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## Convex and Semi-Nonnegative Matrix Factorizations (2008)

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Citations: | 108 - 10 self |

### Citations

4166 | Latent dirichlet allocation - Blei, Ng, et al. |

1640 |
Learning the parts of object by non-negative matrix factorization
- Lee, Seung
- 1999
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Citation Context ...nt work in machine learning has focused on matrix factorizations that directly target some of the special features of statistical data analysis. In particular, nonnegative matrix factorization (NMF) (=-=Lee and Seung, 1999-=-, 2001) focuses on the analysis of data matrices whose elements are nonnegative, a common occurrence in data sets derived from 1stext and images. Moreover, NMF yields nonnegative factors, which can be... |

1214 | Algorithms for non-negative matrix factorization
- LEE, SEUNG
- 2001
(Show Context)
Citation Context ...ate given in Eq. (8) for fixed F . Proof. We write J(H) as J(H) = Tr(−2H T B + + 2H T B − + H T A + H − H T A − H) (15) where A = F T F , B = F T X, and H = G. We use the auxiliary function approach (=-=Lee and Seung, 2001-=-). A function Z(H, ˜ H) is called an auxiliary function of J(H) if it is satisfies for any H, ˜ H. Define Z(H, ˜ H) ≥ J(H), Z(H, H) = J(H), (16) H (t+1) = arg min Z(H, H H (t) ). (17) By construction,... |

1191 | Probabilistic latent semantic indexing - Hofmann - 1999 |

515 |
Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values
- Paatero, Tapper
- 1994
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Citation Context ...from the point of view of interpretability. The scope of research on NMF has grown rapidly in recent years. NMF has been shown to be useful in a variety of applied settings, including environmetrics (=-=Paatero and Tapper, 1994-=-), chemometrics (Xie et al., 1999), pattern recognition (Li et al., 2001), multimedia data analysis (Cooper and Foote, 2002), text mining (Xu et al., 2003; Pauca et al., 2004) and DNA gene expression ... |

487 | Non-negative matrix factorization with sparseness constraints - Hoyer - 2004 |

318 | Document clustering based on non-negative matrix factorization - Xu, Liu, et al. - 2003 |

271 | A Direct Formulation for Sparse PCA Using Semidefinite Programming - d’Aspremont, Ghaoui, et al. |

264 | Sparse principal component analysis - Zou, Hastie, et al. - 2006 |

251 | Maximum-margin matrix factorization
- Srebro, Rennie, et al.
- 2004
(Show Context)
Citation Context ...commodate a variety of objective functions (Dhillon and Sra, 2005; Ding et al., 2006) and a variety of data analysis problems, including classification (Sha et al., 2003) and collaborative filtering (=-=Srebro et al., 2005-=-). A number of studies have focused on further developing computational methodologies for NMF (Hoyer, 2004; Berry et al., 2006; Li and Ma, 2004). Finally, researchers have begun to explore some of the... |

197 | Algorithms and applications for approximate nonnegative matrix factorization. Computational Statistics and Data Analysis
- Berry, Browne, et al.
- 2006
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Citation Context ...including classification (Sha et al., 2003) and collaborative filtering (Srebro et al., 2005). A number of studies have focused on further developing computational methodologies for NMF (Hoyer, 2004; =-=Berry et al., 2006-=-; Li and Ma, 2004). Finally, researchers have begun to explore some of the relationships between matrix factorizations and K-means clustering (Ding et al., 2005); as we emphasize in the current paper,... |

194 | Spectral relaxation for k-means clustering
- Zha, Ding, et al.
- 2001
(Show Context)
Citation Context ...inimization. The minimization problem thus becomes max G T G=I Tr(G T KG), where K is either a linear kernel X T X or 〈φ(X), φ(X)〉. It is known that this is identical to (kernel-) K-means clustering (=-=Zha et al., 2002-=-). ⊓– In the definitions of NMF, Semi-NMF, Convex-NMF, Cluster-NMF and Kernel-NMF, G is not restricted to be orthogonal; these NMF variants are soft versions of K-means clustering. 6 Experiments We fi... |

193 | Learning spatially localized parts-based representations
- Li, Hou, et al.
- 2001
(Show Context)
Citation Context ...n rapidly in recent years. NMF has been shown to be useful in a variety of applied settings, including environmetrics (Paatero and Tapper, 1994), chemometrics (Xie et al., 1999), pattern recognition (=-=Li et al., 2001-=-), multimedia data analysis (Cooper and Foote, 2002), text mining (Xu et al., 2003; Pauca et al., 2004) and DNA gene expression analysis (Brunet et al., 2004). Algorithmic extensions of NMF have been ... |

193 | On convergence properties of the EM algorithm for gaussian mixtures - Xu, Jordan - 1996 |

190 | When does non-negative matrix factorization give a correct decomposition into parts - Donoho, Stodden - 2003 |

170 |
Metagenes and molecular pattern discovery using matrix factorization
- Brunet, Tamayo, et al.
- 2004
(Show Context)
Citation Context ...trics (Xie et al., 1999), pattern recognition (Li et al., 2001), multimedia data analysis (Cooper and Foote, 2002), text mining (Xu et al., 2003; Pauca et al., 2004) and DNA gene expression analysis (=-=Brunet et al., 2004-=-). Algorithmic extensions of NMF have been developed to accommodate a variety of objective functions (Dhillon and Sra, 2005; Ding et al., 2006) and a variety of data analysis problems, including class... |

153 | On the equivalence of nonnegative matrix factorization and spectral clustering
- Ding, He, et al.
- 2005
(Show Context)
Citation Context ...al methodologies for NMF (Hoyer, 2004; Berry et al., 2006; Li and Ma, 2004). Finally, researchers have begun to explore some of the relationships between matrix factorizations and K-means clustering (=-=Ding et al., 2005-=-); as we emphasize in the current paper, this relationship has implications for the interpretability of matrix factors. Our goal in this paper is to expand the repertoire of nonnegative matrix factori... |

96 | Generalized nonnegative matrix approximations with bregman divergences - Dhillon, Sra - 2006 |

92 | Sparse non-negative matrix factorizations via alternating non-negativity constrained least squares for microarray data analysis - Kim, Park - 2007 |

81 | Multiplicative updates for nonnegative quadratic programming
- Sha, Lin, et al.
(Show Context)
Citation Context ...mic extensions of NMF have been developed to accommodate a variety of objective functions (Dhillon and Sra, 2005; Ding et al., 2006) and a variety of data analysis problems, including classification (=-=Sha et al., 2003-=-) and collaborative filtering (Srebro et al., 2005). A number of studies have focused on further developing computational methodologies for NMF (Hoyer, 2004; Berry et al., 2006; Li and Ma, 2004). Fina... |

68 | Relation between plsa and nmf and implications - Gaussier, Goutte - 2005 |

59 |
Text mining using non-negative matrix factorizations
- Pauca, Shahnaz, et al.
- 2004
(Show Context)
Citation Context ...ing environmetrics (Paatero and Tapper, 1994), chemometrics (Xie et al., 1999), pattern recognition (Li et al., 2001), multimedia data analysis (Cooper and Foote, 2002), text mining (Xu et al., 2003; =-=Pauca et al., 2004-=-) and DNA gene expression analysis (Brunet et al., 2004). Algorithmic extensions of NMF have been developed to accommodate a variety of objective functions (Dhillon and Sra, 2005; Ding et al., 2006) a... |

51 | SVD based initialization: A head start for nonnegative matrix factorization - Boutsidis, Gallopoulos |

47 | A general model for clustering binary data - Li - 2005 |

39 | Unsupervised learning by convex and conic coding - Lee, Seung - 1997 |

38 | Summarizing video using non-negative similarity matrix factorization - Cooper, Foote |

34 |
Nonnegative matrix factorization and probabilistic latent semantic indexing: Equivalence chi-square statistic, and a hybrid method
- Ding, Li, et al.
- 1999
(Show Context)
Citation Context ...; Pauca et al., 2004) and DNA gene expression analysis (Brunet et al., 2004). Algorithmic extensions of NMF have been developed to accommodate a variety of objective functions (Dhillon and Sra, 2005; =-=Ding et al., 2006-=-) and a variety of data analysis problems, including classification (Sha et al., 2003) and collaborative filtering (Srebro et al., 2005). A number of studies have focused on further developing computa... |

34 | Low-rank approximations with sparse factors II: Penalized methods with discrete Newton-like iterations - Zhang, Zha, et al. |

19 | K-means clustering and principal component analysis - Ding, He |

10 | On an equivalence between - Girolami, Kabán |

9 | S (2004) IFD: iterative feature and data clustering
- Li, Ma
(Show Context)
Citation Context ...tion (Sha et al., 2003) and collaborative filtering (Srebro et al., 2005). A number of studies have focused on further developing computational methodologies for NMF (Hoyer, 2004; Berry et al., 2006; =-=Li and Ma, 2004-=-). Finally, researchers have begun to explore some of the relationships between matrix factorizations and K-means clustering (Ding et al., 2005); as we emphasize in the current paper, this relationshi... |

8 | Ensemble Non-negative Matrix Factorization Methods for Clustering Protein-Protein Interactions - Greene, Cagney, et al. |

4 |
Positive matrix factorization applied to a curve resolution problem
- Xie, Hopke, et al.
- 1999
(Show Context)
Citation Context ...y. The scope of research on NMF has grown rapidly in recent years. NMF has been shown to be useful in a variety of applied settings, including environmetrics (Paatero and Tapper, 1994), chemometrics (=-=Xie et al., 1999-=-), pattern recognition (Li et al., 2001), multimedia data analysis (Cooper and Foote, 2002), text mining (Xu et al., 2003; Pauca et al., 2004) and DNA gene expression analysis (Brunet et al., 2004). A... |