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.
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