Showing papers by "Courant Institute of Mathematical Sciences published in 2007"

An Extension Problem Related to the Fractional Laplacian

Abstract: The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

Sparse Feature Learning for Deep Belief Networks

Abstract: Unsupervised learning algorithms aim to discover the structure hidden in the data, and to learn representations that are more suitable as input to a supervised machine than the raw input. Many unsupervised methods are based on reconstructing the input from the representation, while constraining the representation to have certain desirable properties (e.g. low dimension, sparsity, etc). Others are based on approximating density by stochastically reconstructing the input from the representation. We describe a novel and efficient algorithm to learn sparse representations, and compare it theoretically and experimentally with a similar machine trained probabilistically, namely a Restricted Boltzmann Machine. We propose a simple criterion to compare and select different unsupervised machines based on the trade-off between the reconstruction error and the information content of the representation. We demonstrate this method by extracting features from a dataset of handwritten numerals, and from a dataset of natural image patches. We show that by stacking multiple levels of such machines and by training sequentially, high-order dependencies between the input observed variables can be captured.

Correlation between neural spike trains increases with firing rate.

Abstract: Deciphering a 'neural code' usually requires measurement of either the rate of spike (electrical impulses) production or the spike synchrony. However, these two measures are not independent, as higher rates are associated with higher synchrony. It is further shown that the connection between rate and synchrony enhances information coding. Populations of neurons in the retina1,2,3, olfactory system4, visual5 and somatosensory6 thalamus, and several cortical regions7,8,9,10 show temporal correlation between the discharge times of their action potentials (spike trains). Correlated firing has been linked to stimulus encoding9, attention11, stimulus discrimination4, and motor behaviour12. Nevertheless, the mechanisms underlying correlated spiking are poorly understood2,3,13,14,15,16,17,18,19,20, and its coding implications are still debated13,16,21,22. It is not clear, for instance, whether correlations between the discharges of two neurons are determined solely by the correlation between their afferent currents, or whether they also depend on the mean and variance of the input. We addressed this question by computing the spike train correlation coefficient of unconnected pairs of in vitro cortical neurons receiving correlated inputs. Notably, even when the input correlation remained fixed, the spike train output correlation increased with the firing rate, but was largely independent of spike train variability. With a combination of analytical techniques and numerical simulations using ‘integrate-and-fire’ neuron models we show that this relationship between output correlation and firing rate is robust to input heterogeneities. Finally, this overlooked relationship is replicated by a standard threshold-linear model, demonstrating the universality of the result. This connection between the rate and correlation of spiking activity links two fundamental features of the neural code.

OpenFst: a general and efficient weighted finite-state transducer library

Abstract: We describe OpenFst, an open-source library for weighted finite-state transducers (WFSTs). OpenFst consists of a C++ template library with efficient WFST representations and over twenty-five operations for constructing, combining, optimizing, and searching them. At the shell-command level, there are corresponding transducer file representations and programs that operate on them. OpenFst is designed to be both very efficient in time and space and to scale to very large problems. This library has key applications speech, image, and natural language processing, pattern and string matching, and machine learning. We give an overview of the library, examples of its use, details of its design that allow customizing the labels, states, and weights and the lazy evaluation of many of its operations. Further information and a download of the OpenFst library can be obtained from http://www.openfst.org.

Rectification of the Bias in the Wavelet Power Spectrum

Abstract: This paper addresses a bias problem in the estimate of wavelet power spectra for atmospheric and oceanic datasets. For a time series comprised of sine waves with the same amplitude at different frequencies the conventionally adopted wavelet method does not produce a spectrum with identical peaks, in contrast to a Fourier analysis. The wavelet power spectrum in this definition, that is, the transform coefficient squared (to within a constant factor), is equivalent to the integration of energy (in physical space) over the influence period (time scale) the series spans. Thus, a physically consistent definition of energy for the wavelet power spectrum should be the transform coefficient squared divided by the scale it associates. Such adjusted wavelet power spectrum results in a substantial improvement in the spectral estimate, allowing for a comparison of the spectral peaks across scales. The improvement is validated with an artificial time series and a real coastal sea level record. Also examined is the previous example of the wavelet analysis of the Nino-3 SST data.

Orientational order and instabilities in suspensions of self-locomoting rods.

Abstract: The orientational order and dynamics in suspensions of self-locomoting slender rods are investigated numerically. In agreement with previous theoretical predictions, nematic suspensions of swimming particles are found to be unstable at long wavelengths as a result of hydrodynamic fluctuations. Nevertheless, a local nematic ordering is shown to persist over short length scales and to have a significant impact on the mean swimming speed. The consequences of the large-scale orientational disorder for particle dispersion are also discussed.

Qualitative network models and genome-wide expression data define carbon/nitrogen-responsive molecular machines in Arabidopsis

Abstract: Background: Carbon (C) and nitrogen (N) metabolites can regulate gene expression in Arabidopsis thaliana. Here, we use multinetwork analysis of microarray data to identify molecular networks regulated by C and N in the Arabidopsis root system. Results: We used the Arabidopsis whole genome Affymetrix gene chip to explore global gene expression responses in plants exposed transiently to a matrix of C and N treatments. We used ANOVA analysis to define quantitative models of regulation for all detected genes. Our results suggest that about half of the Arabidopsis transcriptome is regulated by C, N or CN interactions. We found ample evidence for interactions between C and N that include genes involved in metabolic pathways, protein degradation and auxin signaling. To provide a global, yet detailed, view of how the cell molecular network is adjusted in response to the CN treatments, we constructed a qualitative multinetwork model of the Arabidopsis metabolic and regulatory molecular network, including 6,176 genes, 1,459 metabolites and 230,900 interactions among them. We integrated the quantitative models of CN gene regulation with the wiring diagram in the multinetwork, and identified specific interacting genes in biological modules that respond to C, N or CN treatments. Conclusion: Our results indicate that CN regulation occurs at multiple levels, including potential post-transcriptional control by microRNAs. The network analysis of our systematic dataset of CN treatments indicates that CN sensing is a mechanism that coordinates the global and coordinated regulation of specific sets of molecular machines in the plant cell.

Infinite time aggregation for the critical Patlak-Keller-Segel model in ℝ2

Abstract: We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R2. Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8 Π/Χ. Actually, we prove that solutions blow-up as a delta dirac at the center of mass when t→∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2 blow-up as t→∞ if initially bounded.

The Integument of Water-walking Arthropods: Form and Function

Abstract: We develop a coherent view of the form and function of the integument of water-walking insects and spiders by reviewing biological work on the subject in light of recent advances in surface science. Particular attention is given to understanding the complex nature of the interaction between water-walking arthropods and the air–water surface. We begin with a discussion of the fundamental principles of surface tension and the wetting of a solid by a fluid. These basic concepts are applied to rationalize the form of various body parts of water-walking arthropods according to their function. Particular attention is given to the influence of surface roughness on water-repellency, a critical feature of water-walkers that enables them to avoid entrapment at the interface, survive the impact of raindrops and breathe if submerged. The dynamic roles of specific surface features in thrust generation, drag reduction and anchoring on the free surface are considered. New imaging techniques that promise important insights into this class of problems are discussed. Finally, we highlight the interplay between the biology, physics and engineering communities responsible for the rapid recent advances in the biomimetic design of smart, water-repellent surfaces.

Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems

Abstract: In a recent paper the authors developed and tested two novel computational algorithms for predicting the mean linear response of a chaotic dynamical system to small changes in external forcing via the fluctuation-dissipation theorem (FDT): the short-time FDT (ST-FDT), and the hybrid Axiom A FDT (hA-FDT). Unlike the earlier work in developing fluctuation-dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, these two new methods are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. These two algorithms take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian (qG-FDT) approximation of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. It has been discovered that the ST-FDT algorithm is an extremely precise linear response approximation for short response times, but numerically unstable for longer response times. On the other hand, the hA-FDT method is numerically stable for all times, but is less accurate for short times. Here we develop blended linear response algorithms, by combining accurate prediction of the ST-FDT method at short response times with numerical stability of qG-FDT and hA-FDT methods at longer response times. The new blended linear response algorithms are tested on the nonlinear Lorenz 96 model with 40 degrees of freedom, chaotic behaviour, forcing, dissipation, and mimicking large-scale features of real-world geophysical models in a wide range of dynamical regimes varying from weakly to strongly chaotic, and to fully turbulent. The results below for the blended response algorithms have a high level of accuracy for the linear response of both mean state and variance throughout all the different chaotic regimes of the 40-mode model. These results point the way towards the potential use of the blended response algorithms in operational long-term climate change projection.

Connecting solvation shell structure to proton transport kinetics in hydrogen-bonded networks via population correlation functions.

Abstract: A theory based on population correlation functions is introduced for connecting solvation topologies and microscopic mechanisms to transport kinetics of charge defects in hydrogen-bonded networks. The theory is tested on the hydrated proton by extracting a comprehensive set of relaxation times, lifetimes, and rates from ab initio molecular dynamics simulations and comparing to recent femtosecond experiments. When applied to the controversial case of the hydrated hydroxide ion, the theory predicts that only one out of three proposed transport models is consistent with known experimental data.

Toward closed-loop optimization of deep brain stimulation for Parkinson's disease: Concepts and lessons from a computational model

Abstract: Deep brain stimulation (DBS) of the subthalamic nucleus with periodic, high-frequency pulse trains is an increasingly standard therapy for advanced Parkinson's disease. Here, we propose that a closed-loop global optimization algorithm may identify novel DBS waveforms that could be more effective than their high-frequency counterparts. We use results from a computational model of the Parkinsonian basal ganglia to illustrate general issues relevant to eventual clinical or experimental tests of such an algorithm. Specifically, while the relationship between DBS characteristics and performance is highly complex, global search methods appear able to identify novel and effective waveforms with convergence rates that are acceptably fast to merit further investigation in laboratory or clinical settings.

Birth control for giants

Abstract: The standard Erdős-Renyi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n/2 rounds as one time unit, a phase transition occurs at time t = 1 when a giant component (one of size constant times n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory. We define a broad class of graph evolutions in which at each round one chooses one of two random edges {v 1, v 2}, {v 3, v 4} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a differential equation for S(t). There is a critical time t c so that S(t) → ∞ as t approaches t c from below. We show that the discrete random process asymptotically follows the differential equation for all subcritical t t c we show the existence of a giant component, so that t = t c may be fairly considered a phase transition. Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which we can either delay or accelerate the birth of the giant component.

Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices

Abstract: We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e−V(x) where V is a polynomial, V(x) = κ2mx2m + · · ·, κ2m > 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2 and 1.4 below. For the same class of weights, a proof of universality in the bulk of the spectrum is given in [12] for the unitary ensembles and in [9] for the orthogonal and symplectic ensembles. Our starting point in the unitary case is [12], and for the orthogonal and symplectic cases we rely on our recent work [9], which in turn depends on the earlier work of Widom [46] and Tracy and Widom [42]. As in [9], the uniform Plancherel-Rotach-type asymptotics for the orthogonal polynomials found in [12] plays a central role. The formulae in [46] express the correlation kernels for β = 1, 4 as a sum of a Christoffel-Darboux (CD) term, as in the case β = 2, together with a correction term. In the bulk scaling limit [9], the correction term is of lower order and does not contribute to the limiting form of the correlation kernel. By contrast, in the edge scaling limit considered here, the CD term and the correction term contribute to the same order: this leads to additional technical difficulties over and above [49]. © 2006 Wiley Periodicals, Inc.

Euclidean distortion and the sparsest cut

Abstract: We prove that every $n$-point metric space of negative type (and, in particular, every $n$-point subset of $L_1$) embeds into a Euclidean space with distortion $O(\sqrt{\log n} \cdot\log \log n)$, a result which is tight up to the iterated logarithm factor. As a consequence, we obtain the best known polynomial-time approximation algorithm for the Sparsest Cut problem with general demands. Namely, if the demand is supported on a subset of size $k$, we achieve an approximation ratio of $O(\sqrt{\log k}\cdot \log \log k)$.

Magnitude-preserving ranking algorithms

Abstract: This paper studies the learning problem of ranking when one wishes not just to accurately predict pairwise ordering but also preserve the magnitude of the preferences or the difference between ratings, a problem motivated by its key importance in the design of search engines, movie recommendation, and other similar ranking systems. We describe and analyze several algorithms for this problem and give stability bounds for their generalization error, extending previously known stability results to non-bipartite ranking and magnitude of preference-preserving algorithms. We also report the results of experiments comparing these algorithms on several datasets and compare these results with those obtained using an algorithm minimizing the pairwise misranking error and standard regression.

Stretch-coil transition and transport of fibers in cellular flows.

Abstract: It is shown that a slender elastic fiber moving in a Stokesian fluid can be susceptible to a buckling instability---termed the ``stretch-coil'' instability---when moving in the neighborhood of a hyperbolic stagnation point of the flow. When the stagnation point is part of an extended cellular flow, it is found that immersed fibers can move as random walkers across time-independent closed-streamline flow. It is also found that the flow is segregated into transport regions around hyperbolic stagnation points and their manifolds, and closed entrapment regions around elliptic points.

Nonlinear non-local elliptic equation modelling electrostatic actuation

Abstract: The nonlinear non-local elliptic equation governing the deflection of charged plates in electrostatic actuators is studied under the pinned and the clamped boundary conditions. Results concerning the existence, construction and approximation, and behaviour of classical and singular solutions with respect to the variation of physical parameters of the equation in various situations are presented.

Action minimization and sharp‐interface limits for the stochastic Allen‐Cahn equation

Abstract: We study the action minimization problem which is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharp-interface limit is related to (but di erent from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct “most likely switching pathways,” depending on the relative costs of nucleation and propagation of interfaces. This competition is captured by the limit of the action functional, which we derive formally and propose as the natural candidate for the -limit. Guided by the reduced functional, we prove upper and lower bounds for the minimal action which agree on the level of scaling. (This is a preprint of an article accepted for publication in Comm. Pure App. Math, October 2005.)

Commutation relations for Schramm-Loewner evolutions

Abstract: Schramm-Loewner evolutions (SLEs) describe a one-parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this paper we are interested in questions pertaining to the definition of several SLEs in a domain (i.e., several random curves). In particular, we derive infinitesimal commutation conditions, discuss some elementary solutions, study integrability conditions following from commutation, and show how to lift these infinitesimal relations to global relations in simple cases. The situation in multiply connected domains is also discussed. © 2007 Wiley Periodicals, Inc.

On the number of interior peak solutions for a singularly perturbed Neumann problem

Abstract: We consider the following singularly perturbed Neumann problem: where Δ = Σ ∂2/∂x is the Laplace operator, ϵ > 0 is a constant, Ω is a bounded, smooth domain in ℝN with its unit outward normal ν, and f is superlinear and subcritical. A typical f is f(u) = up where 1 < p < +∞ when N = 2 and 1 < p < (N + 2)/(N − 2) when N ≥ 3. We show that there exists an ϵ0 > 0 such that for 0 < ϵ < ϵ0 and for each integer K bounded by where αN, Ω, f is a constant depending on N, Ω, and f only, there exists a solution with K interior peaks. (An explicit formula for αN, Ω, f is also given.) As a consequence, we obtain that for ϵ sufficiently small, there exists at least [αN, Ωf/ϵN (|ln ϵ|)N] number of solutions. Moreover, for each m ∈ (0, N) there exist solutions with energies in the order of ϵN−m. © 2006 Wiley Periodicals, Inc.

CMV: The unitary analogue of Jacobi matrices

Abstract: We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices studied recently by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behavior of this system. © 2006 Wiley Periodicals, Inc.

On a micro‐macro model for polymeric fluids near equilibrium

Abstract: In this paper, we study a micro-macro model for polymeric fluid. The system involves coupling between the macroscopic momentum equation and a microscopic evolution equation describing the combined effects of the microscopic potential and thermofluctuation. We employ an energetic variation procedure to explore the relation between the macroscopic transport of the particles and the induced elastic stress due to the microscopic structure. For the initial data not far from the equilibrium, we prove the global existence and uniqueness of classical solutions to the system. © 2006 Wiley Periodicals, Inc.

Critical Analysis of Dimension Reduction by a Moment Closure Method in a Population Density Approach to Neural Network Modeling

Abstract: Computational techniques within the population density function (PDF) framework have provided time-saving alternatives to classical Monte Carlo simulations of neural network activity. Efficiency of the PDF method is lost as the underlying neuron model is made more realistic and the number of state variables increases. In a detailed theoretical and computational study, we elucidate strengths and weaknesses of dimension reduction by a particular moment closure method (Cai, Tao, Shelley, & McLaughlin, 2004; Cai, Tao, Rangan, & McLaughlin, 2006) as applied to integrate-and-fire neurons that receive excitatory synaptic input only. When the unitary postsynaptic conductance event has a single-exponential time course, the evolution equation for the PDF is a partial differential integral equation in two state variables, voltage and excitatory conductance. In the moment closure method, one approximates the conditional kth centered moment of excitatory conductance given voltage by the corresponding unconditioned moment. The result is a system of k coupled partial differential equations with one state variable, voltage, and k coupled ordinary differential equations. Moment closure at k = 2 works well, and at k = 3 works even better, in the regime of high dynamically varying synaptic input rates. Both closures break down at lower synaptic input rates. Phase-plane analysis of the k = 2 problem with typical parameters proves, and reveals why, no steady-state solutions exist below a synaptic input rate that gives a firing rate of 59 s-1 in the full 2D problem. Closure at k = 3 fails for similar reasons. Low firing-rate solutions can be obtained only with parameters for the amplitude or kinetics (or both) of the unitary postsynaptic conductance event that are on the edge of the physiological range. We conclude that this dimension-reduction method gives ill-posed problems for a wide range of physiological parameters, and we suggest future directions.