2019/05/31 by Junjun Pan, Nicolas Gillis
Computer Science · Engineering · Mathematics · #Algorithm #Artificial intelligence #Blind Source Separation Techniques #Combinatorics #Computer science #Curse of dimensionality #Eigenvalues and eigenvectors #Face and Expression Recognition #Factorization #Heuristic #Mathematics #Matrix (chemical analysis) #Matrix decomposition #Non-negative matrix factorization #Pattern recognition (psychology) #Physics #Rank (graph theory) #Separable space #Sparse and Compressive Sensing Techniques #Statistics #cs.CV #cs.LG #math.OC #stat.ML
paper · pdf · doi:10.1109/tpami.2019.2956046
published as IEEE Trans. on Pattern Analysis and Machine Intelligence 43 (5), pp. 1546-1561, 2021 · 31 pages, 12 figures, 4 tables. We have added discussions about the identifiability of the model, we have modified the first synthetic experiment, we have clarified some aspects of the contribution
arxiv created 2019/10/15 · openalex publication_date 2019/11/26 · arxiv updated 2021/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 a data matrix M and a factorization rank r, NMF looks for a nonnegative matrix W with r columns and a nonnegative matrix H with r rows such that M ≈ WH. NMF is NP-hard to solve in general. However, it can be computed efficiently under the separability assumption which requires that the basis vectors appear as data points, that is, that there exists an index set K such that W = M(:,K). In this article, we generalize the separability assumption. We only require that for each rank-one factor W(:,k)H(k,:) for k=1,2,…,r, either W(:,k) = M(:,j) for some j or H(k,:) = M(i,:) for some i. We refer to the corresponding problem as generalized separable NMF (GS-NMF). We discuss some properties of GS-NMF and propose a convex optimization model which we solve using a fast gradient method. We also propose a heuristic algorithm inspired by the successive projection algorithm. To verify the effectiveness of our methods, we compare them with several state-of-the-art separable NMF and standard NMF algorithms on synthetic, document and image data sets.