Gaussian processes for classification: mean field algorithms:mean-field algorithms

We derive a mean field algorithm for binary classification with Gaussian processes which is based on the TAP approach originally proposed in Statistical Physics of disordered systems. The theory also yields an approximate leave-one-out estimator for the generalization error which is computed with no extra computational cost. We show that from the TAP approach, it is possible to derive both a simpler 'naive' mean field theory and support vector machines (SVM) as limiting cases. For both mean field algorithms and support vectors machines, simulation results for three small benchmark data sets are presented. They show 1. that one may get state of the art performance by using the leave-one-out estimator for model selection and 2. the built-in leave-one-out estimators are extremely precise when compared to the exact leave-one-out estimate. The latter result is a taken as a strong support for the internal consistency of the mean field approach.

Gaussian processes and SVM: Mean field results and leave-one-out

In this chapter, we elaborate on the well-known relationship between Gaussian processes (GP) and Support Vector Machines (SVM). Secondly, we present approximate solutions for two computational problems arising in GP and SVM. The first one is the calculation of the posterior mean for GP classifiers using a `naive' mean field approach. The second one is a leave-one-out estimator for the generalization error of SVM based on a linear response method. Simulation results on a benchmark dataset show similar performances for the GP mean field algorithm and the SVM algorithm. The approximate leave-one-out estimator is found to be in very good agreement with the exact leave-one-out error.

Online approximations for wind-field models

We study online approximations to Gaussian process models for spatially distributed systems. We apply our method to the prediction of wind fields over the ocean surface from scatterometer data. Our approach combines a sequential update of a Gaussian approximation to the posterior with a sparse representation that allows to treat problems with a large number of observations.

A mean field theory of coded CDMA systems

We present a mean field theory of code-division multiple access (CDMA) systems with error-control coding. On the basis of the relation between the free energy and mutual information, we obtain an analytical expression of the maximum spectral efficiency of the coded CDMA system, from which a mean field description of the coded CDMA system is provided in terms of a bank of scalar Gaussian channels whose variances in general vary at different code symbol positions. Regular low-density parity-check (LDPC)-coded CDMA systems are also discussed as an example of the coded CDMA systems.

Generalized mean field approximation for parallel dynamics of the Ising model

The dynamics of the non-equilibrium Ising model with parallel updates is investigated using a generalized mean field approximation that incorporates multiple two-site correlations at any two time steps, which can be obtained recursively. The proposed method shows significant improvement in predicting local system properties compared to other mean field approximation techniques, particularly in systems with symmetric interactions. Results are also evaluated against those obtained from Monte Carlo simulations. The method is also employed to obtain parameter values for the kinetic inverse Ising modeling problem, where couplings and local field values of a fully connected spin system are inferred from data. © 2014 IOP Publishing Ltd and SISSA Medialab srl.

Mean field methods for Gaussian process classifiers

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Mean field evolution in an open quantum system

In this thesis, we consider N quantum particles coupled to collective thermal quantum environments. The coupling is energy conserving and scaled in the mean field way. There is no direct interaction between the particles, they only interact via the common reservoir. It is well known that an initially disentangled state of the N particles will remain disentangled at times in the limit N -> [infinity]. In this thesis, we evaluate the η-body reduced density matrix (tracing over the reservoirs and the N - η remaining particles). We identify the main disentangled part of the reduced density matrix and obtain the first order correction term in 1/N. We show that this correction term is entangled. We also estimate the speed of convergence of the reduced density matrix as N -> [infinity]. Our model is exactly solvable and it is not based on numerical approximation.

Random Access over Multi-Packet Reception Channels: A Mean-Field Approach

Thesis (Ph.D.)--University of Washington, 2016-06

Enhancing the numerical performance of fire field models (CMS Paper)

This paper describes two new techniques designed to enhance the performance of fire field modelling software. The two techniques are "group solvers" and automated dynamic control of the solution process, both of which are currently under development within the SMARTFIRE Computational Fluid Dynamics environment. The "group solver" is a derivation of common solver techniques used to obtain numerical solutions to the algebraic equations associated with fire field modelling. The purpose of "group solvers" is to reduce the computational overheads associated with traditional numerical solvers typically used in fire field modelling applications. In an example, discussed in this paper, the group solver is shown to provide a 37% saving in computational time compared with a traditional solver. The second technique is the automated dynamic control of the solution process, which is achieved through the use of artificial intelligence techniques. This is designed to improve the convergence capabilities of the software while further decreasing the computational overheads. The technique automatically controls solver relaxation using an integrated production rule engine with a blackboard to monitor and implement the required control changes during solution processing. Initial results for a two-dimensional fire simulation are presented that demonstrate the potential for considerable savings in simulation run-times when compared with control sets from various sources. Furthermore, the results demonstrate the potential for enhanced solution reliability due to obtaining acceptable convergence within each time step, unlike some of the comparison simulations.

The importance of different timings of excitatory and inhibitory pathways in neural field models

In this paper we consider a neural field model comprised of two distinct populations of neurons, excitatory and inhibitory, for which both the velocities of action potential propagation and the time courses of synaptic processing are different. Using recently-developed techniques we construct the Evans function characterising the stability of both stationary and travelling wave solutions, under the assumption that the firing rate function is the Heaviside step. We find that these differences in timing for the two populations can cause instabilities of these solutions, leading to, for example, stationary breathers. We also analyse $quot;anti-pulses,$quot; a novel type of pattern for which all but a small interval of the domain (in moving coordinates) is active. These results extend previous work on neural fields with space dependent delays, and demonstrate the importance of considering the effects of the different time-courses of excitatory and inhibitory neural activity.

Variational mean-field approach to the double-exchange model

It has been recently shown that the double exchange Hamiltonian, with weak antiferromagnetic interactions, has a richer variety of first- and second-order transitions than previously anticipated, and that such transitions are consistent with the magnetic properties of manganites. Here we present a thorough discussion of the variational mean-field approach that leads to these results. We also show that the effect of the Berry phase turns out to be crucial to produce first-order paramagnetic-ferromagnetic transitions near half filling with transition temperatures compatible with the experimental situation. The computation relies on two crucial facts: the use of a mean-field ansatz that retains the complexity of a system of electrons with off-diagonal disorder, not fully taken into account by the mean-field techniques, and the small but significant antiferromagnetic superexchange interaction between the localized spins.

Special Issue on Spatial Moment Techniques for Modelling Biological Processes

Over the last two decades, there has been an increasing awareness of, and interest in, the use of spatial moment techniques to provide insight into a range of biological and ecological processes. Models that incorporate spatial moments can be viewed as extensions of mean-field models. These mean-field models often consist of systems of classical ordinary differential equations and partial differential equations, whose derivation, at some point, hinges on the simplifying assumption that individuals in the underlying stochastic process encounter each other at a rate that is proportional to the average abundance of individuals. This assumption has several implications, the most striking of which is that mean-field models essentially neglect any impact of the spatial structure of individuals in the system. Moment dynamics models extend traditional mean-field descriptions by accounting for the dynamics of pairs, triples and higher n-tuples of individuals. This means that moment dynamics models can, to some extent, account for how the spatial structure affects the dynamics of the system in question.

Constraints on the symmetry energy and neutron skins from pygmy resonances in Ni-68 and Sn-132

Correlations between the behavior of the nuclear symmetry energy, the neutron skins, and the percentage of energy-weighted sum rule (EWSR) exhausted by the pygmy dipole resonance (PDR) in Ni-68 and Sn-132 are investigated by using different random phase approximation (RPA) models for the dipole response, based on a representative set of Skyrme effective forces plus meson-exchange effective Lagrangians. A comparison with the experimental data has allowed us to constrain the value of the derivative of the symmetry energy at saturation. The neutron skin radius is deduced under this constraint.

Towards more accurate and reliable predictions for nuclear applications

The need for nuclear data far from the valley of stability, for applications such as nuclear as- trophysics or future nuclear facilities, challenges the robustness as well as the predictive power of present nuclear models. Most of the nuclear data evaluation and prediction are still performed on the basis of phenomenological nuclear models. For the last decades, important progress has been achieved in funda- mental nuclear physics, making it now feasible to use more reliable, but also more complex microscopic or semi-microscopic models in the evaluation and prediction of nuclear data for practical applications. In the present contribution, the reliability and accuracy of recent nuclear theories are discussed for most of the relevant quantities needed to estimate reaction cross sections and beta-decay rates, namely nuclear masses, nuclear level densities, gamma-ray strength, fission properties and beta-strength functions. It is shown that nowadays, mean-field models can be tuned at the same level of accuracy as the phenomenological mod- els, renormalized on experimental data if needed, and therefore can replace the phenomenogical inputs in the prediction of nuclear data. While fundamental nuclear physicists keep on improving state-of-the-art models, e.g. within the shell model or ab initio models, nuclear applications could make use of their most recent results as quantitative constraints or guides to improve the predictions in energy or mass domain that will remain inaccessible experimentally.

Axonal velocity distributions in neural feld equations

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.