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.
We propose a new method for modeling the temporal correlation in videos, based on local transforms realized by Lie group operators. A large class of transforms can be theoretically described by these operators; however, we propose to learn from natural movies a subset of transforms that are statistically relevant for video representation. The proposed transformation modeling is further exploited to remove inter-view redundancy, i.e., as the prediction step of video encoding. Since the Lie group transformation coefficients are continuous, a quantization step is necessary for each transform. Therefore, we derive theoretical bounds on the distortion due to coefficient quantization. The experimental results demonstrate that the new prediction method with learned transforms leads to better rate-distortion performance at higher bit-rates, and competitive performance at lower bit-rates, compared to the standard prediction based on block-based motion estimation.
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