Graph-based quadratic optimization: A fast evolutionary approach

Quadratic optimization lies at the very heart of many structural pattern recognition and computer vision problems, such as graph matching, object recognition, image segmentation, etc., and it is therefore of crucial importance to devise algorithmic solutions that are both efficient and effective. As it turns out, a large class of quadratic optimization problems can be formulated in terms of so-called “standard quadratic programs” (StQPs), which ask for finding the extrema of a quadratic polynomial over the standard simplex. Computationally, the standard approach for attacking this class of problems is to use replicator dynamics, a well-known family of algorithms from evolutionary game theory inspired by Darwinian selection processes. Despite their effectiveness in finding good solutions in a variety of applications, however, replicator dynamics suffer from being computationally expensive, as they require a number of operations per step which grows quadratically with the dimensionality of the problem being solved. In order to avoid this drawback, in this paper we propose a new population game dynamics (InImDyn) which is motivated by the analogy with infection and immunization processes within a population of “players.” We prove that the evolution of our dynamics is governed by a quadratic Lyapunov function, representing the average population payoff, which strictly increases along non-constant trajectories and that local solutions of StQPs are asymptotically stable (i.e., attractive) points. Each step of InImDyn is shown to have a linear time/space complexity, thereby allowing us to use it as a more efficient alternative to standard approaches for solving StQPs and related optimization problems. Indeed, we demonstrate experimentally that InImDyn is orders of magnitude faster than, and as accurate as, replicator dynamics on various applications ranging from tree matching to image registration, matching and segmentation.     

Approximating the maximum weight clique using replicator dynamics

Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (a clique) having the largest total weight. This is a generalization of the problem of finding the maximum cardinality clique of an unweighted graph, which is the special case of the MWCP when all vertex weights are equal. The problem is NP-hard for arbitrary graphs, and so is the problem of approximating it within a constant factor. We present a parallel, distributed heuristic for approximating the MWCP based on dynamics principles. It centers around a continuous characterization of the MWCP (a purely combinatorial problem), and lets it be formulated in terms of continuous quadratic programming. One drawback is the presence of spurious solutions, and we present their characterizations. To avoid them we introduce a regularized continuous formulation of the MWCP and show how it completely solves the problem. The formulation naturally maps onto a parallel, distributed computational network whose dynamical behavior is governed by the replicator equations. These are dynamical systems introduced in evolutionary game theory and population genetics to model evolutionary processes on a macroscopic scale. We present theoretical results which guarantee that the solutions provided by our clique finding replicator network are actually those sought. Experimental results confirm the effectiveness of the proposed approach.     

TESTING FOR WHITE NOISE AGAINST MULTIMODAL SPECTRAL ALTERNATIVES

Abstract. The test for white noise proposed by Reschenhofer ( Biometrika 76 (1989) 629–32) is adaptive in that the form of its test statistic depends on the data. It proceeds (i) by using the real part of the finite Fourier transform of the data for the selection of the form of the test statistic and (ii) by applying the selected test statistic to the imaginary part.
In this paper it is shown that independently of using information contained in the real part to construct a test statistic for the imaginary part, information contained in the imaginary part can be used to construct a test statistic for the real part. Because of the independence of these tests, they can easily be combined.
Then a further test is proposed which again is adaptive but where both the real and imaginary parts contribute simultaneously to a single test statistic.