on the estimation of an average rigid body motion this paper investigates the definition and the estimation of the frechet mean of a random rigid body motion in super p the sample space se p contains objects me r t where r is a p p rotation matrix and t is a p 1 translation vector this work is motivated by applications in biomechanics where the posture of a joint at a given time is expressed as mise 3 the rigid body displacement needed to map a system of axes on one segment of the joint to a similar system on the other segment this posture can also be reported as m super 1e r super t r super tt by interchanging the role of the two segments several definitions of a frechet mean for a random motion are proposed using weighted least squares distances a special emphasis is given to a frechet mean that is equivariant with respect to the inverse transform this means that if p is the frechet mean for m then p super 1 is the frechet mean for m super 1 where m is a random se p object the sampling properties of moment estimators of the frechet means are studied in a large concentration setting where the scatter of the random ms around their mean value p is small and as the sample size goes to some simple exponential family models for se p data that generalize downs 1972 fisher von mises matrix distribution for rotation matrices are introduced and the least squares mean values for these distributions are calculated asymptotic comparisons between the estimators presented in this work are carried out for a particular model when pe2 a numerical example involving the motion of the ankle is presented to illustrate the methodology copyright 2012 oxford university press

a unified convergence analysis of block successive minimization methods for nonsmooth optimization the block coordinate descent bcd method is widely used for minimizing a continuous function f of several block variables at each iteration of this method a single block of variables is optimized while the remaining variables are held fixed to ensure the convergence of the bcd method the subproblem of each block variable needs to be solved to its unique global optimal unfortunately this requirement is often too restrictive for many practical scenarios in this paper we study an alternative inexact bcd approach which updates the variable blocks by successively minimizing a sequence of approximations of f which are either locally tight upper bounds of f or strictly convex local approximations of f the main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods especially for the cases where the objective functions are either nondifferentiable or nonconvex our results unify and extend the existing convergence results

smoothness index guided bayesian inference for determining joint posterior probability distributions of anti symmetric real laplace wavelet parameters for identification of different bearing faults abstract rolling element bearings are one of the most common components used in machines and they are used to support rotating shafts unexpected bearing failures may cause machine breakdown which results in economic loss detection of bearing faults is crucial to prevent bearing failures vibration based signal processing methods have been proven to be effective in identifying different bearing faults among different signal processing methods wavelet analysis is widely investigated because it is able to highlight the similarity between wavelet functions with different wavelet parameters and impulses caused by bearing faults in wavelet analysis two topics are of great concern the first is how to choose a suitable wavelet mother function for bearing fault diagnosis in recent studies an anti symmetric real laplace wavelet or impulse response wavelet has been experimentally proven to have a high similarity with impulses caused by bearing faults therefore anti symmetric real laplace wavelet is chosen as the wavelet mother function in this paper the second is how to determine the optimal wavelet parameters so as to enhance the ability of wavelet analysis based on the anti symmetric real laplace wavelet smoothness index based bayesian inference is proposed in this paper to establish joint posterior probability density functions of wavelet parameters which reflect graphical relationships between wavelet parameters the smoothness index is chosen because it is not only able to quantify bearing fault signals but also has upper and lower bounds compared with other metrics such as wavelet entropy shannon entropy kurtosis sparsity measurement etc for bayesian inference a general particle filter is adopted to iteratively calculate and update joint posterior probability density functions of wavelet parameters once the joint posterior probability density functions of wavelet parameters are available the optimal wavelet parameters are determined and an optimal wavelet filtering is conducted to extract bearing fault signatures real case studies are investigated to illustrate how the proposed method works the results show that the proposed method can determine joint posterior probability density functions of wavelet parameters and is effective in identifying different bearing faults

orientation statistics without parametric assumptions summary maximum likelihood estimation using the matrix von mises fisher distribution in orientation statistics leads to unacceptably complicated likelihood equations essentially because of the inconvenient form of the normalizing constant in the probability distribution for the case of 3 x 2 or 3 x 3 orientations the main cases of practical importance an alternative approach is developed here by transforming the data into unsigned four dimensional directions and using known results on the sampling properties of the spectral decomposition of the resulting sample moment of inertia matrix it is demonstrated that the necessary computations are relatively simple by applying some of the techniques to a set of vectorcardiogram data

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

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

leave one out approach for matrix completion primal and dual analysis in this paper we introduce a powerful technique based on leave one out analysis to the study of low rank matrix completion problems using this technique we develop a general approach for obtaining fine grained entrywise bounds for iterative stochastic procedures in the presence of probabilistic dependency we demonstrate the power of this approach in analyzing two of the most important algorithms for matrix completion i the non convex approach based on projected gradient descent pgd for a rank constrained formulation also known as the singular value projection algorithm and ii the convex relaxation approach based on nuclear norm minimization nnm using this approach we establish the first convergence guarantee for the original form of pgd without regularization or sample splitting and in particular shows that it converges linearly in the infinity norm for nnm we use this approach to study a fictitious iterative procedure that arises in the dual analysis our results show that nnm recovers an d by d rank r matrix with mathcal o mu r log mu r d log d observed entries this bound has optimal dependence on the matrix dimension and is independent of the condition number to the best of our knowledge none of previous sample complexity results for tractable matrix completion algorithms satisfies these two properties simultaneously

iterative regularization and nonlinear inverse scale space applied to wavelet based denoising in this paper we generalize the iterative regularization method and the inverse scale space method recently developed for total variation tv based image restoration to wavelet based image restoration this continues our earlier joint work with others where we applied these techniques to variational based image restoration obtaining significant improvement over the rudin osher fatemi tv based restoration here we apply these techniques to soft shrinkage and obtain the somewhat surprising result that a the iterative procedure applied to soft shrinkage gives firm shrinkage and converges to hard shrinkage and b that these procedures enhance the noise removal capability both theoretically in the sense of generalized bregman distance and for some examples experimentally in terms of the signal to noise ratio leaving less signal in the residual

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

an efficient inexact symmetric gauss seidel based majorized admm for high dimensional convex composite conic programming in this paper we propose an inexact multi block admm type first order method for solving a class of high dimensional convex composite conic optimization problems to moderate accuracy the design of this method combines an inexact 2 block majorized semi proximal admm and the recent advances in the inexact symmetric gauss seidel sgs technique for solving a multi block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block variable one distinctive feature of our proposed method the sgs imspadmm is that it only needs one cycle of an inexact sgs method instead of an unknown number of cycles to solve each of the subproblems involved with some simple and implementable error tolerance criteria the cost for solving the subproblems can be greatly reduced and many steps in the forward sweep of each sgs cycle can often be skipped which further contributes to the efficiency of the proposed method global convergence as well as the iteration complexity in the non ergodic sense is established preliminary numerical experiments on some high dimensional linear and convex quadratic sdp problems with a large number of linear equality and inequality constraints are also provided the results show that for the vast majority of the tested problems the sgs imspadmm is 2 3 times faster than the directly extended multi block admm with the aggressive step length of 1 618 which is currently the benchmark among first order methods for solving multi block linear and quadratic sdp problems though its convergence is not guaranteed

saddlepoint approximations for the normalizing constant of fisher bingham distributions on products of spheres and stiefel manifolds in an earlier paper kume wood 2005 showed how the normalizing constant of the fisher bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation in this sequel we extend the approach to a more general setting and derive saddlepoint approximations for the normalizing constants of multicomponent fisher bingham distributions on cartesian products of spheres and fisher bingham distributions on stiefel manifolds in each case the approximation for the normalizing constant is essentially a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic forms in normal variables both first order and second order saddlepoint approximations are considered computational algorithms numerical results and theoretical properties of the approximations are presented in the challenging high dimensional settings considered in this paper the saddlepoint approximations perform very well in all examples considered some key words directional data fisher matrix distribution kent distribution orientation statistics

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

vibration signal classification by wavelet packet energy flow manifold learning abstract this paper proposes a new study to explore the wavelet packet energy wpe flow characteristics of vibration signals by using the manifold learning technique this study intends to discover the nonlinear manifold information from the wpe flow map of vibration signals to characterize and discriminate different classes a new feature called wpe manifold feature is achieved by three main steps first the wavelet packet transform wpt is conducted to decompose multi class signals into a library of time frequency subspaces second the wpe is calculated in each subspace to produce a feature vector for each signal and finally low dimensional manifold features carrying class information are extracted from the wpe library for either training or testing samples by using the manifold learning algorithm the new feature reveals the nonlinear wpe flow structure among various redundant time frequency subspaces it combines the benefits of time frequency characteristics and nonlinear information and hence exhibits valuable properties for vibration signal classification the effectiveness and the merits of the proposed method are confirmed by case studies on vibration analysis based machine fault classification

vibration signal component separation by iteratively using basis pursuit and its application in mechanical fault detection abstract using different transform basis the vibration signal components such as impulses harmonics and modulated components can be effectively separated by basis pursuit for multicomponent vibration signals this paper proposes a new vibration signal component separation approach by iteratively using basis pursuit the signal is firstly denoised by basis pursuit denoising and the impulsive component can be separated by using identity matrix and redundant fourier basis then signal components with different frequency ranges can be separated by using short time fourier transform basis with different window lengths and the components with high frequency are separated after each iteration to choose the best basis from the dictionary the separation coefficient is proposed for evaluating the separating performance of the transform basis the proposed approach empirical mode decomposition emd and matching pursuit are respectively applied to analyzing a simulated faulty vibration and a simulated faulty gear vibration signal the comparison results demonstrate its superiority to emd and matching pursuit in separation accuracy finally the proposed approach is used for detecting the faults of a rotor and rolling bearings the results further show that this proposed method can be effectively applied to mechanical fault diagnosis

detection of faults in rotating machinery using periodic time frequency sparsity abstract this paper addresses the problem of extracting periodic oscillatory features in vibration signals for detecting faults in rotating machinery to extract the feature we propose an approach in the short time fourier transform stft domain where the periodic oscillatory feature manifests itself as a relatively sparse grid to estimate the sparse grid we formulate an optimization problem using customized binary weights in the regularizer where the weights are formulated to promote periodicity in order to solve the proposed optimization problem we develop an algorithm called augmented lagrangian majorization minimization algorithm which combines the split augmented lagrangian shrinkage algorithm salsa with majorization minimization mm and is guaranteed to converge for both convex and non convex formulation as examples the proposed approach is applied to simulated data and used as a tool for diagnosing faults in bearings and gearboxes for real data and compared to some state of the art methods the results show that the proposed approach can effectively detect and extract the periodical oscillatory features

objective bayesian meta analysis for sparse discrete data this paper presents a bayesian model for meta analysis of sparse discrete binomial data which are out of the scope of the usual hierarchical normal random effect models treatment effectiveness data are often of this type the crucial linking distribution between the effectiveness conditional on the healthcare center and the unconditional effectiveness is constructed from specific bivariate classes of distributions with given marginals this assures coherency between the marginal and conditional prior distributions utilized in the analysis further we impose a bivariate class of priors that is able to accommodate a wide range of heterogeneity degrees between the multicenter clinical trials involved applications to real multicenter data are given and compared with previous meta analysis copyright 2014 john wiley sons ltd

using directional statistics to test hypotheses regarding rigid body attitude comparison to univariate and multivariate cardan angle tests abstract rigid body attitude and single joint kinematics are typically expressed using three cardan angles which represent rotations in anatomical planes it was recently shown in the biomechanics literature that cardan angles inaccurately estimate true mean attitude due to an important mathematical inadequacy attitude under representation at least four quantities are needed to unambiguously specify attitude directional statistics which is the multivariate generalization of univariate circular statistics solves this problem using four dimensional unit vectors and the mathematics of hyperspherical geometry the purpose of this study was to compare the results of directional analysis to the results of uni and multi variate cardan analysis for representative joint kinematic data during gait we analyzed hip knee and pelvis data from three open datasets and report exemplary results for knee kinematics in v cut vs side shuffle tasks we also conducted monte carlo simulations using synthetic data with precisely controlled true angular effects to systematically compare directional and cardan analyses results show that directional analysis yielded considerably smaller p values p 0 03 than cardan analysis p 0 055 for the exemplary dataset simulation results confirmed that directional analysis is considerably more powerful i e much more able to detect true angular effects than both uni and multi variate cardan analysis these results suggest that directional statistics should be used to analyse attitude including 3d joint kinematics to avoid false negatives

a directional model for the estimation of the rotation axes of the ankle joint this article is motivated by the estimation of the directions of the two rotation axes of the ankle these axes carry information on individual ankles they are useful in the construction of biomechanical models and the treatment of orthopedic problems in biomechanics the rotation axes of the ankle often are estimated using optimization techniques this work investigates a statistical model for carrying out the estimation the data set for analysis is a time ordered sequence of 3 3 rotation matrixes giving the ankle s orientations as the foot moves with respect to the lower leg these rotation matrixes are assumed to follow fisher von mises distributions the predicted values for the observed rotations feature four angles for the orientation of two rotation axes and two series of time varying rotation angles about the two axes maximum likelihood estimators of the parameters are derived approximations to their sampling distributions are obtained when the errors are clustered around the identity matrix

in vivo determination of the anatomical axes of the ankle joint complex an optimization approach this study investigates the feasibility of a subject specific three dimensional model of the ankle joint complex for kinematic and dynamic analysis of movement the ankle joint complex was modelled as a three segment system connected by two ideal highe joints the talocrural and the subtalar joint a mathematical formulation was developed to express the three dimensional translation and rotation between the foot and shank segments as a function of the two joint angles and 12 model parameters describing the locations of the joint axes an optimization method was used to fit the model parameters to three dimensional kinematic data of foot and shank markers obtained during test movements throughout the entire physiological range of motion of the ankle joint the movement of the talus segment which cannot be measured non invasively is not necessary for the analysis this optimization method was used to determine the position and orientation of the joint axes in 14 normal subjects after optimization the discrepancy between the best fitting model and actual marker kinematics was between 1 and 3 mm for all subjects the predicted inclination of the subtalar joint axis from the horizontal plane was 37 4 2 7 and the medial deviation was 18 0 16 2 the lateral side of the talucrural axis was directed slightly posteriorly 6 8 8 1 and inclined downward by 7 0 5 4 these results are similar to previously reported typical results from anatomical in vitro studies reproducibility was evaluated by repeated testing of one subject which resulted in variations of about one fifth of the standard deviation within the group the inclination of the subtalar joint axis was significantly correlated to the arch height and a radiographic tarsal index it is concluded that this optimization method provides the opportunity to incorporate inter individual anatomical differences into kinematic and dynamic analysis of the ankle joint complex this allows a more functional interpretation of kinematic data and more realistic estimates of internal forces

modeling and inference for measured crystal orientations and a tractable class of symmetric distributions for rotations in three dimensions electron backscatter diffraction ebsd is a technique used in materials science to study the microtexture of metals producing data that measure the orientations of crystals in a specimen we examine the precision of such data based on a useful class of distributions on orientations in three dimensions as represented by 3 3 orthogonal matrices with positive determinants although such modeling has received attention in the statistical literature the approach taken typically has been based on general special manifold considerations and the resulting methodology may not be easily accessible to nonspecialists we take a more direct modeling approach beginning from a simple intuitively appealing mechanism for generating random orientations specifically in three dimensional space the resulting class of distributions has many desirable properties including directly interpretable parameters and relatively simple theory we investigate the basic properties of the entire class and one sample quasi likel

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

majorization minimization algorithms for wavelet based image restoration standard formulations of image signal deconvolution under wavelet based priors regularizers lead to very high dimensional optimization problems involving the following difficulties the non gaussian heavy tailed wavelet priors lead to objective functions which are nonquadratic usually nondifferentiable and sometimes even nonconvex the presence of the convolution operator destroys the separability which underlies the simplicity of wavelet based denoising this paper presents a unified view of several recently proposed algorithms for handling this class of optimization problems placing them in a common majorization minimization mm framework one of the classes of algorithms considered when using quadratic bounds on nondifferentiable log priors shares the infamous singularity issue si of iteratively re weighted least squares irls algorithms the possibility of having to handle infinite weights which may cause both numerical and convergence issues in this paper we prove several new results which strongly support the claim that the si does not compromise the usefulness of this class of algorithms exploiting the unified mm perspective we introduce a new algorithm resulting from using bounds for nonconvex regularizers the experiments confirm the superior performance of this method when compared to the one based on quadratic majorization finally an experimental comparison of the several algorithms reveals their relative merits for different standard types of scenarios

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

on robust regression with high dimensional predictors we study regression m estimates in the setting where p the number of covariates and n the number of observations are both large but p n we find an exact stochastic representation for the distribution of argmin p i 1 n y i x i at fixed p and n under various assumptions on the objective function and our statistical model a scalar random variable whose deterministic limit r can be studied when p n 0 plays a central role in this representation we discover a nonlinear system of two deterministic equations that characterizes r interestingly the system shows that r depends on through proximal mappings of as well as various aspects of the statistical model underlying our study several surprising results emerge in particular we show that when p n is large enough least squares becomes preferable to least absolute deviations for double exponential errors

a correction for axis misalignment in the joint angle curves representing knee movement in gait analysis this paper derives a simple mathematical model relating changes in the orientations of the two cartesian coordinate systems involved in recording knee movement and the varus valgus and the internal external rotation angles for describing the knee s motion rotation matrices are given for changing the orientations of the two cartesian coordinate systems in such a way that the quadratic variations in the varus valgus and in the external internal angles are minimal these estimated rotation matrices are used to correct for axis misalignment the correction is calibrated by considering the impact of the new orientation of the thigh cartesian coordinate system on the hip joint angles the procedure is applied to kinematic data collected on normal subjects the uncertainty about the specification of the thigh cartesian coordinate system is shown to explain some of the between subject variability in the varus valgus and in the internal external rotation angles curves

a least squares algorithm for the equiform transformation from spatial marker co ordinates the present paper describes an algorithm for estimating the translation vector and the rotation matrix of a moving body from noisy measurements on the spatial co ordinates of at least three non collinear markers a sensitivity analysis of the estimated parameters and of the helical axis is presented in terms of characteristics of the marker distribution the implementation of the proposed algorithm in a fortran subroutine is appended

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

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

entrywise eigenvector analysis of random matrices with low expected rank recovering low rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning such as in factor analysis community detection ranking matrix completion among others while a large variety of bounds are available for average errors between empirical and population statistics of eigenvectors few results are tight for entrywise analyses which are critical for a number of problems such as community detection this paper investigates entrywise behaviors of eigenvectors for a large class of random matrices whose expectations are low rank which helps settle the conjecture in abbe bandeira and hall 2014 that the spectral algorithm achieves exact recovery in the stochastic block model without any trimming or cleaning steps the key is a first order approximation of eigenvectors under the ell infty norm begin equation u k approx frac au k lambda k end equation where u k and u k are eigenvectors of a random matrix a and its expectation mathbb e a respectively the fact that the approximation is both tight and linear in a facilitates sharp comparisons between u k and u k in particular it allows for comparing the signs of u k and u k even if u k u k infty is large the results are further extended to perturbations of eigenspaces yielding new ell infty type bounds for synchronization mathbb z 2 spiked wigner model and noisy matrix completion

incoherence optimal matrix completion this paper considers the matrix completion problem we show that it is not necessary to assume joint incoherence which is a standard but unintuitive and restrictive condition that is imposed by previous studies this leads to a sample complexity bound that is orderwise optimal with respect to the incoherence parameter as well as to the rank r and the matrix dimension n up to a log factor as a consequence we improve the sample complexity of recovering a semidefinite matrix from o nr 2 log 2 n to o nr log 2 n and the highest allowable rank from theta sqrt n log n to theta n log 2 n the key step in proof is to obtain new bounds in terms of the ell infty 2 norm defined as the maximum of the row and column norms of a matrix to illustrate the applicability of our techniques we discuss extensions to singular value decomposition projection structured matrix completion and semisupervised clustering for which we provide orderwise improvements over existing results finally we turn to the closely related problem of low rank plus sparse matrix decomposition we show that the joint incoherence condition is unavoidable here for polynomial time algorithms conditioned on the planted clique conjecture this means it is intractable in general to separate a rank omega sqrt n positive semidefinite matrix and a sparse matrix interestingly our results show that the standard and joint incoherence conditions are associated respectively with the information statistical and computational aspects of the matrix decomposition problem