Dimension of the manifold on which my data sits

Suppose that I have data points, in the form of vectors with binary entries. We create a metric space, or Vietoris-Rips complex, using the Hamming distance between the data points.

I would like to imagine that my data points naturally sit on some manifold, which I want to somehow reconstruct (at least, its dimension).

What are the tools/techniques/references/keywords for such data analysis?

Thank you!