My question may seem CS-related at first, but it's essentially mathematical, so, please, bear with me.
I'm doing Neural Architecture Search (NAS) by varying the number of layers and neurons per layer for a neural network (connections are feed-forward, throughout), and then training it on a fixed task to test its performance. The end goal is to perform topological data analysis (TDA) on the best and worst architectures (graphs) to spot the persistent structures in a good/bad architecture.
My question is: am I right in my assumption that TDA could spot such persistent structures? If so, which metric should I use? Just the connection weights?
P.S.: My background in (computational) topology is quite minimal.