Comment by Intrinisical-AI

Absolutely — you’re on the right track! The intuition of the "mound" is actually quite powerful.

Let’s imagine the embedding space as a 2D surface with a third hidden dimension — say, curvature pointing ‘up’ toward a peak (like a saddle or a bump).

Then:

Two vectors pointing in the same compass direction (e.g., “north”) might diverge or converge depending on their local curvature.

If you try to compare them using Euclidean cosine similarity, you’d be blind to the underlying geometry.

What you'd want is to "parallel transport" one vector along the surface toward the other — adjusting its direction to the curvature — and then compare them.

In high-curvature regions, even small angular differences can grow rapidly. In flatter regions, directionality might dominate. So yes, context (i.e., local geometry and density) changes everything.

Right now I'm testing some of this with reranking signals based on Ricci curvature + soft k-NN graphs (preserving density), and trying out simple geodesic-aware losses. But it’s still early days.

Your comment actually gave me a better mental image of what happens when you’re “far from the mound” — maybe curvature becomes negligible, and naive methods recover. Thanks for that.