Designing labeled graph classifiers by exploiting the R\'enyi entropy of the dissimilarity representation

Representing patterns as labeled graphs is becoming increasingly common in the broad field of computational intelligence. Accordingly, a wide repertoire of pattern recognition tools, such as classifiers and knowledge discovery procedures, are nowadays available and tested for various datasets of labeled graphs. However, the design of effective learning procedures operating in the space of labeled graphs is still a challenging problem, especially from the computational complexity viewpoint. In this paper, we present a major improvement of a general-purpose classifier for graphs, which is conceived on an interplay between dissimilarity representation, clustering, information-theoretic techniques, and evolutionary optimization algorithms. The improvement focuses on a specific key subroutine devised to compress the input data. We prove different theorems which are fundamental to the setting of the parameters controlling such a compression operation. We demonstrate the effectiveness of the resulting classifier by benchmarking the developed variants on well-known datasets of labeled graphs, considering as distinct performance indicators the classification accuracy, computing time, and parsimony in terms of structural complexity of the synthesized classification models. The results show state-of-the-art standards in terms of test set accuracy and a considerable speed-up for what concerns the computing time.Comment: Revised versio

How Many Pairwise Preferences Do We Need to Rank A Graph Consistently?

We consider the problem of optimal recovery of true ranking of $n$ items from a randomly chosen subset of their pairwise preferences. It is well known that without any further assumption, one requires a sample size of $\Omega(n^2)$ for the purpose. We analyze the problem with an additional structure of relational graph $G([n],E)$ over the $n$ items added with an assumption of \emph{locality}: Neighboring items are similar in their rankings. Noting the preferential nature of the data, we choose to embed not the graph, but, its \emph{strong product} to capture the pairwise node relationships. Furthermore, unlike existing literature that uses Laplacian embedding for graph based learning problems, we use a richer class of graph embeddings---\emph{orthonormal representations}---that includes (normalized) Laplacian as its special case. Our proposed algorithm, {\it Pref-Rank}, predicts the underlying ranking using an SVM based approach over the chosen embedding of the product graph, and is the first to provide \emph{statistical consistency} on two ranking losses: \emph{Kendall's tau} and \emph{Spearman's footrule}, with a required sample complexity of $O(n^2 \chi(\bar{G}))^{\frac{2}{3}}$ pairs, $\chi(\bar{G})$ being the \emph{chromatic number} of the complement graph $\bar{G}$. Clearly, our sample complexity is smaller for dense graphs, with $\chi(\bar G)$ characterizing the degree of node connectivity, which is also intuitive due to the locality assumption e.g. $O(n^\frac{4}{3})$ for union of $k$-cliques, or $O(n^\frac{5}{3})$ for random and power law graphs etc.---a quantity much smaller than the fundamental limit of $\Omega(n^2)$ for large $n$. This, for the first time, relates ranking complexity to structural properties of the graph. We also report experimental evaluations on different synthetic and real datasets, where our algorithm is shown to outperform the state-of-the-art methods.Comment: In Thirty-Third AAAI Conference on Artificial Intelligence, 201

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