Approximate L0 constrained non-negative matrix and tensor factorization

Non-negative matrix factorization (NMF), i.e. V ap WH where both V, W and H are non-negative has become a widely used blind source separation technique due to its part based representation. The NMF decomposition is not in general unique and a part based representation not guaranteed. However, imposing sparseness both improves the uniqueness of the decomposition and favors part based representation. Sparseness in the form of attaining as many zero elements in the solution as possible is appealing from a conceptional point of view and corresponds to minimizing reconstruction error with an L₀ norm constraint. In general, solving for a given L₀ norm is an NP hard problem thus convex relaxation to regularization by the L₁ norm is often considered, i.e., minimizing (1/2||V - WH||_F ² + lambda||H||₁).An open problem is to control the degree of sparsity lambda imposed. We here demonstrate that a full regularization path for the L₁ norm regularized least squares NMF for fixed W can be calculated at the cost of an ordinary least squares solution based on a modification of the least angle regression and selection (LARS) algorithm forming a non-negativity constrained LARS (NLARS). With the full regularization path, the L₁ regularization strength lambda that best approximates a given L₀ can be directly accessed and in effect used to control the sparsity of H. The MATLAB code for the NLARS algorithm is available for download.