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.

该论文提出了一种方法，可以将 Lie 群适配到高维数据集上，通过一个特定的模糊算子使推理可以逃脱局部最小值，鼓励发现稀疏的、最小距离的状态之间的变换，同时演示了如何在自然图像补丁和自然视频序列上进行变换学习和推理。