Learning dictionaries suitable for sparse coding instead of using engineeredbases has proven effective in a variety of image processing tasks. This paperstudies the optimization of dictionaries on image data where the representationis enforced to be explicitly sparse with respect to a smooth, normalizedsparseness measure. This involves the computation of Euclidean projections ontolevel sets of the sparseness measure. While previous algorithms for thisoptimization problem had at least quasi-linear time complexity, here the firstalgorithm with linear time complexity and constant space complexity isproposed. The key for this is the mathematically rigorous derivation of acharacterization of the projection's result based on a soft-shrinkage function.This theory is applied in an original algorithm called Easy Dictionary Learning(EZDL), which learns dictionaries with a simple and fast-to-computeHebbian-like learning rule. The new algorithm is efficient, expressive andparticularly simple to implement. It is demonstrated that despite itssimplicity, the proposed learning algorithm is able to generate a rich varietyof dictionaries, in particular a topographic organization of atoms or separableatoms. Further, the dictionaries are as expressive as those of benchmarklearning algorithms in terms of the reproduction quality on entire images, andresult in an equivalent denoising performance. EZDL learns approximately 30 %faster than the already very efficient Online Dictionary Learning algorithm,and is therefore eligible for rapid data set analysis and problems with vastquantities of learning samples.
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