Is topological data analysis useful for low dimensional data?

Question

I often read how Topological Data Analysis (TDA) is useful especially for highly dimensional data. But, what about (apparently) low dimensional ones?

Example: consider measuring the resistance of a sample over a period of time. This is a 1-D data set. Can TDA be useful here?

Answer 1

Depends.

If you have a system where the resistance depends on the previous state(s) then you can use sliding windows (aka the Takens embedding) to embed the seemingly 1-dimensional signal into a larger state space. TDA will be useful in understanding the state space of the system (using persistence or Mapper).

You may consider this cheating because I turned your 1-dim problem into a (potentially) high dimensional problem.