36 research outputs found
Free energies of Boltzmann Machines: self-averaging, annealed and replica symmetric approximations in the thermodynamic limit
Restricted Boltzmann machines (RBMs) constitute one of the main models for
machine statistical inference and they are widely employed in Artificial
Intelligence as powerful tools for (deep) learning. However, in contrast with
countless remarkable practical successes, their mathematical formalization has
been largely elusive: from a statistical-mechanics perspective these systems
display the same (random) Gibbs measure of bi-partite spin-glasses, whose
rigorous treatment is notoriously difficult. In this work, beyond providing a
brief review on RBMs from both the learning and the retrieval perspectives, we
aim to contribute to their analytical investigation, by considering two
distinct realizations of their weights (i.e., Boolean and Gaussian) and
studying the properties of their related free energies. More precisely,
focusing on a RBM characterized by digital couplings, we first extend the
Pastur-Shcherbina-Tirozzi method (originally developed for the Hopfield model)
to prove the self-averaging property for the free energy, over its quenched
expectation, in the infinite volume limit, then we explicitly calculate its
simplest approximation, namely its annealed bound. Next, focusing on a RBM
characterized by analogical weights, we extend Guerra's interpolating scheme to
obtain a control of the quenched free-energy under the assumption of replica
symmetry: we get self-consistencies for the order parameters (in full agreement
with the existing Literature) as well as the critical line for ergodicity
breaking that turns out to be the same obtained in AGS theory. As we discuss,
this analogy stems from the slow-noise universality. Finally, glancing beyond
replica symmetry, we analyze the fluctuations of the overlaps for an estimate
of the (slow) noise affecting the retrieval of the signal, and by a stability
analysis we recover the Aizenman-Contucci identities typical of glassy systems.Comment: 21 pages, 1 figur
Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs. Diluted regime
We study the linear eigenvalue statistics of large random graphs in the
regimes when the mean number of edges for each vertex tends to infinity. We
prove that for a rather wide class of test functions the fluctuations of linear
eigenvalue statistics converges in distribution to a Gaussian random variable
with zero mean and variance which coincides with "non gaussian" part of the
Wigner ensemble variance.Comment: 19 page
Phoneme-retrieval; voice recognition; vowels recognition
A phoneme-retrieval technique is proposed, which is due to the particular way
of the construction of the network. An initial set of neurons is given. The
number of these neurons is approximately equal to the number of typical
structures of the data. For example if the network is built for voice retrieval
then the number of neurons must be equal to the number of characteristic
phonemes of the alphabet of the language spoken by the social group to which
the particular person belongs. Usually this task is very complicated and the
network can depend critically on the samples used for the learning. If the
network is built for image retrieval then it works only if the data to be
retrieved belong to a particular set of images. If the network is built for
voice recognition it works only for some particular set of words. A typical
example is the words used for the flight of airplanes. For example a command
like the "airplane should make a turn of 120 degrees towards the east" can be
easily recognized by the network if a suitable learning procedure is used.Comment: 10 page
Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs
We consider the adjacency matrix of a large random graph and study
fluctuations of the function
with .
We prove that the moments of fluctuations normalized by in the limit
satisfy the Wick relations for the Gaussian random variables. This
allows us to prove central limit theorem for and then extend
the result on the linear eigenvalue statistics of any
function which increases, together with its
first two derivatives, at infinity not faster than an exponential.Comment: 22 page
Scattering of Lower Hybrid Waves in a Magnetized Plasma
In this paper, the Maxwell equations for the electric field in a cold magnetized plasma in the half-space of x≥0 cm are solved. The boundary conditions for the electric field include a pointwise source at the plane x=0 cm, the derivatives of the electric field that are zero statV/cm2 at x=0 cm, and the field with all its derivatives that are zero at infinity. The solution is explored in terms of the Laplace transform in x and the Fourier transform in y-z directions. The expressions of the field components are obtained by the inverse Laplace transform and the inverse Fourier transform. The saddle-point technique and power expansion have been used for evaluating the inverse Fourier transform. The model represents the propagation of a lower hybrid wave generated by a pointwise antenna located at the boundary of the plasma. Here, the antenna is the boundary condition. The validation of the model is performed assuming that the electric field component Ey=0 statV/cm and by comparing it with the model of electromagnetic waves generated by a local small antenna located near the boundary of a tokamak, and an experiment is suggested
Magnetic Force-Free Theory: Nonlinear Case
In this paper, a theory of force-free magnetic field useful for explaining the formation of convex closed sets, bounded by a magnetic separatrix in the plasma, is developed. This question is not new and has been addressed by many authors. Force-free magnetic fields appear in many laboratory and astrophysical plasmas. These fields are defined by the solution of the problem âĂB=ÎB with some field conditions BâΩ on the boundary âΩ of the plasma region. In many physical situations, it has been noticed that Î is not constant but may vary in the domain Ω giving rise to many different interesting physical situations. We set Î=Î(Ï) with Ï being the poloidal magnetic flux function. Then, an analytic method, based on a first-order expansion of Ï with respect to a small parameter α, is developed. The GradâShafranov equation for Ï is solved by expanding the solution in the eigenfunctions of the zero-order operator. An analytic expression for the solution is obtained deriving results on the transition through resonances, the amplification with respect to the gun inflow. Thus, the formation of spheromaks or protosphera structure of the plasma is determined in the case of nonconstant Î