  Autoencoders are a deep learning model for representation learning. When
trained to minimize the distance between the data and its reconstruction,
linear autoencoders (LAEs) learn the subspace spanned by the top principal
directions but cannot learn the principal directions themselves. In this paper,
we prove that $L_2$-regularized LAEs are symmetric at all critical points and
learn the principal directions as the left singular vectors of the decoder. We
smoothly parameterize the critical manifold and relate the minima to the MAP
estimate of probabilistic PCA. We illustrate these results empirically and
consider implications for PCA algorithms, computational neuroscience, and the
algebraic topology of learning.
