Given a nonnegative matrix X and a factorization rank r, nonnegative matrix factorization (NMF) approximates the matrix X as the product of a nonnegative matrix W with r columns and a nonnegative matrix H with r rows such that X=WH. NMF has become a standard linear dimensionality reduction technique in data mining and machine learning. Although it has been extensively studied in the last 20 years, many questions remain open. In this talk, we address two such questions. The first one is about the uniqueness of NMF decompositions, also known as the identifiability, which is crucial in many applications. We provide a new model and algorithm based on sparsity assumptions that guarantee the uniqueness of the NMF decomposition. The second problem is the generalization of NMF to non-linear models. We consider the linear-quadratic NMF (LQ-NMF) model that adds as basis elements the component-wise product of the columns of W, that is, W(:,j).*W(:,k) for all j,k where .* is the component-wise product. We show that LQ-NMF can be solved in polynomial time, even in the presence of noise, under the separability assumption which requires the presence of the columns of W as columns of X. We illustrate these new results on the blind unmixing of hyperspectral images.
This is joint work with Maryam Abdolali, Christophe Kervazo and Nicolas Dobigeon.