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## Multiclass total variation clustering (2013)

Venue: | In Advances in Neural Information Processing Systems (NIPS |

Citations: | 8 - 2 self |

### Citations

3784 | Normalized cuts and image segmentation
- Shi, Malik
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Citation Context ...ach NMF algorithm. All non-recursive algorithms (LSD [1], NMFR [19], MTV) received two types of initial data: (a) the deterministic data used in [19]; (b) a random procedure leveraging normalized-cut =-=[16]-=-. Procedure (b) first selects one data point uniformly at random from each computed NCut cluster, then sets fr equal to one at the data point drawn from the rth cluster and zero otherwise. We then pro... |

436 | A first-order primal-dual algorithm for convex problems with applications to imaging,
- Chambolle, Pock
- 2011
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Citation Context ...f total variation terms subject to a convex constraint, we can readily adapt these algorithms to compute the second step of our algorithm efficiently. In this work we use the primal-dual algorithm of =-=[6]-=- with acceleration. This relies on a proper uniformly convex formulation of the proximal minimization, which we detail completely in the Appendix. The primal-dual algorithm we use to compute proxT k+δ... |

335 |
A lower bound for the smallest eigenvalue of the Laplacian, in “Problems
- Cheeger
- 1970
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Citation Context ...i,j}1≤i,j≤N denote the non-negative, symmetric similarity matrix. Each entry wij of W encodes the similarity, or lack thereof, between a pair of vertices. The classical balanced-cut (or, Cheeger cut) =-=[7, 8]-=- asks for a partition of V = A ∪ Ac into two disjoint sets that minimizes the set energy Bal(A) := Cut(A,Ac) min{|A|, |Ac|} = ∑ xi∈A,xj∈Ac wij min{|A|, |Ac|} . (1) A simple rationale motivates this mo... |

189 |
Spectral Graph Theory, volume 92
- Chung
- 1997
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Citation Context ...i,j}1≤i,j≤N denote the non-negative, symmetric similarity matrix. Each entry wij of W encodes the similarity, or lack thereof, between a pair of vertices. The classical balanced-cut (or, Cheeger cut) =-=[7, 8]-=- asks for a partition of V = A ∪ Ac into two disjoint sets that minimizes the set energy Bal(A) := Cut(A,Ac) min{|A|, |Ac|} = ∑ xi∈A,xj∈Ac wij min{|A|, |Ac|} . (1) A simple rationale motivates this mo... |

153 | On the equivalence of nonnegative matrix factorization and spectral clustering
- Ding, He, et al.
- 2005
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Citation Context ... − f(xj)|2 denotes the spectral relaxation of Cut(A,Ac); it equals 〈f, Lf〉 if L denotes the unnormalized graph Laplacian matrix. Thus problem (Prlx2) relates to spectral clustering (and therefore NMF =-=[9]-=-) with a positivity constraint. Note that the only difference between (P-rlx2) and (P-rlx) is that the exponent 2 appears in ‖ · ‖Lap while the exponent 1 appears in the total variation. This simple d... |

58 |
A finite algorithm for finding the projection of a point onto the canonical simplex of Rn ,
- Michelot
- 1986
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Citation Context ...ts. This requires computing the projection projC(F ) exactly at each inner iteration. The overall algorithm remains efficient provided we can compute this projection quickly. When C = Σ the algorithm =-=[14]-=- performs the required projection in at most R steps. When C = Σ ∩ Λ the computational effort actually decreases, since in this case the projection consists of a simplex projection on the unlabeled po... |

38 | Spectral clustering based on the graph p-laplacian
- Bühler, Hein
- 2009
(Show Context)
Citation Context ...is well-known. Several previous works have proven that the relaxation is exact in the two-class case; that is, the total variation solution coincides with the solution of the original NP-hard problem =-=[8, 18, 3, 5]-=-. Figure 2 illustrates the result of the difference between total variation and NMF relaxations on the data set OPTDIGITS, which contains 5620 images of handwritten numerical digits. Figure 2(a) shows... |

30 | An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse
- Hein, Buehler
(Show Context)
Citation Context ...n the other hand, has a solution that closely matches the solution of the original discrete NP-hard problem. Ideas from the image processing literature have recently motivated a new set of algorithms =-=[17, 18, 11, 12, 4, 15, 3, 2, 13, 10]-=- that can obtain tighter relaxations than those used by NMF and spectral clustering. These new algorithms all rely on the concept of total variation. Total variation techniques promote the formation o... |

23 | Total variation and Cheeger cuts
- Szlam, Bresson
- 2010
(Show Context)
Citation Context ...n the other hand, has a solution that closely matches the solution of the original discrete NP-hard problem. Ideas from the image processing literature have recently motivated a new set of algorithms =-=[17, 18, 11, 12, 4, 15, 3, 2, 13, 10]-=- that can obtain tighter relaxations than those used by NMF and spectral clustering. These new algorithms all rely on the concept of total variation. Total variation techniques promote the formation o... |

22 | Diffuse interface models on graphs for classification of high dimensional data,” Multiscale Modeling - Bertozzi, Flenner - 2012 |

22 | Beyond spectral clustering - tight relaxations of balanced graph cuts - HEIN, SETZER |

15 | Constrained 1-spectral clustering.
- Rangapuram, Hein
- 2012
(Show Context)
Citation Context ...n the other hand, has a solution that closely matches the solution of the original discrete NP-hard problem. Ideas from the image processing literature have recently motivated a new set of algorithms =-=[17, 18, 11, 12, 4, 15, 3, 2, 13, 10]-=- that can obtain tighter relaxations than those used by NMF and spectral clustering. These new algorithms all rely on the concept of total variation. Total variation techniques promote the formation o... |

10 | Convergence and energy landscape for cheeger cut clustering
- BRESSON, LAURENT, et al.
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10 | An MBO scheme on graphs for segmentation and image processing, UCLA
- Merkurjev, Kostic, et al.
- 2012
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10 | Total variation-based graph clustering algorithm for Cheeger ratio cuts
- Szlam, Bresson
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9 | E.: Clustering by nonnegative matrix factorization using graph random walk
- Yang, Hao, et al.
- 2012
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Citation Context ...ly adapts to handle either unsupervised or transductive clustering tasks. The results significantly outperform previous total variation algorithms and compare well against state-of-the-art approaches =-=[19, 20, 1]-=-. We name our approach Multiclass Total Variation clustering (MTV-clustering). 2 The Multiclass Balanced-Cut Model Given a weighted graph G = (V,W ) we let V = {x1, . . . ,xN} denote the vertex set an... |

5 | Fast multiclass segmentation using diffuse interface methods on graphs, arXiv preprint arXiv:1302.3913
- Garcia-Cardona, Merkurjev, et al.
- 2013
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3 |
Amol Kapila, and Maryam Fazel. Clustering by leftstochastic matrix factorization
- Arora, Gupta
(Show Context)
Citation Context ...ly adapts to handle either unsupervised or transductive clustering tasks. The results significantly outperform previous total variation algorithms and compare well against state-of-the-art approaches =-=[19, 20, 1]-=-. We name our approach Multiclass Total Variation clustering (MTV-clustering). 2 The Multiclass Balanced-Cut Model Given a weighted graph G = (V,W ) we let V = {x1, . . . ,xN} denote the vertex set an... |

3 | Multi-Class Transductive Learning based on `1 - Bresson, Tai, et al. - 2012 |