## Optimal cluster preserving embedding of nonmetric proximity data (2003)

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Venue: | IEEE Trans. Pattern Analysis and Machine Intelligence |

Citations: | 42 - 4 self |

### BibTeX

@ARTICLE{Roth03optimalcluster,

author = {Volker Roth and Julian Laub and Motoaki Kawanabe and Joachim M. Buhmann},

title = {Optimal cluster preserving embedding of nonmetric proximity data},

journal = {IEEE Trans. Pattern Analysis and Machine Intelligence},

year = {2003},

volume = {25},

pages = {2003}

}

### Years of Citing Articles

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### Abstract

Abstract—For several major applications of data analysis, objects are often not represented as feature vectors in a vector space, but rather by a matrix gathering pairwise proximities. Such pairwise data often violates metricity and, therefore, cannot be naturally embedded in a vector space. Concerning the problem of unsupervised structure detection or clustering, in this paper, a new embedding method for pairwise data into Euclidean vector spaces is introduced. We show that all clustering methods, which are invariant under additive shifts of the pairwise proximities, can be reformulated as grouping problems in Euclidian spaces. The most prominent property of this constant shift embedding framework is the complete preservation of the cluster structure in the embedding space. Restating pairwise clustering problems in vector spaces has several important consequences, such as the statistical description of the clusters by way of cluster prototypes, the generic extension of the grouping procedure to a discriminative prediction rule, and the applicability of standard preprocessing methods like denoising or dimensionality reduction. Index Terms—Clustering, pairwise proximity data, cost function, embedding, MDS. 1

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Citation Context ... PCA, we can also apply any other standard method for dimensionality reduction or visualization, such as projection pursuit [5], local linear embedding (LLE) [13], Isomap [18] or Self-organizing maps [7]. We now have presented various aspects of constant shift embedding. The next section contains a detailed analysis of the dierences to the classical MDS approach. 3.2 Comparison between MDS and co... |

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Citation Context ...ering cost function and the classical k-means grouping criterion in the embedding space. 2 PROXIMITY-BASED CLUSTERING Unsupervised grouping or clustering aims at extracting hidden structure from data =-=[4]-=-. The term data refers to both a set of objects and a set of corresponding object representations resulting from some physical measurement process. Different types of object representations are possib... |

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Citation Context ...cond case we are given a n n pairwise proximity matrix. The problem of grouping vectorial data has been widely studied in the literature, and many clustering algorithms have been proposed (see e.g. [=-=1]-=-[3]). One of the most popular methods is k-means clustering. It derives a set of k prototype vectors which quantize the data set with minimal quantization error. Partitioning proximity data is conside... |

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Citation Context ...the second case, we are given a n n pairwise proximity matrix. The problem of grouping vectorial data has been widely studied in the literature, and many clustering algorithms have been proposed [4], =-=[5]-=-. One of the most popular methods is k-means clustering. It derives a set of k prototypesROTH ET AL.: OPTIMAL CLUSTER PRESERVING EMBEDDING OF NONMETRIC PROXIMITY DATA 1541 vectors which quantize the d... |

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Citation Context ...ralized version S c is identical and unique. We now state the following: Theorem 3.1. D derives from a squared Euclidian distance if and only if S c is positive semi-denite. Proof. [19] referring to [=-=20-=-] or the following simple argument: ()) Since D derives from a squared Euclidian distance, we can take vectors x 1 ; : : : x n 2 R d (d 6 n 1) which satisfy D ij = kx i x j k 2 . Then, D c ij = D ij ... |

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Citation Context ...y a complex alignment algorithm. This procedure yields a matrix gathering the pairwise relations between the original objects, which may be the starting point of intelligent data analysis, see, e.g., =-=[3]-=- for an example of such a procedure in the field of image retrieval. We like to stress here that such a matrix is by no means naturally related to the common viewpoint of objects being embedded in som... |

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Citation Context ...t given the exactly reconstructed vectors in R n 2 found by loss-free kernel PCA, we can also apply any other standard method for dimensionality reduction or visualization, such as projection pursuit =-=[5]-=-, local linear embedding (LLE) [13], Isomap [18] or Self-organizing maps [7]. We now have presented various aspects of constant shift embedding. The next section contains a detailed analysis of the di... |

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Citation Context ...ortion of the distance D in MDS is measured by several criteria such as SSTRESS(D; ~ D) = tr(D ~ D) 2 ; (19) STRAIN(D; ~ D) = tr n Q(D ~ D)Q(D ~ D) o ; (20) where D and ~ D are distance matrices, cf. =-=[8]-=-. In the following we will compare the distortions of MDS and constant shift method by the measure STRAIN. This measure can be transformed as STRAIN(D; ~ D) = tr(QDQ Q ~ DQ) 2 = 4tr(S c ~ S c ) 2 : Le... |

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Citation Context ...-dimensional representation of data such that the distortion of the pairwise dissimilarities Dij is minimal with respect to some cost function. One widely used cost function is the SSTRESS criterion, =-=[7]-=-: J Xn i;j1 !ij d 2 ij D2 ij 2 ; ð1Þ where dij kxi xjk are the transformed distances in lowdimensional space, and !ij are weights. Typically, these weights read: 1 !ij nðn 1ÞD2 ij ; !ij 1 P k;... |

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Citation Context ... This subset of clustering methods includes e.g. graph-theoretic approaches like several variations of Cut criteria [16], and many methods derived from an axiomatization of pairwise cost functions in =-=[12]-=-. From a theoretical viewpoint, cost-based clustering methods are interesting insofar, as many properties of the grouping solution can be derived by analyzing invariance properties of the cost functio... |

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Citation Context ...n-like.” The heuristic FASTA scoring method [23] was used for computing pairwise alignment scores which, in turn, were length-corrected (a Bayesian approach for correcting local alignments, following =-=[24]-=-) and normalized to the length of the alignment. From the pair-scores Sij, we derived dissimilarities by setting Dij Sii þ Sjj 2Sij. 3 The eigenvalue spectrum of the centered matrix Sc shows some hi... |

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Citation Context ...scuss the relations of our embedding with graphtheoretic clustering methods (section 4). 2 Proximity-based clustering Unsupervised grouping or clustering aims at extracting hidden structure from data =-=[3-=-]. The term data refers to both a set of objects and a set of corresponding object representations resulting from some physical measurement process. Dierent types of object representations are possibl... |

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Citation Context ...ise Clustering cost function reads: H pc 1 X 2 k Pn Pn i1 j1 1 Mi Mj Dij Pn l1 Ml : ð3Þ The optimal assignments ^M are obtained by minimizing Hpc . The minimization itself is an NP hard problem =-=[11]-=-, and some approximation heuristics have been proposed: In [10], a mean field annealing framework has been presented (see the discussion in Section 4 of this work for some comments and new results on ... |

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Citation Context ...ect large dierences between the models considered. This may explain the somewhat surprising results of a large-scale comparison study of graph partitioning algorithms for image segmentation tasks in [=-=17]-=-. 5 Discussion and Conclusion We have introduced an optimal embedding procedure for pairwise clustering by means of constant shift embedding. For the class of shift-invariant clustering methods, it ou... |

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Citation Context ...ng member of SD, since the following theorem holds. Theorem 1. D derives from a squared Euclidian distance, i.e., Dij kxi xjk 2 , if and only if Sc is positive semidefinite. Proof. [12] referring to =-=[13]-=-. tu For general dissimilarities, Sc will be indefinite. By shifting its diagonal elements, however, we can transform it into a positive semidefinite matrix: The following lemma states that, for any m... |

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Citation Context ..., the dierences between Averaged Association, Averaged Cut and Normalized Cut become vanishingly small. In such situations, all three methods can be reasonably well approximated by k-means (see also [9]). A graph G = (V; E) can be partitioned into disjoint sets A ; = 1; : : : ; k by removing edges: S k =1 A = V; A \ A = ; for 6= . Following [16], we dene the dissimilarity between the ... |