efficiency of the logistic regression and cox proportional hazards models in longitudinal studies both logistic regression and cox proportional hazards models are used widely in longitudinal epidemiologic studies for analysing the relationship between several risk factors and a time related dichotomous event the two models yield similar estimates of regression coefficients in studies with short follow up and low incidence of event occurrence further with just one dichotomous covariate and identical censoring times for all subjects the asymptotic relative efficiency of the two models is very close to 1 unless the duration of follow up is extended we generalize this result to several qualitative or quantitative covariates this was motivated by the analysis of mortality data from a study where all subjects are followed up during the same fixed period without loss except by death logistic and cox models were applied to these data similar results were obtained for the two models in shorter periods of follow up of five years or less but not in longer periods of ten years or more where the survival rate was lower

sle and virasoro representations fusion we continue the study of null vector equations in relation with partition functions of systems of schramm loewner evolutions sles by considering the question of fusion starting from n commuting sles seeded at distinct points the partition function satisfies n null vector equations at level 2 we show how to obtain higher level null vector equations by coalescing the seeds one by one as an example we extend schramm s formula for the position of a marked bulk point relatively to a chordal sle trace to an arbitrary number of sle strands the argument combines input from representation theory the study of verma modules for the virasoro algebra with regularity estimates themselves based on hypoellipticity and stochastic flow arguments

topological interference management through index coding this paper studies linear interference networks both wired and wireless with no channel state information at the transmitters except a coarse knowledge of the end to end one hop topology of the network that only allows a distinction between weak zero and significant nonzero channels and no further knowledge of the channel coefficients realizations the network capacity wired and degrees of freedom dof wireless are found to be bounded above by the capacity of an index coding problem for which the antidote graph is the complement of the given interference graph the problems are shown to be equivalent under linear solutions an interference alignment perspective is then used to translate the existing index coding solutions into the wired network capacity and wireless network dof solutions as well as to find new and unified solutions to different classes of all three problems

toward a conformal field theory for schramm loewner evolutions we discuss the partition function point of view for chordal schramm loewner evolutions and their relationship with correlation functions in conformal field theory both are closely related to crossing probabilities and interfaces in critical models in two dimensional statistical mechanics we gather and supplement previous results with different perspectives point out remaining difficulties and suggest directions for future studies

pliable index coding we formulate a new variant of the index coding problem where instead of demanding a specific message clients are pliable and are interested in receiving any t messages that they do not have we term this problem pliable index coding or picod t we prove that with this formulation although some instances of the problem become simple in general the problem of finding the optimal linear code remains np hard however we show that it is possible to construct pliable index codes that are substantially smaller than index codes in many cases if there are n clients the server has m messages and each client has a side information set of cardinality s leq m t we show that o min t log n t log 2 n broadcast transmissions are sufficient to satisfy all the clients for t 1 this is an exponential improvement over the n messages required in index coding in the worst case for m n in addition for t gg log 2 n the number of broadcast transmissions required is only linearly dependent on t we generalize the results to instances where the side information sets are not necessarily of equal cardinality when m o n delta for some constant delta 0 we show that the codes of size o min t log 2 n t log n log 3 n are sufficient in general we also consider the scenario when the server only knows the cardinality of the side information sets of the clients and each client is interested in receiving any t messages that it does not have we term this formulation oblivious pliable index coding or ob picod t if the cardinalities of side information sets of all the clients is s with s leq m t then we show that min s t m s messages are both sufficient and necessary for linear codes finally we develop efficient heuristic approximation algorithms for picod t and show through simulations on the random instances of picod t that they perform well in practice

asymptotic interference alignment for optimal repair of mds codes in distributed storage the high repair bandwidth cost of n k maximum distance separable mds erasure codes has motivated a new class of codes that can reduce repair bandwidth over that of conventional mds codes in this paper we address n k d exact repair mds codes which allow for any single failed node to be repaired exactly with access to any arbitrary set of d survivor nodes we show the existence of exact repair mds codes that achieve minimum repair bandwidth matching the cut set lower bound for arbitrary admissible n k d i e k d n 1 moreover we extend our results to show the optimality of our codes for multiple node failure scenarios in which an arbitrary set of r n k failed nodes needs to repaired our approach is based on asymptotic interference alignment proposed by cadambe and jafar as a byproduct we also characterize the capacity of a class of multisource nonmulticast networks

a polynomial time algorithm for pliable index coding in pliable index coding we consider a server with m messages and n clients where each client has as side information a subset of the messages we seek to minimize the number of broadcast transmissions so that each client can recover any one unknown message she does not already have previous work has shown that the pliable index coding problem is np hard and requires at most mathcal o log 2 n broadcast transmissions which indicates exponential savings over the conventional index coding that requires in the worst case mathcal o n transmissions in this paper building on a decoding criterion that we propose we first design a deterministic polynomial time algorithm that can realize the exponential benefits by achieving in the worst case a performance upper bounded by mathcal o log 2 n broadcast transmissions we extend our algorithm to the t requests case where each client requires t unknown messages that she does not have and show that our algorithm requires at most mathcal o t log n log 2 n broadcast transmissions we construct lower bound instances that require at least omega log n transmissions for linear pliable index coding and at least omega t log n transmissions for the t requests case indicating that both our upper and lower bounds are polynomials of log n and differ within a factor of mathcal o log n we provide a probabilistic analysis over random instances and show that the required number of transmissions is almost surely theta log n as compared with the theta n log n for index coding in addition we show that these upper and lower bounds also hold for vector pliable index coding in the worst case instances and the random graph instances implying that vector coding does not provide benefits in terms of these bounds our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to 50 less transmissions

index coding an interference alignment perspective author s maleki h cadambe vr jafar sa abstract the index coding problem is studied from an interference alignment perspective providing new results as well as new insights into and generalizations of previously known results an equivalence is established between the capacity of multiple unicast index coding where each message is desired by exactly one receiver and groupcast index coding where a message can be desired by multiple receivers which settles the heretofore open question of insufficiency of linear codes for the multiple unicast index coding problem by equivalence with groupcast settings where this question has previously been answered necessary and sufficient conditions for the achievability of rate half per message in the index coding problem are shown to be a natural consequence of interference alignment constraints and generalizations to feasibility of rate 1 l 1 per message when each destination desires at least l messages are similarly obtained finally capacity optimal solutions are presented to a series of symmetric index coding problems inspired by the local connectivity and local interference characteristics of wireless networks the solutions are based on vector linear coding 1963 2012 ieee

a rate distortion approach to index coding we approach index coding as a special case of rate distortion with multiple receivers each with some side information about the source specifically using techniques developed for the rate distortion problem we provide two upper bounds and one lower bound on the optimal index coding rate the upper bounds involve specific choices of the auxiliary random variables in the best existing scheme for the rate distortion problem the lower bound is based on a new lower bound for the general rate distortion problem the bounds are shown to coincide for a number of groupcast index coding instances including all instances for which the number of decoders does not exceed three

interlinked cycles for index coding generalizing cycles and cliques we consider a graphical approach to index coding as cycles have been shown to provide coding gain cycles and cliques a specific type of overlapping cycles have been exploited in an existing literature in this paper we define a more general form of overlapping cycles called the interlinked cycle mathsf ic structure that generalizes cycles and cliques we propose a scheme called the interlinked cycle cover mathsf icc scheme that leverages mathsf ic structures in digraphs to construct scalar linear index codes we characterize a class of infinitely many digraphs where our proposed scheme is optimal over all linear and non linear index codes consequently for this class of digraphs we indirectly prove that scalar linear index codes are optimal furthermore we show that the mathsf icc scheme can outperform all the existing graph based schemes including partial clique cover and fractional local chromatic number schemes and a random coding scheme namely composite coding for certain graphs

broadcasting with side information bounding and approximating the broadcast rate index coding has received considerable attention recently motivated in part by applications such as fast video on demand and efficient communication in wireless networks and in part by its connection to network coding optimal encoding schemes and efficient heuristics were studied in various settings while also leading to new results for network coding such as improved gaps between linear and non linear capacity as well as hardness of approximation the problem of broadcasting with side information a generalization of the index coding problem begins with a sender and sets of users and messages each user possesses a subset of the messages and desires an additional message from the set the sender wishes to broadcast a message so that on receipt of the broadcast each user can compute her desired message the fundamental parameter of interest is the broadcast rate the average communication cost for sufficiently long broadcasts though there have been many new nontrivial bounds on by bar yossef 2006 lubetzky and stav 2007 alon 2008 and blasiak 2011 there was no known polynomial time algorithm for approximating within a nontrivial factor and the exact value of remained unknown for all nontrivial instances using the information theoretic linear program introduced in blasiak 2011 we give a polynomial time algorithm for recognizing instances with 2 and pinpoint precisely for various classes of graphs e g various cayley graphs of cyclic groups further extending ideas from ramsey theory we give a polynomial time algorithm with a nontrivial approximation ratio for computing finally we provide insight into the quality of previous bounds by giving constructions showing separations between and the respective bounds in particular we construct graphs where is uniformly bounded while its upper bound derived from the naive encoding scheme is polynomially worse

nonintersecting subspaces based on finite alphabets two subspaces of a vector space are here called nonintersecting if they meet only in the zero vector motivated by the design of noncoherent multiple antenna communications systems we consider the following question how many pairwise nonintersecting m sub t dimensional subspaces of an m dimensional vector space v over a field f can be found if the generator matrices for the subspaces may contain only symbols from a given finite alphabet a spl sube f the most important case is when f is the field of complex numbers c then m sub t is the number of antennas if a f gf q it is shown that the number of nonintersecting subspaces is at most q sup m 1 q sup mt 1 and that this bound can be attained if and only if m is divisible by m sub t furthermore these subspaces remain nonintersecting when lifted to the complex field it follows that the finite field case is essentially completely solved in the case when f c only the case m sub t 2 is considered it is shown that if a is a psk configuration consisting of the 2 sup r complex roots of unity the number of nonintersecting planes is at least 2 sup r m 2 and at most 2 sup r m 1 1 the lower bound may in fact be the best that can be achieved

correction of logistic regression relative risk estimates and confidence intervals for systematic within person measurement error errors in the measurement of exposure that are independent of disease status tend to bias relative risk estimates and other measures of effect in epidemiologic studies toward the null value two methods are provided to correct relative risk estimates obtained from logistic regression models for measurement errors in continuous exposures within cohort studies that may be due to either random unbiased within person variation or to systematic errors for individual subjects these methods require a separate validation study to estimate the regression coefficient lambda relating the surrogate measure to true exposure in the linear approximation method the true logistic regression coefficient beta is estimated by beta lambda where beta is the observed logistic regression coefficient based on the surrogate measure in the likelihood approximation method a second order taylor series expansion is used to approximate the logistic function enabling closed form likelihood estimation of beta confidence intervals for the corrected relative risks are provided that include a component representing error in the estimation of lambda based on simulation studies both methods perform well for true odds ratios up to 3 0 for higher odds ratios the likelihood approximation method was superior with respect to both bias and coverage probability an example is provided based on data from a prospective study of dietary fat intake and risk of breast cancer and a validation study of the questionnaire used to assess dietary fat intake

insufficiency of linear coding in network information flow it is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet it is also known that this result does not generalize to arbitrary networks there are several examples in the literature of solvable networks with no scalar linear solution over any finite field however each example has a linear solution for some vector dimension greater than one it has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension we provide a counterexample to this conjecture we also show that if a network has no linear solution over any finite field then it has no linear solution over any finite commutative ring with identity our counterexample network has no linear solution even in the more general algebraic context of modules which includes as special cases all finite rings and abelian groups furthermore we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field exactly 10 greater so the network is not even asymptotically linearly solvable it follows that even for more general versions of linearity such as convolutional coding filter bank coding or linear time sharing the network has no linear solution

coding for errors and erasures in random network coding the problem of error control in random linear network coding is considered a ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic motivated by the property that linear network coding is vector space preserving information transmission is modeled as the injection into the network of a basis for a vector space v and the collection by the receiver of a basis for a vector space u a metric on the projective geometry associated with the packet space is introduced and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space v capu is sufficiently large if the dimension of each codeword is restricted to a fixed integer the code forms a subset of a finite field grassmannian or equivalently a subset of the vertices of the corresponding grassmann graph sphere packing and sphere covering bounds as well as a generalization of the singleton bound are provided for such codes finally a reed solomon like code construction related to gabidulin s construction of maximum rank distance codes is described and a sudan style ldquolist 1rdquo minimum distance decoding algorithm is provided

collinearity diagnosis for a relative risk regression analysis an application to assessment of diet cancer relationship in epidemiological studies in epidemiologic studies two forms of collinear relationships between the intake of major nutrients high correlations and the relative homogeneity of the diet can yield unstable and not easily interpreted regression estimates for the effect of diet on disease risk this paper presents tools for assessing the magnitude and source of the corresponding collinear relationships among the estimated coefficients for relative risk regression models i show how to extend three tools condition indices variance decomposition proportions and standard inflation factors for diagnosing collinearity in standard regression models to likelihood and partial likelihood estimation for logistic and proportional hazards models this extension is based on the analogue role of the information matrix in such analyses and the cross product matrix in the standard linear model i apply the methodology to relative risk models that relate crude intakes on the log scale and nutrient densities to breast cancer cases in the nhanes i follow up study the three diagnostic tools provide complementary evidence of the existence of a strong collinearity in all models that is due largely to homogeneity of the population with respect to our risk scale for the crude intakes the analysis suggests that the non significant relative risks for the crude intakes in these models may be due to their involvement in collinear relationships while the nonsignificant relative risks for the nutrient densities are far less affected by multicollinearity

an equivalence between network coding and index coding we show that the network coding and index coding problems are equivalent this equivalence holds in the general setting which includes linear and nonlinear codes specifically we present a reduction that maps a network coding instance to an index coding instance while preserving feasibility i e the network coding instance has a feasible solution if and only if the corresponding index coding instance is feasible in addition we show that one can determine the capacity region of a given network coding instance with colocated sources by studying the capacity region of a corresponding index coding instance previous connections between network and index coding were restricted to the linear case

error correction for index coding with side information a problem of index coding with side information was first considered by birk and kol in 1998 in this study a generalization of index coding scheme where transmitted symbols are subject to errors is studied error correcting methods for such a scheme and their parameters are investigated in particular the following question is discussed given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols what is the shortest length of a linear index code such that every receiver is able to recover the required information this question turns out to be a generalization of the problem of finding a shortest length error correcting code with a prescribed error correcting capability in the classical coding theory the singleton bound and two other bounds referred to as the bound and the bound for the optimal length of a linear error correcting index code ecic are established for large alphabets a construction based on concatenation of an optimal index code with a maximum distance separable classical code is shown to attain the singleton bound for smaller alphabets however this construction may not be optimal a random construction is also analyzed it yields another inexplicit bound on the length of an optimal linear ecic further the problem of error correcting decoding by a linear ecic is studied it is shown that in order to decode correctly the desired symbol the decoder is required to find one of the vectors belonging to an affine space containing the actual error vector the syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error correcting capability of the corresponding ecic finally the notion of static ecic which is suitable for use with a family of instances of an index coding problem is introduced several bounds on the length of static ecics are derived and constructions for static ecics are discussed connections of these codes to weakly resilient boolean functions are established

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

the intractability of computing the minimum distance of a code it is shown that the problem of computing the minimum distance of a binary linear code is np hard and the corresponding decision problem is np complete this result constitutes a proof of the conjecture of berlekamp mceliece and van tilborg 1978 extensions and applications of this result to other problems in coding theory are discussed

on the index coding problem and its relation to network coding and matroid theory the index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad hoc networks an instance of the index coding problem includes a sender that holds a set of information messages x x1 xk and a set of receivers r each receiver x h in r needs to obtain a message x x and has prior side information consisting of a subset h of x the sender uses a noiseless communication channel to broadcast encoding of messages in x to all clients the objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the demands of all the receivers in this paper we analyze the relation between the index coding problem the more general network coding problem and the problem of finding a linear representation of a matroid in particular we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem specifically we show that vector linear codes outperform scalar linear index codes and that vector linear codes are insufficient for achieving the optimum number of transmissions

topological interference management with transmitter cooperation interference networks with no channel state information at the transmitter except for the knowledge of the connectivity graph have been recently studied under the topological interference management framework in this paper we consider a similar problem with topological knowledge but in a distributed broadcast channel setting i e a network where transmitter cooperation is enabled we show that the topological information can also be exploited in this case to strictly improve the degrees of freedom dof as long as the network is not fully connected which is a reasonable assumption in practice achievability schemes from graph theoretic and interference alignment perspectives are proposed together with outer bounds built upon generator sequence the concept of compound channel settings and the relation to index coding we characterize the symmetric dof for the so called regular networks with constant number of interfering links and identify the sufficient and or necessary conditions for the arbitrary network topologies to achieve a certain amount of symmetric dof

index coding with side information motivated by a problem of transmitting supplemental data over broadcast channels birk and kol infocom 1998 we study the following coding problem a sender communicates with n receivers r1 rn he holds an input x 0 01l n and wishes to broadcast a single message so that each receiver ri can recover the bit xi each ri has prior side information about x induced by a directed graph grain nodes ri knows the bits of a in the positions j i j is an edge of g g is known to the sender and to the receivers we call encoding schemes that achieve this goal indexcodes for 0 1 n with side information graph g in this paper we identify a measure on graphs the minrank which exactly characterizes the minimum length of linear and certain types of nonlinear index codes we show that for natural classes of side information graphs including directed acyclic graphs perfect graphs odd holes and odd anti holes minrank is the optimal length of arbitrary index codes for arbitrary index codes and arbitrary graphs we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph this bound holds even for randomized codes but has been shown not to be tight

on the security of index coding with side information security aspects of the index coding with side information icsi problem are investigated building on the results of bar yossef 2006 the properties of linear index codes are further explored the notion of weak security considered by bhattad and narayanan 2005 in the context of network coding is generalized to block security it is shown that the linear index code based on a matrix l whose column space code c l has length n minimum distance d and dual distance d is d 1 t block secure and hence also weakly secure if the adversary knows in advance t d 2 messages and is completely insecure if the adversary knows in advance more than n d messages strong security is examined under the conditions that the adversary 1 possesses t messages in advance 2 eavesdrops at most transmissions 3 corrupts at most transmissions we prove that for sufficiently large q an optimal linear index code which is strongly secure against such an adversary has length q 2 here q is a generalization of the min rank over fq of the side information graph for the icsi problem in its original formulation in the work of bar yossef et al

schramm s formula and the green s function for multiple sle we construct martingale observables for systems of multiple sle curves by applying screening techniques within the cft framework recently developed by kang and makarov extended to admit multiple sles we illustrate this approach by rigorously establishing explicit expressions for the green s function and schramm s formula in the case of two curves growing towards infinity in the special case when the curves are fused and start at the same point some of the formulas we prove were predicted in the physics literature

loop erased walks and total positivity we consider matrices whose elements enumerate weights of walks in planar directed weighted graphs not necessarily acyclic these matrices are totally nonnegative more precisely all their minors are formal power series in edge weights with nonnegative coefficients a combinatorial explanation of this phenomenon involves loop erased walks applications include total positivity of hitting matrices of brownian motion in planar domains

network coding in node constrained line and star networks line and star networks with both node and edge constraints are studied in the network coding framework for line networks the capacity region of the general multiple multicast problem is established the coding theorem is based on a binary linear coding scheme while the converse requires new upper bounds that improve on standard cut based bounds for star networks the multiple unicast problem is examined capacity upper bounds are derived and a simple linear coding scheme is proposed which is based on the combinatorial optimization problem of cycle packing in directed graphs the optimality of this scheme is established for a broad class of demands the connection of node constrained network coding in star networks and index coding with side information is discussed and used to partially characterize the optimal linear code for general rates

nonlinear index coding outperforming the linear optimum the following source coding problem was introduced by birk and kol a sender holds a word x isin 0 1 n and wishes to broadcast a codeword to n receivers rn rn the receiver ri is interested in xi and has prior side information comprising some subset of the n bits this corresponds to a directed graph g on n vertices where i j is an edge ri ri knows the bit xj an index code for g is an encoding scheme which enables each ri to always reconstruct xi given his side information the minimal word length of an index code was studied by bar yossef birk jayram and kol focs 06 they introduced a graph parameter minrk2 g which completely characterizes the length of an optimal linear index code for g they showed that in various cases linear codes attain the optimal word length and conjectured that linear index coding is in fact always optimal in this work we disprove the main conjecture of bar yossef birk jayram and kol in the following strong sense for any epsiv 0 and sufficiently large n there is an n vertex graph g so that every linear index code for g requires codewords of length at least nepsiv and yet a nonlinear index code for g has a word length of ne this is achieved by an explicit construction which extends alon s variant of the celebrated ramsey construction of frankl and wilson in addition we study optimal index codes in various less restricted natural models and prove several related properties of the graph parameter minrk g

some remarks on sle bubbles and schramm s two point observable simmons and cardy recently predicted a formula for the probability that the chordal sle8 3 path passes to the left of two points in the upper half plane in this paper we give a rigorous proof of their formula starting from this result we derive explicit expressions for several natural connectivity functions for sle8 3 bubbles conditioned to be of macroscopic size by passing to a limit with such a bubble we construct a certain chordal restriction measure and in this way obtain a proof of a formula for the probability that two given points are between two commuting sle8 3 paths the one point version of this result has been predicted by gamsa and cardy finally we derive an integral formula for the second moment of the area of an sle8 3 bubble conditioned to have radius 1 we evaluate the area integral numerically and relate its value to a hypothesis that the area follows the airy distribution