How Many Dissimilarity/Kernel Self Organizing Map Variants Do We Need?

In numerous applicative contexts, data are too rich and too complex to be represented by numerical vectors. A general approach to extend machine learning and data mining techniques to such data is to really on a dissimilarity or on a kernel that measures how different or similar two objects are. This approach has been used to define several variants of the Self Organizing Map (SOM). This paper reviews those variants in using a common set of notations in order to outline differences and similarities between them. It discusses the advantages and drawbacks of the variants, as well as the actual relevance of the dissimilarity/kernel SOM for practical applications

Developments in the theory of randomized shortest paths with a comparison of graph node distances

There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into account the global structure of the graph. In this article, we develop the theory of one family of graph node distances, known as the randomized shortest path dissimilarity, which has its foundation in statistical physics. We show that the randomized shortest path dissimilarity can be easily computed in closed form for all pairs of nodes of a graph. Moreover, we come up with a new definition of a distance measure that we call the free energy distance. The free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it defines a metric, in addition to which it satisfies the graph-geodetic property. The derivation and computation of the free energy distance are also straightforward. We then make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time, or resistance distance. This comparison focuses on the applicability of the distances in graph node clustering and classification. The comparison, in general, shows that the parametrized distances perform well in the tasks. In particular, we see that the results obtained with the free energy distance are among the best in all the experiments.Comment: 30 pages, 4 figures, 3 table

Graph ambiguity

In this paper, we propose a rigorous way to define the concept of ambiguity in the domain of graphs. In past studies, the classical definition of ambiguity has been derived starting from fuzzy set and fuzzy information theories. Our aim is to show that also in the domain of the graphs it is possible to derive a formulation able to capture the same semantic and mathematical concept. To strengthen the theoretical results, we discuss the application of the graph ambiguity concept to the graph classification setting, conceiving a new kind of inexact graph matching procedure. The results prove that the graph ambiguity concept is a characterizing and discriminative property of graphs. (C) 2013 Elsevier B.V. All rights reserved

Analysis of a data matrix and a graph: Metagenomic data and the phylogenetic tree

In biological experiments researchers often have information in the form of a graph that supplements observed numerical data. Incorporating the knowledge contained in these graphs into an analysis of the numerical data is an important and nontrivial task. We look at the example of metagenomic data---data from a genomic survey of the abundance of different species of bacteria in a sample. Here, the graph of interest is a phylogenetic tree depicting the interspecies relationships among the bacteria species. We illustrate that analysis of the data in a nonstandard inner-product space effectively uses this additional graphical information and produces more meaningful results.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS402 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org