Results 1  10
of
16
Simultaneously Structured Models with Application to Sparse and Lowrank Matrices
, 2014
"... The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank matrices, among others. In various applications in signal p ..."
Abstract

Cited by 41 (5 self)
 Add to MetaCart
The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and lowrank. Often norms that promote each individual structure are known, and allow for recovery using an orderwise optimal number of measurements (e.g., `1 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multiobjective optimization with these norms, then we can do no better, orderwise, than an algorithm that exploits only one of the present structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e. not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and lowrank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the `1 and nuclear norms requires many more measurements. This proves an orderwise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structureinducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and lowrank tensor completion.
Sparsity constrained nonlinear optimization: Optimality conditions and algorithms, arXiv preprint arXiv:1203.4580
, 2012
"... This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and coordinatewise optimality. These conditions are then used to de ..."
Abstract

Cited by 31 (8 self)
 Add to MetaCart
(Show Context)
This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and coordinatewise optimality. These conditions are then used to derive three numerical algorithms aimed at finding points satisfying the resulting optimality criteria: the iterative hard thresholding method and the greedy and partial sparsesimplex methods. The first algorithm is essentially a gradient projection method while the remaining two algorithms are of coordinate descent type. The theoretical convergence of these methods and their relations to the derived optimality conditions are studied. The algorithms and results are illustrated by several numerical examples. 1
Phase Retrieval with Application to Optical Imaging
, 2015
"... The problem of phase retrieval, i.e., the recovery of a function given the magnitude of its ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
The problem of phase retrieval, i.e., the recovery of a function given the magnitude of its
CPRL – An Extension of Compressive Sensing to the Phase Retrieval Problem
"... While compressive sensing (CS) has been one of the most vibrant research fields in the past few years, most development only applies to linear models. This limits its application in many areas where CS could make a difference. This paper presents a novel extension of CS to the phase retrieval proble ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
While compressive sensing (CS) has been one of the most vibrant research fields in the past few years, most development only applies to linear models. This limits its application in many areas where CS could make a difference. This paper presents a novel extension of CS to the phase retrieval problem, where intensity measurements of a linear system are used to recover a complex sparse signal. We propose a novel solution using a lifting technique – CPRL, which relaxes the NPhard problem to a nonsmooth semidefinite program. Our analysis shows that CPRL inherits many desirable properties from CS, such as guarantees for exact recovery. We further provide scalable numerical solvers to accelerate its implementation. 1
Vectorial Phase Retrieval of 1D Signals
"... Abstract—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 sign ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract—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. Index Terms—Convex relaxation, 1D phase retrieval, signal recovery from modulus Fourier measurements, statistical model selection. I. INTRODUCTION AND MAIN RESULTS
AN ALGORITHM FOR EXACT SUPERRESOLUTION AND PHASE RETRIEVAL
"... We explore a fundamental problem of superresolving a signal of interest from a few measurements of its lowpass magnitudes. We propose a 2stage tractable algorithm that, in the absence of noise, admits perfect superresolution of an rsparse signal from 2r2 − 2r + 2 lowpass magnitude measurements ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
We explore a fundamental problem of superresolving a signal of interest from a few measurements of its lowpass magnitudes. We propose a 2stage tractable algorithm that, in the absence of noise, admits perfect superresolution of an rsparse signal from 2r2 − 2r + 2 lowpass magnitude measurements. The spike locations of the signal can assume any value over a continuous disk, without increasing the required sample size. The proposed algorithm first employs a conventional superresolution algorithm (e.g. the matrix pencil approach) to recover unlabeled sets of signal correlation coefficients, and then applies a simple sorting algorithm to disentangle and retrieve the true parameters in a deterministic manner. Our approach can be adapted to multidimensional spike models and random Fourier sampling by replacing its first step with other harmonic retrieval algorithms.
Speckle correlation resolution enhancement of widefield fluorescence imaging
"... Highresolution fluorescence imaging is essential in nanoscience and biological sciences. Due to the diffraction limit, conventional imaging systems can only resolve structures larger than 200 nm. Here, we introduce a new fluorescence imaging method that enhances the resolution by using a highindex ..."
Abstract
 Add to MetaCart
Highresolution fluorescence imaging is essential in nanoscience and biological sciences. Due to the diffraction limit, conventional imaging systems can only resolve structures larger than 200 nm. Here, we introduce a new fluorescence imaging method that enhances the resolution by using a highindex scattering medium as an imaging lens. Simultaneously, we achieve a wide field of view. We develop a new image reconstruction algorithm that converges even for complex object structures. We collect twodimensional fluorescence images of a collection of 100 nm diameter dyedoped nanospheres, and demonstrate a deconvolved Abbe resolution of 116 nm with a field of view of 10μm × 10 μm. Our method is robust against optical aberrations and stage drifts, and therefore is well suited to image nanostructures with high resolution under ambient conditions. © 2015 Optical Society of America
Simultaneously Structured Models with Application to Sparse and Lowrank Matrices
, 2014
"... The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank matrices, among others. In various applications in signal p ..."
Abstract
 Add to MetaCart
The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and lowrank. Often norms that promote each individual structure are known, and allow for recovery using an orderwise optimal number of measurements (e.g., `1 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multiobjective optimization with these norms, then we can do no better, orderwise, than an algorithm that exploits only one of the present structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e. not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and lowrank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the `1 and nuclear norms requires many more measurements. This proves an orderwise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structureinducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and lowrank tensor completion.