An Unsupervised Algorithm For Learning Lie Group Transformations

Abstract

We present several theoretical contributions which allow Lie groups to be fit to high dimensional datasets. Transformation operators are represented in their eigen-basis, reducing the computational complexity of parameter estimation to that of training a linear transformation model. A transformation specific "blurring" operator is introduced that allows inference to escape local minima via a smoothing of the transformation space. A penalty on traversed manifold distance is added which encourages the discovery of sparse, minimal distance, transformations between states. Both learning and inference are demonstrated using these methods for the full set of affine transformations on natural image patches. Transformation operators are then trained on natural video sequences. It is shown that the learned video transformations provide a better description of inter-frame differences than the standard motion model based on rigid translation.