an adaptive level set method for nondifferentiable constrained image recovery the formulation of a wide variety of image recovery problems leads to the minimization of a convex objective over a convex set representing the constraints derived from a priori knowledge and consistency with the observed signals in previous years nondifferentiable objectives have become popular due in part to their ability to capture certain features such as sharp edges they also arise naturally in minimax inconsistent set theoretic recovery problems at the same time the issue of developing reliable numerical algorithms to solve such convex programs in the context of image recovery applications has received little attention we address this issue and propose an adaptive level set method for nondifferentiable constrained image recovery the asymptotic properties of the method are analyzed and its implementation is discussed numerical experiments illustrate applications to total variation and minimax set theoretic image restoration and denoising problems

model reduction of systems with zeros interlacing the poles the problem of reducing a system with zeros interlacing the poles zip on the real axis is considered it is proved that many model reduction methods such as the balanced truncation balanced residualization suboptimal and optimal hankel approximations inherit the zip property properties of the hankel singular values of zip systems are also listed

non linear programming maximize sinr for designing spreading sequence signal to interference plus noise ratio sinr is an important index for wireless communications this paper shows a method to derive spreading sequences utilized in code division multiple access systems as the solutions of the non linear programming maximize sinr to this end we consider a frequency selective wide sense stationary uncorrelated scattering channel and obtain the worst case of sinr in the course of the evaluation we derive an expression of sinr whose main terms consist of periodic correlation terms and odd periodic correlation terms also we show a relation between sinr and mean square correlations with our expression we obtain the problem maximize sinr since this problem is not convex we obtain the relaxed problem with a semidefinite relaxation technique from this relaxed problem we obtain the approximated solutions and evaluate the bit error rate for the solutions

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

on lipschitz analysis and lipschitz synthesis for the phase retrieval problem in this paper we prove two results regarding reconstruction from magnitudes of frame coefficients the so called phase retrieval problem first we show that phase retrievability as an algebraic property implies that nonlinear maps are bi lipschitz with respect to appropriate metrics on the quotient space second we prove that reconstruction can be performed using lipschitz continuous maps specifically we show that when nonlinear analysis maps h r m are injective with x hx fki m 1 and x hx fki 2 m 1 where f1 fm is a frame for a hilbert space h and

fast unit modulus least squares with applications in beamforming unit modulus least squares uls problems arise in many applications including phase only beamforming sensor network localization synchronization phase retrieval and radar code design uls formulations can always be recast as unit modulus quadratic programs to which semidefinite relaxation sdr can be applied and is often the state of the art approach however sdr lifts the problem dimension i e the number of variables from n to n 2 which drastically increases the memory burden and computational cost when the problem size is already large e g when designing phase only beamformer weights for massive multiple input multiple output systems this paper focuses on scalable first order algorithms for the uls problem and some of its variants it advocates using simple gradient projection gp as a starting point for solving the uls problem establishes global convergence of gp to a karush kuhn tucker point for this np hard problem and bounds its iteration complexity then it proposes uls extensions tailored to reflect practical beamformer design objectives bringing in and exploiting new degrees of freedom to improve the beampattern designs simple variants of gp are proposed to deal with these extended uls problems simulations are used to showcase the effectiveness of the proposed algorithms in both the plain uls problem and in the context of phase only beamforming

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

mathematical considerations for the problem of fourier transform phase retrieval from magnitude in this paper we deal with the problem of retrieving a finite extent function from the magnitude of its fourier transform this so called phase retrieval problem will first be posed under its different underlying models we will present a brief review of the main results known in this area for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution several important issues which are yet unresolved will be pointed out and discussed we will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real valued sequence x from the magnitude of the output of a linear distortion hx j j 1 cdots n a number of important results will be obtained for this general setting by means of algebraic geometric techniques in particular the problems of the existence of a solution for phase retrieval number of feasible solutions stability of the essential

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

methods for reconstruction of 2 d sequences from fourier transform magnitude the gerchberg saxton gs algorithm and its generalizations have been the main tools for phase retrieval unfortunately it has been observed that the reconstruction using these algorithms does not always converge to the correct result even if the desired solution satisfies the uniqueness condition in this paper we propose a new deautocorrelation algorithm and a few auxiliary techniques we recommend that a combination of the iterative fourier transform ift algorithm with our new algorithm and techniques can improve the probability of success of phase retrieval a pragmatic procedure is illustrated different reconstruction examples that are difficult to reconstructed using the single ift algorithm are used to show the robustness and effectiveness of the new combination of algorithms if the given fourier modulus data contain no noise it is sometimes possible to get a perfect reconstruction even when the signal to noise ratio snr of the fourier modulus data is only 10 db a meaningful result remains reachable for our examples a concept concerning the intrinsic ambiguity of phase retrieval is suggested we emphasize the necessity of verification of the solution since the available phase retrieval algorithms are incompetent for distinguishing between an intrinsically ambiguous solution and the true solution

unique factorization in invariant power series rings abstract let g be a finite group k a perfect field and v a finite dimensional kg module we let g act on the power series k v by linear substitutions and address the question of when the invariant power series k v g form a unique factorization domain we prove that for a permutation module for a p group in characteristic p the answer is always positive on the other hand if g is a cyclic group of order p k has characteristic p and v is an indecomposable kg module of dimension r with 1 r p we show that the invariant power series form a unique factorization domain if and only if r is equal to 1 2 p 1 or p this contradicts a conjecture of peskin

some computational problems arising in adaptive optics imaging systems abstract recently there has been growing interest and progress in using numerical linear algebra techniques in adaptive optics imaging control computations real time adaptive optics is a means for enhancing the resolution of ground based optical telescopes beyond the limits previously imposed by the turbulent atmosphere an adaptive optics system automatically corrects for light distortions caused by the medium of transmission the system measures the characteristics of the phase of the arriving wavefront and corrects for the degradations by means of one or more deformable mirrors controlled by special purpose computers no attempt is made in this paper to give a comprehensive survey of recent numerical linear applications in optical imaging rather two fairly representative applications are discussed in some detail the following research topics in the area of adaptive optics control systems each involving the formulation and numerical solution of difficult problems in numerical linear algebra are described 1 jacobi like eigenvalue computations for multiple bandwidth deformable mirror control methods and 2 covariance matrix computations for performance modeling of adaptive optics systems using fast hankel transforms

global registration of multiple point clouds using semidefinite programming consider n points in r d and m local coordinate systems that are related through unknown rigid transforms for each point we are given possibly noisy measurements of its local coordinates in some of the coordinate systems alternatively for each coordinate system we observe the coordinates of a subset of the points the problem of estimating the global coordinates of the n points up to a rigid transform from such measurements comes up in distributed approaches to molecular conformation and sensor network localization and also in computer vision and graphics the least squares formulation of this problem although nonconvex has a well known closed form solution when m 2 based on the singular value decomposition svd however no closed form solution is known for m geq 3 in this paper we demonstrate how the least squares formulation can be relaxed into a convex program namely a semidefinite program sdp by setting up connections between the uniqueness of this sdp and results fr

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

hermitian completely positive matrices abstract in this paper we introduce the hermitian completely positive hcp matrix which has a hermitian completely positive hcp decomposition with all real and imaginary parts of the decomposition vectors being nonnegative some properties of the hermitian completely positive matrix are given a semidefinite algorithm is also proposed for checking whether a hermitian matrix is hcp or not if a matrix is not hcp a certificate for it can be obtained if it is an hcp decomposition can be obtained

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

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

on the robustness of symmetric systems this note concerns the robustness of lti symmetric systems under symmetric or diagonal perturbations it is shown that for symmetric systems the robust stability condition for dynamic perturbations given by the small gain theorem is also necessary if the perturbation matrix is assumed to be diagonal but the robust stability condition for parametric perturbations given by the real stability radius is no longer necessary if the perturbation matrix is merely assumed to be symmetric

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

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

optical beam shaping and diffraction free waves a variational approach abstract we investigate the problem of shaping radially symmetric annular beams into desired intensity patterns along the optical axis within the fresnel approximation we show that this problem can be expressed in a variational form equivalent to the one arising in phase retrieval using the uncertainty principle we prove various rigorous lower bounds on the functional these lower bounds estimate the l 2 error for the beam shaping problem in terms of the design parameters we also use the method of stationary phase to construct a natural ansatz for a minimizer in the short wavelength limit we illustrate the implications of our results by applying the method of stationary phase coupled with the gerchberg saxton algorithm to beam shaping problems arising in the remote delivery of beams and pulses

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

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

invariant scattering convolution networks a wavelet scattering network computes a translation invariant image representation which is stable to deformations and preserves high frequency information for classification it cascades wavelet transform convolutions with nonlinear modulus and averaging operators the first network layer outputs sift type descriptors whereas the next layers provide complementary invariant information that improves classification the mathematical analysis of wavelet scattering networks explains important properties of deep convolution networks for classification a scattering representation of stationary processes incorporates higher order moments and can thus discriminate textures having the same fourier power spectrum state of the art classification results are obtained for handwritten digits and texture discrimination with a gaussian kernel svm and a generative pca classifier

rings of invariants of modular p groups which are hypersurfaces abstract for l a finite non modular group whose invariants form a polynomial ring and h a subgroup of l containing the derived group of l nakajima found necessary and sufficient conditions on h for its invariant ring s h to be a hypersurface in a crucial step of his proof he showed that if s h is a hypersurface then between h and l there is a group g with polynomial invariant ring such that s h s g b for g a finite modular p group over f p with polynomial invariant ring and h a subgroup of g containing the derived group of g we find necessary and sufficient conditions on h to ensure that s h s g b

on two dimensional hilbert integral equations generalized minimum phase signals and phase retrieval one dimensional 1 d causal signals admit hilbert integral relations between the real and imaginary parts of their fourier spectra for 1 d minimum phase signals the log magnitude and phase spectra also admit such hilbert relations in this paper we extend these results to 2 d signals we first establish the hilbert integral equations for 2 d first quadrant signals for continuous domain 2 d signals we present the partial hilbert transform relations between the real and imaginary parts of the spectrum for the discrete domain counterpart we show that the partial hilbert transform does not suffice which motivates us to introduce the composite hilbert transform considering the problem of phase retrieval we show that in both 1 d 2 d and continuous discrete domains there exists a generalized class of signals which subsumes the well known class of 1 d minimum phase signals with a rational transfer function for which hilbert integral relations exist between the log magnitude and phase spectra we introduce them under the nomenclature of generalized minimum phase signals for this class of signals phase retrieval is possible by using the hilbert transform without the requirement of a rational transfer function furthermore we show that a generalized minimum phase signal admits a stable convolutional inverse which also belongs to the same class simulation results demonstrating accurate reconstruction of 1 d and 2 d generalized minimum phase signals from their magnitude spectra are presented to support the theoretical developments

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

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

a spatial branch and cut method for nonconvex qcqp with bounded complex variables we develop a spatial branch and cut approach for nonconvex quadratically constrained quadratic programs with bounded complex variables cqcqp linear valid inequalities are added at each node of the search tree to strengthen semidefinite programming relaxations of cqcqp these valid inequalities are derived from the convex hull description of a nonconvex set of 2 times 2 positive semidefinite hermitian matrices subject to a rank one constraint we propose branching rules based on an alternative to the rank one constraint that allows for local measurement of constraint violation closed form bound tightening procedures are used to reduce the domain of the problem we apply the algorithm to solve the alternating current optimal power flow problem with complex variables as well as the box constrained quadratic programming problem with real variables

phase retrieval from incomplete magnitude information via total variation regularization the phase retrieval problem has drawn considerable attention as many optical detection devices can only measure magnitudes of the fourier transform of the underlying object signal or image this paper addresses the phase retrieval problem from incomplete data where only partial magnitudes of fourier transform are obtained in particular we consider structured illuminated patterns in holography and find that noninteger values used in designing such patterns often yield better reconstruction than the conventional integer valued ones furthermore we demonstrate theoretically and numerically that three diffracted sets of complete magnitude data are sufficient to recover the object to compensate for incomplete information we incorporate a total variation regularization a priori to guarantee that the reconstructed image satisfies some desirable properties the proposed model can be solved efficiently by an alternative directional multiplier method with provable convergence numerical experiments valid

phasecode fast and efficient compressive phase retrieval based on sparse graph codes we consider the problem of recovering a complex signal x in mathbb c n from m intensity measurements of the form a i mathrm h x 1 leq i leq m where a i mathrm h is the i th row of measurement matrix a in mathbb c m times n our main focus is on the case where the measurement vectors are unconstrained and where x is exactly k sparse or the so called general compressive phase retrieval problem we introduce phasecode a novel family of fast and efficient algorithms that are based on a sparse graph coding framework we show that in the noiseless case the phasecode algorithm can recover an arbitrarily close to one fraction of the k nonzero signal components using only slightly more than 4k measurements when the support of the signal is uniformly random with the order optimal time and memory complexity of theta k 1 it is known that the fundamental limit for the number of measurements in compressive phase retrieval problem is 4k o k for the more difficult problem of recovering the signal exactly and with no assumptions on its support distribution this shows that under mild relaxation of the conditions our algorithm is the first constructive capacity approaching compressive phase retrieval algorithm in fact our algorithm is also order optimal in complexity and memory furthermore we show that for any signal x phasecode can recover a random 1 p fraction of the nonzero components of x with high probability where p can be made arbitrarily close to zero with sample complexity m c p k where c p is a small constant depending on p that can be precisely calculated with optimal time and memory complexity as a result assuming that the nonzero components of x are lower bounded by theta 1 and upper bounded by theta k gamma for some positive constant gamma we are able to provide a strong ell 1 guarantee for the estimated signal hat x as follows hat x x 1 leq p x 1 1 o 1 where p can be made arbitrarily close to zero as one instance the phasecode algorithm can provably recover with high probability a random 1 10 7 fraction of the significant signal components using at most m 14k measurements next motivated by some important practical classes of optical systems we consider a fourier friendly constrained measurement setting and show that its performance matches that of the unconstrained setting when the signal is sparse in the fourier domain with uniform support in the fourier friendly setting that we consider the measurement matrix is constrained to be a cascade of fourier matrices corresponding to optical lenses and diagonal matrices corresponding to diffraction mask patterns finally we tackle the compressive phase retrieval problem in the presence of noise where measurements are in the form of y i a i mathrm h x 2 w i and w i is the additive noise to the i th measurement we assume that the signal is quantized and each nonzero component can take l m possible magnitudes and l p possible phases we consider the regime where k beta n delta delta in 0 1 we use the same architecture of phasecode for the noiseless case and robustify it using two schemes the almost linear scheme and the sublinear scheme we prove that with high probability the almost linear scheme recovers x with sample complexity theta k log n and computational complexity theta l m l p n log n and the sublinear scheme recovers x with sample complexity theta k log 3 n and computational complexity theta l m l p k log 3 n throughout we provide extensive simulation results that validate the practical power of our proposed algorithms for the sparse unconstrained and fourier friendly measurement settings for noiseless and noisy scenarios 1 here we define the notation mathcal o cdot theta cdot and omega cdot we have f mathcal o g if and only if there exists a constant c 1 0 such that left f g right f theta g if and only if there exist two constants c 1 c 2 0 such that c 1 and f omega g if and only if there exists a constant c 1 0 such that left f g right c 1

an approximate approach to h sup 2 optimal model reduction deals with the problem of computing an h sub 2 optimal reduced order model for a given stable multivariable linear system by way of orthogonal projection the problem is formulated as that of minimizing the h sub 2 model reduction cost over the stiefel manifold so that the stability constraint on reduced order models is automatically satisfied and thus totally avoided in the new problem formulation the closed form expression for the gradient of the cost over the manifold is derived from which a gradient flow results as an ordinary differential equation ode a number of nice properties about such a flow are established furthermore two explicit iterative convergent algorithms are developed from the flow one has a constant step size and the other has a varying step size and is much more efficient both of them inherit the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model reduction cost is decreasing to minima along the iterates a procedure for closing the gap between the original and modified problem is proposed in the symmetric case the two problems are shown to be equivalent numerical examples are presented to illustrate the effectiveness of the proposed algorithms as well as convergence

vectorial phase retrieval of 1 d signals reconstruction of signals from measurements of their spectral intensities also known as the phase retrieval problem is of fundamental importance in many scientific fields in this paper we present a novel framework denoted as vectorial phase retrieval for reconstruction of pairs of signals from spectral intensity measurements of the two signals and of their interference we show that this new framework can alleviate some of the theoretical and computational challenges associated with classical phase retrieval from a single signal first we prove that for compactly supported signals in the absence of measurement noise this new setup admits a unique solution next we present a statistical analysis of vectorial phase retrieval and derive a computationally efficient algorithm to solve it finally we illustrate via simulations that our algorithm can accurately reconstruct signals even at considerable noise levels

iterative techniques for minimum phase signal reconstruction from phase or magnitude in this paper we develop iterative algorithms for reconstructing a minimum phase sequence from the phase or magnitude of its fourier transform these iterative solutions involve repeatedly imposing a causality constraint in the time domain and incorporating the known phase or magnitude function in the frequency domain this approach is the basis of a new means of computing the hilbert transform of the log magnitude or phase of the fourier transform of a minimum phase sequence which does not require phase unwrapping finally we discuss the potential use of this iterative computation in determining samples of the unwrapped phase of a mixed phase sequence

signal reconstruction from the magnitude of subspace components we consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems in the deterministic setting a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions that require at least a quadratic number of subspaces moreover we address reconstruction under the erasure of a subset of the norms using the concepts of p fusion frames and list decoding we propose an algorithm that outputs a finite list of candidate signals one of which is the correct one in the random setting we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program

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

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

sensitivity analysis in linear programming and semidefinite programming using interior point methods we analyze perturbations of the right hand side and the cost parameters in linear programming lp and semidefinite programming sdp we obtain tight bounds on the perturbations that allow interior point methods to recover feasible and near optimal solutions in a single interior point iteration for the unique nondegenerate solution case in lp we show that the bounds obtained using interior point methods compare nicely with the bounds arising from using the optimal basis we also present explicit bounds for sdp using the monteiro zhang family of search directions and specialize them to the aho h k m and nt directions

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

deep scattering spectrum a scattering transform defines a locally translation invariant representation which is stable to time warping deformation it extends mfcc representations by computing modulation spectrum coefficients of multiple orders through cascades of wavelet convolutions and modulus operators second order scattering coefficients characterize transient phenomena such as attacks and amplitude modulation a frequency transposition invariant representation is obtained by applying a scattering transform along log frequency state the of art classification results are obtained for musical genre and phone classification on gtzan and timit databases respectively

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

hypersurfaces in modular invariant theory let the finite group g act linearly on the n dimensional vector space v over the field k of characteristic p 0 suppose h g is a normal subgroup of index l a prime number in this paper we shall study the relationship between the two invariant rings k v h and k v g as a corollary of our main result we get that if k v g is a polynomial algebra and k v h is factorial then k v h is a graded hypersurface algebra

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

smoothing technique and its applications in semidefinite optimization in this paper we extend the smoothing technique nesterov in math program 103 1 127 152 2005 nesterov in unconstrained convex mimimization in relative scale 2003 onto the problems of semidefinite optimization for that we develop a simple framework for estimating a lipschitz constant for the gradient of some symmetric functions of eigenvalues of symmetric matrices using this technique we can justify the lipschitz constants for some natural approximations of maximal eigenvalue and the spectral radius of symmetric matrices we analyze the efficiency of the special gradient type schemes on the problems of minimizing the maximal eigenvalue or the spectral radius of the matrix which depends linearly on the design variables we show that in the first case the number of iterations of the method is bounded by o 1 epsilon where epsilon is the required absolute accuracy of the problem in the second case the number of iterations is bounded by 4 delta sqrt 1 delta r ln r where is the required relative accuracy and r is the maximal rank of corresponding linear matrix inequality thus the latter method is a fully polynomial approximation scheme

on positive realness negative imaginariness and h control of state space symmetric systems this paper studies positive realness and negative imaginariness of state space symmetric systems first a necessary and sufficient condition is provided for a state space symmetric system to be respectively positive real and negative imaginary then an explicit formulation for the infimum of h norm under a mixed h and negative imaginary performance and an explicit solution for the associated optimal static output feedback control gain are developed based on symmetric negative imaginary theorem finally three examples are provided to illustrate the main results of the paper

complex quadratic optimization and semidefinite programming in this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the hermitian complex form a special case of the problem that we study corresponds to the max 3 cut model used in a recent paper of goemans and williamson j comput system sci 68 2004 pp 442 470 we first develop a closed form formula to compute the probability of a complex valued normally distributed bivariate random vector to be in a given angular region this formula allows us to compute the expected value of a randomized with a specific rounding rule solution based on the optimal solution of the complex semidefinite programming relaxation problem in particular we present an m 2 1 cos frac 2 pi m 8 pi approximation algorithm and then study the limit of that model in which the problem remains np hard we show that if the objective is to maximize a positive semidefinite hermitian form then the randomization rounding procedure guarantees a worst case performance ratio of pi 4 approx 0 7854 which is better than the ratio of 2 pi approx 0 6366 for its counterpart in the real case due to nesterov furthermore if the objective matrix is real valued positive semidefinite with nonpositive off diagonal elements then the performance ratio improves to 0 9349

a new decentralised controller design method for a class of strongly interconnected systems abstractin this paper two interconnected structures are first discussed under which some closed loop subsystems must be unstable to make the whole interconnected system stable which can be viewed as a kind of strongly interconnected systems then comparisons with small gain theorem are discussed and large gain interconnected characteristics are shown a new approach for the design of decentralised controllers is presented by determining the lyapunov function structure previously which allows the existence of unstable subsystems by fully utilising the orthogonal space information of input matrix some new understandings are presented for the construction of lyapunov matrix this new method can deal with decentralised state feedback static output feedback and dynamic output feedback controllers in a unified framework furthermore in order to reduce the design conservativeness and deal with robustness a new robust decentralised controller design method is given by combining with the parameter depend

excitation retrieval of microwave linear arrays from phaseless far field data a methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed the approach combines tools from convex optimization to solve the phase retrieval problem and a simple measurement procedure to mitigate the non uniqueness of the solution numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed they show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem in addition the proposed approach that only calls for readily available routines is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases

rings of invariants of certainp groups over the fieldfp abstract we use nakajima s j algebra85 1983 253 286 characterization ofp groups with polynomial rings of invariants over the fieldfpto prove that maximal proper subgroups of such groups have rings of invariants that are hypersurfaces we give an explicit construction for the ring of invariants of such a group as a module over the ring of invariants of the total group

model reduction for state space symmetric systems in this paper the model reduction problem for state space symmetric systems is investigated first it is shown that several model reduction methods such as balanced truncation balanced truncation which preserves the dc gain optimal and suboptimal hankel norm approximations inherit the state space symmetric property furthermore for single input and single output siso state space symmetric systems we prove that the h norm of its transfer functions can be calculated via two simple formulas moreover the siso state space symmetric systems are equivalent to systems with zeros interlacing the poles zip under mild conditions

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

optimal symmetric h sub 2 controllers for systems with collocated sensors and actuators this note addresses the problem of designing the optimal symmetric h 2 controller for a plant with collocated sensors and actuators but with possibly asymmetric disturbance injection and performance specification for such a control problem the generalized plant has a symmetric block representing the transfer matrix from the control input to the measured output a complete solution to the optimal h 2 control problem with the symmetric structural constraint is given in terms of the optimal solution to a standard h 2 model matching problem without the constraint

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

variational analysis applied to the problem of optical phase retrieval we apply nonsmooth analysis to a well known optical inverse problem phase retrieval the phase retrieval problem arises in many different modalities of electromagnetic imaging and has been studied in the optics literature for over forty years the state of the art for this problem in two dimensions involves iterated projections for solving a nonconvex feasibility problem despite widespread use of these algorithms current mathematical theory cannot explain their success at the heart of projection algorithms is a nonconvex nonsmooth optimization problem we obtain some insight into these algorithms by applying techniques from nonsmooth analysis in particular we show that the weak closure of the set of directions toward the projection generate the subdifferential of the corresponding squared set distance function following a pattern of proof described in f h clarke yu s ledyaev r j stern and p r wolenski nonsmooth analysis and control theory springer verlag new york 1998 this result is generalized to provide conditions under which the subdifferential of an integral function equals the integral of the subdifferential

hard constrained inconsistent signal feasibility problems we consider the problem of synthesizing feasible signals in a hilbert space in the presence of inconsistent convex constraints some of which must imperatively be satisfied this problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones the resulting convex optimization problem is analyzed and numerical solution schemes are presented along with convergence results the proposed formalism and its algorithmic framework unify and extend existing approaches to inconsistent signal feasibility problems an application to signal synthesis is demonstrated

finding best approximation pairs relative to a convex and prox regular set in a hilbert space we study the convergence of an iterative projection reflection algorithm originally proposed for solving what are known as phase retrieval problems in optics there are two features that frustrate any analysis of iterative methods for solving the phase retrieval problem nonconvexity and infeasibility the algorithm that we developed called relaxed averaged alternating reflections raar was designed primarily to address infeasibility though our strategy has advantages for nonconvex problems as well in the present work we investigate the asymptotic behavior of the raar algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a hilbert space with empty intersection and for the problem of finding points that achieve a local minimum distance between one closed convex set and a closed prox regular set also possibly nonintersecting the nonconvex theory includes and expands prior results limited to convex sets with nonempty intersection to place the raar algorithm in context we develop parallel statements about the standard alternating projections algorithm and gradient descent all of the various algorithms are unified as instances of iterated averaged alternating proximal reflectors applied to a sum of regularized maximal monotone mappings

laplacian eigenvalues and the maximum cut problem we introduce and study an eigenvalue upper bound g on the maximum cut mc g of a weighted graph the function g has several interesting properties that resemble the behaviour of mc g the following results are presented we show that is subadditive with respect to amalgam and additive with respect to disjoint sum and 1 sum we prove that g is never worse that 1 131 mc g for a planar or more generally a weakly bipartite graph with nonnegative edge weights we give a dual characterization of g and show that g is computable in polynomial time with an arbitrary precision

probabilistic analysis of semidefinite relaxation for binary quadratic minimization we consider semidefinite programming relaxation sdr of a binary quadratic minimization problem this np hard problem arises naturally in the maximum likelihood detection of discrete signals for digital communications we analyze the average performance of the sdr algorithm for a class of randomly generated binary quadratic minimization problems although the sdr worst case approximation ratio is unbounded for this np hard problem our analysis shows that sdr can provide in polynomial time a provably near optimal solution achieving a constant factor approximation of the optimal objective value in probability moreover this constant factor remains bounded with increasing problem size our proof is based on an asymptotic analysis of karush kuhn tucker optimality conditions using random matrix theory

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

decentralized control of symmetric systems abstract this paper is concerned with the stabilizability problem of symmetric systems by means of decentralized controllers it is shown that the set of decentralized fixed modes of a symmetric system is equal to the set of uncontrollable and unobservable modes of the system and implies that the stabilizability of a state space symmetric system by means of a decentralized controller is equivalent to the stabilizability for the system by means of a centralized controller

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

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

extended matrix cube theorems with applications to µ theory in control in this paper we study semi infinite systems of linear matrix inequalities which are generically np hard for these systems we introduce computationally tractable approximations and derive quantitative guarantees of their quality as applications we discuss the problem of maximizing a hermitian quadratic form over the complex unit cube and the problem of bounding the complex structured singular value with the help of our complex matrix cube theorem we demonstrate that the standard scaling upper bound on m is a tight upper bound on the largest level of structured perturbations of the matrixm for which all perturbed matrices share a common lyapunov certificate for the discrete time stability

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

a deterministic theory for exact non convex phase retrieval in this paper we analyze the non convex framework of wirtinger flow wf for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory via a perspective in the lifted domain we show that the wf iterates converge to a true solution with fully deterministic arguments under a single condition on the lifted forward model to this end a geometric relationship between between the accuracy of spectral initialization and the validity of the regularity condition is derived in particular we determine that a certain concentration property on the spectral matrix must hold uniformly with a sufficiently tight constant this culminates into a sufficient condition that is equivalent to a restricted isometry type property over rank 1 positive semi definite matrices and amounts to a less stringent requirement on the lifted forward model than those of prominent low rank matrix recovery methods in the literature we characterize the performance limits of our framework in terms of the tightness of the concentration property via novel bounds on the convergence rate and on the signal to noise ratio such that the theoretical guarantees are valid using the spectral initialization at the proper sample complexity

phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms we propose strongly consistent algorithms for reconstructing the characteristic function 1k of an unknown convex body k in r n from possibly noisy measurements of the modulus of its fourier transform c 1k this represents a complete theoretical solution to the phase retrieval problem for characteristic functions of convex bodies the approach is via the closely related problem of reconstructing k from noisy measurements of its covariogram the function giving the volume of the intersection of k with its translates in the many known situations in which the covariogram determines a convex body up to reflection in the origin and when the position of the body is fixed our algorithms use o k n noisy covariogram measurements to construct a convex polytope pk that approximates k or its reflection k in the origin by recent uniqueness results this applies to all planar convex bodies all three dimensional convex polytopes and all symmetric and most in the sense of baire category arbitrary convex bodies in all dimensions two methods are provided and both are shown to be strongly consistent in the sense that almost surely the minimum of the hausdorff distance between pk and k tends to zero as k tends to infinity

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

some applications of small gain theorem to interconnected systems based on small gain theorem decentralized control of interconnected systems is studied linear matrix inequality conditions are given for stabilizability of interconnected systems interconnections can be used to improve system robustness in the sense of small gain theorem an example is given to illustrate the results finally interconnected systems composed of symmetric subsystems are studied

an interior point perspective on sensitivity analysis in semidefinite programming we study the asymptotic behavior of the interior point bounds arising from the work of yildirim and todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition bounds we introduce a weaker notion of nondegeneracy and discuss its implications for perturbations of the right hand side vector or the cost matrix we show that the interior point bounds evaluated on the central path using the monteiro zhang family of search directions converge as the duality gap tends to zero to the symmetrized version of the optimal partition bounds under mild nondegeneracy assumptions furthermore our analysis does not assume strict complementarity as long as the central path converges to the analytic center in a relatively controlled manner we also show that the same convergence results carry over to iterates lying in an appropriate very narrow central path neighborhood if the nesterov todd direction is used to evaluate the interior point bounds we extend our results to the case of simultaneous perturbations of the right hand side vector and the cost matrix we also provide examples illustrating that our assumptions in general cannot be weakened

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

optical wavefront reconstruction theory and numerical methods optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in nasa s hubble space telescope hst nasa s success with reconstruction algorithms on the hst has led to an effort to develop software that can aid and in some cases replace complicated expensive and error prone hardware among the many applications is hst s replacement the next generation space telescope ngst indent this work details the theory of optical wavefront reconstruction reviews some numerical methods for this problem and presents a novel numerical technique that we call extended least squares we compare the performance of these numerical methods for potential inclusion in prototype ngst optical wavefront reconstruction software we begin with a tutorial on rayleigh sommerfeld diffraction theory