bounds for the loss in probability of correct classification under model based approximation in many pattern recognition classification problem the true class conditional model and class probabilities are approximated for reasons of reducing complexity and or of statistical estimation the approximated classifier is expected to have worse performance here measured by the probability of correct classification we present an analysis valid in general and easily computable formulas for estimating the degradation in probability of correct classification when compared to the optimal classifier an example of an approximation is the naive bayes classifier we show that the performance of the naive bayes depends on the degree of functional dependence between the features and labels we provide a sufficient condition for zero loss of performance too

nested lattice codes for gaussian relay networks with interference in this paper we consider a class of single source multicast relay networks we assume that all outgoing channels of a node in the network to its neighbors are orthogonal while the incoming signals from its neighbors can interfere with each other we first focus on gaussian relay networks with interference and find an achievable rate using a lattice coding scheme we show that the achievable rate of our scheme is within a constant bit gap from the information theoretic cut set bound where the constant depends only on the network topology but not on the transmit power noise variance and channel gains this is similar to a recent result by avestimehr diggavi and tse who showed an approximate capacity characterization for general gaussian relay networks however our achievability uses a structured code instead of a random one using the idea used in the gaussian case we also consider a linear finite field symmetric network with interference and characterize its capacity using a linear coding scheme

achieving awgn channel capacity with lattice gaussian coding we propose a new coding scheme using only one lattice that achieves the 1 2 log 1 snr capacity of the additive white gaussian noise awgn channel with lattice decoding which is provable for signal to noise ratio snr e at present the scheme applies a discrete gaussian distribution over an awgn good lattice but otherwise does not require a shaping lattice or dither thus it significantly simplifies the default lattice coding scheme of erez and zamir which involves a quantization good lattice as well as an awgn good lattice using the flatness factor we show that the error probability of the proposed scheme under minimum mean square error lattice decoding is almost the same as that of erez and zamir for any rate up to the awgn channel capacity we introduce the notion of good constellations which carry almost the same mutual information as that of continuous gaussian inputs we also address the implementation of gaussian shaping for the proposed lattice gaussian coding scheme

a fast encoding method for lattice codes and quantizers in an earlier paper the authors described a very fast method which for the root lattices a n d n e n their duals and certain other lattices finds the closest lattice point to an arbitrary point of the underlying space if the lattices are used as codes for a gaussian channel the algorithm provides a fast decoding procedure or if they are used as vector quantizers the algorithm performs the analog to digital conversion efficiently the present paper offers a solution to the inverse problem for the same lattices the encoding problem for channel codes or the digital to analog part of quantizing namely given an integer k to find the kth code vector and to the closely related problem of finding the index k of a given code vector

lattices which are good for almost everything we define an ensemble of lattices and show that for asymptotically high dimension most of its members are simultaneously good as sphere packings sphere coverings additive white gaussian noise awgn channel codes and mean squared error mse quantization codes these lattices are generated by applying construction a to a random linear code over a prime field of growing size i e by lifting the code to spl ropf sup n

corrected proof of de buda s theorem lattice channel codes an error in de buda s see ieee j select areas commun vol 7 no 6 p 893 1989 proof on the asymptotic optimality of lattice channel codes is pointed out and corrected using a modification of de buda s approach comments are given on the correct interpretation and the limitations of this result

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

generalized compute compress and forward compute and forward cf harnesses interference in wireless communications by exploiting structured coding the key idea of cf is to compute integer combinations of code words from multiple source nodes rather than to decode individual code words by treating others as noise compute compress and forward ccf can further enhance the network performance by introducing compression operations at receivers in this paper we develop a more general compression framework termed generalized ccf gccf where the compression function involves the selection of message segments over finite fields we show that gccf achieves a broader compression rate region than ccf we also compare our compression rate region with the fundamental slepian wolf sw region we show that gccf is optimal in the sense of achieving the minimum total compression rate we also establish the criteria under which gccf achieves the sw region in addition we consider a two hop relay network employing the gccf scheme we formulate a sum rate maximization problem and develop an approximate algorithm to solve the problem numerical results are presented to demonstrate the performance superiority of gccf over ccf and other schemes

the approximate capacity of the many to one and one to many gaussian interference channels recently etkin tse and wang found the capacity region of the two user gaussian interference channel to within 1 bit s hz a natural goal is to apply this approach to the gaussian interference channel with an arbitrary number of users we make progress towards this goal by finding the capacity region of the many to one and one to many gaussian interference channels to within a constant number of bits the result makes use of a deterministic model to provide insight into the gaussian channel the deterministic model makes explicit the dimension of signal level a central theme emerges the use of lattice codes for alignment of interfering signals on the signal level

sphere bound achieving coset codes and multilevel coset codes a simple sphere bound gives the best possible tradeoff between the volume per point of an infinite array l and its error probability on an additive white gaussian noise awgn channel it is shown that the sphere bound can be approached by a large class of coset codes or multilevel coset codes with multistage decoding including certain binary lattices these codes have structure of the kind that has been found to be useful in practice capacity curves and design guidance for practical codes are given exponential error bounds for coset codes are developed generalizing poltyrev s 1994 bounds for lattices these results are based on the channel coding theorems of information theory rather than the minkowski hlawka theorem of lattice theory

semianalytic boundary element solution of groundwater seepage problems a numerical solution is developed for the analysis of groundwater flow problems using semianalytic time integration procedures and the boundary element method the secondary reduction method developed earlier is an integral part of this computation process compared to finite difference procedures which are commonly used for time integration in boundary element applications the semianalytic computation yields more efficient and accurate results the proposed method is tested here for two dimensional confined flow applications numerical results obtained are verified using analytical procedures that are available in the literature and are compared with other boundary element solutions

on error exponents of modulo lattice additive noise channels modulo lattice additive noise mlan channels appear in the analysis of structured binning codes for costa s dirty paper channel and of nested lattice codes for the additive white gaussian noise awgn channel in this paper we derive a new lower bound on the error exponents of the mlan channel with a proper choice of the shaping lattice and the scaling parameter the new lower bound coincides with the random coding lower bound on the error exponents of the awgn channel at the same signal to noise ratio snr in the sphere packing and straight line regions this result implies that at least for rates close to channel capacity 1 writing on dirty paper is as reliable as writing on clean paper and 2 lattice encoding and decoding suffer no loss of error exponents relative to the optimal codes with maximum likelihood decoding for the awgn channel

nearest neighbor decoding for additive non gaussian noise channels we study the performance of a transmission scheme employing random gaussian codebooks and nearest neighbor decoding over a power limited additive non gaussian noise channel we show that the achievable rates depend on the noise distribution only via its power and thus coincide with the capacity region of a white gaussian noise channel with signal and noise power equal to those of the original channel the results are presented for single user channels as well as multiple access channels and are extended to fading channels with side information at the receiver

compute and forward harnessing interference through structured codes interference is usually viewed as an obstacle to communication in wireless networks this paper proposes a new strategy compute and forward that exploits interference to obtain significantly higher rates between users in a network the key idea is that relays should decode linear functions of transmitted messages according to their observed channel coefficients rather than ignoring the interference as noise after decoding these linear equations the relays simply send them towards the destinations which given enough equations can recover their desired messages the underlying codes are based on nested lattices whose algebraic structure ensures that integer combinations of codewords can be decoded reliably encoders map messages from a finite field to a lattice and decoders recover equations of lattice points which are then mapped back to equations over the finite field this scheme is applicable even if the transmitters lack channel state information

a simple proof for the existence of good pairs of nested lattices this paper provides a simplified proof for the existence of nested lattice codebooks allowing to achieve the capacity of the additive white gaussian noise channel as well as the optimal rate distortion tradeoff for a gaussian source the proof is self contained and relies only on basic probabilistic and geometrical arguments an ensemble of nested lattices that is different and more elementary than the one used in the previous proofs is introduced this ensemble is based on lifting different subcodes of a linear code to the euclidean space using construction a in addition to being simpler the analysis is less sensitive to the assumption that the additive noise is gaussian in particular for additive ergodic noise channels it is shown that the achievable rates of the nested lattice coding scheme depend on the noise distribution only via its power similarly the nested lattice source coding scheme attains the same rate distortion tradeoff for all ergodic sources with the same second moment

achieving 1 2 log 1 snr on the awgn channel with lattice encoding and decoding we address an open question regarding whether a lattice code with lattice decoding as opposed to maximum likelihood ml decoding can achieve the additive white gaussian noise awgn channel capacity we first demonstrate how minimum mean square error mmse scaling along with dithering lattice randomization techniques can transform the power constrained awgn channel into a modulo lattice additive noise channel whose effective noise is reduced by a factor of spl radic 1 snr snr for the resulting channel a uniform input maximizes mutual information which in the limit of large lattice dimension becomes 1 2 log 1 snr i e the full capacity of the original power constrained awgn channel we then show that capacity may also be achieved using nested lattice codes the coarse lattice serving for shaping via the modulo lattice transformation the fine lattice for channel coding we show that such pairs exist for any desired nesting ratio i e for any signal to noise ratio snr furthermore for the modulo lattice additive noise channel lattice decoding is optimal finally we show that the error exponent of the proposed scheme is lower bounded by the poltyrev exponent

lattice strategies for the dirty multiple access channel in costa s dirty paper channel gaussian random binning is able to eliminate the effect of interference which is known at the transmitter and thus achieve capacity we examine a generalization of the dirty paper problem to a multiple access channel mac setup where structured lattice based binning seems to be necessary to achieve capacity in the dirty mac two additive interference signals are present one known to each transmitter but none to the receiver the achievable rates using costa s gaussian binning vanish if both interference signals are strong in contrast it is shown that lattice strategies lattice precoding can achieve positive rates independent of the interference power furthermore in some cases which depend on the noise variance and power constraints high dimensional lattice strategies are in fact optimal in particular they are optimal in the limit of high snr where the capacity region of the dirty mac with strong interference approaches that of a clean mac whose power is governed by the minimum of the users powers rather than their sum the rate gap at high snr between lattice strategies and optimum rather than gaussian random binning is conjectured to be 1 2 log2 e 6 0 254 bit thus the doubly dirty mac is another instance of a network setting like the korner marton problem where linear structured coding is potentially better than random binning

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

the approximate sum capacity of the symmetric gaussian k user interference channel interference alignment has emerged as a powerful tool in the analysis of multiuser networks despite considerable recent progress the capacity region of the gaussian k user interference channel is still unknown in general in part due to the challenges associated with alignment on the signal scale using lattice codes this paper develops a new framework for lattice interference alignment based on the compute and forward approach within this framework each receiver decodes by first recovering two or more linear combinations of the transmitted codewords with integer valued coefficients and then solving these linear combinations for its desired codeword for the special case of symmetric channel gains this framework is used to derive the approximate sum capacity of the gaussian interference channel up to an explicitly defined outage set of the channel gains the key contributions are the capacity lower bounds for the weak through strong interference regimes where each receiver should jointly decode its own codeword along with part of the interfering codewords as part of the analysis it is shown that decoding k linear combinations of the codewords can approach the sum capacity of the k user gaussian multiple access channel up to a gap of no more than k 2 log k bits

averaging bounds for lattices and linear codes general random coding theorems for lattices are derived from the minkowski hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out a new version of the minkowski hlawka theorem itself is obtained as the limit for p spl rarr spl infin of a simple lemma for linear codes over gf p used with p level amplitude modulation the relation between the combinatorial packing of solid bodies and the information theoretic soft packing with arbitrarily small but positive overlap is illuminated the soft packing results are new when specialized to the additive white gaussian noise channel they reduce to a version of the de buda poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1 2 log sub 2 p n