  Autoencoders are among the earliest introduced nonlinear models for
unsupervised learning. Although they are widely adopted beyond research, it has
been a longstanding open problem to understand mathematically the feature
extraction mechanism that trained nonlinear autoencoders provide.
  In this work, we make progress in this problem by analyzing a class of
two-layer weight-tied nonlinear autoencoders in the mean field framework. Upon
a suitable scaling, in the regime of a large number of neurons, the models
trained with stochastic gradient descent are shown to admit a mean field
limiting dynamics. This limiting description reveals an asymptotically precise
picture of feature learning by these models: their training dynamics exhibit
different phases that correspond to the learning of different principal
subspaces of the data, with varying degrees of nonlinear shrinkage dependent on
the $\ell_{2}$-regularization and stopping time. While we prove these results
under an idealized assumption of (correlated) Gaussian data, experiments on
real-life data demonstrate an interesting match with the theory.
  The autoencoder setup of interests poses a nontrivial mathematical challenge
to proving these results. In this setup, the "Lipschitz" constants of the
models grow with the data dimension $d$. Consequently an adaptation of previous
analyses requires a number of neurons $N$ that is at least exponential in $d$.
Our main technical contribution is a new argument which proves that the
required $N$ is only polynomial in $d$. We conjecture that $N\gg d$ is
sufficient and that $N$ is necessarily larger than a data-dependent intrinsic
dimension, a behavior that is fundamentally different from previously studied
setups.
