  A fundamental operation in many vision tasks, including motion understanding,
stereopsis, visual odometry, or invariant recognition, is establishing
correspondences between images or between images and data from other
modalities. We present an analysis of the role that multiplicative interactions
play in learning such correspondences, and we show how learning and inferring
relationships between images can be viewed as detecting rotations in the
eigenspaces shared among a set of orthogonal matrices. We review a variety of
recent multiplicative sparse coding methods in light of this observation. We
also review how the squaring operation performed by energy models and by models
of complex cells can be thought of as a way to implement multiplicative
interactions. This suggests that the main utility of including complex cells in
computational models of vision may be that they can encode relations not
invariances.
