Sparse Separable Nonnegative Matrix Factorization
This work addresses blind source separation problems in domains like image processing, but it is incremental as it builds on existing separable NMF methods by adding sparsity constraints.
The paper tackles the problem of underdetermined blind source separation, such as multispectral image unmixing, by proposing sparse separable nonnegative matrix factorization (SSNMF), which combines separability and sparsity assumptions; it proves SSNMF is NP-complete and introduces an algorithm that recovers true sources in noiseless settings, as demonstrated on synthetic data and a multispectral image.
We propose a new variant of nonnegative matrix factorization (NMF), combining separability and sparsity assumptions. Separability requires that the columns of the first NMF factor are equal to columns of the input matrix, while sparsity requires that the columns of the second NMF factor are sparse. We call this variant sparse separable NMF (SSNMF), which we prove to be NP-complete, as opposed to separable NMF which can be solved in polynomial time. The main motivation to consider this new model is to handle underdetermined blind source separation problems, such as multispectral image unmixing. We introduce an algorithm to solve SSNMF, based on the successive nonnegative projection algorithm (SNPA, an effective algorithm for separable NMF), and an exact sparse nonnegative least squares solver. We prove that, in noiseless settings and under mild assumptions, our algorithm recovers the true underlying sources. This is illustrated by experiments on synthetic data sets and the unmixing of a multispectral image.