Abstract: Despite being very successful within the pattern recognition and machine learning community, graph-based methods are often unusable because of the lack of mathematical operations defined in graph domain. Graph embedding, which maps graphs to a vectorial space, has been proposed as a way to tackle these difficulties enabling the use of standard machine learning techniques. However, it is well known that graph embedding functions usually suffer from the loss of structural information. In this paper, we consider the hierarchical structure of a graph as a way to mitigate this loss of information. The hierarchical structure is constructed by topologically clustering the graph nodes and considering each cluster as a node in the upper hierarchical level. Once this hierarchical structure is constructed, we consider several configurations to define the mapping into a vector space given a classical graph embedding, in particular, we propose to make use of the stochastic graphlet embedding (SGE). Broadly speaking, SGE produces a distribution of uniformly sampled low-to-high-order graphlets as a way to embed graphs into the vector space. In what follows, the coarse-to-fine structure of a graph hierarchy and the statistics fetched by the SGE complements each other and includes important structural information with varied contexts. Altogether, these two techniques substantially cope with the usual information loss involved in graph embedding techniques, obtaining a more robust graph representation. This fact has been corroborated through a detailed experimental evaluation on various benchmark graph datasets, where we outperform the state-of-the-art methods.

Abstract: The combination of the output of classifiers has been one of the strategies used to improve classification rates in general purpose classification systems. Some of the most common approaches can be explained using the Bayes' formula. In this paper, we tackle the problem of the combination of classifiers using a non-Bayesian probabilistic framework. This approach permits us to derive two linear combination rules that minimize misclassification rates under some constraints on the distribution of classifiers. In order to show the validity of this approach we have compared it with other popular combination rules from a theoretical viewpoint using a synthetic data set, and experimentally using two standard databases: the MNIST handwritten digit database and the GREC symbol database. Results on the synthetic data set show the validity of the theoretical approach. Indeed, results on real data show that the proposed methods outperform other common combination schemes.

Abstract: Median graphs have been presented as a useful tool for capturing the essential information of a set of graphs. Nevertheless, computation of optimal solutions is a very hard problem. In this work we present a new and more efficient optimal algorithm for the median graph computation. With the use of a particular cost function that permits the definition of the graph edit distance in terms of the maximum common subgraph, and a prediction function in the backtracking algorithm, we reduce the size of the search space, avoiding the evaluation of a great amount of states and still obtaining the exact median. We present a set of experiments comparing our new algorithm against the previous existing exact algorithm using synthetic data. In addition, we present the first application of the exact median graph computation to real data and we compare the results against an approximate algorithm based on genetic search. These experimental results show that our algorithm outperforms the previous existing exact algorithm and in addition show the potential applicability of the exact solutions to real problems.