on the span of hadamard products of vectors abstract let a1 ak be positive semidefinite matrices and b1 bk arbitrary complex matrices of order n we show that span a 1 x a 2 x a k x x c n range a 1 a 2 a k and span b 1 x 1 b 2 x 2 b k x k x j c n range b 1 b 1 b 2 b 2 b k b k where means the hadamard product this generalizes two recent results of sun du and liu

modelling quasicrystals at positive temperature we consider a two dimensional lattice model of equilibrium statistical mechanics using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 wang tiles this model has uncountably many ground state configurations all of which are nonperiodic the question addressed in this paper is whether nonperiodicity persists at low but positive temperature we present arguments mostly numerical that this is indeed the case in particular we define an appropriate order parameter prove that it is identically zero at high temperatures and show by monte carlo simulation that it is nonzero at low temperatures

edge exchangeable models for interaction networks abstractmany modern network datasets arise from processes of interactions in a population such as phone calls email exchanges co authorships and professional collaborations in such interaction networks the edges comprise the fundamental statistical units making a framework for edge labeled networks more appropriate for statistical analysis in this context we initiate the study of edge exchangeable network models and explore its basic statistical properties several theoretical and practical features make edge exchangeable models better suited to many applications in network analysis than more common vertex centric approaches in particular edge exchangeable models allow for sparse structure and power law degree distributions both of which are widely observed empirical properties that cannot be handled naturally by more conventional approaches our discussion culminates in the hollywood model which we identify here as the canonical family of edge exchangeable distributions the hollywood model i

analytical approach to network inference investigating degree distribution when the network is reconstructed two types of errors can occur false positive and false negative errors about the presence or absence of links in this paper the influence of these two errors on the vertex degree distribution is analytically analyzed moreover an analytic formula of the density of the biased vertex degree distribution is found in the inverse problem we find a reliable procedure to reconstruct analytically the density of the vertex degree distribution of any network based on the inferred network and estimates for the false positive and false negative errors based on e g simulation studies

on additive nilpotency of jacobian matrices of polynomial maps abstract let h c n c n be a polynomial map jh the jacobian matrix of h and v h the linear subspace of m n c spanned by j h c n the set of evaluated jacobian matrices of h we describe the dimension of v h through a power linear gorni zampieri mate of h and give a numerical algorithm for computing this dimension and determining whether jh is additive nilpotent i e whether v h is a nilpotent subspace

stochastic blockmodels and community structure in networks stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks most blockmodels however ignore variation in vertex degree making them unsuitable for applications to real world networks which typically display broad degree distributions that can significantly affect the results here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks we also propose a heuristic algorithm for community detection using this objective function or its non degree corrected counterpart and show that the degree corrected version dramatically outperforms the uncorrected one in both real world and synthetic networks

stability analysis of diagonally equipotent matrices diagonally equipotent matrices are diagonally dominant matrices for which dominance is never strict in any coordinate they appear e g as laplacian matrices of signed graphs we show in this paper that for this class of matrices it is possible to provide a complete characterization of the stability properties based only on the signs of the entries of the matrices

truncated power method for sparse eigenvalue problems this paper considers the sparse eigenvalue problem which is to extract dominant largest sparse eigenvectors with at most k non zero components we propose a simple yet effective solution called truncated power method that can approximately solve the underlying nonconvex optimization problem a strong sparse recovery result is proved for the truncated power method and this theory is our key motivation for developing the new algorithm the proposed method is tested on applications such as sparse principal component analysis and the densest k subgraph problem extensive experiments on several synthetic and real world data sets demonstrate the competitive empirical performance of our method

on the estimation performance and convergence rate of the generalized power method for phase synchronization an estimation problem of fundamental interest is that of phase or angular synchronization in which the goal is to recover a collection of phases or angles using noisy measurements of relative phases or angle offsets it is known that in the gaussian noise setting the maximum likelihood estimator mle is an optimal solution to a nonconvex quadratic optimization problem and can be found with high probability using semidefinite programming sdp provided that the noise power is not too large in this paper we study the estimation and convergence performance of a recently proposed low complexity alternative to the sdp based approach namely the generalized power method gpm our contribution is twofold first we show that the sequence of estimation errors associated with the gpm iterates is bounded above by a decreasing sequence as a corollary we show that all iterates achieve an estimation error that is on the same order as that of an mle our result holds under the least restrictive assumpti

quadratic linear keller maps abstract it is proved that a quadratic linear keller map c n c n is linearly triangularizable if its rank is at most 2 or corank at most 1

the jacobian conjecture linear triangularization for homogeneous polynomial maps in dimension three let k be a field of characteristic zero and f k3 k3 a polynomial map of the form f x h where h is homogeneous of degree d 2 we show that the jacobian conjecture is true for such mappings more precisely we show that if jh is nilpotent there exists an invertible linear map t such that t 1ht 0 h2 x1 h3 x1 x2 where the hi are homogeneous of degree d as a consequence of this result we show that all generalized druzkowski mappings f x h x1 l1d xn lnd where li are linear forms for all i and d 2 are linearly triangularizable if jh is nilpotent and rkjh 3

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

sufficient conditions for schur and hurwitz diagonal stability of complex interval matrices abstract let c a j b be a complex interval matrix where a b r n n are real interval matrices and j 2 1 denote by d the set of positive definite diagonal n n matrices the schur respectively hurwitz diagonal stability of c abbreviated as sds respectively hds is defined in terms of the stein respectively lyapunov inequality associated with a generic matrix c c the inequality must have a common solution q d for all c c the paper develops two types of results for the sds respectively hds analysis which exploit different properties of c each type of results presents i a sufficient condition relying on the solvability of a matrix norm inequality sds analysis or a matrix measure inequality hds analysis ii a construction procedure for a concrete matrix that satisfies the inequality at i and concomitantly yields a common solution q d for some particular cases both necessary and sufficient conditions can be derived the schur and hurwitz approaches are developed in parallel for concision reasons as well as for emphasizing the similarities numerical examples illustrate the applicability of the theoretical results

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

thermodynamics of a tiling model a specific two dimensional tiling model composed of the so called wang tiles has been studied at finite temperature using monte carlo numerical simulations in the absence of any thermal bath the wang tiles provide the opportunity of building a very large number of non periodic tilings we can construct a local hamiltonian such that only perfectly matched tilings are ground states with zero energy this hamiltonian has a very large degeneracy the thermodynamic behaviour of such a system seems to show a continuous phase transition at non zero temperature an order parameter with non trivial features is proposed under the critical temperature the model exhibits ageing properties the fluctuation dissipation theorem is violated

inferring overlapping community structure with degree corrected block model abstract recent research has shown great interest in statistical inference methods for community detection not only in models and algorithms but also in the detectability in this paper we propose a fast community detection algorithm based on the degree corrected block model by introducing a parameter to select the candidate solutions our algorithm is able to detect overlapping communities experiments on a range of networks have achieved state of the art results moreover we show that the algorithm based on the degree corrected block model also suffers the detectability limitation which is in accord with the most recent research on the detectability threshold

a modularity based spectral method for simultaneous community and anti community detection abstract in a graph or complex network communities and anti communities are node sets whose modularity attains extremely large values positive and negative respectively we consider the simultaneous detection of communities and anti communities by looking at spectral methods based on various matrix based definitions of the modularity of a vertex set invariant subspaces associated to extreme eigenvalues of these matrices provide indications on the presence of both kinds of modular structure in the network the localization of the relevant invariant subspaces can be estimated by looking at particular matrix angles based on frobenius inner products

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

phase transitions in semidefinite relaxations statistical inference problems arising within signal processing data mining and machine learning naturally give rise to hard combinatorial optimization problems these problems become intractable when the dimensionality of the data is large as is often the case for modern datasets a popular idea is to construct convex relaxations of these combinatorial problems which can be solved efficiently for large scale datasets semidefinite programming sdp relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs it has been observed several times that when the statistical noise is small enough sdp relaxations correctly detect the underlying combinatorial structures in this paper we develop asymptotic predictions for several detection thresholds as well as for the estimation error above these thresholds we study some classical sdp relaxations for statistical problems motivated by graph synchronization and community detection in networks we map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions our results clarify the effectiveness of sdp relaxations in solving high dimensional statistical problems

eigenvalue interlacing for certain classes of matrices with real principal minors abstract two common properties of z matrices and hermitian matrices are considered 1 the eigenvalue interlacing property i e the two smallest real eigenvalues of a matrix are interlaced by the smallest real eigenvalue of any principal submatrix of order one less 2 the positive glp property i e if a matrix has a positive sequence of generalized leading principal minors then all the principal minors of the matrix are positive the relationship between these properties as well as related properties is examined in general

noisy matrix completion understanding statistical guarantees for convex relaxation via nonconvex optimization this paper studies noisy low rank matrix completion given partial and noisy entries of a large low rank matrix the goal is to estimate the underlying matrix faithfully and efficiently arguably o

on the convergence of ica algorithms with symmetric orthogonalization independent component analysis ica problem is often posed as the maximization minimization of an objective cost function under a unitary constraint which presumes the prewhitening of the observed mixtures the parallel adaptive algorithms corresponding to this optimization setting where all the separators are jointly trained are typically implemented by a gradient based update of the separation matrix followed by the so called symmetrical orthogonalization procedure to impose the unitary constraint this article addresses the convergence analysis of such algorithms which has been considered as a difficult task due to the complication caused by the minimum frobenius or induced 2 norm distance mapping step we first provide a general characterization of the stationary points corresponding to these algorithms furthermore we show that fixed point algorithms employing symmetrical orthogonalization are monotonically convergent for convex objective functions we later generalize this convergence result for nonconvex objective functions at the last part of the article we concentrate on the kurtosis objective function as a special case we provide a new set of critical points based on householder reflection and we also provide the analysis for the minima maxima saddle point classification of these critical points

the linear dependence problem for power linear maps abstract let bl l 1 k be m nl complex matrices and let x l c n l l 1 k be complex vector variables we show that the components of the map h b 1 x 1 d 1 b k x k d k are linearly dependent over c if and only if det b 1 b 1 d 1 b k b k d k 0 where means the hadamard product x and x d denote the conjugate transpose and the dth hadamard power of a matrix x respectively connections are established between the homogenous dependence problem hdp n d which arises in the study of the jacobian conjecture and the dependence problem for power linear maps pldp n d an algorithm is given to compute counterexamples to pldp n d from those to hdp n d and counterexamples to pldp n 3 are obtained for all n 67

recent directions in matrix stability abstract matrix stability has been intensively investigated in the past two centuries we review work that has been done in this topic focusing on the great progress that has been achieved in the last decade or two we start with classical stability criteria of lyapunov routh and hurwitz and lienard and chipart we then study recently proven sufficient conditions for stability with particular emphasis on p matrices we investigate conditions for the existence of a stable scaling for a given matrix we review results on other types of matrix stability such as d stability additive d stability and lyapunov diagonal stability we discuss the weak principal submatrix rank property shared by lyapunov diagonally semistable matrices we also discuss the uniqueness of lyapunov scaling factors maximal lyapunov scaling factors cones of real positive semidefinite matrices and their applications to matrix stability and inertia preserving matrices

clustering partially observed graphs via convex optimization this paper considers the problem of clustering a partially observed unweighted graph i e one where for some node pairs we know there is an edge between them for some others we know there is no edge and for the remaining we do not know whether or not there is an edge we want to organize the nodes into disjoint clusters so that there is relatively dense observed connectivity within clusters and sparse across clusters we take a novel yet natural approach to this problem by focusing on finding the clustering that minimizes the number of disagreements i e the sum of the number of observed missing edges within clusters and observed present edges across clusters our algorithm uses convex optimization its basis is a reduction of disagreement minimization to the problem of recovering an unknown low rank matrix and an unknown sparse matrix from their partially observed sum we evaluate the performance of our algorithm on the classical planted partition stochastic block model our main theorem provides sufficient conditions for the success of our algorithm as a function of the minimum cluster size edge density and observation probability in particular the results characterize the tradeoff between the observation probability and the edge density gap when there are a constant number of clusters of equal size our results are optimal up to logarithmic factors

community detection in sparse networks via grothendieck s inequality we present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs the method is based on grothendieck s inequality unlike the previous uses of this inequality that lead to constant relative accuracy we achieve any given relative accuracy by leveraging randomness we illustrate the method with the problem of community detection in sparse networks those with bounded average degrees we demonstrate that even in this regime various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices the method is general it can be applied to a variety of stochastic models of networks and semidefinite programs

statistical ensembles for economic networks economic networks share with other social networks the fundamental property of sparsity it is well known that the maximum entropy techniques usually employed to estimate or simulate weighted networks produce unrealistic dense topologies at the same time strengths should not be neglected since they are related to core economic variables like supply and demand to overcome this limitation the exponential bosonic model has been previously extended in order to obtain ensembles where the average degree and strength sequences are simultaneously fixed conditional geometric model in this paper a new exponential model which is the network equivalent of boltzmann ideal systems is introduced and then extended to the case of joint degree strength constraints conditional poisson model finally the fitness of these alternative models is tested against a number of networks while the conditional geometric model generally provides a better goodness of fit in terms of log likelihoods the conditional poisson model could nevertheless be preferred whenever it provides a higher similarity with original data if we are interested instead only in topological properties the simple bernoulli model appears to be preferable to the correlated topologies of the two more complex models

gradient descent with random initialization fast global convergence for nonconvex phase retrieval this paper considers the problem of solving systems of quadratic equations namely recovering an object of interest varvec x natural in mathbb r n from m quadratic equations samples y i varvec a i top varvec x natural 2 1 le i le m this problem also dubbed as phase retrieval spans multiple domains including physical sciences and machine learning we investigate the efficacy of gradient descent or wirtinger flow designed for the nonconvex least squares problem we prove that under gaussian designs gradient descent when randomly initialized yields an epsilon accurate solution in o big log n log 1 epsilon big iterations given nearly minimal samples thus achieving near optimal computational and sample complexities at once this provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval without the need of i carefully designed initialization ii sample splitting or iii sophisticated saddle point escaping schemes all of these are achieved by exploiting the statistical models in analyzing optimization algorithms via a leave one out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data

on the existence of block diagonal solutions to lyapunov and mathcal h _ infty riccati inequalities in this note we describe sufficient conditions when block diagonal solutions to lyapunov and mathcal h infty riccati inequalities exist in order to derive our results we define a new type of comparison systems which are positive and are computed using the state space matrices of the original possibly nonpositive systems computing the comparison system involves only the calculation of mathcal h infty norms of its subsystems we show that the stability of this comparison system implies the existence of block diagonal solutions to lyapunov and riccati inequalities furthermore our proof is constructive and the overall framework allows the computation of block diagonal solutions to these matrix inequalities with linear algebra and linear programming numerical examples illustrate our theoretical results

optimal rates of convergence for noisy sparse phase retrieval via thresholded wirtinger flow this paper considers the noisy sparse phase retrieval problem recovering a sparse signal mathbf x in mathbb r p from noisy quadratic measurements y j mathbf a j mathbf x 2 varepsilon j j 1 ldots m with independent sub exponential noise varepsilon j the goals are to understand the effect of the sparsity of mathbf x on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates adaptively inspired by the wirtinger flow ieee trans inform theory61 2015 1985 2007 proposed for non sparse and noiseless phase retrieval a novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the mathbf a j s are independent standard gaussian random vectors provided that the sample size is sufficiently large compared to the sparsity of mathbf x

random tilings with quasicrystal order transfer matrix approach the random tiling of the plane by a set of objects e g rhombi related by rotational e g tenfold symmetries is a paradigm for the formation of quasiperiodic order due to entropy such tilings are mapped to a higher dimensional space where they form hypersurfaces analogous to the interfaces in a solid on solid model the author argues that the fluctuations of the hypersurface should be described by a gradient squared free energy of entropic origin this implies quasi long range order in d 2 he shows how the random tiling can be decomposed into layers defines a transfer matrix and give prescriptions for using this method to determine numerically the stiffness of the gradient free energy

growing networks of overlapping communities with internal structure we introduce an intuitive model that describes both the emergence of community structure and the evolution of the internal structure of communities in growing social networks the model comprises two complementary mechanisms one mechanism accounts for the evolution of the internal link structure of a single community and the second mechanism coordinates the growth of multiple overlapping communities the first mechanism is based on the assumption that each node establishes links with its neighbors and introduces new nodes to the community at different rates we demonstrate that this simple mechanism gives rise to an effective maximal degree within communities this observation is related to the anthropological theory known as dunbar s number i e the empirical observation of a maximal number of ties which an average individual can sustain within its social groups the second mechanism is based on a recently proposed generalization of preferential attachment to community structure appropriately called structural preferential attachment spa the combination of these two mechanisms into a single model spa allows us to reproduce a number of the global statistics of real networks the distribution of community sizes of node memberships and of degrees the spa model also predicts a three qualitative regimes for the degree distribution within overlapping communities and b strong correlations between the number of communities to which a node belongs and its number of connections within each community we present empirical evidence that support our findings in real complex networks

f divergence inequalities this paper develops systematic approaches to obtain f divergence inequalities dealing with pairs of probability measures defined on arbitrary alphabets functional domination is one such approach where special emphasis is placed on finding the best possible constant upper bounding a ratio of f divergences another approach used for the derivation of bounds among f divergences relies on moment inequalities and the logarithmic convexity property which results in tight bounds on the relative entropy and bhattacharyya distance in terms of chi 2 divergences a rich variety of bounds are shown to hold under boundedness assumptions on the relative information special attention is devoted to the total variation distance and its relation to the relative information and relative entropy including reverse pinsker inequalities as well as on the e gamma divergence which generalizes the total variation distance pinsker s inequality is extended for this type of f divergence a result which leads to an inequality linking the relative entropy and relative information spectrum integral expressions of the renyi divergence in terms of the relative information spectrum are derived leading to bounds on the renyi divergence in terms of either the variational distance or relative entropy

block diagonal semistability factors and lyapunov semistability of block triangular matrices abstract the concept of lyapunov diagonal semi stability is generalized to the block diagonal case unifying the theory of lyapunov diagonal stability and lyapunov stability the corresponding generalizations of the concepts of maximal lyapunov scaling factors are applied to study the lyapunov semistability of block triangular matrices

low rank matrix recovery from errors and erasures this paper considers the recovery of a low rank matrix from an observed version that simultaneously contains both 1 erasures most entries are not observed and 2 errors values at a constant fraction of unknown locations are arbitrarily corrupted we provide a new unified performance guarantee on when minimizing nuclear norm plus l1 norm succeeds in exact recovery our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns by specializing this one single result in different ways we recover up to poly log factors as corollaries all the existing results in exact matrix completion and exact sparse and low rank matrix decomposition our unified result also provides the first guarantees for 1 recovery when we observe a vanishing fraction of entries of a corrupted matrix and 2 deterministic matrix completion

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

maximal lyapunov scaling factors and their applications in the study of lyapunov diagonal semistability of block triangular matrices abstract for a lyapunov diagonally semistable matrix a we introduce the concept of maximal lyapunov scaling factor that is a lyapunov scaling factor d of a such that the range of ad da t is of maximal dimension we investigate properties of maximal lyapunov scaling factors and apply them in proving a characterization for block triangular lyapunov diagonally semistable matrices in terms of lyapunov diagonal semistability of the diagonal blocks the latter result is used to prove that a block triangular matrix is lyapunov diagonally semistable if and only if every 2 2 block principal submatrix of it is lyapunov diagonally semistable

information recovery from pairwise measurements this paper is concerned with jointly recovering n node variables left x i right 1 leq i leq n from a collection of pairwise difference measurements imagine we acquire a few observations taking the form of x i x j the observation pattern is represented by a measurement graph mathcal g with an edge set mathcal e such that x i x j is observed if and only if i j in mathcal e to account for noisy measurements in a general manner we model the data acquisition process by a set of channels with given input output transition measures employing information theoretic tools applied to channel decoding problems we develop a unified framework to characterize the fundamental recovery criterion which accommodates general graph structures alphabet sizes and channel transition measures in particular our results isolate a family of minimum channel divergence measures to characterize the degree of measurement corruption which together with the size of the minimum cut of mathcal g dictates the feasibility of exact information recovery for various homogeneous graphs the recovery condition depends almost only on the edge sparsity of the measurement graph irrespective of other graphical metrics alternatively the minimum sample complexity required for these graphs scales like n log n mathsf hel 1 2 min for certain information metric mathsf hel 1 2 min defined in the main text as long as the alphabet size is not super polynomial in n we apply our general theory to three concrete applications including the stochastic block model the random corruption model and the haplotype assembly problem our theory leads to orderwise tight recovery conditions for all these scenarios

lyapunov diagonal semistability of real h matrices we characterize lyapunov diagonally stable real h matrices and those real h matrices which are lyapunov diagonally semistable but not lyapunov diagonally stable called lyapunov diagonally near stable the latter characterization is given in terms of the principal submatrix rank property defined here we apply our results to the numerical abscissas of real matrices one of our main tools is a slight strengthening of classical results of ostrowski which we derive from a fundamental theorem of wielandt

solving systems of random quadratic equations via truncated amplitude flow this paper presents a new algorithm termed truncated amplitude flow taf to recover an unknown vector x from a system of quadratic equations of the form y i langle a i x rangle 2 where a i s are given random measurement vectors this problem is known to be np hard in general we prove that as soon as the number of equations is on the order of the number of unknowns taf recovers the solution exactly up to a global unimodular constant with high probability and complexity growing linearly with both the number of unknowns and the number of equations our taf approach adapts the amplitude based empirical loss function and proceeds in two stages in the first stage we introduce an orthogonality promoting initialization that can be obtained with a few power iterations stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations which are able to handle the rather challenging nonconvex and nonsmooth amplitude based objective function in particular when vectors x and a i s are real valued our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations furthermore even under the same initialization the proposed amplitude based refinement outperforms existing wirtinger flow variants corroborating the superior performance of taf over state of the art algorithms

achieving exact cluster recovery threshold via semidefinite programming the binary symmetric stochastic block model deals with a random graph of n vertices partitioned into two equal sized clusters such that each pair of vertices is independently connected with probability p within clusters and q across clusters in the asymptotic regime of p a log n n and q b log n n for fixed a b and n to infty we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one resolving a conjecture of abbe et al furthermore we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to n

universally consistent vertex classification for latent positions graphs in this work we show that using the eigen decomposition of the adjacency matrix we can consistently estimate feature maps for latent position graphs with positive definite link function provided that the latent positions are i i d from some distribution f we then consider the exploitation task of vertex classification where the link

sharp nonasymptotic bounds on the norm of random matrices with independent entries we obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results if xx is the n nn n symmetric matrix with xij n 0 b2ij xij n 0 bij2 we show that this bound is optimal in the sense that a matching lower bound holds under mild assumptions and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum analogous results are obtained for rectangular matrices and for more general sub gaussian or heavy tailed distributions of the entries and we derive tail bounds in addition to bounds on the expected norm the proofs are based on a combination of the moment method and geometric functional analysis techniques as an application we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse wigner matrices we also recover a result of seginer on the norm of rademacher matrices

nonconvex phase synchronization we estimate n phases angles from noisy pairwise relative phase measurements the task is modeled as a nonconvex least squares optimization problem it was recently shown that this problem can be solved in polynomial time via convex relaxation under some conditions on the noise in this paper under similar but more restrictive conditions we show that a modified version of the power method converges to the global optimum this is simpler and empirically faster than convex approaches empirically they both succeed in the same regime further analysis shows that in the same noise regime as previously studied second order necessary optimality conditions for this quadratically constrained quadratic program are also sufficient despite nonconvexity

on the range of a hadamard power of a positive semidefinite matrix abstract for any n n complex matrix a and any integer k 1 we prove that the span of image of the map x ax k is equal to the range of aa k where x and x k denote the conjugate transpose and the k th hadamard power of a matrix x respectively this settles a conjecture due to gorni and tutaj gasinska related to the study of the jacobian conjecture

quadratic linear keller maps of nilpotency index three let f x h cn cn be a homogeneous polynomial map of degree d 2 and fa y ay d cn cn a power linear map such that f and fa are a generalized gorni zampieri pair we discuss the relation between the nilpotency indices of jh and j ay d and we show that f is linearly triangularizable if and only if fa is linearly triangularizable as a consequence we show that a quadratic linear keller map fa y ay 2 with nilpotency index three i e j ay 2 3 0 is linearly triangularizable