on polyhedral approximations of the positive semidefinite cone it is well known that state of the art linear programming solvers are more efficient than their semidefinite programming counterparts and can scale to much larger problem sizes this leads us to co

identifiability in blind deconvolution with subspace or sparsity constraints blind deconvolution bd the resolution of a signal and a filter given their convolution arises in many applications without further constraints bd is ill posed in practice subspace or sparsity constraints have been imposed to reduce the search space and have shown some empirical success however the existing theoretical analysis on uniqueness in bd is rather limited in an effort to address the still open question we derive sufficient conditions under which two vectors can be uniquely identified from their circular convolution subject to subspace or sparsity constraints these sufficient conditions provide the first algebraic sample complexities for bd we first derive a sufficient condition that applies to almost all bases or frames for bd of vectors in mathbb c n with two subspace constraints of dimensions m 1 and m 2 the required sample complexity is n geq m 1 m 2 then we impose a sub band structure on one basis and derive a sufficient condition that involves a relaxed sample complexity n geq m 1 m 2 1 which we show to be optimal we present the extensions of these results to bd with sparsity constraints or mixed constraints with the sparsity level replacing the subspace dimension the cost for the unknown support in this case is an extra factor of 2 in the sample complexity

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

spectral behavior of contractive noise we study the behavior of the spectra corresponding to quantum systems subjected to a contractive noise i e the environment reduces the accessible phase space of the system but the total probability is conserved we find that the number of long lived resonances grows as a power law in h but surprisingly there is no relationship between the exponent of this power law and the fractal dimension of the corresponding classical attractor this is in disagreement with the predictions of the fractal weyl law which has been established for open systems where the probability is lost under the effect of a projective opening

convex optimization approaches for blind sensor calibration using sparsity we investigate a compressive sensing framework in which the sensors introduce a distortion to the measurements in the form of unknown gains we focus on blind calibration using measures performed on multiple unknown but sparse signals and formulate the joint recovery of the gains and the sparse signals as a convex optimization problem we divide this problem in 3 subproblems with different conditions on the gains specifially i gains with different amplitude and the same phase ii gains with the same amplitude and different phase and iii gains with different amplitude and phase in order to solve the first case we propose an extension to the basis pursuit optimization which can estimate the unknown gains along with the unknown sparse signals for the second case we formulate a quadratic approach that eliminates the unknown phase shifts and retrieves the unknown sparse signals an alternative form of this approach is also formulated to reduce complexity and memory requirements and provide scalability with respect to the number of input signals finally for the third case we propose a formulation that combines the earlier two approaches to solve the problem the performance of the proposed algorithms is investigated extensively through numerical simulations which demonstrates that simultaneous signal recovery and calibration is possible with convex methods when sufficiently many unknown but sparse calibrating signals are provided

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

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

gauge optimization and duality gauge functions significantly generalize the notion of a norm and gauge optimization as defined by r m freund math programming 38 1987 pp 47 67 seeks the element of a convex set that is minimal with respect to a gauge function this conceptually simple problem can be used to model a remarkable array of useful problems including a special case of conic optimization and related problems that arise in machine learning and signal processing the gauge structure of these problems allows for a special kind of duality framework this paper explores the duality framework proposed by freund and proposes a particular form of the problem that exposes some useful properties of the gauge optimization framework such as the variational properties of its value function and yet maintains most of the generality of the abstract form of gauge optimization

a nonlinear programming algorithm for solving semidefinite programs via low rank factorization in this paper we present a nonlinear programming algorithm for solving semidefinite programs sdps in standard form the algorithm s distinguishing feature is a change of variables that replaces the symmetric positive semidefinite variable x of the sdp with a rectangular variable r according to the factorization x rr t the rank of the factorization i e the number of columns of r is chosen minimally so as to enhance computational speed while maintaining equivalence with the sdp fundamental results concerning the convergence of the algorithm are derived and encouraging computational results on some large scale test problems are also presented

nonstandard analysis and jump conditions for converging shock waves nonstandard analysis is an area of modern mathematics that studies abstract number systems containing both infinitesimal and infinite numbers this article applies nonstandard analysis to derive jump conditions for one dimensional converging shock waves in a compressible inviscid perfect gas it is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval predistributions of the heaviside function and the dirac delta measure are introduced to model the flow parameters across a shock wave the equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the flow parameters

weyl asymptotics from closed to open systems we present microwave experiments on the symmetry reduced five disk billiard studying the transition from a closed to an open system the measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied for the closed system this counting function shows the weyl asymptotic with a leading exponent equal to 2 by opening the system successively this exponent decreases smoothly to a noninteger value for the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed the results can be interpreted as a first experimental indication for the fractal weyl conjecture for resonances

identifiability in bilinear inverse problems with applications to subspace or sparsity constrained blind gain and phase calibration bilinear inverse problems bips the resolution of two vectors given their image under a bilinear mapping arise in many applications without further constraints bips are usually ill posed in practice the properties of natural signals are exploited to solve bips for example subspace constraints or sparsity constraints are imposed to reduce the search space these approaches have shown some success in practice however there are few results on uniqueness in bips for most bips the fundamental question of under what condition the problem admits a unique solution is yet to be answered for example blind gain and phase calibration bgpc is a structured bip which arises in many applications including inverse rendering in computational relighting albedo estimation with unknown lighting blind phase and gain calibration in sensor array processing and multichannel blind deconvolution mbd it is interesting to study the uniqueness of such problems in this paper we define identifiability of a bip up to a group of transformations we derive necessary and sufficient conditions for such identifiability i e the conditions under which the solutions can be uniquely determined up to the transformation group these conditions take the form of dividing the identifiability of the pair of unknown variables into the individual identifiability of each variable although verifying these individual conditions requires problem specific procedures this framework is universally applicable to all bips applying these results to bgpc we derive sufficient conditions for unique recovery under several scenarios including subspace joint sparsity and sparsity models for bgpc with joint sparsity or sparsity constraints we develop a procedure to compute the transformation groups corresponding to inherent ambiguities we also give necessary conditions in the form of tight lower bounds on sample complexities and demonstrate the tightness of these bounds by numerical experiments the results for bgpc not only demonstrate the application of the proposed general framework for identifiability analysis but are also of interest in their own right

path space measure for the 3 1 dimensional dirac equation in momentum space nonstandard analysis is used to construct a measure over paths for the path integral solution to the dirac equation in 3 1 dimension paths are considered in momentum space because the green function in the configuration space contains a derivative of function which keeps us from assigning a measure over paths the solution is obtained not only as the standard part of a nonstandard path sum with respect to a nonstandard measure but also as a standard path integral with respect to a standard measure extracted from the nonstandard one the result is an extension of gaveau s work b gaveau j funct anal 58 310 319 1984

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

role of short periodic orbits in quantum maps with continuous openings we apply a recently developed semiclassical theory of short periodic orbits to the continuously open quantum tribaker map in this paradigmatic system the trajectories are partially bounced back according to continuous reflectivity functions this is relevant in many situations that include optical microresonators and more complicated boundary conditions in a perturbative regime the shortest periodic orbits belonging to the classical repeller of the open map a cantor set given by a region of exactly zero reflectivity prove to be extremely robust in supporting a set of long lived resonances of the continuously open quantum maps moreover for steplike functions a significant reduction in the number needed is obtained similarly to the completely open situation this happens despite a strong change in the spectral properties when compared to the discontinuous reflectivity case in order to give a more realistic interpretation of these results we compare with a fresnel type reflectivity function

phase retrievable operator valued frames and representations of quantum channels abstract we examine some connections among phase retrievable not necessarily self adjoint operator valued frames projective group representation frames and representations of quantum channels we first present some characterizations of phase retrievable frames for general operator systems acting on both finite and infinite dimensional hilbert spaces which generalize the known results for vector valued frames fusion frames and frames of hermitian matrices for an irreducible projective unitary representation of a finite group the image system is automatically phase retrievable and moreover it is a point wisely tight operator valued frames we generalize this notion to more general operator valued frames and prove that point wise tight operator valued frames are exactly the ones that are right equivalent to operator valued tight frames for an operator system that represent a quantum channel we show that phase retrievability of the system is independent of the choices of the representations of the quantum channel

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

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

characterization of analytic wavelet transforms and a new phaseless reconstruction algorithm we obtain a characterization of all wavelets leading to analytic wavelet transforms wt the characterization is obtained as a byproduct of the theoretical foundations of a new method for wavelet phase reconstruction from magnitude only coefficients the cornerstone of our analysis is an expression of the partial derivatives of the continuous wt which results in phase magnitude relationships similar to the short time fourier transform setting and valid for the generalized family of cauchy wavelets we show that the existence of such relations is equivalent to analyticity of the wt up to a multiplicative weight and a scaling of the mother wavelet the implementation of the new phaseless reconstruction method is considered in detail and compared to previous methods it is shown that the proposed method provides significant performance gains and a great flexibility regarding accuracy versus complexity in addition we discuss the relation between scalogram reassignment operators and the wavelet transform phase gradient and present an observation on the phase around zeros of the wt

spectral matrix completion by cyclic projection and application to sound source reconstruction from non synchronous measurements a fundamental limitation of the inverse acoustic problem is determined by the size of the array and the microphone density a solution to achieve large array and or high microphone density is to scan the object of interest by moving sequentially a small prototype array which is referred to as sequential measurements in comparison to a large array and or high microphone density array that can acquire simultaneously all the information of the spectral matrix in particular all cross spectra sequential measurements can only acquire a block diagonal spectral matrix while the cross spectra between the sequential measurements remain unknown due to the missing phase relationships between consecutive positions nevertheless these unknown cross spectra are necessary for sound source reconstruction the objective of this paper is to propose an algorithm to recover the missing elements of the spectral matrix in the case where the acoustical field is highly coherent this issue is shown to boil down to a matrix completion problem subject to given constraint of hermitian symmetry measurements fitting reduced rank and spatial continuity of the sound field a cyclic projection cp algorithm is proposed in this work to find an optimal solution at the intersection between three predefined sets the proposed method is analyzed through numerical simulations of diverse setups and is also validated experimentally

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

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

the space of state vectors a hyperfinite approach we present a version of the formalism of dirac based on nonstandard analysis allowing us to deal with state vectors and operators using the resources of finite dimensional linear algebra the space of state vectors is a nonstandard hilbert space with hyperfinite dimension which includes all square integrable functions together with vectors representing states of definite position or momentum every vector is normalizable even when its norm is infinite observables are represented by hermitian operators which are always hyper bounded and defined on the whole space the connection with the standard theory is established by postulating the existence of hyper observables and nonstandard states each observable in the usual sense appears as a kind of standard scale approximation of some hyper observable we show that the probabilistic predictions are consistent with those of the standard theory consistency extends to time evolution in the sense that if an initial nonstandard state is near standard then the state after a finite time shall be infinitely near the standard state obtained through the schrodinger equation

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

fractal weyl law for open quantum chaotic maps we study a semiclassical quantization of poincar e maps arising in scattering problems with fractal hyperbolic trapped sets the main application is the proof of a fractal weyl upper bound for the number of resonances scattering poles in small domains near the real axis this result includes the case of several convex hard obstacles satisfying a no eclipse condition

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

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 through regularization for seismic problems abstractthe most advanced data processing imaging algorithms aim at optimizing a suitably defined cost function consisting of a data misfit term posed in the data domain and a regularization term for strong phase errors in the data however such algorithms fail to achieve an acceptable solution because they force us to match the incorrect phase information to remedy this deficiency we replaced the time domain misfit term by a fourier magnitude fidelity term to recover the desired signal from just the amplitude spectrum of the observations such a problem is known as phase retrieval and it is a subject of interest for those who cannot measure the phase information about the system under study the proposed phase retrieval problem is solved by a fast iterative shrinkage thresholding algorithm and it is used to tackle two common phase problems arising in seismic processing data recovery in the presence of residual static shifts and deconvolution with a missing wavelet phase under sparsity constrain

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

unstable periodic orbits and semiclassical quantisation the bohr sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic unstable orbits are isolated it provides a semiclassical euler factorisation for the functional determinant of the quantal hamiltonian in contrast to the hadamard infinite product over the eigenvalues by which the exact determinant is defined

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

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

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 retrieval from very few measurements abstract in many applications signals are measured according to a linear process but the phases of these measurements are often unreliable or not available to reconstruct the signal one must perform a process known as phase retrieval this paper focuses on completely determining signals with as few intensity measurements as possible and on efficient phase retrieval algorithms from such measurements for the case of complex m dimensional signals we construct a measurement ensemble of size 4 m 4 which yields injective intensity measurements this is conjectured to be the smallest such ensemble for the case of real signals we devise a theory of almost injective intensity measurements and we characterize such ensembles later we show that phase retrieval from m 1 almost injective intensity measurements is np hard indicating that computationally efficient phase retrieval must come at the price of measurement redundancy

statistical analysis of compressive low rank tomography with random measurements we consider the statistical problem of compressive estimation of low rank states r d with random basis measurements where r d are the rank and dimension of the state respectively we investigate whether for a fixed sample size n the estimation error associated with a compressive measurement setup is close to that of the setting where a large number of bases are measured we generalise and extend previous results and show that the mean square error mse associated with the frobenius norm attains the optimal rate rd n with only o rlogd random basis measurements for all states an important tool in the analysis is the concentration of the fisher information matrix fim we demonstrate that although a concentration of the mse follows from a concentration of the fim for most states the fim fails to concentrate for states with eigenvalues close to zero we analyse this phenomenon in the case of a single qubit and demonstrate a concentration of the mse about its optimal despite a lack of concentration of the fim for states close to the boundary of the bloch sphere we also consider the estimation error in terms of a different metric the quantum infidelity we show that a concentration in the mean infidelity minf does not exist uniformly over all states highlighting the importance of loss function choice specifically we show that for states that are nearly pure the minf scales as 1 n but the constant converges to zero as the number of settings is increased this demonstrates a lack of compressive recovery for nearly pure states in this metric

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

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

phase retrieval without small ball probability assumptions in the context of the phase retrieval problem it is known that certain natural classes of measurements such as fourier measurements and random bernoulli measurements do not lead to the unique reconstruction of all possible signals even in combination with certain practically feasible random masks to avoid this difficulty the analysis is often restricted to measurement ensembles or masks that satisfy a small ball probability condition in order to ensure that the reconstruction is unique this paper shows a complementary result for random bernoulli measurements there is still a large class of signals that can be reconstructed uniquely namely those signals that are non peaky in fact this result is much more general it holds for random measurements sampled from any subgaussian distribution mathcal d without any small ball conditions this is demonstrated in two ways 1 a proof of stability and uniqueness and 2 a uniform recovery guarantee for the phaselift algorithm in all of these cases the number of measurements m approaches the information theoretic lower bound finally for random bernoulli measurements with erasures it is shown that phaselift achieves uniform recovery of all signals including peaky ones

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

performance of uniform and sparse non uniform samplers in presence of modeling errors a cramer rao bound based study this paper evaluates the performance of uniform and sparse nonuniform sampling techniques namely nested and coprime sampling for line spectrum estimation in presence of nonideal conditions such as perturbation in sampling instants and limited data for computing statistical averages coprime and nested sampling are well known deterministic sampling techniques that operate at rates significantly lower than nyquist and yet allow perfect reconstruction of the spectra of wide sense stationary signals however theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling and ability to perfectly compute statistical expectations this paper studies the performance of coprime and nested samplers when these assumptions are violated using a general grid based signal model that applies to both spatial and temporal line spectrum estimation the effect of perturbations in sampling instants is evaluated by deriving fundamental cramer rao bounds crb for line spectrum estimation with perturbed samplers for the first time simplified expressions for the fisher information matrix for perturbed coprime and nested samplers are derived which explicitly highlight the role of coarray even in presence of perturbations it is possible to resolve o m 2 spectral lines under appropriate conditions on the size of the grid the effect of finite data on the crb is also studied and necessary and sufficient conditions are derived to ensure that the crb decreases monotonically to zero with the number of measurements even when there are more sources than sensors finally the theoretical results derived in this paper are supported by extensive numerical experiments

sparse phase retrieval uniqueness guarantees and recovery algorithms the problem of signal recovery from its fourier transform magnitude is of paramount importance in various fields of engineering and has been around for more than 100 years due to the absence of phase information some form of additional information is required in order to be able to uniquely identify the signal of interest in this paper we focus our attention on discrete time sparse signals of length n we first show that if the discrete fourier transform dimension is greater than or equal to 2n then almost all signals with aperiodic support can be uniquely identified by their fourier transform magnitude up to time shift conjugate flip and global phase then we develop an efficient two stage sparse phase retrieval algorithm tspr which involves identifying the support i e the locations of the nonzero components of the signal using a combinatorial algorithm and identifying the signal values in the support using a convex algorithm we show that tspr can provably recover most o n 1 2 epsilon sparse signals up to a time shift conjugate flip and global phase we also show that for most o n 1 4 epsilon sparse signals the recovery is robust in the presence of measurement noise these recovery guarantees are asymptotic in nature numerical experiments complement our theoretical analysis and verify the effectiveness of tspr

density functional approach to quantum lattice systems for quantum lattice systems we consider the problem of characterizing the set of single particle densities which come from the ground state eigenspace of somen particle hamiltonian of the form h 0 sum nolimits i 1 n v x i whereh0 is a fixed bounded operator representing the kinetic and interaction energies we show that the conditions on are that it be strictly positive properly normalized and consistent with the pauli principle our results are valid for both finite and infinite lattices and for either bosons or fermions the coulomb interaction may be included inh0 if the lattice dimension is 2 we also characterize those single particle densities which come from the gibbs states of such hamiltonians at finite temperature in addition to the conditions stated above must satisfy a finite entropy condition

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

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

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

solving phaselift by low rank riemannian optimization methods for complex semidefinite constraints a framework phaselift was recently proposed to solve the phase retrieval problem in this framework the problem is solved by optimizing a cost function over the set of complex hermitian positive semidefinite matrices this approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite hermitian matrices where the desired minimizers are known to have low rank this paper considers an approach based on an alternative cost function defined on a union of appropriate manifolds it is related to the original cost function in a manner that preserves the ability to find a global minimizer and is significantly more efficient computationally a rank based optimality condition for stationary points is given and optimization algorithms based on state of the art riemannian optimization and dynamically reducing rank are proposed empirical evaluations are performed using the phaselift problem the new approach is shown to be an effective method of phase retrieval with

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

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

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

coarse grained spin density functional theory infinite volume limit via the hyperfinite coarse grained spin density functional theory sdft is a version of sdft which works with number spin densities specified to a limited resolution averages over cells of a regular spatial partition and external potentials constant on the cells this coarse grained setting facilitates a rigorous investigation of the mathematical foundations which goes well beyond what is currently possible in the conventional formulation problems of existence uniqueness and regularity of representing potentials in the coarse grained sdft setting are here studied using techniques of robinsonian nonstandard analysis every density which is nowhere spin saturated is v representable and the set of representing potentials is the functional derivative in an appropriate generalized sense of the lieb internal energy functional quasi continuity and closure properties of the set valued representing potentials map are also established the extent of possible non uniqueness is similar to that found in non rigorous studies

undersampled sparse phase retrieval via majorization minimization in the undersampled phase retrieval problem the goal is to recover an n dimensional complex valued signal from only m intensity measurements without phase information this inverse system is not only nonconvex but also underdetermined in this paper we propose to exploit the sparsity in the original signal and develop two low complexity algorithms with superior performance based on the majorization minimization framework the proposed algorithms are preferred to existing benchmark methods since at each iteration a simple convex surrogate problem is solved with a closed form solution that monotonically decreases the objective function value when the unknown signal is sparse in the standard basis the first algorithm c prime can produce a stationary point of the corresponding nonconvex phase retrieval problem when the unknown signal is not sparse in the standard basis the second algorithm sc prime can find a coordinatewise stationary point of the more challenging phase retrieval problem through sparse coding experimental results validate that the proposed algorithms have higher successful recovery rate and less normalized mean square error than existing up to date methods under the same setting

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

pointwise analysis of scalar fields a nonstandard approach a new nonstandard analytical approach to quantum fields is presented which gives a mathematical foundation for manipulating pointwise defined quantum fields in our approach a field operator x is not a standard operator valued distribution but a nonstandard operator valued function then formal expressions containing e g x 2 can be understood literally and shown to be well defined in the free field cases we show that the wightman functions are explicitly calculated with the pointwise field without any regularization e g wick product our notion of pointwise fields is applied also to the path integral formalisms of scalar fields we show that some of physicists naive expressions of lagrangian path integral formulas can be rigorously justified

statistical guarantees for the em algorithm from population to sample based analysis we develop a general framework for proving rigorous guarantees on the performance of the em algorithm and a variant known as gradient em our analysis is divided into two parts a treatment of these algorithms at the population level in the limit of infinite data followed by results that apply to updates based on a finite set of samples first we characterize the domain of attraction of any global maximizer of the population likelihood this characterization is based on a novel view of the em updates as a perturbed form of likelihood ascent or in parallel of the gradient em updates as a perturbed form of standard gradient ascent leveraging this characterization we then provide non asymptotic guarantees on the em and gradient em algorithms when applied to a finite set of samples we develop consequences of our general theory for three canonical examples of incompletedata problems mixture of gaussians mixture of regressions and linear regression with covariates missing completely at random in each case our theory guarantees that with a suitable initialization a relatively small number of em or gradient em steps will yield with high probability an estimate that is within statistical error of the mle we provide simulations to confirm this theoretically predicted behavior

identifiability and stability in blind deconvolution under minimal assumptions blind deconvolution bd arises in many applications without assumptions on the signal and the filter bd does not admit a unique solution in practice subspace or sparsity assumptions have shown the ability to reduce the search space and yield the unique solution however existing theoretical analysis on uniqueness in bd is rather limited in an earlier paper we provided the first algebraic sample complexities for bd that hold for lebesgue almost all bases or frames we showed that for bd of a pair of vectors in mathbb c n with subspace constraints of dimensions m 1 and m 2 respectively a sample complexity of n geq m 1 m 2 is sufficient this result is suboptimal since the number of degrees of freedom is merely m 1 m 2 1 we provided analogous results with similar suboptimality for bd with sparsity or mixed subspace and sparsity constraints in this paper taking advantage of the recent progress on the information theoretic limits of unique low rank matrix recovery we finally bridge this gap and derive an optimal sample complexity result for bd with generic bases or frames we show that for bd of an arbitrary pair respectively all pairs of vectors in mathbb c n with sparsity constraints of sparsity levels s 1 and s 2 a sample complexity of n s 1 s 2 respectively n 2 s 1 s 2 is sufficient we also present analogous results for bd with subspace constraints or mixed constraints with the subspace dimension replacing the sparsity level last but not least in all the above scenarios if the bases or frames follow a probabilistic distribution specified in this paper the recovery is not only unique but also stable against small perturbations in the measurements under the same sample complexities

hohenberg kohn theorem and non v representable densities abstract in the density functional formalism of hohenberg and kohn1 the variation is only allowed over the one particle densities which are pure state v representable ps v representable levy2 and lieb3 proved that not every ensemble v representable e v representable density is ps v representable since we show that the hohenberg kohn formalism can be extended to a variation over e v representable densities for degenerated ground states levy s and lieb s result is not a counterexample to the universality of the hohenberg kohn theorem the question whether every n representable density is e v representable has remained open so far presenting examples of non e v representable densities we answer this question in the negative thus the value of levy s functional4 for the calculation of ground state energies is obvious since this functional only requires the n representability of the densities therefore we transfer two approaches for the calculation of excited state energies into the framework of levy s formalism

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

beyond v representability local one body hamiltonians for arbitrary densities the v representability problem arising in density functional theory is this for an particle system of interacting identical particles can every one particle density of finite intrinsic energy be realized as a ground state density upon modifying the hamiltonian by addition of an appropriate external one body potential v the answer is known to be no we show that the answer is yes if the inter particle interaction is bounded and allowed modifications are broadened from one body potentials to local one body operators the required operator is obtained from a non standard potential in the sense of non standard analysis which is a simple function defined on a regular grid of infinitesimal finesse at a vague conceptual level therefore the result is not far from universal v representability

confidence region of singular subspaces for low rank matrix regression low rank matrix regression refers to the instances of recovering a low rank matrix based on specially designed measurements and the corresponding noisy outcomes numerous statistical methods have been developed over the recent decade for efficiently reconstructing the unknown low rank matrices it is often interesting in certain applications to estimate the unknown singular subspaces in this paper we revisit the low rank matrix regression model and introduce a two step procedure to construct confidence regions of the singular subspaces we investigate distributions of the joint projection distance between the empirical singular subspaces and the unknown true singular subspaces we prove asymptotical normality of the joint projection distance with data dependent centering and normalization when r 3 2 m 1 m 2 3 2 o n log n where m 1 m 2 denote the matrix row and column sizes r is the rank and n is the number of independent random measurements consequently data dependent confidence regions of the true singular subspaces are established which attain pre determined confidence levels asymptotically additionally non asymptotic convergence rates are also established numerical results are presented to show the merits of our methods

convex and nonconvex formulations for mixed regression with two components minimax optimal rates we consider the mixed regression problem with two components under adversarial and stochastic noise we give a convex optimization formulation that provably recovers the true solution as well as a nonconvex formulation that works under more general settings and remains tractable upper bounds are provided on the recovery errors for both arbitrary noise and stochastic noise models we also give matching minimax lower bounds up to log factors showing that our algorithm is information theoretical optimal in a precise sense our results represent the first tractable algorithm guaranteeing successful recovery with tight bounds on recovery errors and sample complexity moreover we pinpoint the statistical cost of mixtures our minimax optimal results indicate that the mixture poses a fundamentally more difficult problem in the low snr regime where the learning rate changes

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

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

iterative hard thresholding for low rank recovery from rank one projections abstract a novel algorithm for the recovery of low rank matrices acquired via compressive linear measurements is proposed and analyzed the algorithm a variation on the iterative hard thresholding algorithm for low rank recovery is designed to succeed in situations where the standard rank restricted isometry property fails e g in case of subexponential unstructured measurements or of subgaussian rank one measurements the stability and robustness of the algorithm are established based on distinctive matrix analytic ingredients and its performance is substantiated numerically

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

fractal weyl law for three dimensional chaotic hard sphere scattering systems the fractal weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller we provide the first test for the fractal weyl law for a three dimensional open scattering system for the four sphere billiard we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition we test the fractal weyl law for various symmetry subspaces and sphere to sphere separations

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