  We present a pipeline for characterizing and constraining initial conditions
in cosmology via persistent homology. The cosmological observable of interest
is the cosmic web of large scale structure, and the initial conditions in
question are non-Gaussianities (NG) of primordial density perturbations. We
compute persistence diagrams and derived statistics for simulations of dark
matter halos with Gaussian and non-Gaussian initial conditions. For
computational reasons and to make contact with experimental observations, our
pipeline computes persistence in sub-boxes of full simulations and simulations
are subsampled to uniform halo number. We use simulations with large NG
($f_{\rm NL}^{\rm loc}=250$) as templates for identifying data with mild NG
($f_{\rm NL}^{\rm loc}=10$), and running the pipeline on several cubic volumes
of size $40~(\textrm{Gpc/h})^{3}$, we detect $f_{\rm NL}^{\rm loc}=10$ at
$97.5\%$ confidence on $\sim 85\%$ of the volumes for our best single
statistic. Throughout we benefit from the interpretability of topological
features as input for statistical inference, which allows us to make contact
with previous first-principles calculations and make new predictions.
