  Denoising autoencoders (DAEs) have proven useful for unsupervised
representation learning, but a thorough theoretical understanding is still
lacking of how the input noise influences learning. Here we develop theory for
how noise influences learning in DAEs. By focusing on linear DAEs, we are able
to derive analytic expressions that exactly describe their learning dynamics.
We verify our theoretical predictions with simulations as well as experiments
on MNIST and CIFAR-10. The theory illustrates how, when tuned correctly, noise
allows DAEs to ignore low variance directions in the inputs while learning to
reconstruct them. Furthermore, in a comparison of the learning dynamics of DAEs
to standard regularised autoencoders, we show that noise has a similar
regularisation effect to weight decay, but with faster training dynamics. We
also show that our theoretical predictions approximate learning dynamics on
real-world data and qualitatively match observed dynamics in nonlinear DAEs.
