applications and generalizations of fisher hartwig asymptotics fisher hartwig asymptotics refers to the large n form of a class of toeplitz determinants with singular generating functions this class of toeplitz determinants occurs in the study of the spin spin correlations for the two dimensional ising model and the ground state density matrix of the impenetrable bose gas amongst other problems in mathematical physics we give a new application of the original fisher hartwig formula to the asymptotic decay of the ising correlations above tc while the study of the bose gas density matrix leads us to generalize the fisher hartwig formula to the asymptotic form of random matrix averages over the classical groups and the gaussian and laguerre unitary matrix ensembles another viewpoint of our generalizations is that they extend to hankel determinants the fisher hartwig asymptotic form known for toeplitz determinants

feature extraction by non parametric mutual information maximization we present a method for learning discriminative feature transforms using as criterion the mutual information between class labels and transformed features instead of a commonly used mutual information measure based on kullback leibler divergence we use a quadratic divergence measure which allows us to make an efficient non parametric implementation and requires no prior assumptions about class densities in addition to linear transforms we also discuss nonlinear transforms that are implemented as radial basis function networks extensions to reduce the computational complexity are also presented and a comparison to greedy feature selection is made

algebraic bethe ansatz for singular solutions the bethe equations for the isotropic periodic spin 1 2 heisenberg chain with n sites have solutions containing i 2 that are singular both the corresponding energy and the algebraic bethe ansatz vector are divergent such solutions must be carefully regularized we consider a regularization involving a parameter that can be determined using a generalization of the bethe equations these generalized bethe equations provide a practical way of determining which singular solutions correspond to eigenvectors of the model

complex wishart matrices and conductance in mesoscopic systems exact results it is noted that the hypothesis of independent random complex elements for the off diagonal blocks say m2 and m3 of the transfer matrix describing conductance in a mesoscopic wire allows the eigenvalue distribution of the matrix product m3 m3 or m2 m2 to be computed exactly exact expressions in terms of double wronskian and toeplitz determinants are derived for the distribution of the smallest and second smallest eigenvalue of this and similar random matrix products

smallest eigenvalue distribution of the fixed trace laguerre beta ensemble in this paper we study the entanglement of the reduced density matrix of a bipartite quantum system in a random pure state it transpires that this involves the computation of the smallest eigenvalue distribution of the fixed trace laguerre ensemble of n n random matrices we showed that for finite n the smallest eigenvalue distribution may be expressed in terms of jack polynomials furthermore based on the exact results we found a limiting distribution when the smallest eigenvalue is suitably scaled with n followed by a large n limit our results turn out to be the same as the smallest eigenvalue distribution of the classical laguerre ensembles without the fixed trace constraint this suggests in a broad sense the global constraint does not influence local correlations at least in the large n limit consequently we have solved an open problem the determination of the smallest eigenvalue distribution of the reduced density matrix obtained by tracing out the environmental degrees of freedom for a bipartite quantum system of unequal dimensions

large deviation eigenvalue density for the soft edge laguerre and jacobi β ensembles we analyze the eigenvalue density for the laguerre and jacobi ensembles in the cases that the corresponding exponents are extensive in particular we obtain the asymptotic expansion up to terms o 1 in the large deviation regime outside the limiting interval of support as found in recent studies of the large deviation density for the gaussian ensemble and laguerre ensemble with fixed exponent there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge

beyond fano s inequality bounds on the optimal f score ber and cost sensitive risk and their implications fano s inequality lower bounds the probability of transmission error through a communication channel applied to classification problems it provides a lower bound on the bayes error rate and motivates the widely used infomax principle in modern machine learning we are often interested in more than just the error rate in medical diagnosis different errors incur different cost hence the overall risk is cost sensitive two other popular criteria are balanced error rate ber and f score in this work we focus on the two class problem and use a general definition of conditional entropy including shannon s as a special case to derive upper lower bounds on the optimal f score ber and cost sensitive risk extending fano s result as a consequence we show that infomax is not suitable for optimizing f score or cost sensitive risk in that it can potentially lead to low f score and high risk for cost sensitive risk we propose a new conditional entropy formulation which avoids this inconsistency in addition we consider the common practice of using a threshold on the posterior probability to tune performance of a classifier as is widely known a threshold of 0 5 where the posteriors cross minimizes error rate we derive similar optimal thresholds for f score and ber

origin of the singular bethe ansatz solutions for the heisenberg xxz spin chain we investigate symmetry properties of the bethe ansatz wave functions for the heisenberg xxz spin chain the xxz hamiltonian commutes simultaneously with the shift operator t and the lattice inversion operator v in the space of 1 with the eigenvalue of t we show that the bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of t and v with quantum number 1 1 where is the eigenvalue of v therefore the bethe ansatz wave functions should be singular for nondegenerate eigenstates of the hamiltonian with quantum number 1 1 it is also shown that such states exist in any nontrivial down spin number sector and that the number of them diverges exponentially with the chain length c 2000 published by elsevier science b v all rights reserved

the beta wishart ensemble we introduce a broken arrow matrix model for the wishart ensemble which improves on the traditional bidiagonal model by generalizing to non identity covariance parameters we prove that its joint eigenvalue density involves the correct hypergeometric function of two matrix arguments and a continuous parameter 0 if we choose 1 2 4 we recover the classical wishart ensembles of general covariance over the reals complexes and quaternions jack polynomials are often defined as the eigenfunctions of the laplace beltrami operator we prove that jack polynomials are in addition eigenfunctions of an integral operator defined as an average over a dependent measure on the sphere when combined with an identity due to stanley we derive a definition of jack polynomials an efficient numerical algorithm is also presented for simulations the algorithm makes use of secular equation software for broken arrow matrices currently unavailable in the popular technical computing languages the simulations are matched against the cdfs for the extreme eigenvalues the techniques here suggest that arrow and broken arrow matrices can play an important role in theoretical and computational random matrix theory including the study of corners processes we provide a number of simulations illustrating the extreme eigenvalue distributions that are likely to be useful for applications we also compare the n answer for all with the free probability prediction

multidimensional density shaping by sigmoids an estimate of the probability density function of a random vector is obtained by maximizing the output entropy of a feedforward network of sigmoidal units with respect to the input weights classification problems can be solved by selecting the class associated with the maximal estimated density newton s optimization method applied to the estimated density yields a recursive estimator for a random variable or a random sequence a constrained connectivity structure yields a linear estimator which is particularly suitable for real time prediction a gaussian nonlinearity yields a closed form solution for the network s parameters which may also be used for initializing the optimization algorithm when other nonlinearities are employed a triangular connectivity between the neurons and the input which is naturally suggested by the statistical setting reduces the number of parameters applications to classification and forecasting problems are demonstrated

plurality and resemblance in fmri data analysis abstract we apply nine analytic methods employed currently in imaging neuroscience to simulated and actual bold fmri signals and compare their performances under each signal type starting with baseline time series generated by a resting subject during a null hypothesis study we compare method performance with embedded focal activity in these series of three different types whose magnitudes and time courses are simple convolved with spatially varying hemodynamic responses and highly spatially interactive we then apply these same nine methods to bold fmri time series from contralateral primary motor cortex and ipsilateral cerebellum collected during a sequential finger opposition study paired comparisons of results across methods include a voxel specific concordance correlation coefficient for reproducibility and a resemblance measure that accommodates spatial autocorrelation of differences in activity surfaces receiver operating characteristic curves show considerable model differences in ranges less than 10 significance level false positives and greater than 80 power true positives concordance and resemblance measures reveal significant differences between activity surfaces in both data sets these measures can assist researchers by identifying groups of models producing similar and dissimilar results and thereby help to validate consolidate and simplify reports of statistical findings a pluralistic strategy for fmri data analysis can uncover invariant and highly interactive relationships between local activity foci and serve as a basis for further discovery of organizational principles of the brain results also suggest that a pluralistic empirical strategy coupled formally with substantive prior knowledge can help to uncover new brain behavior relationships that may remain hidden if only a single method is employed

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

completeness of solutions of bethe s equations we consider the bethe equations for the isotropic spin 1 2 heisenberg quantum spin chain with periodic boundary conditions we formulate a conjecture for the number of solutions with pairwise distinct roots of these equations in terms of numbers of so called singular or exceptional solutions using homotopy continuation methods we find all such solutions of the bethe equations for chains of length up to 14 the numbers of these solutions are in perfect agreement with the conjecture we also discuss an indirect method of finding solutions of the bethe equations by solving the baxter t q equation we briefly comment on implications for thermodynamical computations based on the string hypothesis

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

information divergence and risk for binary experiments we unify f divergences bregman divergences surrogate regret bounds proper scoring rules cost curves roc curves and statistical information we do this by systematically studying integral and variational representations of these objects and in so doing identify their representation primitives which all are related to cost sensitive binary classification as well as developing relationships between generative and discriminative views of learning the new machinery leads to tight and more general surrogate regret bounds and generalised pinsker inequalities relating f divergences to variational divergence the new viewpoint also illuminates existing algorithms it provides a new derivation of support vector machines in terms of divergences and relates maximum mean discrepancy to fisher linear discriminants

recursion for the smallest eigenvalue density of β wishart laguerre ensemble the statistics of the smallest eigenvalue of wishart laguerre ensemble is important from several perspectives the smallest eigenvalue density is typically expressible in terms of determinants or pfaffians these results are of utmost significance in understanding the spectral behavior of wishart laguerre ensembles and among other things unveil the underlying universality aspects in the asymptotic limits however obtaining exact and explicit expressions by expanding determinants or pfaffians becomes impractical if large dimension matrices are involved for the real matrices beta 1 edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices very recently an analogous recurrence scheme has been obtained for the complex matrices beta 2 in the present work we extend this to beta wishart laguerre ensembles for the case when exponent alpha in the associated laguerre weight function lambda alpha e beta lambda 2 is a non negative integer while beta is positive real this also gives access to the smallest eigenvalue density of fixed trace beta wishart laguerre ensemble as well as moments for both cases moreover comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly exact evaluations for large values of n the matrix dimension and alpha also enable us to compare with tracy widom density and large deviation results of katzav and castillo we also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the wigner smith matrix and are relevant to the problem of quantum chaotic scattering

information maximization and independent component analysis is there a difference this article provides a detailed and rigorous analysis of the two commonly used methods for redundancy reduction linear independent component analysis ica posed as a direct minimization of a suitably chosen redundancy measure and information maximization infomax of a continuous stochastic signal transmitted through an appropriate nonlinear network the article shows analytically that ica based on the kullbackleibler information as a redundancy measure and infomax lead to the same solution if the parameterization of the output nonlinear functions in the latter method is sufficiently rich furthermore this work discusses the alternative redundancy measures not based on the kullback leibler information distance the practical issues of applying ica and infomax are also discussed and illustrated on the problem of extracting statistically independent factors from a linear pixel by pixel mixture of images

on the relationship between the separability measures and the bayes probability of error the information measures as a special class of efficiency measures of muiticategory information systems and their relations to the bayes probability of error p b have been recently defined and investigated another class of efficiency measures called the separability measures is introduced in this paper the relationship between any separability measure and p b is determined for any 2 leq q where q denotes the number of categories in a multicategory information system as before epsilon 0 and epsilon m criteria are proposed as the similarity measures between the separability measures and p b the problems of determination for any 2 leq q of all the separability measures with minimal epsilon 0 and epsilon m criteria called epsilon 0 optimal and epsilon m optimal are defined and completely solved respectively the minimal values of epsilon 0 and epsilon m criteria are evaluated as well it is proved that the information measures are for each 3 leq q more similar to p b than the separability measures with respect to the minimal values of both epsilon 0 and epsilon m criteria it is pointed out that the average conditional quadratic entropy is not only an information measure but also a separability measure which is for each 3 leq q optimal and very close to epsilon m optimal separability measures with respect to the epsilon m criterion

nonuniversal correlations for random matrix ensembles the eigenvalue correlations of the generalized gaussian and laguerre random matrix ensembles are calculated exactly the fluctuations are shown to be nonuniversal in certain intervals of the spectrum a physical example from quantum transport where such nonuniversal effects occur is discussed

random matrix ensembles with an effective extensive external charge recent theoretical studies of chaotic scattering have encountered ensembles of random matrices in which the eigenvalue probability density function contains a one body factor with an exponent proportional to the number of eigenvalues two such ensembles have been encountered an ensemble of unitary matrices specified by the so called poisson kernel and the laguerre ensemble of positive definite matrices here we consider various properties of these ensembles jack polynomial theory is used to prove a reproducing property of the poisson kernel and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the dyson circular ensemble for the laguerre ensemble the scaled global density is calculated exactly for all even values of the parameter while for random matrices with unitary symmetry the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class

on the relationship between the efficiency measures of multicategory information systems the concepts of efficiency measures of multicategory information systems and similarity measures between two efficiency measures are introduced and developed the relationship between two efficiency measures is defined by the corresponding set d and the appropriate marginal functions are introduced an earlier work of ben bassat s dealing with the ranking of features according to feature selection criteria is generalized and refined consequently a general similarity measure between two efficiency measures is established it is analyzed in the case of monotonic marginal functions and especially when the set d is convex some important sufficient conditions for d to be convex are also given finally two other similarity measures are defined and interpreted

bethe algebra of homogeneous xxx heisenberg model has simple spectrum we show that the algebra of commuting hamiltonians of the homogeneous xxx heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two dimensional mathfrak gl 2 modules as a byproduct we show that there exist exactly binom n l binom n l 1 two dimensional vector subspaces v subset mathbb c u with a basis f g in v such that deg f l deg g n l 1 and f u g u 1 f u 1 g u u 1 n

entropy analysis and new constructions of biometric key generation systems in this paper a new soft two level approach for the generation of multiple and revocable biometric keys adapted to the analog nature of biometric signals is proposed it consists of a novel randomized soft code offset construction for the euclidean metric at the first level and a code redundancy construction for the hamming metric possibly based on a reed solomon code at the second level the shannon entropy analysis shows that the new construction achieves optimal security with respect to biometric template protection whereas the previously proposed constructions for the euclidean metric are shown to be insecure in the multiple key setting in addition a general code offset construction for the hamming metric is analyzed in terms of the shannon entropy instead of the usual min entropy this allows a nice characterization of the codes that provide biometric template protection in the multiple key scenario as well as a further differentiation among these codes with respect to the achievable security level

fluid dynamic models for bhattacharyya based discriminant analysis classical discriminant analysis attempts to discover a low dimensional subspace where class label information is maximally preserved under projection canonical methods for estimating the subspace optimize an information theoretic criterion that measures the separation between the class conditional distributions unfortunately direct optimization of the information theoretic criteria is generally non convex and intractable in high dimensional spaces in this work we propose a novel tractable algorithm for discriminant analysis that considers the class conditional densities as interacting fluids in the high dimensional embedding space we use the bhattacharyya criterion as a potential function that generates forces between the interacting fluids and derive a computationally tractable method for finding the low dimensional subspace that optimally constrains the resulting fluid flow we show that this model properly reduces to the optimal solution for homoscedastic data as well as for heteroscedastic gaussian distributions with equal means we also extend this model to discover optimal filters for discriminating gaussian processes and provide experimental results and comparisons on a number of datasets

application of the τ function theory of painleve equations to random matrices pv piii the lue jue and cue with denoting an average with respect to the eigenvalue pdf for the laguerre unitary ensemble the object of our study is e n i a x l i l 0 i l for i 0 s and i s where x i l 1 for l i and x i l 0 other wise using okamoto s development of the theory of the painleve v equation it is shown that e n i a is a function associated with the hamiltonian therein and so can be characterized as the solution of a certain second order second degree differential equation or in terms of the solution of certain difference equations the cases 0 and 2 are of particular interest because they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue in the case i s e n i a is simply related to an average in the jacobi unitary ensemble and this in turn is simply related to certain averages over the orthogonal group the unitary symplectic group and the circular unitary ensemble the latter integrals are of interest for their combinatorial content also considered are the hard edge and soft edge scaled limits of e n i a in particular in the hard edge scaled limit it is shown that the limiting quantity e hard 0 s a can be evaluated as a function associated with the hamiltonian in okamoto s theory of the painleve iii equation

a fast fixed point algorithm for independent component analysis we introduce a novel fast algorithm for independent component analysis which can be used for blind source separation and feature extraction we show how a neural network learning rule can be transformed into a fixedpoint iteration which provides an algorithm that is very simple does not depend on any user defined parameters and is fast to converge to the most accurate solution allowed by the data the algorithm finds one at a time all nongaussian independent components regardless of their probability distributions the computations can be performed in either batch mode or a semiadaptive manner the convergence of the algorithm is rigorously proved and the convergence speed is shown to be cubic some comparisons to gradient based algorithms are made showing that the new algorithm is usually 10 to 100 times faster sometimes giving the solution in just a few iterations

quantifying stimulus discriminability a comparison of information theory and ideal observer analysis performance in sensory discrimination tasks is commonly quantified using either information theory or ideal observer analysis these two quantitative frameworks are often assumed to be equivalent for example higher mutual information is said to correspond to improved performance of an ideal observer in a stimulus estimation task to the contrary drawing on and extending previous results we show that five information theoretic quantities entropy response conditional entropy specific information equivocation and mutual information violate this assumption more positively we show how these information measures can be used to calculate upper and lower bounds on ideal observer performance and vice versa the results show that the mathematical resources of ideal observer analysis are preferable to information theory for evaluating performance in a stimulus discrimination task we also discuss the applicability of information theory to questions that ideal observer analysis cannot address

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

asymptotics of tracy widom distributions and the total integral of a painleve ii function the tracy widom distribution functions involve integrals of a painleve ii function starting from positive infinity in this paper we express the tracy widom distribution functions in terms of integrals starting from minus infinity there are two consequences of these new representations the first is the evaluation of the total integral of the hastings mcleod solution of the painleve ii equation the second is the evaluation of the constant term of the asymptotic expansions of the tracy widom distribution functions as the distribution parameter approaches minus infinity for the gue tracy widom distribution function this gives an alternative proof of the recent work of deift its and krasovsky the constant terms for the goe and gse tracy widom distribution functions are new

on the relationship between the information measures and the bayes probability of error the concept of efficiency measures of multicategory information systems as well as the concepts of the relationship and the similarity measures between two efficiency measures have been recently introduced and developed in this paper the concave measures as a general class of efficiency measures and the information measures as a special class of concave measures are defined and investigated the relationship between any concave measure information measure in particular and the bayes probability of error p b is determined for any 2 leq q where q denotes the number of categories in a multicategory information system the so called epsilon 0 and epsilon m criteria are proposed as the similarity measures between the information measures and p b the problems of determination for any 2 leq q of all the information measures with minimal epsilon 0 and epsilon m criteria called epsilon 0 optimal and epsilon m optimal are formulated and completely solved respectively additionally the minimal values of epsilon 0 and epsilon m criteria are evaluated as well it is pointed out that the well known average conditional quadratic entropy is for all 2 leq q very close to the epsilon 0 optimal and epsilon m optimal information measures with respect to epsilon 0 and epsilon m criteria respectively

laguerre ensembles of random matrices nonuniversal correlation functions the eigenvalue correlation functions of the laguerre ensembles exponential ensembles of real symmetric and self dual quaternion random matrices are calculated it is shown that in certain intervals of the spectrum the renormalized correlations are not universal a physical example in the transmission spectrum of a disordered conductor where such nonuniversality occurs is given

correlation functions of the bc calogero sutherland model the bc type calogero sutherland model csm is an integrable extension of the ordinary a type csm that possesses a reflection symmetry point the bc cs mi sr elated to the chiral classes of random matrix ensembles rmes in exactly the same way as the a csm is related to the dyson classes we first develo pt he fermionic replica model formalism suitable to treat all chiral rmes by exploiting generalized colour flavour transformation we then extend the method to find the exact asymptotics of the bc csm density profile consistency of our result with the c 1g aussian conformal field theory description is verified the emerging friedel oscillations structure and sum rules are discussed in details we also compute the distribution of the particle nearest to the reflection point

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

comment on relations between entropy and error probability for original paper see feder and merhav ieee trans inform theory vol 40 p 259 66 1994 january the present authors discuss the upper and lower bounds on the equivocation in terms of the bayes error probability

universality of the stochastic bessel operator we establish universality at the hard edge for general beta ensembles assuming that the background potential v is a polynomial such that x mapsto v x 2 is strongly convex beta ge 1 and the dimension difference parameter a ge 0 the method rests on the corresponding tridiagonal matrix models showing that their appropriate continuum scaling limit is given by the stochastic bessel operator as conjectured in edelman and sutton j stat phys 127 1121 1165 2007 and rigorously established in ramirez and rider commun math phys 288 887 906 2009 the latter characterizes the hard edge in the case of linear potential and all beta the classical beta laguerre ensembles

on the distributions of the lengths of the longest monotone subsequences in random words we consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length n from an alphabet of k letters in the limit as k these become the corresponding distributions for permutations on n letters we find toeplitz determinant representations for the exponential generating functions on n of these distribution functions and show that they are expressible in terms of solutions of painleve v equations we show further that in the weakly increasing case the generating unction gives the distribution of the smallest eigenvalue in the k k laguerre random matrix ensemble and that the distribution itself has after centering and normalizing an n limit which is equal to the distribution function for the largest eigenvalue in the gaussian unitary ensemble of k k hermitian matrices of trace zero

conditional likelihood maximisation a unifying framework for information theoretic feature selection we present a unifying framework for information theoretic feature selection bringing almost two decades of research on heuristic filter criteria under a single theoretical interpretation this is in response to the question what are the implicit statistical assumptions of feature selection criteria based on mutual information to answer this we adopt a different strategy than is usual in the feature selection literature instead of trying to define a criterion we derive one directly from a clearly specified objective function the conditional likelihood of the training labels while many hand designed heuristic criteria try to optimize a definition of feature relevancy and redundancy our approach leads to a probabilistic framework which naturally incorporates these concepts as a result we can unify the numerous criteria published over the last two decades and show them to be low order approximations to the exact but intractable optimisation problem the primary contribution is to show that common heuristics for information based feature selection including markov blanket algorithms as a special case are approximate iterative maximisers of the conditional likelihood a large empirical study provides strong evidence to favour certain classes of criteria in particular those that balance the relative size of the relevancy redundancy terms overall we conclude that the jmi criterion yang and moody 1999 meyer et al 2008 provides the best tradeoff in terms of accuracy stability and flexibility with small data samples

statistical analysis of the product high order ambiguity function the high order ambiguity function haf was introduced for the estimation of polynomial phase signals pps embedded in noise since the haf is a nonlinear operator it suffers from noise masking effects and from the appearance of undesired cross terms and possibly spurious harmonics in the presence of multicomponent mc signals the product haf phaf was then proposed as a way to improve the performance of the haf in the presence of noise and to solve the ambiguity problem in this correspondence we derive a statistical analysis of the phaf in the presence of additive white gaussian noise awgn valid for high signal to noise ratio snr and a finite number of data samples the analysis is carried out in detail for single component pps but the multicomponent case is also discussed error propagation phenomena implicit in the recursive structure of the phaf based estimator are explicitly taken into account the analysis is validated by simulation results for both single and multicomponent ppss

criteria for multichannel signal separation we consider the problem in which we want to separate two or more signals that are coupled to each other through an unknown multiple input multiple output linear system channel we prove that the signals can be decoupled or separated using only the condition that they are statistically independent and find even weaker sufficient conditions involving their cross polyspectra by imposing these conditions on the reconstructed signals we obtain a class of criteria for signal separation these criteria are universal in the sense that they do not require any prior knowledge or information concerning the nature of the source signals they may be communication signals or speech signals or any other 1 d or multidimensional signals e g images computationally efficient algorithms for implementing the proposed criteria that only involve the iterative solution to a linear least squares problem are presented

increasing subsequences and the hard to soft edge transition in matrix ensembles our interest is in the cumulative probabilities pr l t l for the maximum length of increasing subsequences in poissonized ensembles of random permutations random fixed point free involutions and reversed random fixed point free involutions it is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary orthogonal and symplectic symmetry respectively the gap probabilities can be written as a sum over correlations for certain determinantal point processes from these expressions a proof can be given that the limiting form of pr l t l in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary orthogonal and symplectic symmetry respectively thereby reclaiming theorems due to baik deift johansson and baik rains

wishart and anti wishart random matrices we provide a compact exact representation for the distribution of the matrix elements of the wishart type random matrices for any finite number of rows and columns of without any large n approximations in particular we treat the case when the wishart type random matrix contains redundant non random information which is a new result this representation is of interest for a procedure for reconstructing the redundant information hidden in wishart matrices with potential applications to numerous models based on biological social and artificial intelligence networks

statistical independence and novelty detection with information preserving nonlinear maps according to barlow 1989 feature extraction can be understood as finding a statistically independent representation of the probability distribution underlying the measured signals the search for a statistically independent representation can be formulated by the criterion of minimal mutual information which reduces to decorrelation in the case of gaussian distributions if nongaussian distributions are to be considered minimal mutual information is the appropriate generalization of decorrelation as used in linear principal component analyses pca we also generalize to nonlinear transformations by only demanding perfect transmission of information this leads to a general class of nonlinear transformations namely symplectic maps conservation of information allows us to consider only the statistics of single coordinates the resulting factorial representation of the joint probability distribution gives a density estimation we apply this concept to the real world problem of electrical motor fault detection treated as a novelty detection task

blind source separation by sparse decomposition in a signal dictionary the blind source separation problem is to extract the underlying source signals from a set of linear mixtures where the mixing matrix is unknown this situation is common in acoustics radio medical signal and image processing hyperspectral imaging and other areas we suggest a two stage separation process a priori selection of a possibly overcomplete signal dictionary for instance a wavelet frame or a learned dictionary in which the sources are assumed to be sparsely representable followed by unmixing the sources by exploiting the their sparse representability we consider the general case of more sources than mixtures but also derive a more efficient algorithm in the case of a nonovercomplete dictionary and an equal numbers of sources and mixtures experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques

exact quantum numbers of collapsed and non collapsed two string solutions in the spin 1 2 heisenberg spin chain every solution of the bethe ansatz equations baes is characterized by a set of quantum numbers by which we can evaluate it numerically however no general rule is known how to give quantum numbers for the physical solutions of bae for the spin 1 2 xxx chain we rigorously derive all the quantum numbers for the complete set of the bethe ansatz eigenvectors in the two down spin sector with any chain length n here we obtain them both for real and complex solutions we also show that all the solutions associated with them are distinct consequently we prove the completeness of the bethe ansatz and give an exact expression for the number of real solutions which correspond to collapsed bound state solutions i e two string solutions in the sector in terms of gauss symbol moreover we prove in the sector the scheme conjectured by takahashi for solving bae systematically we also suggest that by applying the present method we can derive the quantum numbers for the spin 1 2 xxz chain

on the bayes optimality of f measure maximizers the f measure which has originally been introduced in information retrieval is nowadays routinely used as a performance metric for problems such as binary classification multi label classification and structured output prediction optimizing this measure is a statistically and computationally challenging problem since no closed form solution exists adopting a decision theoretic perspective this article provides a formal and experimental analysis of different approaches for maximizing the f measure we start with a bayes risk analysis of related loss functions such as hamming loss and subset zero one loss showing that optimizing such losses as a surrogate of the f measure leads to a high worst case regret subsequently we perform a similar type of analysis for f measure maximizing algorithms showing that such algorithms are approximate while relying on additional assumptions regarding the statistical distribution of the binary response variables furthermore we present a new algorithm which is not only computationally efficient but also bayes optimal regardless of the underlying distribution to this end the algorithm requires only a quadratic with respect to the number of binary responses number of parameters of the joint distribution we illustrate the practical performance of all analyzed methods by means of experiments with multi label classification problems

on the convexity of some divergence measures based on entropy functions three measures of divergence between vectors in a convex set of a n dimensional real vector space are defined in terms of certain types of entropy functions and their convexity property is studied among other results a classification of the entropies of degree alpha is obtained by the convexity of these measures these results have applications in information theory and biological studies

random matrix averages and the impenetrable bose gas in dirichlet and neumann boundary conditions the density matrix for the impenetrable bose gas in dirichlet and neumann boundary conditions can be written in terms of l 1n cos 1 cos l cos 2 cos l where the average is with respect to the eigenvalue probability density function for random unitary matrices from the classical groups sp n and o 2n respectively in the large n limit log gas considerations imply that the average factorizes into the product of averages of the form l 1n cos cos l by changing variables this average in turn is a special case of the function of t obtained by averaging l 1n t xl 2q over the jacobi unitary ensemble from random matrix theory the latter task is accomplished by a duality formula from the theory of selberg correlation integrals and the large n asymptotic form is obtained the corresponding large n asymptotic form of the density matrix is used via the exact solution of a particular integral equation to compute the asymptotic form of the low lying effective single particle states and their occ

exact results and universal asymptotics in the laguerre random matrix ensemble for various values of the parameters in the laguerre random matrix ensemble the distribution of the smallest eigenvalue and the scaled n level distribution function are calculated exactly in terms of generalized hypergeometric functions in certain cases these functions are expressed as multidimensional integrals from which asymptotic formulas are calculated and predictions of universal behavior verified

convergent algorithm for sensory receptive field development an unsupervised developmental algorithm for linear maps is derived which reduces the pixel entropy using the measure introduced in previous work at every update and thus removes pairwise correlations between pixels since the measure of pixel entropy has only a global minimum the algorithm is guaranteed to converge to the minimum entropy map such optimal maps have recently been shown to possess cognitively desirable properties and are likely to be used by the nervous system to organize sensory information the algorithm derived here turns out to be one proposed by goodall for pairwise decorrelation it is biologically plausible since in a neural network implementation it requires only data available locally to a neuron in training over ensembles of two dimensional input signals with the same spatial power spectrum as natural scenes networks develop output neurons with center surround receptive fields similar to those of ganglion cells in the retina some technical issues pertinent to developmental algorithms of this sort such as symmetry fixing are also discussed

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing