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75
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 ..."
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Cited by 8 (2 self)
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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
Alternating Projections and DouglasRachford for Sparse Affine Feasibility
"... The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NPcomplete problem that is typically solved numerically via convex heuristics or nicelybehaved nonconvex relaxations. In this work we consider elementary methods base ..."
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Cited by 7 (1 self)
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The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NPcomplete problem that is typically solved numerically via convex heuristics or nicelybehaved nonconvex relaxations. In this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics. It has been shown recently that, locally, the fundamental method of alternating projections must converge linearly to a solution to the sparse feasibility problem with an affine constraint. In this paper we apply different analytical tools that allow us to show global linear convergence of alternating projections under familiar constraint qualifications. These analytical tools can also be applied to other algorithms. This is demonstrated with the prominent DouglasRachford algorithm where we establish local linear convergence of this method applied to the sparse affine feasibility problem.
Compressed modes for variational problems in mathematics and physics
 Proceedings of the National Academy of Sciences
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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 ..."
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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
Deep Scattering Spectrum
 TRANSACTIONS ON SIGNAL PROCESSING
, 2014
"... A scattering transform defines a locally translation invariant representation which is stable to timewarping deformations. It extends MFCC representations by computing modulation spectrum coefficients of multiple orders, through cascades of wavelet convolutions and modulus operators. Secondorder s ..."
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Cited by 5 (0 self)
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A scattering transform defines a locally translation invariant representation which is stable to timewarping deformations. It extends MFCC representations by computing modulation spectrum coefficients of multiple orders, through cascades of wavelet convolutions and modulus operators. Secondorder scattering coefficients characterize transient phenomena such as attacks and amplitude modulation. A frequency transposition invariant representation is obtained by applying a scattering transform along logfrequency. Statetheofart classification results are obtained for musical genre and phone classification on GTZAN and TIMIT databases, respectively.
Alternating Projection, Ptychographic Imaging and Phase Synchronization. ArXiv eprints
, 2014
"... Abstract. We demonstrate necessary and sufficient conditions of the global convergence of the alternating projection algorithm to a unique solution up to a global phase factor. Additionally, for the ptychographic imaging problem, we discuss phase synchronization and connection graph Laplacian, and ..."
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Abstract. We demonstrate necessary and sufficient conditions of the global convergence of the alternating projection algorithm to a unique solution up to a global phase factor. Additionally, for the ptychographic imaging problem, we discuss phase synchronization and connection graph Laplacian, and show how to construct an accurate initial guess to accelerate convergence speed to handle the big imaging data in the coming new light source era. 1.
SIGNAL RECONSTRUCTION FROM THE MAGNITUDE OF SUBSPACE COMPONENTS
"... Abstract. 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 ..."
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Abstract. 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 pfusion 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. 1.
Phase retrieval from very few measurements
, 2013
"... 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 inte ..."
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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 Mdimensional signals, we construct a measurement ensemble of size 4M − 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 NPhard, indicating that computationally efficient phase retrieval must come at the price of measurement redundancy.