quadratic optimization with orthogonality constraint explicit łojasiewicz exponent and linear convergence of retraction based line search and stochastic variance reduced gradient methods the problem of optimizing a quadratic form over an orthogonality constraint qp oc for short is one of the most fundamental matrix optimization problems and arises in many applications in this paper we characterize the growth behavior of the objective function around the critical points of the qp oc problem and demonstrate how such characterization can be used to obtain strong convergence rate results for iterative methods that exploit the manifold structure of the orthogonality constraint i e the stiefel manifold to find a critical point of the problem specifically our primary contribution is to show that the ojasiewicz exponent at any critical point of the qp oc problem is 1 2 such a result is significant as it expands the currently very limited repertoire of optimization problems for which the ojasiewicz exponent is explicitly known moreover it allows us to show in a unified manner and for the first time that a large family of retraction based line search methods will converge linearly to a critical point of the qp oc problem then as our secondary contribution we propose a stochastic variance reduced gradient svrg method called stiefel svrg for solving the qp oc problem and present a novel ojasiewicz inequality based linear convergence analysis of the method an important feature of stiefel svrg is that it allows for general retractions and does not require the computation of any vector transport on the stiefel manifold as such it is computationally more advantageous than other recently proposed svrg type algorithms for manifold optimization

stability and vibration of thick laminated composite sector plates this study presents a simple analytical formulation for the eigenvalue problem of buckling and free vibration analysis of shear deformable laminated sector plates made up of cylindrically orthotropic layers the non axisymmetric formulation in cylindrical coordinates is discretized in space domain in terms of two dimensional chebyshev polynomials several combinations of simply supported clamped and free edge conditions are considered convergence study has been carried out and the obtained results are compared with the results of laminated square plates and isotropic sector plates available in literature extensive results pertaining to critical buckling loads and natural frequencies are presented effects of boundary conditions number of layers moduli ratio rotary and in plane inertia plate thickness sector angle and annularity are studied

an analytical solution for the free vibration analysis of rectangular plates resting on symmetrically distributed point supports abstract a general analytical solution is obtained for the free vibration of rectangular plates supported on their lateral surface by symmetrically distributed point supports highly accurate eigenvalues are provided for a square plate supported at various points along the diagonal results are compared with earlier findings obtained by a finite element method it is shown that the method may be extended to solve various other problems the tabulation of eigenvalues appears to be the most comprehensive yet made for a square plate with point supports on the diagonals

prime phase retrieval via majorization minimization this paper considers the phase retrieval problem in which measurements consist of only the magnitude of several linear measurements of the unknown e g spectral components of a time sequence we develop low complexity algorithms with superior performance based on the majorization minimization mm framework the proposed algorithms are referred to as prime phase retrieval via the majorization minimization technique they are preferred to existing benchmark methods since at each iteration a simple surrogate problem is solved with a closed form solution that monotonically decreases the original objective function in total three algorithms are proposed using different majorization minimization techniques experimental results validate that our algorithms outperform existing methods in terms of successful recovery and mean square error under various settings

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

quantum tomography under prior information we provide a detailed analysis of the question how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information we show that if the prior information restricts the possible states to a set of lower dimensionality then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states conversely we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the minkowski dimension of the set we apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints in particular we show that the minimal number of measurement outcomes povm elements necessary to identify all pure states in a d dimensional hilbert space is 4d 3 c d d for some c d in 1 2 and d being the number of ones appearing in the binary expansion of d 1

plate vibration under irregular internal supports this paper studies the problem of plate vibration under complex and irregular internal support conditions such a problem has its widely spread industrial applications and has not been addressed in the literature yet partially due to the numerical difficulties a novel computational method discrete singular convolution dsc is introduced for solving this problem the dsc algorithm exhibits controllable accuracy for approximations and shows excellent flexibility in handling complex geometries boundary conditions and internal support conditions convergence and comparison studies are carried out to check the validity and accuracy of the dsc method case studies are considered to the combination of a few different boundary conditions and irregular internal supports the latter are generated by using an image processing algorithm completely independent verifications are conducted by using the established pb 2 ritz method which is available for two relatively simpler support patterns the morphology of the first few eigenmodes is found to be localized to largest support free spatial regions

a unified approach for the solution of the fokker planck equation this paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the fokker planck equation the unified features of the discrete singular convolution algorithm are discussed it is demonstrated that different implementations of the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point three benchmark stochastic systems the repulsive wong process the black scholes equation and a genuine nonlinear model are employed to illustrate the robustness and to test the accuracy of the present approach for the solution of the fokker planck equation via a time dependent method an additional example the incompressible euler equation is used to further validate the present approach for more difficult problems numerical results indicate that the present unified approach is robust and accurate for solving the fokker planck equation

phaseless recovery using the gauss newton method in this paper we propose a gauss newton algorithm to recover an n dimensional signal from its phaseless measurements the algorithm has two stages in the first stage the algorithm obtains a good initialization by calculating the eigenvector corresponding to the largest eigenvalue of a hermitian matrix in the second stage the algorithm solves an optimization problem iteratively using the gauss newton method our initialization method makes full use of all measurements and provides a good initial guess as long as the number of random measurements is o n for real valued signals we prove that a resampled version of gauss newton iterations converges to the global optimal solution quadratically with o n log n random measurements numerical experiments show that the gauss newton method has better empirical performance than other algorithms such as the wirtinger flow algorithm and altmin phase algorithm

consensus admm for general quadratically constrained quadratic programming nonconvex quadratically constrained quadratic programming qcqp problems have numerous applications in signal processing machine learning and wireless communications albeit the general qcqp is np hard and several interesting special cases are np hard as well this paper proposes a new algorithm for general qcqp the problem is first reformulated in consensus optimization form to which the alternating direction method of multipliers can be applied the reformulation is done in such a way that each of the subproblems is a qcqp with only one constraint qcqp 1 which is efficiently solvable irrespective of non convexity the core components are carefully designed to make the overall algorithm more scalable including efficient methods for solving qcqp 1 memory efficient implementation parallel distributed implementation and smart initialization the proposed algorithm is then tested in two applications multicast beamforming and phase retrieval the results indicate superior performance over prior state of the art methods

role of sparsity and structure in the optimization landscape of non convex matrix sensing in this work we study the optimization landscape of the non convex matrix sensing problem that is known to have many local minima in the worst case since the existing results are related to the notion of restricted isometry property rip that cannot directly capture the underlying structure of a given problem they can hardly be applied to real world problems where the amount of data is not exorbitantly high to address this issue we develop the notion of kernel structure property to obtain necessary and sufficient conditions for the inexistence of spurious local solutions for any class of matrix sensing problems over a given search space this notion precisely captures the underlying sparsity and structure of the problem based on tools in conic optimization we simplify the conditions for a certain class of problems to show their satisfaction and apply them to data analytics for power systems

first order methods almost always avoid strict saddle points we establish that first order methods avoid strict saddle points for almost all initializations our results apply to a wide variety of first order methods including manifold gradient descent block coordinate descent mirror descent and variants thereof the connecting thread is that such algorithms can be studied from a dynamical systems perspective in which appropriate instantiations of the stable manifold theorem allow for a global stability analysis thus neither access to second order derivative information nor randomness beyond initialization is necessary to provably avoid strict saddle points

on the free flexural vibration of rectangular plates with straight or curved internal line supports abstract the rayleigh ritz method is used with simple polynomials as admissible functions to obtain the eigenvalue equation for the free vibration of a rectangular plate which has an internal line support lying along a line describable in terms of a general polynomial numerical and graphical results are presented for plates with internal supports lying along straight lines for which comparison results also exist and for plates with the supports lying along different types of curves including a central circular support

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

fast and guaranteed blind multichannel deconvolution under a bilinear system model we consider the multichannel blind deconvolution problem where we observe the output of multiple channels that are all excited with the same unknown input from these observations we wish to estimate the impulse responses of each of the channels we show that this problem is well posed if the channels follow a bilinear model where the ensemble of channel responses is modeled as lying in a low dimensional subspace but with each channel modulated by an independent gain under this model we show how the channel estimates can be found by minimizing a quadratic function over a non convex set we analyze two methods for solving this non convex program and provide performance guarantees for each the first is a method of alternating eigenvectors that breaks the program down into a series of eigenvalue problems the second is a truncated power iteration which can roughly be interpreted as a method for finding the largest eigenvector of a symmetric matrix with the additional constraint that it adheres to our bilinear model as with most non convex optimization algorithms the performance of both of these algorithms is highly dependent on having a good starting point we show how such a starting point can be constructed from the channel measurements our performance guarantees are non asymptotic and provide a sufficient condition on the number of samples observed per channel in order to guarantee channel estimates of certain accuracy our analysis uses a model with a generic subspace that is drawn at random and we show the performance bounds hold with high probability mathematically the key estimates are derived by quantifying how well the eigenvectors of certain random matrices approximate the eigenvectors of their mean we also present a series of numerical results demonstrating that the empirical performance is consistent with the presented theory

on global convergence of gradient descent algorithms for generalized phase retrieval problem abstract in this paper we study the generalized phase retrieval problem to recover a signal x c n from the measurements y r a r x 2 r 1 2 m the problem can be reformulated as a least squares minimization problem although the cost function is nonconvex the global convergence of gradient descent algorithms from a random initialization is studied when m is large enough we improve the known result of the local convergence from a spectral initialization when the signal x is real valued we prove that the cost function is local convex near the solution x to accelerate the gradient descent we review and apply several efficient line search methods with exact line search stepsize we also perform a comparative numerical study of the line search methods and the alternative projection method numerical simulations demonstrate the superior ability of lbfgs algorithm than other algorithms

vibration of annular sector plates from three dimensional analysis accurate three dimensional elasticity solutions are presented for the free vibration analysis of annular sector plates with arbitrary but continuous boundary conditions the ritz procedure is used to minimize the integral energy functional which is formulated in a cylindrical polar coordinate system in the derivation of the governing eigenvalue equation the vibratory characteristics of annular sector plates with variations in boundary condition sector angle and thickness are examined the accuracy of the method is validated through appropriate convergence and comparison studies the first known results presented here provide a basis for understanding the physics of vibrations of annular sector plates and can be used to assess numerical results obtained from various refined two dimensional theories

influence of boundary stress singularities on the vibration of clamped and simply supported sectorial plates with arbitrary radial edge conditions abstract this paper extends previous studies made for sectorial plates having re entrant i e interior corners causing stress singularities to provide accurate frequencies when the circular edge is either clamped or simply supported an extensive review of the literature is also given herein spanning nearly the past two decades explaining the free vibration characteristics of sectorial plates in this work the classical ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements these sets include 1 mathematically complete algebraic trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained and 2 corner functions which account for the bending moment singularities at the re entrant vertex corner of the radial edges having arbitrary edge conditions extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of non dimensional frequencies for sectorial plates having either a clamped or hinged circumferential edge and various combinations of clamped hinged and free conditions on the radial edges accurate to at least four significant figure frequencies and normalized contours of the transverse vibratory displacement are presented for the spectra of sector angles 90 180 semi circular 270 300 330 350 355 360 complete circular causing a re entrant vertex corner of the radial edges for sector angles of 360 a clamped clamped clamped hinged clamped free hinged free or free free radial crack ensues one general observation is the substantial reduction in the first six frequencies as the sector angle increases for all plates except in the first two modes of plates having free free radial edges

phaseless near field far field transformation utilizing combinations of probe signals phaseless near field far field transformations based on nonconvex optimization suffer from local stationary points in order to avoid the corresponding suboptimal solutions the incorporation of combinations of probe signals into the cost functional is proposed and investigated the nonconvex optimization based on the wirtinger calculus relies on forward and adjoint radiation operator evaluations via the fast irregular antenna field transformation algorithm fiafta and is performed by a memory limited quasi newton method this allows the solution of large scale problems with utilization of the full flexibility of the fiafta in terms of measurement probe correction and arbitrary sample locations results for synthetic and measured near field data reveal significant improvements when the knowledge of probe signal combinations is utilized especially in applications with a large number of unknowns

misspecified nonconvex statistical optimization for sparse phase retrieval existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models to address this issue we take a first step towards taming model misspecification by studying the high dimensional sparse phase retrieval problem with misspecified link functions in particular we propose a simple variant of the thresholded wirtinger flow algorithm that given a proper initialization linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions we further provide extensive numerical experiments to support our theoretical findings

sparse phase retrieval via truncated amplitude flow this paper develops a novel algorithm termed sparse truncated amplitude flow sparta to reconstruct a sparse signal from a small number of magnitude only measurements it deals with what is also known as sparse phase retrieval pr which is np hard in general and emerges in many science and engineering applications upon formulating sparse pr as an amplitude based nonconvex optimization task sparta works iteratively in two stages in stage one the support of the underlying sparse signal is recovered using an analytically well justified rule and subsequently a sparse orthogonality promoting initialization is obtained via power iterations restricted on the support and in the second stage the initialization is successively refined by means of hard thresholding based gradient type iterations sparta is a simple yet effective scalable and fast sparse pr solver on the theoretical side for any n dimensional k sparse k ll n signal boldsymbol x with minimum in modulus nonzero entries on the order of 1 sqrt k vert boldsymbol x vert 2 sparta recovers the signal exactly up to a global unimodular constant from about k 2 log n random gaussian measurements with high probability furthermore sparta incurs computational complexity on the order of k 2n log n with total runtime proportional to the time required to read the data which improves upon the state of the art by at least a factor of k finally sparta is robust against additive noise of bounded support extensive numerical tests corroborate markedly improved recovery performance and speedups of sparta relative to existing alternatives

on fienup methods for sparse phase retrieval alternating minimization or fienup methods have a long history in phase retrieval we provide new insights related to the empirical and theoretical analysis of these algorithms when used with fourier measurements and combined with convex priors in particular we show that fienup methods can be viewed as performing alternating minimization on a regularized nonconvex least squares problem with respect to amplitude measurements furthermore we prove that under mild additional structural assumptions on the prior semialgebraicity the sequence of signal estimates has a smooth convergent behavior toward a critical point of the nonconvex regularized least squares objective finally we propose an extension to fienup techniques based on a projected gradient descent interpretation and acceleration using inertial terms we demonstrate experimentally that this modification combined with an ell 1 prior constitutes a competitive approach for sparse phase retrieval

non linear transient analysis of moderately thick laminated composite plates the non linear transient analysis of the shear deformable laminated composite plates subjected to step ramp and sinusoidal loading is presented the clamped simply supported free and their combinations non levy type of boundary conditions are considered the formulation is based on the mindlin first order shear deformation theory and von karman non linearity the methodology of the solution is based on the chebyshev series technique houbolt time marching scheme and quadratic extrapolation technique are used for the temporal discretization and linearization respectively the effects of magnitude and duration of loading rotary inertia in plane inertia b a a h lamination scheme and boundary conditions on the central displacement response are studied typical results are presented in dimensionless graphical forms for different parameters and loading conditions

phaseequal convex phase retrieval via alternating direction method of multipliers the problem of estimating an unknown complex signal from its magnitude only measurements is a classical problem known as phase retrieval this problem arises naturally because in some applications it is difficult or costly to measure the phase of the measurements in this paper we propose a new convex formulation referred to as phaseequal for solving the phase retrieval problem similar to phasemax t goldstein and c studer 2018 by goldstein and studer iee tit 2018 phaseequal works in the natural signal space and thus is computationally efficient we then extend phaseequal to the sparse signal case with the proposed convex formulation termed as compressed phaseequal different from existing convex compressed phase retrieval methods the proposed compressed phaseequal formulation does not involve any regularization parameter and thus is free of the parameter tuning issue which is always tricky in practice order wise recovery conditions for phaseequal and its sparse version i e compressed phaseequal are analyzed our theoretical results show that phaseequal resp compressed phaseequal achieves perfect recovery with mathcal o n resp mathcal o k text log frac n k magnitude measurements provided that a well correlated reference vector is available where k and n denote the number of nonzero entries in the complex sparse signal and the dimension of the signal respectively simulation results are provided to illustrate the effectiveness of the proposed methods

3 d vibration analysis of annular sector plates using the chebyshev ritz method abstract the three dimensional free vibration of annular sector plates with various boundary conditions is studied by means of the chebyshev ritz method the analysis is based on the three dimensional small strain linear elasticity theory the product of chebyshev polynomials satisfying the necessary boundary conditions is selected as admissible functions in such a way that the governing eigenvalue equation can be conveniently derived through an optimization process by the ritz method the boundary functions guarantee the satisfaction of the geometric boundary conditions of the plates and the chebyshev polynomials provide the robustness for numerical calculation the present study provides a full vibration spectrum for the thick annular sector plates which cannot be given by the two dimensional 2 d theories such as the mindlin theory comprehensive numerical results with high accuracy are systematically produced which can be used as benchmark to evaluate other numerical methods the effect of radius ratio thickness ratio and sector angle on natural frequencies of the plates with a sector angle from 120 to 360 is discussed in detail the three dimensional vibration solutions for plates with a re entrant sector angle larger than 180 and shallow helicoidal shells sector angle larger than 360 with a small helix angle are presented for the first time

a note on iterative fourier transform phase reconstruction from magnitude in this correspondence a well known iterative procedure for fourier transform phase reconstruction from magnitude usually referred to as the gerchberg saxton fienup procedure is revisited the convergence of this algorithm has raised some controversy in the recent literature in this correspondence we first point out a potential source of error in the numerical implementation of the algorithm then we present a conjecture which would explain why this algorithm sometimes fails

phase retrieval via sparse wirtinger flow abstract phase retrieval pr problem is an inverse problem which arises in various applications based on the wirtinger flow method an algorithm utilizing the sparsity priority called swf sparse wirtinger flow is proposed in this paper to deal with the pr problem firstly the support of the signal is estimated besides the initialization is evaluated based on this support then the evaluation is updated by the hard thresholding method from this initialization we prove that for any k sparse signal with length n swf has a linear convergence with o k 2 log n phaseless gaussian random measurements to get e accuracy the computational complexity of swf is o k 3 n log n log 1 e numerical tests also demonstrate that swf has a higher recovery rate than other algorithms compared especially when we have no prior information about sparsity k moreover swf is robust to the noise

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

vibrations of composite plates with intermediate line supports abstract free vibrations of rectangular composite plates with arbitrary support conditions along the edges and an arbitrary number of intermediate line supports are investigated using the rayleigh ritz method a procedure is presented for generating polynomial approximation functions that satisfy the essential boundary conditions and zero displacement constraints along each line support and the stiffness and mass matrix coefficients are obtained by exact integration plates that are continuous over many supports are analyzed and the effects lay up and material properties are investigated an exact solution is found for predicting some of the vibration modes of plates with simple supports along the edges and equal length spans in each direction in this case the fundamental natural frequency is always predicted exactly and simple results are obtained for finding which lay up maximizes the natural frequency of the plate

sparse and low rank tensor estimation via cubic sketchings in this paper we propose a general framework for sparse and low rank tensor estimation from cubic sketchings a two stage non convex implementation is developed based on sparse tensor decomposition and thresholded gradient descent which ensures exact recovery in the noiseless case and stable recovery in the noisy case with high probability the non asymptotic analysis sheds light on an interplay between optimization error and statistical error the proposed procedure is shown to be rate optimal under certain conditions as a technical by product novel high order concentration inequalities are derived for studying high moment sub gaussian tensors an interesting tensor formulation illustrates the potential application to high order interaction pursuit in high dimensional linear regression

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

harmonic mean iteratively reweighted least squares for low rank matrix recovery we propose a new iteratively reweighted least squares irls algorithm for the recovery of a matrix x cd1 d2 of rank r min d1 d2 from incomplete linear observations solving a sequence of low complexity linear problems the easily implementable algorithm which we call harmonic mean iteratively reweighted least squares hm irls optimizes a non convex schatten p quasi norm penalization to promote low rankness and carries three major strengths in particular for the matrix completion setting first we observe a remarkable global convergence behavior of the algorithm s iterates to the low rank matrix for relevant interesting cases for which any other state of the art optimization approach fails the recovery secondly hm irls exhibits an empirical recovery probability close to 1 even for a number of measurements very close to the theoretical lower bound r d1 d2 r i e already for significantly fewer linear observations than any other tractable approach in the literature thirdly hm irls exhibits a locally superlinear rate of convergence of order 2 p if the linear observations fulfill a suitable null space property while for the first two properties we have so far only strong empirical evidence we prove the third property as our main theoretical result

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

sublinear time algorithms for compressive phase retrieval in the problem of compressed phase retrieval the goal is to reconstruct a sparse or approximately k sparse vector x in mathbb c n given access to y phi x where v denotes the vector obtained from taking the absolute value of v in mathbb c n coordinate wise in this paper we present sublinear time algorithms for a few for each variants of the compressive phase retrieval problem which are akin to the variants considered for the classical compressive sensing problem in theoretical computer science our algorithms use pure combinatorial techniques and near optimal number of measurements

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

on data increasing in phase retrieval via quadratic inversion flattening manifold and local minima in this article we address the question of traps in phase retrieval via quadratic inversion first we clarify the genesis of local minima from a geometrical perspective by showing how the shape of the manifold of data and the position of the data point are related to the presence of local minima in the objective functional later we recall a mathematical condition for the lack of traps afterward we illustrate how the ratio between the dimension of data space m and the number of real unknowns n r impacts on the manifold of data and consequently on the presence of local minima in particular we show that if the dimension of data space is high enough the manifold of data is sufficiently large and no traps appear in the functional in the last part of this article with reference to a particular unknown sequence we establish the minimum value of the ratio m n r such that no traps should appear in the functional in this condition phase retrieval via quadratic inversion exhibits a global convergence behavior consequently the actual solution of the problem can be reached regardless of the initial guess which can be chosen completely at random

measurement matrix design for phase retrieval based on mutual information in phase retrieval problems a signal of interest soi is reconstructed based on the magnitude of a linear transformation of the soi observed with additive noise the linear transform is typically referred to as a measurement matrix many works on phase retrieval assume that the measurement matrix is a random gaussian matrix which in the noiseless scenario with sufficiently many measurements guarantees invertability of the transformation between the soi and the observations up to an inherent phase ambiguity however in many practical applications the measurement matrix corresponds to an underlying physical setup and is therefore deterministic possibly with structural constraints in this paper we study the design of deterministic measurement matrices based on maximizing the mutual information between the soi and the observations we characterize necessary conditions for the optimality of a measurement matrix and analytically obtain the optimal matrix in the low signal to noise ratio regime practical methods for designing general measurement matrices and masked fourier measurements are proposed simulation tests demonstrate the performance gain achieved by the suggested techniques compared to random gaussian measurements for various phase recovery algorithms

chebyshev series solution to non linear boundary value problems in rectangular domain abstract the von karman equations governing the behavior of moderately large deformations of rectangular plates are expressed in displacement field a methodology based on chebyshev polynomials approximation to analyze the non linear boundary value problems in rectangular domain is developed these non linear partial differential equations of motion are linearized using quadratic extrapolation techniques the inertia and dissipative terms are evaluated by employing houbolt implicit time marching scheme the spatial discretization of the differential equations generates incompatibility viz greater number of equations than the unknowns the multiple linear regression analysis based on the least square error norm is employed to overcome the incompatibility and a compatible solution is obtained convergence study has been carried out the clamped and simply supported immovable rectangular plates subjected to static and dynamic loadings are analyzed results have been compared with the results obtained by other numerical and analytical methods

exact phase retrieval in principal shift invariant spaces we address the problem of phase retrieval from fourier transform magnitude spectrum for continuous time signals that lie in a shift invariant space spanned by integer shifts of a generator kernel the phase retrieval problem for such signals is formulated as one of reconstructing the combining coefficients in the shift invariant basis expansion we develop sufficient conditions on the coefficients and the bases to guarantee exact phase retrieval by which we mean reconstruction up to a global phase factor we present a new class of discrete domain signals that are not necessarily minimum phase but allow for exact phase retrieval from their fourier magnitude spectra we also establish hilbert transform relations between log magnitude and phase spectra for this class of discrete signals it turns out that the corresponding continuous domain counterparts need not satisfy a hilbert transform relation notwithstanding the continuous domain signals can be reconstructed from their fourier magnitude spectra we validate the reconstruction guarantees through simulations for some important classes of signals such as bandlimited signals and piecewise smooth signals we also present an application of the proposed phase retrieval technique for artifact free signal reconstruction in frequency domain optical coherence tomography fdoct

multichannel sparse blind deconvolution on the sphere multichannel blind deconvolution is the problem of recovering an unknown signal f and multiple unknown channels x i from their circular convolution y i x i circledast f i 1 2 dots n we consider the case where the x i s are sparse and convolution with f is invertible our nonconvex optimization formulation solves for a filter h on the unit sphere that produces sparse output y i circledast h under some technical assumptions we show that all local minima of the objective function correspond to the inverse filter of f up to an inherent sign and shift ambiguity and all saddle points have strictly negative curvatures this geometric structure allows successful recovery of f and x i using a simple manifold gradient descent mgd algorithm the same approach is also applicable to blind gain and phase calibration with a fourier sensing matrix our algorithm and analysis require fewer assumptions than previous algorithms for the same problem our theoretical findings are complemented by numerical experiments which demonstrate superior performance of the proposed approach over the previous methods empirically our algorithm has low computation cost converging in a small number of iterations and low memory footprint solving only for the inverse filter of f

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

differential quadrature method for vibration analysis of shear deformable annular sector plates this paper presents differential quadrature solutions for free vibration analysis of moderately thick annular sector plates based on the mindlin first order shear deformation theory numerical characteristics of the differential quadrature method are illustrated through solving selected annular sector plates with different boundary conditions relative thickness ratios inner to outer radius ratios and various sector angles parametric studies in terms of the vibration frequency parameters are thoroughly investigated

free vibration analysis of mindlin sector plates numerical solutions by differential quadrature method abstract in this paper the free vibration analysis of moderately thick sector plates based on mindlin s first order shear deformation theory is examined the solutions to this problem are determined using the differential quadrature dq method for numerical evaluation the dq formulations for free vibration of the mindlin sector plates in two dimensional polar co ordinate system are derived to demonstrate the accuracy of the dq method convergence and comparison studies are carried out in parametric study the first eight frequency parameters are computed for sector plates with different boundary conditions relative thickness ratios and sector angles 30 360 the effects of sector angle and relative thickness ratio on the frequency parameters are discussed

complete dictionary recovery over the sphere ii recovery by riemannian trust region method we consider the problem of recovering a complete i e square and invertible matrix a 0 from y in mathbb r n times p with y boldsymbol a 0 x 0 provided x 0 is sufficiently sparse this recovery problem is central to theoretical understanding of dictionary learning which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning we give the first efficient algorithm that provably recovers a 0 when x 0 has o left n right nonzeros per column under suitable probability model for x 0 our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint and hence is naturally phrased in the language of manifold optimization in a companion paper we have showed that with high probability our nonconvex formulation has no spurious local minimizers and around any saddle point the objective function has a negative directional curvature in this paper we take advantage of the particular geometric structure and describe a riemannian trust region algorithm that provably converges to a local minimizer with from arbitrary initializations such minimizers give excellent approximations to the rows of x 0 the rows are then recovered by a linear programming rounding and deflation

stationary graph processes and spectral estimation stationarity is a cornerstone property that facilitates the analysis and processing of random signals in the time domain although time varying signals are abundant in nature in many practical scenarios the information of interest resides in more irregular graph domains this lack of regularity hampers the generalization of the classical notion of stationarity to graph signals this paper proposes a definition of weak stationarity for random graph signals that takes into account the structure of the graph where the random process takes place while inheriting many of the meaningful properties of the classical time domain definition provided that the topology of the graph can be described by a normal matrix stationary graph processes can be modeled as the output of a linear graph filter applied to a white input this is shown equivalent to requiring the correlation matrix to be diagonalized by the graph fourier transform a fact that is leveraged to define a notion of power spectral density psd properties of the graph psd are analyzed and a number of methods for its estimation are proposed this includes generalizations of nonparametric approaches such as periodograms window based average periodograms and filter banks as well as parametric approaches using moving average autoregressive and arma processes graph stationarity and graph psd estimation are investigated numerically for synthetic and real world graph signals

phase retrieval using alternating minimization phase retrieval problems involve solving linear equations but with missing sign or phase for complex numbers information more than four decades after it was first proposed the seminal error reduction algorithm of gerchberg and saxton and fienup is still the popular choice for solving many variants of this problem the algorithm is based on alternating minimization i e it alternates between estimating the missing phase information and the candidate solution despite its wide usage in practice no global convergence guarantees for this algorithm are known in this paper we show that a resampling variant of this approach converges geometrically to the solution of one such problem finding a vector bf x from bf y bf a where bf y vert bf a t bf x vert and vert bf z vert denotes a vector of element wise magnitudes of bf z under the assumption that bf a is gaussian empirically we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem which are based on lifting to a convex matrix problem in sample complexity and robustness to noise however it is much more efficient and can scale to large problems analytically for a resampling version of alternating minimization we show geometric convergence to the solution and sample complexity that is off by log factors from obvious lower bounds we also establish close to optimal scaling for the case when the unknown vector is sparse our work represents the first theoretical guarantee for alternating minimization albeit with resampling for any variant of phase retrieval problems in the non convex setting

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

exact analytical solutions for free vibrations of thick sectorial plates with simply supported radial edges abstract the first known exact analytical solutions are derived for the free vibrations of thick mindlin sectorial plates having simply supported radial edges and arbitrary conditions along the circular edge the general solutions to the mindlin differential equations of motion contain noninteger order ordinary and modified bessel functions of the first and second kinds and six arbitrary constants of integration by exercising a careful limiting process three regularity conditions at the vertex of the radial edges are invoked to yield three equations of constraint among the six constants for sector angles exceeding 180 re entrant corners three additional linearly independent equations among the six constants are obtained by satisfying the three boundary conditions along the circular edge frequency determinant equations are derived for mindlin sectorial plates with circular boundaries which are clamped simply supported or free nondimensional frequency parameters are presented for over a wide range of salient and re entrant sector angles 30 360 and thickness to radius ratios of 0 1 0 2 and 0 4 frequency results obtained for mindlin sectorial plates are compared to those determined for classically thin sectorial plates and the results are found to be considerably different than those derived from thin plate theory particularly for the fundamental frequencies of plates having sector angles slightly greater than 180 when the circular boundary is free the frequencies for 360 sectorial plates i e circular plates having a hinged crack are compared with those for complete circular ones

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

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

free vibration and buckling of unsymmetric cross ply laminated plates using a refined theory abstract the free vibration and buckling of unsymmetric cross ply plates with a variety of boundary conditions are analyzed by using a refined shear deformation theory used in reference 18 this theory incorporates tranverse shear deformation and transverse normal stress effects higher order effects are accounted for and the static conditions on the external bounding planes of the plate are satisfied the results obtained are compared with their classical and first order transverse shear deformation counterparts and with results available in the literature excellent agreement is found with simply supported edge condition results in the literature further evidence is provided that classical laminated plate theory over predicts both the eigenfrequencies and the buckling loads

structured signal recovery from quadratic measurements breaking sample complexity barriers via nonconvex optimization this paper concerns the problem of recovering an unknown but structured signal x in mathbb r n from m quadratic measurements of the form y r left langle a r x rangle right 2 for r 1 2 ldots m we focus on the under determined setting where the number of measurements is significantly smaller than the dimension of the signal m we formulate the recovery problem as a nonconvex optimization problem where prior structural information about the signal is enforced through constrains on the optimization variables we prove the projected gradient descent when initialized in a neighborhood of the desired signal converges to the unknown signal at a linear rate these results hold for any closed constraint set convex or non convex providing convergence guarantees to the global optimum even when the objective function and constraint set are nonconvex furthermore these results hold with a number of measurements that are only a constant factor away from the minimal number of measurements required to uniquely identify the unknown signal our results provide the first provably tractable algorithm for this data poor regime breaking local sample complexity barriers that have emerged in this paper in this paper we utilize and further develop powerful tools for uniform convergence of empirical processes that may have broader implications for rigorous understanding of constrained nonconvex optimization heuristics

vibration analysis of corner supported mindlin plates of arbitrary shape using the lagrange multiplier method abstract this paper presents the first known solutions of the problem of free flexural vibration of corner supported mindlin plates of arbitrary shape a hybrid numerical approach combining the rayleigh ritz method and the lagrange multiplier method has been developed to solve the plate vibration problem the algorithm uses the pb 2 shape functions to account for different geometries and lagrange multipliers to impose zero lateral deflection constraints at plate corners the method of solution is applicable to arbitrarily shaped plates with corner supports in this paper however only triangular skew and annular sector plates are chosen for the purpose of demonstration some comparison studies for corner supported thin square plates are made to verify the accuracy of the derived solutions

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

vibration of annular sector mindlin plates with internal radial line and circumferential arc supports an energy approach is presented for the free vibration analysis of thick annular sector plates with internal line arc supports this has been a topic of practical interest but there appears to have been no previous work reported based on the stationary energy principle and the recently proposed pb 2 rayleigh ritz method the governing eigenvalue equation is derived for internally supported thick annular sector plates of different radii relative thickness ratios sector angles and with various end conditions along the radial and circumferential edges the proposed solution algorithm involves 1 sets of mathematically complete two dimensional polynomials p 2 used as the admissible displacement and rotational functions and 2 a basic function b formed from the piecewise expressions for the boundary and internal line arc supports these functions automatically satisfy all the kinematic boundary conditions of the four edges and internal supports the numerical procedure has been demonstrated by several example plate problems convergence and comparison studies for simple examples are used to verify the accuracy of the present method

reconstruction of signals from their autocorrelation and cross correlation vectors with applications to phase retrieval and blind channel estimation we consider the problem of reconstructing two signals from the autocorrelation and cross correlation measurements this inverse problem is a fundamental one in signal processing and arises in many applications including phase retrieval and blind channel estimation in a typical phase retrieval setup only the autocorrelation measurements are obtainable we show that when the measurements are obtained using three simple masks phase retrieval reduces to the aforementioned reconstruction problem the classic solution to this problem is based on finding common factors between the z transforms of the autocorrelation and cross correlation vectors this solution has enjoyed limited practical success mainly due to the fact that it is not sufficiently stable in the noisy setting in this paper inspired by the success of convex programming in provably and stably solving various quadratic constrained problems we develop a semidefinite programming based algorithm and provide theoretical guarantees in particular we show that almost all signals can be uniquely recovered by this algorithm up to a global phase comparative numerical studies demonstrate that the proposed method significantly outperforms the classic method in the noisy setting

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

globally convergent levenberg marquardt method for phase retrieval in this paper we consider a nonlinear least squares model for the phase retrieval problem since the hessian matrix may not be positive definite and the gauss newton gn matrix is singular at any optimal solution we propose a modified levenberg marquardt lm method where the hessian is substituted by a summation of the gn matrix and a regularization term similar to the well known wirtinger flow wf algorithm under certain assumptions we start from an initial point provably close to the set of the global optimal solutions global linear convergence and local quadratic convergence to the global solution set are established by estimating the smallest nonzero eigenvalues of the gn matrix establishing local error bound properties and constructing a modified regularization condition the computational cost becomes tractable if a preconditioned conjugate gradient method is applied to solve the lm equation inexactly specifically the pre conditioner is constructed from the expectation of the lm coefficient matrix by assuming the independence between the measurements and iteration point the initialization can be dropped by slightly modifying the algorithm preliminary numerical experiments show that our algorithm is robust and it is often faster than the wf method on both random examples and natural image recovery

majorization minimization algorithms in signal processing communications and machine learning this paper gives an overview of the majorization minimization mm algorithmic framework which can provide guidance in deriving problem driven algorithms with low computational cost a general introduction of mm is presented including a description of the basic principle and its convergence results the extensions acceleration schemes and connection to other algorithmic frameworks are also covered to bridge the gap between theory and practice upperbounds for a large number of basic functions derived based on the taylor expansion convexity and special inequalities are provided as ingredients for constructing surrogate functions with the pre requisites established the way of applying mm to solving specific problems is elaborated by a wide range of applications in signal processing communications and machine learning

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

bilinear compressed sensing under known signs via convex programming we consider the bilinear inverse problem of recovering two vectors boldsymbol x in mathbb r l and boldsymbol w in mathbb r l from their entrywise product we consider the case where boldsymbol x and boldsymbol w have known signs and are sparse with respect to known dictionaries of size k and n respectively here k and n may be larger than smaller than or equal to l we introduce ell 1 branchhull which is a convex program posed in the natural parameter space and does not require an approximate solution or initialization in order to be stated or solved under the assumptions that boldsymbol x and boldsymbol w satisfy a comparable effective sparsity condition and are s 1 and s 2 sparse with respect to a random dictionary we present a recovery guarantee in a noisy case we show that ell 1 branchhull is robust to small dense noise with high probability if the number of measurements satisfy l geq omega s 1 s 2 log 2 k n numerical experiments show that the scaling constant in the theorem is not too large we also introduce variants of ell 1 branchhull for the purposes of tolerating noise and outliers and for the purpose of recovering piecewise constant signals we provide an admm implementation of these variants and show they can extract piecewise constant behavior from real images

phase retrieval via reweighted amplitude flow this paper deals with finding an n dimensional solution boldsymbol x to a system of quadratic equations of the form y i langle boldsymbol a i boldsymbol x rangle 2 for 1 leq i leq m which is also known as the generalized phase retrieval problem for this np hard problem a novel approach is developed for minimizing the amplitude based least squares empirical loss which starts with a weighted maximal correlation initialization obtainable through a few power or lanczos iterations followed by successive refinements based on a sequence of iteratively reweighted gradient iterations the two stages initialization and gradient flow distinguish themselves from prior contributions by the inclusion of a fresh re weighting regularization procedure for certain random measurement models the novel scheme is shown to be able to recover the true solution boldsymbol x in time proportional to reading the data lbrace boldsymbol a i y i rbrace 1 leq i leq m this holds with high probability and without extra assumption on the signal vector boldsymbol x to be recovered provided that the number m of equations is some constant c 0 times the number n of unknowns in the signal vector namely m cn empirically the upshots of this contribution are first almost text 100 perfect signal recovery in the high dimensional say n geq 2000 regime given only an information theoretic limit number of noiseless equations namely m 2n 1 in the real gaussian case and second nearly optimal statistical accuracy in the presence of additive noise of bounded support finally substantial numerical tests using both synthetic data and real images corroborate markedly improved recovery performance and computational efficiency of the novel scheme relative to the state of the art approaches

coded aperture design for solving the phase retrieval problem in x ray crystallography abstract x ray crystallography is an experimental technique used in material analysis that allows to measure the atomic positions of the elements present in a crystal this technique is based on the x ray diffraction patterns that provide electronic and elastic properties of the crystal of interest thus the crystal can be uniquely identified by means of the phase of its diffraction patterns that are also used to analyze the material of interest the phase of the x ray cannot be directly measured however it can be recovered from the intensity of diffraction patterns a recent work has shown that the phase signal can be recovered more efficiently when the acquisition architecture includes an optical element known as coded aperture such that the underlying signal is recovered from coded diffraction patterns a coded aperture is an element that modulates the x ray diffraction patterns by blocking some x ray beams the structure and the number of coded projections are crucial inasmuch they determine the quality and the acquisition time of the x ray signal this paper presents the analysis of a coded x ray crystallography system and the design of the spatial structure of the coded aperture such that the images are recovered with high psnr peak signal to noise ratio using the minimum number of coded projections the simulations indicate that the designed coded apertures obtain a reduction of up to 50 in the number of coded projections and an increase in the psnr of up to 2 db when the results are compared with the reconstructed images by using random non designed coded aperture structures all simulations were carried out on a set of diffraction pattern images obtained by using the saxs waxs x ray crystallography software to simulated the diffraction patterns of a real crystal structure called rhombic dodecahedron

phase recovery maxcut and complex semidefinite programming phase retrieval seeks to recover a signal x in mathbb c p x c p from the amplitude a x a x of linear measurements ax in mathbb c n a x c n we cast the phase retrieval problem as a non convex quadratic program over a complex phase vector and formulate a tractable relaxation called phasecut similar to the classical maxcut semidefinite program we solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in gerchberg and saxton optik 35 237 246 1972 where each iteration is a matrix vector product numerical results show the performance of this approach over three different phase retrieval problems in comparison with greedy phase retrieval algorithms and matrix completion formulations

on the 2d phase retrieval problem the recovery of a signal from the magnitude of its fourier transform also known as phase retrieval is of fundamental importance in many scientific fields it is well known that due to the loss of fourier phase the problem in one dimensional 1d is ill posed without further constraints there is no unique solution to the problem in contrast uniqueness up to trivial ambiguities very often exists in higher dimensions with mild constraints on the input in this paper we focus on the 2d phase retrieval problem and provide insight into this uniqueness property by exploring the connection between the 2d and 1d formulations in particular we show that 2d phase retrieval can be cast as a 1d problem with additional constraints which limit the solution space we then prove that only one additional constraint is sufficient to reduce the many feasible solutions in the 1d setting to a unique solution for almost all signals these results allow to obtain an analytical approach with combinatorial complexity to solve the 2d phase retrieval problem when it is unique

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

complete dictionary recovery over the sphere i overview and the geometric picture we consider the problem of recovering a complete i e square and invertible matrix a 0 from y in mathbb r n times p with y a 0 x 0 provided x 0 is sufficiently sparse this recovery problem is central to theoretical understanding of dictionary learning which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning we give the first efficient algorithm that provably recovers a 0 when x 0 has o left n right nonzeros per column under suitable probability model for x 0 in contrast prior results based on efficient algorithms either only guarantee recovery when x 0 has o sqrt n zeros per column or require multiple rounds of semidefinite programming relaxation to work when x 0 has o n nonzeros per column our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint in this paper we provide a geometric characterization of the objective landscape in particular we show that the problem is highly structured with high probability 1 there are no spurious local minimizers and 2 around all saddle points the objective has a negative directional curvature this distinctive structure makes the problem amenable to efficient optimization algorithms in a companion paper we design a second order trust region algorithm over the sphere that provably converges to a local minimizer from arbitrary initializations despite the presence of saddle points

low rank positive semidefinite matrix recovery from corrupted rank one measurements we study the problem of estimating a low rank positive semidefinite psd matrix from a set of rank one measurements using sensing vectors composed of i i d standard gaussian entries which are possibly corrupted by arbitrary outliers this problem arises from applications such as phase retrieval covariance sketching quantum space tomography and power spectrum estimation we first propose a convex optimization algorithm that seeks the psd matrix with the minimum ell 1 norm of the observation residual the advantage of our algorithm is that it is free of parameters therefore eliminating the need for tuning parameters and allowing easy implementations we establish that with high probability a low rank psd matrix can be exactly recovered as soon as the number of measurements is large enough even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes moreover the recovery is also stable against bounded noise with the additional information of an upper bound of the rank of the psd matrix we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy

scalable solvers of random quadratic equations via stochastic truncated amplitude flow a novel approach termed stochastic truncated amplitude flow staf is developed to reconstruct an unknown n dimensional real complex valued signal boldsymbol x from m phaseless quadratic equations of the form psi i langle boldsymbol a i boldsymbol x rangle this problem also known as phase retrieval from magnitude only information is np hard in general adopting an amplitude based nonconvex formulation staf leads to an iterative solver comprising two stages s1 orthogonality promoting initialization through a stochastic variance reduced gradient algorithm and s2 a series of iterative refinements of the initialization using stochastic truncated gradient iterations both stages involve a single equation per iteration thus rendering staf a simple scalable and fast approach amenable to large scale implementations that are useful when n is large when lbrace boldsymbol a i rbrace i 1 m are independent gaussian staf provably recovers exactly any boldsymbol x in mathbb r n exponentially fast based on order of n quadratic equations staf is also robust in the presence of additive noise of bounded support simulated tests involving real gaussian lbrace boldsymbol a i rbrace vectors demonstrate that staf empirically reconstructs any boldsymbol x in mathbb r n exactly from about 2 3n magnitude only measurements outperforming state of the art approaches and narrowing the gap from the information theoretic number of equations m 2n 1 extensive experiments using synthetic data and real images corroborate markedly improved performance of staf over existing alternatives

an analytical solution for vibrations of a polarly orthotropic mindlin sectorial plate with simply supported radial edges abstract this paper presents the first known analytical solution for vibrations of a polarly orthotropic mindlin sectorial plate with simply supported radial edges the solution is a series solution constructed using the frobenius method and exactly satisfies not only the boundary conditions along the radial and circular edges but also the regularity conditions at the vertex of the radial edges the moment and shear force singularities at the vertex are exactly considered in the solution the correctness of the proposed solution is confirmed by comparing non dimensional frequencies of isotropic plates obtained from the present solution with published data obtained from a closed form solution this paper also investigates the effects of elastic and shear moduli on the vibration frequencies of the sectorial plates with free or fix boundary conditions along the circumferential edge a study is also carried out about the influence of elastic and shear moduli on the moment and shear force singularities at the plate origin r 0 for different vertex angles

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

phase retrieval by alternating minimization with random initialization we consider the phase retrieval problem where the goal is to reconstruct an n dimensional complex vector from its phaseless scalar products with m sensing vectors independently sampled from complex normal distributions we show that if m geq mn 3 2 log 7 2 n for some m 0 then the classical algorithm of alternating minimization with random initialization succeeds with high probability as n m rightarrow infty this is a step toward proving the conjecture in which conjectures that the algorithm succeeds when m o n the analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc

low rank phase retrieval we develop two iterative algorithms for solving the low rank phase retrieval lrpr problem lrpr refers to recovering a low rank matrix boldsymbol x from magnitude only phaseless measurements of random linear projections of its columns both methods consist of a spectral initialization step followed by an iterative algorithm to maximize the observed data likelihood we obtain sample complexity bounds for our proposed initialization approach to provide a good approximation of the true boldsymbol x when the rank is low enough these bounds are significantly lower than what existing single vector phase retrieval algorithms need via extensive experiments we show that the same is also true for the proposed complete algorithms