Projected Wirtinger Gradient Descent for Low-Rank Hankel Matrix Completion in Spectral Compressed Sensing

This paper considers reconstructing a spectrally sparse signal from a small number of randomly observed time-domain samples. The signal of interest is a linear combination of complex sinusoids at $R$ distinct frequencies. The frequencies can assume any continuous values in the normalized frequency domain $[0,1)$. After converting the spectrally sparse signal recovery into a low rank structured matrix completion problem, we propose an efficient feasible point approach, named projected Wirtinger gradient descent (PWGD) algorithm, to efficiently solve this structured matrix completion problem. We further accelerate our proposed algorithm by a scheme inspired by FISTA. We give the convergence analysis of our proposed algorithms. Extensive numerical experiments are provided to illustrate the efficiency of our proposed algorithm. Different from earlier approaches, our algorithm can solve problems of very large dimensions very efficiently.Comment: 12 page

Exploiting the structure effectively and efficiently in low rank matrix recovery

Low rank model arises from a wide range of applications, including machine learning, signal processing, computer algebra, computer vision, and imaging science. Low rank matrix recovery is about reconstructing a low rank matrix from incomplete measurements. In this survey we review recent developments on low rank matrix recovery, focusing on three typical scenarios: matrix sensing, matrix completion and phase retrieval. An overview of effective and efficient approaches for the problem is given, including nuclear norm minimization, projected gradient descent based on matrix factorization, and Riemannian optimization based on the embedded manifold of low rank matrices. Numerical recipes of different approaches are emphasized while accompanied by the corresponding theoretical recovery guarantees

Spectral Compressed Sensing via Projected Gradient Descent

Let $x\in\mathbb{C}^n$ be a spectrally sparse signal consisting of $r$ complex sinusoids with or without damping. We consider the spectral compressed sensing problem, which is about reconstructing $x$ from its partial revealed entries. By utilizing the low rank structure of the Hankel matrix corresponding to $x$, we develop a computationally efficient algorithm for this problem. The algorithm starts from an initial guess computed via one-step hard thresholding followed by projection, and then proceeds by applying projected gradient descent iterations to a non-convex functional. Based on the sampling with replacement model, we prove that $O(r^2\log(n))$ observed entries are sufficient for our algorithm to achieve the successful recovery of a spectrally sparse signal. Moreover, extensive empirical performance comparisons show that our algorithm is competitive with other state-of-the-art spectral compressed sensing algorithms in terms of phase transitions and overall computational time

Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation

Low-rank modeling plays a pivotal role in signal processing and machine learning, with applications ranging from collaborative filtering, video surveillance, medical imaging, to dimensionality reduction and adaptive filtering. Many modern high-dimensional data and interactions thereof can be modeled as lying approximately in a low-dimensional subspace or manifold, possibly with additional structures, and its proper exploitations lead to significant reduction of costs in sensing, computation and storage. In recent years, there is a plethora of progress in understanding how to exploit low-rank structures using computationally efficient procedures in a provable manner, including both convex and nonconvex approaches. On one side, convex relaxations such as nuclear norm minimization often lead to statistically optimal procedures for estimating low-rank matrices, where first-order methods are developed to address the computational challenges; on the other side, there is emerging evidence that properly designed nonconvex procedures, such as projected gradient descent, often provide globally optimal solutions with a much lower computational cost in many problems. This survey article will provide a unified overview of these recent advances on low-rank matrix estimation from incomplete measurements. Attention is paid to rigorous characterization of the performance of these algorithms, and to problems where the low-rank matrix have additional structural properties that require new algorithmic designs and theoretical analysis.Comment: To appear in IEEE Signal Processing Magazin

Spectral Compressed Sensing via CANDECOMP/PARAFAC Decomposition of Incomplete Tensors

We consider the line spectral estimation problem which aims to recover a mixture of complex sinusoids from a small number of randomly observed time domain samples. Compressed sensing methods formulates line spectral estimation as a sparse signal recovery problem by discretizing the continuous frequency parameter space into a finite set of grid points. Discretization, however, inevitably incurs errors and leads to deteriorated estimation performance. In this paper, we propose a new method which leverages recent advances in tensor decomposition. Specifically, we organize the observed data into a structured tensor and cast line spectral estimation as a CANDECOMP/PARAFAC (CP) decomposition problem with missing entries. The uniqueness of the CP decomposition allows the frequency components to be super-resolved with infinite precision. Simulation results show that the proposed method provides a competitive estimate accuracy compared with existing state-of-the-art algorithms

Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent

Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and then optimize these factors directly via simple iterative methods such as gradient descent and alternating minimization. Despite nonconvexity, recent literatures have shown that these simple heuristics in fact achieve linear convergence when initialized properly for a growing number of problems of interest. However, upon closer examination, existing approaches can still be computationally expensive especially for ill-conditioned matrices: the convergence rate of gradient descent depends linearly on the condition number of the low-rank matrix, while the per-iteration cost of alternating minimization is often prohibitive for large matrices. The goal of this paper is to set forth a competitive algorithmic approach dubbed Scaled Gradient Descent (ScaledGD) which can be viewed as pre-conditioned or diagonally-scaled gradient descent, where the pre-conditioners are adaptive and iteration-varying with a minimal computational overhead. With tailored variants for low-rank matrix sensing, robust principal component analysis and matrix completion, we theoretically show that ScaledGD achieves the best of both worlds: it converges linearly at a rate independent of the condition number of the low-rank matrix similar as alternating minimization, while maintaining the low per-iteration cost of gradient descent. Our analysis is also applicable to general loss functions that are restricted strongly convex and smooth over low-rank matrices. To the best of our knowledge, ScaledGD is the first algorithm that provably has such properties over a wide range of low-rank matrix estimation tasks

Data Driven Tight Frame for Compressed Sensing MRI Reconstruction via Off-the-Grid Regularization

Recently, the finite-rate-of-innovation (FRI) based continuous domain regularization is emerging as an alternative to the conventional on-the-grid sparse regularization for the compressed sensing (CS) due to its ability to alleviate the basis mismatch between the true support of the shape in the continuous domain and the discrete grid. In this paper, we propose a new off-the-grid regularization for the CS-MRI reconstruction. Following the recent works on two dimensional FRI, we assume that the discontinuities/edges of the image are localized in the zero level set of a band-limited periodic function. This assumption induces the linear dependencies among the Fourier samples of the gradient of the image, which leads to a low rank two-fold Hankel matrix. We further observe that the singular value decomposition of a low rank Hankel matrix corresponds to an adaptive tight frame system which can represent the image with sparse canonical coefficients. Based on this observation, we propose a data driven tight frame based off-the-grid regularization model for the CS-MRI reconstruction. To solve the nonconvex and nonsmooth model, a proximal alternating minimization algorithm with a guaranteed global convergence is adopted. Finally, the numerical experiments show that our proposed data driven tight frame based approach outperforms the existing approaches

Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization

This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the theoretical support of this approach is still far from optimal in the noisy setting, falling short of explaining its empirical success. We make progress towards demystifying the practical efficacy of convex relaxation vis-\`a-vis random noise. When the rank and the condition number of the unknown matrix are bounded by a constant, we demonstrate that the convex programming approach achieves near-optimal estimation errors --- in terms of the Euclidean loss, the entrywise loss, and the spectral norm loss --- for a wide range of noise levels. All of this is enabled by bridging convex relaxation with the nonconvex Burer-Monteiro approach, a seemingly distinct algorithmic paradigm that is provably robust against noise. More specifically, we show that an approximate critical point of the nonconvex formulation serves as an extremely tight approximation of the convex solution, thus allowing us to transfer the desired statistical guarantees of the nonconvex approach to its convex counterpart

Quantized Spectral Compressed Sensing: Cramer-Rao Bounds and Recovery Algorithms

Efficient estimation of wideband spectrum is of great importance for applications such as cognitive radio. Recently, sub-Nyquist sampling schemes based on compressed sensing have been proposed to greatly reduce the sampling rate. However, the important issue of quantization has not been fully addressed, particularly for high-resolution spectrum and parameter estimation. In this paper, we aim to recover spectrally-sparse signals and the corresponding parameters, such as frequency and amplitudes, from heavy quantizations of their noisy complex-valued random linear measurements, e.g. only the quadrant information. We first characterize the Cramer-Rao bound under Gaussian noise, which highlights the trade-off between sample complexity and bit depth under different signal-to-noise ratios for a fixed budget of bits. Next, we propose a new algorithm based on atomic norm soft thresholding for signal recovery, which is equivalent to proximal mapping of properly designed surrogate signals with respect to the atomic norm that motivates spectral sparsity. The proposed algorithm can be applied to both the single measurement vector case, as well as the multiple measurement vector case. It is shown that under the Gaussian measurement model, the spectral signals can be reconstructed accurately with high probability, as soon as the number of quantized measurements exceeds the order of K log n, where K is the level of spectral sparsity and $n$ is the signal dimension. Finally, numerical simulations are provided to validate the proposed approaches

Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees

Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the low-rank matrix, and to run projected gradient descent on the nonconvex factorized optimization problem. The goal of this problem is to provide a general theoretical framework for understanding when such methods work well, and to characterize the nature of the resulting fixed point. We provide a simple set of conditions under which projected gradient descent, when given a suitable initialization, converges geometrically to a statistically useful solution. Our results are applicable even when the initial solution is outside any region of local convexity, and even when the problem is globally concave. Working in a non-asymptotic framework, we show that our conditions are satisfied for a wide range of concrete models, including matrix regression, structured PCA, matrix completion with real and quantized observations, matrix decomposition, and graph clustering problems. Simulation results show excellent agreement with the theoretical predictions