fast exact search in hamming space with multi index hashing there is growing interest in representing image data and feature descriptors using compact binary codes for fast near neighbor search although binary codes are motivated by their use as direct indices addresses into a hash table codes longer than 32 bits are not being used as such as it was thought to be ineffective we introduce a rigorous way to build multiple hash tables on binary code substrings that enables exact k nearest neighbor search in hamming space the approach is storage efficient and straight forward to implement theoretical analysis shows that the algorithm exhibits sub linear run time behavior for uniformly distributed codes empirical results show dramatic speedups over a linear scan baseline for datasets of up to one billion codes of 64 128 or 256 bits

the stochastic rayleigh diffusion model statistical inference and computational aspects applications to modelling of real cases abstract in this paper we consider a stochastic system modelling by the stochastic rayleigh diffusion process and we discuss theoretical aspects of the latter and establish a statistical methodology to adjust it to real cases particulary in the field of biometry and related areas the rayleigh process according to the definition of c giorno a nobile l ricciardi l sacerdote some remarks on the rayleigh process journal of applied probability 23 1986 398 408 is examined from the perspective of the corresponding nonlinear stochastic differential equation and from its associated probability density function we obtain the corresponding mean functions trend function and conditional trend function which depend of kummer functions the drift parameters are estimated by maximum likelihood on the basis of continuous sampling of the process and they are calculated by computational methods we propose numerical approximations for the diffusion coefficient from an extension of the m chesney j elliot estimating the instantaneous volatility and covariance of risky assets applied stochastic models and data analysis 11 1995 51 58 procedure to the case of nonlinear stochastic differential equations and we establish also computational procedures and simulation algorithm that are applied to obtened simulated paths of the fitted process the proposed methodology is applied to two studies carried out in andalusia spain on females and males life expectancy at birth between 1944 and 2001

a hill series approach to wavefunction nodes the hill series method is shown to have several advantages for treating the schrodinger equation when the wavefunction obeys the dirichlet boundary condition r 0 a novel combination of the virial theorem the hill series method and a finite difference method is used to find the node positions of the wavefunction for several excited states of perturbed oscillator and coulombic systems and to find critical radii for the hydrogen atom

green s operator for hamiltonians with coulomb plus polynomial potentials the hamiltonian of a coulomb plus polynomial potential in the coulomb sturmian basis has an infinite symmetric band matrix structure a band matrix can always be considered as a block tridiagonal matrix so the corresponding green s operator can be given as a matrix valued continued fraction as examples we calculate green s operator for the coulomb plus linear and quadratic confinement potential problems and determine the energy levels

escape rates in periodically driven markov processes we present an approximate analytical expression for the escape rate of time dependent driven stochastic processes with an absorbing boundary such as the driven leaky integrateand fire model for neural spiking the novel approximation i s based on a discrete state markovian modeling of the full long time dynamics with time dependent rates it is valid in a wide parameter regime beyond the restraining limits of weak driving linear response and or weak noise the scheme is carefully tested and yields excellent agreement with three different numerical methods based on the langevin equation the fokker planck equation and an integral equation

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

what is the goal of sensory coding a number of recent attempts have been made to describe early sensory coding in terms of a general information processing strategy in this paper two strategies are contrasted both strategies take advantage of the redundancy in the environment to produce more effective representations the first is described as a compact coding scheme a compact code performs a transform that allows the input to be represented with a reduced number of vectors cells with minimal rms error this approach has recently become popular in the neural network literature and is related to a process called principal components analysis pca a number of recent papers have suggested that the optimal compact code for representing natural scenes will have units with receptive field profiles much like those found in the retina and primary visual cortex however in this paper it is proposed that compact coding schemes are insufficient to account for the receptive field properties of cells in the mammalian visual pathway in contrast it is proposed that the visual system is near to optimal in representing natural scenes only if optimality is defined in terms of sparse distributed coding in a sparse distributed code all cells in the code have an equal response probability across the class of images but have a low response probability for any single image in such a code the dimensionality is not reduced rather the redundancy of the input is transformed into the redundancy of the firing pattern of cells it is proposed that the signature for a sparse code is found in the fourth moment of the response distribution i e the kurtosis in measurements with 55 calibrated natural scenes the kurtosis was found to peak when the bandwidths of the visual code matched those of cells in the mammalian visual cortex codes resembling wavelet transforms are proposed to be effective because the response histograms of such codes are sparse i e show high kurtosis when presented with natural scenes it is proposed that the structure of the image that allows sparse coding is found in the phase spectrum of the image it is suggested that natural scenes to a first approximation can be considered as a sum of self similar local functions the inverse of a wavelet possible reasons for why sensory systems would evolve toward sparse coding are presented

inference based similarity search in randomized montgomery domains for privacy preserving biometric identification similarity search is essential to many important applications and often involves searching at scale on high dimensional data based on their similarity to a query in biometric applications recent vulnerability studies have shown that adversarial machine learning can compromise biometric recognition systems by exploiting the biometric similarity information existing methods for biometric privacy protection are in general based on pairwise matching of secured biometric templates and have inherent limitations in search efficiency and scalability in this paper we propose an inference based framework for privacy preserving similarity search in hamming space our approach builds on an obfuscated distance measure that can conceal hamming distance in a dynamic interval such a mechanism enables us to systematically design statistically reliable methods for retrieving most likely candidates without knowing the exact distance values we further propose to apply montgomery multiplication for generating search indexes that can withstand adversarial similarity analysis and show that information leakage in randomized montgomery domains can be made negligibly small our experiments on public biometric datasets demonstrate that the inference based approach can achieve a search accuracy close to the best performance possible with secure computation methods but the associated cost is reduced by orders of magnitude compared to cryptographic primitives

experimental measurement of the persistence exponent of the planar ising model using a twisted nematic liquid crystal system exhibiting planar ising model dynamics we have measured the scaling exponent ensuremath theta which characterizes the time evolution p t ensuremath sim t ensuremath ensuremath theta of the probability p t that the local order parameter has not switched its state by the time t for 0 4 sec 200 sec following the phase quench the system exhibits scaling behavior and measured over this interval ensuremath theta 0 19 ifmmode pm else textpm fi 0 03 in good agreement with theoretical analysis and numerical simulations

boundary conditions for stochastic solutions of the convection diffusion equation stochastic methods offer an attractively simple solution to complex transport controlled problems and have a wide range of physical chemical and biological applications stochastic methods do not suffer from the numerical diffusion that plagues grid based methods but they typically lose accuracy in the vicinity of interfacial boundaries in this work we introduce some ideas and algorithms that can be used to implement boundary conditions in stochastic simulations of the convection diffusion equation with accuracies comparable to the bulk phase the algorithms have been tested in two dimensional channel flows over a range of peclet numbers and compared with independent finite difference calculations

on the area under a continuous time brownian motion till its first passage time the area swept out under a one dimensional brownian motion till its first passage time is analysed using a fokker planck technique we obtain an exact expression for the area distribution for the zero drift case and provide various asymptotic results for the nonzero drift case emphasizing the critical nature of the behaviour in the limit of vanishing drift the results offer important insights into the asymptotic behaviour of a number of discrete models we also provide a succinct derivation for the distribution of the maximum displacement observed during a first passage

the distribution of the area under a bessel excursion and its moments a bessel excursion is a bessel process that begins at the origin and first returns there at some given time t we study the distribution of the area under such an excursion which recently found application in the context of laser cooling the area a scales with the time as a sim t 3 2 independent of the dimension d but the functional form of the distribution does depend on d we demonstrate that for d 1 the distribution reduces as expected to the distribution for the area under a brownian excursion known as the airy distribution deriving a new expression for the airy distribution in the process we show that the distribution is symmetric in d 2 with nonanalytic behavior at d 2 we calculate the first and second moments of the distribution as well as a particular fractional moment we also analyze the analytic continuation from d 2 in the limit where d rightarrow 4 from below this analytically continued distribution is described by a one sided levy alpha stable distribution with index 2 3 and a scale factor proportional to 4 d t 3 2

statistics of the occupation time of renewal processes we present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time according to the nature of the distribution of time intervals the probability density functions of these random variables have very different scalings in time we analyze successively the cases where this distribution is narrow where it is broad with index 1 and finally where it is broad with index 1 2 the methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes see joint paper by g de smedt c godreche and j m luck 26

power laws and stretched exponentials in a noisy finite time singularity model we discuss the influence of white noise on a generic dynamical finite time singularity model for a single degree of freedom we find that the noise effectively resolves the finite time singularity and replaces it by a first passage time or absorbing state distribution with a peak at the singularity and a long time tail exhibiting power law or stretched exponential behavior the study might be of relevance in the context of hydrodynamics on a nanometer scale in material physics and in biophysics

algebraic evaluation of matrix elements in the laguerre function basis the laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron structure theory with applications in hadronic and nuclear theory as well while similar in form to the coulomb bound state eigenfunctions from the schrodinger eigenproblem or the coulomb sturmian functions from a related sturm liouville problem the laguerre functions unlike these former functions constitute a complete discrete orthonormal set for square integrable functions in three dimensions we construct the su 1 1 so 3 dynamical algebra for the laguerre functions and apply the ideas of factorization or supersymmetric quantum mechanics to derive shift operators for these functions we use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators involving powers of the radial coordinate and radial derivative in the laguerre function basis we illustrate how matrix elements for more general spherical tensor operators in three dimensional space such as the gradient may then be constructed from these radial matrix elements

a class of fokker planck equations with logarithmic factors in diffusion and drift terms the x2 ln x dependence of the diffusion coefficient in the fokker planck equation is retrieved by means of symmetry arguments exact solutions of the equation with logarithmic factors in coefficients are presented algebraic and log algebraic solutions are found for some values of exponents they seem to be analogues on an interval of log normal distributions

damped finite time singularity driven by noise we consider the combined influence of linear damping and noise on a dynamical finite time singularity model for a single degree of freedom we find that the noise effectively resolves the finite time singularity and replaces it by a first passage time distribution or absorbing state distribution with a peak at the singularity and a long time tail the damping introduces a characteristic cross over time in the early time regime the probability distribution and first passage time distribution show a power law behavior with scaling exponent depending on the ratio of the nonlinear coupling strength to the noise strength in the late time regime the behavior is controlled by the damping the study might be of relevance in the context of hydrodynamics on a nanometer scale in material physics and in biophysics

modeling the shape of the scene a holistic representation of the spatial envelope in this paper we propose a computational model of the recognition of real world scenes that bypasses the segmentation and the processing of individual objects or regions the procedure is based on a very low dimensional representation of the scene that we term the spatial envelope we propose a set of perceptual dimensions naturalness openness roughness expansion ruggedness that represent the dominant spatial structure of a scene then we show that these dimensions may be reliably estimated using spectral and coarsely localized information the model generates a multidimensional space in which scenes sharing membership in semantic categories e g streets highways coasts are projected closed together the performance of the spatial envelope model shows that specific information about object shape or identity is not a requirement for scene categorization and that modeling a holistic representation of the scene informs about its probable semantic category

product quantization for nearest neighbor search this paper introduces a product quantization based approach for approximate nearest neighbor search the idea is to decompose the space into a cartesian product of low dimensional subspaces and to quantize each subspace separately a vector is represented by a short code composed of its subspace quantization indices the euclidean distance between two vectors can be efficiently estimated from their codes an asymmetric version increases precision as it computes the approximate distance between a vector and a code experimental results show that our approach searches for nearest neighbors efficiently in particular in combination with an inverted file system results for sift and gist image descriptors show excellent search accuracy outperforming three state of the art approaches the scalability of our approach is validated on a data set of two billion vectors

superaging correlation function and ergodicity breaking for brownian motion in logarithmic potentials we consider an overdamped brownian particle moving in a confining asymptotically logarithmic potential which supports a normalized boltzmann equilibrium density we derive analytical expressions for the two time correlation function and the fluctuations of the time averaged position of the particle for large but finite times we characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential and we support our predictions with extensive langevin simulations while the boltzmann measure is used to obtain stationary correlation functions we show how the non normalizable infinite covariant density is related to the superaging behavior

symmetry classification and exact solutions of the one dimensional fokker planck equation with arbitrary coefficients of drift and diffusion symmetry properties of the one dimensional fokker planck equations with arbitrary coefficients of drift and diffusion are investigated it is proved that the group symmetry of these equations can be one two four or six parametric and corresponding criteria are obtained the changes of the variables reducing fokker planck equations to the heat equation and schrodinger equation with certain potentials are determined using the substructure of the invariance algebra of the rayleigh process equation we obtain the differential invariants and construct some classes of the exact solutions of this equation

green s matrix from jacobi matrix hamiltonian we propose two ways for determining the green s matrix for problems admitting hamiltonians that have infinite symmetric tridiagonal i e jacobi matrix form on some basis representation in addition to the recurrence relation coming from the jacobi matrix the first approach also requires the matrix elements of the green s operator between the first elements of the basis in the second approach the recurrence relation is solved directly by continued fractions and the solution is continued analytically to the whole complex plane both approaches are illustrated with the non trivial but calculable example of the d dimensional coulomb green s matrix we give the corresponding formulas for the d dimensional harmonic oscillator as well

stochastic calculus for uncoupled continuous time random walks the continuous time random walk ctrw is a pure jump stochastic process with several applications not only in physics but also in insurance finance and economics a definition is given for a class of stochastic integrals driven by a ctrw which includes the it and stratonovich cases an uncoupled ctrw with zero mean jumps is a martingale it is proved that as a consequence of the martingale transform theorem if the ctrw is a martingale the it integral is a martingale too it is shown how the definition of the stochastic integrals can be used to easily compute them by monte carlo simulation the relations between a ctrw its quadratic variation its stratonovich integral and its it integral are highlighted by numerical calculations when the jumps in space of the ctrw have a symmetric levy alpha stable distribution and its waiting times have a one parameter mittag leffler distribution remarkably these distributions have fat tails and an unbounded quadratic variation in the diffusive limit of vanishing scale parameters the probability density of this kind of ctrw satisfies the space time fractional diffusion equation fde or more in general the fractional fokker planck equation which generalizes the standard diffusion equation solved by the probability density of the wiener process and thus provides a phenomenologic model of anomalous diffusion we also provide an analytic expression for the quadratic variation of the stochastic process described by the fde and check it by monte carlo

what does the retina know about natural scenes by examining the experimental data on the statistical properties of natural scenes together with retinal contrast sensitivity data we arrive at a first principle theoretical hypothesis for the purpose of retinal processing and its relationship to an animal s environment we argue that the retinal goal is to transform the visual input as much as possible into a statistically independent basis as the first step in creating a redundancy reduced representation in the cortex as suggested by barlow the extent of this whitening of the input is limited however by the need to suppress input noise our explicit theoretical solutions for the retinal filters also show a simple dependence on mean stimulus luminance they predict an approximate weber law at low spatial frequencies and a de vries rose law at high frequencies assuming that the dominant source of noise is quantum we generate a family of contrast sensitivity curves as a function of mean luminance this family is compared to psychophysical data

ldahash improved matching with smaller descriptors sift like local feature descriptors are ubiquitously employed in computer vision applications such as content based retrieval video analysis copy detection object recognition photo tourism and 3d reconstruction feature descriptors can be designed to be invariant to certain classes of photometric and geometric transformations in particular affine and intensity scale transformations however real transformations that an image can undergo can only be approximately modeled in this way and thus most descriptors are only approximately invariant in practice second descriptors are usually high dimensional e g sift is represented as a 128 dimensional vector in large scale retrieval and matching problems this can pose challenges in storing and retrieving descriptor data we map the descriptor vectors into the hamming space in which the hamming metric is used to compare the resulting representations this way we reduce the size of the descriptors by representing them as short binary strings and learn descriptor invariance from examples we show extensive experimental validation demonstrating the advantage of the proposed approach

power law tail distributions and nonergodicity we establish an explicit correspondence between ergodicity breaking in a system described by power law tail distributions and the divergence of the moments of these distributions

reaction kinetics of cluster impurities we study the density of clustered immobile reactants in the diffusion controlled single species annihilation an initial state in which these impurities occupy a subspace of codimension d leads to a substantial enhancement of their survival probability the smoluchowski rate theory suggests that the codimensionality plays a crucial role in determining the long time behavior the system undergoes a transition at d 2 for d 2 above this critical codimension d 2 the subspace decays indefinitely at the critical codimension inverse logarithmic decay occurs ni t log t a d d above the critical codimension the decay is algebraic ni t t a d d in general the exponents governing the long time behavior depend on the dimension as well as the codimension

self similar gaussian processes for modeling anomalous diffusion we study some gaussian models for anomalous diffusion which include the time rescaled brownian motion two types of fractional brownian motion and models associated with fractional brownian motion based on the generalized langevin equation gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean square displacement to vary with t ensuremath alpha 0l ensuremath alpha l2 however these processes have different properties thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism although the two versions of fractional brownian motion and time rescaled brownian motion all have the same probability distribution function the slepian theorem can be used to compare their first passage time distributions which are different finally in order to model anomalous diffusion with a variable exponent ensuremath alpha t it is necessary to consider the multifractional extensions of these gaussian processes

first passage and first exit times of a bessel like stochastic process we study a stochastic process x t which is a particular case of the rayleigh process and whose square is the bessel process with various applications in physics chemistry biology economics finance and other fields the stochastic differential equation is dx t nd x t dt 2d dw t where w t is the wiener process the drift term can arise from a logarithmic potential or from taking x t as the norm of a multidimensional random walk due to the singularity of the drift term for x t 0 different natures of boundary at the origin arise depending on the real parameter n entrance exit and regular for each of them we calculate analytically and numerically the probability density functions of first passage times or first exit times nontrivial behavior is observed in the case of a regular boundary