A Comparison between Two Representatives of a Set of Graphs: Median vs. Barycenter Graph

Bardají, Itziar; Ferrer, Miquel; Sanfeliu, Alberto

doi:10.1007/978-3-642-14980-1_14

Cited by 6 publications

(8 citation statements)

References 17 publications

(21 reference statements)

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“…However, it is possible to extend the applicability of the k-means algorithm to a wider class of input domains, by suitably redefining the concept of cluster representative. A well-known technique to model a (finite) input set of arbitrarily defined objects is the so-called Minimum Sum Of Distances (MinSOD) representative [3,7,17]. The representative element is selected as the pattern of the modeled set that minimizes the sum of the dissimilarity values.…”

Section: The K-means Algorithm and Related Initialization Techniquesmentioning

confidence: 99%

Two density-based k-means initialization algorithms for non-metric data clustering

Bianchi

Livi

Rizzi

2015

Pattern Anal Applic

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In this paper, we propose a density-based clusters' representatives selection algorithm that identifies the most central patterns from the dense regions in the dataset. The method, which has been implemented using two different strategies, is applicable to input spaces with no trivial geometry. Our approach exploits a probability density function built through the Parzen estimator, which relies on a (not necessarily metric) dissimilarity measure. Being a representatives extractor a general-purpose algorithm, our method is obviously applicable in different contexts. However, to test the proposed procedure, we specifically consider the problem of initializing the k-means algorithm. We face problems defined on standard real-valued vectors, labeled graphs, and finally sequences of real-valued vectors and sequences of characters. The obtained results demonstrate the effectiveness of the proposed representative selection method with respect to other state-of-the-art solutions

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Section: The K-means Algorithm and Related Initialization Techniquesmentioning

confidence: 99%

Two density-based k-means initialization algorithms for non-metric data clustering

Bianchi

Livi

Rizzi

2015

Pattern Anal Applic

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“…The work by Jiang et al simultaneously triggered two directions of research. The first direction focused on devising algorithms for minimizing different formulations of the sample Fréchet function [2,26,27,28,30,31,41,64,67]. The second direction developed prototype-based clustering algorithms partly resulting in novel ways to compute a sample mean [6,8,11,29,39,72].…”

Section: Related Workmentioning

confidence: 99%

“…Then M minimizes the right hand side of Equation (3). Since a midpoint of X and Y exists, a minimizer satisfies Equation (2). Thus, M is a midpoint.…”

Section: C3 Proof Of Corollary 36mentioning

confidence: 99%

Properties of the Sample Mean in Graph Spaces and the Majorize-Minimize-Mean Algorithm

Jain¹

2015

Preprint

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One of the most fundamental concepts in statistics is the concept of sample mean. Properties of the sample mean that are well-defined in Euclidean spaces become unwieldy or even unclear in graph spaces. Open problems related to the sample mean of graphs include: non-existence, non-uniqueness, statistical inconsistency, lack of convergence results of mean algorithms, non-existence of midpoints, and disparity to midpoints. We present conditions to resolve all six problems and propose a Majorize-Minimize-Mean (MMM) Algorithm. Experiments on graph datasets representing images and molecules show that the MMM-Algorithm best approximates a sample mean of graphs compared to six other mean algorithms.

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“…It is easy to understand that these two definitions of representative of a cluster and the sample-to-cluster dissimilarity strictly depend on the domain of the problem. For example, if X=G, where G is a set of graphs, the problem of deriving a representative graph is known as the set median graph computation [1], [5], [6].…”

Section: A Clustering Structured Data By K-meansmentioning

confidence: 99%

“…The element in the cluster that minimizes the Sum Of Distances (SOD) between itself and the other elements is a natural candidate for representing the cluster where more powerful abstraction techniques are not available [4]- [6]. The determination of the SOD-minimizing element can be seen as a meta-algorithm, since it only refers to the definition of a dissimilarity measure between samples.…”

Section: Introductionmentioning

confidence: 99%

On the Problem of Modeling Structured Data with the MinSOD Representative

Vescovo¹,

Livi²,

Massimo³

et al. 2014

IJCTE

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Abstract-The possibility to face pattern recognition problems directly on structured domains (e.g., multimedia data, strings, graphs) is fundamental to the effective solution of many interesting applications. In this paper, we deal with a clustering problem defined in the string domain, focusing on the problem of cluster representation in data domains where only a dissimilarity measure can be fixed. To this aim, we adopt the MinSOD (Minimum Sum of Distances) cluster representation technique, which defines the representative as the element of the cluster minimizing the sum of dissimilarities from all the other elements in the considered set. Since the precise computation of the MinSOD have a high computational cost, we propose a suboptimal procedure consisting in computing the representative of the cluster considering only a reduced pool of samples, instead of the whole set of objects in the cluster. We have carried out some tests in order to ascertain the sensitivity of the clustering procedure with respect to the number of samples in the pool used to compute the MinSOD. Results show a good robustness of the proposed procedure. The implementations are available as part of the SPARE library, which is available as an open source project.

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A Comparison between Two Representatives of a Set of Graphs: Median vs. Barycenter Graph

Cited by 6 publications

References 17 publications

Two density-based k-means initialization algorithms for non-metric data clustering

Two density-based k-means initialization algorithms for non-metric data clustering

Properties of the Sample Mean in Graph Spaces and the Majorize-Minimize-Mean Algorithm

On the Problem of Modeling Structured Data with the MinSOD Representative

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