  We introduce Hades, an unsupervised algorithm to detect singularities in
data. This algorithm employs a kernel goodness-of-fit test, and as a
consequence it is much faster and far more scaleable than the existing
topology-based alternatives. Using tools from differential geometry and optimal
transport theory, we prove that Hades correctly detects singularities with high
probability when the data sample lives on a transverse intersection of
equidimensional manifolds. In computational experiments, Hades recovers
singularities in synthetically generated data, branching points in road network
data, intersection rings in molecular conformation space, and anomalies in
image data.
