We consider the general question of estimating decay of correlations for non-uniformly expanding maps, for classes of observables which are much larger than the usual class of Holder continuous functions. Our results give new estimates for many non-uniformly expanding systems, including Manneville-Pomeau maps, many one-dimensional systems with critical points, and Viana maps. In many situations, we also obtain a Central Limit Theorem for a much larger class of observables than usual. Our main tool is an extension of the coupling method introduced by L.-S. Young for estimating rates of mixing on certain non-uniformly expanding tower maps.
Let us now mention some other results concerning estimates on decay of correlations for non-H "older observables. Most of these are stated in the context of one-sided finite alphabet shift maps, or subshifts of finite type. (For a comprehensive discussion of shift maps and equilibrium measures, we suggest the book of Baladi, @cite_6 .) Shift maps are relatively simple dynamical systems, but are often used to more complicated systems via a semi-conjugacy, in much the same way that each of the examples we consider can be represented by a suitable (see ). Where a system @math being coded has an invariant measure @math which is absolutely continuous with respect to Lebesgue measure, @math is an equilibrium measure for the potential @math , where @math is the Jacobian with respect to Lebesgue measure. Most results for shift maps work with an equilibruim measure given by a potential @math which is H "older continuous (in terms of the usual metric on shift spaces - two sequences are said to be distance @math apart if they agree for exactly the first @math symbols). This assumption corresponds to assuming good distortion for @math .
@cite_6
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Consider a Markov chain on an infinite tree T = (V,E) rooted at . In such a chain, once the initial root state ( ) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let µj denote the resulting measure for ( ) = j. µj is defined on configurations = ( (x))x2V 2 A V , where A is some finite set. Let µ n denote the restriction of µ to the sigma-algebra generated by the variables (x) where x is at distance exactly n from . Letting n = maxi,j2A dTV (µ n ,µ n ), where dTV denotes total variation distance, we say that the reconstruction problem is solvable if liminfn!1 n > 0. Reconstruction solvability roughly means that the n’th level of the tree contains a non-vanishing amount of information on the root of the tree, as n ! 1. In this paper we study the problem of robust reconstruction. Let be a non degenerate distribution on A and > 0. Let be chosen according to µ n and 0 be obtained from by letting for each node independently (v) = 0 (v) with probability 1 and 0 (v) to be an independent sample from otherwise. We denote by µ n [, ] the resulting measure on 0 . The measure µ n [, ] is a perturbation of the measure µ n . Letting n(, ) = maxi,j2A dTV (µ n [, ],µ n [, ]), we say that the reconstruction problem is -robust-solvable if liminfn!1 n(, ) > 0 for all 0 1, where 2(M) is the second largest eigenvalue of M (in absolute value), then for all non-degenerate , the reconstruction problem is -robust-solvable. We prove a converse and show that the reconstruction problem is not -robust-solvable if B| 2(M)| 2 < 1. This proves a conjecture by the second author and Peres. We also consider other models of noise and general trees. We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly @math fixed points. There is a unique totally nonnegative fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). We introduce a parameter q, and show that the fixed points of a q-deformation of the cyclic shift map are precisely the critical points of the mirror-symmetric superpotential @math on Gr(k,n). This follows from results of Rietsch about the quantum cohomology ring of Gr(k,n). We survey many other diverse contexts which feature moment curves and the cyclic shift map. This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset @math , associated with the Euclidean metric, with points in the cube @math and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in @math . Our methods rely on quantizing fast Johnson-Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our quantization methods utilize noise-shaping, and include Sigma-Delta schemes and distributed noise-shaping schemes. The resulting approximation errors decay polynomially and exponentially fast in @math , depending on the embedding method. This dramatically outperforms the current decay rates associated with binary embeddings and Hamming distances. Additionally, it is the first such binary embedding result that applies to fast Johnson-Lindenstrauss maps while preserving @math norms. Second, we again consider noise-shaping schemes, albeit this time to quantize compressed sensing measurements arising from bounded orthonormal ensembles and partial circulant matrices. We show that these methods yield a reconstruction error that again decays with the number of measurements (and bits), when using convex optimization for reconstruction. Specifically, for Sigma-Delta schemes, the error decays polynomially in the number of measurements, and it decays exponentially for distributed noise-shaping schemes based on beta encoding. These results are near optimal and the first of their kind dealing with bounded orthonormal systems. Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ|I|. We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ and λ< λc=(Δ-1)Δ-1 (Δ-2)Δ. This improves on the previously known general bound of λ ≤ (2 ‾ Δ-2). The new regime includes the interesting case of λ=1 (uniform weights) and Δ ≤ 5. The previous bound required Δ ≤ 4 for uniform approximate counting and there is evidence that for Δ ≥ 6 the problem is hard. Note that λc is the critical activity for uniqueness of the Gibbs measure on the regular tree of degree Δ, i.e., for λ ≤ λc the probability that the root is in the independent set is asymptotically independent of the configuration on the leaves far below. Indeed, our analysis is focused on establishing decay of correlations with distance in the above weighted distribution. We show that on any graph of maximum degree Δ correlations decay with distance at least as fast as they do on the regular tree of the same degree. This resolves an open conjecture in statistical physics. Our comparison of a general graph with the tree uses an algorithmic argument yielding the approximation scheme mentioned above. Also, by existing arguments, establishing decay of correlations for all graphs and λ<λc gives that the Glauber dynamics is rapidly mixing in this regime. However, the implication from decay of correlations to rapid mixing of the dynamics is only known to hold for graphs of subexponential growth, and hence, our result regarding the Glauber dynamics is limited to this class of graphs. We consider a Glauber dynamics reversible with respect to the two-dimensional Ising model in a finite square of sideL, in the absence of an external field and at large inverse temperature β. We first consider the gap in the spectrum of the generator of the dynamics in two different cases: with plus and open boundary conditions. We prove that, when the symmetry under global spin flip is broken by the boundary conditions, the gap is much larger than the case in which the symmetry is present. For this latter we compute exactly the asymptotics of −(1 βL) log(gap) asL→∞ and show that it coincides with the surface tension along one of the coordinate axes. As a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin-spin time correlation in the infinite-volume plus phase. Our results establish a connection between the dynamical large deviations and those of the equilibrium Gibbs measure studied by Shlosman in the framework of the rigorous description of the Wulff shape for the Ising model. Finally we show that, in the case of open boundary conditions, it is possible to rescale the time withL in such a way that, asL→∞, the finite-dimensional distributions of the time-rescaled magnetization converge to those of a symmetric continuous-time Markov chain on the two-state space −m*(β),m*(β) ,m*(β) being the spontaneous magnetization. Our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric Markov chains on complicated graphs, developed by Jerrum and Sinclair in their Markov chain approach to hard computational problems, and the idea of introducing “block Glauber dynamics” instead of the standard single-site dynamics, in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium. We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+?j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I?I?Rd,f?TdI?I?Rd,f?Td, p,q?Rnp,q?Rn, t?T1t?T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0<e«10<e«1, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O(1)O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of and . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I?I?RdI?I?Rd. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from and . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31]—notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in and Many methods for planning under uncertainty operate in the belief space, i.e., the set of probability distributions over states. Although the problem is computationally hard, recent advances have shown that belief-space planning is becoming practical for many medium size problems. Some of the most successful methods utilize sampling and often rely on distances between beliefs to partially guide the search process. This paper deals with the question of what is a suitable distance function for belief space planning, which despite its importance remains unanswered. This work indicates that the rarely used Wasserstein distance (also known as Earth Mover’s Distance ( ( EMD ))) is a more suitable metric than the commonly used ( L1 ) and Kullback–Leibler ( ( KL )) for belief-space planning. Simulation results on Non-Observable Markov Decision Problems, i.e., the simplest class of belief-space planning, indicate that as the problem becomes more complex, the differences on the effectiveness of different distance functions become quite prominent. In fact, in state spaces with more than 4 dimensions, by just replacing ( L1 ) or ( KL ) distance with ( EMD ), the problems become from virtually unsolvable to solvable within a reasonable time frame. Furthermore, preliminary results on Partially Observable Markov Decision Processes indicate that point-based solvers with ( EMD ) use a smaller number of samples to generate policies with similar qualities, compared to those with ( L1 ) and ( KL ). This paper also shows that ( EMD ) carries the Lipschitz continuity of the state’s cost function to Lipschitz continuity of the expected cost of the beliefs. Such a continuity property is often critical for convergence to optimal solutions. The aim of this paper is to show that spatial coupling can be viewed not only as a means to build better graphical models, but also as a tool to better understand uncoupled models. The starting point is the observation that some asymptotic properties of graphical models are easier to prove in the case of spatial coupling. In such cases, one can then use the so-called interpolation method to transfer known results for the spatially coupled case to the uncoupled one. Our main use of this framework is for Low-density parity check (LDPC) codes, where we use interpolation to show that the average entropy of the codeword conditioned on the observation is asymptotically the same for spatially coupled as for uncoupled ensembles. We give three applications of this result for a large class of LDPC ensembles. The first one is a proof of the so-called Maxwell construction stating that the MAP threshold is equal to the area threshold of the BP GEXIT curve. The second is a proof of the equality between the BP and MAP GEXIT curves above the MAP threshold. The third application is the intimately related fact that the replica symmetric formula for the conditional entropy in the infinite block length limit is exact. This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (a.k.a. the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in BFM86 ). We then proceed to construct optimal Ramsey partitions, and use them to show that for every @math , every @math -point metric space has a subset of size @math which embeds into Hilbert space with distortion @math . This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor BLMN05 , in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in TZ05 . Namely, we show that for every @math point metric space @math , and @math , there exists an @math -approximate distance oracle whose storage requirement is @math , and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking. New phase transition phenomena have recently been discovered for the stochastic block model, for the special case of two non-overlapping symmetric communities. This gives raise in particular to new algorithmic challenges driven by the thresholds. This paper investigates whether a general phenomenon takes place for multiple communities, without imposing symmetry. In the general stochastic block model @math , @math vertices are split into @math communities of relative size @math , and vertices in community @math and @math connect independently with probability @math . This paper investigates the partial and exact recovery of communities in the general SBM (in the constant and logarithmic degree regimes), and uses the generality of the results to tackle overlapping communities. The contributions of the paper are: (i) an explicit characterization of the recovery threshold in the general SBM in terms of a new divergence function @math , which generalizes the Hellinger and Chernoff divergences, and which provides an operational meaning to a divergence function analog to the KL-divergence in the channel coding theorem, (ii) the development of an algorithm that recovers the communities all the way down to the optimal threshold and runs in quasi-linear time, showing that exact recovery has no information-theoretic to computational gap for multiple communities, in contrast to the conjectures made for detection with more than 4 communities; note that the algorithm is optimal both in terms of achieving the threshold and in having quasi-linear complexity, (iii) the development of an efficient algorithm that detects communities in the constant degree regime with an explicit accuracy bound that can be made arbitrarily close to 1 when a prescribed signal-to-noise ratio (defined in term of the spectrum of @math ) tends to infinity. In this paper we study the effect on a dynamical system of small random perturbations of the type of white noise: where is the -dimensional Wiener process and as . We are mainly concerned with the effect of these perturbations on long time-intervals that increase with the decreasing . We discuss two problems: the first is the behaviour of the invariant measure of the process as , and the second is the distribution of the position of a trajectory at the first time of its exit from a compact domain. An important role is played in these problems by an estimate of the probability for a trajectory of not to deviate from a smooth function by more than during the time . It turns out that the main term of this probability for small and has the form , where is a certain non-negative functional of . A function , the minimum of over the set of all functions connecting and , is involved in the answers to both the problems. By means of we introduce an independent of perturbations relation of equivalence in the phase-space. We show, under certain assumption, at what point of the phase-space the invariant measure concentrates in the limit. In both the problems we approximate the process in question by a certain Markov chain; the answers depend on the behaviour of on graphs that are associated with this chain. Let us remark that the second problem is closely related to the behaviour of the solution of a Dirichlet problem with a small parameter at the highest derivatives. When studying some process or development in different subjects or units-be it biological, chemical or physical-we usually see a typical pattern, common to all curves. Yet there is variation both in amplitude and dynamics between curves. Following some ideas of structural analysis introduced by Kneip and Gasser, we study a method-dynamic time warping with a proper cost function-for estimating the shift or warping function from one curve to another to align the two functions. For some models this method can identify the true shift functions if the data are noise free. Noisy data are smoothed by a nonparametric function estimate such as a kernel estimate. It is shown that the proposed estimator is asymptotically normal and converges to the true shift function as the sample size per subject goes to infinity. Some simulation results are presented to illustrate the performance of this method. By introducing a quantitative degree of commutativity'' in terms of the angle between spin observables we present two tight quantitative trade-off relations in the case of two qubits. First, for entangled states, between the degree of commutativity of local observables and the maximal amount of violation of the Bell inequality: if both local angles increase from zero to @math (i.e., the degree of local commutativity decreases), the maximum violation of the Bell inequality increases. Secondly, a converse trade-off relation holds for separable states: if both local angles approach @math the maximal value obtainable for the correlations in the Bell inequality decreases and thus the non-violation increases. As expected, the extremes of these relations are found in the case of anticommuting local observables where, respectively, the bounds of @math and @math hold for the expectation value of the Bell operator. The trade-off relations show that noncommmutativity gives a more than classical result'' for entangled states, whereas a less than classical result'' is obtained for separable states. The experimental relevance of the trade-off relation for separable states is that it provides an experimental test for two qubit entanglement. Its advantages are twofold: in comparison to violations of Bell inequalities it is a stronger criterion and in comparison to entanglement witnesses it needs to make less strong assumptions about the observables implemented in the experiment. Summary In this paper we propose, survey and compare some classes of probability densities that may be used to represent partial prior information, to model either prior ignorance or Bayesian sensitivity analysis. We distinguish two types of models appropriate for two different situations: near ignorance models which are suitable in problems where there is little prior information, and neighbourhood models, which can be used to 'robustify' a strict Bayesian analysis in problems where there is substantial prior information about location. We argue that, especially for the first situation, a reasonable class of prior densities is not the same as a class of reasonable prior densities. We discuss various desiderata for a 'reasonable' class, including coherence and sensible dependence of inferences on sample size. The translation invariant models studied here are classes of conjugate priors, classes of double exponential densities and a neighbourhood of the uniform prior. Of the neighbourhood models we examine examples of E-contamination neighbourhoods (previously studied by Huber, Berger and Berliner) and intervals of measures (DeRobertis and Hartigan). We illustrate the models in the simple problem of constructing credible intervals for an unknown normal mean. Of the models studied in detail, a translation-invariant class of double exponential priors is favoured for modelling little prior information, and a type of interval of measures seems most suitable for robust Bayesian analysis. We consider the non-parametric estimation of a function that is observed in white noise after convolution with a boxcar, the indicator of an interval ( a; a). In a recent paper Johnstone, Kerkyacharian, Picard and Raimondo (2004) have developed a wavelet deconvolution method (called WaveD) that can be used for " boxcar kernels. For example, WaveD can be tuned to achieve near optimal rates over Besov spaces when a is a Badly Approximable (BA) irrational number. While the set of all BA's contains quadratic irrationals, e.g., a = p 5, it has Lebesgue measure zero. In this paper we derive two tuning scenarios of WaveD that are valid for all" boxcar convolutions (i.e., when a 2 A where A is a full Lebesgue measure set). We propose (i) a tuning inspired from Minimax theory over Besov spaces; (ii) a tuning derived from Maxiset theory providing similar rates as for WaveD in the BA widths setting. Asymptotic theory nds that (i) in the worst case scenario, departures from the BA assumption aect WaveD convergence rates, at most, by log factors; (ii) the Maxiset tuning, which yields smaller thresholds, is superior to the Minimax tuning over a whole range of Besov sub-scales. Our asymptotic results are illustrated in an extensive simulation of boxcar convolution observed in white noise. In this paper we propose a novel approach for detecting interest points invariant to scale and affine transformations. Our scale and affine invariant detectors are based on the following recent results: (1) Interest points extracted with the Harris detector can be adapted to affine transformations and give repeatable results (geometrically stable). (2) The characteristic scale of a local structure is indicated by a local extremum over scale of normalized derivatives (the Laplacian). (3) The affine shape of a point neighborhood is estimated based on the second moment matrix. Our scale invariant detector computes a multi-scale representation for the Harris interest point detector and then selects points at which a local measure (the Laplacian) is maximal over scales. This provides a set of distinctive points which are invariant to scale, rotation and translation as well as robust to illumination changes and limited changes of viewpoint. The characteristic scale determines a scale invariant region for each point. We extend the scale invariant detector to affine invariance by estimating the affine shape of a point neighborhood. An iterative algorithm modifies location, scale and neighborhood of each point and converges to affine invariant points. This method can deal with significant affine transformations including large scale changes. The characteristic scale and the affine shape of neighborhood determine an affine invariant region for each point. We present a comparative evaluation of different detectors and show that our approach provides better results than existing methods. The performance of our detector is also confirmed by excellent matching resultss the image is described by a set of scale affine invariant descriptors computed on the regions associated with our points. Abstract Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles yield stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4 M − 4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M = 2 , 3 . Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer–Rao lower bounds to identify stability with stronger versions of the injectivity characterizations. We showed in this work how the Hassanat distance metric enhances the performance of the nearest neighb our classifiers. The results demonstrate the superiority of this distanc e metric over the traditional and most-used distanc es, such as Manhattan distance and Euclidian distance. Moreover, we prove d that the Hassanat distance metric is invariant to data scale, noise and outliers. Throughout this work, it is clearly notab le that both ENN and IINC performed very well with the distance investigated, as their accuracy increased significantly by 3.3 a nd 3.1 respectively, with no significant advantage of the ENN over the IINC in terms of accuracy. Correspondingly, it can be no ted from our results that there is no optimal algor ithm that can solve all reallife problems perfectly; this is supported by the n o-free-lunch theorem. Abstract This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho. Abstract We consider boundary value problems for semilinear hyperbolic systems of the type ∂ t u j + a j ( x , λ ) ∂ x u j + b j ( x , λ , u ) = 0 , x ∈ ( 0 , 1 ) , j = 1 , … , n , with smooth coefficient functions a j and b j such that b j ( x , λ , 0 ) = 0 for all x ∈ [ 0 , 1 ] , λ ∈ R , and j = 1 , … , n . We state conditions for Hopf bifurcation, i.e., for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence on λ of time-periodic solutions bifurcating from the zero stationary solution. Furthermore, we derive a formula which determines the bifurcation direction. The proof is done by means of a Liapunov–Schmidt reduction procedure. For this purpose, Fredholm properties of the linearized system and implicit function theorem techniques are used.