We exploit the biconvex nature of the Euclidean non-negative matrix factorization (NMF) optimization problem to derive optimization schemes based on sequential quadratic and second order cone programming. We show that for ordinary NMF, our approach performs as well as existing state-of-the-art algorithms, while for sparsity-constrained NMF, as recently proposed by P. O. Hoyer in JMLR 5 (2004), it outperforms previous methods. In addition, we show how to extend NMF learning within the same optimization framework in order to make use of class membership information in supervised learning problems.

Example-based learning of codes that statistically encode general image classes is of vital importance for computational vision. Recently non negative matrix factorization (NMF) was suggested to provide image code that was both sparse and localized, in contrast to established non local methods like PCA. In this paper, we adopt and generalize this approach to develop a novel learning framework that allows to efficiently compute sparsity-controlled invariant image codes by a well defined sequence of convex conic programs. Applying the corresponding parameter-free algorithm to various image classes results in semantically relevant and transformation-invariant image representations that are remarkably robust against noise and quantization

Building on recent progress in modeling filter response statistics of natural images we integrate a statistical model into a variational framework for image segmentation. Incorporated in a sound probabilistic distance measure the model drives level sets toward meaningful segmentations of complex textures and natural scenes. Since each region comprises two model parameters only the approach is computationally efficient and enables the application of variational segmentation to a considerably larger class of real-world images. We validate the statistical basis of our approach on thousands of natural images and demonstrate that our model outperforms recent variational segmentation methods based on second-order statistics.