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We introduce Hades, an unsupervised algorithm to detect singularities indata. This algorithm employs a kernel goodness-of-fit test, and as aconsequence it is much faster and far more scaleable than the existingtopology-based alternatives. Using tools from differential geometry and optimaltransport theory, we prove that Hades correctly detects singularities with highprobability when the data sample lives on a transverse intersection ofequidimensional manifolds. In computational experiments, Hades recoverssingularities in synthetically generated data, branching points in road networkdata, intersection rings in molecular conformation space, and anomalies inimage data.

HADES: Fast Singularity Detection with Local Measure Comparison