Learning the Irreducible Representations of Commutative Lie Groups
This work addresses representation learning for data with symmetries, such as images under rotations and translations, offering a novel approach but appearing incremental in its application to known groups.
The paper tackles the problem of learning invariant-equivariant and disentangled representations from data with symmetries, using a new probabilistic model for compact commutative Lie groups, and demonstrates that the learned invariant representation achieves high effectiveness for classification.
We present a new probabilistic model of compact commutative Lie groups that produces invariant-equivariant and disentangled representations of data. To define the notion of disentangling, we borrow a fundamental principle from physics that is used to derive the elementary particles of a system from its symmetries. Our model employs a newfound Bayesian conjugacy relation that enables fully tractable probabilistic inference over compact commutative Lie groups -- a class that includes the groups that describe the rotation and cyclic translation of images. We train the model on pairs of transformed image patches, and show that the learned invariant representation is highly effective for classification.