functional compression through graph coloring motivated by applications to sensor networks and privacy preserving databases we consider the problem of functional compression the objective is to separately compress possibly correlated discrete sources such that an arbitrary but fixed deterministic function of those sources can be computed given the compressed data from each source we consider both the lossless and lossy computation of a function specifically we present results of the rate regions for three instances of the problem where there are two sources 1 lossless computation where one source is available at the decoder 2 under a special condition lossless computation where both sources are separately encoded and 3 lossy computation where one source is available at the decoder for all of these instances we present a layered architecture for distributed coding first preprocess data at each source using colorings of certain characteristic graphs and then use standard distributed source coding a la slepian and wolfs scheme to compress them for the first instance our results extend the approach developed by orlitsky and roche 2001 in the sense that our scheme requires simpler structure of coloring rather than independent sets as in the previous case as an intermediate step to obtain these results we obtain an asymptotic characterization of conditional graph coloring for an or product of graphs generalizing a result of korner 1973 which should be of interest in its own right

corrections to the kalman yakubovich popov lemma for pritchard salamon systems abstract in this note we correct the result in the paper the kalman yakubovich popov lemma for pritchard salamon systems 3 there was a gap in the proof which can be bridged but only by assuming that the system is exactly controllable

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

indirect and direct gaussian distributed source coding problems we consider the distributed source coding system of l correlated gaussian sources y l l 1 2 ldots l which are noisy observations of correlated gaussian remote sources x k k 1 2 ldots k we assume that y l t y 1 y 2 ldots y l is an observation of the source vector x k t x 1 x 2 ldots x k having the form y l ax k n l where a is a l times k matrix and n l t n 1 n 2 ldots n l is a vector of l independent gaussian random variables also independent of x k in this system l correlated gaussian observations are separately compressed by l encoders and sent to the information processing center we study the remote source coding problem where the decoder at the center attempts to reconstruct the remote source x k we consider three distortion criteria based on the covariance matrix of the estimation error on x k for each of those three criteria we derive explicit inner and outer bounds of the rate distortion region next in the case of k l and a i l we study the multiterminal source coding problem where the decoder wishes to reconstruct the observation y l x l n l to investigate this problem we shall establish a result that provides a strong connection between the remote source coding problem and multiterminal source coding problem using this result we derive several new partial solutions to the multiterminal source coding problem

successive encoding of correlated sources the encoding of a discrete memoryless multiple source x i y i i 1 infty for reconstruction of a sequence z i i 1 infty with z i f x i y i i 1 2 cdots is considered we require that the encoding should be such that x i i 1 infty is encoded first without any consideration of y i i 1 infty while in a second part of the encoding this latter sequence is encoded based on knowledge of the outcome of the first encoding the resulting scheme is called successive encoding we find general outer and inner bounds for the corresponding set of achievable rates along with a complete single letter characterization for the special case h x i z i y i 0 comparisons with the slepian wolf problem and the ahlswede korner wyner side information problem are carried out

multiterminal source coding with high resolution we consider separate encoding and joint decoding of correlated continuous information sources subject to a difference distortion measure we first derive a multiterminal extension of the shannon lower bound for the rate region then we show that this shannon outer bound is asymptotically tight for small distortions these results imply that the loss in the sum of the coding rates due to the separation of the encoders vanishes in the limit of high resolution furthermore lattice quantizers followed by slepian wolf lossless encoding are asymptotically optimal we also investigate the high resolution rate region in the remote coding case where the encoders observe only noisy versions of the sources for the quadratic gaussian case we establish a separation result to the effect that multiterminal coding aimed at reconstructing the noisy sources subject to the rate constraints followed by estimation of the remote sources from these reconstructions is optimal under certain regularity conditions on the structure of the coding scheme

the gaussian ceo problem for scalar sources with arbitrary memory in this paper we consider the achievable sum rate distortion tradeoff for the gaussian central estimation officer ceo problem with a scalar source having arbitrary memory we describe how the arbitrary memory problem can be fully characterized by using known results for the vector ceo problem and then we formulate the variational problem of minimizing the sum rate subject to a distortion constraint to solve the problem we extend the conventional lagrange method and show that if the solution exists it should consist of a zero part and a non zero part where the non zero part is determined by solving a set of euler equations by calculating the second variation of the min sum rate problem a sufficient condition is also found that can be used to determine if the necessary solution results in the minimal sum rate the special case of two terminals is examined in detail and it is shown that an analytical solution is possible in this case analysis and discussion with examples are provided to illustrate the theoretical results the general solution obtained in this paper is shown to be compatible with the previous results for cases such as the problem of rate evaluation for sources without memory

distributed source coding using abelian group codes a new achievable rate distortion region a distributed source coding problem with a joint distortion criterion that depends on the sources and the reconstruction is considered in this work while the prevalent trend in information theory has been to prove achievability results using shannon s random coding arguments using structured random codes offer rate gains over unstructured random codes for many problems motivated by this a new achievable rate distortion region an inner bound to the performance limit is presented for this problem for discrete memoryless sources based on good structured random nested codes built over abelian groups for certain sources and distortion functions the new rate region is shown to be strictly bigger than the berger tung rate region which has been the best known achievable rate region for this problem till now this is done using numerical plots achievable rates for single user source coding using abelian group codes are also obtained as a corollary of the main coding theorem it is shown that nested linear codes achieve the shannon rate distortion function in the arbitrary discrete memoryless case

concordance homomorphisms from knot floer homology abstract we modify the construction of knot floer homology to produce a one parameter family of homologies thfk for knots in s 3 these invariants can be used to give homomorphisms from the smooth concordance group c to z giving bounds on the four ball genus and the concordance genus of knots we give some applications of these homomorphisms

coordination capacity we develop elements of a theory of cooperation and coordination in networks rather than considering a communication network as a means of distributing information or of reconstructing random processes at remote nodes we ask what dependence can be established among the nodes given the communication constraints specifically in a network with communication rates ri j between the nodes we ask what is the set of all achievable joint distributions p x1 xm of actions at the nodes of the network several networks are solved including arbitrarily large cascade networks distributed cooperation can be the solution to many problems such as distributed games distributed control and establishing mutual information bounds on the influence of one part of a physical system on another

distributed source coding using syndromes discus design and construction we address the problem of compressing correlated distributed sources i e correlated sources which are not co located or which cannot cooperate to directly exploit their correlation we consider the related problem of compressing a source which is correlated with another source that is available only at the decoder this problem has been studied in the information theory literature under the name of the slepian wolf 1973 source coding problem for the lossless coding case and as rate distortion with side information for the lossy coding case we provide a constructive practical framework based on algebraic trellis codes dubbed as distributed source coding using syndromes discus that can be applicable in a variety of settings simulation results are presented for source coding of independent and identically distributed i i d gaussian sources with side information available at the decoder in the form of a noisy version of the source to be coded our results reveal the promise of this approach using trellis based quantization and coset construction the performance of the proposed approach is 2 5 db from the wyner ziv 1976 bound

connected floer homology of covering involutions using the covering involution on the double branched cover of s 3 branched along a knot and adapting ideas of hendricks manolescu and hendricks hom lidman we define new knot concordance invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots

on the sum rate of gaussian multiterminal source coding new proofs and results we show that the lower bound on the sum rate of the direct and indirect gaussian multiterminal source coding problems can be derived in a unified manner by exploiting the semidefinite partial order of the distortion covariance matrices associated with the minimum mean squared error mmse estimation and the so called reduced optimal linear estimation through which an intimate connection between the lower bound and the berger tung upper bound is revealed we give a new proof of the minimum sum rate of the indirect gaussian multiterminal source coding problem i e the gaussian ceo problem for the direct gaussian multiterminal source coding problem we derive a general lower bound on the sum rate and establish a set of sufficient conditions under which the lower bound coincides with the berger tung upper bound we show that the sufficient conditions are satisfied for a class of sources and distortion constraints in particular they hold for arbitrary positive definite source covariance matrices in the high resolution regime in contrast with the existing proofs the new method does not rely on shannon s entropy power inequality

data compression of stereopairs two fundamentally different techniques for compressing stereopairs are discussed the first technique called disparity compensated transform domain predictive coding attempts to minimize the mean square error between the original stereopair and the compressed stereopair the second technique called mixed resolution coding is a psychophysically justified technique that exploits known facts about human stereovision to code stereopairs in a subjectively acceptable manner a method for assessing the quality of compressed stereopairs is also presented it involves measuring the ability of an observer to perceive depth in coded stereopairs it was found that observers generally perceived objects to be further away in compressed stereopairs than they did in originals it is proved that the rate distortion limit for coding stereopairs cannot in general be achieved by a coder that first codes and decodes the right picture sequence independently of the left picture sequence and then codes and decodes the left picture sequence given the decoded right picture sequence

the non regular ceo problem we consider the ceo problem for non regular source distributions such as uniform or truncated gaussian a group of agents observe independently corrupted versions of data and transmit coded versions over rate limited links to a ceo the ceo then estimates the underlying data based on the received coded observations agents are not allowed to convene before transmitting their observations this formulation is motivated by the practical problem of a firm s ceo estimating non regular beliefs about a sequence of events before acting on them agents observations are modeled as jointly distributed with the underlying data through a given conditional probability density function we study the asymptotic behavior of the minimum achievable mean squared error distortion at the ceo in the limit when the number of agents l and the sum rate r tend to infinity we establish a 1 r 2 convergence of the distortion an intermediate regime of performance between the exponential behavior in discrete ceo problems berger zhang and viswanathan 1996 and the 1 r behavior in gaussian ceo problems viswanathan and berger 1997 achievability is proved by a layered architecture with scalar quantization distributed entropy coding and midrange estimation the converse is proved using the bayesian chazan zakai ziv bound

distributed rate distortion with common components we describe a scheme for rate distortion with distributed encoding in which the sources to be compressed contain a common component we show that this scheme is optimal in some situations and that it strictly improves upon existing schemes which do not make full use of common components this establishes that independent quantization followed by independent binning is not optimal for the two encoder problem with a distortion constraint on one source we also show that independent quantization and binning is suboptimal for the three encoder problem in which the goal is to reproduce one of the sources losslessly this provides a counterexample that is fundamentally different from one provided earlier by korner and marton the proofs rely on the binary analogue of the entropy power inequality and the existence of a rate loss for the binary symmetric wyner ziv problem

vector gaussian two terminal source coding we derive a lower bound on each supporting line of the rate region of the vector gaussian two terminal ceo problem which is a special case of the indirect vector gaussian two terminal source coding problem the key technical ingredient is a new extremal inequality it is shown that the lower bound coincides with the berger tung upper bound in the high resolution regime similar results are derived for the direct vector gaussian two terminal source coding problem

on the sum rate loss of quadratic gaussian multiterminal source coding this work studies the sum rate loss of quadratic gaussian multiterminal source coding i e the difference between the minimum sum rates of distributed encoding and joint encoding both with joint decoding of correlated gaussian sources subject to mse distortion constraints on individual sources it is shown that under the nondegraded assumption i e all target distortions are simultaneously achievable by a gaussian berger tung scheme the supremum of the sum rate loss of distributed encoding over joint encoding of l jointly gaussian sources increases almost linearly in the number of sources l with an asymptotic slope of 0 1083 bit per sample per source as l goes to infinity this result is obtained even though we currently do not have the full knowledge of the minimum sum rate for the distributed encoding case the main idea is to upper bound the minimum sum rate of multiterminal source coding by that achieved by parallel gaussian test channels while lower bounding the minimum sum rate of joint encoding by a reverse water filling solution to a relaxed joint encoding problem of the same set of gaussian sources with a sum distortion constraint that equals the sum of the individual target distortions we show that under the nondegraded assumption the supremum difference between the upper bound for distributed encoding and the lower bound for joint encoding is achieved in the bi eigen equal variance with equal distortion case in which both bounds are known to be tight

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

a universal scheme for wyner ziv coding of discrete sources we consider the wyner ziv wz problem of lossy compression where the decompressor observes a noisy version of the source whose statistics are unknown a new family of wz coding algorithms is proposed and their universal optimality is proven compression consists of sliding window processing followed by lempel ziv lz compression while the decompressor is based on a modification of the discrete universal denoiser dude algorithm to take advantage of side information the new algorithms not only universally attain the fundamental limits but also suggest a paradigm for practical wz coding the effectiveness of our approach is illustrated with experiments on binary images and english text using a low complexity algorithm motivated by our class of universally optimal wz codes

rate region of the quadratic gaussian two encoder source coding problem we determine the rate region of the quadratic gaussian two encoder source coding problem this rate region is achieved by a simple architecture that separates the analog and digital aspects of the compression furthermore this architecture requires higher rates to send a gaussian source than it does to send any other source with the same covariance our techniques can also be used to determine the sum rate of some generalizations of this classical problem our approach involves coupling the problem to a quadratic gaussian ldquoceo problem rdquo

encoding of correlated observations an important class of engineering problems involves sensing an environment and making estimates based on the phenomena sensed in the traditional model of this problem the sensors observations are available to the estimator without alteration there is growing interest in em distributed sensing systems in which several observations are communicated to the estimator over channels of limited capacity the observations must be separately encoded so that the target can be estimated with minimum distortion two questions are addressed for a special case of this problem wherein there are two sensors which observe noisy data and communicate with a single estimator 1 if the encoder is unlimited in complexity what communication rates and distortions can be achieved 2 if the encoder must be a quantizer a mapping of a single observation sample into a digital output how can it be designed for good performance the first question is treated by the techniques of information theory it is proved that a given combination of rates and distortion is achievable if there exist degraded versions of the observations that satisfy certain formulas the second question is treated by two approaches in the first the outputs of the quantizers undergo a second stage of encoding which exploits their correlation to reduce the output rate algorithms which design the second stage are presented and tested the second approach is based on the em distributional distance a measure of dissimilarity between two probability distributions an algorithm to modify a quantizer for increased distributional distance is derived and tested

universal quantization for separate encodings and joint decoding of correlated sources we consider the multi user lossy source coding problem for continuous alphabet sources in a previous work ziv proposed a single user universal coding scheme which uses uniform quantization with dither followed by a lossless source encoder entropy coder in this paper we generalize ziv s scheme to the multi user setting for this generalized universal scheme upper bounds are derived on the redundancies defined as the differences between the actual rates and the closest corresponding rates on the boundary of the rate region for the mean square error distortion measure it is shown that this scheme can achieve redundancies of no more than 0 754 b per sample for each user these bounds are obtained without the knowledge of the multi user rate region which is an open problem in general as a direct consequence of these results the inner and outer bounds on the rate distortion achievable region are obtained

wyner ziv theory for a general function of the correlated sources corresp a source coding problem is considered for the wyner ziv type system where the decoder is required to estimate the value of some function of the encoder input and the side information the rate distortion function is established for this system and for some binary cases parametric expressions are obtained to enable numerical calculations

bounds on the extreme eigenvalues of positive definite toeplitz matrices easily computable bounds on the extreme eigenvalues of positive semidefinite psd toeplitz matrices are presented the bounds are especially suitable for matrices of relatively small dimension the bounds are derived for the wider class of psd hermitian matrices and interpreted via the levinson durbin algorithm for toeplitz matrices as a by product of this derivation an order recursive algorithm for the eigenvector eigenvalue decomposition is obtained and certain properties of the eigenvalues distribution are revealed

the likelihood encoder for lossy compression a likelihood encoder is studied in the context of lossy source compression the analysis of the likelihood encoder is based on the soft covering lemma it is demonstrated that the use of a likelihood encoder together with the soft covering lemma yields simple achievability proofs for classical source coding problems the cases of the point to point rate distortion function the rate distortion function with side information at the decoder i e the wyner ziv problem and the multi terminal source coding inner bound i e the berger tung problem are examined in this paper furthermore a non asymptotic analysis is used for the point to point case to examine the upper bound on the excess distortion provided by this method the likelihood encoder is also related to a recent alternative technique using the properties of random binning

comparison theorems for infinite dimensional riccati equations abstract comparison theorems are proved for the algebraic riccati equation in the framework of infinite dimensional hilbert spaces these theorems extend well known results for the finite dimensional case

lattice based robust distributed source coding for three correlated sources the problem of robust distributed source coding for three correlated sources is studied in this work a lattice based scheme is proposed and the analysis of its performance is provided in the high resolution regime special attention is paid to the degenerate case where the three sources are gaussian and identical in this case our scheme is shown to achieve within an asymptotic gap of 0 069 bits in terms of rate per description from the information theoretic limit of quadratic symmetric gaussian multiple description coding with central and individual decoders when the side distortion and the ratio between the central and side distortions both approach 0

weak variable length slepian wolf coding with linked encoders for mixed sources coding problems for correlated information sources were first investigated by slepian and wolf they considered the data compression system called the sw system where two sequences emitted from correlated sources are separately encoded to codewords and sent to a single decoder which has to output the original sequence pairs with a small probability or error in this paper we investigate the coding problem of a modified sw system allowing two encoders to communicate with zero rate first we consider the fixed length coding and clarify that the admissible rate region for general sources is equal to that of the original sw system next we investigate the variable length coding having the asymptotically vanishing probability of error we clarify the admissible rate region for mixed sources characterized by two ergodic sources and show that this region is strictly wider than that for fixed length codes further we investigate the universal coding problem for memoryless sources in the system and show that the sw system with linked encoders has much more flexibility than the original sw system

links of rational singularities l spaces and lo fundamental groups we prove that the link of a complex normal surface singularity is an l space if and only if the singularity is rational this via a result of hanselman et al taut foliations on graph manifolds 2015 arxiv 1508 0591 proving the conjecture of boyer et al math ann 356 4 1213 1245 2013 shows that a singularity link is not rational if and only if its fundamental group is left orderable if and only if it admits a coorientable taut foliation

source coding theory for a triangular communication system a rate distortion problem is considered for a triangular communication system tcs which has one encoder f and two decoders g sub x and g sub y the encoder f maps correlated source outputs x sup k y sup k to two codewords w sub x and w sub y which are sent to g sub x and g sub y respectively the decoders g sub x and g sub y can communicate with each other via rate constrained channels as many times as they need and g sub x reproduces x spl circ sup k while g sub y reproduces y spl circ sup k the admissible rate distortion region is determined for this tcs furthermore the relations between the tcs and the gray wyner 1974 system wyner s 1974 common information yamamoto s cascade communication system 1981 and the successive refinement system are discussed

two way source coding with a helper consider the two way rate distortion problem in which a helper sends a common limited rate message to both users based on side information at its disposal we characterize the region of achievable rates and distortions when the markov relation helper user 1 user 2 holds the main insight of the result is that in order to achieve the optimal rate the helper may use a binning scheme as in wyner ziv where the side information at the decoder is the further user namely user 2 we derive these regions explicitly for the gaussian sources with square error distortion analyze a tradeoff between the rate from the helper and the rate from the source and examine a special case where the helper has the freedom to send different messages at different rates to the encoder and the decoder the converse proofs use a technique for verifying markov relations via undirected graphs

multiterminal source coding achievable rates and reliability some open problems concerning classical multiterminal configuration in which two correlated sources x and y are encoded separately and decoded by common decoder with respect to a fidelity criterion for messages of both sources are resolved the results of berger and yeung 1989 are expanded in two directions first we determine the rate distortion region r spl delta of achievable rates r sub x r sub y for a given spl delta spl delta sub x spl delta sub y with spl delta sub x the permissible distortion level for reproduction of x and spl delta sub y for y thus we give a full solution of the problem noted by kaspi and berger 1982 they constructed the inner bound for r spl delta and solved by berger and yeung for the case of one distortion second we introduce the notion of rate reliability distortion region for the multiterminal source encoding problem more precisely the region of admissible rates of codes ensuring exponential decrease with a given exponent e 0 of the probability of exceeding the distortion levels spl delta sub x or spl delta sub y the inner and outer bounds for this region r e spl delta are constructed the analytical representation obtained for r spl delta proved to be somewhat simpler than that considered by kaspi and berger and by berger and yeung ours uses only variables pertaining to the problem statement with no auxiliary ones

a method for proving multiterminal source coding theorems the two step method used by wyner and ziv to prove the wyner ziv theorem is extended to prove sliding block source coding theorems for coding a general finite alphabet ergodic multiterminal source with respect to a single letter fidelity criterion the first step replaces every stochastic encoding in the network by a deterministic sliding block encoding the second step involves using the slepian wolf theorem to adjust the sliding block encodings so that they will have the desired rate while introducing little additional distortion the method is applied to give quick proofs of sliding block versions of theorems of berger kaspi and tung since the method obtains sliding block coders directly without first obtaining block coders the block coding results can be obtained as an easy corollary the direct methods for proving results on block coding are more difficult and do not imply the corresponding results on sliding block coding indeed in the multiterminal case it is in general not known how to construct good sliding block coders from good block coders

holomorphic disks and three manifold invariants properties and applications in 27 we introduced floer homology theories hf y s hf y s hf y t hf y s and hf red y s associated to closed oriented three manifolds y equipped with a spiny structures s spin c y in the present paper we give calculations and study the properties of these invariants the calculations suggest a conjectured relationship with seiberg witten theory the properties include a relationship between the euler characteristics of hf and turaev s torsion a relationship with the minimal genus problem thurston norm and surgery exact sequences we also include some applications of these techniques to three manifold topology

the slice ribbon conjecture for 3 stranded pretzel knots we determine the smooth concordance order of the 3 stranded pretzel knots p p q r with p q r odd we show that each one of finite order is in fact ribbon thereby proving the slice ribbon conjecture for this family of knots as corollaries we give new proofs of results first obtained by fintushel stern and casson gordon

a new sufficient condition for sum rate tightness in quadratic gaussian multiterminal source coding this paper considers the quadratic gaussian multiterminal mt source coding problem and provides a new sufficient condition for the berger tung bt sum rate bound to be tight the converse proof utilizes a set of virtual remote sources given which the observed sources are block independent with a maximum block size of 2 the given mt source coding problem is then related to a set of two terminal problems with matrix distortion constraints for which a new lower bound on the sum rate is given by formulating a convex optimization problem over all distortion matrices a sufficient condition is derived for the optimal bt scheme to satisfy the subgradient based karush kuhn tucker condition the subset of the quadratic gaussian mt problem satisfying our new sufficient condition subsumes all previously known tight cases and our proof technique opens a new direction for more general partial solutions

distributed source coding of correlated gaussian remote sources we consider the distributed source coding system for l correlated memoryless gaussian remote sources specified with l correlated gaussian random variables xi i 1 2 l we deal with the case where each of l distributed encoders cannot directly observe the source xi but its noisy version yi xi ni here ni i 1 2 l are independent additive l gaussian noises also independent of xi i 1 2 l on this coding system the determination problem of the rate distortion region remains open in general in this paper we derive explicit outer and inner bounds of the rate distortion region we further find an explicit sufficient condition for those two bounds to match we also study the sum rate part of the rate distortion region when the correlation has some symmetrical property we derive an explicit upper bound of the sum rate part from the inner bounds of the rate distortion region on a lower bound we derive a new explicit bound by making full use of the symmetrical property of the correlation we further derive an explicit necessary and sufficient condition for the lower bound to coincide with the upper bound

two way source coding with a fidelity criterion let x i y i i 1 2 cdots be a sequence of independent identically distributed bivariate random variables and consider the following communication situation the x component of the process is observed at some location say a while the y component is observed at a different location say b the x y component of the process has to be reproduced at location b a it is desired to find the amount of information that should be exchanged between the two locations so that the distortions incurred will not exceed some predetermined tolerance a single letter characterization of a certain region cal k subset cal r 4 of rates and distortions is given this region contains all and only the achievable rates and distortions for the special case where block coding is used and the information is conveyed through a one way link that can switch direction only k times per block

dissipativity and the lur e problem for parabolic boundary control systems dissipativity is studied for a class of boundary control systems whose free evolution is described by a holomorphic semigroup it is proved that the system is dissipative if and only if there exists a bounded selfadjoint solution of the corresponding linear operator inequality in addition the lur e problem is solved under particular assumptions

on network functional compression in this thesis we consider different aspects of the functional compression problem in functional compression the computation of a function or some functions of sources is desired at the receiver s the rate region of this problem has been considered in the literature under certain restrictive assumptions in chapter 2 of this thesis we consider this problem for an arbitrary tree network and asymptotically lossless computations in particular for one stage tree networks we compute a rate region and for an arbitrary tree network we derive a rate lower bound based on the graph entropy we introduce a new condition on colorings of source random variables characteristic graphs called the coloring connectivity condition c c c we show that unlike the condition mentioned in doshi et al this condition is necessary and sufficient for any achievable coding scheme based on colorings we also show that unlike entropy graph entropy does not satisfy the chain rule for one stage trees with correlated sources and general trees with independent sources we propose a modularized coding scheme based on graph colorings to perform arbitrarily closely to the derived rate lower bound we show that in a general tree network case with independent sources to achieve the rate lower bound intermediate nodes should perform some computations however for a family of functions and random variables called chain rule proper sets it is sufficient to have intermediate nodes act like relays to perform arbitrarily closely to the rate lower bound in chapter 3 of this thesis we consider a multi functional version of this problem with side information where the receiver wants to compute several functions with different side information random variables and zero distortion our results are applicable to the case with several receivers computing different desired functions we define a new concept named multi functional graph entropy which is an extension of graph entropy defined by k6rner we show that the minimum achievable rate for this problem is equal to conditional multi functional graph entropy of the source random variable given the side information we also propose a coding scheme based on graph colorings to achieve this rate in these proposed coding schemes one needs to compute the minimum entropy coloring a coloring random variable which minimizes the entropy of a characteristic graph in general finding this coloring is an np hard problem however in chapter 4 we show that depending on the characteristic graph s structure there are some interesting cases where finding the minimum entropy coloring is not np hard but tractable and practical in one of these cases we show that by having a non zero joint probability condition on random variables distributions for any desired function finding the minimum entropy coloring can be solved in polynomial time in another case we show that if the desired function is a quantization function this problem is also tractable we also consider this problem in a general

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

multiterminal source encoding with one distortion criterion the authors unify earlier investigations concerning the encoding of two correlated sources x sub k y sub k by means of separate encoders decoding is done by a single decoder which receives the outputs from both encoders the reconstruction of x sub k is required to be perfect in the usual shannon sense the authors determine the admissible rate region r d where d is the distortion of the reconstruction of y sub k the binary hamming case is investigated explicitly

universal coding for correlated sources with linked encoders on the coding for correlated sources we extend the slepian wolf 1973 data compression system called the sw system to define a new system called the swl system where two separate encoders of the sw system are mutually linked determining the optimal error exponent for all rates inside the admissible rate region remains an open problem for the sw system we completely solve this problem for the swl system and show that the optimal exponents can be achieved by universal codes furthermore it is shown that the linkage of two encoders does not extend the admissible rate region and does not even improve the exponent of correct decoding outside this region the zero error data transmission problem for the swl system is also considered we determine the zero error rate region the admissible rate region under the condition that the decoding error probability is strictly zero and show that this region can be attained by universal codes furthermore we make it clear that the linkage of encoders enlarges the zero error rate region it is interesting to note that the above results for the swl system correspond in some sense to the previous results for the discrete memoryless channel with feedback

the sum rate of vector gaussian multiple description coding with tree structured covariance distortion constraints a single letter lower bound on the sum rate of multiple description coding with tree structured distortion constraints is established by generalizing ozarow s celebrated converse argument through the introduction of auxiliary random variables that form a markov tree for the quadratic vector gaussian case this lower bound is shown to be achievable by an extended el gamal cover scheme yielding a complete characterization of the minimum sum rate

an improved outer bound for multiterminal source coding we prove a new outer bound on the rate distortion region for the multiterminal source coding problem this bound subsumes the best outer bound in the literature and improves upon it strictly in some cases the improved bound enables us to obtain a new conclusive result for the binary erasure version of the ceo problem the bound recovers many of the converse results that have been established for special cases of the problem including the recent one for the gaussian two encoder problem

lattices for distributed source coding jointly gaussian sources and reconstruction of a linear function consider a pair of correlated gaussian sources x 1 x 2 two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder the decoder is interested in reconstructing a linear combination of x 1 and x 2 to within a mean square distortion of d we obtain an inner bound to the optimal rate distortion region for this problem a portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components x 1 and x 2 first this results in a better rate region for certain parameter values our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory we then consider the case of linear reconstruction of k sources and provide an inner bound to the optimal rate distortion region some parts of the inner bound are achieved using the following coding structure lattice vector quantization followed by ldquocorrelatedrdquo lattice structured binning

lattice quantization with side information codes asymptotics and applications in sensor networks in this paper we consider the problem of rate distortion with side information available only at the decoder for the case of jointly gaussian source x and side information y and mean squared error distortion wyner proved in 1976 that the rate distortion function for this problem is identical to the conditional rate distortion function r x y assuming the side information y is available at the encoder in this paper we construct a structured class of asymptotically optimal quantizers for this problem under the assumption of high correlation between source x and side information y we show there exist quantizers within our class whose performance comes arbitrarily close to wyner s bound as an application illustrating the relevance of the high correlation asymptotics we also explore the use of these quantizers in the context of a problem of data compression for sensor networks in a setup involving a large number of devices collecting highly correlated measurements within a confined area an important feature of our formulation is that although the per node throughput of the network tends to zero as network size increases so does the amount of information generated by each transmitter this is a situation likely to be encountered often in practice which allows us to cast under new and more optimistic light some negative results on the transport capacity of large scale wireless networks

the kalman yakubovich popov lemma for pritchard salamon systems in this paper we generalize the kalman yakubovich popov lemma to the pritchard salamon class of infinite dimensional systems i e systems determined by semigroups of operators on a hilbert space with unbounded input and output operators at the same time we prove a comparison theorem for riccati equations of the same class

the sum rate bound for a new class of quadratic gaussian multiterminal source coding problems in this paper we show tightness of the berger tung bt sum rate bound for a new class of quadratic gaussian multiterminal mt source coding problems dubbed bi eigen equal variance with equal distortion beev ed where the l l source covariance matrix has equal diagonal elements with two distinct eigenvalues and the l target distortions are equal let k k l be the number of repetitions of the larger eigenvalue the beev covariance structure allows us to connect k i i d virtual gaussian sources with the l given mt sources via an l k semiorthogonal transform whose rows have equal euclidean norm plus additive i i d gaussian noises resulting in the two sets of sources being mutually conditional i i d by relating the given mt source coding problem to a generalized gaussian ceo problem with the k virtual sources as remote sources and the l mt sources as observations we obtain a lower bound on the mt sum rate and show its achievability by bt schemes under the equal distortion constraints our beev ed class of quadratic gaussian mt source coding problems subsumes both the positive symmetric case considered by wagner et al and the negative symmetric case other examples including a subclass of sources with be circulant symmetric covariance matrices and equal distortion constraints are also provided to highlight tightness of the sum rate bound

optimality and approximate optimality of source channel separation in networks we consider the source channel separation architecture for lossy source coding in communication networks it is shown that the separation approach is optimal in two general scenarios and is approximately optimal in a third scenario the two scenarios for which separation is optimal complement each other the first is when the memoryless sources at source nodes are arbitrarily correlated each of which is to be reconstructed at possibly multiple destinations within certain distortions but the channels in this network are synchronized orthogonal and memoryless point to point channels the second is when the memoryless sources are mutually independent each of which is to be reconstructed only at one destination within a certain distortion but the channels are general including multi user channels such as multiple access broadcast interference and relay channels possibly with feedback the third scenario for which we demonstrate approximate optimality of source channel separation generalizes the second scenario by allowing each source to be reconstructed at multiple destinations with different distortions for this case the loss from optimality using the separation approach can be upper bounded when a difference distortion measure is taken and in the special case of quadratic distortion measure this leads to universal constant bounds

distributed compression of linear functions partial sum rate tightness and gap to optimal sum rate we consider the problem of distributed compression of the difference z y1 cy2 of two jointly gaussian sources y1 and y2 under an mse distortion constraint d on z the rate region for this problem is unknown if the correlation coefficient and the weighting factor c satisfy c 0 inspired by ahlswede and han s scheme for the problem of distributed compression of the modulo 2 sum of two binary sources we first propose a hybrid random structured coding scheme that is capable of saving the sum rate over both the random quantize and bin qb coding scheme and krithivasan and pradhan s structured lattice coding scheme the main idea is to use a random coding component in the first layer to adjust the source correlation so that the structured coding component in the second layer can be more efficient with the outputs from the first layer as decoder side information we then provide a new sum rate lower bound for the problem in hand by connecting it to the gaussian two terminal source coding problem with covariance matrix distortion constraint our lower bound not only improves existing bounds in many cases but also allows us to prove sum rate tightness of the qb scheme when c is either relatively small or large and d is larger than some threshold furthermore our lower bound enables us to show that our new hybrid scheme performs within two b s from the optimal sum rate for all values of c and d

the wyner ziv problem with multiple sources this paper provides bounds on the rate distortion region for the distributed compression scenario where two or more sources are compressed separately for a decoder that has access to side information conclusive rate distortion results are found for the case where the sources are conditionally independent given the side information

the rate distortion function for the quadratic gaussian ceo problem a new multiterminal source coding problem called the ceo problem was presented and investigated by berger zhang and viswanathan recently viswanathan and berger have investigated an extension of the ceo problem to gaussian sources and call it the quadratic gaussian ceo problem they considered this problem from a statistical viewpoint deriving some interesting results in this paper we consider the quadratic gaussian ceo problem from a standpoint of multiterminal rate distortion theory we regard the ceo problem as a certain multiterminal remote source coding problem with infinitely many separate encoders whose observations are conditionally independent if the remote source is given this viewpoint leads us to a complete solution to the problem we determine the tradeoff between the total amount of rate and squared distortion deriving an explicit formula of the rate distortion function the derived function has the form of a sum of two nonnegative functions one is a classical rate distortion function for single gaussian source and the other is a new rate distortion function which dominates the performance of the system for a relatively small distortion it follows immediately from our result that the conjecture of viswanathan and berger on the asymptotic behavior of minimum squared distortion for large rates is true

on the generalized gaussian ceo problem this paper considers a distributed source coding dsc problem where l encoders observe noisy linear combinations of k correlated remote gaussian sources and separately transmit the compressed observations to the decoder to reconstruct the remote sources subject to a sum distortion constraint this dsc problem is referred to as the generalized gaussian ceo problem since it can be viewed as a generalization of the quadratic gaussian ceo problem where the number of remote source k 1 first we provide a new outer region obtained using the entropy power inequality and an equivalent argument in the sense of having the same rate distortion region and berger tung inner region among a certain class of generalized gaussian ceo problems we then give two sufficient conditions for our new outer region to match the inner region achieved by berger tung schemes where the second matching condition implies that in the low distortion regime the berger tung inner rate region is always tight while in the high distortion regime the same region is tight if a certain condition holds the sum rate part of the outer region is also studied and shown to meet the berger tung sum rate upper bound under a certain condition which is obtained using the karush kuhn tucker conditions of the underlying convex semidefinite optimization problem and is in general weaker than the aforesaid two for rate region tightness

duality of a source coding problem and the semi deterministic broadcast channel with rate limited cooperation the wyner ahlswede korner wak empirical coordination problem where the encoders cooperate via a finite capacity one sided link is considered the coordination capacity region is derived by combining several source coding techniques such as wyner ziv coding binning and superposition coding furthermore a semi deterministic sd broadcast channel bc with one sided decoder cooperation is considered duality principles relating the two problems are presented and the capacity region for the sd bc setting is derived the direct part follows from an achievable region for a general bc that is tight for the sd scenario a converse is established by using telescoping identities the sd bc is shown to be operationally equivalent to a class of relay bcs and the correspondence between their capacity regions is established the capacity region of the sd bc is transformed into an equivalent region that is shown to be dual to the admissible region of the wak problem in the sense that the information measures defining the corner points of both regions coincide achievability and converse proofs for the equivalent region are provided for the converse we use a probabilistic construction of auxiliary random variables that depends on the distribution induced by the codebook several examples illustrate the results

rate distortion theory for gaussian multiterminal source coding systems with several side informations at the decoder in this paper we consider the separate coding problem for l 1 correlated gaussian memoryless sources we deal with the case where l sources work as partial side information at the decoder for the reconstruction of the remaining source the determination problem of the rate distortion region for this system is the so called many help one problem and it has been known as a highly challenging problem for almost 20 years in this paper we give a partial solution to this problem we determine the rate distortion region in the case where the l sources working as partial side information are conditionally independent if the remaining source we wish to reconstruct is given the additive white gaussian noise ceo problem is a special case of this we also discuss the relation of the result previous results of ours

reliability in source coding with side information we study error exponents for source coding with side information both achievable exponents and converse bounds are obtained for the following two cases lossless source coding with coded information and lossy source coding with full side information wyner ziv these results recover and extend several existing results on source coding error exponents and are tight in some circumstances our bounds have a natural interpretation as a two player game between nature and the code designer with nature seeking to minimize the exponent and the code designer seeking to maximize it in the wyner ziv problem our analysis exposes a tension in the choice of test channel with the optimal test channel balancing two competing error events the gaussian and binary erasure cases are examined in detail

design of consensus and adaptive consensus filters for distributed parameter systems this work establishes an abstract framework that considers the distributed filtering of spatially varying processes using a sensor network it is assumed that the sensor network consists of groups of sensors each of which provides a number of state measurements from sensing devices that are not necessarily identical and which only transmit their information to their own sensor group a modification to the local spatially distributed filters provides the non adaptive case of spatially distributed consensus filters which penalize the disagreement amongst themselves in a dynamic manner a subsequent modification to this scheme incorporates the adaptation of the consensus gains in the disagreement terms of all local filters both the well posedness of these two consensus spatially distributed filters and the convergence of the associated observation errors to zero in appropriate norms are presented their performance is demonstrated on three different examples of a diffusion partial differential equation with point measurements

a simple proof of the entropy power inequality this correspondence gives a simple proof of shannon s entropy power inequality epi using the relationship between mutual information and minimum mean square error mmse in gaussian channels

vector gaussian multiterminal source coding we derive an outer bound of the rate region of the vector gaussian l terminal ceo problem by establishing a lower bound on each supporting hyperplane of the rate region to this end we prove a new extremal inequality by exploiting the connection between differential entropy and fisher information as well as some fundamental estimation theoretic inequalities it is shown that the outer bound matches the berger tung inner bound in the high resolution regime we then derive a lower bound on each supporting hyperplane of the rate region of the direct vector gaussian l terminal source coding problem by coupling it with the ceo problem through a limiting argument the tightness of this lower bound in the high resolution regime and the weak dependence regime is also proved

the distributed karhunen loeve transform the karhunen loeve transform klt is a key element of many signal processing and communication tasks many recent applications involve distributed signal processing where it is not generally possible to apply the klt to the entire signal rather the klt must be approximated in a distributed fashion this paper investigates such distributed approaches to the klt where several distributed terminals observe disjoint subsets of a random vector we introduce several versions of the distributed klt first a local klt is introduced which is the optimal solution for a given terminal assuming all else is fixed this local klt is different and in general improves upon the marginal klt which simply ignores other terminals both optimal approximation and compression using this local klt are derived two important special cases are studied in detail namely the partial observation klt which has access to a subset of variables but aims at reconstructing them all and the conditional klt which has access to side information at the decoder we focus on the jointly gaussian case with known correlation structure and on approximation and compression problems then the distributed klt is addressed by considering local klts in turn at the various terminals leading to an iterative algorithm which is locally convergent sometimes reaching a global optimum depending on the overall correlation structure for compression it is shown that the classical distributed source coding techniques admit a natural transform coding interpretation the transform being the distributed klt examples throughout illustrate the performance of the proposed distributed klt this distributed transform has potential applications in sensor networks distributed image databases hyper spectral imagery and data fusion

an on line universal lossy data compression algorithm via continuous codebook refinement i basic results a new on line universal lossy data compression algorithm is presented for finite memoryless sources with unknown statistics its performance asymptotically approaches the fundamental rate distortion limit the codebook is generated on the fly and continuously adapted by simple rules there is no separate codebook training or codebook transmission candidate codewords are randomly generated according to an arbitrary and possibly suboptimal distribution through a carefully designed gold washing or information theoretic sieve mechanism good codewords and only good codewords are promoted to permanent status with high probability we also determine the rate at which our algorithm approaches the fundamental limit

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

rate distortion for correlated sources with partially separated encoders we resolve some open problems concerning the encoding of a pair of correlated sources with respect to a fidelity criterion two encoders are used one encoder observes only the output of the first source while the other is supplied both with the second source output and with partial information about the first source at some predetermined rate a general coding theorem is proved which established that cal s ast a certain region defined in terms of single letter information theoretic quantities is an inner bound to the region of all attainable vectors of rates and distortions for certain cases of interest the converse is proved too thereby establishing the rate distortion region for these cases

capacity of a class of multicast tree networks in this paper we characterize the capacity of a new class of discrete memoryless multicast networks having a tree topology for achievability a novel coding scheme is constructed where some relays employ a combination of decode and forward and compress and forward and the other relays perform a random binning such that codebook constructions and relay operations are independent for each node and do not depend on the network topology for converse a new technique of iteratively manipulating inequalities exploiting the tree topology is used this class of multicast tree networks includes the class of diamond networks studied by kang and ulukus as a special case

a semi continuous version of the berger yeung problem we present a continuous dual of multiterminal source encoding with one distortion criterion the berger yeung 1989 problem a continuous source x is encoded with high resolution d sub x spl rarr 0 with the aid of a helper i e a correlated discrete or continuous source y that is encoded separately subject to some arbitrary distortion criterion d sub y we find the asymptotic form of the set of achievable coding rates r d sub x d sub y of the x and y encoders as d sub x spl rarr 0 two extreme cases of our result provide high resolution interpretations to the classical work of wyner 1975 1978 ahlswede 1975 korner 1975 and ziv 1976 on source coding with side information

sending a bivariate gaussian source over a gaussian mac with unidirectional conferencing encoders we consider the transmission of a memoryless bivariate gaussian source over a two user additive gaussian multiple access channel with unidirectional conferencing encoders here prior to each transmission block encoder 1 which observes the first source component is allowed to communicate with encoder 2 which observes the second source component via a unidirectional noise free bit pipe of given capacity the main results of this paper are sufficient conditions and a necessary condition for the achievability of a distortion pair expressed as a function of the channel sf snr and of the source correlation the main sufficient condition is obtained by an extension of the vector quantizer scheme suggested by lapidoth tinguely for the case without conferencing to the case with unidirectional conference in the high sf snr regime and when the capacity of the conference channel is unlimited these necessary and sufficient conditions are shown to agree we evaluate the precise high sf snr asymptotics for a subset of distortion pairs when the capacity of the conference channel is unlimited in which case we show that a separation based scheme attains these optimal distortion pairs however with symmetric average power constraints and fixed conferencing capacity at high sf snr the latter separation based scheme is shown to be suboptimal

absolutely graded floer homologies and intersection forms for four manifolds with boundary abstract in ozsvath and szabo holomorphic triangles and invariants for smooth four manifolds math sg 0110169 2001 we introduced absolute gradings on the three manifold invariants developed in ozsvath and szabo holomorphic disks and topological invariants for closed three manifolds math sg 0101206 ann of math 2001 to appear coupled with the surgery long exact sequences we obtain a number of three and four dimensional applications of this absolute grading including strengthenings of the complexity bounds derived in ozsvath and szabo holomorphic disks and three manifold invariants properties and applications math sg 0105202 ann of math 2001 to appear restrictions on knots whose surgeries give rise to lens spaces and calculations of hf for a variety of three manifolds moreover we show how the structure of hf constrains the exoticness of definite intersection forms for smooth four manifolds which bound a given three manifold in addition to these new applications the techniques also provide alternate proofs of donaldson s diagonalizability theorem and the thom conjecture for cp 2

holomorphic triangles and invariants for smooth four manifolds abstract in earlier articles the authors introduced invariants for closed oriented three manifolds defined using a variant of lagrangian floer homology in the symmetric products of riemann surfaces the aim of this article is to introduce invariants of oriented smooth four manifolds built using these floer homology groups this four dimensional theory also endows the corresponding three dimensional theories with additional structure an absolute grading of certain of its floer homology groups

multiterminal source coding under logarithmic loss we consider the classical two encoder multiterminal source coding problem where distortion is measured under logarithmic loss we provide a single letter description of the achievable rate distortion region for all discrete memoryless sources with finite alphabets by doing so we also give the rate distortion region for the m encoder ceo problem also under logarithmic loss several applications and examples are given

minimum phase infinite dimensional second order systems in general better performance can be achieved with a controlled minimum phase system than a controlled nonminimum phase system we show that a wide class of second order infinite dimensional systems with either velocity or position measurements are minimum phase the results are illustrated by two examples

the ceo problem multiterminal source coding we consider a new problem in multiterminal source coding motivated by the following decentralized communication estimation task a firm s chief executive officer ceo is interested in the data sequence x t sub t 1 sup spl infin which cannot be observed directly perhaps because it represents tactical decisions by a competing firm the ceo deploys a team of l agents who observe independently corrupted versions of x t sub t 1 sup spl infin because x t is only one among many pressing matters to which the ceo must attend the combined data rate at which the agents may communicate information about their observations to the ceo is limited to say r bits per second if the agents were permitted to confer and pool their data then in the limit as l spl rarr spl infin they usually would be able to smooth out their independent observation noises entirely then they could use their r bits per second to provide the ceo with a representation of x t with fidelity d r where d spl middot is the distortion rate function of x t in particular with such data pooling d can be made arbitrarily small if r exceeds the entropy rate h of x t suppose however that the agents are not permitted to convene agent i having to send data based solely on his own noisy observations y sub i t we show that then there does not exist a finite value of r for which even infinitely many agents can make d arbitrarily small furthermore in this isolated agents case we determine the asymptotic behavior of the minimal error frequency in the limit as l and then r tend to infinity