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13.3. !1 regularization: basics 437

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B, the coe!cients gradually “turn on”. But for any value between 0 and Bmax = ||wOLS ||1,the solution is sparse.4

Remarkably, it can be shown that the solution path is a piecewise linear function of B (Efronet al. 2004). That is, there are a set of critical values of B where the active set of non-zerocoe!cients changes. For values of B between these critical values, each non-zero coe!cientincreases or decreases in a linear fashion. This is illustrated in Figure 13.8(a). Furthermore,one can solve for these critical values analytically. This is the basis of the LARS algorithm(Efron et al. 2004), which stands for “least angle regression and shrinkage” (see Section 13.4.2for details). Remarkably, LARS can compute the entire regularization path for roughly the same

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Sparse representations for image restorationK-SVD: [Elad and Aharon, 2006]

Figure: Dictionary trained on a noisy version of the imageboat.


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Sparse representations for image restorationGrayscale vs color image patches


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Matrix Factorization Problems and Dictionary LearningFaces

(d) PCA (e) NNMF (f) DL


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