Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow

This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal $x \in \mathbb{R}^p$ from noisy quadratic measurements $y_j = (a_j' x )^2 + \epsilon_j$, $j=1, \ldots, m$, with independent sub-exponential noise $\epsilon_j$. The goals are to understand the effect of the sparsity of $x$ on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates. Inspired by the Wirtinger Flow [12] proposed for noiseless and non-sparse phase retrieval, a novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the $a_j$'s are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of $x$.Comment: 28 pages, 4 figure

Fourier Phase Retrieval with a Single Mask by Douglas-Rachford Algorithm

Douglas-Rachford (DR) algorithm is analyzed for Fourier phase retrieval with a single random phase mask. Local, geometric convergence to a unique fixed point is proved with numerical demonstration of global convergence

Sparse Phase Retrieval: Convex Algorithms and Limitations

We consider the problem of recovering signals from their power spectral densities. This is a classical problem referred to in literature as the phase retrieval problem, and is of paramount importance in many fields of applied sciences. In general, additional prior information about the signal is required to guarantee unique recovery as the mapping from signals to power spectral densities is not one-to-one. In this work, we assume that the underlying signals are sparse. Recently, semidefinite programming (SDP) based approaches were explored by various researchers. Simulations of these algorithms strongly suggest that signals upto O(n^(1/2−ϵ) sparsity can be recovered by this technique. In this work, we develop a tractable algorithm based on reweighted ℓ_1-minimization that recovers a sparse signal from its power spectral density for significantly higher sparsities, which is unprecedented. We also discuss the limitations of the existing SDP algorithms and provide a combinatorial algorithm which requires significantly fewer ”phaseless” measurements to guarantee recovery

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 over 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 work, we focus our attention on discrete-time sparse signals (of length n). We first show that, if the DFT 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: (i) identifying the support, i.e., the locations of the non-zero components, of the signal using a combinatorial algorithm (ii) identifying the signal values in the support using a convex algorithm. We show that TSPR can provably recover most O(n^(1/2-ϵ)-sparse signals (up to a timeshift, conjugate-flip and global phase). We also show that, for most O(n^(1/4-ϵ)-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

Structured Sub-Nyquist Sampling with Applications in Compressive Toeplitz Covariance Estimation, Super-Resolution and Phase Retrieval

Sub-Nyquist sampling has received a huge amount of interest in the past decade. In classical compressed sensing theory, if the measurement procedure satisfies a particular condition known as Restricted Isometry Property (RIP), we can achieve stable recovery of signals of low-dimensional intrinsic structures with an order-wise optimal sample size. Such low-dimensional structures include sparse and low rank for both vector and matrix cases. The main drawback of conventional compressed sensing theory is that random measurements are required to ensure the RIP property. However, in many applications such as imaging and array signal processing, applying independent random measurements may not be practical as the systems are deterministic. Moreover, random measurements based compressed sensing always exploits convex programs for signal recovery even in the noiseless case, and solving those programs is computationally intensive if the ambient dimension is large, especially in the matrix case. The main contribution of this dissertation is that we propose a deterministic sub-Nyquist sampling framework for compressing the structured signal and come up with computationally efficient algorithms. Besides widely studied sparse and low-rank structures, we particularly focus on the cases that the signals of interest are stationary or the measurements are of Fourier type. The key difference between our work from classical compressed sensing theory is that we explicitly exploit the second-order statistics of the signals, and study the equivalent quadratic measurement model in the correlation domain. The essential observation made in this dissertation is that a difference/sum coarray structure will arise from the quadratic model if the measurements are of Fourier type. With these observations, we are able to achieve a better compression rate for covariance estimation, identify more sources in array signal processing or recover the signals of larger sparsity. In this dissertation, we will first study the problem of Toeplitz covariance estimation. In particular, we will show how to achieve an order-wise optimal compression rate using the idea of sparse arrays in both general and low-rank cases. Then, an analysis framework of super-resolution with positivity constraint is established. We will present fundamental robustness guarantees, efficient algorithms and applications in practices. Next, we will study the problem of phase-retrieval for which we successfully apply the sparse array ideas by fully exploiting the quadratic measurement model. We achieve near-optimal sample complexity for both sparse and general cases with practical Fourier measurements and provide efficient and deterministic recovery algorithms. In the end, we will further elaborate on the essential role of non-negative constraint in underdetermined inverse problems. In particular, we will analyze the nonlinear co-array interpolation problem and develop a universal upper bound of the interpolation error. Bilinear problem with non-negative constraint will be considered next and the exact characterization of the ambiguous solutions will be established for the first time in literature. At last, we will show how to apply the nested array idea to solve real problems such as Kriging. Using spatial correlation information, we are able to have a stable estimate of the field of interest with fewer sensors than classic methodologies. Extensive numerical experiments are implemented to demonstrate our theoretical claims