  A basic problem in machine learning is to find a mapping $f$ from a low
dimensional latent space $\mathcal{Y}$ to a high dimensional observation space
$\mathcal{X}$. Modern tools such as deep neural networks are capable to
represent general non-linear mappings. A learner can easily find a mapping
which perfectly fits all the observations. However, such a mapping is often not
considered as good, because it is not simple enough and can overfit. How to
define simplicity? We try to make a formal definition on the amount of
information imposed by a non-linear mapping $f$. Intuitively, we measure the
local discrepancy between the pullback geometry and the intrinsic geometry of
the latent space. Our definition is based on information geometry and is
independent of the empirical observations, nor specific parameterizations. We
prove its basic properties and discuss relationships with related machine
learning methods.
