  When a three-dimensional object moves relative to an observer, a change
occurs on the observer's image plane and in the visual representation computed
by a learned model. Starting with the idea that a good visual representation is
one that transforms linearly under scene motions, we show, using the theory of
group representations, that any such representation is equivalent to a
combination of the elementary irreducible representations. We derive a striking
relationship between irreducibility and the statistical dependency structure of
the representation, by showing that under restricted conditions, irreducible
representations are decorrelated. Under partial observability, as induced by
the perspective projection of a scene onto the image plane, the motion group
does not have a linear action on the space of images, so that it becomes
necessary to perform inference over a latent representation that does transform
linearly. This idea is demonstrated in a model of rotating NORB objects that
employs a latent representation of the non-commutative 3D rotation group SO(3).
