ℓ1-K-SVD: A robust dictionary learning algorithm with simultaneous update

•We propose an algorithm, which we refer to as ℓ1-K-SVD, to learn data-adaptive dictionaries in the presence of non-Gaussian noise. The fundamental idea behind the algorithm is to replace the usual ℓ2-norm-based data-fidelity metric with ℓ2-norm, and minimize it using iteratively reweighted least-squares (IRLS).•In the dictionary update stage of ℓ1-K-SVD, we adopt a simultaneous updating strategy similar to K-SVD, that is found to result in faster convergence.•We elucidate how the proposed idea can be extended to minimize the ℓp data error, where 0

We develop a new dictionary learning algorithm called the ℓ1-K-SVD, by minimizing the ℓ1 distortion on the data term. The proposed formulation corresponds to maximum a posteriori estimation assuming a Laplacian prior on the coefficient matrix and additive noise, and is, in general, robust to non-Gaussian noise. The ℓ1 distortion is minimized by employing the iteratively reweighted least-squares algorithm. The dictionary atoms and the corresponding sparse coefficients are simultaneously estimated in the dictionary update step. Experimental results show that ℓ1-K-SVD results in noise-robustness, faster convergence, and higher atom recovery rate than the method of optimal directions, K-SVD, and the robust dictionary learning algorithm (RDL), in Gaussian as well as non-Gaussian noise. For a fixed value of sparsity, number of dictionary atoms, and data dimension, ℓ1-K-SVD outperforms K-SVD and RDL on small training sets. We also consider the generalized ℓp,0