Compressive Phase Retrieval Research Articles

Compressive phase retrieval: Optimal sample complexity with deep generative priors

AbstractAdvances in compressive sensing (CS) provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able to achieve optimal sample complexity. This has created an open problem in compressive phase retrieval: under generic, phaseless linear measurements, are there tractable reconstruction algorithms that succeed with optimal sample complexity? Meanwhile, progress in machine learning has led to the development of new data‐driven signal priors in the form of generative models, which can outperform sparsity priors with significantly fewer measurements. In this work, we resolve the open problem in compressive phase retrieval and demonstrate that generative priors can lead to a fundamental advance by permitting optimal sample complexity by a tractable algorithm. We additionally provide empirics showing that exploiting generative priors in phase retrieval can significantly outperform sparsity priors. These results provide support for generative priors as a new paradigm for signal recovery in a variety of contexts, both empirically and theoretically. The strengths of this paradigm are that (1) generative priors can represent some classes of natural signals more concisely than sparsity priors, (2) generative priors allow for direct optimization over the natural signal manifold, which is intractable under sparsity priors, and (3) the resulting non‐convex optimization problems with generative priors can admit benign optimization landscapes at optimal sample complexity, perhaps surprisingly, even in cases of nonlinear measurements.

Heavy-Ball-Based Hard Thresholding Pursuit for Sparse Phase Retrieval Problems

We introduce a novel iterative algorithm, termed the Heavy-Ball-Based Hard Thresholding Pursuit for sparse phase retrieval problem (SPR-HBHTP), to reconstruct a sparse signal from a small number of magnitude-only measurements. Our algorithm is obtained via a natural combination of the Hard Thresholding Pursuit for sparse phase retrieval (SPR-HTP) and the classical Heavy-Ball (HB) acceleration method. The robustness and convergence for the proposed algorithm were established with the help of the restricted isometry property. Furthermore, we prove that our algorithm can exactly recover a sparse signal with overwhelming probability in finite steps whenever the initialization is in the neighborhood of the underlying sparse signal, provided that the measurement is accurate. Extensive numerical tests show that SPR-HBHTP has a markedly improved recovery performance and runtime compared to existing alternatives, such as the Hard Thresholding Pursuit for sparse phase retrieval problem (SPR-HTP), the SPARse Truncated Amplitude Flow (SPARTA), and Compressive Phase Retrieval with Alternating Minimization (CoPRAM).

Inertial Proximal ADMM for Separable Multi-Block Convex Optimizations and Compressive Affine Phase Retrieval

Inertial Proximal ADMM for Separable Multi-Block Convex Optimizations and Compressive Affine Phase Retrieval

Robust amplitude method with <inline-formula><tex-math id="M1">$ L_{1/2} $</tex-math></inline-formula>-regularization for compressive phase retrieval

Compressive phase retrieval aims to recover a sparse signal from the amplitude of its linear measurements when the number of measurements required is far less than the dimension of the signal. In real applications, the measurements are often corrupted by outliers and asymmetrical distribution noise. In this paper, we introduce a novel method for compressive phase retrieval which consists of least absolute deviation loss function and an $ L_{1/2}- $regularization term. It is a nonconvex, nonsmooth, and non-Lipschitz optimization problem. We design an efficient alternating direction method of multipliers(ADMM) to solve the problem and establish its convergence. Extensive numerical experiments demonstrate that the proposed method has higher success probability when the number of measurements is relatively less and is robust to outliers, dense bounded noise as well as Laplace noise.

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.

Using Black-Box Compression Algorithms for Phase Retrieval

Compressive phase retrieval refers to the problem of recovering a structured $n$ -dimensional complex-valued vector from its phase-less under-determined linear measurements. The non-linearity of the measurement process makes designing theoretically-analyzable efficient phase retrieval algorithms challenging. As a result, to a great extent, existing recovery algorithms only take advantage of simple structures such as sparsity and its convex generalizations. The goal of this article is to move beyond simple models through employing compression codes. Such codes are typically developed to take advantage of complex signal models to represent the signals as efficiently as possible. In this work, it is shown how an existing compression code can be treated as a black box and integrated into an efficient solution for phase retrieval. First, COmpressive PhasE Retrieval (COPER) optimization, a computationally-intensive compression-based phase retrieval method, is proposed. COPER provides a theoretical framework for studying compression-based phase retrieval. The number of measurements required by COPER is connected to $\kappa $ , the $\alpha $ -dimension (closely related to the rate-distortion dimension) of a given family of compression codes. To finds the solution of COPER, an efficient iterative algorithm called gradient descent for COPER (GD-COPER) is proposed. It is proven that under some mild conditions on the initialization and the compression code, if the number of measurements is larger than $ C \kappa ^{2} \log ^{2}~n$ , where $C$ is a constant, GD-COPER obtains an accurate estimate of the input vector in polynomial time. In the simulation results, JPEG2000 is integrated in GD-COPER to confirm the state-of-the-art performance of the resulting algorithm on real-world images.

Partially Coherent Compressive Phase Retrieval for Millimeter-Wave Massive MIMO Channel Estimation

Channel estimation is challenging for millimeter-wave (mmWave) massive MIMO systems. Leveraging the spatial sparsity of mmWave channels, compressive sensing (CS) based channel estimation been extensively studied. However, the mmWave hardware imperfections may introduce random phase distortions to the received pilots, which makes the conventional CS methods fail to estimate the channel. The compressive phase retrieval (CPR) method can be employed to solve this challenging problem. In this paper, we exploit the partial coherence in hybrid mmWave systems, i.e., the pilots sent from different radio frequency (RF) chains share the same phase distortion in the same time frame, while the phase distortions are different across different time frames. Based on this property, we propose an on-grid partially coherent CPR (PC-CPR) algorithm for mmWave channel estimation in the presence of severe phase distortions. Unlike the existing coherent channel estimation schemes that require perfect phase information or the noncoherent channel estimation schemes that ignore the phases of measurements, the proposed on-grid PC-CPR algorithm exploits the partial coherence property to estimate the sparse angle-domain channel vector. Furthermore, to solve the resolution limitation of the on-grid PC-CPR algorithm, we propose an off-grid PC-CPR algorithm that directly estimates the parameters of channel paths. The proposed partially coherent channel estimation framework subsumes the existing coherent and noncoherent channel estimation methods as special cases. Simulation results show that under the presence of random phase distortions, the proposed PC-CPR algorithms outperform noncoherent channel estimation methods with higher reliability and lower pilot overhead by leveraging the partial coherence.

Fast Target Detection via Template Matching in Compressive Phase Retrieval

Template matching (TM) is a common methodology for target detection (TD) which allows detecting a target based on cross-correlation analysis between a reference pattern and the scene. State-of-the-art TD approaches do not consider the optical phase of the target as a discriminant in the detection process, because to recover the phase involves solving a computationally demanding inverse problem known as phase retrieval (PR). However, in applications such as microscopy and optical imaging, the optical phase contains valuable information that describes the shape and depth of the object. This work proposes a method for fast TD via TM, which considers the optical phase of the object in the reference pattern as a discriminant in a setup that records coded phaseless measurements (CPM). Specifically, the proposed TD methodology is established for far-field imaging. This approach consists of two steps: (i) fast approximation of the optical field from CPM based on compressive PR, including its optical phase information; (ii) cross-correlation analysis to detect the target using its optical phase. The approximation of the optical field considering its phase is performed by low-pass-filtering the leading eigenvector of a designed matrix, overcoming traditional approaches in terms of relative error. Since no explicit TD methodology that includes the optical phase as a discriminant exists in the literature, the proposed approach is compared to a method that reconstructs the optical field and then performs the detection step. Numerical results suggest that the proposed methodology detects a target under noisy scenarios using up to 75% fewer measurements in the tested datasets. Also, the proposed TD using the filtered spectral method reduces the detection time in up to 79% in the tested datasets, compared to a methodology that requires the reconstruction of the phase.

Sample-Efficient Algorithms for Recovering Structured Signals From Magnitude-Only Measurements

We consider the problem of recovering a signal $ \mathbf {x^{*}}\in \mathbb {R}^{n}$ , from magnitude-only measurements, $y_{i} = \left |{ \left \langle{ { \mathbf {a}_{i}}, \mathbf {x^{*}} }\right \rangle }\right | $ for $i=\{1,2,\ldots ,m\}$ . This is a stylized version of the classical phase retrieval problem , and is a fundamental challenge in nano- and bio-imaging systems, astronomical imaging, and speech processing. It is well-known that the above problem is ill-posed, and therefore some additional assumptions on the signal and/or the measurements are necessary. In this paper, we consider the case where the underlying signal $ \mathbf {x^{*}}$ is $s$ -sparse. For this case, we develop a novel recovery algorithm that we call Compressive Phase Retrieval with Alternating Minimization , or CoPRAM . Our algorithm is simple and is obtained via a natural combination of the classical alternating minimization approach for phase retrieval with the CoSaMP algorithm for sparse recovery. Despite its simplicity, we prove that our algorithm achieves a sample complexity of $ \mathcal {O}\left ({{s^{2} \log n}}\right )$ with Gaussian measurements $ { \mathbf {a}_{i}}$ , which matches the best known existing results; moreover, it also demonstrates linear convergence in theory and practice. An appealing feature of our algorithm is that it requires no extra tuning parameters other than the signal sparsity level $s$ . Moreover, we show that our algorithm is robust to noise. The quadratic dependence of sample complexity on the sparsity level is sub-optimal, and we demonstrate how to alleviate this via additional assumptions beyond sparsity. First, we study the (practically) relevant case where the sorted coefficients of the underlying sparse signal exhibit a power law decay. In this scenario, we show that the CoPRAM algorithm achieves a sample complexity of $ \mathcal {O}\left ({{s \log n}}\right )$ , which is close to the information-theoretic limit. We then consider the case where the underlying signal $ \mathbf {x^{*}}$ arises from structured sparsity models. We specifically examine the case of block-sparse signals with uniform block size of $b$ and block sparsity $k=s/b$ . For this problem, we design a recovery algorithm that we call Block CoPRAM that further reduces the sample complexity to $ \mathcal {O}\left ({{ks \log n}}\right )$ . For sufficiently large block lengths of $b=\Theta (s)$ , this bound equates to $ \mathcal {O}\left ({{s \log n}}\right )$ . To our knowledge, our approach constitutes the first family of linearly convergent algorithms for signal recovery from magnitude-only Gaussian measurements that exhibit a sub-quadratic dependence on the signal sparsity level.

Compressive Phase Retrieval Realized by Combining Generalized Approximate Message Passing with Cartoon-Texture Model

Compressive Phase Retrieval Realized by Combining Generalized Approximate Message Passing with Cartoon-Texture Model

Compressive Phase Retrieval via Reweighted Amplitude Flow

The problem of reconstructing a sparse signal vector from magnitude-only measurements (a.k.a., compressive phase retrieval), emerges naturally in diverse applications, but it is NP-hard in general. Building on recent advances in nonconvex optimization, this paper puts forth a new algorithm that is termed compressive reweighted amplitude flow and abbreviated as CRAF, for compressive phase retrieval. Specifically, CRAF operates in two stages. The first stage seeks a sparse initial guess via a new spectral procedure. In the second stage, CRAF implements a few hard thresholding based iterations using reweighted gradients. When there are sufficient measurements, CRAF provably recovers the underlying signal vector exactly with high probability under suitable conditions. Moreover, its sample complexity coincides with that of the state-of-the-art procedures. Finally, substantial simulated tests showcase remarkable performance of the new spectral initialization, as well as improved exact recovery relative to competing alternatives.

Robustness properties of dimensionality reduction with Gaussian random matrices

In this paper, motivated by the results in compressive phase retrieval, we study the robustness properties of dimensionality reduction with Gaussian random matrices having arbitrarily erased rows. We first study the robustness property against erasure for the almost norm preservation property of Gaussian random matrices by obtaining the optimal estimate of the erasure ratio for a small given norm distortion rate. As a consequence, we establish the robustness property of Johnson-Lindenstrauss lemma and the robustness property of restricted isometry property with corruption for Gaussian random matrices. Secondly, we obtain a sharp estimate for the optimal lower and upper bounds of norm distortion rates of Gaussian random matrices under a given erasure ratio. This allows us to establish the strong restricted isometry property with the almost optimal restricted isometry property (RIP) constants, which plays a central role in the study of phaseless compressed sensing. As a byproduct of our results, we also establish the robustness property of Gaussian random finite frames under erasure.

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}$ .

Secret shared multiple-image encryption based on row scanning compressive ghost imaging and phase retrieval in the Fresnel domain

A multiple-image encryption method is proposed that is based on row scanning compressive ghost imaging, (t, n) threshold secret sharing, and phase retrieval in the Fresnel domain. In the encryption process, after wavelet transform and Arnold transform of the target image, the ciphertext matrix can be first detected using a bucket detector. Based on a (t, n) threshold secret sharing algorithm, the measurement key used in the row scanning compressive ghost imaging can be decomposed and shared into two pairs of sub-keys, which are then reconstructed using two phase-only mask (POM) keys with fixed pixel values, placed in the input plane and transform plane 2 of the phase retrieval scheme, respectively; and the other POM key in the transform plane 1 can be generated and updated by the iterative encoding of each plaintext image. In each iteration, the target image acts as the input amplitude constraint in the input plane. During decryption, each plaintext image possessing all the correct keys can be successfully decrypted by measurement key regeneration, compression algorithm reconstruction, inverse wavelet transformation, and Fresnel transformation. Theoretical analysis and numerical simulations both verify the feasibility of the proposed method.

Fast Compressive Phase Retrieval under Bounded Noise

We study the problem of recovering a t-sparse real vector from m quadratic equations yi=(ai*x)^2 with noisy measurements yi's. This is known as the problem of compressive phase retrieval, and has been widely applied to X-ray diffraction imaging, microscopy, quantum mechanics, etc. The challenge is to design a a) fast and b) noise-tolerant algorithms with c) near-optimal sample complexity. Prior work in this direction typically achieved one or two of these goals, but none of them enjoyed the three performance guarantees simultaneously. In this work, with a particular set of sensing vectors ai's, we give a provable algorithm that is robust to any bounded yet unstructured deterministic noise. Our algorithm requires roughly O(t) measurements and runs in O(tn*log (1/epsilon)) time, where epsilon is the error. This result advances the state-of-the-art work, and guarantees the applicability of our method to large datasets. Experiments on synthetic and real data verify our theory.

Fast Phase Retrieval from Local Correlation Measurements

We develop a fast phase retrieval method which can utilize a large class of local phaseless correlation-based measurements in order to recover a given signal ${\bf x} \in \mathbb{C}^d$ (up to an unknown global phase) in near-linear $\mathcal{O} \left( d \log^4 d \right)$-time. Accompanying theoretical analysis proves that the proposed algorithm is guaranteed to deterministically recover all signals ${\bf x}$ satisfying a natural flatness (i.e., non-sparsity) condition for a particular choice of deterministic correlation-based measurements. A randomized version of these same measurements is then shown to provide nonuniform probabilistic recovery guarantees for arbitrary signals ${\bf x} \in \mathbb{C}^d$. Numerical experiments demonstrate the method's speed, accuracy, and robustness in practice -- all code is made publicly available. Finally, we conclude by developing an extension of the proposed method to the sparse phase retrieval problem; specifically, we demonstrate a sublinear-time compressive phase retrieval algorithm which is guaranteed to recover a given $s$-sparse vector ${\bf x} \in \mathbb{C}^d$ with high probability in just $\mathcal{O}(s \log^5 s \cdot \log d)$-time using only $\mathcal{O}(s \log^4 s \cdot \log d)$ magnitude measurements. In doing so we demonstrate the existence of compressive phase retrieval algorithms with near-optimal linear-in-sparsity runtime complexities.

Stable Signal Recovery from Phaseless Measurements

The aim of this paper is to study the stability of the $\ell_1$ minimization for the compressive phase retrieval and to extend the instance-optimality in compressed sensing to the real phase retrieval setting. We first show that the $m={\mathcal O}(k\log(N/k))$ measurements is enough to guarantee the $\ell_1$ minimization to recover $k$-sparse signals stably provided the measurement matrix $A$ satisfies the strong RIP property. We second investigate the phaseless instance-optimality with presenting a null space property of the measurement matrix $A$ under which there exists a decoder $\Delta$ so that the phaseless instance-optimality holds. We use the result to study the phaseless instance-optimality for the $\ell_1$ norm. The results build a parallel for compressive phase retrieval with the classical compressive sensing.

Compressive Phase Retrieval via Generalized Approximate Message Passing

In phase retrieval, the goal is to recover a signal x ∈ C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> from the magnitudes of linear measurements Ax ∈ C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sup> . While recent theory has established that M ≈ 4N intensity measurements are necessary and sufficient to recover generic x, there is great interest in reducing the number of measurements through the exploitation of sparse \mbi x, which is known as compressive phase retrieval. In this work, we detail a novel, probabilistic approach to compressive phase retrieval based on the generalized approximate message passing (GAMP) algorithm. We then present a numerical study of the proposed PR-GAMP algorithm, demonstrating its excellent phase-transition behavior, robustness to noise, and runtime. Our experiments suggest that approximately M ≥ 2 Klog <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (N/K) intensity measurements suffice to recover K-sparse Bernoulli-Gaussian signals for \mbi A with i.i.d Gaussian entries and K ≪ N. Meanwhile, when recovering a 6678-sparse 65536-pixel grayscale image from 32768 randomly masked and blurred Fourier intensity measurements at 30 dB measurement SNR, PR-GAMP achieved an output SNR of no less than 28 dB in all of 100 random trials, with a median runtime of only 7.3 seconds. Compared to the recently proposed CPRL, sparse-Fienup, and GESPAR algorithms, our experiments suggest that PR-GAMP has a superior phase transition and orders-of-magnitude faster runtimes as the sparsity and problem dimensions increase.

Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming

Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear program, namely, ℓ1-minimization, guarantees recovery of sparse parameter signals even when the system is underdetermined. In this paper, we consider a more challenging problem: when the phase of the output measurements from a linear system is omitted. Using a lifting technique, we show that even though the phase information is missing, the sparse signal can be recovered exactly by solving a semidefinite program when the sampling rate is sufficiently high. This is an interesting finding since the exact solutions to both sparse signal recovery and phase retrieval are combinatorial. The results extend the type of applications that compressive sensing can be applied to those where only output magnitudes can be observed. We demonstrate the accuracy of the algorithms through extensive simulation and a practical experiment.