Sebastian Risau-Gusman

Deterministic walks in random networks: an application to thesaurus graphs

In a landscape composed of N randomly distributed sites in Euclidean space, a walker (“tourist”) goes to the nearest one that has not been visited in the last τ steps. This procedure leads to trajectories composed of a transient part and a final cyclic attractor of period p. The tourist walk presents a simple scaling with respect to τ and can be performed in a wide range of networks that can be viewed as ordinal neighborhood graphs. As an example, we show that graphs defined by thesaurus dictionaries share some of the statistical properties of low-dimensional (d=2) Euclidean graphs and are easily distinguished from random link networks which correspond to the d→∞ limit. This approach furnishes complementary information to the usual clustering coefficient and mean minimum separation length.

Exploratory behavior, trap models, and glass transitions

A random walk is performed on a disordered landscape composed of N sites randomly and uniformly distributed inside a d-dimensional hypercube. The walker hops from one site to another with probability proportional to exp[-betaE(D)], where beta=1/T is the inverse of a formal temperature and E(D) is an arbitrary cost function which depends on the hop distance D. Analytic results indicate that, if E(D)=D(d) and N--> infinity, there exists a glass transition at beta(d)=pi(d/2)/[(d/2)Gamma(d/2)]. Below T(d), the average trapping time diverges and the system falls into an out-of-equilibrium regime with aging phenomena. A Lévy flight scenario and applications of exploratory behavior are considered.

Escaping from cycles through a glass transition.

A random walk is performed over a disordered media composed of N sites random and uniformly distributed inside a d-dimensional hypercube. The walker cannot remain in the same site and hops to one of its n neighboring sites with a transition probability that depends on the distance D between sites according to a cost function E(D). The stochasticity level is parametrized by a formal temperature T. In the case T=0, the walk is deterministic and ergodicity is broken: the phase space is divided in a O(N) number of attractor basins of two-cycles that trap the walker. For d=1, analytic results indicate the existence of a glass transition at T(1)=1/2 as N--> infinity. Below T1, the average trapping time in two-cycles diverges and an out-of-equilibrium behavior appears. Similar glass transitions occur in higher dimensions when the right cost function is chosen. We also present some results for the statistics of distances for Poisson spatial point processes.

Topology, phase transitions, and the spherical model

The topological hypothesis states that phase transitions should be related to changes in the topology of configuration space. The necessity of such changes has already been demonstrated. We characterize exactly the topology of the configuration space of the short range Berlin-Kac spherical model, for spins lying in hypercubic lattices of dimension d. We find a continuum of changes in the topology and also a finite number of discontinuities in some topological functions. We show, however, that these discontinuities do not coincide with the phase transitions which happen for d > or = 3, and conversely, that no topological discontinuity can be associated with them. This is the first short range, confining potential for which the existence of special topological changes are shown not to be sufficient to infer the occurrence of a phase transition.

Generalization properties of finite-size polynomial support vector machines

The learning properties of finite-size polynomial support vector machines are analyzed in the case of realizable classification tasks. The normalization of the high-order features acts as a squeezing factor, introducing a strong anisotropy in the patterns distribution in feature space. As a function of the training set size, the corresponding generalization error presents a crossover, more or less abrupt depending on the distributions anisotropy and on the task to be learned, between a fast-decreasing and a slowly decreasing regime. This behavior corresponds to the stepwise decrease found by Dietrich et al. [Phys. Rev. Lett. 82, 2975 (1999)] in the thermodynamic limit. The theoretical results are in excellent agreement with the numerical simulations.

Statistical mechanics of learning with soft margin classifiers.

We study the typical learning properties of the recently introduced soft margin classifiers (SMCs), learning realizable and unrealizable tasks, with the tools of statistical mechanics. We derive analytically the behavior of the learning curves in the regime of very large training sets. We obtain exponential and power laws for the decay of the generalization error towards the asymptotic value, depending on the task and on general characteristics of the distribution of stabilities of the patterns to be learned. The optimal learning curves of the SMCs, which give the minimal generalization error, are obtained by tuning the coefficient controlling the trade-off between the error and the regularization terms in the cost function. If the task is realizable by the SMC, the optimal performance is better than that of a hard margin support vector machine and is very close to that of a Bayesian classifier.