expanded gdof optimality regime of treating interference as noise in the m times 2 x channel treating interference as noise tin as the most appropriate approach in dealing with interference and the conditions on its optimality has attracted the interest of researchers recently however our knowledge on necessary and sufficient conditions of tin is restricted to a few setups with limited number of users in this paper we study the optimality of tin in terms of the generalized degrees of freedom gdof for a fundamental network namely the m times 2 x channel to this end the achievable gdof of tin with power allocations at the transmitters is studied it turns out that the transmit power allocation maximizing the achievable gdof is given by on off signaling as long as the receivers use tin this leads to two variants of tin namely p2p tin and 2 ic tin while in the first variant the m times 2 x channel is reduced to a point to point p2p channel in the second variant the setup is reduced to a two user interference channel in which the receivers use tin the optimality of these two variants is studied separately to this end novel genie aided upper bounds on the capacity of the x channel are established the conditions on the optimality of p2p tin can be summarized as follows p2p tin is gdof optimal if there exists a dominant multiple access channel or a dominant broadcast channel embedded in the x channel furthermore the necessary and sufficient conditions on the gdof optimality of 2 ic tin are presented interestingly it turns out that operating the m times 2 x channel in the 2 ic tin mode might be still gdof optimal although the conditions given by geng et al are violated however 2 ic tin is sub optimal if there exists a single interferer which causes sufficiently strong interference at both receivers the comparison of the results with the state of the art shows that the gdof optimality of tin is expanded significantldy

information geometry on hierarchy of probability distributions an exponential family or mixture family of probability distributions has a natural hierarchical structure this paper gives an orthogonal decomposition of such a system based on information geometry a typical example is the decomposition of stochastic dependency among a number of random variables in general they have a complex structure of dependencies pairwise dependency is easily represented by correlation but it is more difficult to measure effects of pure triplewise or higher order interactions dependencies among these variables stochastic dependency is decomposed quantitatively into an orthogonal sum of pairwise triplewise and further higher order dependencies this gives a new invariant decomposition of joint entropy this problem is important for extracting intrinsic interactions in firing patterns of an ensemble of neurons and for estimating its functional connections the orthogonal decomposition is given in a wide class of hierarchical structures including both exponential and mixture families as an example we decompose the dependency in a higher order markov chain into a sum of those in various lower order markov chains

kac moody monopoles and periodic instantons it is shown that calorons or periodic instantons are the same as monopoles with the loop group as their structure group their twistor correspondences and spectral data are defined the spectral data is shown to determine the general caloron

learning high dimensional markov forest distributions analysis of error rates the problem of learning forest structured discrete graphical models from i i d samples is considered an algorithm based on pruning of the chow liu tree through adaptive thresholding is proposed it is shown that this algorithm is both structurally consistent and risk consistent and the error probability of structure learning decays faster than any polynomial in the number of samples under fixed model size for the high dimensional scenario where the size of the model d and the number of edges k scale with the number of samples n sufficient conditions on n d k are given for the algorithm to satisfy structural and risk consistencies in addition the extremal structures for learning are identified we prove that the independent resp tree model is the hardest resp easiest to learn using the proposed algorithm in terms of error rates for structure learning

principal inertia components and applications we explore properties and applications of the principal inertia components pics between two discrete random variables x and y the pics lie in the intersection of information and estimation theory and provide a fine grained decomposition of the dependence between x and y moreover the pics describe which functions of x can or cannot be reliably inferred in terms of mmse given an observation of y we demonstrate that the pics play an important role in information theory and they can be used to characterize information theoretic limits of certain estimation problems in privacy settings we prove that the pics are related to the fundamental limits of perfect privacy

renyi resolvability and its applications to the wiretap channel the conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy in contrast to previous works in this paper we use the normalized or unnormalized renyi divergence with the renyi parameter in 0 2 cup infty to measure the level of approximation we also provide asymptotic expressions for normalized renyi divergence when the renyi parameter is larger than or equal to 1 as well as lower and upper bounds for the case when the same parameter is smaller than 1 we characterize the renyi resolvability which is defined as the minimum rate required to ensure that the renyi divergence vanishes asymptotically the renyi resolvabilities are the same for both the normalized and unnormalized divergence cases in addition when the renyi parameter smaller than 1 consistent with the traditional case where the renyi parameter is equal to 1 the renyi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution when the renyi parameter is larger than 1 the renyi resolvability is in general larger than the mutual information the optimal renyi divergence is proven to vanish at least exponentially fast for both of these two cases as long as the code rate is larger than the renyi resolvability the optimal exponential rate of decay for i i d random codes is also characterized exactly we apply these results to the wiretap channel and completely characterize the optimal tradeoff between the rates of the secret and non secret messages when the leakage measure is given by the unnormalized renyi divergence this tradeoff differs from the conventional setting when the leakage is measured by the traditional mutual information

high dimensional structure estimation in ising models local separation criterion we consider the problem of high dimensional ising graphical model selection we propose a simple algorithm for structure estimation based on the thresholding of the empirical conditional variation distances we introduce a novel criterion for tractable graph families where this method is efficient based on the presence of sparse local separators between node pairs in the underlying graph for such graphs the proposed algorithm has a sample complexity of n omega j min 2 log p where p is the number of variables and j min is the minimum absolute edge potential in the model we also establish nonasymptotic necessary and sufficient conditions for structure estimation

sklyanin algebras and hilbert schemes of points we construct projective moduli spaces for torsion free sheaves on noncommutative projective planes these moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual commutative projective plane p2 the generic noncommutative plane corresponds to the sklyanin algebra s skl e constructed from an automorphism of infinite order on an elliptic curve e p2 in this case the fine moduli space of line bundles over s with first chern class zero and euler characteristic 1 n provides a symplectic variety that is a deformation of the hilbert scheme of n points on p2 e

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

universal and composite hypothesis testing via mismatched divergence for the universal hypothesis testing problem where the goal is to decide between the known null hypothesis distribution and some other unknown distribution hoeffding proposed a universal test in the nineteen sixties hoeffding s universal test statistic can be written in terms of kullback leibler k l divergence between the empirical distribution of the observations and the null hypothesis distribution in this paper a modification of hoeffding s test is considered based on a relaxation of the k l divergence referred to as the mismatched divergence the resulting mismatched test is shown to be a generalized likelihood ratio test glrt for the case where the alternate distribution lies in a parametric family of distributions characterized by a finite dimensional parameter i e it is a solution to the corresponding composite hypothesis testing problem for certain choices of the alternate distribution it is shown that both the hoeffding test and the mismatched test have the same asymptotic performance in terms of error exponents a consequence of this result is that the glrt is optimal in differentiating a particular distribution from others in an exponential family it is also shown that the mismatched test has a significant advantage over the hoeffding test in terms of finite sample size performance for applications involving large alphabet distributions this advantage is due to the difference in the asymptotic variances of the two test statistics under the null hypothesis

forest density estimation we study graph estimation and density estimation in high dimensions using a family of density estimators based on forest structured undirected graphical models for density estimation we do not assume the true distribution corresponds to a forest rather we form kernel density estimates of the bivariate and univariate marginals and apply kruskal s algorithm to estimate the optimal forest on held out data we prove an oracle inequality on the excess risk of the resulting estimator relative to the risk of the best forest for graph estimation we consider the problem of estimating forests with restricted tree sizes we prove that finding a maximum weight spanning forest with restricted tree size is np hard and develop an approximation algorithm for this problem viewing the tree size as a complexity parameter we then select a forest using data splitting and prove bounds on excess risk and structure selection consistency of the procedure experiments with simulated data and microarray data indicate that the methods are a practical alternative to gaussian graphical models

non realizable cr structures the examples given by nirenberg in 5 and 6 show that not all three dimensional cr manifolds may be realized as real hypersurfaces in 1172 these examples are perturbations of the cr structure on the sphere in this paper we extend his construction and show that any strictly pseudo convex realizable cr structure of dimension 3 may be perturbed so as to obtain a non realizable one this is done in sect 1 in sect 2 we show non realizability for higher dimensional structures provided the levi form is non degenerate and has only one positive eigenvalue in our last section we summarize cartan s approach to cr geometry from this viewpoint it was important to decide if nirenberg s example was really limited to perturbations of the sphere we have made this paper self contained and have provided proofs of several known results for instance lemma 1 l and the first part of lemma 1 5 were proved by lewy 2 and lemma 1 2 is the well known observation that a strictly pseudo convex hypersurface is convex in some coordinate system

connected graded gorenstein algebras with enough normal elements we generalize 12 1 1 and 1 2 to the following situation theorem 1 let a be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of a contains a homogeneous normal element of positive degree then 1 a is auslander gorenstein and cohen macaulay 2 a has a quasi frobenius quotient ring 3 every minimal prime ideal p is graded andgkdim a p d 4 if moreover a has finite global dimension then a is a domain and a maximal order in its quotient division ring to prove the above we need the following result which is a generalization of 3 2 46 ii theorem 2 let a be a connected graded noetherian as gorenstein algebra of injective dimension d then 1 the last term of the minimal injective resolution of aais isomorphic to a shift of a 2 for every noetherian graded a module m extd m a is finite dimensional over k

remarks on functional canonical variates alternating least squares methods and ace we discuss properties of some data analytic methods which are intimately related to each other alternating least squares als correspondence analysis and more recently breiman and friedman s ace algorithm the application of these methods to regression produces nonparametric estimators of nonlinear transformations both of the response and the predictors we point out some anomalies as well as some curiosities in the mathematics of these methods and we relate them to some areas in computer aided tomography projection pursuit regression and nonlinear devices in the theory of noise

on nichols algebras over basic hopf algebras this is a contribution to the classification of finite dimensional hopf algebras over an algebraically closed field mathbb k of characteristic 0 concretely we show that a finite dimensional hopf algebra whose hopf coradical is basic is a lifting of a nichols algebra of a semisimple yetter drinfeld module and we explain how to classify nichols algebras of this kind we provide along the way new examples of nichols algebras and hopf algebras with finite gelfand kirillov dimension

high dimensional gaussian graphical model selection walk summability and local separation criterion we consider the problem of high dimensional gaussian graphical model selection we identify a set of graphs for which an efficient estimation algorithm exists and this algorithm is based on thresholding of empirical conditional covariances under a set of transparent conditions we establish structural consistency or sparsistency for the proposed algorithm when the number of samples n jmin 2 log p where p is the number of variables and jmin is the minimum absolute edge potential of the graphical model the sufficient conditions for sparsistency are based on the notion of walk summability of the model and the presence of sparse local vertex separators in the underlying graph we also derive novel non asymptotic necessary conditions on the number of samples required for sparsistency

eigenvalues of fourth order singular sturm liouville boundary value problems abstract in this paper by establishing a new comparison theorem and constructing upper and lower solutions some sufficient conditions of existence of positive solutions for the following nonlinear fourth order singular sturm liouville eigenvalue problem 1 p t p t u t f t u t 0 1 u 0 u 1 0 u 0 lim t 0 p t u t 0 u 1 lim t 1 p t u t 0 are established due to the schauder s fixed point theorem for large enough where 0 0 f and p can be singular at t 0 and or 1 moreover f can also be singular at u 0 in addition some peculiar cases are discussed and some further results are obtained

central extensions of current groups in two dimensions the paper is devoted to generalization of the theory of loop groups to the two dimensional case to every complex riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex lie group g by the jacobian of this surface this extension is topologically nontrivial as in the loop group case orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalg bundles over the surface when the surface is the torus elliptic curve classification of coadjoint orbits is related to linear difference equations in one variable and to classification of conjugacy classes in the loop group we study integral orbits for which the kirillov kostant form is a curvature form for some principal torus bundle the number of such orbits for a given level is finite as in the loop group case conjecturedly they correspond to analogues of integrable modules occurring in conformal field theory

a coordinate system for gaussian networks this paper investigates network information theory problems where the external noise is gaussian distributed in particular the gaussian broadcast channel with coherent fading and the gaussian interference channel are considered it is shown that in these problems non gaussian code ensembles can achieve higher rates than the gaussian ones it is also shown that the strong shamai laroia conjecture on the gaussian isi channel does not hold in order to analyze non gaussian code ensembles over gaussian networks a geometrical tool using the hermite polynomials is proposed this tool provides a coordinate system to analyze a class of non gaussian input distributions that are invariant over gaussian networks

simulation based computation of information rates for channels with memory the information rate of finite state source channel models can be accurately estimated by sampling both a long channel input sequence and the corresponding channel output sequence followed by a forward sum product recursion on the joint source channel trellis this method is extended to compute upper and lower bounds on the information rate of very general channels with memory by means of finite state approximations further upper and lower bounds can be computed by reduced state methods

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

the efficiency of investment information we investigate how the description of a correlated information v improves the investment in the stock market x the objective is to maximize the growth rate of wealth in repeated investments we find a single letter characterization of the incremental growth rate spl delta r the maximum increase in growth rate when v is described to the investor at rate r the incremental growth rate specialized to the horse race market is related to source coding with side information of wyner and ahlswede kormer we provide two horse race examples jointly binary and jointly gaussian the initial efficiency spl delta 0 is the maximum possible increase in the growth rate per bit of description we show that the initial efficiency is related to the dependency between v and the market in particular for the horse race market the initial efficiency is the square of the hirschfeld gebelein renyi maximal correlation between v and x this provides a connection with the hypercontraction of the markov operator of ahlswede and gacs for the general market the initial efficiency is 1 when the side information v is equal to the stock market outcome x

feedback capacity of the gaussian interference channel to within 2 bits we characterize the capacity region to within 2 bits s hz and the symmetric capacity to within 1 bit s hz for the two user gaussian interference channel ic with feedback we develop achievable schemes and derive a new outer bound to arrive at this conclusion one consequence of the result is that feedback provides multiplicative gain at high signal to noise ratio the gain becomes arbitrarily large for certain channel parameters this finding is in contrast to point to point and multiple access channels where feedback provides no gain and only bounded additive gain respectively the result makes use of a linear deterministic model to provide insights into the gaussian channel this deterministic model is a special case of the el gamal costa deterministic model and as a side generalization we establish the exact feedback capacity region of this general class of deterministic ics

automorphisms and ideals of the weyl algebra let a 1 be the first weyl algebra and let g be its automorphism group we study the natural action of g on the space of isomorphism classes of right ideals of a 1 equivalently of finitely generated rank 1 torsion free right a 1 modules we show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety our results are strikingly similar to those for the commutative algebra of polynomials in two variables however we do not know of any general principle that would allow us to predict this in advance as a key step in the proof we obtain a new description of the bispectral involution of w1 we also make some comments on the group g from the viewpoint of shafarevich s theory of infinite dimensional algebraic groups

interference as noise friend or foe this paper shows that for the two user gaussian interference channel g ic treating interference as noise without time sharing tinnots achieves the closure of the capacity region to within either a constant gap or to within a gap of the order o log ln min mathsf s mathsf i gamma up to a set of lebesgue measure gamma in 0 1 where mathsf s is the largest signal to noise ratio on the direct links and mathsf i is the largest interference to noise ratio on the cross links as a consequence tinnots is optimal from a generalized degrees of freedom gdof perspective for all channel gains except for a subset of zero measure tinnots with gaussian inputs is known to be optimal within 1 2 bit for a subset of the weak interference regime rather surprisingly this paper shows that tinnots is gdof optimal in all parameter regimes even in the strong and very strong interference regimes where joint decoding of gaussian inputs is optimal for approximate optimality of tinnots in all parameter regimes it is critical to use non gaussian inputs this paper thus proposes to use mixed inputs as channel inputs for the g ic where a mixed input is the sum of a discrete and a gaussian random variable interestingly with reference to the han kobayashi achievable scheme the discrete part of a mixed input is shown to effectively behave as a common message in the sense that although treated as noise its effect on the achievable rate region is as if it were jointly decoded together with the desired messages at a non intended receiver the practical implication is that a discrete interfering input is a friend while an gaussian interfering input is in general a foe this paper also discusses other practical implications of the proposed tinnots scheme with mixed inputs since tinnots requires neither explicit joint decoding nor time sharing the results of this paper are applicable to a variety of oblivious or asynchronous channels such as the block asynchronous g ic which is not an information stable channel and the g ic with partial codebook knowledge at one or more receivers

robust estimation of latent tree graphical models inferring hidden states with inexact parameters latent tree graphical models are widely used in computational biology signal and image processing and network tomography here we design a new efficient estimation procedure for latent tree models including gaussian and discrete reversible models that significantly improves on previous sample requirement bounds our techniques are based on a new hidden state estimator that is robust to inaccuracies in estimated parameters more precisely we prove that latent tree models can be estimated with high probability in the so called kesten stigum regime with o log2n samples where n is the number of nodes

generalized entropy power inequalities and monotonicity properties of information new families of fisher information and entropy power inequalities for sums of independent random variables are presented these inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables for an arbitrary collection of subsets as a consequence a simple proof of the monotonicity of information in central limit theorems is obtained both in the setting of independent and identically distributed i i d summands as well as in the more general setting of independent summands with variance standardized sums

on incidence between strata of the hilbert scheme of points on p 2 the hilbert scheme of n points in the projective plane has a natural stratification obtained from the associated hilbert series in general the precise inclusion relation between the closures of the strata is still unknown guerimand ph d thesis universite de nice 2002 studied this problem for strata whose hilbert series are as close as possible preimposing a certain technical condition he obtained necessary and sufficient conditions for the incidence of such strata in this paper we present a new approach based on deformation theory to guerimand s result this allows us to show that the technical condition is not necessary

on communication through a gaussian channel with an mmse disturbance constraint this paper considers a gaussian channel with one transmitter and two receivers the goal is to maximize the communication rate at the intended primary receiver subject to a disturbance constraint at the unintended secondary receiver the disturbance is measured in terms of the minimum mean square error mmse of the interference that the transmission to the primary receiver inflicts on the secondary receiver this paper presents a new upper bound for the problem of maximizing the mutual information subject to an mmse constraint the new bound holds for vector inputs of any length and recovers a previously known limiting when the length of the vector input tends to infinity expression from the work of bustin et al the key technical novelty is a new upper bound on the mmse this bound allows one to bound the mmse for all signal to noise ratio snr values below a certain snr at which the mmse is known which corresponds to the disturbance constraint the bound also complements the single crossing point property of the mmse that upper bounds the mmse for all snr values above a certain value at which the mmse value is known the mmse upper bound provides a refined characterization of the phase transition phenomenon which manifests in the limit as the length of the vector input goes to infinity as a discontinuity of the mmse for the problem at hand for vector inputs of size n 1 a matching lower bound to within an additive gap of order o log log 1 sf mmse where sf mmse is the disturbance constraint is shown by means of the mixed inputs technique recently introduced by dytso et al

efficient calculation of spectral density functions for specific classes of singular sturm liouville problems new families of approximations to sturm liouville spectral density functions are derived for cases where the potential function has one of several specific forms this particular form dictates the type of expansion functions used in the approximation error bounds for the residuals are established for each case in the case of power potentials the approximate solutions of an associated terminal value problem at are shown to be asymptotic power series expansions of the exact solution numerical algorithms have been implemented and several examples are given demonstrating the utility of the approach

the intersymbol interference channel lower bounds on capacity and channel precoding loss the discrete time additive gaussian intersymbol interference isi channel with i i d not necessarily gaussian input signals is considered several new and old lower bounds on the capacity are derived in a unified manner by assuming different front end receiver filters in particular the sampled whitened matched filter swmf and the minimum mean squared error decision feedback equalizer mmse dfe filter the features of the bounds are demonstrated and compared in several examples with binary and quaternary input signals it is also shown that the effect of an ideal post cursor or tail cancellation in an information preserving context depends primarily on the front end filter while as is well known ideal post cursor cancellation at the output of the swmf decreases the information the opposite trend is seen when an mmse dfe front filter is considered this observation reflects the basic theoretical obstacles in precoding i e ideal post cursor cancellation in the presence of a pre cursor it is used to assess the inherent loss in terms of information rates as compared to the rates achievable with the hypothetical ideal post cursor cancellation associated with any post cursor cancellation technique such as precoding dfe or other variants when operating in synergy with the mmse dfe front end filter the effect of the front end filter on an ideally interleaved precoded coded modulation system is also addressed

some noncommutative minimal surfaces abstract in the ongoing programme to classify noncommutative projective surfaces connected graded noetherian domains of gelfand kirillov dimension three a natural question is to determine the minimal models within any birational class in this paper we show that the generic noncommutative projective plane corresponding to the three dimensional sklyanin algebra as well as noncommutative analogues of p 1 p 1 and the more general van den bergh quadrics satisfy very strong minimality conditions translated into an algebraic question where one is interested in a maximality condition we prove the following result theorem a let r be a sklyanin algebra or a van den bergh quadric that is infinite dimensional over its centre and let a r be any connected graded noetherian maximal order with the same graded quotient ring as r then up to taking veronese rings a is isomorphic to r let t be an elliptic algebra that is the coordinate ring of a noncommutative surface containing an elliptic curve then under an appropriate homological condition we prove that every connected graded noetherian overring of t is obtained by blowing down finitely many lines line modules of self intersection 1

residue complex for sklyanin algebras of dimension three we describe the residue complex for three dimensional sklyanin algebras which are the interesting special cases of quantum polynomial rings in three variables in particular we show that the multiplicities of the point modules in the residue complex are all one just as in the classical case of commutative polynomial rings in three variables we explain why the residue complex in the quantum case has the same multiplicities of point modules as that in the commutative case even if there are fewer point modules in the quantum case by pointing out two quantum anomalies

weak convergence analysis of asymptotically optimal hypothesis tests in recent years solutions to various hypothesis testing problems in the asymptotic setting have been proposed using the results from large deviation theory such tests are optimal in terms of appropriately defined error exponents for the practitioner however error probabilities in the finite sample size setting are more important in this paper we show how results on weak convergence of the test statistic can be used to obtain better approximations for the error probabilities in the finite sample size setting while this technique is popular among statisticians for common tests we demonstrate its applicability for several recently proposed asymptotically optimal tests including tests for robust goodness of fit homogeneity tests outlier hypothesis testing and graphical model estimation

rate distortion for lossy in network linear function computation and consensus distortion accumulation and sequential reverse water filling we consider the problem of distributed lossy linear function computation in a tree network we examine two cases 1 data aggregation only one sink node computes and 2 consensus all nodes compute the same function by quantifying the accumulation of information loss in distributed computing we obtain fundamental limits on network computation rate as a function of incremental distortions and hence incremental loss of information along the edges of the network the above characterization based on quantifying distortion accumulation offers an improvement over classical cut set type techniques which are based on overall distortions instead of incremental distortions this quantification of information loss qualitatively resembles information dissipation in cascaded channels 2 surprisingly this accumulation effect of distortion happens even at infinite blocklength combining this observation with an inequality on the dominance of mean square quantities over relative entropy quantities we obtain outer bounds on the rate distortion function that are tighter than classical cut set bounds by a difference which can be arbitrarily large in both data aggregation and consensus we also obtain inner bounds on the optimal rate using random gaussian coding which differ from the outer bounds by mathcal o sqrt d where d is the overall distortion the obtained inner and outer bounds can provide insights on rate bit allocations for both the data aggregation problem and the consensus problem we show that for tree networks the rate allocation results have a mathematical structure similar to classical reverse water filling for parallel gaussian sources apart from data aggregation and distributed consensus the distortion accumulation analysis framework is also applicable in large scale data summarization through histograms and linear sketching e g word counting tasks for document summarization

a large deviation analysis of the maximum likelihood learning of markov tree structures the problem of maximum likelihood ml estimation of discrete tree structured distributions is considered chow and liu established that ml estimation reduces to the construction of a maximum weight spanning tree using the empirical mutual information quantities as the edge weights using the theory of large deviations we analyze the exponent associated with the error probability of the event that the ml estimate of the markov tree structure differs from the true tree structure given a set of independently drawn samples by exploiting the fact that the output of ml estimation is a tree we establish that the error exponent is equal to the exponential rate of decay of a single dominant crossover event we prove that in this dominant crossover event a non neighbor node pair replaces a true edge of the distribution that is along the path of edges in the true tree graph connecting the nodes in the non neighbor pair using ideas from euclidean information theory we then analyze the scenario of ml estimation in the very noisy learning regime and show that the error exponent can be approximated as a ratio which is interpreted as the signal to noise ratio snr for learning tree distributions we show via numerical experiments that in this regime our snr approximation is accurate

computing the spectral function for singular sturm liouville problems algorithms for computing sturm liouville spectral density functions are developed based on several mathematical characterizations convergence and error bounds are derived and methods are tested on several examples the results are compared with those from the existing software package sledge

learning gaussian tree models analysis of error exponents and extremal structures the problem of learning tree structured gaussian graphical models from independent and identically distributed i i d samples is considered the influence of the tree structure and the parameters of the gaussian distribution on the learning rate as the number of samples increases is discussed specifically the error exponent corresponding to the event that the estimated tree structure differs from the actual unknown tree structure of the distribution is analyzed finding the error exponent reduces to a least squares problem in the very noisy learning regime in this regime it is shown that the extremal tree structure that minimizes the error exponent is the star for any fixed set of correlation coefficients on the edges of the tree if the magnitudes of all the correlation coefficients are less than 0 63 it is also shown that the tree structure that maximizes the error exponent is the markov chain in other words the star and the chain graphs represent the hardest and the easiest structures to learn in the class of tree structured gaussian graphical models this result can also be intuitively explained by correlation decay pairs of nodes which are far apart in terms of graph distance are unlikely to be mistaken as edges by the maximum likelihood estimator in the asymptotic regime

a spectral transform method for singular sturm liouville problems with applications to energy diffusion in plasma physics we develop a spectrally accurate numerical method to compute solutions of a model pde used in plasma physics to describe diffusion in velocity space due to fokker planck collisions the solution is represented as a discrete and continuous superposition of normalizable and nonnormalizable eigenfunctions via the spectral transform associated with a singular sturm liouville operator we present a new algorithm for computing the spectral density function of the operator that uses chebyshev polynomials to extrapolate the value of the titchmarsh weyl m function from the complex upper half plane to the real axis the eigenfunctions and density function are rescaled and a new formula for the limiting value of the m function is derived to avoid amplification of roundoff errors when the solution is reconstructed the complexity of the algorithm is also analyzed showing that the cost of computing the spectral density function at a point grows less rapidly than any fractional inverse power of the desired acc

polynomial singular value decompositions of a family of source channel models in this paper we show that the conditional expectation operators corresponding to a family of source channel models defined by natural exponential families with quadratic variance functions and their conjugate priors have orthonormal polynomials as singular vectors these models include the gaussian channel with gaussian source the poisson channel with gamma source and the binomial channel with beta source to derive the singular vectors of these models we prove and employ the equivalent condition that their conditional moments are strictly degree preserving polynomials

learning latent tree graphical models we study the problem of learning a latent tree graphical model where samples are available only from a subset of variables we propose two consistent and computationally efficient algorithms for learning minimal latent trees that is trees without any redundant hidden nodes unlike many existing methods the observed nodes or variables are not constrained to be leaf nodes our algorithms can be applied to both discrete and gaussian random variables and our learned models are such that all the observed and latent variables have the same domain state space our first algorithm recursive grouping builds the latent tree recursively by identifying sibling groups using so called information distances one of the main contributions of this work is our second algorithm which we refer to as clgrouping clgrouping starts with a pre processing procedure in which a tree over the observed variables is constructed this global step groups the observed nodes that are likely to be close to each other in the true latent tree thereby guiding subsequent recursive grouping or equivalent procedures such as neighbor joining on much smaller subsets of variables this results in more accurate and efficient learning of latent trees we also present regularized versions of our algorithms that learn latent tree approximations of arbitrary distributions we compare the proposed algorithms to other methods by performing extensive numerical experiments on various latent tree graphical models such as hidden markov models and star graphs in addition we demonstrate the applicability of our methods on real world data sets by modeling the dependency structure of monthly stock returns in the s p index and of the words in the 20 newsgroups data set

shifted poisson structures and moduli spaces of complexes abstract in this paper we study the moduli stack of complexes of vector bundles with chain isomorphisms over a smooth projective variety x via derived algebraic geometry we prove that if x is a calabi yau variety of dimension d then this moduli stack has a 1 d shifted poisson structure in the case d 1 we construct a natural foliation of the moduli stack by 0 shifted symplectic substacks we show that our construction recovers various known poisson structures associated to complex elliptic curves including the poisson structure on hilbert scheme of points on elliptic quantum projective planes studied by nevins and stafford and the poisson structures on the moduli spaces of stable triples over an elliptic curves considered by one of us we also relate the latter poisson structures to the semi classical limits of the elliptic sklyanin algebras studied by feigin and odesskii

wyner s common information under renyi divergence measures we study a generalized version of wyner s common information problem also coined the distributed source simulation problem the original common information problem consists in understanding the minimum rate of the common input to independent processors to generate an approximation of a joint distribution when the distance measure used to quantify the discrepancy between the synthesized and target distributions is the normalized relative entropy our generalization involves changing the distance measure to the unnormalized and normalized renyi divergences of order alpha 1 s in 0 2 we show that the minimum rate needed to ensure the renyi divergences between the distribution induced by a code and the target distribution vanishes remains the same as the one in wyner s setting except when the order alpha 1 s 0 this implies that wyner s common information is rather robust to the choice of distance measure employed as a byproduct of the proofs used to the establish the above results the exponential strong converse for the common information problem under the total variation distance measure is established

ideal classes of three dimensional artin schelter regular algebras abstract we determine the possible hilbert functions of graded rank one torsion free modules over three dimensional artin schelter regular algebras it turns out that as in the commutative case they are related to castelnuovo functions from this we obtain an intrinsic proof that the space of torsion free rank one modules on a non commutative p 2 is connected a different proof of this fact based on deformation theoretic methods and the known commutative case has recently been given by nevins and stafford sklyanin algebras and hilbert schemes of points math ag 0310045 for the weyl algebra it was proved by wilson invent math 133 1 1998 1 41

euclidean information theory of networks in this paper we extend the information theoretic framework that was developed in earlier works to multi hop network settings for a given network we construct a novel deterministic model that quantifies the ability of the network in transmitting private and common messages across users based on this model we formulate a linear optimization problem that explores the throughput of a multi layer network thereby offering the optimal strategy as to what kind of common messages should be generated in the network to maximize the throughput with this deterministic model we also investigate the role of feedback for multi layer networks from which we identify a variety of scenarios in which feedback can improve transmission efficiency our results provide fundamental guidelines as to how to coordinate cooperation between users to enable efficient information exchanges across them

generic noncommutative surfaces abstract we study a class of noncommutative surfaces and their higher dimensional analogs which come from generic subalgebras of twisted homogeneous coordinate rings of projective space such rings provide answers to several open questions in noncommutative projective geometry specifically these rings r are the first known graded algebras over a field k which are noetherian but not strongly noetherian in other words r k b is not noetherian for some choice of commutative noetherian extension ring b this answers a question of artin small and zhang the rings r are also maximal orders but they do not satisfy all of the conditions of artin and zhang in particular they satisfy the 1 condition but not i for i 2 answering a question of stafford and zhang and a question of stafford and van den bergh finally we show that the noncommutative scheme r proj has finite global dimension

solution of shannon s problem on the monotonicity of entropy it is shown that if x1 x2 are independent and identically distributed square integrable random variables then the entropy of the normalized sum ent x1 xn over n is an increasing function of n this resolves an old problem which goes back to 6 7 5 the result also has a version for non identically distributed random variables or random vectors

new characterizations of spectral density functions for singular sturm liouville problems a generalization is given for a characterization of the spectral density function of weyl and titchmarsh for a singular sturm liouville problem having absolutely continuous spectrum in 0 a recurrent formulation is derived that generates a family of approximations based on this scheme proofs of convergence for these new approximations are supplied and a numerical method is implemented the computational results show more rapid rates of convergence which are in accord with the theoretical rates

homological properties of sklyanin algebras to a pair consisting of an elliptic curve and a point on it odeskii and feigin associate certain quadratic algebras sklyanin algebras having the hilbert series of a polynomial algebra in this paper we show that sklyanin algebras have good homological properties and we obtain some information about their so called linear modules we also show how the construction by odeskii and feigin may be generalized so as to yield other sklyanin type algebras

phase transition in the sample complexity of likelihood based phylogeny inference reconstructing evolutionary trees from molecular sequence data is a fundamental problem in computational biology stochastic models of sequence evolution are closely related to spin systems that have been extensively studied in statistical physics and that connection has led to important insights on the theoretical properties of phylogenetic reconstruction algorithms as well as the development of new inference methods here we study maximum likelihood a classical statistical technique which is perhaps the most widely used in phylogenetic practice because of its superior empirical accuracy at the theoretical level except for its consistency that is the guarantee of eventual correct reconstruction as the size of the input data grows much remains to be understood about the statistical properties of maximum likelihood in this context in particular the best bounds on the sample complexity or sequence length requirement of maximum likelihood that is the amount of data required for correct reconstruction are exponential in the number n of tips far from known lower bounds based on information theoretic arguments here we close the gap by proving a new upper bound on the sequence length requirement of maximum likelihood that matches up to constants the known lower bound for some standard models of evolution more specifically for the r state symmetric model of sequence evolution on a binary phylogeny with bounded edge lengths we show that the sequence length requirement behaves logarithmically in n when the expected amount of mutation per edge is below what is known as the kesten stigum threshold in general the sequence length requirement is polynomial in n our results imply moreover that the maximum likelihood estimator can be computed efficiently on randomly generated data provided sequences are as above our main technical contribution which may be of independent interest relates the total variation distance between the leaf state distributions of two trees with a notion of combinatorial distance between the trees in words we show in a precise quantitative manner that the more different two evolutionary trees are the easier it is to distinguish their output

stability for an abelian category abstract the main goal of the article is to give the definition of algebraic stability that would permit us to consider stability not only for algebraic vector bundles or torsion free coherent sheaves but for the abelian category of coherent sheaves or for whatever abelian category we present an axiomatic description of the algebraic stability on an abelian category and prove some general results then the stability for coherent sheaves on a projective variety is constructed which generalizes gieseker stability stabilities for graded modules and for quiver representations are also discussed the constructions could be used for other abelian categories as well

noncommutative graded domains with quadratic growth letk be an algebraically closed field and letr be a finitely generated connected gradedk algebra which is a domain of gelfand kirillov dimension two write the graded quotient ringq r ofr asd z z 1 for some automorphism of the division ringd we prove thatd is a finitely generated field extension ofk of transcendence degree one moreover we describer in terms of geometric data ifr is generated in degree one then up to a finite dimensional vector space r is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curvey this implies in particular thatr is noetherian thatr is primitive when and thatr is a finite module over its centre when ifr is not generated in degree one thenr will still be noetherian and primitive if has infinite order butr need not be noetherian when has finite order

on the two user interference channel with lack of knowledge of the interference codebook at one receiver in multiuser information theory it is often assumed that every node in the network possesses all codebooks used in the network this assumption may be impractical in distributed ad hoc cognitive or heterogeneous networks this paper considers the two user interference channel with one oblivious receiver ic or i e one receiver lacks knowledge of the interfering cookbook whereas the other receiver knows both codebooks this paper asks whether and if so how much the channel capacity of the ic or is reduced compared with that of the classical ic where both receivers know all codebooks a novel outer bound is derived and shown to be achievable to within a gap for the class of injective semideterministic ic ors the gap is shown to be zero for injective fully deterministic ic ors an exact capacity result is shown for the general memoryless ic or when the nonoblivious receiver experiences very strong interference for the linear deterministic ic or that models the gaussian noise channel at high snr nonindependent identically distributed bernoulli 1 2 input bits are shown to achieve points not achievable by i i d bernoulli 1 2 input bits used in the same achievability scheme for the real valued gaussian ic or the gap is shown to be at most 1 2 bit per channel use even though the set of optimal input distributions for the derived outer bound could not be determined toward understanding the gaussian ic or an achievability strategy is evaluated in which the input alphabets at the nonoblivious transmitter are a mixture of discrete and gaussian random variables where the cardinality of the discrete part is appropriately chosen as a function of the channel parameters surprisingly as the oblivious receiver intuitively should not be able to jointly decode the intended and interfering messages whose codebook is unavailable it is shown that with this choice of input the capacity region of the symmetric gaussian ic or is to within 1 2 log 12 e 3 34 bits per channel use per user of an outer bound for the classical gaussian ic with full codebook knowledge at both receivers

on universal hypotheses testing via large deviations a prototype problem in hypotheses testing is discussed the problem of deciding whether an i i d sequence of random variables has originated from a known source p sub 1 or an unknown source p sub 2 is considered the exponential rate of decrease in type ii probability of error under a constraint on the minimal rate of decrease in type i probability of error is chosen for a criterion of optimality using large deviations estimates a decision rule that is based on the relative entropy of the empirical measure with respect to p sub 1 is proposed in the case of discrete random variables this approach yields weaker results than the combinatorial approach used by hoeffding 1965 however it enables the analysis to be extended to the general case of r sup n valued random variables finally the results are extended to the case where p sub 1 is an unknown parameter dependent distribution that is known to belong to a set of distributions p sup 0 sub 1 theta in theta

chernoff information between gaussian trees abstract in this paper we deal with gaussian graphical classifiaction problem we aim to provide a systematic study of the relationship between chernoff information and topological as well as algebraic properties of the corresponding gaussian graphs for the underlying graphical testing problems we first show the relationship between chernoff information and generalized eigenvalues of the associated covariance matrices it is then proved that chernoff information between two gaussian trees sharing certain local subtree structures can be transformed into that of two smaller trees in this way we provide a sequence of equivalent gaussian tree pairs in terms of chernoff information under our proposed grafting operations bottleneck gaussian trees namely gaussian trees connected by one such operation can thus be simplified into two 3 node gaussian trees thereafter we provide a thorough study about how chernoff information changes when small differences are accumulated into bigger ones via concatenated grafting operations as well as partial ordering in the end we propose an optimal linear dimension reduction method based on generalized eigenvalues for the purpose of classification which is proved to achieve maximum chernoff information among all linear transformations

a controlled sensing approach to graph classification the problem of classifying graphs with respect to connectivity via partial observations of nodes is posed as a composite hypothesis testing problem with controlled sensing an observation at a node is a subset of edges incident to the node on the complete graph drawn according to a probability model which is a function of a fixed underlying graph connectivity is measured through average node degree and is classified with respect to a threshold a simple approximation of the controlled sensing test is derived and simulated on erdos renyi graphs to characterize the error probabilities as a function of the expected stopping times the test is also experimentally validated on a real world example of the social structure of long tailed manakins it is shown that the proposed test achieves favorable tradeoffs between the classification error and the number of measurements furthermore the test outperforms existing approaches especially at low target error rates in addition the proposed test achieves the optimal error exponent

a bound on the financial value of information it is shown that each bit of information at most doubles the resulting wealth in the general stock market setup this information bound on the growth of wealth is actually attained for certain probability distributions on the market investigated by j kelly 1956 the bound is shown to be a special case of the result that the increase in exponential growth of wealth achieved with true knowledge of the stock market distribution f over that achieved with incorrect knowledge g is bounded above by the entropy of f relative to g

cohomology of schematic algebras letrbe a positively graded noetheriank algebra and let proj r r grbe the quotient category of the category of gradedr modulesr grmodulo those of finite length ifris schematic cf f van oystaeyen and l willaert j pure appl algebra104 1995 109 122 and has finite global dimension we show that the cohomology groups ofr as defined in m artin and j j zhang adv in math 109 1994 228 287 are finite dimensional furthermore we introduce cech cohomology groups for schematic algebras and prove that they coincide with the usual graded cohomology groups these generalized cech cohomology groups are still manageable we illustrate this with an example

minimax estimation of functionals of discrete distributions we propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters and elaborate on the case of discrete distributions where the support size s is unknown and may be comparable with or even much larger than the number of observations n we treat the respective regions where the functional is nonsmooth and smooth separately in the nonsmooth regime we apply an unbiased estimator for the best polynomial approximation of the functional whereas in the smooth regime we apply a bias corrected version of the maximum likelihood estimator mle we illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures 1 the entropy h p sum i 1 s p i ln p i and 2 f alpha p sum i 1 s p i alpha alpha 0 we obtain the minimax l 2 rates for estimating these functionals in particular we demonstrate that our estimator achieves the optimal sample complexity n asymp s ln s for entropy estimation we also demonstrate that the sample complexity for estimating f alpha p 0 is n asymp s 1 alpha ln s which can be achieved by our estimator but not the mle for 1 we show the minimax l 2 rate for estimating f alpha p is n ln n 2 alpha 1 for infinite support size while the maximum l 2 rate for the mle is n 2 alpha 1 for all the above cases the behavior of the minimax rate optimal estimators with n samples is essentially that of the mle plug in rule with n ln n samples which we term effective sample size enlargement we highlight the practical advantages of our schemes for the estimation of entropy and mutual information we compare our performance with various existing approaches and demonstrate that our approach reduces running time and boosts the accuracy moreover we show that the minimax rate optimal mutual information estimator yielded by our framework leads to significant performance boosts over the chow liu algorithm in learning graphical models the wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable

large deviations of estimators the performance of a sequence of estimators t n of g theta can be measured by its inaccuracy rate lim inf n rightarrow infty n 1 log mathbb p theta t n g theta varepsilon for fixed varepsilon 0 optimality of consistent estimators operatorname wrt the inaccuracy rate is investigated it is shown that for exponential families in standard representation with a convex parameter space the maximum likelihood estimator is optimal if the parameter space is not convex which occurs for instance in curved exponential families in general no optimal estimator exists for the location problem the inaccuracy rate of m estimators is established if the underlying density is sufficiently smooth an optimal m estimator is obtained within the class of translation equivariant estimators tail behaviour of location estimators is studied a connection is made between gross error and inaccuracy rate optimality

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data

extension of an entropy property for binary input memoryless symmetric channels the channel output entropy property introduced by a d wyner and j ziv ibid vol it 19 p 769 762 nov 1973 for a binary symmetric channel is extended to arbitrary memoryless symmetric channels with binary inputs and discrete or continuous outputs this yields lower bounds on the achievable information rates of these channels under constrained binary inputs using the interpretation of entropy as a measure of order and randomness the authors deduce that output sequences of memoryless symmetric channels induced by binary inputs are of a higher degree of randomness if the redundancy of the input binary sequence is spread in memory rather than in one dimensional asymmetry it is of interest to characterize the general class of schemes for which this interpretation holds

ideal classes of three dimensional sklyanin algebras abstract in this paper we classify graded reflexive ideals up to isomorphism and shift in certain three dimensional artin schelter regular algebras this classification is similar to the classification of right ideals in the first weyl algebra a problem that was completely settled recently the situation we consider is substantially more complicated however

lower bounds and approximations for the information rate of the isi channel we consider the discrete time intersymbol interference isi channel model with additive gaussian noise and fixed independent identically distributed inputs in this setting we investigate the expression put forth by shamai and laroia as a conjectured lower bound for the input output mutual information after application of a minimum mean square error decision feedback equalizer receiver a low signal to noise ratio snr expansion is used to prove that the conjectured bound does not hold under general conditions and to characterize inputs for which it is particularly ill suited one such input is used to construct a counterexample indicating that the shamai laroia expression does not always bound even the achievable rate of the channel thus excluding a natural relaxation of the original conjectured bound however this relaxed bound is then shown to hold for any finite entropy input and isi channel when the snr is sufficiently high we derive two conditions under which the relaxed bound holds involving compound channel capacity and quasiconvexity arguments finally new simple bounds for the achievable rate are proven and compared with other known bounds information estimation relations and estimation theoretic bounds play a key role in establishing our results