometry of data: the missing metric tensor and the Stein score [Part II]," in Terra Incognita, 12/11/2024, https://blog.christianperone.com/2024/11/the-geometry-of-data-part-ii/.

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by Christian S. Perone
Machine Learning, Math

Thoughts on Riemannian metrics and its connection with diffusion/score matching [Part I]

Different gaussian curvature surfaces. Image by Nicoguaro.

We are so used to Euclidean geometry that we often overlook the significance of curved geometries and the methods for measuring things that don’t reside on orthonormal bases. Just as understanding physics and the curvature of spacetime requires Riemannian geometry, I believe a profound comprehension of Machine Learning (ML) and data is also not possible without it. There is an increasing body of research that integrates differential geometry into ML. Unfortunately, the term “geometric deep learning” has predominantly become associated with graphs. However, modern geometry offers much more than just graph-related applications in ML.

I was reading the excellent article from Sander Dieleman about different perspectives on diffusion, so I thought it would be cool to try to contribute a bit with a new perspective.

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by Christian S. Perone