  Methods of performing anomaly detection on high-dimensional data sets are
needed, since algorithms which are trained on data are only expected to perform
well on data that is similar to the training data. There are theoretical
results on the ability to detect if a population of data is likely to come from
a known base distribution, which is known as the goodness-of-fit problem.
One-sample approaches to this problem offer significant computational
advantages for online testing, but require knowing a model of the base
distribution. The ability to correctly reject anomalous data in this setting
hinges on the accuracy of the model of the base distribution. For high
dimensional data, learning an accurate-enough model of the base distribution
such that anomaly detection works reliably is very challenging, as many
researchers have noted in recent years. Existing methods for the one-sample
goodness-of-fit problem do not account for the fact that a model of the base
distribution is learned. To address that gap, we offer a theoretically
motivated approach to account for the density learning procedure. In
particular, we propose training an ensemble of density models, considering data
to be anomalous if the data is anomalous with respect to any member of the
ensemble. We provide a theoretical justification for this approach, proving
first that a test on typicality is a valid approach to the goodness-of-fit
problem, and then proving that for a correctly constructed ensemble of models,
the intersection of typical sets of the models lies in the interior of the
typical set of the base distribution. We present our method in the context of
an example on synthetic data in which the effects we consider can easily be
seen.
