Analysis of Linsker's simulations of Hebbian rules

Linsker has reported the development of center-surround receptive fields and oriented receptive fields in simulations of a Hebb-type equation in a linear network. The dynamics of the learning rule are analyzed in terms of the eigenvectors of the covariance matrix of cell activities. Analytic and computational results for Linsker's covariance matrices, and some general theorems, lead to an explanation of the emergence of center-surround and certain oriented structures. We estimate criteria for the parameter regime in which center-surround structures emerge

The Role of Constraints in Hebbian Learning

Models of unsupervised, correlation-based (Hebbian) synaptic plasticity are typically unstable: either all synapses grow until each reaches the maximum allowed strength, or all synapses decay to zero strength. A common method of avoiding these outcomes is to use a constraint that conserves or limits the total synaptic strength over a cell. We study the dynamic effects of such constraints. Two methods of enforcing a constraint are distinguished, multiplicative and subtractive. For otherwise linear learning rules, multiplicative enforcement of a constraint results in dynamics that converge to the principal eigenvector of the operator determining unconstrained synaptic development. Subtractive enforcement, in contrast, typically leads to a final state in which almost all synaptic strengths reach either the maximum or minimum allowed value. This final state is often dominated by weight configurations other than the principal eigenvector of the unconstrained operator. Multiplicative enforcement yields a “graded” receptive field in which most mutually correlated inputs are represented, whereas subtractive enforcement yields a receptive field that is “sharpened” to a subset of maximally correlated inputs. If two equivalent input populations (e.g., two eyes) innervate a common target, multiplicative enforcement prevents their segregation (ocular dominance segregation) when the two populations are weakly correlated; whereas subtractive enforcement allows segregation under these circumstances. These results may be used to understand constraints both over output cells and over input cells. A variety of rules that can implement constrained dynamics are discussed

Diffraction-limited CCD imaging with faint reference stars

By selecting short exposure images taken using a CCD with negligible readout noise we obtained essentially diffraction-limited 810 nm images of faint objects using nearby reference stars brighter than I=16 at a 2.56 m telescope. The FWHM of the isoplanatic patch for the technique is found to be 50 arcseconds, providing ~20% sky coverage around suitable reference stars.Comment: 4 page letter accepted for publication in Astronomy and Astrophysic

Error correcting code using tree-like multilayer perceptron

An error correcting code using a tree-like multilayer perceptron is proposed. An original message \mbi{s}^0 is encoded into a codeword \boldmath{y}_0 using a tree-like committee machine (committee tree) or a tree-like parity machine (parity tree). Based on these architectures, several schemes featuring monotonic or non-monotonic units are introduced. The codeword \mbi{y}_0 is then transmitted via a Binary Asymmetric Channel (BAC) where it is corrupted by noise. The analytical performance of these schemes is investigated using the replica method of statistical mechanics. Under some specific conditions, some of the proposed schemes are shown to saturate the Shannon bound at the infinite codeword length limit. The influence of the monotonicity of the units on the performance is also discussed.Comment: 23 pages, 3 figures, Content has been extended and revise

Comprehensive cosmographic analysis by Markov Chain Method

We study the possibility to extract model independent information about the dynamics of the universe by using Cosmography. We intend to explore it systematically, to learn about its limitations and its real possibilities. Here we are sticking to the series expansion approach on which Cosmography is based. We apply it to different data sets: Supernovae Type Ia (SNeIa), Hubble parameter extracted from differential galaxy ages, Gamma Ray Bursts (GRBs) and the Baryon Acoustic Oscillations (BAO) data. We go beyond past results in the literature extending the series expansion up to the fourth order in the scale factor, which implies the analysis of the deceleration, q_{0}, the jerk, j_{0} and the snap, s_{0}. We use the Markov Chain Monte Carlo Method (MCMC) to analyze the data statistically. We also try to relate direct results from Cosmography to dark energy (DE) dynamical models parameterized by the Chevalier-Polarski-Linder (CPL) model, extracting clues about the matter content and the dark energy parameters. The main results are: a) even if relying on a mathematical approximate assumption such as the scale factor series expansion in terms of time, cosmography can be extremely useful in assessing dynamical properties of the Universe; b) the deceleration parameter clearly confirms the present acceleration phase; c) the MCMC method can help giving narrower constraints in parameter estimation, in particular for higher order cosmographic parameters (the jerk and the snap), with respect to the literature; d) both the estimation of the jerk and the DE parameters, reflect the possibility of a deviation from the LCDM cosmological model.Comment: 24 pages, 7 figure

Kramers-Kronig, Bode, and the meaning of zero

The implications of causality, as captured by the Kramers-Kronig relations between the real and imaginary parts of a linear response function, are familiar parts of the physics curriculum. In 1937, Bode derived a similar relation between the magnitude (response gain) and phase. Although the Kramers-Kronig relations are an equality, Bode's relation is effectively an inequality. This perhaps-surprising difference is explained using elementary examples and ultimately traces back to delays in the flow of information within the system formed by the physical object and measurement apparatus.Comment: 8 pages; American Journal of Physics, to appea

Multi-Objective Supervised Learning

Copyright © 2008 Springer-Verlag Berlin Heidelberg. The final publication is available at link.springer.comBook title: Multiobjective Problem Solving from NatureExtended version of the 2006 workshop paper presented at the Workshop on Multiobjective Problem-Solving from Nature, 9th International Conference on Parallel Problem Solving from Nature (PPSN IX), Reykjavik, Iceland, 9-13 September 2006; see: http://hdl.handle.net/10871/11785This chapter sets out a number of the popular areas in multiobjective supervised learning. It gives empirical examples of model complexity optimization and competing error terms, and presents the recent advances in multi-class receiver operating characteristic analysis enabled by multiobjective optimization. It concludes by highlighting some specific areas of interest/concern when dealing with multiobjective supervised learning problems, and sets out future areas of potential research

The Statistical Physics of Regular Low-Density Parity-Check Error-Correcting Codes

A variation of Gallager error-correcting codes is investigated using statistical mechanics. In codes of this type, a given message is encoded into a codeword which comprises Boolean sums of message bits selected by two randomly constructed sparse matrices. The similarity of these codes to Ising spin systems with random interaction makes it possible to assess their typical performance by analytical methods developed in the study of disordered systems. The typical case solutions obtained via the replica method are consistent with those obtained in simulations using belief propagation (BP) decoding. We discuss the practical implications of the results obtained and suggest a computationally efficient construction for one of the more practical configurations.Comment: 35 pages, 4 figure

Planck priors for dark energy surveys

Although cosmic microwave background (CMB) anisotropy data alone cannot constrain simultaneously the spatial curvature and the equation of state of dark energy, CMB data provide a valuable addition to other experimental results. However computing a full CMB power spectrum with a Boltzmann code is quite slow; for instance if we want to work with many dark energy and/or modified gravity models, or would like to optimize experiments where many different configurations need to be tested, it is possible to adopt a quicker and more efficient approach. In this paper we consider the compression of the projected Planck CMB data into four parameters, R (scaled distance to last scattering surface), l_a (angular scale of sound horizon at last scattering), Omega_b h^2 (baryon density fraction) and n_s (powerlaw index of primordial matter power spectrum), all of which can be computed quickly. We show that, although this compression loses information compared to the full likelihood, such information loss becomes negligible when more data is added. We also demonstrate that the method can be used for scalar field dark energy independently of the parametrisation of the equation of state, and discuss how this method should be used for other kinds of dark energy models.Comment: 8 pages, 3 figures, 4 table

Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs

Belief propagation -- a powerful heuristic method to solve inference problems involving a large number of random variables -- was recently generalized to quantum theory. Like its classical counterpart, this algorithm is exact on trees when the appropriate independence conditions are met and is expected to provide reliable approximations when operated on loopy graphs. In this paper, we benchmark the performances of loopy quantum belief propagation (QBP) in the context of finite-tempereture quantum many-body physics. Our results indicate that QBP provides reliable estimates of the high-temperature correlation function when the typical loop size in the graph is large. As such, it is suitable e.g. for the study of quantum spin glasses on Bethe lattices and the decoding of sparse quantum error correction codes.Comment: 5 pages, 4 figure