group symmetric robust covariance estimation in this paper we consider tyler s robust covariance rm m estimator under group symmetry constraints we assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group referred to as group symmetry examples of group symmetric structures include circulant perhermitian and proper quaternion matrices we introduce a group symmetric version of tyler s estimator styler and provide an iterative fixed point algorithm to compute it the classical results claim that at least n p 1 sample points in general position are necessary to ensure the existence and uniqueness of tyler s estimator where p is the ambient dimension we show that the styler requires significantly less samples in some groups even two samples are enough to guarantee its existence and uniqueness in addition in the case of elliptical populations we provide high probability bounds on the error of the styler these too quantify the advantage of exploiting the symmetry structure finally these theoretical results are supported by numerical simulations

connected components of irreducible maps and 1d quantum phases we investigate elementary topological properties of sets of completely positive cp maps that arise in quantum perron frobenius theory we prove that the set of primitive cp maps of fixed kraus rank is path connected and we provide a complete classification of the connected components of irreducible cp maps at given kraus rank and fixed peripheral spectrum in terms of a multiplicity index these findings are then applied to analyse 1d quantum phases by studying equivalence classes of translational invariant matrix product states that correspond to the connected components of the respective cp maps our results extend the previously obtained picture in that they do not require blocking of physical sites they lead to analytic paths and they allow us to decompose into ergodic components and to study the breaking of translational symmetry

time varying autoregressive moving average models for covariance estimation we consider large scale covariance estimation using a small number of samples in applications where there is a natural ordering between the random variables the two classical approaches to this problem rely on banded covariance and banded inverse covariance structures corresponding to time varying moving average ma and autoregressive ar models respectively motivated by this analogy to spectral estimation and the well known modeling power of autoregressive moving average arma processes we propose a novel time varying arma covariance structure similarly to known results in the context of ar and ma we address the completion of an arma covariance matrix from its main band and its estimation based on random samples finally we examine the advantages of our proposed methods using numerical experiments

efficiency test of pseudorandom number generators using random walks markov chain monte carlo methods and computer simulations usually use long sequences of random numbers generated by deterministic rules so called pseudorandom number generators their efficiency depends on the convergence rate to the stationary distribution and the quality of random numbers used for simulations various methods have been employed to measure the convergence rate to the stationary distribution but the effect of random numbers has not been much discussed we present how to test the efficiency of pseudorandom number generators using random walks

computing monodromy via continuation methods on random riemann surfaces we consider a riemann surface x defined by a polynomial f x y of degree d whose coefficients are chosen randomly hence we can suppose that x is smooth that the discriminant x of f has d d 1 simple roots and that 0 0 i e the corresponding fiber has d distinct points y1 yd when we lift a loop 0 c by a continuation method we get d paths in x connecting y1 yd hence defining a permutation of that set this is called monodromy here we present experimentations in maple to get statistics on the distribution of transpositions corresponding to loops around each point of multiplying families of neighbor transpositions we construct permutations and the subgroups of the symmetric group they generate this allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups assuming that these two conjectures are true we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions

adaptive detection of point like targets in spectrally symmetric interference we address adaptive radar detection of targets embedded in ground clutter dominated environments characterized by a symmetrically structured power spectral density at the design stage we leverage on the spectrum symmetry for the interference to come up with decision schemes capable of capitalizing the a priori information on the covariance structure to this end we prove that the detection problem at hand can be formulated in terms of real variables and then we apply design procedures relying on the glrt the rao test and the wald test specifically the estimates of the unknown parameters under the target presence hypothesis are obtained through an iterative optimization algorithm whose convergence and quality guarantee is thoroughly proved the performance analysis both on simulated and on real radar data confirms the superiority of the considered architectures over their conventional counterparts which do not take advantage of the clutter spectral symmetry

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

cutoff for the ising model on the lattice introduced in 1963 glauber dynamics is one of the most practiced and extensively studied methods for sampling the ising model on lattices it is well known that at high temperatures the time it takes this chain to mix in l 1 on a system of size n is o logn whether in this regime there is cutoff i e a sharp transition in the l 1 convergence to equilibrium is a fundamental open problem if so as conjectured by peres it would imply that mixing occurs abruptly at c o 1 logn for some fixed c 0 thus providing a rigorous stopping rule for this mcmc sampler however obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple markov chains already for the one dimensional ising model showing cutoff is a longstanding open problem

l2 lower bounds for a special class of random walks we investigate thel 2 speed of convergence to stationarity for a certain class of random walks on a compact connected lie group we give a lower bound on the number of stepsk necessary such that thek fold convolution power of the original step distribution has anl 2 density our method uses work by heckman on the asymptotics of multiplicities along a ray of representations several examples are presented

contractivity of positive and trace preserving maps under lp norms we provide a complete picture of contractivity of trace preserving positive maps with respect to p norms we show that for p 1 contractivity holds in general if and only if the map is unital when the domain is restricted to the traceless subspace of hermitian matrices then contractivity is shown to hold in the case of qubits for arbitrary p 1 and in the case of qutrits if and only if p 1 in all noncontractive cases best possible bounds on the p norms are derived

the χ2 divergence and mixing times of quantum markov processes we introduce quantum versions of the 2 divergence provide a detailed analysis of their properties and apply them in the investigation of mixing times of quantum markov processes an approach similar to the one presented in the literature for classical markov chains is taken to bound the trace distance from the steady state of a quantum processes a strict spectral bound to the convergence rate can be given for time discrete as well as for time continuous quantum markov processes furthermore the contractive behavior of the 2 divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm in this context we analyze different versions of quantum detailed balance and finally give a geometric conductance bound to the convergence rate for unital quantum markov processes

perturbation bounds for quantum markov processes and their fixed points we investigate the stability of quantum markov processes with respect to perturbations of their transition maps in the first part we introduce a condition number that measures the sensitivity of fixed points of a quantum channel to perturbations we establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map in the second part we consider quantum markov processes that converge to a unique stationary state and we analyze the stability of the evolution at finite times in this way we obtain a linear relation between the mixing time of a quantum markov process and the sensitivity of its fixed point with respect to perturbations of the transition map

wave packet pseudomodes of twisted toeplitz matrices the pseudospectra of banded nonsymmetric toeplitz or circulant matrices with varying coefficients are considered such matrices are characterized by a symbol that depends on both position x and wave number k it is shown that when a certain winding number or twist condition is satisfied related to hormander s commutator condition for partial differential equations e pseudoeigenvectors of such matrices for exponentially small values of e exist in the form of localized wave packets the symbol need not be smooth with respect to x just differentiable at a point or less c 2004 wiley periodicals inc

improvements on khrapchenko s theorem abstract we present an improvement of khrapchenko s theorem which gives lower bounds for the size of boolean formulae our main theorem gives a better lower bound than the original khrapchenko s theorem or at least the same although we know of no function where it gives an improvement factor larger than two this lower bound is the largest eigenvalue of a certain matrix associated with the formula moreover we give an approximation of this bound which is easier to compute and is never smaller than the bound given by khrapchenko s theorem

the l 2 cutoff for reversible markov processes abstract we consider the problem of proving the existence of an l 2 cutoff for families of ergodic markov processes started from given initial distributions and associated with reversible more generally normal markov semigroups this includes classical examples such as families of finite reversible markov chains and brownian motion on compact riemannian manifolds we give conditions that are equivalent to the existence of an l 2 cutoff and describe the l 2 cutoff time in terms of the spectral decomposition this is illustrated by several examples including the ehrenfest process and the biased p q random walk on the non negative integers both started from an arbitrary point

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

detection of multivariate cyclostationarity this paper derives an asymptotic generalized likelihood ratio test glrt and an asymptotic locally most powerful invariant test lmpit for two hypothesis testing problems 1 is a vector valued random process cyclostationary cs or is it wide sense stationary wss 2 is a vector valued random process cs or is it nonstationary our approach uses the relationship between a scalar valued cs time series and a vector valued wss time series for which the knowledge of the cycle period is required this relationship allows us to formulate the problem as a test for the covariance structure of the observations the covariance matrix of the observations has a block toeplitz structure for cs and wss processes by considering the asymptotic case where the covariance matrix becomes block circulant we are able to derive its maximum likelihood ml estimate and thus an asymptotic glrt moreover using wijsman s theorem we also obtain an asymptotic lmpit these detectors may be expressed in terms of the loeve spectrum the cyclic spectrum and the power spectral density establishing how to fuse the information in these spectra for an asymptotic glrt and lmpit this goes beyond the state of the art where it is common practice to build detectors of cyclostationarity from ad hoc functions of these spectra

commutation relations and markov chains it is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain markov chains examples studied include random walk on irreducible representations a local random walk on partitions whose stationary distribution is the ewens distribution and some birth death chains

tight estimates for convergence of some non stationary consensus algorithms the present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete time multi agent systems in the systems considered here both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time under the hypothesis that some spanning tree structure is preserved along time and that some non zero minimal weight of the information transfer along this tree is guaranteed an estimate of the contraction rate is given the latter is expressed explicitly as the spectral radius of some matrix depending upon the tree depth and the lower bounds on the weights

how many shuffles to randomize a deck of cards a celebrated theorem of aldous bayer and diaconis asserts that it takes 3 2 log 2 n riffle shuffles to randomize a deck of n cards asymptotically as n and that the randomization occurs abruptly according to a 9cut off phenomenon9 these results depend upon measuring randomness by a quantity known as the total variation distance if randomness is measured by uncertainty or entropy in the sense of information theory the behaviour is different it takes only log 2 n shuffles to reduce the information to a proportion arbitrarily close to zero and 3 2 log 2 n to reduce it to an arbitrarily small number of bits at 3 2 log 2 n shuffles ca 0 0601 bits remain independently of n

on convex complexity measures khrapchenko s classical lower bound n 2 on the formula size of the parity function f can be interpreted as designing a suitable measure of sub rectangles of the combinatorial rectangle f 1 0 xf 1 1 trying to generalize this approach we arrived at the concept of convex measures we prove the negative result that convex measures are bounded by o n 2 and show that several measures considered for proving lower bounds on the formula size are convex we also prove quadratic upper bounds on a class of measures that are not necessarily convex

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

cutting and shuffling with diffusion evidence for cut offs in interval exchange maps low dimensional dynamical systems are fruitful models for mixing in fluid and granular flows we study a one dimensional discontinuous dynamical system termed cutting and shuffling of a line segment and we present a comprehensive computational study of its finite time mixing properties including the effect of diffusion to explore a large parameter space we introduce fit functions for two mixing metrics of choice the number of cutting interfaces a standard quantity in dynamical systems theory of interval exchange transformations and a mixing norm a more physical measure of mixing we compute averages of the mixing metrics across different permutations shuffling protocols showing that the latter averages are a robust descriptor of mixing for any permutation choice if the decay of the normalized mixing norm is plotted against the number of map iterations rescaled by the characteristic e folding time then universality emerges mixing norm decay curves across all cutting and shuffling protocols collapse onto a single stretched exponential profile next we predict this critical number of iterations using the average length of unmixed subsegments of continuous color during cutting and shuffling and a batchelor scale type diffusion argument this prediction called a stopping time for finite markov chains compares well with the e folding time of the stretched exponential fit finally we quantify the effect of diffusion on cutting and shuffling through a peclet number a dimensionless inverse diffusivity showing that the system transitions more sharply from an unmixed initial state to a mixed final state as the peclet number becomes large our numerical investigation of cutting and shuffling of a line segment in the presence of diffusion thus presents evidence for the latter phenomenon known as a cut off for finite markov chains in interval exchange maps

generalized ensemble and tempering simulations a unified view from the underlying master equations we derive one dimensional stochastic processes that describe generalized ensemble simulations as well as tempering simulated and parallel simulations the representations obtained are either in the form of a one dimensional fokker planck equation or a hopping process on a one dimensional chain in particular we discuss the conditions under which these representations are valid approximate markovian descriptions of the random walk in order parameter or control parameter space they allow a unified discussion of the stationary distribution on as well as of the stationary flow across each space we demonstrate that optimizing the flow is equivalent to minimizing the first passage time for crossing the space and discuss the consequences of our results for optimizing simulations finally we point out the limitations of these representations under conditions of broken ergodicity

accuracy limitations and the measurement of errors in the stochastic simulation of chemically reacting systems this paper introduces the concept of distribution distance for the measurement of errors in exact and approximate methods for stochastic simulation of chemically reacting systems two types of distance are discussed the kolmogorov distance and the histogram distance the self distance an important property of monte carlo methods that quantifies the accuracy limitation at a given resolution for a given number of realizations is defined and studied estimation formulas are established for the histogram and the kolmogorov self distance these formulas do not depend on the distribution of the samples and thus show a property of the monte carlo method itself numerical results demonstrate that the formulas are very accurate application of these results to two problems of current interest in the simulation of biochemical systems is discussed

exact solution for a class of random walk on the hypercube a class of families of markov chains defined on the vertices of the n dimensional hypercube n 0 1 n is studied the single step transition probabilities pn ij with i j n are given by p n ij frac 1 alpha d ij 2 alpha n where 0 1 and dij is the hamming distance between i and j this corresponds to flip independently each component of the vertex with probability frac 1 alpha 2 alpha the m step transition matrix p n ij m is explicitly computed in a close form the class is proved to exhibit cutoff a model independent result about the vanishing of the first m terms of the expansion in of p n ij m is also proved

a matrix theory for finite group actions we introduce a new matrix theory to investigate finite group actions on spaces given a finite group action we associate it with a family of orbit matrices the spectral radius of an action is also introduced it is shown that the spectral raduis is bounded below by a constant depending only on some geometric invariants of the underlying riemannian manifolds the relation between the eigenspaces of orbit matrices and regular representations of finite groups are also investigated in particular we obtain that the eigenvalues of orbit matrices reveal some structures of the groups 1 introduction in this note we shall discuss the relation between pairwise distances of orbit points of finite groups acting effectively and isometrically on compact riemannian manifolds and their group structures our approach is to consider corresponding matrices of these pairwise distances and then relate the eigenspaces of these matrices to the regular representation of the groups the motivation of these ideas traces back to the papers cl 2 r w2 3 to get a sense of how the pairwise distances of the orbit points can reveal the structure of the group itself let us consider the following two simple examples

the shrinkage exponent of de morgan formulas is 2 we prove that if we hit a de morgan formula of size l with a random restriction from rp then the expected remaining size is at most o p 2 log frac 1 p 3 2 l p sqrt l as a corollary we obtain an omega n 3 o 1 formula size lower bound for an explicit function in p this is the strongest known lower bound for any explicit function in np

cutoff phenomena for random walks on random regular graphs the cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite markov chains to equilibrium many natural families of chains are believed to exhibit cutoff and yet establishing this fact is often extremely challenging an important such family of chains is the random walk on g n d a random d regular graph on n vertices it is well known that almost every such graph for d 3 is an expander and even essentially ramanujan implying a mixing time of o logn according to a conjecture of peres the simple random walk on g n d for such d should then exhibit cutoff with high probability whp as a special case of this durrett conjectured that the mixing time of the lazy random walk on a random 3 regular graph is whp 6 o 1 log2n in this work we confirm the above conjectures and establish cutoff in total variation its location and its optimal window both for simple and for non backtracking random walks on g n d namely for any fixed d 3 the simple random walk on g n d whp has cutoff at d d 2 logd 1n with window order logn surprisingly the non backtracking random walk on g n d whp has cutoff already at logd 1n with constant window order we further extend these results to g n d for any d no 1 that grows with n beyond which the mixing time is o 1 where we establish concentration of the mixing time on one of two consecutive integers

invariant adaptive detection of range spread targets under structured noise covariance the invariance principle is adopted to develop an exhaustive study for adaptive detection of range spread targets in gaussian noise sharing a block diagonal covariance structure for this problem the usual generalized likelihood ratio principle is intractable in this paper we first determine the largest group of affine transformations that does not alter the decision problem then a maximal invariant identified by this group is derived which can characterize the totality of the invariant detectors and extends the existing results for the point target case a theoretical performance analysis of the maximal invariant is also given finally we propose two classes of invariant detectors which are distinguished by whether a constant false alarm rate cfar is maintained numerical experiments are provided for a comparison of the proposed detectors where a tradeoff between the cfarness and the improved performance has been observed and studied

stochastic sampling algorithms for state estimation of jump markov linear systems jump markov linear systems are linear systems whose parameters evolve with time according to a finite state markov chain given a set of observations our aim is to estimate the states of the finite state markov chain and the continuous in space states of the linear system the computational cost in computing conditional mean or maximum a posteriori map state estimates of the markov chain or the state of the jump markov linear system grows exponentially in the number of observations we present three globally convergent algorithms based on stochastic sampling methods for state estimation of jump markov linear systems the cost per iteration is linear in the data length the first proposed algorithm is a data augmentation da scheme that yields conditional mean state estimates the second proposed scheme is a stochastic annealing sa version of da that computes the joint map sequence estimate of the finite and continuous states finally a metropolis hastings da scheme based on sa is designed to yield the map estimate of the finite state markov chain convergence results of the three above mentioned stochastic algorithms are obtained computer simulations are carried out to evaluate the performances of the proposed algorithms the problem of estimating a sparse signal developing from a neutron sensor based on a set of noisy data from a neutron sensor and the problem of narrow band interference suppression in spread spectrum code division multiple access cdma systems are considered

on the cover time and mixing time of random geometric graphs the cover time and mixing time of graphs has much relevance to algorithmic applications and has been extensively investigated recently with the advent of ad hoc and sensor networks an interesting class of random graphs namely random geometric graphs has gained new relevance and its properties have been the subject of much study a random geometric graph g n r is obtained by placing n points uniformly at random on the unit square and connecting two points iff their euclidean distance is at most r the phase transition behavior with respect to the radius r of such graphs has been of special interest we show that there exists a critical radius r o p t such that for any r r o p tg n r has optimal cover time of q nlogn with high probability and importantly r o p email protected r c o n where r c o n denotes the critical radius guaranteeing asymptotic connectivity moreover since a disconnected graph has infinite cover time there is a phase transition and the corresponding threshold width is o r c o n on the other hand the radius required for rapid mixing r r a p i email protected r c o n and in particular r r a p i email protected 1 poly logn we are able to draw our results by giving a tight bound on the electrical resistance and conductance of g n r via certain constructed flows

invariance theory for adaptive detection in interference with group symmetric covariance matrix this paper describes the adaptive detection of point like targets in a gaussian environment assuming a group symmetric structure for the interference covariance matrix this special configuration enforces block sparsity and permits splitting of the original observation space into lower dimensional subspaces each characterized by its own nuisance parameters hence the principle of invariance is invoked to select decision rules enjoying some symmetry defined by a suitable group of transformations which leave the decision problem unaltered all the invariant tests can be expressed as functions of a maximal invariant statistic whose derivation is among the key results of this paper finally some common design criteria are applied to come up with adaptive architectures that share invariance and their behavior is assessed to highlight the interplay between sample size and detection performance in comparison with conventional decision schemes

siegmund duality with applications to the neutral moran model conditioned on never being absorbed we first consider the classical neutral moran model with two alleles whose fate is either to become extinct or to reach fixation we study an ergodic version of the moran model obtained by conditioning it to never hit the boundaries making use of a doob transform we call it the recurrent moran model we show that the siegmund dual of the recurrent moran process exists and is a substochastic birth and death chain conditioning this process to exit in its natural absorbing state we construct a process with a unique absorbing state which is intertwined to the original recurrent moran process the time needed for the intertwined process to first hit its absorbing state is related to the time needed to reach stationarity for the recurrent moran process using spectral information on the intertwined chain we extract limiting information on this first hitting time that shows that there is no abrupt relaxation to equilibrium for the recurrent moran chain this makes use of the relation between duality and intertwining and strong stationary times other related transition times of the recurrent moran chain are also briefly investigated namely the first return time to the ground state and the expected time needed to move from one end to the other end of the state space

abrupt convergence and escape behavior for birth and death chains we link two phenomena concerning the asymptotical behavior of stochastic processes i abrupt convergence or cut off phenomenon and ii the escape behavior usually associated to exit from metastability the former is characterized by convergence at asymptotically deterministic times while the convergence times for the latter are exponentially distributed we compare and study both phenomena for discrete time birth and death chains on with drift towards zero in particular this includes energy driven evolutions with energy functions in the form of a single well under suitable drift hypotheses we show that there is both an abrupt convergence towards zero and escape behavior in the other direction furthermore as the evolutions are reversible the law of the final escape trajectory coincides with the time reverse of the law of cut off paths thus for evolutions defined by one dimensional energy wells with sufficiently steep walls cut off and escape behavior are related by time inversion

convergence rates for markov chains this is an expository paper that presents various ideas related to nonasymptotic rates of convergence for markov chains such rates are of great importance for stochastic algorithms that are widely used in statistics and in computer science they also have applications to analysis of card shuffling and other areas in this paper we attempt to describe various mathematical techniques that have been used to bound such rates of convergence in particular we describe eigenvalue analysis random walks on groups coupling and minorization conditions connections are made to modern areas of research wherever possible elements of linear algebra probability theory group theory and measure theory are used but efforts are made to keep the presentation elementary and accessible

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

application of shemesh theorem to quantum channels completely positive maps are useful in modeling the discrete evolution of quantum systems spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel we discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its kraus operators a a1 ak by applying the shemesh and amitsur levitzki theorems to analyse the structure of the algebra a a1 ak one can predict the asymptotic dynamics for a class of operations completely positive maps are useful in modeling the discrete evolution of quantum systems spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel we discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its kraus operators a a1 ak by applying the shemesh and amitsur levitzki theorems to analyse the structure of the algebra a a1 ak one can predict the asymptotic dynamics for a class of operations

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

a maximal invariant framework for adaptive detection with structured and unstructured covariance matrices we introduce a framework for exploring array detection problems in a reduced dimensional space by exploiting the theory of invariance in hypothesis testing this involves calculating a low dimensional basis set of functions called the maximal invariant the statistics of which are often tractable to obtain thereby making analysis feasible and facilitating the search for tests with some optimality property using this approach we obtain a locally most powerful invariant test for the unstructured covariance case and show that all invariant tests can be expressed in terms of the previously published kelly s generalized likelihood ratio glrt and robey s adaptive matched filter amf test statistics applying this framework to structured covariance matrices corresponding to stochastic interferers in a known subspace for which the glrt is unavailable we obtain the maximal invariant and propose several new invariant detectors that are shown to perform as well or better than existing ad hoc detectors these invariant tests are unaffected by most nuisance parameters hence the variation in the level of performance is sharply reduced this framework facilitates the search for such tests even when the usual glrt is unavailable

reflection couplings and contraction rates for diffusions we consider contractivity for diffusion semigroups w r t kantorovich l 1 wasserstein distances based on appropriately chosen concave functions these distances are inbetween total variation and usual wasserstein distances it is shown that by appropriate explicit choices of the underlying distance contractivity with rates of close to optimal order can be obtained in several fundamental classes of examples where contractivity w r t standard wasserstein distances fails applications include overdamped langevin diffusions with locally non convex potentials products of these processes and systems of weakly interacting diffusions both of mean field and nearest neighbour type

universality of cutoff for the ising model on any locally finite geometry the stochastic ising model is known to be contractive when the inverse temperature beta is small enough via classical results of dobrushin and of holley in the 1970s by a general principle proposed by peres the dynamics is then expected to exhibit cutoff however so far cutoff for the ising model has been confirmed mainly for lattices heavily relying on amenability and log sobolev inequalities without these cutoff was unknown at any fixed beta 0 no matter how small even in basic examples such as the ising model on a binary tree or a random regular graph we use the new framework of information percolation to show that in any geometry there is cutoff for the ising model at high enough temperatures precisely on any sequence of graphs with maximum degree d the ising model has cutoff provided that beta kappa d for some absolute constant kappa a result which up to the value of kappa is best possible moreover the cutoff location is established as the time at which the sum of squared magnetizations drops to 1 and the cutoff window is o 1 just as when beta 0 finally the mixing time from almost every initial state is not more than a factor of 1 varepsilon beta faster then the worst one with varepsilon beta to0 as beta to0 whereas the uniform starting state is at least 2 varepsilon beta times faster

hilbert s projective metric in quantum information theory we introduce and apply hilbert s projective metric in the context of quantum information theory the metric is induced by convex cones such as the sets of positive separable or positive partial transpose operators it provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone preserving operations the results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels for instance improving ruskai s trace norm contraction inequality a new duality between distinguishability measures and base norms is provided for two given pairs of quantum states we show that the contraction of hilbert s projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other inequalities between hilbert s projective metric and the chernoff bound the

entropy driven cutoff phenomena in this paper we present in the context of diaconis paradigm a general method to detect the cutoff phenomenon we use this method to prove cutoff in a variety of models some already known and others not yet appeared in literature including a non reversible random walk on a cylindrical lattice all the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon in the case of birth and death chains this mechanism is fairly well understood our work is an effort to generalize this picture to more general systems such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one dimensional problem in those situations the drift may be looked for by means of a suitable partitioning of the state space into classes using a statistical mechanics language it is then possible to set up a kind of energy entropy competition between the weight and the size of the classes under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease

kinetics of ring formation we study reversible polymerization of rings in this stochastic process two monomers bond and as a consequence two disjoint rings may merge into a compound ring or a single ring may split into two fragment rings this aggregation fragmentation process exhibits a percolation transition with a finite ring phase in which all rings have microscopic length and a giant ring phase where macroscopic rings account for a finite fraction of the entire mass interestingly while the total mass of the giant rings is a deterministic quantity their total number and their sizes are stochastic quantities the size distribution of the macroscopic rings is universal although the span of this distribution increases with time moreover the average number of giant rings scales logarithmically with system size we introduce a card shuffling algorithm for efficient simulation of the ring formation process and we present numerical verification of the theoretical predictions

fast consensus by the alternating direction multipliers method the alternating direction multipliers method admm has been recently proposed as a practical and efficient algorithm for distributed computing we discuss its applicability to the average consensus problem in this paper by carefully relaxing admm augmentation coefficients we are able to analytically investigate its properties and to propose simple and strict analytical bounds these provide a clear indication on how to choose system parameters for optimized performance we prove both analytically and via simulations that the proposed approach exhibits convergence speed between the best in the literature classical and optimized solutions while providing the most powerful resilience to noise

cutoff at the entropic time for sparse markov chains we study convergence to equilibrium for a large class of markov chains in random environment the chains are sparse in the sense that in every row of the transition matrix p the mass is essentially concentrated on few entries moreover the random environment is such that rows of p are independent and such that the entries are exchangeable within each row this includes various models of random walks on sparse random directed graphs the models are generally non reversible and the equilibrium distribution is itself unknown in this general setting we establish the cutoff phenomenon for the total variation distance to equilibrium with mixing time given by the logarithm of the number of states times the inverse of the average row entropy of p as an application we consider the case where the rows of p are i i d random vectors in the domain of attraction of a poisson dirichlet law with index 0 1 our main results are based on a detailed analysis of the weight of the trajectory followed by the walker this approach offers an interpretation of cutoff as an instance of the concentration of measure phenomenon

on circularity circularity is an assumption that was originally introduced for the definition of the probability distribution function of complex normal vectors however this concept can be extended in various ways for nonnormal vectors the first purpose of the paper is to introduce and compare some possible definitions of circularity from these definitions it is also possible to introduce the concept of circular signals and to study whether or not the spectral representation of stationary signals introduces circular components therefore the relationships between circularity and stationarity are analyzed in detail finally the theory of linear mean square estimation for complex signals exhibits some connections with circularity and it is shown that without this assumption the estimation theory must be reformulated

random walk on upper triangular matrices mixes rapidly we present an upper bound o n 2 for the mixing time of a simple random walk on upper triangular matrices we show that this bound is sharp up to a constant and find tight bounds on the eigenvalue gap we conclude by applying our results to indicate that the asymmetric exclusion process on a circle indeed mixes more rapidly than the corresponding symmetric process

glrt based detection estimation for undersampled training conditions for sensors where the number of available independent identically distributed training samples t is less than the number of antenna array elements m we propose nondegenerate properly normalized likelihood ratio lr tests both standard and scale invariant to support detection estimation of m point sources m t in white noise based on a generalized likelihood ratio test glrt approach we demonstrate that these tests can detect music specific ldquooutliersrdquo in the direction of arrival doa estimation of closely spaced independent sources caused by insufficient training volume and or signal to noise ratio snr we then compare the performance of the introduced lrs to other test statistics available in this undersampled regime we show that a search for solutions that increase the introduced lr allows us to replace the detected outliers by proper doa estimates this ldquopredict and curerdquo process leverages the snr ldquogaprdquo between music breakdown and breakdown of maximum likelihood estimation itself the resultant lr maximization makes the associated covariance model statistically ldquoas likelyrdquo as the true covariance matrix and removes the vast percentage of outliers in certain scenarios

quantitative approximations of evolving probability measures and sequential markov chain monte carlo methods we study approximations of evolving probability measures by an interacting particle system the particle system dynamics is a combination of independent markov chain moves and importance sampling resampling steps under global regularity conditions we derive non asymptotic error bounds for the particle system approximation in a few simple examples including high dimensional product measures bounds with explicit constants of feasible size are obtained our main motivation are applications to sequential mcmc methods for monte carlo integral estimation

spectral convergence bounds for classical and quantum markov processes we introduce a new framework that yields spectral bounds on norms of functions of transition maps for finite homogeneous markov chains the techniques employed work for bounded semigroups in particular for classical as well as for quantum markov chains and they do not require additional assumptions like detailed balance irreducibility or aperiodicity we use the method in order to derive convergence bounds that improve significantly upon known spectral bounds the core technical observation is that power boundedness of transition maps of markov chains enables a wiener algebra functional calculus in order to upper bound any norm of any holomorphic function of the transition map finally we discuss how general detailed balance conditions for quantum markov processes lead to spectral convergence bounds

rao test for adaptive detection in gaussian interference with unknown covariance matrix this paper deals with the problem of detecting a signal known up to a scaling factor in the presence of gaussian disturbance with unknown covariance matrix a new detector based on the rao test criterion is introduced and its invariance properties and constant false alarm rate cfar behavior are studied at the analysis stage the performance of the new receiver is assessed also in comparison with some classical adaptive radar detectors both in the matched as well as in the mismatched signal case remarkably the rao test may achieve a matched detection performance which is commensurate with that of the generalized likelihood ratio test glrt based detectors if a sufficient number of training data is available moreover it also exhibits better rejection capabilities of mismatched signals than the counterparts in the last part of the work a two stage detector whose second stage coincides with the rao test is devised it represents a suitable means to restore the detection performance of the plain rao test in the presence of a small number of training data finally the performance of the aforementioned two stage processor is analyzed in closed form and the cfar behavior is proved

polarimetric detection in compound gaussian clutter with kronecker structured covariance matrix in this paper we consider polarimetric adaptive detection in compound gaussian clutter whose covariance matrix cm has a kronecker structure we derive the cramer rao bound of a kronecker structured cm estimate and analyze the constant false alarm rate property of the adaptive subspace matched filter detector that uses kronecker structured estimates we provide a general expression for the average signal to clutter ratio loss scrl as a function of the mean square error of the covariance estimate the aforementioned expression is helpful in determining how many samples are required in order to achieve a desired average scrl level in practical scenarios based on that expression we show that the required sample size can be effectively reduced by exploiting the kronecker structure of the clutter cm we also derive the asymptotic detection performance of the adaptive subspace matched filter the analysis of scrl and detection performance can be extended to more general scenarios especially when the maximum likelihood estimate of the structured cm involves solving fixed point equations numerical simulations validate the merits of the proposed methods

generalized forward backward subaperture smoothing techniques for sample starved stap a major issue in space time adaptive processing stap for moving target indicator mti radar is the so called sample support problem often the available sample support for estimating the requisite interference covariance matrix is inadequate thereby precluding stap beamforming utilizing many adaptive degrees of freedom dofs although deterministic rank reduction methods can reduce sample support requirements they are invariably suboptimal from a signal to interference plus noise ratio sinr standpoint a new generalized subspatial and subtemporal aperture smoothing method employing forward and backward data vectors is introduced to overcome the data deficiency problem it is shown that multiplicative improvement in data samples can be obtained at the expense of negligible loss in space time aperture of the steering vector

adaptive radar detection of a subspace signal embedded in subspace structured plus gaussian interference via invariance this paper deals with adaptive radar detection of a subspace signal competing with two sources of interference the former is gaussian with unknown covariance matrix and accounts for the joint presence of clutter plus thermal noise the latter is structured as a subspace signal and models coherent pulsed jammers impinging on the radar antenna the problem is solved via the principle of invariance which is based on the identification of a suitable group of transformations leaving the considered hypothesis testing problem invariant a maximal invariant statistic which completely characterizes the class of invariant decision rules and significantly compresses the original data domain as well as its statistical characterization are determined thus the existence of the optimum invariant detector is addressed together with the design of practically implementable invariant decision rules at the analysis stage the performance of some receivers belonging to the new invariant class is established through the use of analytic expressions

performance analysis of forward backward matched filterbank spectral estimators the problem of complex spectral estimation is of great interest in many applications this paper studies the general class of the forward backward matched filterbank mafi spectral estimators including the widely used capon as well as the more recently introduced amplitude and phase estimation of a sinusoid apes methods in particular we show by means of a higher order expansion technique that the one dimensional 1 d capon estimator underestimates the true spectrum whereas the 1 d apes method is unbiased we also show that the bias of the forward backward capon is half that of the forward only capon to within a second order approximation furthermore we show that these results can be extended to the two dimensional 2 d capon and apes estimators numerical examples are also presented to demonstrate quantitatively the properties of and the relation between these mafi estimators

random walks that find perfect objects and the lovasz local lemma we give an algorithmic local lemma by establishing a sufficient condition for the uniform random walk on a directed graph to reach a sink quickly our work is inspired by moser s entropic method proof of the lovasz local lemma lll for satisfiability and completely bypasses the probabilistic method formulation of the lll in particular our method works when the underlying state space is entirely unstructured similarly to moser s argument the key point is that the inevitability of reaching a sink is established by bounding the entropy of the walk as a function of time

a general steady state distribution based stopping criteria for finite length genetic algorithms we propose two general stopping criteria for finite length simple genetic algorithms based on steady state distributions and empirically investigate the impact of mutation rate string length crossover rate and population size on their convergence our first stopping criterion is based on the second largest eigenvalue of the genetic algorithm transition matrix and the second stopping criterion is based on minorization conditions

the emergence of typical entanglement in two party random processes we investigate the entanglement within a system undergoing a random local process we find that there is initially a phase of very fast generation and spread of entanglement at the end of this phase the entanglement is typically maximal in oliveira et al 2007 phys rev lett 98 130502 we proved that the maximal entanglement is reached to a fixed arbitrary accuracy within o n3 steps where n is the total number of qubits here we provide a detailed and more pedagogical proof we demonstrate that one can use the so called stabilizer gates to simulate this process efficiently on a classical computer furthermore we discuss three ways of identifying the transition from the phase of rapid spread of entanglement to the stationary phase i the time when saturation of the maximal entanglement is achieved ii the cutoff moment when the entanglement probability distribution is practically stationary and iii the moment block entanglement exhibits volume scaling we furthermore investigate the mixed state and multipartite setting numerically we find that the mutual information appears to behave similarly to the quantum correlations and that there is a well behaved phase space flow of entanglement properties towards an equilibrium we describe how the emergence of typical entanglement can be used to create a much simpler tripartite entanglement description the results form a bridge between certain abstract results concerning typical also known as generic entanglement relative to an unbiased distribution on pure states and the more physical picture of distributions emerging from random local interactions

upper bound on the characters of the symmetric groups for balanced young diagrams and a generalized frobenius formula we study asymptotics of an irreducible representation of the symmetric group sn corresponding to a balanced young diagram a young diagram with at most rows and columns for some fixed constant c in the limit as n tends to infinity we show that there exists a constant d which depends only on c with a property that where p denotes the length of a permutation the minimal number of factors necessary to write p as a product of transpositions our main tool is an analogue of the frobenius character formula which holds true not only for cycles but for arbitrary permutations

invariant tests for spatial stationarity using covariance structure we propose new tests for validating the ula model based on the received array data these tests use the structure of the covariance matrix of the data to decide whether the received data is spatially stationary invariance principles are used to ensure that the test has constant significance equivalent to a cfar test when the model is indeed true by eliminating the dependence on the covariance in addition a modification of the test is proposed for verifying stationarity within a specified subspace and heuristics are presented for distinguishing propagation effects from deviation from the ula model

random walks and hyperplane arrangements let be the set of chambers of a real hyperplane arrangement we study a random walk on introduced by bidigare hanlon and rockmore this includes various shuffling schemes used in computer science biology and card games it also includes random walks on zonotopes and zonotopal tilings we find the stationary distributions of these markov chains give good bounds on the rate of convergence to stationarity and prove that the transition matrices are diagonalizable the results are extended to oriented matroids

regeneration in markov chain samplers abstract markov chain sampling has recently received considerable attention in particular in the context of bayesian computation and maximum likelihood estimation this article discusses the use of markov chain splitting originally developed for the theoretical analysis of general state space markov chains to introduce regeneration into markov chain samplers this allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance the approach is applied to several samplers including certain metropolis samplers that can be used on their own or in hybrid samplers and is illustrated in several examples

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc