  Many machine learning techniques incorporate identity-preserving
transformations into their models to generalize their performance to previously
unseen data. These transformations are typically selected from a set of
functions that are known to maintain the identity of an input when applied
(e.g., rotation, translation, flipping, and scaling). However, there are many
natural variations that cannot be labeled for supervision or defined through
examination of the data. As suggested by the manifold hypothesis, many of these
natural variations live on or near a low-dimensional, nonlinear manifold.
Several techniques represent manifold variations through a set of learned Lie
group operators that define directions of motion on the manifold. However,
these approaches are limited because they require transformation labels when
training their models and they lack a method for determining which regions of
the manifold are appropriate for applying each specific operator. We address
these limitations by introducing a learning strategy that does not require
transformation labels and developing a method that learns the local regions
where each operator is likely to be used while preserving the identity of
inputs. Experiments on MNIST and Fashion MNIST highlight our model's ability to
learn identity-preserving transformations on multi-class datasets.
Additionally, we train on CelebA to showcase our model's ability to learn
semantically meaningful transformations on complex datasets in an unsupervised
manner.
