Learning by a nerual net in a noisy environment - The pseudo-inverse solution revisited

A recurrent neural net is described that learns a set of patterns in the presence of noise. The learning rule is of Hebbian type, and, if noise would be absent during the learning process, the resulting final values of the weights would correspond to the pseudo-inverse solution of the fixed point equation in question. For a non-vanishing noise parameter, an explicit expression for the expectation value of the weights is obtained. This result turns out to be unequal to the pseudo-inverse solution. Furthermore, the stability properties of the system are discussed.Comment: 16 pages, 3 figure

Probing the basins of attraction of a recurrent neural network

A recurrent neural network is considered that can retrieve a collection of patterns, as well as slightly perturbed versions of this `pure' set of patterns via fixed points of its dynamics. By replacing the set of dynamical constraints, i.e., the fixed point equations, by an extended collection of fixed-point-like equations, analytical expressions are found for the weights w_ij(b) of the net, which depend on a certain parameter b. This so-called basin parameter b is such that for b=0 there are, a priori, no perturbed patterns to be recognized by the net. It is shown by a numerical study, via probing sets, that a net constructed to recognize perturbed patterns, i.e., with values of the connections w_ij(b) with b unequal zero, possesses larger basins of attraction than a net made with the help of a pure set of patterns, i.e., with connections w_ij(b=0). The mathematical results obtained can, in principle, be realized by an actual, biological neural net.Comment: 17 pages, LaTeX, 2 figure

Conserving Approximations in Time-Dependent Density Functional Theory

In the present work we propose a theory for obtaining successively better approximations to the linear response functions of time-dependent density or current-density functional theory. The new technique is based on the variational approach to many-body perturbation theory (MBPT) as developed during the sixties and later expanded by us in the mid nineties. Due to this feature the resulting response functions obey a large number of conservation laws such as particle and momentum conservation and sum rules. The quality of the obtained results is governed by the physical processes built in through MBPT but also by the choice of variational expressions. We here present several conserving response functions of different sophistication to be used in the calculation of the optical response of solids and nano-scale systems.Comment: 11 pages, 4 figures, revised versio

Combining Hebbian and reinforcement learning in a minibrain model

A toy model of a neural network in which both Hebbian learning and reinforcement learning occur is studied. The problem of `path interference', which makes that the neural net quickly forgets previously learned input-output relations is tackled by adding a Hebbian term (proportional to the learning rate $\eta$) to the reinforcement term (proportional to $\rho$) in the learning rule. It is shown that the number of learning steps is reduced considerably if $1/4 < \eta/\rho < 1/2$, i.e., if the Hebbian term is neither too small nor too large compared to the reinforcement term

Measuring the impact of observations on the predictability of the Kuroshio Extension in a shallow-water model

In this paper sequential importance sampling is used to assess the impact of observations on a ensemble prediction for the decadal path transitions of the Kuroshio Extension (KE). This particle filtering approach gives access to the probability density of the state vector, which allows us to determine the predictive power — an entropy based measure — of the ensemble prediction. The proposed set-up makes use of an ensemble that, at each time, samples the climatological probability distribution. Then, in a post-processing step, the impact of different sets of observations is measured by the increase in predictive power of the ensemble over the climatological signal during one-year. The method is applied in an identical-twin experiment for the Kuroshio Extension using a reduced-gravity shallow water model. We investigate the impact of assimilating velocity observations from different locations during the elongated and the contracted meandering state of the KE. Optimal observations location correspond to regions with strong potential vorticity gradients. For the elongated state the optimal location is in the first meander of the KE. During the contracted state of the KE it is located south of Japan, where the Kuroshio separates from the coast

Cepheid Parallaxes and the Hubble Constant

Revised Hipparcos parallaxes for classical Cepheids are analysed together with 10 HST-based parallaxes (Benedict et al.). In a reddening-free V,I relation we find that the coefficient of logP is the same within the uncertainties in our Galaxy as in the LMC, contrary to some previous suggestions. Cepheids in the inner region of NGC4258 with near solar metallicities (Macri et al.) confirm this result. We obtain a zero-point for the reddening-free relation and apply it to Cepheids in galaxies used by Sandage et al. to calibrate the absolute magnitudes of SNIa and to derive the Hubble constant. We revise their result from 62 to 70+/-5 km/s/Mpc. The Freedman et al. 2001 value is revised from 72 to 76+/-8 km/s/Mpc. These results are insensitive to Cepheid metallicity corrections. The Cepheids in the inner region of NGC4258 yield a modulus of 29.22+/-0.03(int) compared with a maser-based modulus of 29.29+/-0.15. Distance moduli for the LMC, uncorrected for any metallicity effects, are; 18.52+/-0.03 from a reddening-free relation in V,I; 18.47+/-0.03 from a period-luminosity relation at K; 18.45+/-0.04 from a period-luminosity-colour relation in J,K. Adopting a metallicity correction in V,I from Marci et al. leads to a true LMC modulus of 18.39+/-0.05.Comment: 9 pages, 1 figure, on-line material from [email protected]. Accepted for MNRA