Miquel Ferrer

Generalized median graph computation by means of graph embedding in vector spaces

The median graph has been presented as a useful tool to represent a set of graphs. Nevertheless its computation is very complex and the existing algorithms are restricted to use limited amount of data. In this paper we propose a new approach for the computation of the median graph based on graph embedding. Graphs are embedded into a vector space and the median is computed in the vector domain. We have designed a procedure based on the weighted mean of a pair of graphs to go from the vector domain back to the graph domain in order to obtain a final approximation of the median graph. Experiments on three different databases containing large graphs show that we succeed to compute good approximations of the median graph. We have also applied the median graph to perform some basic classification tasks achieving reasonable good results. These experiments on real data open the door to the application of the median graph to a number of more complex machine learning algorithms where a representative of a set of graphs is needed.

Graph-Based k-Means Clustering: A Comparison of the Set Median versus the Generalized Median Graph

In this paper we propose the application of the generalized median graph in a graph-based k-means clustering algorithm. In the graph-based k-means algorithm, the centers of the clusters have been traditionally represented using the set median graph. We propose an approximate method for the generalized median graph computation that allows to use it to represent the centers of the clusters. Experiments on three databases show that using the generalized median graph as the clusters representative yields better results than the set median graph.

Median graph: A new exact algorithm using a distance based on the maximum common subgraph

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.

Median graphs: A genetic approach based on new theoretical properties

Given a set of graphs, the median graph has been theoretically presented as a useful concept to infer a representative of the set. However, the computation of the median graph is a highly complex task and its practical application has been very limited up to now. In this work we present two major contributions. On one side, and from a theoretical point of view, we show new theoretical properties of the median graph. On the other side, using these new properties, we present a new approximate algorithm based on the genetic search, that improves the computation of the median graph. Finally, we perform a set of experiments on real data, where none of the existing algorithms for the median graph computation could be applied up to now due to their computational complexity. With these results, we show how the concept of the median graph can be used in real applications and leaves the box of the only-theoretical concepts, demonstrating, from a practical point of view, that can be a useful tool to represent a set of graphs.

Generalized median string computation by means of string embedding in vector spaces

Highlights? Generalized median string is an important concept in structural pattern recognition. ? We present an approximate method for the np-hard generalized median string problem. ? Superior performance is demonstrated against other methods in a series of experiments. In structural pattern recognition the median string has been established as a useful tool to represent a set of strings. However, its exact computation is complex and of high computational burden. In this paper we propose a new approach for the computation of median string based on string embedding. Strings are embedded into a vector space and the median is computed in the vector domain. We apply three different inverse transformations to go from the vector domain back to the string domain in order to obtain a final approximation of the median string. All of them are based on the weighted mean of a pair of strings. Experiments show that we succeed to compute good approximations of the median string.

A generic framework for median graph computation based on a recursive embedding approach

The median graph has been shown to be a good choice to obtain a representative of a set of graphs. However, its computation is a complex problem. Recently, graph embedding into vector spaces has been proposed to obtain approximations of the median graph. The problem with such an approach is how to go from a point in the vector space back to a graph in the graph space. The main contribution of this paper is the generalization of this previous method, proposing a generic recursive procedure that permits to recover the graph corresponding to a point in the vector space, introducing only the amount of approximation inherent to the use of graph matching algorithms. In order to evaluate the proposed method, we compare it with the set median and with the other state-of-the-art embedding-based methods for the median graph computation. The experiments are carried out using four different databases (one semi-artificial and three containing real-world data). Results show that with the proposed approach we can obtain better medians, in terms of the sum of distances to the training graphs, than with the previous existing methods.

Synthesis of median spectral graph

In pattern recognition, median computation is an important technique for capturing the important information of a given set of patterns but it has the main drawback of its exponential complexity. Moreover, the Spectral Graph techniques can be used for the fast computation of the approximate graph matching error, with a considerably reduced execution complexity. In this paper, we merge both methods to define the Median Spectral Graphs. With the use of the Spectral Graph theories, we find good approximations of median graph. Experiments on randomly generated graphs demonstrate that this method works well and it is robust against noise.

Improving bipartite graph matching by assessing the assignment confidence

We substantially reduce the overestimation of the approximate graph edit distance.We define several heuristics in order to quantify the confidence of individual node assignments.We define a generic approach that can be applied with various search techniques Due to the ability of graphs to represent properties of entities and binary relations at the same time, a growing interest in this representation formalism can be observed in various fields of pattern recognition. The availability of a distance measure is a basic requirement for pattern recognition. For graphs, graph edit distance is still one of the most popular distance measures. In the present paper we substantially improve the distance accuracy of a recent framework for the approximation of graph edit distance. The basic idea of our novel approach is to manipulate the initial assignment returned by the approximation algorithm such that the individual assignments are ordered according to their individual confidence. Next, the individual assignments are post processed in this specific order. In an experimental evaluation we show that the order of the assignments plays a crucial role for the resulting distance accuracy. Moreover, we empirically verify that our novel generalization is able to generate approximations which are very near to the exact edit distance (in contrast with the original framework).

Approximation of Graph Edit Distance in Quadratic Time

The basic idea of a recent graph matching framework is to reduce the problem of graph edit distance (GED) to an instance of a linear sum assignment problem (LSAP). The optimal solution for this simplified GED problem can be computed in cubic time and is eventually used to derive a suboptimal solution for the original GED problem. Yet, for large scale graphs and/or large scale graph sets the cubic time complexity remains a severe handicap of this procedure. Therefore, we propose to use suboptimal algorithms – with quadratic rather than cubic time complexity – for solving the underlying LSAP. In particular, we introduce several greedy assignment algorithms for approximating GED. In an experimental evaluation we show that there is great potential for further speeding up the GED computation. Moreover, we empirically confirm that the distances obtained by this procedure remain sufficiently accurate for graph based pattern classification.

Greedy Graph Edit Distance

In pattern recognition and data mining applications, where the underlying data is characterized by complex structural relationships, graphs are often used as a formalism for object representation. Yet, the high representational power and flexibility of graphs is accompanied by a significant increase of the complexity of many algorithms. For instance, exact computation of pairwise graph dissimilarity, i.e.i?źdistance, can be accomplished in exponential time complexity only. A previously introduced approximation framework reduces the problem of graph comparison to an instance of a linear sum assignment problem which allows graph dissimilarity computation in cubic time. The present paper introduces an extension of this approximation framework that runs in quadratic time. We empirically confirm the scalability of our extension with respect to the run time, and moreover show that the quadratic approximation leads to graph dissimilarities which are sufficiently accurate for graph based pattern classification.