Batch kernel SOM and related Laplacian methods for social network analysis

Large graphs are natural mathematical models for describing the structure of the data in a wide variety of fields, such as web mining, social networks, information retrieval, biological networks, etc. For all these applications, automatic tools are required to get a synthetic view of the graph and to reach a good understanding of the underlying problem. In particular, discovering groups of tightly connected vertices and understanding the relations between those groups is very important in practice. This paper shows how a kernel version of the batch Self Organizing Map can be used to achieve these goals via kernels derived from the Laplacian matrix of the graph, especially when it is used in conjunction with more classical methods based on the spectral analysis of the graph. The proposed method is used to explore the structure of a medieval social network modeled through a weighted graph that has been directly built from a large corpus of agrarian contracts

Advances in dissimilarity-based data visualisation

Gisbrecht A. Advances in dissimilarity-based data visualisation. Bielefeld: Universitätsbibliothek Bielefeld; 2015

Relational topographic Maps

Hammer B, Hasenfuss A. Relational topographic Maps. IfI Technical reports. Clausthal-Zellerfeld: Clausthal University of Technology; 2007

Relational topographic maps

Hasenfuss A, Hammer B. Relational topographic maps. In: Berthold MR, Shawe-Taylor J, Lavrac N, eds. Advances in Intelligent Data Analysis VII, Proceedings of the 7th International Symposium on Intelligent Data Analysis. Vol 4723. Berlin: Springer; 2007: 93-105

Topographic Mapping of Large Dissimilarity Data Sets

Hammer B, Hasenfuss A. Topographic Mapping of Large Dissimilarity Data Sets. Neural Computation. 2010;22(9):2229-2284.Topographic maps such as the self-organizing map (SOM) or neural gas (NG) constitute powerful data mining techniques that allow simultaneously clustering data and inferring their topological structure, such that additional features, for example, browsing, become available. Both methods have been introduced for vectorial data sets; they require a classical feature encoding of information. Often data are available in the form of pairwise distances only, such as arise from a kernel matrix, a graph, or some general dissimilarity measure. In such cases, NG and SOM cannot be applied directly. In this article, we introduce relational topographic maps as an extension of relational clustering algorithms, which offer prototype-based representations of dissimilarity data, to incorporate neighborhood structure. These methods are equivalent to the standard (vectorial) techniques if a Euclidean embedding exists, while preventing the need to explicitly compute such an embedding. Extending these techniques for the general case of non-Euclidean dissimilarities makes possible an interpretation of relational clustering as clustering in pseudo-Euclidean space. We compare the methods to well-known clustering methods for proximity data based on deterministic annealing and discuss how far convergence can be guaranteed in the general case. Relational clustering is quadratic in the number of data points, which makes the algorithms infeasible for huge data sets. We propose an approximate patch version of relational clustering that runs in linear time. The effectiveness of the methods is demonstrated in a number of examples

Topographic mapping of large dissimilarity data sets

Hammer B, Hasenfuss A. Topographic Mapping of Large Dissimilarity Data Sets. Neural Computation. 2010;22(9):2229-2284.Topographic maps such as the self-organizing map (SOM) or neural gas (NG) constitute powerful data mining techniques that allow simultaneously clustering data and inferring their topological structure, such that additional features, for example, browsing, become available. Both methods have been introduced for vectorial data sets; they require a classical feature encoding of information. Often data are available in the form of pairwise distances only, such as arise from a kernel matrix, a graph, or some general dissimilarity measure. In such cases, NG and SOM cannot be applied directly. In this article, we introduce relational topographic maps as an extension of relational clustering algorithms, which offer prototype-based representations of dissimilarity data, to incorporate neighborhood structure. These methods are equivalent to the standard (vectorial) techniques if a Euclidean embedding exists, while preventing the need to explicitly compute such an embedding. Extending these techniques for the general case of non-Euclidean dissimilarities makes possible an interpretation of relational clustering as clustering in pseudo-Euclidean space. We compare the methods to well-known clustering methods for proximity data based on deterministic annealing and discuss how far convergence can be guaranteed in the general case. Relational clustering is quadratic in the number of data points, which makes the algorithms infeasible for huge data sets. We propose an approximate patch version of relational clustering that runs in linear time. The effectiveness of the methods is demonstrated in a number of examples