  Recently proposed consistency-based Semi-Supervised Learning (SSL) methods
such as the $\Pi$-model, temporal ensembling, the mean teacher, or the virtual
adversarial training, have advanced the state of the art in several SSL tasks.
These methods can typically reach performances that are comparable to their
fully supervised counterparts while using only a fraction of labelled examples.
Despite these methodological advances, the understanding of these methods is
still relatively limited. In this text, we analyse (variations of) the
$\Pi$-model in settings where analytically tractable results can be obtained.
We establish links with Manifold Tangent Classifiers and demonstrate that the
quality of the perturbations is key to obtaining reasonable SSL performances.
Importantly, we propose a simple extension of the Hidden Manifold Model that
naturally incorporates data-augmentation schemes and offers a framework for
understanding and experimenting with SSL methods.
