Noisy regression and classification with continuous multilayer networks

We investigate zero temperature Gibbs learning for two classes of unrealizable rules which play an important role in practical applications of multilayer neural networks with differentiable activation functions: classification problems and noisy regression problems. Considering one step of replica symmetry breaking, we surprisingly find that for sufficiently large training sets the stable state is replica symmetric even though the target rule is unrealizable. Further, the classification problem is shown to be formally equivalent to the noisy regression problem.Comment: 7 pages, including 2 figure

Field-free two-direction alignment alternation of linear molecules by elliptic laser pulses

We show that a linear molecule subjected to a short specific elliptically polarized laser field yields postpulse revivals exhibiting alignment alternatively located along the orthogonal axis and the major axis of the ellipse. The effect is experimentally demonstrated by measuring the optical Kerr effect along two different axes. The conditions ensuring an optimal field-free alternation of high alignments along both directions are derived.Comment: 5 pages, 4 color figure

Ionisation by quantised electromagnetic fields: The photoelectric effect

In this paper we explain the photoelectric effect in a variant of the standard model of non relativistic quantum electrodynamics, which is in some aspects more closely related to the physical picture, than the one studied in [BKZ]: Now we can apply our results to an electron with more than one bound state and to a larger class of electron-photon interactions. We will specify a situation, where ionisation probability in second order is a weighted sum of single photon terms. Furthermore we will see, that Einstein's equality $$E_{kin}=h\nu-\bigtriangleup E>0$$ for the maximal kinetic energy $E_{kin}$ of the electron, energy $h\nu$ of the photon and ionisation gap $\bigtriangleup E$ is the crucial condition for these single photon terms to be nonzero.Comment: 59 pages, LATEX2

Asymptotic Level Density of the Elastic Net Self-Organizing Feature Map

Whileas the Kohonen Self Organizing Map shows an asymptotic level density following a power law with a magnification exponent 2/3, it would be desired to have an exponent 1 in order to provide optimal mapping in the sense of information theory. In this paper, we study analytically and numerically the magnification behaviour of the Elastic Net algorithm as a model for self-organizing feature maps. In contrast to the Kohonen map the Elastic Net shows no power law, but for onedimensional maps nevertheless the density follows an universal magnification law, i.e. depends on the local stimulus density only and is independent on position and decouples from the stimulus density at other positions.Comment: 8 pages, 10 figures. Link to publisher under http://link.springer.de/link/service/series/0558/bibs/2415/24150939.ht

Non-analytic microscopic phase transitions and temperature oscillations in the microcanonical ensemble: An exactly solvable 1d-model for evaporation

We calculate exactly both the microcanonical and canonical thermodynamic functions (TDFs) for a one-dimensional model system with piecewise constant Lennard-Jones type pair interactions. In the case of an isolated $N$-particle system, the microcanonical TDFs exhibit (N-1) singular (non-analytic) microscopic phase transitions of the formal order N/2, separating N energetically different evaporation (dissociation) states. In a suitably designed evaporation experiment, these types of phase transitions should manifest themselves in the form of pressure and temperature oscillations, indicating cooling by evaporation. In the presence of a heat bath (thermostat), such oscillations are absent, but the canonical heat capacity shows a characteristic peak, indicating the temperature-induced dissociation of the one-dimensional chain. The distribution of complex zeros (DOZ) of the canonical partition may be used to identify different degrees of dissociation in the canonical ensemble.Comment: version accepted for publication in PRE, minor additions in the text, references adde

Collision of One-Dimensional Nonlinear Chains

We investigate one-dimensional collisions of unharmonic chains and a rigid wall. We find that the coefficient of restitution (COR) is strongly dependent on the velocity of colliding chains and has a minimum value at a certain velocity. The relationship between COR and collision velocity is derived for low-velocity collisions using perturbation methods. We found that the velocity dependence is characterized by the exponent of the lowest unharmonic term of interparticle potential energy

Scaling properties of granular materials

Given an assembly of viscoelastic spheres with certain material properties, we raise the question how the macroscopic properties of the assembly will change if all lengths of the system, i.e. radii, container size etc., are scaled by a constant. The result leads to a method to scale down experiments to lab-size.Comment: 4 pages, 2 figure

Phase transitions in soft-committee machines

Equilibrium statistical physics is applied to layered neural networks with differentiable activation functions. A first analysis of off-line learning in soft-committee machines with a finite number (K) of hidden units learning a perfectly matching rule is performed. Our results are exact in the limit of high training temperatures. For K=2 we find a second order phase transition from unspecialized to specialized student configurations at a critical size P of the training set, whereas for K > 2 the transition is first order. Monte Carlo simulations indicate that our results are also valid for moderately low temperatures qualitatively. The limit K to infinity can be performed analytically, the transition occurs after presenting on the order of N K examples. However, an unspecialized metastable state persists up to P= O (N K^2).Comment: 8 pages, 4 figure

Statistical mechanics of mutual information maximization

An unsupervised learning procedure based on maximizing the mutual information between the outputs of two networks receiving different but statistically dependent inputs is analyzed (Becker S. and Hinton G., Nature, 355 (1992) 161). By exploiting a formal analogy to supervised learning in parity machines, the theory of zero-temperature Gibbs learning for the unsupervised procedure is presented for the case that the networks are perceptrons and for the case of fully connected committees

Microcanonical quantum fluctuation theorems

Previously derived expressions for the characteristic function of work performed on a quantum system by a classical external force are generalized to arbitrary initial states of the considered system and to Hamiltonians with degenerate spectra. In the particular case of microcanonical initial states explicit expressions for the characteristic function and the corresponding probability density of work are formulated. Their classical limit as well as their relations to the respective canonical expressions are discussed. A fluctuation theorem is derived that expresses the ratio of probabilities of work for a process and its time reversal to the ratio of densities of states of the microcanonical equilibrium systems with corresponding initial and final Hamiltonians.From this Crooks-type fluctuation theorem a relation between entropies of different systems can be derived which does not involve the time reversed process. This entropy-from-work theorem provides an experimentally accessible way to measure entropies.Comment: revised and extended versio