design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity this article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices under polynomial growth assumptions we prove that the energy functionals f varepsilon stored in the deformation of an varepsilon scaling of a stochastic lattice converge to a continuous energy functional when varepsilon goes to zero in particular the limiting energy functional is of integral type and deterministic if the lattice is ergodic we also generalize to systems and nonlinear settings well known results on stochastic homogenization of discrete elliptic equations as an application of the main result we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics we finally address some mechanical properties of the limiting models such as frame invariance isotropy and natural states

learning eigenfunctions links spectral embedding and kernel pca in this letter we show a direct relation between spectral embedding methods and kernel principal components analysis and how both are special cases of a more general learning problem learning the principal eigenfunctions of an operator defined from a kernel and the unknown data generating density whereas spectral embedding methods provided only coordinates for the training points the analysis justifies a simple extension to out of sample examples the nystrom formula for multidimensional scaling mds spectral clustering laplacian eigenmaps locally linear embedding lle and isomap the analysis provides for all such spectral embedding methods the definition of a loss function whose empirical average is minimized by the traditional algorithms the asymptotic expected value of that loss defines a generalization performance and clarifies what these algorithms are trying to learn experiments with lle isomap spectral clustering and mds show that this out of sample embedding formula generalizes well with a level of error comparable to the effect of small perturbations of the training set on the embedding

typical performance of regular low density parity check codes over general symmetric channels typical performance of low density parity check ldpc codes over a general binary input output symmetric memoryless channel is investigated using methods of statistical mechanics relationship between the free energy in statistical mechanics approach and the mutual information used in the information theory literature is established within a general framework gallager and mackay neal codes are studied as specific examples of ldpc codes it is shown that basic properties of these codes known for particular channels including their potential to saturate shannon s bound hold for general symmetric channels the binary input additive white gaussian noise channel and the binary input laplace channel are considered as specific channel models

random parking euclidean functionals and rubber elasticity we study subadditive functions of the random parking model previously analyzed by the second author in particular we consider local functions s of subsets of mathbb r d and of point sets that are almost subadditive in their first variable denoting by the random parking measure in mathbb r d and by r the random parking measure in the cube qr r r d we show under some natural assumptions on s that there exists a constant overline s in mathbb r such that lim r to infty frac s q r xi q r lim r to infty frac s q r xi r q r overline s almost surely if zeta mapsto s q r zeta is the counting measure of zeta in qr then we retrieve the result by the second author on the existence of the jamming limit the present work generalizes this result to a wide class of almost subadditive functions in particular classical euclidean optimization problems as well as the discrete model for rubber previously studied by alicandro cicalese and the first author enter this class of functions in the case of rubber elasticity this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit as well as a generalization to stochastic networks generated on bounded sets

gaussian limits for multidimensional random sequential packing at saturation consider the random sequential packing model with infinite input and in any dimension when the input consists of non zero volume convex solids we show that the total number of solids accepted over cubes of volume is asymptotically normal as we provide a rate of approximation to the normal and show that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized gaussian field the method of proof involves showing that the collection of accepted solids satisfies the weak spatial dependence condition known as stabilization

analyticity of homogenized coefficients under bernoulli perturbations and the clausius mossotti formulas this paper is concerned with the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a bernoulli perturbation introducing a new family of energy estimates that combine probability and physical spaces we prove the analyticity of the perturbed homogenized coefficients with respect to the bernoulli parameter our approach holds under the minimal assumptions of stationarity and ergodicity both in the scalar and vector cases and gives analytical formulas for each derivative that essentially coincide with the so called cluster expansion used by physicists in particular the first term yields the celebrated electric and elastic clausius mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium this work constitutes the first general proof of these formulas in the case of random inclusions

self compensation technique for simplified belief propagation algorithm the min sum algorithm is the most common method to simplify the belief propagation algorithm for decoding low density parity check ldpc codes however there exists a performance gap between the min sum and belief propagation algorithms due to nonlinear approximation in this paper a self compensation technique using dynamic normalization is thus proposed to improve the approximation accuracy the proposed scheme scales the min sum algorithm by a dynamic factor that can be derived theoretically from order statistics moreover applying the proposed technique to several ldpc codes for dvb s2 system the average signal to noise ratio degradation which results from approximation inaccuracy and quantization error is reduced to 0 2 db not only does it enhance the error correcting capability of the min sum algorithm but the proposed self compensation technique also preserves a modest hardware cost after realized with 0 13 mum standard cell library the dynamic normalization requires about 100 additional gates for each check node unit in the min sum algorithm

random parking sequential adsorption and the jamming limit identical cars are dropped sequentially from above into a large parking lot each car is positioned uniformly at random subject to non overlap with its predecessors until jamming occurs there have been many studies of the limiting mean coverage as the parking lot becomes large but no complete proof that such a limit exists until now

reconstruction of a constitutive law for rubber from in silico experiments using ogden s laws this article deals with the following data assimilation problem construct an analytical approximation of a numerical constitutive law in three dimensional nonlinear elasticity more precisely we are concerned with a micro macro model for rubber macroscopic quantities of interest such as the piola kirchhoff stress tensor can be approximated for any value of the strain gradient by numerically solving a nonlinear pde this procedure is however computationally demanding hence although conceptually satisfactory this physically based model is of no direct practical use the aim of this article is to circumvent this difficulty by proposing a numerical strategy to reconstruct from in silico experiments an accurate analytical proxy for the micro macro constitutive law

parity check density versus performance of binary linear block codes new bounds and applications the moderate complexity of low density parity check ldpc codes under iterative decoding is attributed to the sparseness of their parity check matrices it is therefore of interest to consider how sparse parity check matrices of binary linear block codes can be a function of the gap between their achievable rates and the channel capacity this issue was addressed by sason and urbanke and it is revisited in this paper the remarkable performance of ldpc codes under practical and suboptimal decoding algorithms motivates the assessment of the inherent loss in performance which is attributed to the structure of the code or ensemble under maximum likelihood ml decoding and the additional loss which is imposed by the suboptimality of the decoder these issues are addressed by obtaining upper bounds on the achievable rates of binary linear block codes and lower bounds on the asymptotic density of their parity check matrices as a function of the gap between their achievable rates and the channel capacity these bounds are valid under ml decoding and hence they are valid for any suboptimal decoding algorithm the new bounds improve on previously reported results by burshtein and by sason and urbanke and they hold for the case where the transmission takes place over an arbitrary memoryless binary input output symmetric mbios channel the significance of these information theoretic bounds is in assessing the tradeoff between the asymptotic performance of ldpc codes and their decoding complexity per iteration under message passing decoding they are also helpful in studying the potential achievable rates of ensembles of ldpc codes under optimal decoding by comparing these thresholds with those calculated by the density evolution technique one obtains a measure for the asymptotic suboptimality of iterative decoding algorithms

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

efficient encoding of low density parity check codes low density parity check ldpc codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy constrained random code ensembles and iterative decoding algorithms we consider the encoding problem for ldpc codes more generally we consider the encoding problem for codes specified by sparse parity check matrices we show how to exploit the sparseness of the parity check matrix to obtain efficient encoders for the 3 6 regular ldpc code for example the complexity of encoding is essentially quadratic in the block length however we show that the associated coefficient can be made quite small so that encoding codes even of length n spl sime 100000 is still quite practical more importantly we show that optimized codes actually admit linear time encoding

approximating the llr distribution for a class of soft output mimo detectors we present approximations of the llr distribution for a class of fixed complexity soft output mimo detectors such as the optimal soft detector and the soft output via partial marginalization detector more specifically in a mimo awgn setting we approximate the llr distribution conditioned on the transmitted signal and the channel matrix with a gaussian mixture model gmm our main results consist of an analytical expression of the gmm model including the number of modes and their corresponding parameters and a proof that in the limit of high snr this llr distribution converges in probability towards a unique gaussian distribution

stochastic homogenization of nonconvex unbounded integral functionals with convex growth we consider the well trodden ground of the problem of the homogenization of random integral functionals when the integrand has standard growth conditions the qualitative theory is well understood when it comes to unbounded functionals that is when the domain of the integrand is not the whole space and may depend on the space variable there is no satisfactory theory in this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth we first prove that if the integrand is convex and has p growth from below with p d the dimension then it admits homogenization regardless of growth conditions from above this result that crucially relies on the existence and sublinearity at infinity of correctors is also new in the periodic case in the case of nonconvex integrands we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two sided estimate by a convex integrand the domain of which may depend on the space variable that itself admits homogenization this result is of interest to the rigorous derivation of rubber elasticity from polymer physics which involves the stochastic homogenization of such unbounded functionals

maximizing the spectral efficiency of coded cdma under successive decoding we investigate the spectral efficiency achievable by random synchronous code division multiple access cdma with quaternary phase shift keying qpsk modulation and binary error control codes in the large system limit where the number of users the spreading factor and the code block length go to infinity for given codes we maximize spectral efficiency assuming a minimum mean square error mmse successive stripping decoder for the cases of equal rate and equal power users in both cases the maximization of spectral efficiency can be formulated as a linear program and admits a simple closed form solution that can be readily interpreted in terms of power and rate control we provide examples of the proposed optimization methods based on off the shelf low density parity check ldpc codes and we investigate by simulation the performance of practical systems with finite code block length

asymptotic enumeration methods for analyzing ldpc codes we show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of low density parity check ldpc codes in particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular ldpc code ensembles we then consider the binary erasure channel bec using these estimates we derive lower bounds on the error exponent under iterative decoding of ldpc codes used over the bec both regular and irregular code structures are considered these bounds are compared to the corresponding bounds when optimal maximum likelihood ml decoding is applied

raptor codes on binary memoryless symmetric channels in this paper we will investigate the performance of raptor codes on arbitrary binary input memoryless symmetric channels bimscs in doing so we generalize some of the results that were proved before for the erasure channel we will generalize the stability condition to the class of raptor codes this generalization gives a lower bound on the fraction of output nodes of degree 2 of a raptor code if the error probability of the belief propagation decoder converges to zero using information theoretic arguments we will show that if a sequence of output degree distributions is to achieve the capacity of the underlying channel then the fraction of nodes of degree 2 in these degree distributions has to converge to a certain quantity depending on the channel for the class of erasure channels this quantity is independent of the erasure probability of the channel but for many other classes of bimscs this fraction depends on the particular channel chosen this result has implications on the universality of raptor codes for classes other than the class of erasure channels in a sense that will be made more precise in the paper we will also investigate the performance of specific raptor codes which are optimized using a more exact version of the gaussian approximation technique

ldpc codes for non coherent block fading channels with correlation analysis and design ldpc codes are analyzed and optimized for a noncoherent block fading channel in which there is correlation between the blocks this extends related work in the literature which considered independent blocks low complexity estimation and estimation decoding algorithms based on kalman smoothing are developed and density evolution is used to analyze and optimize the resulting code results indicate that significant gains can be realized by taking inter block correlation into account compared to architectures in which the blocks are assumed to be independent

constructing short length irregular ldpc codes with low error floor trapping sets tss are known to cause error floors in regular and irregular low density parity check ldpc codes by avoiding major error contributing tss during the code construction process codes with low error floors can effectively be built in it has been shown that tss labeled as w u are considered as being equivalent under the automorphism of the graph and are therefore contributing equally to the error floor however tss with the same label w u are not identical in general particularly for the case of irregular ldpc codes in this paper we introduce a parameter e that can identify the number of distinguishable cycles in the connected subgraph induced by an elementary trapping set further we propose a code construction algorithm namely the progressive edge growth approximate minimum cycle set extrinsic message degree peg acse method that aims to avoid small elementary trapping sets etss particularly detrimental etss we also develop theorems evaluating the minimum possible etss formed by peg construction algorithms in general we compare the characteristics of the codes built using the proposed method and those built using peg only or peg approximate minimum cycle extrinsic message degree peg ace methods results from simulations show that the codes constructed using the proposed peg acse method produce lower error rates particularly at the high signal to noise snr region compared with codes constructed using other peg based algorithms

hierarchical and high girth qc ldpc codes we present an approach to designing capacity approaching high girth low density parity check ldpc codes that are friendly to hardware implementation and compatible with some desired input code structure defined using a protograph the approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasi cyclic hqc ldpc codes whereas the parity check matrices of standard quasi cyclic qc ldpc codes are composed of circulant submatrices those of hqc ldpc codes are composed of a hierarchy of circulant submatrices that are in turn constructed from circulant submatrices and so on through some number of levels next we present a girth maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high girth hqc ldpc code subject to bounds imposed by the fact that hqc codes are still quasi cyclic finally we discuss how certain characteristics of a code protograph will lead to inevitable short cycles and show that these short cycles can be eliminated using a squashing procedure that results in a high girth qc ldpc code although not a hierarchical one we illustrate our approach with three design examples of qc ldpc codes two girth 10 codes of rates 1 3 and 0 45 and one girth 8 code of rate 0 7 all of which are obtained from protographs of one sided spatially coupled codes

nonlinear stochastic homogenization and ergodic theory on etudie l homogeneisation stochastique non lineaire d une classe de fonctionnelles integrales on relie le probleme a la theorie ergodique

comparison of constructions of irregular gallager codes the low density parity check codes whose performance is closest to the shannon limit are gallager codes based on irregular graphs we compare alternative methods for constructing these graphs and present two results first we find a super poisson construction which gives a small improvement in empirical performance over a random construction second whereas gallager codes normally take n sup 2 time to encode we investigate constructions of regular and irregular gallager codes that allow more rapid encoding and have smaller memory requirements in the encoder we find that these fast encoding gallager codes have equally good performance

integer forcing linear receivers linear receivers are often used to reduce the implementation complexity of multiple antenna systems in a traditional linear receiver architecture the receive antennas are used to separate out the codewords sent by each transmit antenna which can then be decoded individually although easy to implement this approach can be highly suboptimal when the channel matrix is near singular this paper develops a new linear receiver architecture that uses the receive antennas to create an effective channel matrix with integer valued entries rather than attempting to recover transmitted codewords directly the decoder recovers integer combinations of the codewords according to the entries of the effective channel matrix the codewords are all generated using the same linear code which guarantees that these integer combinations are themselves codewords provided that the effective channel is full rank these integer combinations can then be digitally solved for the original codewords this paper focuses on the special case where there is no coding across transmit antennas and no channel state information at the transmitter s which corresponds either to a multiuser uplink scenario or to single user v blast encoding in this setting the proposed integer forcing linear receiver significantly outperforms conventional linear architectures such as the zero forcing and linear minimum mean squared error receiver in the high signal to noise ratio regime the proposed receiver attains the optimal diversity multiplexing tradeoff for the standard multiple input multiple output mimo channel with no coding across transmit antennas it is further shown that in an extended mimo model with interference the integer forcing linear receiver achieves the optimal generalized degrees of freedom

algebraic construction of quasi cyclic ldpc codes for the awgn and erasure channels this paper is concerned with construction of quasi cyclic qc low density parity check ldpc codes for three different types of channels the additive white gaussian noise the binary random erasure and the binary burst erasure channels two algebraic methods for systematic construction of qc ldpc codes are presented codes constructed perform well over all three types of channels

superposition coding for side information channels we present simple practical codes designed for the binary and gaussian dirty paper channels we show that the dirty paper decoding problem can be transformed into an equivalent multiple access decoding problem for which we apply superposition coding our concept is a generalization of the nested lattices approach of zamir shamai and erez in a theoretical setting our constructions are capable of achieving capacity using random component codes and maximum likelihood decoding we also present practical implementations of the constructions and simulation results for both dirty paper channels our results for the gaussian dirty paper channel are on par with the best known results for nested lattices we discuss the binary dirty tape channel for which we present a simple effective coding technique finally we propose a framework for extending our approach to general gel fand pinsker channels

analysis of low density parity check codes for the gilbert elliott channel density evolution analysis of low density parity check ldpc codes in memoryless channels is extended to the gilbert elliott ge channel which is a special case of a large class of channels with hidden markov memory in a procedure referred to as estimation decoding the sum product algorithm spa is used to perform ldpc decoding jointly with channel state detection density evolution results show and simulation results confirm that such decoders provide a significantly enlarged region of successful decoding within the ge parameter space compared with decoders that do not exploit the channel memory by considering a variety of ways in which a ge channel may be degraded it is shown how knowledge of the decoding behavior at a single point of the ge parameter space may be extended to a larger region within the space thereby mitigating the large complexity needed in using density evolution to explore the parameter space point by point using the ge channel as a straightforward example we conclude that analysis of estimation decoding for ldpc codes is feasible in channels with memory and that such analysis shows large potential gains

a physical layer rateless code for wireless channels in this paper we propose a physical layer rateless code for wireless channels a novel rateless encoding scheme is developed to overcome the high error floor problem caused by the low density generator matrix ldgm like encoding scheme in conventional rateless codes this is achieved by providing each symbol with approximately equal protection in the encoding process an extrinsic information transfer exit chart based optimization approach is proposed to obtain a robust check node degree distribution which can achieve near capacity performances for a wide range of signal to noise ratios snr simulation results show that under the same channel conditions and transmission overheads the bit error rate ber performance of the proposed scheme considerably outperforms the existing rateless codes in additive white gaussian noise awgn channels particularly at low ber regions

design of repeat accumulate codes for iterative detection and decoding extrinsic information transfer exit charts are used to design systematic and nonsystematic repeat accumulate ra codes for iterative detection and decoding the convergence problems of nonsystematic ra codes are solved by introducing a biregular or doped layer of check nodes as examples such nonsystematic codes are designed for multi input multi output mimo fading channels and are shown to operate close to capacity

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

pilot symbol assisted coded transmission over the block noncoherent awgn channel in this paper pilot symbol assisted transmission in conjunction with high performance coding over the block independent noncoherent additive white gaussian noise channel is investigated several approximate iterative receivers are proposed which either perform carrier phase estimation separately from detection or joint carrier phase estimation decoding in an iterative fashion the performance of the proposed receivers is analyzed using density evolution the power allocation to the pilot symbol is quantified and it is shown that an optimal allocation scheme exists that minimizes the overall information bit signal to noise ratio required for error free communication this optimal power allocation which could be utilized in code design is found to be sensitive to the channel coherence interval as well as to the particular receiver used in addition a simple upper bound on the performance of any receiver that performs joint iterative carrier phase estimation and detection is derived the obtained results are compared with the simulated performance of the proposed receivers

density weighted nystrom method for computing large kernel eigensystems the nystrom method is a well known sampling based technique for approximating the eigensystem of large kernel matrices however the chosen samples in the nystrom method are all assumed to be of equal importance which deviates from the integral equation that defines the kernel eigenfunctions motivated by this observation we extend the nystrom method to a more general density weighted version we show that by introducing the probability density function as a natural weighting scheme the approximation of the eigensystem can be greatly improved an efficient algorithm is proposed to enforce such weighting in practice which has the same complexity as the original nystrom method and hence is notably cheaper than several other alternatives experiments on kernel principal component analysis spectral clustering and image segmentation demonstrate the encouraging performance of our algorithm

decoding algorithms for nonbinary ldpc codes over gf q in this letter we address the problem of decoding nonbinary low density parity check ldpc codes over finite fields gf q with reasonable complexity and good performance in the first part of the letter we recall the original belief propagation bp decoding algorithm and its fourier domain implementation we show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding in the second part of the letter we introduce a simplified decoder which is inspired by the min sum decoder for binary ldpc codes we called this decoder extended min sum ems we show that it is possible to greatly reduce the computational complexity of the check node processing by computing approximate reliability measures with a limited number of values in a message by choosing appropriate correction factors or offsets we show that the ems decoder performance is quite good and in some cases better than the regular bp decoder the optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution our simulations on ultra sparse codes over very high order fields show that nonbinary ldpc codes are promising for applications which require low frame error rates for small or moderate codeword lengths the ems decoder is a good candidate for practical hardware implementations of such codes

ldpc codes for flat rayleigh fading channels with channel side information in this paper we design capacity approaching low density parity check ldpc codes in the low signal to noise ratio snr regime for flat rayleigh fading channels with channel side information at transmitter and receiver we use the structure advocated by caire et al which uses a single codebook with dynamic power allocation the extrinsic information transfer exit function method is used to design the ldpc codes which approach the channel capacities we also study the exit function properties of various demappers

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

on the design of fast convergent ldpc codes for the bec an optimization approach the complexity performance trade off is a fundamental aspect of the design of low density parity check ldpc codes in this paper we consider ldpc codes for the binary erasure channel bec use code rate for performance metric and number of decoding iterations to achieve a certain residual erasure probability for complexity metric we first propose a quite accurate approximation of the number of iterations for the bec moreover a simple but efficient utility function corresponding to the number of iterations is developed using the aforementioned approximation and the utility function two optimization problems w r t complexity are formulated to find the code degree distributions we show that both optimization problems are convex in particular the problem with the proposed approximation belongs to the class of semi infinite problems which are computationally challenging to be solved however the problem with the proposed utility function falls into the class of semi definite programming sdp and thus the global solution can be found efficiently using available sdp solvers numerical results reveal the superiority of the proposed code design compared to existing code designs from literature

quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics we study quantitatively the effective large scale behavior of discrete elliptic equations on the lattice mathbb z d with random coefficients the theory of stochastic homogenization relates the random stationary and ergodic field of coefficients with a deterministic matrix of effective coefficients this is done via the corrector problem which can be viewed as a highly degenerate elliptic equation on the infinite dimensional space of admissible coefficient fields in this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a spectral gap estimate w r t a glauber dynamics on coefficient fields as it is the case for independent and identically distributed coefficients as a main result we prove an optimal decay in time of the semigroup associated with the corrector problem i e of the generator of the process called random environment as seen from the particle as a corollary we recover existence of stationary correctors in dimensions d 2 and prove new optimal estimates for regularized versions of the corrector in dimensions d ge 2 we also give a self contained proof of a new estimate on the gradient of the parabolic variable coefficient green s function which is a crucial analytic ingredient in our approach as an application of these results we prove the first and optimal estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients

the spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators we are concerned with the numerical solution of the eigenvalue problem t varphi lambda varphi varphi ne 0 where t is a linear operator in a bana ch space t may represent a bounded integral operator or a closed differential operator with bounded inverse the linear operator t and its approximation t n are defined in the same space perturbation theory is then a suitable framework for our problem we present in the first part a systematic study of the various notions of convergence t n to t which imply the convergence of the eigenvalues with preservation of the multiplicities the results are then applied to practical methods for integral and differential operators in the second part we present convergence rates analytic perturbation theory is used to refine on the computed eigenelements of an integral operator and to produce localization results on the eigenelements finally superconvergence results are discussed both for integral and differential operators

large scale eigenvector approximation via hilbert space embedding nystrom the nystrom method approximates eigenvectors of a given kernel matrix by randomly sampling subset of data previous researches focus on good kernel approximation while the quality of eigenvector approximation is rarely explored in online eigenvector approximation method one can minimize the kernel approximation error to guarantee a good eigenvector approximation however in this work we paradoxically prove that for batch approximation methods like nystrom it is no longer true this unexpected discovery opens a question what criterion should we use in nystrom to generate a decent eigenvector approximation to address this problem we propose a novel criterion named hilbert space embedding hse nystrom criterion which directly minimizes the eigenvector approximation error the proposed hse criterion provides a general framework to approximate eigenvectors within linear time and space complexity we then show that we can rediscover many successful nystrom methods with the proposed criterion including k means nystrom and density nystrom to further demonstrate the power of our criterion we actually design a novel algorithm to approximate eigenvectors of laplacian matrices based on the proposed criterion with better accuracy among existing linear complexity methods we demonstrate the efficiency and efficacy of our proposal in numerical experiments highlightsa new criterion for nystrom method to find accurate eigenvector approximation the conventional criterion does not work well in eigenvector approximation we need to adjust kernel width together with average weight adaptively propose a new criterion to adaptively optimize the parameters efficiently

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

on the nystrom method for approximating a gram matrix for improved kernel based learning a problem for many kernel based methods is that the amount of computation required to find the solution scales as o n3 where n is the number of training examples we develop and analyze an algorithm to compute an easily interpretable low rank approximation to an n n gram matrix g such that computations of interest may be performed more rapidly the approximation is of the form gk cwk ct where c is a matrix consisting of a small number c of columns of g and wk is the best rank k approximation to w the matrix formed by the intersection between those c columns of g and the corresponding c rows of g an important aspect of the algorithm is the probability distribution used to randomly sample the columns we will use a judiciously chosen and data dependent nonuniform probability distribution let 2 and f denote the spectral norm and the frobenius norm respectively of a matrix and let gk be the best rank k approximation to g we prove that by choosing o k e4 columns g cwk ct g gk e i 1n gii2 both in expectation and with high probability for both 2 f and for all k 0 k rank w this approximation can be computed using o n additional space and time after making two passes over the data from external storage the relationships between this algorithm other related matrix decompositions and the nystrom method from integral equation theory are discussed

design of low density parity check codes for modulation and detection a coding and modulation technique is studied where the coded bits of an irregular low density parity check ldpc code are passed directly to a modulator at the receiver the variable nodes of the ldpc decoder graph are connected to detector nodes and iterative decoding is accomplished by viewing the variable and detector nodes as one decoder the code is optimized by performing a curve fitting on extrinsic information transfer charts design examples are given for additive white gaussian noise channels as well as multiple input multiple output mimo fading channels where the receiver but not the transmitter knows the channel for the mimo channels the technique operates within 1 25 db of capacity for various antenna configurations and thereby outperforms a scheme employing a parallel concatenated turbo code by wide margins when there are more transmit than receive antennas

hilbert space embeddings and metrics on probability measures a hilbert space embedding for probability measures has recently been proposed with applications including dimensionality reduction homogeneity testing and independence testing this embedding represents any probability measure as a mean element in a reproducing kernel hilbert space rkhs a pseudometric on the space of probability measures can be defined as the distance between distribution embeddings we denote this as k indexed by the kernel function k that defines the inner product in the rkhs we present three theoretical properties of k first we consider the question of determining the conditions on the kernel k for which k is a metric such k are denoted characteristic kernels unlike pseudometrics a metric is zero only when two distributions coincide thus ensuring the rkhs embedding maps all distributions uniquely i e the embedding is injective while previously published conditions may apply only in restricted circumstances e g on compact domains and are difficult to check our conditions are straightforward and intuitive integrally strictly positive definite kernels are characteristic alternatively if a bounded continuous kernel is translation invariant on d then it is characteristic if and only if the support of its fourier transform is the entire d second we show that the distance between distributions under k results from an interplay between the properties of the kernel and the distributions by demonstrating that distributions are close in the embedding space when their differences occur at higher frequencies third to understand the nature of the topology induced by k we relate k to other popular metrics on probability measures and present conditions on the kernel k under which k metrizes the weak topology

on the evaluation of the polyanskiy poor verdu converse bound for finite block length coding in awgn a tight converse bound to the channel coding rate in the finite block length regime and under additive white gaussian noise conditions was recently proposed by polyanskiy poor verdu ppv the bound is a generalization of a number of other classical results and it was also claimed to be equivalent to shannon s 1959 cone packing bound in this paper we investigate methods for a reliable numerical evaluation of the bound which is troublesome even for not too large values of the block length n by compactly expressing the polyanskiy poor and verdu ppv converse bound in terms of non central chi squared distributions and by evaluating those through an integral expression and a corresponding series expansion which exploit a method proposed by temme as a result a robust evaluation method and new insights on the bound s asymptotics as well as new approximate expressions are obtained