The shortest form of this question is:

How much TDA can be done with tSNE?

Specifically, I'm referring to the application of TDA to clustering data, so, think along the lines of Ayasdi's implementation:

My understanding is that TDA constructs simplical complexes on a continuum of scales, to then find persistent components. This is an oversimplification because there is also work done by the Mapper algorithm to recognize persistent homologies.

Whereas, tSNE is a 2-D stochastic embedding, which assumes two separate distributions: a gaussian distribution that generates neighbors in high dimensions, and a Cauchy distribution in 2 dimensions, and then constructs an embedding that preserves distances as best as possible between the original space and the embedded space. **The analogy to persistent components can be made in our choice of the tSNE perplexity parameter**, which specifies the width of each Gaussian distribution, and can therefore give us a simplical complex where edges connect pairs of points that generate each other with some minimal probability.

I would think then that tSNE is equivalent to TDA if we also compute persistent homologies?