Why does this matter? This research shows that AI can leverage abstract math like category theory to better model language. It reveals deeper semantic structures in LLMs. It pushes the boundaries of AI evaluation (beyond metrics like perplexity). This is awesome math-AI research, blending logic, geometry, and meaning.

Pₖ Is All You Need 🧐

@protosphinx

@gabrielpeyre Thanks Gabriel! This is much needed.

How do we mathematically build the shapes that emerge in a filtration? Enter Čech and Vietoris-Rips complexes! Čech Complex: •Imagine placing a ball around each point in your dataset. •If balls overlap, the points are connected (form edges, triangles, etc.). •It captures the true topology of the union of balls. Vietoris-Rips Complex: •A simplified version: connect points if their pairwise distance is below a threshold. •Easier to compute but less precise than Čech. Both evolve as the scale (radius) increases, forming the backbone of persistent homology. Next: How do we extract topological features from these complexes?

@cutezu_ Awesome. Can I do an X post/thread on this paper like the others I have done? Persistent homology is one of my fav topics. 😅

What’s exciting is the universality of this approach: - From edge detection in images to molecular modeling in physics - group theory helps neural networks exploit symmetry and learn efficiently. Deep learning + group theory = learnable insights for AI-mathematicians!