Generalized Approximate Survey Propagation for High-Dimensional Estimation

In Generalized Linear Estimation (GLE) problems, we seek to estimate a signal that is observed through a linear transform followed by a component-wise, possibly nonlinear and noisy, channel. In the Bayesian optimal setting, Generalized Approximate Message Passing (GAMP) is known to achieve optimal performance for GLE. However, its performance can significantly degrade whenever there is a mismatch between the assumed and the true generative model, a situation frequently encountered in practice. In this paper, we propose a new algorithm, named Generalized Approximate Survey Propagation (GASP), for solving GLE in the presence of prior or model mis-specifications. As a prototypical example, we consider the phase retrieval problem, where we show that GASP outperforms the corresponding GAMP, reducing the reconstruction threshold and, for certain choices of its parameters, approaching Bayesian optimal performance. Furthermore, we present a set of State Evolution equations that exactly characterize the dynamics of GASP in the high-dimensional limit

Quantum phase transitions of the diluted O(3) rotor model

We study the phase diagram and the quantum phase transitions of a site-diluted two-dimensional O(3) quantum rotor model by means of large-scale Monte-Carlo simulations. This system has two quantum phase transitions, a generic one for small dilutions, and a percolation transition across the lattice percolation threshold. We determine the critical behavior for both transitions and for the multicritical point that separates them. In contrast to the exotic scaling scenarios found in other random quantum systems, all these transitions are characterized by finite-disorder fixed points with power-law scaling. We relate our findings to a recent classification of phase transitions with quenched disorder according to the rare region dimensionality, and we discuss experiments in disordered quantum magnets.Comment: 11 pages, 14 eps figures, final version as publishe

Monte Carlo study of the scaling of universal correlation lengths in three-dimensional O(n) spin models

Using an elaborate set of simulational tools and statistically optimized methods of data analysis we investigate the scaling behavior of the correlation lengths of three-dimensional classical O($n$) spin models. Considering three-dimensional slabs $S^1\times S^1\times\mathbb{R}$, the results over a wide range of $n$ indicate the validity of special scaling relations involving universal amplitude ratios that are analogous to results of conformal field theory for two-dimensional systems. A striking mismatch of the $n\to\infty$ extrapolation of these simulations against analytical calculations is traced back to a breakdown of the identification of this limit with the spherical model.Comment: 18 pages, 9 figures, REVTeX4, slightly shortened, updated critical exponent estimate

Inference of stochastic nonlinear oscillators with applications to physiological problems

A new method of inferencing of coupled stochastic nonlinear oscillators is described. The technique does not require extensive global optimization, provides optimal compensation for noise-induced errors and is robust in a broad range of dynamical models. We illustrate the main ideas of the technique by inferencing a model of five globally and locally coupled noisy oscillators. Specific modifications of the technique for inferencing hidden degrees of freedom of coupled nonlinear oscillators is discussed in the context of physiological applications.Comment: 11 pages, 10 figures, 2 tables Fluctuations and Noise 2004, SPIE Conference, 25-28 May 2004 Gran Hotel Costa Meloneras Maspalomas, Gran Canaria, Spai

Generalized-ensemble simulations and cluster algorithms

The importance-sampling Monte Carlo algorithm appears to be the universally optimal solution to the problem of sampling the state space of statistical mechanical systems according to the relative importance of configurations for the partition function or thermal averages of interest. While this is true in terms of its simplicity and universal applicability, the resulting approach suffers from the presence of temporal correlations of successive samples naturally implied by the Markov chain underlying the importance-sampling simulation. In many situations, these autocorrelations are moderate and can be easily accounted for by an appropriately adapted analysis of simulation data. They turn out to be a major hurdle, however, in the vicinity of phase transitions or for systems with complex free-energy landscapes. The critical slowing down close to continuous transitions is most efficiently reduced by the application of cluster algorithms, where they are available. For first-order transitions and disordered systems, on the other hand, macroscopic energy barriers need to be overcome to prevent dynamic ergodicity breaking. In this situation, generalized-ensemble techniques such as the multicanonical simulation method can effect impressive speedups, allowing to sample the full free-energy landscape. The Potts model features continuous as well as first-order phase transitions and is thus a prototypic example for studying phase transitions and new algorithmic approaches. I discuss the possibilities of bringing together cluster and generalized-ensemble methods to combine the benefits of both techniques. The resulting algorithm allows for the efficient estimation of the random-cluster partition function encoding the information of all Potts models, even with a non-integer number of states, for all temperatures in a single simulation run per system size.Comment: 15 pages, 6 figures, proceedings of the 2009 Workshop of the Center of Simulational Physics, Athens, G

Detecting Generalized Synchronization Between Chaotic Signals: A Kernel-based Approach

A unified framework for analyzing generalized synchronization in coupled chaotic systems from data is proposed. The key of the proposed approach is the use of the kernel methods recently developed in the field of machine learning. Several successful applications are presented, which show the capability of the kernel-based approach for detecting generalized synchronization. It is also shown that the dynamical change of the coupling coefficient between two chaotic systems can be captured by the proposed approach.Comment: 20 pages, 15 figures. massively revised as a full paper; issues on the choice of parameters by cross validation, tests by surrogated data, etc. are added as well as additional examples and figure

Stability of a cubic fixed point in three dimensions. Critical exponents for generic N

The detailed analysis of the global structure of the renormalization-group (RG) flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions (3D) within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary $N$ up to the four-loop order and resummed by means of the generalized Pad$\acute{\rm e}$-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to $N_c=2.89 \pm 0.02$ that agrees well with the estimate obtained on the basis of the five-loop \ve-expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B342, 284 (1995)] resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for $N\ge3$, and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when N=0 is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.Comment: 33 pages, LaTeX, 7 PostScript figures. Final version corrected and added with an Appendix on the six-loop stud