Generalized Median Graphs: Theory and Applications

Abstract

We study the so-called Generalized Median graph problem where the task is to to construct a prototype (i.e., a 'model') from an input set of graphs. The problem finds applications in many vision (e.g., object recognition) and learning problems where graphs are increasingly being adopted as a representation tool. Existing techniques for this problem are evolutionary search based; in this paper, we propose a polynomial time algorithm based on a linear programming formulation. We present an additional bi-level method to obtain solutions arbitrarily close to the optimal in non-polynomial time (in worst case). Within this new framework, one can optimize edit distance functions that capture similarity by considering vertex labels as well as the graph structure simultaneously. In context of our motivating application, we discuss experiments on molecular image analysis problems -the methods will provide the basis for building a topological map of all pairs of the human chromosome.

Gauging Association Patterns of Chromosome Territories via Chromatic Median

“…Distance between two graphs is usually defined as their edit distance. A variant, called Generalized Median Graph (GMG) [18], uses a generalized distance function that considers both vertex labels and edge weights, and produces near optimal solutions in polynomial time. In the context of finding association patterns of chromosome territories, GMG, as well as any other median graph method, has two major limitations.…”

“…Recently, a suitable model based upon integer programming was proposed by Stojkovic et al [20]. Although, it gives better experimental results than GMG in [18], its exponential running time (due to the nature of integer programming formulation) limits its application to only small-size input data sets.…”

“…To alleviate this problem, we take the average similarity (or dissimilarity) between the generated CAP-graph and the input dataset as the quality evaluation criteria. In order to produce an equal-footing comparison of our method with [18] and [20], we provide three values for our output. The objective of our model is to minimize the average Jaccard distance, while [18] shows Jaccard similarity, and [20] shows Sorensen index.…”

“…Table 3 shows all of the three evaluations. The last row shows the improvement of our result (if we denote the Jaccard similarity (Sorensen index) we obtain as s 1 , and the Jaccard similarity (Sorensen index) obtained from [20] or [18] as s 2 , then the improvement would be (s 1 − s 2 )/s 2 ). From Table 3, it is clear that our method outperforms all existing approaches.…”

“…New association patterns discovered. For the association pattern, we compare the edges in our association pattern with those in the graph [22] generated by GMG [18] using the same dataset. Comparison shows that our method can discover all edges (or associations) yielded by GMG.…”