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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 paper, 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.
This article was published in the following journal.
Name: IEEE transactions on pattern analysis and machine intelligence
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The aim of the present study is to evaluate the clinical response of proximal furcations treated with enamel matrix derivative proteins (EMD).
A secreted matrix metalloproteinase that is the predominant proteolytic activity in the enamel matrix. The enzyme has a high specificity for dental enamel matrix protein AMELOGENIN.
A transmembrane domain-containing matrix metalloproteinase. It is synthesized as an inactive zymogen that is activated by the proteolytic action of PROPROTEIN CONVERTASES. Matrix metalloproteinase 16 plays a direct role in the cleavage of proteins in the pericellular environment. In addition it can function indirectly by enzymatically activating the proprotein form of other MATRIX METALLOPROTEINASES such as the zymogen of MATRIX METALLOPROTEINASE 2.
A secreted matrix metalloproteinase that is believed to play a role in EXTRACELLULAR MATRIX remodeling and cell fate determination during normal and pathological processes. Matrix metalloproteinase 11 was originally isolated in primary BREAST NEOPLASMS and may be involved in the process of tumorigenesis.
A transmembrane domain-containing matrix metalloproteinase. It is synthesized as an inactive zymogen that is activated by the action of PROPROTEIN CONVERTASES such as FURIN. Matrix metalloproteinase 14 plays a direct role in the cleavage of proteins in the pericellular environment. In addition it can function indirectly by enzymatically activating the proprotein form of MATRIX METALLOPROTEINASE 15.
A secreted matrix metalloproteinase that plays a physiological role in the degradation of extracellular matrix found in skeletal tissues. It is synthesized as an inactive precursor that is activated by the proteolytic cleavage of its N-terminal propeptide.
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