Dictionary Learning Problem

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Dictionary learning can be formulated as an optimization problem in several ways. We present here the basic form, where the representation error is minimized under the constraint of sparsity, and discuss several views and relations with other data analysis and signal processing problems. We study some properties of the DL problem and their implications for the optimization process and explain the two subproblems that are crucial in DL algorithms: sparse coding and dictionary update. In preparation to algorithms analysis and comparisons, we present the main test problems, dealing with representation error and dictionary recovery; we give all details of test procedures, using either artificial data or images. Finally, as an appetizer for the remainder of the book, we illustrate the wide use of DL algorithms in the context of sparse representations for several applications like denoising, inpainting, compression, compressed sensing, classification.