Joint data alignment up to (lossy) transformations

Joint data alignment is often regarded as a data simplification process. This idea is powerful and general, but raises two delicate issues. First, one must make sure that the use full information about the data is preserved by the alignment process. This is especially important when data are affected by non-invertible transformations, such as those originating from continuous domain deformations in a discrete image lattice. We propose a formulation that explicitly avoids this pitfall. Second, one must choose an appropriate measure of data complexity. We show that standard concepts such as entropy might not be optimal for the task, and we propose alternative measures that reflect the regularity of the codebook space. We also propose a novel and efficient algorithm that allows joint alignment of a large number of samples (tens of thousands of image patches), and does not rely on the assumption that pixels are independent. This is done for the case where the data is postulated to live in an affine subspaces of the embedding space of the raw data. We apply our scheme to learn sparse bases for natural images that discount domain deformations and hence significantly decrease the complexity of codebooks while maintaining the same generative power.