An Unsupervised Algorithm For Learning Lie Group Transformations

We present several theoretical contributions which allow Lie group, or continuous transformation, models to be fit to large high dimensional datasets. We then demonstrate training of Lie group models on natural video. 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. It is shown that the learned video transformations provide a better description of inter-frame differences than the standard motion model, rigid translation.