expectation consistent approximate inference we propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation the approximation can be understood as replacing an average over the original intractable distribution with a tractable one it requires two tractable probability distributions which are made consistent on a set of moments and encode different features of the original intractable distribution in this way we are able to use gaussian approximations for models with discrete or bounded variables which allow us to include non trivial correlations these are neglected in many other methods we test the framework on toy benchmark problems for binary variables on fully connected graphs and 2d grids and compare with other methods such as loopy belief propagation good performance is already achieved by using single nodes as tractable substructures significant improvements are obtained when a spanning tree is used instead

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

lattice basis reduction improved practical algorithms and solving subset sum problems we report on improved practical algorithms for lattice basis reduction we present a variant of the l3 algorithm with deep insertions and a practical algorithm for blockwise korkine zolotarev reduction a concept extending l3 reduction that has been introduced by schnorr 1987 empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a unisys 6000 70 or within a couple of minutes on a sparc 2 computer

a cdma multiuser detection algorithm on the basis of belief propagation an iterative algorithm for the multiuser detection problem that arises in code division multiple access cdma systems is developed on the basis of pearl s belief propagation bp we show that the bp based algorithm exhibits nearly optimal performance in a practical time scale by utilizing the central limit theorem and self averaging property appropriately whereas direct application of bp to the detection problem is computationally difficult and far from practical we further present close relationships of the proposed algorithm to the thouless anderson palmer approach and replica analysis known in spin glass research

random matrix ensemble for the level statistics of many body localization we numerically study the level statistics of the gaussian ensemble these statistics generalize wigner dyson level statistics from the discrete set of dyson indices 1 2 4 to the continuous range 0 the gaussian ensemble covers poissonian level statistics for 0 and provides a smooth interpolation between poissonian and wigner dyson level statistics we establish the physical relevance of the level statistics of the gaussian ensemble by showing near perfect agreement with the level statistics of a paradigmatic model in studies on many body localization over the entire crossover range from the thermal to the many body localized phase in addition we show similar agreement for a related hamiltonian with broken time reversal symmetry

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

expectation propagation detection for high order high dimensional mimo systems modern communications systems use multiple input multiple output mimo and high order qam constellations for maximizing spectral efficiency however as the number of antennas and the order of the constellation grow the design of efficient and low complexity mimo receivers possesses big technical challenges for example symbol detection can no longer rely on maximum likelihood detection or sphere decoding methods as their complexity increases exponentially with the number of transmitters receivers in this paper we propose a low complexity high accuracy mimo symbol detector based on the expectation propagation ep algorithm ep allows approximating iteratively at polynomial time the posterior distribution of the transmitted symbols we also show that our ep mimo detector outperforms classic and state of the art solutions reducing the symbol error rate at a reduced computational complexity

on the complexity of sphere decoding in digital communications sphere decoding has been suggested by a number of authors as an efficient algorithm to solve various detection problems in digital communications in some cases the algorithm is referred to as an algorithm of polynomial complexity without clearly specifying what assumptions are made about the problem structure another claim is that although worst case complexity is exponential the expected complexity of the algorithm is polynomial herein we study the expected complexity where the problem size is defined to be the number of symbols jointly detected and our main result is that the expected complexity is exponential for fixed signal to noise ratio snr contrary to previous claims the sphere radius which is a parameter of the algorithm must be chosen to ensure a nonvanishing probability of solving the detection problem this causes the exponential complexity since the squared radius must grow linearly with problem size the rate of linear increase is however dependent on the noise variance and thus the rate of the exponential function is strongly dependent on the snr therefore sphere decoding can be efficient for some snr and problems of moderate size even though the number of operations required by the algorithm strictly speaking always grows as an exponential function of the problem size

entropy and the shannon mcmillan breiman theorem for beta random matrix ensembles we show that beta ensembles in random matrix theory with generic real analytic potential have the asymptotic equipartition property in addition we prove a central limit theorem for the density of the eigenvalues of these ensembles

periodically forced leaky integrate and fire model the discharge pattern of periodically forced leaky integrate and fire models is studied while previous analyses have been mainly concerned with the response of this model to sinusoidal stimulation our results hold for arbitrary periodic inputs it is shown that for any periodic input the map representing the relation between input phases at consecutive discharge times can be restricted to a piecewise continuous orientation preserving circle map this implies that i the rotation number is well defined and independent of the initial condition and ii in the same way as for sinusoidal forcing other forms of periodic stimuli can evoke only one of four types of response namely phase locking quasiperiodic discharges nonchaotic aperiodic firing and termination of the discharge after a finite number of firings

the fixed trace β hermite ensemble of random matrices and the low temperature distribution of the determinant of an n n β hermite matrix the hermite ensemble he of tridiagonal n n random matrices of dumitriu and edelman 2002 j math phys 43 5830 is a continuum of ensembles in which the reciprocal of the temperature in the 2d electrostatic interpretation of the eigenvalue characteristics can take any value the eigenvalue distributions coincide with those of the classical gaussian ensembles goe gue gse for 1 2 4 a fixed trace hermite ensemble fthe is defined from the he and is used to extend the spherical ensembles of classical symmetries to spherical ensembles at low temperature when for a fixed value of n the asymptotic distributions of reduced determinants dn of random n n h and fth matrices are shown to be standard gaussians accordingly the fluctuations of the potential at the origin ln dn have a generalized gumbel distribution at low temperature for large n and large a ln n variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions

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

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

mimo detection for high order qam based on a gaussian tree approximation this paper proposes a new detection algorithm for mimo communication systems employing high order qam constellations the factor graph that corresponds to this problem is very loopy in fact it is a complete graph hence a straightforward application of the belief propagation bp algorithm yields very poor results our algorithm is based on an optimal tree approximation of the gaussian density of the unconstrained linear system the finite set constraint is then applied to obtain a cycle free discrete distribution simulation results show that even though the approximation is not directly applied to the exact discrete distribution applying the bp algorithm to the cycle free factor graph outperforms current methods in terms of both performance and complexity the improved performance of the proposed algorithm is demonstrated on the problem of mimo detection

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

fluctuations of eigenvalues and second order poincare inequalities linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems the proofs of such results are usually rather difficult involving hard computations specific to the model in question in this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments in the process we introduce a notion of second order poincare inequalities just as ordinary poincare inequalities give variance bounds second order poincare inequalities give central limit theorems the proof of the main result employs stein s method of normal approximation a number of examples are worked out some of which are new one of the new results is a clt for the spectrum of gaussian toeplitz matrices

on invariant 2 2 β ensembles of random matrices abstract we introduce and solve exactly a family of invariant 2 2 random matrices depending on one parameter and we show that rotational invariance and real dyson index are not incompatible properties the probability density for the entries contains a weight function and a multiple trace trace interaction term which corresponds to the representation of the vandermonde squared coupling on the basis of power sums as a result the effective dyson index eff of the ensemble can take any real value in an interval two weight functions gaussian and non gaussian are explored in detail and the connections with ensembles of dumitriu edelman and the so called poisson wigner crossover for the level spacing are respectively highlighted a curious spectral twinning between ensembles of different symmetry classes is unveiled as a consequence the identification between symmetry group orthogonal unitary or symplectic and the exponent of the vandermonde 1 2 4 is shown to be potentially deceptive the proposed technical tool more generically allows for designing actual matrix models which i are rotationally invariant ii have a real dyson index eff iii have a pre assigned confining potential or alternatively level spacing profile the analytical results have been checked through numerical simulations with an excellent agreement eventually we discuss possible generalizations and further directions of research

global spectrum fluctuations for the β hermite and β laguerre ensembles via matrix models we study the global spectrum fluctuations for hermite and laguerre ensembles via the tridiagonal matrix models introduced previously by the present authors j math phys 43 5830 2002 and prove that the fluctuations describe a gaussian process on polynomials we extend our results to slightly larger classes of random matrices

large deviations asymptotics for spherical integrals consider the spherical integral i n dn en exp n tr udnu en dm n u where m n denote the haar measure on the orthogonal group on when 1 and on the unitary group un when 2 and dn en are diagonal real matrices whose spectral measures converge to d e in this paper we prove the existence and represent as solution to a variational problem the limit i d e lim n 2 log i n dn en this limit appears in so called matrix models but also in the evaluation of large deviations of the spectral measure of generalized wishart matrices our technique is based on stochastic calculus large deviations and elements from free probability

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

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

limit theorems for biorthogonal ensembles and related combinatorial identities abstract we study the fluctuations of certain biorthogonal ensembles for which the underlying family p q satisfies a finite term recurrence relation of the form x p x j p x for polynomial linear statistics of such ensembles we reformulate the cumulant method introduced in 53 in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix j in the spirit of 12 we show that the asymptotic fluctuations are described by the right limits of the matrix j moreover whenever the right limit is a laurent matrix we show that the clt is equivalent to soshnikov s main combinatorial lemma we discuss several applications to unitary invariant hermitian random matrices in particular we provide a general central limit theorem clt and a law of large numbers in the one cut regime we also prove a clt for the square singular values of the product of independent complex rectangular ginibre matrices as well as for the laguerre and jacobi biorthogonal ensembles introduced in 7 and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients finally we discuss the connection with the strong szeg limit theorem where this combinatorial method originates

sphere decoding complexity exponent for decoding full rate codes over the quasi static mimo channel in the setting of quasi static multiple input multiple output channels we consider the high signal to noise ratio snr asymptotic complexity required by the sphere decoding sd algorithm for decoding a large class of full rate linear space time codes with sd complexity having random fluctuations induced by the random channel noise and codeword realizations the introduced sd complexity exponent manages to concisely describe the computational reserves required by the sd algorithm to achieve arbitrarily close to optimal decoding performance bounds and exact expressions for the sd complexity exponent are obtained for the decoding of large families of codes with arbitrary performance characteristics for the particular example of decoding the recently introduced threaded cyclic division algebra based codes the only currently known explicit designs that are uniformly optimal with respect to the diversity multiplexing tradeoff the sd complexity exponent is shown to take a particularly concise form as a non monotonic function of the multiplexing gain to date the sd complexity exponent also describes the minimum known complexity of any decoder that can provably achieve a gap to maximum likelihood performance that vanishes in the high snr limit

free probability of type b analytic interpretation and applications in this paper we give an analytic interpretation of free convolution of type b and provide a new formula for its computation this formula allows us to show that free additive convolution of type b is essentially a re casting of conditionally free convolution we put in evidence several aspects of this operation the most significant being its apparition as an intertwiner between derivation and free convolution of type a we also show connections between several limit theorems in type a and type b free probability moreover we show that the analytical picture fits very well with the idea of considering type b random variables as infinitesimal deformations to ordinary non commutative random variables

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

a unified fluctuation formula for one cut β ensembles of random matrices using a coulomb gas approach we compute the generating function of the covariances of power traces for one cut ensembles of random matrices in the limit of large matrix size this formula depends only on the support of the spectral density and is therefore universal for a large class of models this allows us to derive a closed form expression for the limiting covariances of an arbitrary one cut ensemble as particular cases of the main result we consider the classical gaussian wishart and jacobi ensembles for which we derive previously available results as well as new ones within a unified simple framework we also discuss the connections between the problem of trace fluctuations for the gaussian unitary ensemble and the enumeration of planar maps

achieving a vanishing snr gap to exact lattice decoding at a subexponential complexity this study identifies the first lattice decoding solution that achieves in the general outage limited multiple input multiple output mimo setting and in the high rate and high signal to noise ratio limit both a vanishing gap to the error performance of the exact solution of regularized lattice decoding as well as a computational complexity that is subexponential in the number of codeword bits and in the rate the proposed solution employs lenstra lenstra lovasz based lattice reduction lr aided regularized lattice sphere decoding and proper timeout policies these performance and complexity guarantees hold for most mimo scenarios most fading statistics all channel dimensions and all full rate lattice codes in sharp contrast to the aforementioned very manageable complexity the complexity of other standard preprocessed lattice decoding solutions is revealed here to be extremely high specifically this study has quantified the complexity of regularized lattice sphere decoding and has proved that the computational resources required by this decoder to achieve a good rate reliability performance are exponential in the lattice dimensionality and in the number of codeword bits and it in fact matches in common scenarios the complexity of ml based sphere decoders through this sharp contrast this study was able to for the first time rigorously demonstrate and quantify the pivotal role of lr as a special complexity reducing ingredient

large n expansions for the laguerre and jacobi β ensembles from the loop equations the ensembles of random matrix theory with classical weights have many special properties one is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at the general order of the correlator via aomoto s method from the theory of the selberg integral we use aomoto s method to derive the full hierarchy of loop equations for laguerre and jacobi ensembles and use these to systematically construct the explicit form of the 1 n expansion at low orders this allows us to give the explicit form of corrections to the global density and allows various moments to be computed complementing results available in the literature motivated by problems in quantum transport

flexible detection criterion for source enumeration in array processing two new procedures are proposed using random matrix theory to evaluate theoretically the performance of the bayesian information criterion bic for source enumeration in array processing the proposed procedures can predict precisely the estimation results of the bic also they shed light on the behavior of the bic in order to remedy the limitation of the bic that is the bic tends to underestimate the number of sources by one in most cases an extra parameter is introduced into the bic formulation which results in a new criterion for source enumeration referred to as the flexible detection criterion fdc by carefully choosing this parameter the fdc is capable of substantially reducing the probability of underestimation which is validated by theoretical analysis and simulations note that the fdc is consistent and has low computational complexity as the bic

gaussian fluctuations of young diagrams and structure constants of jack characters in this paper we consider a deformation of plancherel measure linked to jack polynomials our main result is the description of the first and second order asymptotics of the bulk of a random young diagram under this distribution which extends celebrated results of vershik kerov logan and shepp for the first order asymptotics and kerov for the second order asymptotics this gives more evidence for the connection with the gaussian beta ensemble already suggested by a work of matsumoto our main tool is a polynomiality result for the structure constants of some quantities that we call jack characters recently introduced by lassalle we believe that this result is also interesting in itself and we give several other applications of it

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized

discrete input two dimensional gaussian channels with memory estimation and information rates via graphical models and statistical mechanics discrete input two dimensional 2d gaussian channels with memory represent an important class of systems which appears extensively in communications and storage in spite of their widespread use the workings of 2d channels are still very much unknown in this work we try to explore their properties from the perspective of estimation theory and information theory at the heart of our approach is a mapping of a 2d channel to an undirected graphical model and inferring its a posteriori probabilities apps using generalized belief propagation gbp the derived probabilities are shown to be practically accurate thus enabling optimal maximum a posteriori map estimation of the transmitted symbols also the shannon theoretic information rates are deduced either via the vector wise shannon mcmillan breiman smb theorem or via the recently derived symbol wise guo shamai verdu gsv theorem our approach is also described from the perspective of statistical mechanics as the graphical model and inference algorithm have their analogues in physics our experimental study based on common channel settings taken from cellular networks and magnetic recording devices demonstrates that under nontrivial memory conditions the performance of this fully tractable gbp estimator is almost identical to the performance of the optimal map estimator it also enables a practically accurate simulation based estimate of the information rate rationalization of this excellent performance of gbp in the 2 d gaussian channel setting is addressed