  A persistent challenge in practical classification tasks is that labeled
training sets are not always available. In particle physics, this challenge is
surmounted by the use of simulations. These simulations accurately reproduce
most features of data, but cannot be trusted to capture all of the complex
correlations exploitable by modern machine learning methods. Recent work in
weakly supervised learning has shown that simple, low-dimensional classifiers
can be trained using only the impure mixtures present in data. Here, we
demonstrate that complex, high-dimensional classifiers can also be trained on
impure mixtures using weak supervision techniques, with performance comparable
to what could be achieved with pure samples. Using weak supervision will
therefore allow us to avoid relying exclusively on simulations for
high-dimensional classification. This work opens the door to a new regime
whereby complex models are trained directly on data, providing direct access to
probe the underlying physics.
