  In machine learning and neuroscience, certain computational structures and
algorithms are known to yield disentangled representations without us
understanding why, the most striking examples being perhaps convolutional
neural networks and the ventral stream of the visual cortex in humans and
primates. As for the latter, it was conjectured that representations may be
disentangled by being flattened progressively and at a local scale. An attempt
at a formalization of the role of invariance in learning representations was
made recently, being referred to as I-theory. In this framework and using the
language of differential geometry, we show that pooling over a group of
transformations of the input contracts the metric and reduces its curvature,
and provide quantitative bounds, in the aim of moving towards a theoretical
understanding on how to disentangle representations.
