An L1 criterion for dictionary learning by subspace identification

We propose an ℓ¹ criterion for dictionary learning for sparse signal representation. Instead of directly searching for the dictionary vectors, our dictionary learning approach identifies vectors that are orthogonal to the subspaces in which the training data concentrate. We study conditions on the coefficients of training data that guarantee that ideal normal vectors deduced from the dictionary are local optima of the criterion. We illustrate the behavior of the criterion on a 2D example, showing that the local minima correspond to ideal normal vectors when the number of training data is sufficient. We conclude by describing an algorithm that can be used to optimize the criterion in higher dimension.