  Autoencoders are a prominent model in many empirical branches of machine
learning and lossy data compression. However, basic theoretical questions
remain unanswered even in a shallow two-layer setting. In particular, to what
degree does a shallow autoencoder capture the structure of the underlying data
distribution? For the prototypical case of the 1-bit compression of sparse
Gaussian data, we prove that gradient descent converges to a solution that
completely disregards the sparse structure of the input. Namely, the
performance of the algorithm is the same as if it was compressing a Gaussian
source - with no sparsity. For general data distributions, we give evidence of
a phase transition phenomenon in the shape of the gradient descent minimizer,
as a function of the data sparsity: below the critical sparsity level, the
minimizer is a rotation taken uniformly at random (just like in the compression
of non-sparse data); above the critical sparsity, the minimizer is the identity
(up to a permutation). Finally, by exploiting a connection with approximate
message passing algorithms, we show how to improve upon Gaussian performance
for the compression of sparse data: adding a denoising function to a shallow
architecture already reduces the loss provably, and a suitable multi-layer
decoder leads to a further improvement. We validate our findings on image
datasets, such as CIFAR-10 and MNIST.
