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This work applies cluster analysis as a unified approach for a wide range of vision applications, thereby combining the research domain of computer vision and that of machine learning. Cluster analysis is the formal study of algorithms and methods for recovering the inherent structure within a given dataset. Many problems of computer vision have precisely this goal, namely to find which visual entities belong to an inherent structure, e.g. in an image or in a database of images. For example, a meaningful structure in the context of image segmentation is a set of pixels which correspond to the same object in a scene. Clustering algorithms can be used to partition the pixels of an image into meaningful parts, which may correspond to different objects. In this work we focus on the problems of image segmentation and image database organization. The visual entities to consider are pixels and images, respectively. Our first contribution in this work is a novel partitional (flat) clustering algorithm. The algorithm uses pairwise representation, where the visual objects (pixels,

In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.

We present a new energy-minimization framework for the graph isomorphism problem that is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid-1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feature of this formulation is that a clear one-to-one correspondence exists between the solutions of the quadratic program and those in the original, combinatorial problem. To solve the program we use the so-called replicator equations—a class of straightforward continuous- and discrete-time dynamical systems developed in various branches of theoretical biology. We show how, despite their inherent inability to escape from local solutions, they nevertheless provide experimental results that are competitive with those obtained using more elaborate mean-field annealing heuristics.

Given a graph G whose adjacency matrix is A, the Motzkin-Strauss formulation of the Maximum-Clique Problem is the quadratic program maxfx T Axjx T e = 1; x 0g. It is well known that the global optimum value of this QP is (1 \Gamma 1=!(G)), where !(G) is the clique number of G. Here, we characterize the following: 1) first order optimality 2) second order optimality 3) local optimality 4) strict local. These characterizations reveal interesting underlying discrete structures, and are polynomial time verifiable. A parametrization of the Motzkin-Strauss QP is then introduced and its properties are investigated. Finally, an extension of the Motzkin-Strauss formulation is provided for the weighted clique number of a graph and this is used to derive a maximin characterization of perfect graphs. 1 Introduction 1.1 The Problem of Interest Let A be the adjacency matrix of a graph G. Consider the Motzkin-Strauss formulation (also called the Motzkin-Strauss QP) of the Maximum Clique Pro...

this paper, it is shown how to take advantage of the properties of these models to approximately solve the maximum clique problem, a well-known intractable optimization problem which has practical applications in various fields. The approach is based on a result by Motzkin and Straus which naturally leads to formulate the problem in a manner that is readily mapped onto a relaxation labeling network. Extensive simulations have demonstrated the validity of the proposed model, both in terms of quality of solutions and speed. Maximum clique problem, relaxation labeling processes, neural networks, optimization. 1 INTRODUCTION

The problem of selecting a subset of relevant features in a potentially overwhelming quantity of data is classic and found in many branches of science including --- examples in computer vision, text processing and more recently bioinformatics are abundant. In this work we present a definition of "relevancy" based on spectral properties of the Affinity (or Laplacian) of the features' measurement matrix. The feature selection process is then based on a continuous ranking of the features defined by a least-squares optimization process. A remarkable property of the feature relevance function is that sparse solutions for the ranking values naturally emerge as a result of a "biased non-negativity" of a key matrix in the process. As a result, a simple least-squares optimization process converges onto a sparse solution, i.e., a selection of a subset of features which form a local maxima over the relevance function. The feature selection algorithm can be embedded in both unsupervised and supervised inference problems and empirical evidence show that the feature selections typically achieve high accuracy even when only a small fraction of the features are relevant.

Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. This is a generalization of the classical problem of finding the maximum cardinality clique of an unweighted graph, which arises as a special case of the MWCP when all the weights associated to the vertices are equal. The problem is known to be NP -hard for arbitrary graphs and, according to recent theoretical results, so is the problem of approximating it within a constant factor. Although there has recently been much interest around neural network algorithms for the unweighted maximum clique problem, no effort has been directed so far towards its weighted counterpart. In this paper, we present a parallel, distributed heuristic for approximating the MWCP based on dynamics principles developed and studied in various branches of mathematical biology. The proposed framework centers aroun...

In this paper, a new heuristic for approximating the maximum clique problem is proposed, based on a detailed analysis of a class of continuous optimization models which yield a complete solution to this NP-hard combinatorial problem. The idea is to alter a regularization parameter iteratively in such a way that an iterative procedure with the updated parameter value would avoid unwanted, inefficient local solutions, i.e., maximal cliques which contain less than the maximum possible number of vertices. The local search procedure is performed with the help of the replicator dynamics, and the regularization parameter is chosen deliberately as to render dynamical instability of the (formerly) stable solutions which we want to discard in order to get an improvement. In this respect, the proposed procedure differs from usual simulated annealing approaches which mostly use a "black-box" cooling schedule. To demonstrate the validity of this approach, we report on the performance applied to sel...

INTRODUCTION Let G = (V; E) be an undirected graph, where V = f1; \Delta \Delta \Delta ; ng is the set of vertices, and E ` V \Theta V is the set of edges. Vertices i and j are called adjacent if they are connected by an edge. A clique of G is a subset of V in which every pair of vertices is adjacent. A clique C is called maximal if no strict superset of C is a clique. The highest-cardinality maximal clique is called a maximum clique. The maximum clique problem is to find a maximum clique in a given graph G. The problem is NP-hard [1], even to approximate well [2]. 0 The authors thank J. Shawe-Taylor for