max norm optimization for robust matrix recovery this paper studies the matrix completion problem under arbitrary sampling schemes we propose a new estimator incorporating both max norm and nuclear norm regularization based on which we can conduct efficient low rank matrix recovery using a random subset of entries observed with additive noise under general non uniform and unknown sampling distributions this method significantly relaxes the uniform sampling assumption imposed for the widely used nuclear norm penalized approach and makes low rank matrix recovery feasible in more practical settings theoretically we prove that the proposed estimator achieves fast rates of convergence under different settings computationally we propose an alternating direction method of multipliers algorithm to efficiently compute the estimator which bridges a gap between theory and practice of machine learning methods with max norm regularization further we provide thorough numerical studies to evaluate the proposed method using both simulated and real datasets

optimal large scale quantum state tomography with pauli measurements quantum state tomography aims to determine the state of a quantum system as represented by a density matrix it is a fundamental task in modern scientific studies involving quantum systems in this paper we study estimation of high dimensional density matrices based on pauli measurements in particular under appropriate notion of sparsity we establish the minimax optimal rates of convergence for estimation of the density matrix under both the spectral and frobenius norm losses and show how these rates can be achieved by a common thresholding approach numerical performance of the proposed estimator is also investigated

a theoretical analysis of noisy sparse subspace clustering on dimensionality reduced data subspace clustering is the problem of partitioning unlabeled data points into a number of clusters so that data points within one cluster lie approximately on a low dimensional linear subspace in many practical scenarios the dimensionality of data points to be clustered is compressed due to the constraints of measurement computation or privacy in this paper we study the theoretical properties of a popular subspace clustering algorithm named sparse subspace clustering ssc and establish formal success conditions of ssc on dimensionality reduced data our analysis applies to the most general fully deterministic model where both underlying subspaces and data points within each subspace are deterministically positioned and also a wide range of dimensionality reduction techniques e g gaussian random projection uniform subsampling and sketching that fall into a subspace embedding framework finally we apply our analysis to a differentially private ssc algorithm and established both privacy and utility guarantees of the proposed method

matrix completion with column manipulation near optimal sample robustness rank tradeoffs this paper considers the problem of matrix completion when some number of the columns are completely and arbitrarily corrupted potentially by a malicious adversary it is well known that standard algorithms for matrix completion can return arbitrarily poor results if even a single column is corrupted one direct application comes from robust collaborative filtering here some number of users are so called manipulators who try to skew the predictions of the algorithm by calibrating their inputs to the system in this paper we develop an efficient algorithm for this problem based on a combination of a trimming procedure and a convex program that minimizes the nuclear norm and the ell 1 2 norm our theoretical results show that given a vanishing fraction of observed entries it is nevertheless possible to complete the underlying matrix even when the number of corrupted columns grows significantly our results hold without any assumptions on the locations or values of the observed entries of the manipulated columns moreover we show by an information theoretic argument that our guarantees are nearly optimal in terms of the fraction of sampled entries on the authentic columns the fraction of corrupted columns and the rank of the underlying matrix our results therefore sharply characterize the tradeoffs between sample robustness and rank in matrix completion

blind deconvolution using convex programming we consider the problem of recovering two unknown vectors w and x of length l from their circular convolution we make the structural assumption that the two vectors are members of known subspaces one with dimension n and the other with dimension k although the observed convolution is nonlinear in both w and x it is linear in the rank 1 matrix formed by their outer product wx this observation allows us to recast the deconvolution problem as low rank matrix recovery problem from linear measurements whose natural convex relaxation is a nuclear norm minimization program we prove the effectiveness of this relaxation by showing that for generic signals the program can deconvolve w and x exactly when the maximum of n and k is almost on the order of l that is we show that if x is drawn from a random subspace of dimension n and w is a vector in a subspace of dimension k whose basis vectors are spread out in the frequency domain then nuclear norm minimization recovers wx without error we discuss this result in the context of blind channel estimation in communications if we have a message of length n which we code using a random l x n coding matrix and the encoded message travels through an unknown linear time invariant channel of maximum length k then the receiver can recover both the channel response and the message when l n k to within constant and log factors

deterministic and probabilistic conditions for finite completability of low tucker rank tensor we investigate the fundamental conditions on the sampling pattern i e locations of the sampled entries for finite completability of a low rank tensor given some components of its tucker rank in order to find the deterministic necessary and sufficient conditions we propose an algebraic geometric analysis on the tucker manifold which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the grassmannian manifold this analysis characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern which is closely related to finite completability of the sampled tensor where finite completability simply means that the number of possible completions of the sampled tensor is finite probabilistic conditions are then studied and a lower bound on the sampling probability is given which guarantees that the proposed deterministic conditions on the sampling patterns for finite completability hold with high probability furthermore using the proposed geometric approach for finite completability we propose a sufficient condition on the sampling pattern that ensures there exists exactly one completion of the sampled tensor

decoupling inequalities for the tail probabilities of multivariate u statistics in this paper we present a decoupling inequality that shows that multivariate u statistics can be studied as sums of conditionally independent random variables this result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration more precisely we get the following result let x j be a sequence of independent random variables on a measurable space mathscr j s and let x j i j 1 ldots k be k independent copies of x i let f i 1i 2 ldots i k be families of functions of k variables taking s times cdots times s into a banach space b cdots then for all n geq k geq 2 t 0 there exist numerical constants c k depending on k only so that p bigg big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 1 i 2 ldots x 1 i k big geq t bigg leq c kp bigg c k big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 2 i 2 ldots x k i k big geq t bigg the reverse bound holds if in addition the following symmetry condition holds almost surely f i 1i 2 cdots i k x i 1 x i 2 ldots x i k f i pi 1 i pi 2 cdots i pi k x i pi 1 x i pi 2 ldots x i pi k for all permutations pi of 1 ldots k

completing low rank matrices with corrupted samples from few coefficients in general basis subspace recovery from the corrupted and missing data is crucial for various applications in signal processing and information theory to complete missing values and detect column corruptions the existing robust matrix completion mc methods mostly concentrate on recovering a low rank matrix from a few corrupted coefficients with respect to standard basis which however does not apply to more general basis e g fourier basis in this paper we prove that the range space of an m times n matrix with rank r can be exactly recovered from a few coefficients with respect to general basis though r and the number of corrupted samples are both as high as o min m n log 3 m n our model covers the previous ones as special cases and robust mc can recover the intrinsic matrix with a higher rank moreover we suggest a universal choice of the regularization parameter which is lambda 1 sqrt log n by our ell 2 1 filtering algorithm which has theoretical guarantees we can further reduce the computational cost of our model as an application we also find that the solutions to extended robust low rank representation and to our extended robust mc are mutually expressible so both our theory and algorithm can be applied to the subspace clustering problem with missing values under certain conditions the experiments verify our theories

convex optimization learning of faithful euclidean distance representations in nonlinear dimensionality reduction classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by euclidean distances in a low dimensional space advanced models such as maximum variance unfolding mvu and minimum volume embedding mve use semi definite programming sdp to reconstruct such faithful representations while those sdp models are capable of producing high quality configuration numerically they suffer two major drawbacks one is that there exist no theoretically guaranteed bounds on the quality of the configuration the other is that they are slow in computation when the data points are beyond moderate size in this paper we propose a convex optimization model of euclidean distance matrices we establish a non asymptotic error bound for the random graph model with sub gaussian noise and prove that our model produces a matrix estimator of high accuracy when the order of the uniform sample size is roughly the degree of freedom of a low rank matrix up to a logarithmic factor our results partially explain why mvu and mve often work well moreover the convex optimization model can be efficiently solved by a recently proposed 3 block alternating direction method of multipliers numerical experiments show that the model can produce configurations of high quality on large data points that the sdp approach would struggle to cope with

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

time discretizations for numerical optimisation of hyperbolic problems abstract we consider a class of numerical schemes for optimal control problems of hyperbolic conservation laws we focus on finite volume schemes using relaxation as a numerical approach to the optimality system in particular we study the arising numerical schemes for the adjoint equation and derive necessary conditions on the time integrator we show that the resulting schemes are in particular asymptotic preserving for both the adjoint and forward equation we furthermore prove that higher order time integrator yields suitable runge kutta schemes the discussion includes the numerically interesting zero relaxation case

a fixed point iterative method for low n rank tensor pursuit the linearly constrained tensor n rank minimization problem is an extension of matrix rank minimization it is applicable in many fields which use the multi way data such as data mining machine learning and computer vision in this paper we adapt operator splitting technique and convex relaxation technique to transform the original problem into a convex unconstrained optimization problem and propose a fixed point iterative method to solve it we also prove the convergence of the method under some assumptions by using a continuation technique we propose a fast and robust algorithm for solving the tensor completion problem which is called fp lrtc fixed point for low n rank tensor completion our numerical results on randomly generated and real tensor completion problems demonstrate that this algorithm is effective especially for easy problems

sublinear time algorithms for approximate semidefinite programming we consider semidefinite optimization in a saddle point formulation where the primal solution is in the spectrahedron and the dual solution is a distribution over affine functions we present an approximation algorithm for this problem that runs in sublinear time in the size of the data to the best of our knowledge this is the first algorithm to achieve this our algorithm is also guaranteed to produce low rank solutions we further prove lower bounds on the running time of any algorithm for this problem showing that certain terms in the running time of our algorithm cannot be further improved finally we consider a non affine version of the saddle point problem and give an algorithm that under certain assumptions runs in sublinear time

poisson matrix recovery and completion we extend the theory of low rank matrix recovery and completion to the case when poisson observations for a linear combination or a subset of the entries of a matrix are available which arises in various applications with count data we consider the usual matrix recovery formulation through maximum likelihood with proper constraints on the matrix m of size d 1 by d 2 and establish theoretical upper and lower bounds on the recovery error our bounds for matrix completion are nearly optimal up to a factor on the order of cal o log d 1 d 2 these bounds are obtained by combining techniques for recovering sparse vectors with compressed measurements in poisson noise those for analyzing low rank matrices as well as those for one bit matrix completion davenport 1 bit matrix completion information and inference information and inference vol 3 no 3 pp 189 223 sep 2014 although these two problems are different in nature the adaptation requires new techniques exploiting properties of the poisson likelihood function and tackling the difficulties posed by the locally sub gaussian characteristic of the poisson distribution our results highlight a few important distinctions of the poisson case compared to the prior work including having to impose a minimum signal to noise requirement on each observed entry and a gap in the upper and lower bounds we also develop a set of efficient iterative algorithms and demonstrate their good performance on synthetic examples and real data

rank minimization over finite fields fundamental limits and coding theoretic interpretations this paper establishes information theoretic limits for estimating a finite field low rank matrix given random linear measurements of it these linear measurements are obtained by taking inner products of the low rank matrix with random sensing matrices necessary and sufficient conditions on the number of measurements required are provided it is shown that these conditions are sharp and the minimum rank decoder is asymptotically optimal the reliability function of this decoder is also derived by appealing to de caen s lower bound on the probability of a union the sufficient condition also holds when the sensing matrices are sparse a scenario that may be amenable to efficient decoding more precisely it is shown that if the n n sensing matrices contain on average nlog n entries the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field analogies are drawn between the aforementioned results and rank metric codes in the coding theory literature in fact we are also strongly motivated by understanding when minimum rank distance decoding of random rank metric codes succeeds to this end we derive minimum distance properties of equiprobable and sparse rank metric codes these distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one

concentration inequalities for matrix martingales in continuous time this paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time these results extend previously established freedman and bernstein inequalities for series of random matrices to the class of continuous time processes our analysis relies on a new supermartingale property of the trace exponential proved within the framework of stochastic calculus we provide also several examples that illustrate the fact that our results allow us to recover easily several formerly obtained sharp bounds for discrete time matrix martingales

coherence pursuit fast simple and robust principal component analysis this paper presents a remarkably simple yet powerful algorithm termed coherence pursuit cop to robust principal component analysis pca as inliers lie in a low dimensional subspace and are mostly correlated an inlier is likely to have strong mutual coherence with a large number of data points by contrast outliers either do not admit low dimensional structures or form small clusters in either case an outlier is unlikely to bear strong resemblance to a large number of data points given that cop sets an outlier apart from an inlier by comparing their coherence with the rest of the data points the mutual coherences are computed by forming the gram matrix of the normalized data points subsequently the sought subspace is recovered from the span of the subset of the data points that exhibit strong coherence with the rest of the data as cop only involves one simple matrix multiplication it is significantly faster than the state of the art robust pca algorithms we derive analytical performance guarantees for cop under different models for the distributions of inliers and outliers in both noise free and noisy settings cop is the first robust pca algorithm that is simultaneously non iterative provably robust to both unstructured and structured outliers and can tolerate a large number of unstructured outliers

a flexible and efficient algorithmic framework for constrained matrix and tensor factorization we propose a general algorithmic framework for constrained matrix and tensor factorization which is widely used in signal processing and machine learning the new framework is a hybrid between alternating optimization ao and the alternating direction method of multipliers admm each matrix factor is updated in turn using admm hence the name ao admm this combination can naturally accommodate a great variety of constraints on the factor matrices and almost all possible loss measures for the fitting computation caching and warm start strategies are used to ensure that each update is evaluated efficiently while the outer ao framework exploits recent developments in block coordinate descent bcd type methods which help ensure that every limit point is a stationary point as well as faster and more robust convergence in practice three special cases are studied in detail non negative matrix tensor factorization constrained matrix tensor completion and dictionary learning extensive simulations and experiments with real data are used to showcase the effectiveness and broad applicability of the proposed framework

completing any low rank matrix provably matrix completion i e the exact and provable recovery of a low rank matrix from a small subset of its elements is currently only known to be possible if the matrix satisfies a restrictive structural constraint known as incoherence on its row and column spaces in these cases the subset of elements is assumed to be sampled uniformly at random in this paper we show that any rank r n by n matrix can be exactly recovered from as few as o nr log2 n randomly chosen elements provided this random choice is made according to a specific biased distribution suitably dependent on the coherence structure of the matrix the probability of any element being sampled should be at least a constant times the sum of the leverage scores of the corresponding row and column moreover we prove that this specific form of sampling is nearly necessary in a natural precise sense this implies that many other perhaps more intuitive sampling schemes fail we further establish three ways to use the above result for the setting when leverage scores are not known a priori a we describe a provably correct sampling strategy for the case when only the column space is incoherent and no assumption or knowledge of the row space is required b we propose a two phase sampling procedure for general matrices that first samples to estimate leverage scores followed by sampling for exact recovery these two approaches assume control over the sampling procedure c by using our main theorem in a reverse direction we provide an analysis showing the advantages of the empirically successful weighted nuclear trace norm minimization approach over the vanilla un weighted formulation given non uniformly distributed observed elements this approach does not require controlled sampling or knowledge of the leverage scores

matrix completion by least square low rank and sparse self representations abstract conventional matrix completion methods are generally based on rank minimization these methods assume that the given matrix is of low rank and the data points are drawn from a single subspace of low dimensionality therefore they are not effective in completing matrices where the data are drawn from multiple subspaces in this paper we establish a novel matrix completion framework that is based on self representation specifically least square low rank and sparse self representations based matrix completion algorithms are provided the underlying idea is that one data point can be efficiently reconstructed by other data points belonging to a common subspace where the missing entries are recovered so as to fit the common subspace the proposed algorithms actually maximize the weighted correlations among the columns of a given matrix we prove that the proposed algorithms are approximations for rank minimization of the incomplete matrix in addition they are able to complete high rank or even full rank matrices when the data are drawn from multiple subspaces comparative studies are conducted on synthetic datasets natural image inpainting tasks temperature prediction task and collaborative filtering tasks the results show that the proposed algorithms often outperform other state of the art algorithms in various tasks

finding a low rank basis in a matrix subspace for a given matrix subspace how can we find a basis that consists of low rank matrices this is a generalization of the sparse vector problem it turns out that when the subspace is spanned by rank 1 matrices the matrices can be obtained by the tensor cp decomposition for the higher rank case the situation is not as straightforward in this work we present an algorithm based on a greedy process applicable to higher rank problems our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation and then computes a matrix with that rank using the method of alternating projections we provide local convergence results and compare our algorithm with several alternative approaches applications include data compression beyond the classical truncated svd computing accurate eigenvectors of a near multiple eigenvalue image separation and graph laplacian eigenproblems

structured matrix completion with applications to genomic data integration abstractmatrix completion has attracted significant recent attention in many fields including statistics applied mathematics and electrical engineering current literature on matrix completion focuses primarily on independent sampling models under which the individual observed entries are sampled independently motivated by applications in genomic data integration we propose a new framework of structured matrix completion smc to treat structured missingness by design specifically our proposed method aims at efficient matrix recovery when a subset of the rows and columns of an approximately low rank matrix are observed we provide theoretical justification for the proposed smc method and derive lower bound for the estimation errors which together establish the optimal rate of recovery over certain classes of approximately low rank matrices simulation studies show that the method performs well in finite sample under a variety of configurations the method is applied to integrate several ovarian can

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

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

accelerating permutation testing in voxel wise analysis through subspace tracking a new plugin for snpm permutation testing is a non parametric method for obtaining the max null distribution used to compute corrected p values that provide strong control of false positives in neuroimaging however the computational burden of running such an algorithm can be significant we find that by viewing the permutation testing procedure as the construction of a very large permutation testing matrix t one can exploit structural properties derived from the data and the test statistics to reduce the runtime under certain conditions in particular we see that t is low rank plus a low variance residual this makes t a good candidate for low rank matrix completion where only a very small number of entries of t 0 35 of all entries in our experiments have to be computed to obtain a good estimate based on this observation we present rapidpt an algorithm that efficiently recovers the max null distribution commonly obtained through regular permutation testing in voxel wise analysis we present an extensive validation on a synthetic dataset and four varying sized datasets against two baselines statistical nonparametric mapping snpm13 and a standard permutation testing implementation referred as naivept we find that rapidpt achieves its best runtime performance on medium sized datasets 50 n 200 with speedups of 1 5 38 vs snpm13 and 20x 1000 vs naivept for larger datasets n 200 rapidpt outperforms naivept 6 200 on all datasets and provides large speedups over snpm13 when more than 10000 permutations 2 15 are needed the implementation is a standalone toolbox and also integrated within snpm13 able to leverage multi core architectures when available

random projections for classification a recovery approach random projection has been widely used in data classification it maps high dimensional data into a low dimensional subspace in order to reduce the computational cost in solving the related optimization problem while previous studies are focused on analyzing the classification performance in the low dimensional space in this paper we consider the recovery problem i e how to accurately recover the optimal solution to the original high dimensional optimization problem based on the solution learned after random projection we present a simple algorithm termed dual random projection which uses the dual solution of the low dimensional optimization problem to recover the optimal solution to the original problem our theoretical analysis shows that with a high probability the proposed algorithm is able to accurately recover the optimal solution to the original problem provided that the data matrix is approximately low rank and or optimal solution is approximately sparse we further show that the proposed algorithm can be applied iteratively to reducing the recovery error exponentially

expected length of the longest probe sequence in hash code searching an investigation ts made of the expected value of the maximum number of accesses needed to locate any element m a hashing file under various colhston resoluuon schemes this differs from usual worst case considerations winch for hashmg would be the largest sequence of accesses for the worst possible file asymptotic expressxons of these expected values are found for full and partly full tables for the open addressing scheme with a clustering free model these values are found to be 0 6315 x n for a full table and logan for a partly full table where n ts the number of records m is the size of the table and a n m for the open addressing scheme which reorders the insertions to minimize the worst case the lower bounds in n 1 077 and a in l a are found for full and partly full tables respectively fmauy for the separate chaining or direct chaining method both expected values are found to be f t n these results show that for these schemes the actual behawor of the worst case in hash tables is quite good on the average

restricted strong convexity and weighted matrix completion optimal bounds with noise we consider the matrix completion problem under a form of row column weighted entrywise sampling including the case of uniformentrywise sampling as a special case we analyze the associated random observation operator and prove that with high probability it satisfies a form of restricted strong convexity with respect to weighted frobenius norm using this property we obtain as corollaries a number of error bounds on matrix completion in the weighted frobenius norm under noisy sampling and for both exact and near low rank matrices our results are based on measures of the spikiness and low rankness of matrices that are less restrictive than the incoherence conditions imposed in previous work our technique involves an m estimator that includes controls on both the rank and spikiness of the solution and we establish non asymptotic error bounds in weighted frobenius norm for recovering matrices lying with lq balls of bounded spikiness using information theoretic methods we show that no algorithm can achieve better estimates up to a logarithmic factor over these same sets showing that our conditions on matrices and associated rates are essentially optimal

compressive multiplexing of correlated signals we present a general architecture for the acquisition of ensembles of correlated signals the signals are multiplexed onto a single line by mixing each one against a different code and then adding them together and the resulting signal is sampled at a high rate we show that if the m signals each band limited to w 2 hz can be approximated by a superposition of r m underlying signals then the ensemble can be recovered by sampling at a rate within a logarithmic factor of rw as compared with the cumulative nyquist rate of mw this sampling theorem shows that the correlation structure of the signal ensemble can be exploited in the acquisition process even though it is unknown a priori the reconstruction of the ensemble is recast as a low rank matrix recovery problem from linear measurements the architectures we are considering impose a certain type of structure on the linear operators although our results depend on the mixing forms being random this imposed structure results in a very different type of random projection than those analyzed in the low rank recovery literature to date

fast and accurate matrix completion via truncated nuclear norm regularization recovering a large matrix from a small subset of its entries is a challenging problem arising in many real applications such as image inpainting and recommender systems many existing approaches formulate this problem as a general low rank matrix approximation problem since the rank operator is nonconvex and discontinuous most of the recent theoretical studies use the nuclear norm as a convex relaxation one major limitation of the existing approaches based on nuclear norm minimization is that all the singular values are simultaneously minimized and thus the rank may not be well approximated in practice in this paper we propose to achieve a better approximation to the rank of matrix by truncated nuclear norm which is given by the nuclear norm subtracted by the sum of the largest few singular values in addition we develop a novel matrix completion algorithm by minimizing the truncated nuclear norm we further develop three efficient iterative procedures tnnr admm tnnr apgl and tnnr admmap to solve the optimization problem tnnr admm utilizes the alternating direction method of multipliers admm while tnnr agpl applies the accelerated proximal gradient line search method apgl for the final optimization for tnnr admmap we make use of an adaptive penalty according to a novel update rule for admm to achieve a faster convergence rate our empirical study shows encouraging results of the proposed algorithms in comparison to the state of the art matrix completion algorithms on both synthetic and real visual datasets

matrix concentration inequalities via the method of exchangeable pairs this paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix the analysis requires a matrix extension of the scalar concentration theory developed by sourav chatterjee using stein s method of exchangeable pairs when applied to a sum of independent random matrices this approach yields matrix generalizations of the classical inequalities due to hoeffding bernstein khintchine and rosenthal the same technique delivers bounds for sums of dependent random matrices and more general matrix valued functions of dependent random variables

randomized robust subspace recovery and outlier detection for high dimensional data matrices this paper explores and analyzes two randomized designs for robust principal component analysis employing low dimensional data sketching in one design a data sketch is constructed using random column sampling followed by low dimensional embedding while in the other sketching is based on random column and row sampling both designs are shown to bring about substantial savings in complexity and memory requirements for robust subspace learning over conventional approaches that use the full scale data a characterization of the sample and computational complexity of both designs is derived in the context of two distinct outlier models namely sparse and independent outlier models the proposed randomized approach can provably recover the correct subspace with computational and sample complexity which depend only weakly on the size of the data only through the coherence parameters the results of the mathematical analysis are confirmed through numerical simulations using both synthetic and real data

high rank matrix completion this paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low rank subspaces this generalizesthe standardlow rankmatrix completion problem to situations in which the matrix rank can be quite high or even full rank since the columns belong to a union of subspaces this problem may also be viewed as a missing data version of the subspace clustering problem let x be an n n matrix whose complete columns lie in a union of at most k subspaces each of rank r 1 a constant depending on the usual incoherence conditions the geometrical arrangement of subspaces and the distribution of columns over the subspaces the result is illustrated with numerical experiments and an application to internet distance matrix completion and topology identification

incoherent tensor norms and their applications in higher order tensor completion in this paper we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion we introduce a class of tensor norms by allowing for different levels of coherence which allows us to leverage the incoherence of a tensor in particular we show that a k th order tensor of multilinear rank r and dimension d times cdots times d can be recovered perfectly from as few as o r k 1 2 d 3 2 r k 1 d log d 2 uniformly sampled entries through an appropriate incoherent nuclear norm minimization our results demonstrate some key differences between completing a matrix and a higher order tensor they not only point to potential room for improvement over the usual nuclear norm minimization but also highlight the importance of explicitly accounting for incoherence when dealing with higher order tensors although our focus is primarily on the theoretical guarantees for nuclear norm minimization such insights may prove useful for understanding performance of other related methods and developing improved practical algorithms

nonconvex rectangular matrix completion via gradient descent without l regularization the analysis of nonconvex matrix completion has recently attracted much attention in the community of machine learning thanks to its computational convenience existing analysis on this problem however usually relies on ell 2 infty projection or regularization that involves unknown model parameters although they are observed to be unnecessary in numerical simulations in this paper we extend the analysis of the vanilla gradient descent for positive semidefinite matrix completion in the literature to the rectangular case and more significantly improve the required sampling rate from o mathrm poly kappa mu 3 r 3 log 3 n n to o mu 2 r 2 kappa 14 log n n our technical ideas and contributions are potentially useful in improving the leave one out analysis in other related problems

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

the spectra of multiplicative attribute graphs abstract a multiplicative attribute graph is a random graph in which vertices are represented by random words of length t in a finite alphabet and the probability of adjacency is a symmetric function t t 0 1 these graphs are a generalization of stochastic kronecker graphs and both classes have been shown to exhibit several useful real world properties we establish asymptotic bounds on the spectra of the adjacency matrix and the normalized laplacian matrix for these two families of graphs under certain mild conditions as an application we examine various properties of the stochastic kronecker graph and the multiplicative attribute graph including the diameter clustering coefficient chromatic number and bounds on low congestion routing

spectral operators of matrices the class of matrix optimization problems mops has been recognized in recent years to be a powerful tool by researchers far beyond the optimization community to model many important ap plications involving structured low rank matrices this trend can be credited to some extent to the exciting developments in the emerging field of compressed sensing the lowner operator which gen erates a matrix valued function by applying a single variable function to each of the singular values of a matrix has played an important role for a long time in solving matrix optimization problems however the classical theory developed for lowner operators has become inadequate in these recent applications the main objective of this paper is to provide some necessary theoretical foundations for designing numerical methods for solving the mop this goal is achieved by introducing and conducting a thorough study on a new class of matrix valued functions coined as spectral operators of matrices several fundamental properties of spectral operators including the well definedness continuity di rectional differentiability frechet differentiability locally lipschitzian continuity order b ouligand differentiability 0 1 order g semismooth 0 1 and the characterization of clarke s generalized jacobian are systematically studied

matrix completion with covariate information this article investigates the problem of matrix completion from the corrupted data when the additional covariates are available despite being seldomly considered in the matrix completion literatu

asymptotic equivalence of quantum state tomography and noisy matrix completion matrix completion and quantum tomography are two unrelated research areas with great current interest in many modern scientiflc studies this paper investigates the statistical relationship between trace regression in matrix completion and quantum state tomography in quantum physics and quantum information science as quantum state tomography and trace regression share the common goal of recovering an unknown matrix it is nature to put them in the le cam paradigm for statistical comparison regarding the two types of matrix inference problems as two statistical experiments we establish their asymptotic equivalence in terms of deflciency distance the equivalence study motivates us to introduce a new trace regression model the asymptotic equivalence provides a sound statistical foundation for applying matrix completion methods to quantum state tomography we investigate the asymptotic equivalence for sparse density matrices and low rank density matrices and demonstrate that sparsity and low rank are not necessarily helpful for achieving the asymptotic equivalence of quantum state tomography and trace regression in particular we show that popular pauli measurements are bad for establishing the asymptotic equivalence for sparse density matrices and low rank density matrices

exact reconstruction of euclidean distance geometry problem using low rank matrix completion the euclidean distance geometry problem arises in a wide variety of applications from determining molecular conformations in computational chemistry to localization in sensor networks when the distance information is incomplete the problem can be formulated as a nuclear norm minimization problem in this paper this minimization program is recast as a matrix completion problem of a low rank r gram matrix with respect to a suitable basis the well known restricted isometry property cannot be satisfied in this scenario instead a dual basis approach is introduced to theoretically analyze the reconstruction problem if the gram matrix satisfies certain coherence conditions with parameter nu the main result shows that the underlying configuration of n points can be recovered with very high probability from o nr nu log 2 n uniformly random samples computationally simple and fast algorithms are designed to solve the euclidean distance geometry problem numerical tests on different 3 d data and protein molecules validate the effectiveness and efficiency of the proposed algorithms

minimax lower bounds for noisy matrix completion under sparse factor models this paper examines fundamental error characteristics for a general class of matrix completion problems where the matrix of interest is a product of two a priori unknown matrices one of which is sparse and the observations are noisy our main contributions come in the form of minimax lower bounds for the expected per element squared error for this problem under several common noise models specifically we analyze scenarios where the corruptions are characterized by additive gaussian noise or additive heavier tailed laplace noise poisson distributed observations and highly quantized e g one bit observations as instances of our general result our results establish that the error bounds derived in soni et al 2016 for complexity regularized maximum likelihood estimators achieve up to multiplicative constants and logarithmic factors the minimax error rates in each of these noise scenarios provided that the nominal number of observations is large enough and the sparse factor has on an average at least one non zero per column

von neumann entropy penalization and low rank matrix estimation we study a problem of estimation of a hermitian nonnegatively definite matrix of unit trace for instance a density matrix of a quantum system based on n i i d measurements x1 y1 xn yn where

strong converse for identification via quantum channels we present a simple proof of the strong converse for identification via discrete memoryless quantum channels based on a novel covering lemma the new method is a generalization to quantum communication channels of ahlswede s 1979 1992 approach to classical channels it involves a development of explicit large deviation estimates to the case of random variables taking values in self adjoint operators on a hilbert space this theory is presented separately in an appendix and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems

on the rank minimization problem over a positive semidefinite linear matrix inequality we consider the problem of minimizing the rank of a positive semidefinite matrix subject to the constraint that an affine transformation of it is also positive semidefinite our method for solving this problem employs ideas from the ordered linear complementarity theory and the notion of the least element in a vector lattice this problem is of importance in many contexts for example in feedback synthesis problems and such an example is also provided

bifurcation into gaps in the essential spectrum in this lecture bifurcation from any boundary point of the essential spectrum is studied for a class of nonlinear operators generalizing results concerning bifurcation from the lowest point of the continuous spectrum the study is motivated by a nonlinear hill s equation as a typical example for a problem with gaps in the purely continuous spectrum