Learning non-structured , overcomplete and sparsifying transform

Transform learning has been introduced and studied in [1],[2], [3] and [4]. An optimal transform learning for structured and overcomplete matrix was proposed in [5]. However, several issues (optimality, convergence and computational complexity) related to learning an incoherent, well-conditioned, non-structured and overcomplete sparsifing transform still remain open. Let X ∈ <N×L be a data matrix, having as columns data samples xi ∈ < , i ∈ I = {1, 2, ..., L}. Assuming a sparsifing transform model [1], then the problem formulation for learning the overcomplete transform matrix A ∈ <M×N , (M > N) has the following form: