A cool concept in topological data analysis is the simplicial complex. It connects points with edges, triangles, and higher order analogs. By gradually increasing each point's neighborhood size it captures data's topological structure (e.g. voids, holes) at different scales.

Thinking of making a blog post about this topic.

If you find this interesting you might find this recent blog I wrote about non-linear dimensionality reduction interesting. Link:

Great! Thanks, I am working on a related topic.

I remember having to code this for the TDA course. It had so many awesome concepts!

TDA has hands down best nomenclature. “Rips Complex” , “simplicial” , “cliques” 😅

Filtrations right? TDA was one of the most interesting courses I had in university :)

Great visual. Some questions. 1. are the partitions unique? 2. Are there provable results that show that the partitions are robust (i.e. you don't discover spurious holes if the dataset changes a little bit)? 3. How fast does the algorithm converge to the "ground state" (i.e. how much data do you need to find all the holes), assuming a certain geometric constraint on the data (manifold assumption, etc)? Are there such results? I'm asking if discovering holes requires less data than discovering a more geometric feature like PCA.

Nice, really cool rendering, classic Vietoris–Rips. Did you use lazy witness or full VR on this? I've been playing with landmark witnesses (de Silva & Carlsson, 2004)

Yep I believe this is a Vietoris–Rips complex! Made from epsilon ball neighborhoods. I’ll have to read about landmark witnesses. I’m only recently exploring this TDA stuff a bit.

Awesome. Love the blog post btw, great interactive visualizations. Looking forward to seeing more!

Waouh that actually very very much what I'm looking for but in 3D xD

Maybe treating it as Rymannian manifold is better.

I’m applying TDA to fMRI-based brain networks using persistent homology, specifically working with time-series measures like dynamic functional connectivity. Since the data are highly dynamic and noisy,

I’d love to hear how people across the broader scientific community think about applying TDA in these kinds of settings.

Very intuitive and easy to understand

Thank you for sharing such impressive work!

Great viz. I've been looking into the same concept. What you are looking into this for?

Dimensionality reduction and related topics are just an interest of mine.

Furthermore, the results tend to be quite sensitive to choices in thresholding and scale (e.g., edge-length cutoffs or sparsity), which makes the biological interpretation of these features feel quite elusive.

Any thoughts to what the holes mean, with regards to all different types of data?

Emergent from adjacency

Curious if insights on simplicial complex can help drive principled approaches on the best noise schedules on training diffusion models. I have thought about the problem for years but still did not found a good answer.

Reminds me of a growing neural gas. They are fun to play around with and make animations for.

How many dimensions do you need?

@grok posso correlacionar com a forma pelo qual o cérebro usa ressonância e topologia para manifestar consciência?