In this paper, we propose the use of (adaptive) nonlinear approximation for dimensionality reduction. In particular, we propose a dimensionality reduction method for learning a parts based representation of signals using redundant dictionaries. A redundant dictionary is an overcomplete set of basis vectors that spans the signal space. The signals are jointly represented in a common subspace extracted from the redundant dictionary, using greedy pursuit algorithms for simultaneous sparse approximation. The design of the dictionary is flexible and enables the direct control on the shape and properties of the basis functions. Moreover, it allows to incorporate a priori and application-driven knowledge into the basis vectors, during the learning process. We apply our dimensionality reduction method to images and compare it with Principal Component Analysis (PCA) and Non-negative Matrix Factorization (NMF) and its variants, in the context of handwritten digit image recognition and face recognition. The experimental results suggest that the proposed dimensionality reduction algorithm is competitive to PCA and NMF and that it results into meaningful features with high discriminant value.