期刊
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
卷 43, 期 5, 页码 1546-1561出版社
IEEE COMPUTER SOC
DOI: 10.1109/TPAMI.2019.2956046
关键词
Nonnegative matrix factorization; separability; algorithms
资金
- Fonds de la Recherche Scientifique-FNRS under EOS Project [O005318F-RG47]
- Fonds Wetenschappelijk Onderzoek-Vlanderen (FWO) under EOS Project [O005318F-RG47]
- European Research Council (ERC) [679515]
Nonnegative matrix factorization (NMF) is a linear dimensionality technique for nonnegative data, which can be efficiently computed under the separability assumption. The algorithm operates by finding data points that contain basis vectors for decomposition.
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 approximate to 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)II(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.
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