optimal phase transitions in compressed sensing compressed sensing deals with efficient recovery of analog signals from linear encodings this paper presents a statistical study of compressed sensing by modeling the input signal as an i i d process with known distribution three classes of encoders are considered namely optimal nonlinear optimal linear and random linear encoders focusing on optimal decoders we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise respectively the optimal phase transition threshold is determined as a functional of the input distribution and compared to suboptimal thresholds achieved by popular reconstruction algorithms in particular we show that gaussian sensing matrices incur no penalty on the phase transition threshold with respect to optimal nonlinear encoding our results also provide a rigorous justification of previous results based on replica heuristics in the weak noise regime

channel capacity under sub nyquist nonuniform sampling this paper investigates the effect of sub nyquist sampling upon the capacity of an analog channel the channel is assumed to be a linear time invariant gaussian channel where perfect channel knowledge is available at both the transmitter and the receiver we consider a general class of right invertible time preserving sampling methods which includes irregular nonuniform sampling and characterize in closed form the channel capacity achievable by this class of sampling methods under a sampling rate and power constraint our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal to noise ratio among all spectral sets of measure equal to the sampling rate this can be attained through filterbank sampling with uniform sampling grid employed at each branch with possibly different rates or through a single branch of modulation and filtering followed by uniform sampling these results reveal that for a large class of channels employing irregular nonuniform sampling sets while are typically complicated to realize in practice does not provide capacity gain over uniform sampling sets with appropriate preprocessing our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain in this scenario which is in contrast to the benefits obtained from random mixing in spectrum blind compressive sampling schemes

ergodic properties of operator semigroups in general weak topologies abstract the aim of this paper is to study ergodic properties i e properties about the limit of cesaro averages of a semigroup of bounded linear operators on a banach space x which is assumed to be continuous and locally integrable in the sense of a certain general weak topology of x then the results are applied to particular examples such as locally strongly integrable semigroups their dual semigroups and the tensor product semigroup of two c 0 semigroups

the essential spectrum of neumann laplacians on some bounded singular domains in the present paper we consider neumann laplacians on singular domains of the type rooms and passages or combs and we show that in typical situations the essential spectrum can be determined from the geometric data moreover given an arbitrary closed subset s of the non negative reals we construct domains s such that the essential spectrum of the neumann laplacian on is just this set s

double variational principle for mean dimension with potential abstract this paper contributes to the mean dimension theory of dynamical systems we introduce a new concept called mean dimension with potential and develop a variational principle for it this is a mean dimension analogue of the theory of topological pressure we consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function we prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property the basic idea of the proof is a dynamicalization of geometric measure theory

a gromov hausdorff framework with diffusion geometry for topologically robust non rigid shape matching in this paper the problem of non rigid shape recognition is studied from the perspective of metric geometry in particular we explore the applicability of diffusion distances within the gromov hausdorff framework while the traditionally used geodesic distance exploits the shortest path between points on the surface the diffusion distance averages all paths connecting the points the diffusion distance constitutes an intrinsic metric which is robust in particular to topological changes such changes in the form of shortcuts holes and missing data may be a result of natural non rigid deformations as well as acquisition and representation noise due to inaccurate surface construction the presentation of the proposed framework is complemented with examples demonstrating that in addition to the relatively low complexity involved in the computation of the diffusion distances between surface points its recognition and matching performances favorably compare to the classical geodesic distances in the presence of topological changes between the non rigid shapes

universal compressed sensing for almost lossless recovery in this paper the problem of developing universal algorithms for noiseless compressed sensing of stochastic processes is studied first renyi s notion of information dimension id is generalized to continuous valued discrete time stationary processes this provides a measure of complexity for such processes and is connected to the rate of measurement sampling rate required for their accurate recovery then based on occam s razor a minimum entropy pursuit mep optimization approach for universal compressed sensing is proposed it is proven that for any stationary process satisfying certain mixing conditions if the sampling rate is larger than the id of the source process mep optimization can reliably recover the source vector almost losslessly without having any prior information about its distribution then a lagrangian type relaxation of mep optimization problem referred to as lagrangian mep is studied it is shown that lagrangian mep is identical to an implementable algorithm proposed by baron and coauthors and for the right choice of parameters has the same asymptotic performance as mep optimization finally it is proven that lagrangian mep is robust to small measurement noise

extensions and generalizations of a theorem of widder and of the theory of symmetric local semigroups the theory of symmetric local semigroups due to a klein and l landau j funct anal 44 1981 121 136 is generalized to semigroups indexed by subsets of rn for n 1 the result implies a similar result of a e nussbaum j funct anal 48 1982 213 223 it is further generalized to semigroups that are symmetric local in some directions and unitary in others the results are used to give a simple proof of a devinatz s duke math j 22 1955 185 192 and n i akhiezer s the classical moment problem and some related questions hafner new york 1965 generalization of a theorem of widder concerning the representation of functions as laplace integrals this result is extended to the representation as a laplace integral of a function taking values in b r the set of bounded linear operators on a hilbert space r also a theorem is proved encompassing both the result of devinatz and akhiezer and bochner s theorem on the representation of positive definite functions as fourier integrals

reconstructing a three dimensional model with arbitrary errors a number of current technologies allow for the determination of interatomic distance information in structures such as proteins and rna thus the reconstruction of a three dimensional set of points using information about its interpoint distances has become a task of basic importance in determining molecular structure the distance measurements one obtains from techniques such as nmr are typically sparse and error prone greatly complicating the reconstruction task many of these errors result in distance measurements that can be safely assumed to lie within certain fixed tolerances but a number of sources of systematic error in these experiments lead to inaccuracies in the data that are very hard to quantify in effect one must treat certain entries of the measured distance matrix as being arbitrarily corrupted the existence of arbitrary errors leads to an interesting sort of error correction problem how many corrupted entries in a distance matrix can be efficiently corrected to produce a consistent three dimensional structure for the case of an n n matrix in which every entry is specified we provide a randomized algorithm running in time o n log n that enumerates all structures consistent with at most 1 2 e n errors per row with high probability in the case of randomly located errors we can correct errors of the same density in a sparse matrix one in which only a fraction of the entries in each row are given for any constant gt 0

from rate distortion theory to metric mean dimension variational principle the purpose of this paper is to point out a new connection between information theory and dynamical systems in the information theory side we consider rate distortion theory which studies lossy data compression of stochastic processes under distortion constraints in the dynamical systems side we consider mean dimension theory which studies how many parameters per iterate we need to describe a dynamical system the main results are new variational principles connecting rate distortion function to metric mean dimension

reflection positivity and monotonicity we prove general reflection positivity results on riemannian manifolds for both scalar fields and dirac fields and comment on applications to quantum field theory as another application we prove the inequality cd cn between dirichlet and neumann covariance operators on a manifold with a reflection

proto value functions a laplacian framework for learning representation and control in markov decision processes this paper introduces a novel spectral framework for solving markov decision processes mdps by jointly learning representations and optimal policies the major components of the framework described in this paper include i a general scheme for constructing representations or basis functions by diagonalizing symmetric diffusion operators ii a specific instantiation of this approach where global basis functions called proto value functions pvfs are formed using the eigenvectors of the graph laplacian on an undirected graph formed from state transitions induced by the mdp iii a three phased procedure called representation policy iteration comprising of a sample collection phase a representation learning phase that constructs basis functions from samples and a final parameter estimation phase that determines an approximately optimal policy within the linear subspace spanned by the current basis functions iv a specific instantiation of the rpi framework using least squares policy iteration lspi as the parameter estimation method v several strategies for scaling the proposed approach to large discrete and continuous state spaces including the nystrom extension for out of sample interpolation of eigenfunctions and the use of kronecker sum factorization to construct compact eigenfunctions in product spaces such as factored mdps vi finally a series of illustrative discrete and continuous control tasks which both illustrate the concepts and provide a benchmark for evaluating the proposed approach many challenges remain to be addressed in scaling the proposed framework to large mdps and several elaboration of the proposed framework are briefly summarized at the end

embedding riemannian manifolds by the heat kernel of the connection laplacian abstract given a class of closed riemannian manifolds with prescribed geometric conditions we introduce an embedding of the manifolds into l 2 based on the heat kernel of the connection laplacian associated with the levi civita connection on the tangent bundle as a result we can construct a distance in this class which leads to a pre compactness theorem on the class under consideration

multi parameter local semi groups of hermetian operators abstract it is proven that if a weakly continuous multi parameter family of hermetian operators in a hilbert space whose domains are not necessarily dense but whose union is dense and which has the semi group property for the parameters belonging to an open convex set in euclidean n space with 0 a limit point then the operators have self adjoint extensions which have a joint spectral representation

construction of a unique self adjoint generator for a symmetric local semigroup abstract a definition is given of a symmetric local semigroup of unbounded operators p t 0 t t for some t 0 on a hilbert space h such that p t is eventually densely defined as t 0 it is shown that there exists a unique unbounded below self adjoint operator h on h such that p t is a restriction of e th as an application it is proven that h0 v is essentially self adjoint where e th0 is an lp contractive semigroup and v is multiplication by a real measurable function such that v l2 e and e v l1 for some e 0

on the information dimension of stochastic processes in 1959 renyi proposed the information dimension and the d dimensional entropy to measure the information content of general random variables this paper proposes a generalization of information dimension to stochastic processes by defining the information dimension rate as the entropy rate of the uniformly quantized stochastic process divided by minus the logarithm of the quantizer step size 1 m in the limit as m to infty it is demonstrated that the information dimension rate coincides with the rate distortion dimension defined as twice the rate distortion function r d of the stochastic process divided by log d in the limit as d downarrow 0 it is further shown that among all multivariate stationary processes with a given matrix valued spectral distribution function sdf the gaussian process has the largest information dimension rate and the information dimension rate of multivariate stationary gaussian processes is given by the average rank of the derivative of the sdf the presented results reveal that the fundamental limits of almost zero distortion recovery via compressible signal pursuit and almost lossless analog compression are different in general

reconstruction of kubo martin schwinger structure from euclidean green functions the explicit form of the poisson kernel for the strip t z c 0 im z is used to derive some conditions for a state on a given c algebra dynamical system a t to be a kubo martin schwinger kms state reflection positivity and generalized periodicity of the euclidean green functions corresponding to are established and used for an analytic continuation to real time reflection positivity is also used for the construction of a cyclic and faithful representation of a t on some hilbert space a sufficient condition for the identification of the constructed dynamics with the modular group of a von neumann algebra generated by the representation is formulated in terms of the euclidean green functions

regularization on graphs with function adapted diffusion processes harmonic analysis and diffusion on discrete data has been shown to lead to state of the art algorithms for machine learning tasks especially in the context of semi supervised and transductive learning the success of these algorithms rests on the assumption that the function s to be studied learned interpolated etc are smooth with respect to the geometry of the data in this paper we present a method for modifying the given geometry so the function s to be studied are smoother with respect to the modified geometry and thus more amenable to treatment using harmonic analysis methods among the many possible applications we consider the problems of image denoising and transductive classification in both settings our approach improves on standard diffusion based methods

estimation in gaussian noise properties of the minimum mean square error consider the minimum mean square error mmse of estimating an arbitrary random variable from its observation contaminated by gaussian noise the mmse can be regarded as a function of the signal to noise ratio snr as well as a functional of the input distribution of the random variable to be estimated it is shown that the mmse is concave in the input distribution at any given snr for a given input distribution the mmse is found to be infinitely differentiable at all positive snr and in fact a real analytic function in snr under mild conditions the key to these regularity results is that the posterior distribution conditioned on the observation through gaussian channels always decays at least as quickly as some gaussian density furthermore simple expressions for the first three derivatives of the mmse with respect to the snr are obtained it is also shown that as functions of the snr the curves for the mmse of a gaussian input and that of a non gaussian input cross at most once over all snrs these properties lead to simple proofs of the facts that gaussian inputs achieve both the secrecy capacity of scalar gaussian wiretap channels and the capacity of scalar gaussian broadcast channels as well as a simple proof of the entropy power inequality in the special case where one of the variables is gaussian

sample distortion for compressed imaging we propose the notion of a sample distortion sd function for independent and identically distributed i i d compressive distributions to fundamentally quantify the achievable reconstruction performance of compressed sensing for certain encoder decoder pairs at a given sampling ratio two lower bounds on the achievable performance and the intrinsic convexity property is derived a zeroing procedure is then introduced to improve non convex sd functions the sd framework is then applied to analyse compressed imaging with a multi resolution statistical image model using both the generalized gaussian distribution and the two state gaussian mixture distribution we subsequently focus on the gaussian encoder bayesian optimal approximate message passing amp decoder pair whose theoretical sd function is provided by the rigorous analysis of the amp algorithm given the image statistics analytic bandwise sample allocation for bandwise independent model is derived as a reverse water filling scheme som and schniter s turbo message passing approach is further deployed to integrate the bandwise sampling with the exploitation of the hidden markov tree structure of wavelet coefficients natural image simulations confirm that with oracle image statistics the sd function associated with the optimized sample allocation can accurately predict the possible compressed sensing gains finally a general sample allocation profile based on average image statistics not only illustrates preferable performance but also makes the scheme practical

w kms structure from multi time euclidean green functions an axiomatic approach to the problem of reconstruction of a dynamics from the euclidean multi time green functions is proposed the existence of a w algebra on which the reconstructed dynamics acts as the canonical group of modular automorphisms is shown a condition under which the reconstruction from the one time and multi time green functions are equivalent is formulated

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

continuous integral kernels for unbounded schrodinger semigroups and their spectral projections abstract by suitably extending a feynman kac formula of simon canad math soc conf proc 28 2000 317 we study one parameter semigroups generated by the negative of rather general schrodinger operators which may be unbounded from below and include a magnetic vector potential in particular a common domain of essential self adjointness for such a semigroup is specified moreover each member of the semigroup is proven to be a maximal carleman operator with a continuous integral kernel given by a brownian bridge expectation the results are used to show that the spectral projections of the generating schrodinger operator also act as carleman operators with continuous integral kernels applications to schrodinger operators with rather general random scalar potentials include a rigorous justification of an integral kernel representation of their integrated density of states a relation frequently used in the physics literature on disordered solids

analytic continuation of local representations of symmetric spaces we prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space g k the problem of passing from the euclidean group to the poincare group appears first to have been addressed and solved this way by klein and landau let g be a lie group k a closed subgroup and an involutive automorphism with k as fixed point subgroup if g k m is the corresponding symmetric lie algebra we form g k im and let g denote the simply connected lie group with g as lie algebra we consider virtual representations of g on a fixed complex hilbert space h adopting the definitions due to j frohlich k osterwalder and e seiler in particular g 1 g possibly unbounded operators for g in a neighborhood of e in g we prove that every such continues analytically to a strongly continuous unitary representation of g on h our theorem extends results due to klein landau frohlich et al and others earlier for special cases previous results were known only for special g k and then only for certain

solving partial differential equations on manifolds from incomplete interpoint distance solutions of partial differential equations pdes on manifolds have provided important applications in different fields in science and engineering existing methods are mainly based on discretization of manifolds as implicit functions triangle meshes or point clouds where the manifold structure is approximated by either zero level set of an implicit function or a set of points in many applications manifolds might be only provided as an interpoint distance matrix with possible missing values this paper discusses a framework to discretize pdes on manifolds represented as incomplete interpoint distance information without conducting a time consuming global coordinates reconstruction we propose a more efficient strategy by discretizing differential operators only based on pointwisely local reconstruction our local reconstruction model is based on the recent advances of low rank matrix completion theory where only a very small random portion of distance information is required this method enables u

spectral representations for unbounded operators with real spectrum soit t un operateur agissant dans un espace de banach reflexif x avec domaine d t et spectre reel alors il existe une mesure spectrale e sur z la variete de semisimplicite pour t telle que 1 pour chaque b r e commute avec chaque u b x qui commute avec t 2 d t z x z r se ds x existe et appartient a z et 3 tx r se ds x pour tout x d t z

recovery from linear measurements with complexity matching universal signal estimation we study the compressed sensing cs signal estimation problem where an input signal is measured via a linear matrix multiplication under additive noise while this setup usually assumes sparsity or compressibility in the input signal during recovery the signal structure that can be leveraged is often not known a priori in this paper we consider universal cs recovery where the statistics of a stationary ergodic signal source are estimated simultaneously with the signal itself inspired by kolmogorov complexity and minimum description length we focus on a maximum a posteriori map estimation framework that leverages universal priors to match the complexity of the source our framework can also be applied to general linear inverse problems where more measurements than in cs might be needed we provide theoretical results that support the algorithmic feasibility of universal map estimation using a markov chain monte carlo implementation which is computationally challenging we incorporate some techniques to accelerate the algorithm while providing comparable and in many cases better reconstruction quality than existing algorithms experimental results show the promise of universality in cs particularly for low complexity sources that do not exhibit standard sparsity or compressibility

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

support recovery with sparsely sampled free random matrices consider a bernoulli gaussian complex n vector whose components are vi xibi with xi c n 0 px and binary bi mutually independent and iid across i this random q sparse vector is multiplied by a square random matrix u and a randomly chosen subset of average size n p p 0 1 of the resulting vector components is then observed in additive gaussian noise we extend the scope of conventional noisy compressive sampling models where u is typically a matrix with iid components to allow u satisfying a certain freeness condition this class of matrices encompasses haar matrices and other unitarily invariant matrices we use the replica method and the decoupling principle of guo and verdu as well as a number of information theoretic bounds to study the input output mutual information and the support recovery error rate in the limit of n we also extend the scope of the large deviation approach of rangan and characterize the performance of a class of estimators encompassing thresholded linear mmse and l1 relaxation

weyl asymptotic formula for the laplacian on domains with rough boundaries we study asymptotic distribution of eigenvalues of the laplacian on a bounded domain in n our main results include an explicit remainder estimate in the weyl formula for the dirichlet laplacian on an arbitrary bounded domain sufficient conditions for the validity of the weyl formula for the neumann laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula in particular we show that the weyl formula holds true for the neumann laplacian on a lip domain whenever d 1 d prove that the remainder in this formula is o d 1 and give an example where the remainder estimate o d 1 is order sharp we use a new version of the variational technique which does not require the extension theorem

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

the generation of weakly integrable semigroups abstract a characterization of the generators of a class of weakly integrable semigroups on a locally convex space is proved it bears a resemblance to the hille yosida phillips theorem on the generators of strongly continuous semigroups on a banach space

from the euclidean group to the poincare group via osterwalder schrader positivity given a continuous representation of the euclidean group inn 1 dimensions together with a covariant system of subspaces which satisfies osterwalder schrader positivity we construct a continuous unitary representation of the orthochronous poincare group inn 1 dimensions satisfying the spectral condition a similar result holds for the covering groups of the euclidean and poincare group

sampling rate distortion consider a discrete memoryless multiple source with m components of which k leq m possibly different sources are sampled at each time instant and jointly compressed in order to reconstruct all the m sources under a given distortion criterion a new notion of sampling rate distortion function is introduced and is characterized first for the case of fixed set sampling next for independent random sampling performed without knowledge of the source outputs it is shown that the sampling rate distortion function is the same regardless of whether or not the decoder is informed of the sequence of sampled sets furthermore memoryless random sampling is considered with the sampler depending on the source outputs and with an informed decoder it is shown that deterministic sampling characterized by a conditional point mass is optimal and suffices to achieve the sampling rate distortion function for memoryless random sampling with an uninformed decoder an upper bound for the sampling rate distortion function is seen to possess a similar property of conditional point mass optimality it is shown by example that memoryless sampling with an informed decoder can outperform strictly any independent random sampler and that memoryless sampling can do strictly better with an informed decoder than without

embeddings of riemannian manifolds with heat kernels and eigenfunctions we show that any closed n dimensional riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points both this time and this number can be bounded in terms of the dimension a lower bound on the ricci curvature the injectivity radius and the volume it follows that the manifold can be embedded by a finite number of eigenfunctions of the laplace operator again this number only depends on the geometric bounds and the dimension in addition both maps can be made arbitrarily close to an isometry in the appendix we derive quantitative estimates of the harmonic radius so that the estimates on the number of eigenfunctions or heat kernels needed can be made quantitative as well 2016 wiley periodicals inc

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

manifold parametrizations by eigenfunctions of the laplacian and heat kernels we use heat kernels or eigenfunctions of the laplacian to construct local coordinates on large classes of euclidean domains and riemannian manifolds not necessarily smooth e g with alpha metric these coordinates are bi lipschitz on large neighborhoods of the domain or manifold with constants controlling the distortion and the size of the neighborhoods that depend only on natural geometric properties of the domain or manifold the proof of these results relies on novel estimates from above and below for the heat kernel and its gradient as well as for the eigenfunctions of the laplacian and their gradient that hold in the non smooth category and are stable with respect to perturbations within this category finally these coordinate systems are intrinsic and efficiently computable and are of value in applications

functional properties of minimum mean square error and mutual information in addition to exploring its various regularity properties we show that the minimum mean square error mmse is a concave functional of the input output joint distribution in the case of additive gaussian noise the mmse is shown to be weakly continuous in the input distribution and lipschitz continuous with respect to the quadratic wasserstein distance for peak limited inputs regularity properties of mutual information are also obtained several applications to information theory and the central limit theorem are discussed

on virtual representations of symmetric spaces and their analytic continuation in this paper we study a problem in group representation theory motivated by relativistic quantum field theory one of the most powerful approaches to constructing models of relativistic quantum fields relies on euclidean field theory a description of quantum field theory in which time is purely imaginary the basic objects of euclidean field theory the euclidean green s or schwinger functions can often be expressed in terms of functional integrals the construction of quantum field models is thereby reduced to quadratures a well known simple example of this method is the reformulation of quantum mechanisms by means of wiener integrals the fundamental solution of the schrodinger equation analytically continued to imaginary time is given by the well known feynman kac formula see e g 15 19 in this approach one faces the problem of analytic continuation back to real time while this problem is elementary in non relativistic quantum mechanics it is rather intricate in relativistic quantum field theory for some class of relativistic quantum field theories a fairly general solution to this problem was given in papers by osterwalder and

stochastically positive structures on weyl algebras the case of quasi free states we consider quasi free stochastically positive ground and thermal states on weyl algebras in the imaginary time formulation in particular we obtain a new derivation of a general form of thermal quasi free state and give conditions when such a state is stochastically positive i e when it defines a periodic stochastic process with respect to imaginary time a so called thermal process then we show that the thermal process completely determines modular structure canonically associated with the quasi free thermal state on weyl algebra we discuss a variety of examples connected with free quantum field theories on globally hyperbolic stationary space times and models of quantum statistical mechanics

fundamental distortion limits of analog to digital compression representing a continuous time signal by a set of samples is a classical problem in signal processing we study this problem under the additional constraint that the samples are quantized or compressed in a lossy manner under a limited bitrate budget to this end we consider a combined sampling and source coding problem in which an analog stationary gaussian signal is reconstructed from its encoded samples these samples are obtained by a set of bounded linear functionals of the continuous time path with a limitation on the average number of samples per unit time given in this setting we provide a full characterization of the minimal distortion in terms of the sampling frequency the bitrate and the signal s spectrum assuming that the signal s energy is not uniformly distributed over its spectral support we show that for each compression bitrate there exists a critical sampling frequency smaller than the nyquist rate such that the distortion in signal reconstruction when sampling at this frequency is minimal our results can be seen as an extension of the classical sampling theorem for bandlimited random processes in the sense that they describe the minimal amount of excess distortion in the reconstruction due to lossy compression of the samples and provide the minimal sampling frequency required in order to achieve this distortion finally we compare the fundamental limits in the combined source coding and sampling problem to the performance of pulse code modulation where each sample is quantized by a scalar quantizer using a fixed number of bits

discrete time high order schemes for viscosity solutions of hamilton jacobi bellman equations a general method for constructing high order approximation schemes for hamilton jacobi bellman equations is given the method is based on a discrete version of the dynamic programming principle we prove a general convergence result for this class of approximation schemes also obtaining under more restrictive assumptions an estimate in l infty of the order of convergence and of the local truncation error the schemes can be applied in particular to the stationary linear first order equation in bbb r n we present several examples of schemes belonging to this class and with fast convergence to the solution

mean dimension mean rank and von neumann luck rank we introduce an invariant called mean rank for any module m of the integral group ring of a discrete amenable group as an analogue of the rank of an abelian group it is shown that the mean dimension of the induced action on the pontryagin dual of m the mean rank of m and the von neumann luck rank of m all coincide as applications we establish an addition formula for mean dimension of algebraic actions prove the analogue of the pontryagin schnirelmnn theorem for algebraic actions and show that for elementary amenable groups with an upper bound on the orders of finite subgroups algebraic actions with zero mean dimension are inverse limits of finite entropy actions

randomly spread cdma asymptotics via statistical physics this paper studies randomly spread code division multiple access cdma and multiuser detection in the large system limit using the replica method developed in statistical physics arbitrary input distributions and flat fading are considered a generic multiuser detector in the form of the posterior mean estimator is applied before single user decoding the generic detector can be particularized to the matched filter decorrelator linear minimum mean square error mmse detector the jointly or the individually optimal detector and others it is found that the detection output for each user although in general asymptotically non gaussian conditioned on the transmitted symbol converges as the number of users go to infinity to a deterministic function of a hidden gaussian statistic independent of the interferers thus the multiuser channel can be decoupled each user experiences an equivalent single user gaussian channel whose signal to noise ratio snr suffers a degradation due to the multiple access interference mai the uncoded error performance e g symbol error rate and the mutual information can then be fully characterized using the degradation factor also known as the multiuser efficiency which can be obtained by solving a pair of coupled fixed point equations identified in this paper based on a general linear vector channel model the results are also applicable to multiple input multiple output mimo channels such as in multiantenna systems

sampling signals with finite rate of innovation the authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation examples of signals with a finite rate of innovation include streams of diracs e g the poisson process nonuniform splines and piecewise polynomials even though these signals are not bandlimited we show that they can be sampled uniformly at or above the rate of innovation using an appropriate kernel and then be perfectly reconstructed thus we prove sampling theorems for classes of signals and kernels that generalize the classic bandlimited and sinc kernel case in particular we show how to sample and reconstruct periodic and finite length streams of diracs nonuniform splines and piecewise polynomials using sinc and gaussian kernels for infinite length signals with finite local rate of innovation we show local sampling and reconstruction based on spline kernels the key in all constructions is to identify the innovative part of a signal e g time instants and weights of diracs using an annihilating or locator filter a device well known in spectral analysis and error correction coding this leads to standard computational procedures for solving the sampling problem which we show through experimental results applications of these new sampling results can be found in signal processing communications systems and biological systems

universal prediction of individual sequences the problem of predicting the next outcome of an individual binary sequence using finite memory is considered the finite state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite state fs predictor it is proven that this fs predictability can be achieved by universal sequential prediction schemes an efficient prediction procedure based on the incremental parsing procedure of the lempel ziv data compression algorithm is shown to achieve asymptotically the fs predictability some relations between compressibility and predictability are discussed and the predictability is proposed as an additional measure of the complexity of a sequence

a feasible method for optimization with orthogonality constraints minimization with orthogonality constraints e g x top x i and or spherical constraints e g vert x vert 2 1 has wide applications in polynomial optimization combinatorial optimization eigenvalue problems sparse pca p harmonic flows 1 bit compressive sensing matrix rank minimization etc these problems are difficult because the constraints are not only non convex but numerically expensive to preserve during iterations to deal with these difficulties we apply the cayley transform a crank nicolson like update scheme to preserve the constraints and based on it develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics the efficiency of the proposed algorithms is demonstrated on a variety of test problems in particular for the maxcut problem it exactly solves a decomposition formulation for the sdp relaxation for polynomial optimization nearest correlation matrix estimation and extreme eigenvalue problems the proposed algorithms run very fast and return solutions no worse than those from their state of the art algorithms for the quadratic assignment problem a gap 0 842 to the best known solution on the largest problem tai256c in qaplib can be reached in 5 min on a typical laptop

on the rate distortion function of random vectors and stationary sources with mixed distributions the asymptotic small distortion behavior of the rate distortion function of an n dimensional source vector with mixed distribution is derived the source distribution is a finite mixture of components such that under each component distribution a certain subset of the coordinates have a discrete distribution while the remaining coordinates have a joint density the expected number of coordinates with a joint density is shown to equal the rate distortion dimension of the source vector also the exact small distortion asymptotic behavior of the rate distortion function of a special but interesting class of stationary information sources is determined

thermal quantum fields with spatially cutoff interactions in 1 1 space time dimensions we construct interacting quantum fields in 1 1 space time dimensions representing charged or neutral scalar bosons at positive temperature and zero chemical potential our work is based on prior work by klein and landau and hoegh krohn generalized path space methods are used to add a spatially cut off interaction to the free system which is described in the araki woods representation it is shown that the interacting kms state is normal w r t the araki woods representation the observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of the free system and the interacting liouvillean is described in terms of the free liouvillean and the interaction

stochastic processes associated with kms states abstract a general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes stochastically positive kms systems periodic os positive stochastic processes periodic positive semigroup structures are defined and shown to be equivalent the objects of the tomita takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven a two sided markov property for the stochastic process is related to a cyclicity property of the kms state perturbations are constructed by the feynman kac nelson formula the general framework is applied to kms states given by density matrices

smallest singular value of a random rectangular matrix we prove an optimal estimate of the smallest singular value of a random sub gaussian matrix valid for all dimensions for an nn matrix a with inde pendent and identically distributed sub gaussian entries the smallest singular value of a is at least of the order p np n 1 with high probability a sharp

metric mean dimension and analog compression wu and verdu developed a theory of almost lossless analog compression where one imposes various regularity conditions on the compressor and the decompressor with the input signal being modelled by a typically infinite entropy stationary stochastic process in this work we consider all stationary stochastic processes with trajectories in a prescribed set of bi infinite sequences and find uniform lower and upper bounds for certain compression rates in terms of metric mean dimension and mean box dimension an essential tool is the recent lindenstrauss tsukamoto variational principle expressing metric mean dimension in terms of rate distortion functions we obtain also lower bounds on compression rates for a fixed stationary process in terms of the rate distortion dimension rates and study several examples

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

sofic mean dimension abstract we introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces these generalize the gromov lindenstrauss weiss mean dimensions for actions of countable amenable groups and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy

a statistical mechanics approach to large system analysis of cdma multiuser detectors we present a theory based on statistical mechanics to evaluate analytically the performance of uncoded fully synchronous randomly spread code division multiple access cdma multiuser detectors with additive white gaussian noise awgn channel under perfect power control and in the large system limit application of the replica method a tool developed in the literature of statistical mechanics allows us to derive analytical expressions for the bit error rate as well as the multiuser efficiency of the individually optimum io and jointly optimum jo multiuser detectors over the whole range of noise levels the information theoretic capacity of the randomly spread cdma channel and the performance of decorrelating and linear minimum mean square error mmse detectors are also derived in the same replica formulation thereby demonstrating validity of the statistical mechanical approach

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

on classical analogues of free entropy dimension abstract we define a classical probability analogue of voiculescu s free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on r n we show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension first it can be viewed as a kind of fractal dimension second if one extends bochner s inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical bochner s inequalities then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in bochner s inequality for a measure regularized by convolution with the gaussian law as the regularization is removed we introduce a free analogue of the bochner inequality and study the related free entropy dimension quantity we show that it is greater or equal to the non microstates free entropy dimension

mean dimension of the dynamical system of brody curves mean dimension is a topological invariant of dynamical systems counting the number of parameters averaged by dynamics brody curves are lipschitz holomorphic maps f mathbb c rightarrow mathbb c p n and they form an infinite dimensional dynamical system gromov started the problem of estimating its mean dimension in 1999 we solve this problem by proving the exact mean dimension formula our formula expresses the mean dimension by the energy density of brody curves as a key novel ingredient we use an information theoretic approach to mean dimension introduced by lindenstrauss and weiss

improving autocorrelation regression for the hurst parameter estimation of long range dependent time series based on golden section search abstract this article presents an improved autocorrelation correlation function acf regression method of estimating the hurst parameter of a time series with long range dependence lrd by using golden section search gss we shall show that the present method is substantially efficient than the conventional acf regression method of h estimation our research uses fractional gaussian noise as a data case but the method introduced is applicable to time series with lrd in general

mmse dimension if n is standard gaussian the minimum mean square error mmse of estimating a random variable x based on snr x n vanishes at least as fast as 1 snr as snr we define the mmse dimension of x as the limit as snr of the product of snr and the mmse mmse dimension is also shown to be the asymptotic ratio of nonlinear mmse to linear mmse for discrete absolutely continuous or mixed distribution we show that mmse dimension equals renyi s information dimension however for a class of self similar singular x e g cantor dis tribution we show that the product of snr and mmse oscillates around information dimension periodically in snr db we also show that these results extend considerably beyond gaussian noise under various technical conditions

reflection positivity and conformal symmetry abstract a reflection positive hilbert space is a triple e e where e is a hilbert space a unitary involution and e a closed subspace on which the hermitian form v w v w is positive semidefinite for a triple g s where g is a lie group an involutive automorphism of g and s a subsemigroup invariant under the involution s s s 1 a unitary representation of g on e e is called reflection positive if g g and s e e this is the first in a series of papers in which we develop a new and systematic approach to reflection positive representations based on reflection positive distributions and reflection positive distribution vectors this approach is most natural to obtain classification results in particular in the abelian case among the tools we develop is a generalization of the bochner schwartz theorem to positive definite distributions on open convex cones we further illustrate our techniques with a non abelian example by constructing reflection positive distribution vectors for complementary series representations of the conformal group o 1 n 1 r of the sphere s n

brain surface conformal parameterization using riemann surface structure in medical imaging parameterized 3 d surface models are useful for anatomical modeling and visualization statistical comparisons of anatomy and surface based registration and signal processing here we introduce a parameterization method based on riemann surface structure which uses a special curvilinear net structure conformal net to partition the surface into a set of patches that can each be conformally mapped to a parallelogram the resulting surface subdivision and the parameterizations of the components are intrinsic and stable their solutions tend to be smooth functions and the boundary conditions of the dirichlet problem can be enforced conformal parameterization also helps transform partial differential equations pdes that may be defined on 3 d brain surface manifolds to modified pdes on a two dimensional parameter domain since the jacobian matrix of a conformal parameterization is diagonal the modified pde on the parameter domain is readily solved to illustrate our techniques we computed parameterizations for several types of anatomical surfaces in 3 d magnetic resonance imaging scans of the brain including the cerebral cortex hippocampi and lateral ventricles for surfaces that are topologically homeomorphic to each other and have similar geometrical structures we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other finally we present an automatic sulcal landmark location algorithm by solving pdes on cortical surfaces the landmark detection results are used as constraints for building conformal maps between surfaces that also match explicitly defined landmarks

exact reconstruction of euclidean distance geometry problem using low rank matrix completion the euclidean distance geometry problem arises in a wide variety of applications from determining molecular conformations in computational chemistry to localization in sensor networks when the distance information is incomplete the problem can be formulated as a nuclear norm minimization problem in this paper this minimization program is recast as a matrix completion problem of a low rank r gram matrix with respect to a suitable basis the well known restricted isometry property cannot be satisfied in this scenario instead a dual basis approach is introduced to theoretically analyze the reconstruction problem if the gram matrix satisfies certain coherence conditions with parameter nu the main result shows that the underlying configuration of n points can be recovered with very high probability from o nr nu log 2 n uniformly random samples computationally simple and fast algorithms are designed to solve the euclidean distance geometry problem numerical tests on different 3 d data and protein molecules validate the effectiveness and efficiency of the proposed algorithms

diffusion semigroups in spaces of continuous functions with mixed topology abstract we study transition semigroups and kolmogorov equations corresponding to stochastic semilinear equations on a hilbert space h it is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on h when endowed with the so called mixed topology as a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups

a proof of the fisher information inequality via a data processing argument the fisher information j x of a random variable x under a translation parameter appears in information theory in the classical proof of the entropy power inequality epi it enters the proof of the epi via the de bruijn identity where it measures the variation of the differential entropy under a gaussian perturbation and via the convolution inequality j x y sup 1 spl ges j x sup 1 j y sup 1 for independent x and y known as the fisher information inequality fii the fii is proved in the literature directly in a rather involved way we give an alternative derivation of the fii as a simple consequence of a data processing inequality for the cramer rao lower bound on parameter estimation

minimum complexity pursuit for universal compressed sensing the nascent field of compressed sensing is founded on the fact that high dimensional signals with simple structure can be recovered accurately from just a small number of randomized samples several specific kinds of structures have been explored in the literature from sparsity and group sparsity to low rankness however two fundamental questions have been left unanswered what are the general abstract meanings of structure and simplicity do there exist universal algorithms for recovering such simple structured objects from fewer samples than their ambient dimension in this paper we address these two questions using algorithmic information theory tools such as the kolmogorov complexity we provide a unified definition of structure and simplicity leveraging this new definition we develop and analyze an abstract algorithm for signal recovery motivated by occam s razor minimum complexity pursuit mcp requires approximately 2 randomized samples to recover a signal of complexity and ambient dimension n we also discuss the performance of the mcp in the presence of measurement noise and with approximately simple signals

sofic mean length abstract given a length function l on the r modules of a unital ring r for each sofic group we define a mean length for every locally l finite r module relative to a bigger r module we establish an addition formula for the mean length we give two applications the first one shows that for any unital left noetherian ring r r is stably direct finite the second one shows that for any z module m the mean topological dimension of the induced action on the pontryagin dual of m coincides with the von neumann luck rank of m

weakly integrable semigroups on locally convex spaces abstract a class of weakly integrable semigroups on locally convex spaces is introduced and studied the results are illustrated by examples of semigroups of unbounded operators on a banach space which include fractional powers of a closed operator and spectral local semigroups

stationary parabolic anderson model and intermittency this paper is devoted to the analysis of the large time behavior of the solutions of the anderson parabolic problem frac partial u partial t kappa delta u xi x u when the potential x is a homogeneous ergodic random field on d our goal is to prove the asymptotic spatial intermittency of the solution and for this reason we analyze the large time properties of all the moments of the positive solutions this provides an extension to the continuous space d of the work done originally by gartner and molchanov in the case of the lattice d in the process of our moment analysis we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized gaussian fields interpolating continuously between the exponent 2 which is found in the case of bona fide continuous gaussian fields x and the exponent 3 2 appearing in the case of a one dimensional white noise finally we also determine the precise almost sure large time asymptotics of the positive solutions

the rate distortion dimension of sets and measures data compression of independent samples drawn from a fractal set is considered the asymptotic ratio of rate to magnitude log distortion characterizes the effective dimension occupied by the underlying distribution this quantity is shown to be identical to renyi s 1959 information dimension for self similar fractal sets this dimension is distribution dependent in sharp contrast with the behavior of absolutely continuous measures the rate distortion dimension of a set is defined as the maximal rate distortion dimension for distributions supported on this set kolmogorov s metric dimension is an upper bound on the rate distortion dimension while the hausdorff dimension is a lower bound examples of sets for which the rate distortion dimension differs from these bounds are provided