  We discuss necessary and sufficient conditions for an auto-encoder to define
a conservative vector field, in which case it is associated with an energy
function akin to the unnormalized log-probability of the data. We show that the
conditions for conservativeness are more general than for encoder and decoder
weights to be the same ("tied weights"), and that they also depend on the form
of the hidden unit activation function, but that contractive training criteria,
such as denoising, will enforce these conditions locally. Based on these
observations, we show how we can use auto-encoders to extract the conservative
component of a vector field.
