On Optimality Conditions for Auto-Encoder Signal Recovery

Auto-Encoders are unsupervised models that aim to learn patterns from observed data by minimizing a reconstruction cost. The useful representations learned are often found to be sparse and distributed. On the other hand, compressed sensing and sparse coding assume a data generating process, where the observed data is generated from some true latent signal source, and try to recover the corresponding signal from measurements. Looking at auto-encoders from this \textit{signal recovery perspective} enables us to have a more coherent view of these techniques. In this paper, in particular, we show that the \textit{true} hidden representation can be approximately recovered if the weight matrices are highly incoherent with unit $ \ell^{2} $ row length and the bias vectors takes the value (approximately) equal to the negative of the data mean. The recovery also becomes more and more accurate as the sparsity in hidden signals increases. Additionally, we empirically demonstrate that auto-encoders are capable of recovering the data generating dictionary when only data samples are given.