High-Dimensional Inference with the generalized Hopfield Model: Principal Component Analysis and Corrections

We consider the problem of inferring the interactions between a set of N binary variables from the knowledge of their frequencies and pairwise correlations. The inference framework is based on the Hopfield model, a special case of the Ising model where the interaction matrix is defined through a set of patterns in the variable space, and is of rank much smaller than N. We show that Maximum Lik elihood inference is deeply related to Principal Component Analysis when the amp litude of the pattern components, xi, is negligible compared to N^1/2. Using techniques from statistical mechanics, we calculate the corrections to the patterns to the first order in xi/N^1/2. We stress that it is important to generalize the Hopfield model and include both attractive and repulsive patterns, to correctly infer networks with sparse and strong interactions. We present a simple geometrical criterion to decide how many attractive and repulsive patterns should be considered as a function of the sampling noise. We moreover discuss how many sampled configurations are required for a good inference, as a function of the system size, N and of the amplitude, xi. The inference approach is illustrated on synthetic and biological data.Comment: Physical Review E: Statistical, Nonlinear, and Soft Matter Physics (2011) to appea

The sparticle spectrum in Minimal gaugino-Gauge Mediation

We compute the sparticle mass spectrum in the minimal four-dimensional construction that interpolates between gaugino mediation and ordinary gauge mediation.Comment: 21 pages, 9 figures; V2: refs. added; V3: some typos correcte

Two-Species Reaction-Diffusion System with Equal Diffusion Constants: Anomalous Density Decay at Large Times

We study a two-species reaction-diffusion model where A+A->0, A+B->0 and B+B->0, with annihilation rates lambda0, delta0 > lambda0 and lambda0, respectively. The initial particle configuration is taken to be randomly mixed with mean densities nA(0) > nB(0), and with the two species A and B diffusing with the same diffusion constant. A field-theoretic renormalization group analysis suggests that, contrary to expectation, the large-time density of the minority species decays at the same rate as the majority when d<=2. Monte Carlo data supports the field theory prediction in d=1, while in d=2 the logarithmically slow convergence to the large-time asymptotics makes a numerical test difficult.Comment: revised version (more figures, claim on exactnes of d=2 treatment removed), 5 pages, 3 figures, RevTex, see related paper Phys. Rev. E, R3787, (1999) or cond-mat/9901147, to appear in Phys. Rev.

Intrinsic limitations of inverse inference in the pairwise Ising spin glass

We analyze the limits inherent to the inverse reconstruction of a pairwise Ising spin glass based on susceptibility propagation. We establish the conditions under which the susceptibility propagation algorithm is able to reconstruct the characteristics of the network given first- and second-order local observables, evaluate eventual errors due to various types of noise in the originally observed data, and discuss the scaling of the problem with the number of degrees of freedom

Criticality in one dimension with inverse square-law potentials

It is demonstrated that the scaled order parameter for ferromagnetic Ising and three-state Potts chains with inverse-square interactions exhibits a universal critical jump, in analogy with the superfluid density in helium films. Renormalization-group arguments are combined with numerical simulations of systems containing up to one million lattice sites to accurately determine the critical properties of these models. In strong contrast with earlier work, compelling quantitative evidence for the Kosterlitz--Thouless-like character of the phase transition is provided.Comment: To appear in Phys. Rev. Let

Analysis of Bidirectional Associative Memory using SCSNA and Statistical Neurodynamics

Bidirectional associative memory (BAM) is a kind of an artificial neural network used to memorize and retrieve heterogeneous pattern pairs. Many efforts have been made to improve BAM from the the viewpoint of computer application, and few theoretical studies have been done. We investigated the theoretical characteristics of BAM using a framework of statistical-mechanical analysis. To investigate the equilibrium state of BAM, we applied self-consistent signal to noise analysis (SCSNA) and obtained a macroscopic parameter equations and relative capacity. Moreover, to investigate not only the equilibrium state but also the retrieval process of reaching the equilibrium state, we applied statistical neurodynamics to the update rule of BAM and obtained evolution equations for the macroscopic parameters. These evolution equations are consistent with the results of SCSNA in the equilibrium state.Comment: 13 pages, 4 figure

On the Spectrum of Direct Gaugino Mediation

In direct gauge mediation, the gaugino masses are anomalously small, giving rise to a split SUSY spectrum. Here we investigate the superpartner spectrum in a minimal version of "direct gaugino mediation." We find that the sfermion masses are comparable to those of the gauginos - even in the hybrid gaugino-gauge mediation regime - if the messenger scale is sufficiently small.Comment: 21 pages, 4 figures; V2: refs. adde

The Non-local Kardar-Parisi-Zhang Equation With Spatially Correlated Noise

The effects of spatially correlated noise on a phenomenological equation equivalent to a non-local version of the Kardar-Parisi-Zhang equation are studied via the dynamic renormalization group (DRG) techniques. The correlated noise coupled with the long ranged nature of interactions prove the existence of different phases in different regimes, giving rise to a range of roughness exponents defined by their corresponding critical dimensions. Finally self-consistent mode analysis is employed to compare the non-KPZ exponents obtained as a result of the long range -long range interactions with the DRG results.Comment: Plain Latex, 10 pages, 2 figures in one ps fil

Superpartner spectrum of minimal gaugino-gauge mediation

We evaluate the sparticle mass spectrum in the minimal four-dimensional construction that interpolates between gaugino and ordinary gauge mediation at the weak scale. We find that even in the hybrid case -- when the messenger scale is comparable to the mass of the additional gauge particles -- both the right-handed as well as the left-handed sleptons are lighter than the bino in the low-scale mediation regime. This implies a chain of lepton production and, consequently, striking signatures that may be probed at the LHC already in the near future.Comment: 8 pages, 3 figures; V2: refs and a few comments added; V3 title change