Recently, I have been studying dimensionality reduction, specifically the UMAP algorithm. However, I cannot grasp the mathematics behind it because I do not understand what a Riemannian manifold is. I have watched several videos and tutorials, but they all seem to require complex math knowledge.

Can someone explain what a Riemannian manifold is in one simple sentence that a typical computer science or first-year STEM student can understand?

Stuck on Riemannian manifolds for UMAP? Don’t worry! Think of it as a bumpy, curved surface in a higher dimensional space. It’s like a non-flat map where distances still matter. That’s the key for UMAP to analyze complex data!

You don’t need an in-depth grasp of Riemannian geometry to use UMAP. It leverages this concept as a theoretical underpinning to examine the connections between data points on the manifold.

A Riemannian manifold is basically a smooth, curved surface, like a bumpy sphere, where you can still measure distances in a way that makes sense. This understanding helps with UMAP’s fancy math for visualizing high-dimensional data in lower dimensions.