I tried analysing data using persistent homology. What is not obvious, although they do admit it in one line of every paper, is that is it susceptible to noise :( So it has to go in the bin even though I really want to know what my manifolds look like!
Well, the thing about noise, from a topological point of view, is that persistent homology simply cannot decide whether something is noise or not --- at least not right away. To be more precise: A lot of what PH does is actually some sort of multi-scale Betti number calculation. The Betti numbers count the number of k-dimensional "holes" in a data set. Their calculation is usually done by something that is called "si…
"Now, the problem about real-world data is that it does not come in the form of a simplicial complex." Not really true. real world data comes from a low dimensional manifold complex + noise.
I thought PH would extract the manifold but I could not get a tractable solution.