interference alignment with finite extensions in partially connected networks in this paper we investigate the interference alignment ia for a class of partially connected multi input multi output heterogeneous networks which consist of l fully connected pico cells j partially connected pico cells and one macro cell in particular we investigate the feasibility of ia conditions with no channel extensions t independent channel extensions and t dependent channel extensions respectively for the single beam case we present three theorems to characterize some necessary conditions which l j t and the channel diversity order overline l must jointly satisfy when ia conditions are feasible these necessary conditions also implicitly impose the constraints on the upper bounds of achievable degrees of freedom with different channel extensions

inner envelopes efficient estimation in multivariate linear regression in this article we propose a new model called the inner envelope model which leads to efficient estimation in the context of multivariate normal linear regression the asymptotic distribution and the consistency of its maximum likelihood estimators are established theoretical results simulation studies and examples all show that the efficiency gains can be substantial relative to standard methods and to the maximum likelihood estimators from the envelope model introduced recently by cook et al 2010 compared to the envelope model the inner envelope model is based on a different construction and it can produce substantial efficiency gains in situations where the envelope model offers no gains in effect inner envelopes open a new frontier to the way in which reducing subspaces can be used to improve efficiency in multivariate problems copyright 2012 oxford university press

generalized interference alignment part i theoretical framework interference alignment ia has attracted enormous research interest as it achieves optimal capacity scaling with respect to signal to noise ratio on interference networks ia has also recently emerged as an effective tool in engineering interference for secrecy protection on wireless wiretap networks however despite the numerous works dedicated to ia two of its fundamental issues i e feasibility conditions and transceiver design are not completely addressed in the literature in this two part paper a generalized interference alignment gia technique is proposed to enhance the ia s capability in secrecy protection a theoretical framework is established to analyze the two fundamental issues of gia in part i and then the performance of gia in large scale stochastic networks is characterized to illustrate how gia benefits secrecy protection in part ii the theoretical framework for gia adopts methodologies from algebraic geometry determines the necessary and sufficient feasibility conditions of gia and generates a set of algorithms for solving the gia problem this framework sets up a foundation for the development and implementation of gia

envelopes and reduced rank regression we incorporate the nascent idea of envelopes cook et al statist sinica 20 927 1010 into reduced rank regression by proposing a reduced rank envelope model which is a hybrid of reduced rank and envelope regressions the proposed model has total number of parameters no more than either of reduced rank regression or envelope regression the resulting estimator is at least as efficient as both existing estimators the methodology of this paper can be adapted to other envelope models such as partial envelopes su cook biometrika 98 133 46 and envelopes in predictor space cook et al j r statist soc b 75 851 77

permanental bounds for nonnegative matrices via decomposition abstract we investigate an old suggestion of a e brouwer we call decomposition for constructing a class of permanental upper bounds for nonnegative matrices a from a single permanental upper bound u b for 0 1 matrices b for certain feasible u which include the minc bregman bound u b m b and the jurkat ryser bound u b j b we can identify the best and worst of these decomposition bounds the best decomposition bound the star bound u a is the only decomposition bound which agrees with u on the 0 1 matrices if u j then u a turns out to be the very bound uj a used by jurkat and ryser to obtain j as a special case if u m then u a is a new upper bound um a we believe its sharpened version u g m a to be the best extant permanental upper bound for nonnegative matrices as well as for 0 1 matrices

minimization of csi feedback dimension for interference alignment in mimo interference multicast networks it is well known that interference alignment ia can achieve substantial theoretical gains in multiple input multiple output mimo networks however the conventional works usually assume the full channel state information csi is available at the transmitter side which would create a overwhelming csi feedback overhead for practical frequency division duplexing fdd wireless systems to implement ia in practice it is highly desirable to reduce the amount of required csi feedback overhead in this paper we consider ia in mimo interference multicast networks under partial csi feedback and we attempt to minimize the csi feedback cost subject to ia feasibility constraints with a given degree of freedom dof requirements first we propose a csi feedback profile to embrace two important csi feedback reduction strategies and we use the metric of csi feedback dimension to quantify the associated csi feedback cost we then formulate the ia conditions under partial csi feedback in mimo interference multicast networks and we derive new ia feasibility conditions under the proposed partial csi feedback framework based on these results we consider the csi feedback dimension minimization subject to the ia feasibility constraints with a given dof requirements in mimo interference multicast networks which is formulated as a combinatorial optimization problem based on the specific problem structure we derive an asymptotically optimal solution for a category of network topologies and we further obtain closed form tradeoff results between dofs and the csi feedback cost for mimo interference multicast networks

envelopes and partial least squares regression summary we build connections between envelopes a recently proposed context for efficient estimation in multivariate statistics and multivariate partial least squares pls regression in particular we establish an envelope as the nucleus of both univariate and multivariate pls which opens the door to pursuing the same goals as pls but using different envelope estimators it is argued that a likelihood based envelope estimator is less sensitive to the number of pls components that are selected and that it outperforms pls in prediction and estimation

distributed low overhead schemes for multi stream mimo interference channels our aim in this paper is to propose fully distributed schemes for transmit and receive filter optimization the novelty of the proposed schemes is that they only require a few forward backward iterations thus causing minimal communication overhead for that purpose we relax the well known leakage minimization problem and then propose two different filter update structures to solve the resulting nonconvex problem though one leads to conventional full rank filters the other results in rank deficient filters that we exploit to gradually reduce the transmit and receive filter rank and greatly speed up the convergence furthermore inspired from the decoding of turbo codes we propose a turbo like structure to the algorithms where a separate inner optimization loop is run at each receiver in addition to the main forward backward iteration in that sense the introduction of this turbo like structure converts the communication overhead required by conventional methods to computational overhead at each receiver a cheap resource allowing us to achieve the desired performance under a minimal overhead constraint finally we show through comprehensive simulations that both proposed schemes hugely outperform the relevant benchmarks especially for large system dimensions

multidimensional permanents of polystochastic matrices abstract a d dimensional matrix is called 1 polystochastic if it is non negative and the sum over each line equals 1 such a matrix that has a single 1 in each line and zeros elsewhere is called a 1 permutation matrix a diagonal of a d dimensional matrix of order n is a choice of n elements no two in the same hyperplane the permanent of a d dimensional matrix is the sum over the diagonals of the product of the elements within the diagonal for a given order n and dimension d the set of 1 polystochastic matrices forms a convex polytope that includes the 1 permutation matrices within its set of vertices for even n and odd d we give a construction for a class of 1 permutation matrices with zero permanent consequently we show that the set of 1 polystochastic matrices with zero permanent contains at least n n 3 2 1 2 o 1 1 permutation matrices and contains a polytope of dimension at least c n 3 2 for fixed c d and even n we also provide counterexamples to a conjecture by taranenko 11 about the location of local extrema of the permanent for odd d we give a construction of 1 permutation matrices that decompose into a convex linear sum of positive diagonals these combine with a theorem of taranenko 11 to provide counterexamples to a conjecture by dow and gibson 4 generalising van der waerden s conjecture to higher dimensions

simultaneous envelopes for multivariate linear regression we introduce envelopes for simultaneously reducing the predictors and the responses in multivariate linear regression so the regression then depends only on estimated linear combinations of x and y we use a likelihood based objective function for estimating envelopes and then propose algorithms for estimation of a simultaneous envelope as well as for basic grassmann manifold optimization the asymptotic properties of the resulting estimator are studied under normality and extended to general distributions we also investigate likelihood ratio tests and information criteria for determining the simultaneous envelope dimensions simulation studies and real data examples show substantial gain over the classical methods like partial least squares canonical correlation analysis and reduced rank regression this article has supplementary material available online

tropical determinant on transportation polytopes abstract let d k l m n be the set of all the integer points in the transportation polytope of k n l n matrices with row sums lm and column sums km in this paper we find the sharp lower bound on the tropical determinant over the set d k l m n this integer piecewise linear programming problem in arbitrary dimension turns out to be equivalent to an integer non linear in fact quadratic optimization problem in dimension two we also compute the sharp upper bound on a modification of the tropical determinant where the maximum over all the transversals in a matrix is replaced with the minimum

a stronger form of the egorychev falikman theorem on permanents if a is a doubly stochastic matrix it is shown that under certain conditions there exist i j such that the matrix obtained by replacing both the ith column of a with their average has a smaller permanent than that of a a result which is stronger than the egorychev falikman theorem on permanents is also proved

feasibility of interference alignment for the mimo interference channel we study vector space interference alignment for the multiple input multiple output interference channel with no time or frequency diversity and no symbol extensions we prove both necessary and sufficient conditions for alignment in particular we characterize the feasibility of alignment for the symmetric three user channel where all users transmit along d dimensions all transmitters have m antennas and all receivers have n antennas as well as feasibility of alignment for the fully symmetric m n channel with an arbitrary number of users an implication of our results is that the total degrees of freedom available in a k user interference channel using only spatial diversity from the multiple antennas is at most 2 this is in sharp contrast to the k 2 degrees of freedom shown to be possible by cadambe and jafar with arbitrarily large time or frequency diversity moving beyond the question of feasibility we additionally discuss computation of the number of solutions using schubert calculus in cases where there are a finite number of solutions

the scaling mean and a law of large permanents abstract in this paper we study two types of means of the entries of a nonnegative matrix the permanental mean which is defined using permanents and the scaling mean which is defined in terms of an optimization problem we explore relations between these two means making use of important results by egorychev and falikman the van der waerden conjecture friedland sinkhorn and others we also define a scaling mean for functions in a much more general context our main result is a law of large permanents a pointwise ergodic theorem for permanental means of dynamically defined matrices that expresses the limit as a functional scaling mean the concepts introduced in this paper are general enough so to include as particular cases certain classical types of means as for example symmetric means and muirhead means as a corollary we reobtain a formula of halasz and szekely for the limit of the symmetric means of a stationary random process

on the likelihood ratio test for envelope models in multivariate linear regression we investigate the likelihood ratio test for a hypothesis regarding the dimension of the envelope of span in a multivariate linear regression model the asymptotic null distribution of the likelihood ratio statistic is obtained as some nuisance parameters approach infinity a saddlepoint approximation is also given for this limiting distribution the accuracy of this approximation and its comparison to the standard chi squared approximation are assessed via simulation the results can be used in a similar test for partial envelope models copyright 2013 oxford university press

on the number of matrices and a random matrix with prescribed row and column sums and 0 1 entries abstract we consider the set r c of all m n matrices having 0 1 entries and prescribed row sums r r 1 r m and column sums c c 1 c n we prove an asymptotic estimate for the cardinality r c via the solution to a convex optimization problem we show that if r c is sufficiently large then a random matrix d r c sampled from the uniform probability measure in r c with high probability is close to a particular matrix z z r c that maximizes the sum of entropies of entries among all matrices with row sums r column sums c and entries between 0 and 1 similar results are obtained for 0 1 matrices with prescribed row and column sums and assigned zeros in some positions

communication over mimo x channels interference alignment decomposition and performance analysis in a multiple antenna system with two transmitters and two receivers a scenario of data communication known as the x channel is studied in which each receiver receives data from both transmitters in this scenario it is assumed that each transmitter is unaware of the other transmitter s data noncooperative scenario this system can be considered as a combination of two broadcast channels from the transmitters points of view and two multiple access channels from the receivers points of view taking advantage of both perspectives two signaling schemes for such a scenario are developed in these schemes some linear filters are employed at the transmitters and at the receivers which decompose the system into either two noninterfering multiple antenna broadcast subchannels or two noninterfering multiple antenna multiple access subchannels the main objective in the design of the filters is to exploit the structure of the channel matrices to achieve the highest multiplexing gain mg it is shown that the proposed noncooperative signaling schemes outperform other known noncooperative schemes in terms of the achievable mg in particular it is shown that in some specific cases the achieved mg is the same as the mg of the system if full cooperation is provided either between the transmitters or between the receivers in the second part of the paper it is shown that by using mixed design schemes rather than decomposition schemes and taking the statistical properties of the interference terms into account the power offset of the system can be improved the power offset represents the horizontal shift in the curve of the sum rate versus the total power in decibels

interference alignment with partial csi feedback in mimo cellular networks interference alignment ia is a linear precoding strategy that can achieve optimal capacity scaling at high snr in interference networks however most existing ia designs require full channel state information csi at the transmitters which could lead to significant csi signaling overhead there are two techniques namely csi quantization and csi feedback filtering that reduce csi feedback overhead in this paper we consider ia processing with csi feedback filtering in mimo cellular networks we introduce a novel metric namely the feedback dimension to quantify the first order csi feedback cost associated with the csi feedback filtering the csi feedback filtering poses several important challenges in ia processing first there is a hidden partial csi knowledge constraint in ia precoder design which cannot be handled using conventional ia design methodology furthermore existing results on the feasibility conditions of ia cannot be applied due to the partial csi knowledge finally it is very challenging to find out how much csi feedback is actually needed to support ia processing we shall address the above challenges and propose a new ia feasibility condition under partial csit knowledge in mimo cellular networks based on this we consider the csi feedback dimension minimization subject to the degrees of freedom requirements and further propose an asymptotically optimal solution and derive closed form trade off results between the csi feedback cost and ia performance in mimo cellular networks

degrees of freedom of mimo cellular networks decomposition and linear beamforming design this paper investigates the symmetric degrees of freedom dof of multiple input multiple output mimo cellular networks with g cells and k users per cell having n antennas at each base station and m antennas at each user in particular we investigate techniques for achievability that are based on either decomposition with asymptotic interference alignment or linear beamforming schemes and show that there are distinct regimes of g k m n where one outperforms the other we first note that both one sided and two sided decomposition with asymptotic interference alignment achieve the same dof we then establish specific antenna configurations under which the dof achieved using decomposition based schemes is optimal by deriving a set of outer bounds on the symmetric dof using these results we completely characterize the optimal dof of any g cell network with single antenna users for linear beamforming schemes we first focus on small networks and propose a structured approach to linear beamforming based on a notion called packing ratios packing ratio describes the interference footprint or shadow cast by a set of transmit beamformers and enables us to identify the underlying structures for aligning interference such a structured beamforming design can be shown to achieve the optimal spatially normalized dof sdof of two cell two user cell network and the two cell three user cell network for larger networks we develop an unstructured approach to linear interference alignment where transmit beamformers are designed to satisfy conditions for interference alignment without explicitly identifying the underlying structures for interference alignment the main numerical insight of this paper is that such an approach appears to be capable of achieving the optimal sdof for mimo cellular networks in regimes where linear beamforming dominates asymptotic decomposition and a significant portion of sdof elsewhere remarkably polynomial identity test appears to play a key role in identifying the boundary of the achievable sdof region in the former case

on feasibility of interference alignment in mimo interference networks we explore the feasibility of interference alignment in signal vector space based only on beamforming for k user mimo interference channels our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry it is well known e g from bezout s theorem that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables following this intuition we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables rigorous connections between feasible and proper systems are made through bernshtein s theorem for the case where each transmitter uses only one beamforming vector the multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems in this case we show that the connection between feasible and proper systems can be further strengthened since the equivalency between feasible and proper systems does not always hold by including standard information theoretic outer bounds in the feasibility analysis

dimension reduction in regression without matrix inversion summary regressions in which the fixed number of predictors p exceeds the number of independent observational units n occur in a variety of scientific fields sufficient dimension reduction provides a promising approach to such problems by restricting attention to d n linear combinations of the original p predictors however standard methods of sufficient dimension reduction require inversion of the sample predictor covariance matrix we propose a method for estimating the central subspace that eliminates the need for such inversion and is applicable regardless of the n p relationship simulations show that our method compares favourably with standard large sample techniques when the latter are applicable we illustrate our method with a genomics application

linear beamformer design for interference alignment via rank minimization this paper proposes a new framework for the design of transmit and receive beamformers for interference alignment ia without symbol extensions in multi antenna cellular networks we consider ia in a g cell network with k users cell n antennas at each base station bs and m antennas at each user the proposed framework is developed by recasting the conditions for ia as two sets of rank constraints one on the rank of interference matrices and the other on the transmit beamformers in the uplink the interference matrix consists of all the interfering vectors received at a bs from the out of cell users in the uplink using these conditions and the crucial observation that the rank of interference matrices under alignment can be determined beforehand this paper develops two sets of algorithms for ia the first part of this paper develops rank minimization algorithms for ia by iteratively minimizing a weighted matrix norm of the interference matrix different choices of matrix norms lead to reweighted nuclear norm minimization rnnm or reweighted frobenius norm minimization rfnm algorithms with significantly different per iteration complexities alternately the second part of this paper devises an alternating minimization am algorithm where the rank deficient interference matrices are expressed as a product of two lower dimensional matrices that are then alternately optimized simulation results indicate that rnnm which has a per iteration complexity of a semidefinite program is effective in designing aligned beamformers for proper feasible systems with or without redundant antennas while rfnm and am which have a per iteration complexity of a quadratic program are better suited for systems with redundant antennas

ergodic interference alignment this paper develops a new communication strategy ergodic interference alignment for the k user interference channel with time varying fading at any particular time each receiver will see a superposition of the transmitted signals plus noise the standard approach to such a scenario results in each transmitter receiver pair achieving a rate proportional to 1 k its interference free ergodic capacity however given two well chosen time indices the channel coefficients from interfering users can be made to exactly cancel by adding up these two observations each receiver can obtain its desired signal without any interference if the channel gains have independent uniform phases this technique allows each user to achieve at least 1 2 its interference free ergodic capacity at any signal to noise ratio prior interference alignment techniques were only able to attain this performance as the signal to noise ratio tended to infinity extensions are given for the case where each receiver wants a message from more than one transmitter as well as the x channel case with two receivers where each transmitter has an independent message for each receiver finally it is shown how to generalize this strategy beyond gaussian channel models for a class of finite field interference channels this approach yields the ergodic capacity region

subspace alignment chains and the degrees of freedom of the three user mimo interference channel we show that the three user mt mr multiple input multiple output mimo interference channel where each transmitter is equipped with mt antennas and each receiver is equipped with mr antennas has m d m n min m 2 1 n 2 1 degrees of freedom dof normalized by time frequency and space dimensions where min mt mr n max mt mr m n m while the dof outer bound of d m n is established for every mt mr value the achievability of d m n dof is established in general subject to a normalization with respect to spatial extensions i e the scaling of the number of antennas at all nodes in particular we show that qd m n dof are achievable for the three user qmt qmr mimo interference channel for some positive integer q which may be seen as a spatial extension factor q is the scaling factor needed to make the value qd m n an integer given spatial extensions the achievability relies only on linear beamforming based interference alignment schemes and requires neither channel extensions nor channel variations in time or frequency in the absence of spatial extensions it is shown through examples how essentially the same interference alignment scheme may be applied over time extensions over either constant or time varying channels the central new insight to emerge from this paper is the notion of subspace alignment chains as the dof bottlenecks the subspace alignment chains are instrumental both in identifying the extra dimensions to be provided by a genie to a receiver for the dof outer bound as well as in the construction of the optimal interference alignment schemes the dof value d m n is a piecewise linear function of m n with either m or n being the bottleneck within each linear segment whereas the other value contains some redundancy i e it can be reduced without reducing the dof the corner points of these piecewise linear segments correspond to two sets a 1 2 2 3 3 4 and b 1 3 3 5 5 7 the set a contains all those values of m n and only those values of m n for which there is redundancy in both m and n i e either can be reduced without reducing the dof the set b contains all those values of m n and only those values of m n for which there is no redundancy in either m or n i e neither can be reduced without reducing the dof because a and b represent settings with maximum and minimum redundancy essentially they are the basis for the dof outer bounds and inner bounds respectively our results settle the question of feasibility of linear interference alignment introduced previously by cenk et al for the three user mt mr mimo interference channel completely for all values of mt mr in particular we show that the linear interference alignment problem mt mr d 3 as defined in previous paper by cenk et al is feasible if and only if d d m n with the exception of the values m n b and only with that exception we show that for every m n value there are proper systems as defined by cenk et al that are not feasible evidently the redundancy contained in all other values of m n manifests itself as superfluous variables that are not discounted in the definition of proper systems thus creating a discrepancy between proper and feasible systems our results show that m n a are the only values for which there is no dof benefit of joint processing among co located antennas at the transmitters or receivers this may also be seen as a consequence of the maximum redundancy in the m n a settings

an iteratively weighted mmse approach to distributed sum utility maximization for a mimo interfering broadcast channel consider the multiple input multiple output mimo interfering broadcast channel whereby multiple base stations in a cellular network simultaneously transmit signals to a group of users in their own cells while causing interference to each other the basic problem is to design linear beamformers that can maximize the system throughput in this paper we propose a linear transceiver design algorithm for weighted sum rate maximization that is based on iterative minimization of weighted mean square error mse the proposed algorithm only needs local channel knowledge and converges to a stationary point of the weighted sum rate maximization problem furthermore the algorithm and its convergence can be extended to a general class of sum utility maximization problem the effectiveness of the proposed algorithm is validated by numerical experiments

partial envelopes for efficient estimation in multivariate linear regression we introduce the partial envelope model which leads to a parsimonious method for multivariate linear regression when some of the predictors are of special interest it has the potential to achieve massive efficiency gains compared with the standard model in the estimation of the coefficients for the selected predictors the partial envelope model is a variation on the envelope model proposed by cook et al 2010 but as it focuses on part of the predictors it has looser restrictions and can further improve the efficiency we develop maximum likelihood estimation for the partial envelope model and discuss applications of the bootstrap an example is provided to illustrate some of its operating characteristics

optimal linear representations of images for object recognition although linear representations are frequently used in image analysis their performances are seldom optimal in specific applications this paper proposes a stochastic gradient algorithm for finding optimal linear representations of images for use in appearance based object recognition using the nearest neighbor classifier a recognition performance function is specified and linear representations that maximize this performance are sought for solving this optimization problem on a grassmann manifold a stochastic gradient algorithm utilizing intrinsic flows is introduced several experimental results are presented to demonstrate this algorithm

on relative convergence properties of principal component analysis algorithms we investigate the convergence properties of two different stochastic approximation algorithms for principal component analysis and analytically explain some commonly observed experimental results in our analysis we use the theory of stochastic approximation and in particular the results of fabian 1968 to explore the asymptotic mean square errors amses of the algorithms this study reveals the conditions under which the algorithms produce smaller amses and also the conditions under which one algorithm has a smaller amse than the other experimental study with multidimensional gaussian data corroborate our analytical findings we next explore the convergence rates of the two algorithms our experiments and an analytical explanation reveals the conditions under which the algorithms converge faster to the solution and also the conditions under which one algorithm converges faster than the other

linear interference alignment in full duplex mimo networks with imperfect csi in this paper we consider a system where full duplex fd base stations bss communicate with half duplex hd downlink dl and uplink ul users in a multi user multi cell network where all nodes are equipped with multiple antennas the introduction of fd bss offers potential to increase spectral efficiency however it also causes a surge in the number of interference links compared with the hd network counterpart here we apply linear interference alignment ia to manage interference in this network under imperfect channel state information csi first we characterize the performance losses incurred with respect to the achievable sum rate and degrees of freedom dofs results show that the general trend in performance loss is mainly determined by how the error scales with the signal to noise ratio snr in particular full ul and dl dof can be achieved even under imperfect csi when the channel error is at least inversely proportional to snr moreover in such cases the sum rate loss is always finite and either goes to zero or is upper bounded by a derived value second we design two linear ia algorithms applicable to the system under consideration these are based on minimizing the mean square error mmse and maximizing the signal to interference plus noise ratio and consider statistical knowledge of the csi error for added robustness the proposed algorithms follow specific design principles that distribute the different interference components amongst the various beamformers and result in unitary receivers and precoders in addition we show that under certain conditions both designs result in identical beamforming solutions even though the mmse algorithm has lower computational complexity finally we also derive the proper condition for ia feasibility in the multi cell system under consideration

interference alignment with asymmetric complex signaling on mimo x channels it is known that with conventional spatial interference alignment i e linear interference alignment without symbol extension the achievable degrees of freedom dof of mimo x channels is no more than lfloor d outer rfloor where d outer denotes the outer bound of dof of mimo x channels in this paper we propose a spatial interference alignment and zero forcing framework in combination with asymmetric complex signaling which can achieve the dof of 1 2 lfloor 2d outer rfloor for arbitrary number of antennas on each terminal note that 1 2 lfloor 2d outer rfloor is equal to or larger than lfloor d outer rfloor the result shows that the technique of asymmetric complex signaling which was originally proposed for single antenna systems can be useful for mimo channels in particular when symbol extensions are not allowed

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

degrees of freedom of the k user m times n mimo interference channel we provide inner bound and outer bound for the total number of degrees of freedom of the k user multiple input multiple output mimo gaussian interference channel with m antennas at each transmitter and n antennas at each receiver if the channel coefficients are time varying and drawn from a continuous distribution the bounds are tight when the ratio max m n min m n r is equal to an integer for this case we show that the total number of degrees of freedom is equal to min m n k if k r and min m n r r 1 k if k r achievability is based on interference alignment we also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some mimo interference channels with constant channel coefficients

brunn minkowski inequalities for contingency tables and integer flows we establish approximate log concavity for a wide family of combinatorially defined integer valued functions examples include the number of non negative integer matrices contingency tables with prescribed row and column sums margins as a function of the margins and the number of integer feasible flows in a network as a function of the excesses at the vertices as a corollary we obtain approximate log concavity for the kostant partition function of type a we also present an indirect evidence that at least some of the considered functions might be genuinely log concave

the steiner triple systems of order 19 using an orderly algorithm the steiner triple systems of order 19 are classied there are 11 084 874 829 pairwise nonisomorphic such designs for each design the order of its automorphism group and the number of pasch congurations it contains are recorded 2 591 of the designs are anti pasch there are three main parts of the classication constructing an initial set of blocks the seeds completing the seeds to triple systems with an algorithm for exact cover and carrying out isomorph rejection of the nal triple systems isomorph rejection is based on the graph canonical labeling software nauty supplemented with a vertex invariant based on pasch congurations the possibility of using the strongly regular block graphs of these designs in the isomorphism tests is utilized the aforementioned value is in fact a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters 57 24 11 9

degrees of freedom for the mimo interference channel in this correspondence we show that the exact number of spatial degrees of freedom dof for a two user nondegenerate full rank channel matrices multiple input multiple output mimo gaussian interference channel with m1 m2 antennas at transmitters 1 2 and n1 n2 antennas at the corresponding receivers and perfect channel knowledge at all transmitters and receivers is min m1 m2 n1 m2 max m1 n2 max m2 n1 a constructive achievability proof shows that zero forcing is sufficient to achieve all the available dof on the two user mimo interference channel we also show through an example of a share and transmit scheme how the gains of transmitter cooperation may be entirely offset by the cost of enabling that cooperation so that the available dof are not increased

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

permanents of d dimensional matrices let a ai1i2 id be an n1 n2 nd matrix over a commutative ring the permanent of a is defined by per a n1i 1ai 2 i 3 i d i where the summation ranges over all one to one functions k from 1 2 n1 to 1 2 nk k 2 3 d in this paper it is shown that a number of properties of permanents of 2 dimensional matrices extend to higher dimensional matrices in particular permanents of nonnegative d dimensional matrices with constant hyperplane sums are investigated the paper concludes by introducing s permanents which generalize the definition above that the permanent becomes the 1 permanent and showing that an s permanent can always be converted into a 1 permanent

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

matrix variate regressions and envelope models summary modern technology often generates data with complex structures in which both response and explanatory variables are matrix valued existing methods in the literature can tackle matrix valued predictors but are rather limited for matrix valued responses we study matrix variate regressions for such data where the response y on each experimental unit is a random matrix and the predictor x can be either a scalar a vector or a matrix treated as non stochastic in terms of the conditional distribution y x we propose models for matrix variate regressions and then develop envelope extensions of these models under the envelope framework redundant variation can be eliminated in estimation and the number of parameters can be notably reduced when the matrix variate dimension is large possibly resulting in significant gains in efficiency the methods proposed are applicable to high dimensional settings

stability of oja s pca subspace rule this paper deals with stability of oja s symmetric algorithm for estimating the principal component subspace of the input data exact conditions are derived for the gain parameter on which the discrete algorithm remains bounded the result is extended for a nonlinear version of oja s algorithm

matrix permanent and quantum entanglement of permutation invariant states we point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states this connection allows us to interpret the permanent as an angle between vectors by employing a recently introduced permanent inequality by carlen et al methods appl anal 13 1 2006 we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way

on the number of interference alignment solutions for the k user mimo channel with constant coefficients in this paper we study the number of different interference alignment ia solutions in a k user multiple input multiple output mimo interference channel when the alignment is performed via beamforming and no symbol extensions are allowed we focus on the case where the number of ia equations matches the number of variables in this situation the number of ia solutions is finite and constant for any channel realization out of a zero measure set and as we prove in this paper it is given by an integral formula that can be numerically approximated using monte carlo integration methods more precisely the number of alignment solutions is the scaled average of the determinant of a certain hermitian matrix related to the geometry of the problem interestingly while the value of this determinant at an arbitrary point can be used to check the feasibility of the ia problem its average properly scaled gives the number of solutions for single beam systems the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented nonetheless our results can be applied to arbitrary interference mimo networks with any number of users antennas and streams per user

parsimonious tensor response regression abstractaiming at abundant scientific and engineering data with not only high dimensionality but also complex structure we study the regression problem with a multidimensional array tensor response and a vector predictor applications include among others comparing tensor images across groups after adjusting for additional covariates which is of central interest in neuroimaging analysis we propose parsimonious tensor response regression adopting a generalized sparsity principle it models all voxels of the tensor response jointly while accounting for the inherent structural information among the voxels it effectively reduces the number of free parameters leading to feasible computation and improved interpretation we achieve model estimation through a nascent technique called the envelope method which identifies the immaterial information and focuses the estimation based upon the material information in the tensor response we demonstrate that the resulting estimator is asymptotically efficient

efficient integration of sufficient dimension reduction and prediction in discriminant analysis abstractsufficient dimension reduction sdr methods are popular model free tools for preprocessing and data visualization in regression problems where the number of variables is large unfortunate

homotopy continuation for spatial interference alignment in arbitrary mimo x networks in this paper we propose an algorithm to design interference alignment ia precoding and decoding matrices for arbitrary mimo x networks the proposed algorithm is rooted in the homotopy continuation techniques commonly used to solve systems of nonlinear equations homotopy methods find the solution of a target system by smoothly deforming the solution of a start system which can be trivially solved unlike previously proposed ia algorithms the homotopy continuation technique allows us to solve the ia problem for both unstructured i e generic and structured channels such as those that arise when time or frequency symbol extensions are jointly employed with the spatial dimension to this end we consider an extended system of bilinear equations that include the standard alignment equations to cancel the interference and a new set of bilinear equations that preserve the desired dimensionality of the signal spaces at the intended receivers we propose a simple method to obtain the start system by randomly choosing a set of precoders and decoders and then finding a set of channels satisfying the system equations which is a linear problem once the start system is available standard prediction and correction techniques are applied to track the solution all the way to the target system we analyze the convergence of the proposed algorithm and prove that for many feasible systems and a sufficiently small continuation parameter the algorithm converges with probability one to a perfect ia solution the simulation results show that the proposed algorithm is able to consistently find solutions achieving the maximum number of degrees of freedom in a variety of mimo x networks with or without symbol extensions further the algorithm provides insights into the feasibility of ia in mimo x networks for which theoretical results are scarce

a note on the determinant and permanent problem abstract in valiant s theory of arithmetic complexity the following question occupies a central position given an integer n what is the minimal m such that the permanent of an n n matrix is the projection of the determinant of an m m matrix more generally for affine linear transformations we ask the similar question what is the minimal m such that the permanent of an n n matrix is the determinant of an m m matrix via affine linear transformation this paper gives the lower bound m 2 n for affine linear transformations the result is an improvement of earlier results by von zur gathen babai and seress it also generalizes a classical theorem of marcus and minc

weighted envelope estimation to handle variability in model selection summaryenvelope methodology can provide substantial efficiency gains in multivariate statistical problems but in some applications the estimation of the envelope dimension can induce selection volatility that may mitigate those gains current envelope methodology does not account for the added variance that can result from this selection in this article we circumvent dimension selection volatility through the development of a weighted envelope estimator theoretical justification is given for our estimator and the validity of the residual bootstrap for estimating its asymptotic variance is established a simulation study and real data analysis illustrate the utility of our weighted envelope estimator

foundations for envelope models and methods envelopes were recently proposed by cook li and chiaromonte as a method for reducing estimative and predictive variations in multivariate linear regression we extend their formulation proposing a general definition of an envelope and a general framework for adapting envelope methods to any estimation procedure we apply the new envelope methods to weighted least squares generalized linear models and cox regression simulations and illustrative data analysis show the potential for envelope methods to significantly improve standard methods in linear discriminant analysis logistic regression and poisson regression supplementary materials for this article are available online

asymptotic theory of overparameterized structural models abstract a theory of overparameterized structural models is presented in such a model some redundant parameters are involved the parameter vector is not identified and the information matrix is not nonsingular the minimum discrepancy function mdf test statistic is shown to have an asymptotic chi squared distribution almost everywhere for a wide class of discrepancy functions asymptotic distribution properties of the mdf estimators are investigated the factor analysis model is discussed as an example

on stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix abstract in applications of signal processing and pattern recognition eigenvectors and eigenvalues of the statistical mean of a random matrix sequence are needed iterative methods are suggested and analyzed in which no sample moments are used convergence is shown by stochastic approximation theory

on the feasibility of linear interference alignment for mimo interference broadcast channels with constant coefficients in this paper we analyze the feasibility of linear interference alignment ia for multi input multi output mimo interference broadcast channel mimo ibc with constant coefficients we pose and prove the necessary conditions of linear ia feasibility for general mimo ibc except for the proper condition we find another necessary condition to ensure a kind of irreducible interference to be eliminated we then prove the necessary and sufficient conditions for a special class of mimo ibc where the numbers of antennas are divisible by the number of data streams per user since finding an invertible jacobian matrix is crucial for the sufficiency proof we first analyze the impact of sparse structure and repeated structure of the jacobian matrix considering that for the mimo ibc the sub matrices of the jacobian matrix corresponding to the transmit and receive matrices have different repeated structure we find an invertible jacobian matrix by constructing the two sub matrices separately we show that for the mimo ibc where each user has one desired data stream a proper system is feasible for symmetric mimo ibc we provide proper but infeasible region of antenna configurations by analyzing the difference between the necessary conditions and the sufficient conditions of linear ia feasibility

indirect multivariate response linear regression we propose a class of estimators of the multivariate response linear regression coefficient matrix that exploits the assumption that the response and predictors have a joint multivariate normal distribution this allows us to indirectly estimate the regression coefficient matrix through shrinkage estimation of the parameters of the inverse regression or the conditional distribution of the predictors given the responses we establish a convergence rate bound for estimators in our class and we study two examples which respectively assume that the inverse regression s coefficient matrix is sparse and rank deficient these estimators do not require that the forward regression coefficient matrix is sparse or has small frobenius norm using simulation studies we show that our estimators outperform competitors

sparse envelope model efficient estimation and response variable selection in multivariate linear regression the envelope model allows efficient estimation in multivariate linear regression in this paper we propose the sparse envelope model which is motivated by applications where some response variables are invariant with respect to changes of the predictors and have zero regression coefficients the envelope estimator is consistent but not sparse and in many situations it is important to identify the response variables for which the regression coefficients are zero the sparse envelope model performs variable selection on the responses and preserves the efficiency gains offered by the envelope model response variable selection arises naturally in many applications but has not been studied as thoroughly as predictor variable selection in this paper we discuss response variable selection in both the standard multivariate linear regression and the envelope contexts in response variable selection even if a response has zero coefficients it should still be retained to improve the estimation efficiency of the nonzero coefficients this is different from the practice in predictor variable selection we establish consistency and the oracle property and obtain the asymptotic distribution of the sparse envelope estimator

the solution of van der waerden s problem for permanents in this paper a solution to van der waerden s problem is presented it is hoped that the solution of this problem will stimulate new and vigorous interest in the theory of permanents and related topics permanents are ideal objects of study having both practical applications and abstract mathematical problems problems involving permanents tend to be easily stated but only solvable using deep methods the theory of permanents as an independent branch of mathematics which has developed parallel to the theory of determinants is far from exhausted many difficult questions must still be resolved in particular the problem of calculation and estimation of permanents for specific classes of matrices is of prime importance the solution of van der waerden s problem makes use of the penetrating force of the method of mixed volumes from combinatorial analysis even hilbert noted that there were connections to be uncovered between the theory of mixed volumes and other areas of mathematics the method of mixed volumes is ideally suited to solving extremal problems and problems of uniqueness and obtaining deep new inequalities it is reasonable to assume that in the future the method of mixed volumes will stand with that of generating functions as one of the basic analytic tools of computational combinatorial analysis

groupwise envelope models for imaging genetic analysis motivated by searching for associations between genetic variants and brain imaging phenotypes the aim of this article is to develop a groupwise envelope model for multivariate linear regression in order to establish the association between both multivariate responses and covariates the groupwise envelope model allows for both distinct regression coefficients and distinct error structures for different groups statistically the proposed envelope model can dramatically improve efficiency of tests and of estimation theoretical properties of the proposed model are established numerical experiments as well as the analysis of an imaging genetic data set obtained from the alzheimer s disease neuroimaging initiative adni study show the effectiveness of the model in efficient estimation data used in preparation of this article were obtained from the alzheimer s disease neuroimaging initiative adni database

on the degrees of freedom achievable through interference alignment in a mimo interference channel consider a k user flat fading mimo interference channel where the kth transmitter or receiver is equipped with mk respectively nk antennas if an exponential in k number of generic channel extensions are used either across time or frequency cadambe and jafar 1 showed that the total achievable degrees of freedom dof can be maximized via interference alignment resulting in a total dof that grows linearly with a even if mk and nk are bounded in this work we consider the case where no channel extension is allowed and establish a general condition that must be satisfied by any degrees of freedom tuple d1 d2 dk achievable through linear interference alignment for a symmetric system with mk m nk n dk d for all k this condition implies that the total achievable dof cannot grow linearly with k and is in fact no more than k m n k 1 we also show that this bound is tight when the number of antennas at each transceiver is divisible by d the number of data streams per user

a feasibility test for linear interference alignment in mimo channels with constant coefficients in this paper we consider the feasibility of linear interference alignment ia for multiple input multiple output mimo channels with constant coefficients for any number of users antennas and streams per user and propose a polynomial time test for this problem combining algebraic geometry techniques with differential topology ones we first prove a result that generalizes those previously published on this topic in particular we consider the input set complex projective space of mimo interference channels the output set precoder and decoder grassmannians and the solution set channels decoders and precoders satisfying the ia polynomial equations not only as algebraic sets but also as smooth compact manifolds using this mathematical framework we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective if that mapping is not surjective then the solution variety projects into the input space in a singular way and the projection is a zero measure set this result naturally yields a simple feasibility test which amounts to checking the rank of a matrix we also provide an exact arithmetic version of the test which proves that testing the feasibility of ia for generic mimo channels belongs to the bounded error probabilistic polynomial complexity class

the feasibility conditions for interference alignment in mimo networks interference alignment ia has attracted great attention in the last few years for its breakthrough performance in interference networks however despite the numerous works dedicated to ia the feasibility conditions of ia remains unclear for most network topologies the ia feasibility analysis is challenging as the ia constraints are sets of high degree polynomials for which no systematic tool to analyze the solvability conditions exists in this work by developing a new mathematical framework that maps the solvability of sets of polynomial equations to the linear independence of their first order terms we propose a sufficient condition that applies to mimo interference networks with general configurations we have further proved that this sufficient condition coincides with the necessary conditions under a wide range of configurations these results further consolidate the theoretical basis of ia

csi feedback reduction for mimo interference alignment interference alignment ia is a linear precoding strategy that can achieve optimal capacity scaling at high snr in interference networks most of the existing ia designs require full channel state information csi at the transmitters which induces a huge csi signaling cost hence it is desirable to improve the feedback efficiency for ia and in this paper we propose a novel ia scheme with a significantly reduced csi feedback to quantify the csi feedback cost we introduce a novel metric namely the feedback dimension this metric serves as a first order measurement of csi feedback overhead due to the partial csi feedback constraint conventional ia schemes can not be applied and hence we develop a novel ia precoder decorrelator design and establish new ia feasibility conditions via dynamic feedback profile design the proposed ia scheme can also achieve a flexible tradeoff between the degree of freedom dof requirements for data streams the antenna resources and the csi feedback cost we show by analysis and simulations that the proposed scheme achieves substantial reductions of csi feedback overhead under the same dof requirement in mimo interference networks

a generalized lieb s theorem and its applications to spectrum estimates for a sum of random matrices in this paper we prove the concavity of the k trace functions a tr k exp h lna 1 k on the convex cone of all positive definite matrices tr k a denotes the k th elementary symmetric polynomial of the eigenvalues of a as an application we use the concavity of these k trace functions to derive tail bounds and expectation estimates on the sum of the k largest or smallest eigenvalues of a sum of random matrices

tropical determinant of integer doubly stochastic matrices abstract let d m n be the set of all the integer points in the m dilate of the birkhoff polytope of doubly stochastic n n matrices in this paper we find the sharp upper bound on the tropical determinant over the set d m n we define a version of the tropical determinant where the maximum over all the transversals in a matrix is replaced with the minimum and then find the sharp lower bound on thus defined tropical determinant over d m n

boolean matrices with prescribed row column sums and stable homogeneous polynomials we prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorithmic and combinatorial applications our main application is the first poly time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor this new algorithm is a particular instance of new polynomial time deterministic algorithms related to the multiple partial differentiation of polynomials given by evaluation oracles

fast subspace tracking and neural network learning by a novel information criterion we introduce a novel information criterion nic for searching for the optimum weights of a two layer linear neural network nn the nic exhibits a single global maximum attained if and only if the weights span the desired principal subspace of a covariance matrix the other stationary points of the nic are unstable saddle points we develop an adaptive algorithm based on the nic for estimating and tracking the principal subspace of a vector sequence the nic algorithm provides a fast on line learning of the optimum weights for the two layer linear nn we establish the connections between the nic algorithm and the conventional mean square error mse based algorithms such as oja s algorithm oja 1989 lmser past apex and gha the nic algorithm has several key advantages such as faster convergence which is illustrated through analysis and simulation

the van der waerden conjecture for mixed discriminants we prove that the mixed discriminant of doubly stochastic n tuples of semidefinite hermitian n n matrices is bounded below by n nn and that this bound is uniquely attained at the n tuple 1ni 1ni this result settles a conjecture posed by r bapat in 1989 we consider various generalizations and applications of this result

on the achievable degrees of freedom of two cell multiuser mimo interference networks in this paper we study the sum degrees of freedom dof of an uplink two cell multiuser mimo interference network with asymmetric number of users in the cells the achievable dof is devised based on a two dimensional space time spreading code framework with linear precoding decoding design and finite channel extension the derivation of the achievable dof is shown related to a rank minimization problem which corresponds to the minimization of the dimension of the interference subspace the problem is solved by the proposed grouping algorithm ga based on aligning interfering signals into a low dimensional subspace as a group and attaining the minimum number of groups the achievable sum dof derived based on the proposed ga is shown to be greater than prior arts and achieves the theoretic upper bound in several cases we also give a closed form expression of the max imum achievable sum dof when there is the maximum number of admissible users in the considered finite diversity environment