Regularized Auto-Encoders Estimate Local Statistics

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 the unknown data generating density. This paper clarifies some of these previous intuitive 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. More precisely, we show that the auto-encoder captures the score (derivative of the logdensity with respect to the input) or the local mean associated with the unknown data-generating density. This is the second result linking denoising auto-encoders and score matching, but in way that is different from previous work, and can be applied to the case when the auto-encoder reconstruction function does not necessarily correspond to the derivative of an energy function. The theorems provided here are completely generic and do not depend on the parametrization of the autoencoder: 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, i.e., one not involving a partition function. Finally, we make the connection to existing sampling algorithms for such autoencoders, based on an MCMC walking near the high-density manifold.