Distance measure used by phom.dist()

[sarahsamorodnitsky]

Hello! I am using the phom.dist() function to compute the distance between persistence diagrams. Can you clarify what distance measure is being computed by this function? Is there a reference/citation/source for the distance measure being computed? I was under the impression phom.dist() returned the Wasserstein distance based on the function naming, but looking at a previous issue (https://github.com/rrrlw/TDAstats/issues/13) I see that that isn't the case.

Thanks!

[corybrunson]

Digging into the code in 'inference.R', it looks like the distance is ~the sum~ a vector of the exponentiated absolute differences between the sorted feature lifespans (rather than birth–death coordinates) within each dimension:

$$Dq(X,Y)[d] = \sum{k=1}^{n_d} \lvert (\ell(x_k)) - (\ell(y_k)) \rvert ^ q$$

where $d$ ranges over dimensions, $n_d$ is the maximum number of $d$-dimensional features of $X$ and $Y$, $\ell(x)$ is the lifespan of feature $x$, and the features $x_k$ and $y_k$ are in descending order of lifespan.

@rrrlw may want to chime in. I'm not sure when the package might be upgraded, but certainly clarifying this, and hopefully providing the Wasserstein distance, will be part of that.

[sarahsamorodnitsky]

Any intuition as to why this distance is recommended/used over the p-Wasserstein distance to compare persistence diagrams? I haven't seen this distance measure in the literature, though my literature search has not been exhaustive.

Thanks again!

[corybrunson]

I didn't contribute to it and i don't know a reference for it. My intuition is that it's much less complicated and expensive, though it would certainly be good to provide an explicit rationale.