[openasocket]

I was literally toying around with some of these ideas today. I had the thought that you could take some time series data, compute the velocities and positions over time, and use that to try and learn the metric tensors of it, assuming the time series data is following a geodesic on some manifold for the most part. And from there you could compute the symmetries and curvature of the system. I thought you could use that to try and define an energy for the system or some other conserved quantities, and then use that as an anomaly detector (change in the energy of the system means there was some outside interference).

[A_N_T_O_N_I_O]

Unbelievable. Your ""toy"" experimentation led you to such a intriganting and curious mechanism to time series forecasting. Regardless of what I tell you, I wouldn't be capable of expressing my interest in your statement. It might be exciting to move forward and dive deeper in your idea. I will definitely jump on it.

[kk58]

I was looking at a similar approach, my question is how do parameterize the manifold and how do you map points on the manifold if it's irregular -think filamentous, so that path trace can be done

And in general this is a question of interest. Are graphs really a good discrete approximation of a manifold?

[A_N_T_O_N_I_O]

I hope you find these two resources useful to address your question. I'm also pretty integrated about the possible answers, but I definitely bet on doing it using graphs. I'm preparing a detailed and concrete explanation on why I guess this conclusion but that one will need to wait for now!

For now as I have already told you, check out these ones.

https://ncatlab.org/nlab/show/manifold

https://www.youtube.com/watch?v=LvmjbXZyoP0

[A_N_T_O_N_I_O]

Would you like to explore this and other ideas related to the field and contemplate some kind of potential collaboration?