Consider an Erd "os-Renyi random graph in which each edge is present independently with probability 1 2, except for a subset @math of the vertices that form a clique (a completely connected subgraph). We consider the problem of identifying the clique, given a realization of such a random graph. The best known algorithm provably finds the clique in linear time with high probability, provided @math dekel2011finding . Spectral methods can be shown to fail on cliques smaller than @math . In this paper we describe a nearly linear time algorithm that succeeds with high probability for @math for any @math . This is the first algorithm that provably improves over spectral methods. We further generalize the hidden clique problem to other background graphs (the standard case corresponding to the complete graph on @math vertices). For large girth regular graphs of degree @math we prove that local' algorithms succeed if @math and fail if @math .
The algorithm we introduce for the case @math is analogous to the linearized BP' algorithm of @cite_24 @cite_9 , and to the approximate message passing (AMP) algorithm of @cite_13 @cite_36 @cite_15 . These ideas have been applied to low-rank approximation in @cite_7 . The present setting poses however several technical challenges with respect to earlier work in this area: @math The entries of the data matrix are not i.i.d.; @math They are non-Gaussian with --in general-- non-zero mean; @math We seek exact recovery instead of estimation; @math The sparsity set @math to be reconstructed scales sublinearly with @math .
@cite_7 @cite_36 @cite_9 @cite_24 @cite_15 @cite_13
2040969041 2610971674 2537629200 2101435685 2067038358 2082029531
Given a high-dimensional data matrix @math , Approximate Message Passing (AMP) algorithms construct sequences of vectors @math , @math , indexed by @math by iteratively applying @math or @math , and suitable non-linear functions, which depend on the specific application. Special instances of this approach have been developed --among other applications-- for compressed sensing reconstruction, robust regression, Bayesian estimation, low-rank matrix recovery, phase retrieval, and community detection in graphs. For certain classes of random matrices @math , AMP admits an asymptotically exact description in the high-dimensional limit @math , which goes under the name of state evolution.' Earlier work established state evolution for separable non-linearities (under certain regularity conditions). Nevertheless, empirical work demonstrated several important applications that require non-separable functions. In this paper we generalize state evolution to Lipschitz continuous non-separable nonlinearities, for Gaussian matrices @math . Our proof makes use of Bolthausen's conditioning technique along with several approximation arguments. In particular, we introduce a modified algorithm (called LAMP for Long AMP) which is of independent interest. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math . In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension @math and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components @math has at most @math non-zero entries. We are particularly interested in the high dimensional regime wherein @math is comparable to, or even much larger than @math . In an influential paper, johnstone2004sparse introduced a simple algorithm that estimates the support of the principal vectors @math by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if @math , and to fail with high probability if @math for two constants @math . Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by KrauthgamerSPCA . On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for @math (assuming @math of the same order as @math ). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as @math much smaller than @math . Recent lower bounds berthet2013computational, ma2015sum suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before. Low-rank approximation is a common tool used to accelerate kernel methods: the @math kernel matrix @math is approximated via a rank- @math matrix @math which can be stored in much less space and processed more quickly. In this work we study the limits of computationally efficient low-rank kernel approximation. We show that for a broad class of kernels, including the popular Gaussian and polynomial kernels, computing a relative error @math -rank approximation to @math is at least as difficult as multiplying the input data matrix @math by an arbitrary matrix @math . Barring a breakthrough in fast matrix multiplication, when @math is not too large, this requires @math time where @math is the number of non-zeros in @math . This lower bound matches, in many parameter regimes, recent work on subquadratic time algorithms for low-rank approximation of general kernels [MM16,MW17], demonstrating that these algorithms are unlikely to be significantly improved, in particular to @math input sparsity runtimes. At the same time there is hope: we show for the first time that @math time approximation is possible for general radial basis function kernels (e.g., the Gaussian kernel) for the closely related problem of low-rank approximation of the kernelized dataset. In sparse recovery we are given a matrix @math (the dictionary) and a vector of the form @math where @math is sparse, and the goal is to recover @math . This is a central notion in signal processing, statistics and machine learning. But in applications such as sparse coding, edge detection, compression and super resolution, the dictionary @math is unknown and has to be learned from random examples of the form @math where @math is drawn from an appropriate distribution --- this is the dictionary learning problem. In most settings, @math is overcomplete: it has more columns than rows. This paper presents a polynomial-time algorithm for learning overcomplete dictionaries; the only previously known algorithm with provable guarantees is the recent work of Spielman, Wang and Wright who gave an algorithm for the full-rank case, which is rarely the case in applications. Our algorithm applies to incoherent dictionaries which have been a central object of study since they were introduced in seminal work of Donoho and Huo. In particular, a dictionary is @math -incoherent if each pair of columns has inner product at most @math . The algorithm makes natural stochastic assumptions about the unknown sparse vector @math , which can contain @math non-zero entries (for any @math ). This is close to the best @math allowable by the best sparse recovery algorithms even if one knows the dictionary @math exactly. Moreover, both the running time and sample complexity depend on @math , where @math is the target accuracy, and so our algorithms converge very quickly to the true dictionary. Our algorithm can also tolerate substantial amounts of noise provided it is incoherent with respect to the dictionary (e.g., Gaussian). In the noisy setting, our running time and sample complexity depend polynomially on @math , and this is necessary. We show how to approximate a data matrix @math with a much smaller sketch @math that can be used to solve a general class of constrained k-rank approximation problems to within @math error. Importantly, this class of problems includes @math -means clustering and unconstrained low rank approximation (i.e. principal component analysis). By reducing data points to just @math dimensions, our methods generically accelerate any exact, approximate, or heuristic algorithm for these ubiquitous problems. For @math -means dimensionality reduction, we provide @math relative error results for many common sketching techniques, including random row projection, column selection, and approximate SVD. For approximate principal component analysis, we give a simple alternative to known algorithms that has applications in the streaming setting. Additionally, we extend recent work on column-based matrix reconstruction, giving column subsets that not only cover' a good subspace for @math , but can be used directly to compute this subspace. Finally, for @math -means clustering, we show how to achieve a @math approximation by Johnson-Lindenstrauss projecting data points to just @math dimensions. This gives the first result that leverages the specific structure of @math -means to achieve dimension independent of input size and sublinear in @math . Motivated by applications in which the data may be formulated as a matrix, we consider algorithms for several common linear algebra problems. These algorithms make more efficient use of computational resources, such as the computation time, random access memory (RAM), and the number of passes over the data, than do previously known algorithms for these problems. In this paper, we devise two algorithms for the matrix multiplication problem. Suppose @math and @math (which are @math and @math , respectively) are the two input matrices. In our main algorithm, we perform @math independent trials, where in each trial we randomly sample an element of @math with an appropriate probability distribution @math on @math . We form an @math matrix @math consisting of the sampled columns of @math , each scaled appropriately, and we form a @math matrix @math using the corresponding rows of @math , again scaled appropriately. The choice of @math and the column and row scaling are crucial features of the algorithm. When these are chosen judiciously, we show that @math is a good approximation to @math . More precisely, we show that @math where @math denotes the Frobenius norm, i.e., @math . This algorithm can be implemented without storing the matrices @math and @math in RAM, provided it can make two passes over the matrices stored in external memory and use @math additional RAM to construct @math and @math . We then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm. In addition, we present a model (the pass-efficient model) in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets. In this model, the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm. The input matrices may be presented in any order of the entries (and not just row or column order), as is the case in many applications where, e.g., the data has been written in by multiple agents. In addition, the input matrices may be presented in a sparse representation, where only the nonzero entries are written. Abstract Irregular sampling and “stable sampling” of band-limited functions have been studied by H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52]. We prove that quasicrystals are sets of stable sampling. To cite this article: B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008). Purpose – The purpose of this paper is to suggest an alternative to the widely used Times Cited criterion for analysing citation networks. The approach involves taking account of the natures of the papers that cite a given paper, so as to differentiate between papers that attract the same number of citations.Design methodology approach – ArticleRank is an algorithm that has been derived from Google's PageRank algorithm to measure the influence of journal articles. ArticleRank is applied to two datasets – a citation network based on an early paper on webometrics, and a self‐citation network based on the 19 most cited papers in the Journal of Documentation – using citation data taken from the Web of Knowledge database.Findings – ArticleRank values provide a different ranking of a set of papers from that provided by the corresponding Times Cited values, and overcomes the inability of the latter to differentiate between papers with the same numbers of citations. The difference in rankings between Times Cited ... We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with @math constraint matrices, each of dimension @math , rank @math , and sparsity @math . The first algorithm assumes an input model where one is given access to entries of the matrices at unit cost. We show that it has run time @math , where @math is the error. This gives an optimal dependence in terms of @math and quadratic improvement over previous quantum algorithms when @math . The second algorithm assumes a fully quantum input model in which the matrices are given as quantum states. We show that its run time is @math , with @math an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically in @math and polynomially in @math . We apply the second SDP solver to the problem of learning a good description of a quantum state with respect to a set of measurements: Given @math measurements and copies of an unknown state @math , we show we can find in time @math a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as @math on the @math measurements, up to error @math . The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest. We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze and generalize the Belief Propagation (BP) approach to computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. We introduce a fractional free energy functional parameterized by a scalar parameter @math , where @math corresponds to the BP limit and @math corresponds to the exclusion principle Mean-Field (MF) limit, and show monotonicity and continuity of the functional with @math . We observe that the optimal value of @math , where the @math -parameterized functional is equal to the exact free energy (defined as the minus log of the permanent), lies in the @math range, with the low and high values from the range producing provable low and upper bounds for the permanent. Our experimental analysis suggests that the optimal @math varies for different ensembles considered but it always lies in the @math interval. Besides, for all ensembles considered the behavior of the optimal @math is highly distinctive, thus offering a lot of practical potential for estimating permanents of non-negative matrices via the fractional free energy functional approach. Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an @math matrix @math and a rank parameter @math . In our first algorithm, @math is chosen, and we let @math , where @math is the Moore-Penrose generalized inverse of @math . In our second algorithm @math , @math , @math are chosen, and we let @math . ( @math and @math are matrices that consist of actual columns and rows, respectively, of @math , and @math is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least @math , @math , where @math is the “best” rank- @math approximation provided by truncating the SVD of @math , and where @math is the Frobenius norm of the matrix @math . The number of columns of @math and rows of @math is a low-degree polynomial in @math , @math , and @math . Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants of these matrix decompositions over the last ten years. However, our two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist. Both of our algorithms are simple and they take time of the order needed to approximately compute the top @math singular vectors of @math . The technical crux of our analysis is a novel, intuitive sampling method we introduce in this paper called “subspace sampling.” In subspace sampling, the sampling probabilities depend on the Euclidean norms of the rows of the top singular vectors. This allows us to obtain provable relative-error guarantees by deconvoluting “subspace” information and “size-of- @math ” information in the input matrix. This technique is likely to be useful for other matrix approximation and data analysis problems. We prove that any real matrix A contains a subset of at most 4k e+ 2k log(k+1) rows whose span “contains” a matrix of rank at most k with error only (1+e) times the error of the best rank-k approximation of A. We complement it with an almost matching lower bound by constructing matrices where the span of any k 2e rows does not “contain” a relative (1+e)-approximation of rank k. Our existence result leads to an algorithm that finds such rank-k approximation in time @math i.e., essentially O(Mk e), where M is the number of nonzero entries of A. The algorithm maintains sparsity, and in the streaming model [12,14,15], it can be implemented using only 2(k+1)(log(k+1)+1) passes over the input matrix and @math additional space. Previous algorithms for low-rank approximation use only one or two passes but obtain an additive approximation. We develop several efficient algorithms for the classical problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input @math matrix @math , this problem asks to find diagonal (scaling) matrices @math and @math (if they exist), so that @math @math -approximates a doubly stochastic, or more generally a matrix with prescribed row and column sums. We address the general scaling problem as well as some important special cases. In particular, if @math has @math nonzero entries, and if there exist @math and @math with polynomially large entries such that @math is doubly stochastic, then we can solve the problem in total complexity @math . This greatly improves on the best known previous results, which were either @math or @math . Our algorithms are based on tailor-made first and second order techniques, combined with other recent advances in continuous optimization, which may be of independent interest for solving similar problems. For any real number @math , we nearly completely characterize the space complexity of estimating @math for @math matrices @math in which each row and each column has @math non-zero entries and whose entries are presented one at a time in a data stream model. Here the @math are the singular values of @math , and when @math , @math is the @math -th power of the Schatten @math -norm. We show that when @math is not an even integer, to obtain a @math -approximation to @math with constant probability, any @math -pass algorithm requires @math bits of space, where @math as @math and @math is a constant independent of @math . However, when @math is an even integer, we give an upper bound of @math bits of space, which holds even in the turnstile data stream model. The latter is optimal up to @math factors. Our results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices @math : the previous best lower bound was @math for @math , @math for @math and @math for @math . We note for @math , while our lower bound for even integers is the same, for other @math in this range our lower bound is @math , which is considerably stronger than the previous @math for small enough constant @math . We obtain similar near-linear lower bounds for Ky-Fan norms, SVD entropy, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to our work. The little Grothendieck problem consists of maximizing @math źijCijxixj for a positive semidefinite matrix C, over binary variables @math xiź ?1 . In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given @math CźRdn?dn a positive semidefinite matrix, the objective is to maximize @math źijtrCijTOiOjT restricting @math Oi to take values in the group of orthogonal matrices @math Od, where @math Cij denotes the (ij)-th @math d?d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices @math Od and show a constant approximation ratio. Our method is based on semidefinite programming. For a given @math dź1, we show a constant approximation ratio of @math źR(d)2, where @math źR(d) is the expected average singular value of a @math d?d matrix with random Gaussian @math N0,1d i.i.d. entries. For @math d=1 we recover the known @math źR(1)2=2 ź approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous little Grothendieck problem over the Unitary Group @math Ud. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio of @math 122. The little Grothendieck problem falls under the larger class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and provides better approximation with matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints, recovering, when @math d=1, the sharp, known ratios. The following problem is considered: given a matrix @math in @math , ( @math rows and @math columns), a vector @math in @math , and @math , compute a vector @math satisfying @math if such exists, such that @math has the fewest number of non-zero entries over all such vectors. It is shown that the problem is NP-hard, but that the well-known greedy heuristic is good in that it computes a solution with at most @math non-zero entries, where @math is the optimum number of nonzero entries at error @math , @math is the matrix obtained by normalizing each column of @math with respect to the @math norm, and @math is its pseudo-inverse. We study the problem of recovering a low-rank matrix @math from linear measurements using an over-parameterized model. We parameterize the rank- @math matrix @math by @math where @math is a square matrix, whereas the number of linear measurements is much less than @math . We show that with @math random linear measurements, the gradient descent on the squared loss, starting from a small initialization, recovers @math approximately in @math iterations. The results solve the conjecture of under the restricted isometry property, and demonstrate that the training algorithm can provide an implicit regularization for non-linear matrix factorization models. We initiate the study of numerical linear algebra in the sliding window model, where only the most recent @math updates in the data stream form the underlying set. Although most existing work in the sliding window model uses the smooth histogram framework, most interesting linear-algebraic problems are not smooth; we show that the spectral norm, vector induced matrix norms, generalized regression, and low-rank approximation are not amenable for the smooth histogram framework. To overcome this challenge, we first give a deterministic algorithm that achieves spectral approximation in the sliding window model that can be viewed as a generalization of smooth histograms, using the Loewner ordering of PSD matrices. We then give algorithms for both spectral approximation and low-rank approximation that are space-optimal up to polylogarithmic factors. Our algorithms are based on a new notion of "reverse online" leverage scores that account for both how unique and how recent a row is, while preserving sparsity so that both our algorithms run in input sparsity runtime, up to lower order factors. We show that our techniques have applications to linear-algebraic problems in other settings. Specifically, we show that our analysis immediately implies an algorithm for low-rank approximation in the online setting that is space-optimal up to logarithmic factors, as well as nearly input sparsity time. We show our deterministic spectral approximation algorithm can be used to handle @math spectral approximation in the sliding window model under a certain assumption on the bit complexity of the entries. Finally, we show that our downsampling framework can be applied to the problem of approximate matrix multiplication and provide upper and lower bounds that are tight up to @math factors.