Generalized Separable Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is a linear dimensionality technique for nonnegative data with applications such as image analysis, text mining, audio source separation, and hyperspectral unmixing. Given MM rank rr, NMF looks WW rr columns HH rows that M ≈ WHM≈WH. NP-hard to solve in general. However, it can be computed efficiently under the separability assumption which requires basis vectors appear points, is, there exists an index set K K W = M(:, )W=M(:,K). In this article, we generalize assumption. We only require each rank-one factor W(:,k)H(k,:)W(:,k)H(k,:) k=1,2,...,rk=1,2,...,r, either W(:,k) M(:,j)W(:,k)=M(:,j) some jj or H(k,:) M(i,:)H(k,:)=M(i,:) ii. refer corresponding problem generalized separable (GS-NMF). discuss properties of GS-NMF propose convex optimization model using fast gradient method. also heuristic algorithm inspired by successive projection algorithm. To verify effectiveness our methods, compare them several state-of-the-art standard algorithms on synthetic, document sets.