  This paper sheds light on the risk landscape of unsupervised least squares in
the context of deep auto-encoding neural nets. We formally establish an
equivalence between unsupervised least squares and principal manifolds. This
link provides insight into the risk landscape of auto--encoding under the mean
squared error, in particular all non-trivial critical points are saddlepoints.
Finding saddlepoints is in itself difficult, overcomplete auto-encoding poses
the additional challenge that the saddlepoints are degenerate. Within this
context we discuss regularization of auto-encoders, in particular bottleneck,
denoising and contraction auto-encoding and propose a new optimization strategy
that can be framed as particular form of contractive regularization.
