Marcello Pelillo

The Maximum Clique Problem

In this paper we present a survey of results concerning algorithms, complexity, and applications of the maximum clique problem. We discuss enumerative and exact algorithms, heuristics, and a variety of other proposed methods. An up to date bibliography on the maximum clique and related problems is also provided.

Dominant Sets and Pairwise Clustering

We develop a new graph-theoretic approach for pairwise data clustering which is motivated by the analogies between the intuitive concept of a cluster and that of a dominant set of vertices, a notion introduced here which generalizes that of a maximal complete subgraph to edge-weighted graphs. We establish a correspondence between dominant sets and the extrema of a quadratic form over the standard simplex, thereby allowing the use of straightforward and easily implementable continuous optimization techniques from evolutionary game theory. Numerical examples on various point-set and image segmentation problems confirm the potential of the proposed approach

An iterative pruning algorithm for feedforward neural networks

The problem of determining the proper size of an artificial neural network is recognized to be crucial, especially for its practical implications in such important issues as learning and generalization. One popular approach for tackling this problem is commonly known as pruning and it consists of training a larger than necessary network and then removing unnecessary weights/nodes. In this paper, a new pruning method is developed, based on the idea of iteratively eliminating units and adjusting the remaining weights in such a way that the network performance does not worsen over the entire training set. The pruning problem is formulated in terms of solving a system of linear equations, and a very efficient conjugate gradient algorithm is used for solving it, in the least-squares sense. The algorithm also provides a simple criterion for choosing the units to be removed, which has proved to work well in practice. The results obtained over various test problems demonstrate the effectiveness of the proposed approach.

A new graph-theoretic approach to clustering and segmentation

We develop a framework for the image segmentation problem based on a new graph-theoretic formulation of clustering. The approach is motivated by the analogies between the intuitive concept of a cluster and that of a dominant set of vertices, a notion that generalizes that of a maximal complete subgraph to edge-weighted graphs. We also establish a correspondence between dominant sets and the extrema of a quadratic form over the standard simplex, thereby allowing us the use of continuous optimization techniques such as replicator dynamics from evolutionary game theory. Such systems are attractive as they can be coded in a few lines of any high-level programming language, can easily be implemented in a parallel network of locally interacting units, and offer the advantage of biological plausibility. We present experimental results on real-world images which show the effectiveness of the proposed approach.

Replicator Equations, Maximal Cliques, and Graph Isomorphism

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 equationsa 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.

Structured class-labels in random forests for semantic image labelling

In this paper we propose a simple and effective way to integrate structural information in random forests for semantic image labelling. By structural information we refer to the inherently available, topological distribution of object classes in a given image. Different object class labels will not be randomly distributed over an image but usually form coherently labelled regions. In this work we provide a way to incorporate this topological information in the popular random forest framework for performing low-level, unary classification. Our paper has several contributions: First, we show how random forests can be augmented with structured label information. In the second part, we introduce a novel data splitting function that exploits the joint distributions observed in the structured label space for learning typical label transitions between object classes. Finally, we provide two possibilities for integrating the structured output predictions into concise, semantic labellings. In our experiments on the challenging MSRC and CamVid databases, we compare our method to standard random forest and conditional random field classification results.

The Dynamics of Nonlinear Relaxation Labeling Processes

We present some new results which definitively explain thebehavior of the classical, heuristic nonlinear relaxation labelingalgorithm of Rosenfeld, Hummel, and Zucker in terms of theHummel-Zucker consistency theory and dynamical systems theory. Inparticular, it is shown that, when a certain symmetry condition is met,the algorithm possesses a Liapunov function which turns out to be (thenegative of) a well-known consistency measure. This follows almostimmediately from a powerful result of Baum and Eagon developed in thecontext of Markov chain theory. Moreover, it is seen that most of theessential dynamical properties of the algorithm are retained when thesymmetry restriction is relaxed. These properties are also shown tonaturally generalize to higher-order relaxation schemes. Someapplications and implications of the presented results are finallyoutlined.

Matching Hierarchical Structures Using Association Graphs

It is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a (derived) “association graph.” However, it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized, i.e. are trees, since maximal cliques are not constrained to preserve the partial order. Here we provide a solution to the problem of matching two trees, by constructing the association graph using the graph-theoretic concept of connectivity. We prove that in the new formulation there is a one-to-one correspondence between maximal cliques and maximal subtree isomorphisms, and show how to solve the matching problem using simple “replicator” dynamical systems developed in theoretical biology. Such continuous solutions to discrete problems can motivate analog and biological implementations. We illustrate the power of the approach by matching articulated and deformed shapes described by shock trees.

Polynomial-time metrics for attributed trees

We address the problem of comparing attributed trees and propose four novel distance measures centered around the notion of a maximal similarity common subtree. The proposed measures are general and defined on trees endowed with either symbolic or continuous-valued attributes and can be applied to rooted as well as unrooted trees. We prove that our measures satisfy the metric constraints and provide a polynomial-time algorithm to compute them. This is a remarkable and attractive property, since the computation of traditional edit-distance-based metrics is, in general, NP-complete, at least in the unordered case. We experimentally validate the usefulness of our metrics on shape matching tasks and compare them with (an approximation of) edit-distance.

Grouping with Asymmetric Affinities: A Game-Theoretic Perspective

Pairwise grouping and clustering approaches have traditionally worked under the assumption that the similarities or compatibilities between the elements to be grouped are symmetric. However, asymmetric compatibilities arise naturally in many areas of computer vision and pattern recognition. Hence, there is a need for a new generic approach to clustering and grouping that can deal with asymmetries in the compatibilities. In this paper, we present a generic framework for grouping and clustering derived from a game-theoretic formalization of the competition between the hypotheses of group membership, and apply it to perceptual grouping. In the proposed approach groups correspond to evolutionary stable strategies, a classic notion in evolutionary game theory. We also provide a combinatorial characterization of the stable strategies, and, hence, of the elements that belong to a group. Experiments show that our approach outperforms both state-of-the-art clustering-based perceptual grouping approacheswith symmetric compatibilities, and other approaches explicitly designed to make use of asymmetric compatibilities.