Thermal Equilibrium with the Wiener Potential: Testing the Replica Variational Approximation

We consider the statistical mechanics of a classical particle in a one-dimensional box subjected to a random potential which constitutes a Wiener process on the coordinate axis. The distribution of the free energy and all correlation functions of the Gibbs states may be calculated exactly as a function of the box length and temperature. This allows for a detailed test of results obtained by the replica variational approximation scheme. We show that this scheme provides a reasonable estimate of the averaged free energy. Furthermore our results shed more light on the validity of the concept of approximate ultrametricity which is a central assumption of the replica variational method.Comment: 6 pages, 1 file LaTeX2e generating 2 eps-files for 2 figures automaticall

On-Line AdaTron Learning of Unlearnable Rules

We study the on-line AdaTron learning of linearly non-separable rules by a simple perceptron. Training examples are provided by a perceptron with a non-monotonic transfer function which reduces to the usual monotonic relation in a certain limit. We find that, although the on-line AdaTron learning is a powerful algorithm for the learnable rule, it does not give the best possible generalization error for unlearnable problems. Optimization of the learning rate is shown to greatly improve the performance of the AdaTron algorithm, leading to the best possible generalization error for a wide range of the parameter which controls the shape of the transfer function.)Comment: RevTeX 17 pages, 8 figures, to appear in Phys.Rev.

Charge Symmetry Breaking and QCD

Charge symmetry breaking (CSB) in the strong interaction occurs because of the difference between the masses of the up and down quarks. The use of effective field theories allows us to follow this influence of confined quarks in hadronic and nuclear systems. The progress in observing and understanding CSB is reviewed with particular attention to the recent successful observations of CSB in measurements involving the production of a single neutral pion and to the related theoretical progress.Comment: 41 pages, 10 figures, for Nov. 2006 edition Annual Review of Nuclear and Particle Physic

Perceptron capacity revisited: classification ability for correlated patterns

In this paper, we address the problem of how many randomly labeled patterns can be correctly classified by a single-layer perceptron when the patterns are correlated with each other. In order to solve this problem, two analytical schemes are developed based on the replica method and Thouless-Anderson-Palmer (TAP) approach by utilizing an integral formula concerning random rectangular matrices. The validity and relevance of the developed methodologies are shown for one known result and two example problems. A message-passing algorithm to perform the TAP scheme is also presented

Statistical Mechanics of Learning: A Variational Approach for Real Data

Using a variational technique, we generalize the statistical physics approach of learning from random examples to make it applicable to real data. We demonstrate the validity and relevance of our method by computing approximate estimators for generalization errors that are based on training data alone.Comment: 4 pages, 2 figure

Generalizing with perceptrons in case of structured phase- and pattern-spaces

We investigate the influence of different kinds of structure on the learning behaviour of a perceptron performing a classification task defined by a teacher rule. The underlying pattern distribution is permitted to have spatial correlations. The prior distribution for the teacher coupling vectors itself is assumed to be nonuniform. Thus classification tasks of quite different difficulty are included. As learning algorithms we discuss Hebbian learning, Gibbs learning, and Bayesian learning with different priors, using methods from statistics and the replica formalism. We find that the Hebb rule is quite sensitive to the structure of the actual learning problem, failing asymptotically in most cases. Contrarily, the behaviour of the more sophisticated methods of Gibbs and Bayes learning is influenced by the spatial correlations only in an intermediate regime of $\alpha$, where $\alpha$ specifies the size of the training set. Concerning the Bayesian case we show, how enhanced prior knowledge improves the performance.Comment: LaTeX, 32 pages with eps-figs, accepted by J Phys

Statistical mechanical analysis of the linear vector channel in digital communication

A statistical mechanical framework to analyze linear vector channel models in digital wireless communication is proposed for a large system. The framework is a generalization of that proposed for code-division multiple-access systems in Europhys. Lett. 76 (2006) 1193 and enables the analysis of the system in which the elements of the channel transfer matrix are statistically correlated with each other. The significance of the proposed scheme is demonstrated by assessing the performance of an existing model of multi-input multi-output communication systems.Comment: 15 pages, 2 figure

Dynamics of on-line Hebbian learning with structurally unrealizable restricted training sets

We present an exact solution for the dynamics of on-line Hebbian learning in neural networks, with restricted and unrealizable training sets. In contrast to other studies on learning with restricted training sets, unrealizability is here caused by structural mismatch, rather than data noise: the teacher machine is a perceptron with a reversed wedge-type transfer function, while the student machine is a perceptron with a sigmoidal transfer function. We calculate the glassy dynamics of the macroscopic performance measures, training error and generalization error, and the (non-Gaussian) student field distribution. Our results, which find excellent confirmation in numerical simulations, provide a new benchmark test for general formalisms with which to study unrealizable learning processes with restricted training sets.Comment: 7 pages including 3 figures, using IOP latex2e preprint class fil

Replica theory for learning curves for Gaussian processes on random graphs

Statistical physics approaches can be used to derive accurate predictions for the performance of inference methods learning from potentially noisy data, as quantified by the learning curve defined as the average error versus number of training examples. We analyse a challenging problem in the area of non-parametric inference where an effectively infinite number of parameters has to be learned, specifically Gaussian process regression. When the inputs are vertices on a random graph and the outputs noisy function values, we show that replica techniques can be used to obtain exact performance predictions in the limit of large graphs. The covariance of the Gaussian process prior is defined by a random walk kernel, the discrete analogue of squared exponential kernels on continuous spaces. Conventionally this kernel is normalised only globally, so that the prior variance can differ between vertices; as a more principled alternative we consider local normalisation, where the prior variance is uniform

Lef1 regulates caveolin expression and caveolin dependent endocytosis, a process necessary for Wnt5a/Ror2 signaling during Xenopus gastrulation

The activation of distinct branches of the Wnt signaling network is essential for regulating early vertebrate development. Activation of the canonical Wnt/β-catenin pathway stimulates expression of β-catenin-Lef/Tcf regulated Wnt target genes and a regulatory network giving rise to the formation of the Spemann organizer. Non-canonical pathways, by contrast, mainly regulate cell polarization and migration, in particular convergent extension movements of the trunk mesoderm during gastrulation. By transcriptome analyses, we found caveolin1, caveolin3 and cavin1 to be regulated by Lef1 in the involuting mesoderm of Xenopus embryos at gastrula stages. We show that caveolins and caveolin dependent endocytosis are necessary for proper gastrulation, most likely by interfering with Wnt5a/Ror2 signaling. Wnt5a regulates the subcellular localization of receptor complexes, including Ror2 homodimers, Ror2/Fzd7 and Ror2/dsh heterodimers in an endocytosis dependent manner. Live-cell imaging revealed endocytosis of Ror2/caveolin1 complexes. In Xenopus explants, in the presence of Wnt5a, these receptor clusters remain stable exclusively at the basolateral side, suggesting that endocytosis of non-canonical Wnt/receptor complexes preferentially takes place at the apical membrane. In support of this blocking endocytosis with inhibitors prevents the effects of Wnt5a. Thus, target genes of Lef1 interfere with Wnt5a/Ror2 signaling to coordinate gastrulation movements