Sparse coding is a proven principle for learning compact representations of images. However, sparse coding by itself often leads to very redundant dictionaries. With images, this often takes the form of similar edge detectors which are replicated many times at various positions, scales and orientations. An immediate consequence of this observation is that the estimation of the dictionary components is not statistically efficient. We propose a factored model in which factors of variation (e.g. position, scale and orientation) are untangled from the underlying Gabor-like filters. There is so much redundancy in sparse codes for natural images that our model requires only a single dictionary element (a Gabor-like edge detector) to outperform standard sparse coding. Our model scales naturally to arbitrary-sized images while achieving much greater statistical efficiency during learning. We validate this claim with a number of experiments showing, in part, superior compression of out-of-sample data using a sparse coding dictionary learned with only a single image.
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