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Fast Inexact Graph Matching with Applications in Statistical Connectomics.

Mon, 12/26/2011 - 15:01 — SomNambul
Authors: Joshua T. Vogelstein, R. Jacob Vogelstein, Carey E. Priebe, Donniell E. Fishkind, John M. Conroy, Louis J. Podrazik, Steven G. Kratzer
Subjects: Optimization and Control
link: http://arxiv.org/abs/1112.5507
Abstract

It is becoming increasingly popular to represent myriad and diverse data sets
as graphs. When the labels of vertices of these graphs are unavailable, graph
matching (GM)---the process of determining which permutation assigns vertices
of one graph to those of another---is a computationally daunting problem. This
work presents an inexact strategy for GM. Specifically, we frame GM as a
quadratic assignment problem, and then relax the feasible region to its convex
hull. We prove that our relaxed optimization function has the same solution as
the original problem, yet it is continuously differentiable. Because the
objective function is not necessarily convex, we consider multiple principled
initializations. Performance exceeds the previous state-of-the-art in all of 16
benchmark tests. Moreover, this approach is fast, scaling cubically with the
number of vertices, requiring only about a minute on a laptop for graphs with a
few hundred vertices. We illustrate this approach via a brain-graph application
(the Caenorhabditis elegans "connectome"). We find that we can find the optimal
solution for nearly every random permutation of the connectome that we sample.
Although this strategy already natively operates on weighted graphs, either
directed or undirected, we propose a number of possible extensions, and make
all code available.

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