  What do auto-encoders learn about the underlying data generating
distribution? Recent work suggests that some auto-encoder variants do a good
job of capturing the local manifold structure of data. This paper clarifies
some of these previous observations by showing that minimizing a particular
form of regularized reconstruction error yields a reconstruction function that
locally characterizes the shape of the data generating density. We show that
the auto-encoder captures the score (derivative of the log-density with respect
to the input). It contradicts previous interpretations of reconstruction error
as an energy function. Unlike previous results, the theorems provided here are
completely generic and do not depend on the parametrization of the
auto-encoder: they show what the auto-encoder would tend to if given enough
capacity and examples. These results are for a contractive training criterion
we show to be similar to the denoising auto-encoder training criterion with
small corruption noise, but with contraction applied on the whole
reconstruction function rather than just encoder. Similarly to score matching,
one can consider the proposed training criterion as a convenient alternative to
maximum likelihood because it does not involve a partition function. Finally,
we show how an approximate Metropolis-Hastings MCMC can be setup to recover
samples from the estimated distribution, and this is confirmed in sampling
experiments.
