arXiv subjects

Jian-Feng Cai

Publications and source records attributed to Jian-Feng Cai.

At least 73 records · Page 4Linked to original sources

arXiv2018

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.

math.NA↗

arXiv2018

Solving systems of phaseless equations via Riemannian optimization with optimal sampling complexity

A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y$. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-$1$ positive semidefinite matrices. Theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. Two key ingredients in the analysis are the restricted well conditioned property and the restricted weak correlation property of the associated truncated linear operator. Empirical evaluations show that our algorithms are competitive with other state-of-the-art first order nonconvex approaches with provable guarantees.

math.NA↗

arXiv2018

Vandermonde Factorization of Hankel Matrix for Complex Exponential Signal Recovery -- Application in Fast NMR Spectroscopy

Many signals are modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. This paper studies the problem of recovering exponential signals from a random subset of samples. We exploit the Vandermonde structure of the Hankel matrix formed by the exponential signal and formulate signal recovery as Hankel matrix completion with Vandermonde factorization (HVaF). A numerical algorithm is developed to solve the proposed model and its sequence convergence is analyzed theoretically. Experiments on synthetic data demonstrate that HVaF succeeds over a wider regime than the state-of-the-art nuclear-normminimization-based Hankel matrix completion method, while has a less restriction on frequency separation than the state-of-the-art atomic norm minimization and fast iterative hard thresholding methods. The effectiveness of HVaF is further validated on biological magnetic resonance spectroscopy data.

eess.SP↗

arXiv2017

Separation-Free Super-Resolution from Compressed Measurements is Possible: an Orthonormal Atomic Norm Minimization Approach

We consider the problem of recovering the superposition of $R$ distinct complex exponential functions from compressed non-uniform time-domain samples. Total Variation (TV) minimization or atomic norm minimization was proposed in the literature to recover the $R$ frequencies or the missing data. However, it is known that in order for TV minimization and atomic norm minimization to recover the missing data or the frequencies, the underlying $R$ frequencies are required to be well-separated, even when the measurements are noiseless. This paper shows that the Hankel matrix recovery approach can super-resolve the $R$ complex exponentials and their frequencies from compressed non-uniform measurements, regardless of how close their frequencies are to each other. We propose a new concept of orthonormal atomic norm minimization (OANM), and demonstrate that the success of Hankel matrix recovery in separation-free super-resolution comes from the fact that the nuclear norm of a Hankel matrix is an orthonormal atomic norm. More specifically, we show that, in traditional atomic norm minimization, the underlying parameter values $\textbf{must}$ be well separated to achieve successful signal recovery, if the atoms are changing continuously with respect to the continuously-valued parameter. In contrast, for the OANM, it is possible the OANM is successful even though the original atoms can be arbitrarily close. As a byproduct of this research, we provide one matrix-theoretic inequality of nuclear norm, and give its proof from the theory of compressed sensing.

cs.IT↗

arXiv2017

Fast Rank One Alternating Minimization Algorithm for Phase Retrieval

The phase retrieval problem is a fundamental problem in many fields, which is appealing for investigation. It is to recover the signal vector $\tilde{x}\in\mathbb{C}^d$ from a set of $N$ measurements $b_n=|f^*_n\tilde{x}|^2,\ n=1,\cdots, N$, where $\{f_n\}_{n=1}^N$ forms a frame of $\mathbb{C}^d$. %It is generally a non-convex minimization problem, which is NP-hard. Existing algorithms usually use a least squares fitting to the measurements, yielding a quartic polynomial minimization. In this paper, we employ a new strategy by splitting the variables, and we solve a bi-variate optimization problem that is quadratic in each of the variables. An alternating gradient descent algorithm is proposed, and its convergence for any initialization is provided. Since a larger step size is allowed due to the smaller Hessian, the alternating gradient descent algorithm converges faster than the gradient descent algorithm (known as the Wirtinger flow algorithm) applied to the quartic objective without splitting the variables. Numerical results illustrate that our proposed algorithm needs less iterations than Wirtinger flow to achieve the same accuracy.

math.OC↗

arXiv2017

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.

math.OC↗

arXiv2017

Hankel Matrix Nuclear Norm Regularized Tensor Completion for $N$-dimensional Exponential Signals

Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired and thus how to recover the full signal becomes an active research topic. But existing approaches can not efficiently recover $N$-dimensional exponential signals with $N\geq 3$. In this paper, we study the problem of recovering N-dimensional (particularly $N\geq 3$) exponential signals from partial observations, and formulate this problem as a low-rank tensor completion problem with exponential factor vectors. The full signal is reconstructed by simultaneously exploiting the CANDECOMP/PARAFAC structure and the exponential structure of the associated factor vectors. The latter is promoted by minimizing an objective function involving the nuclear norm of Hankel matrices. Experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed approach can successfully recover full signals from very limited samples and is robust to the estimated tensor rank.

stat.ML↗

arXiv2017

Accelerated Magnetic Resonance Spectroscopy with Vandermonde Factorization

Multi-dimensional magnetic resonance spectroscopy is an important tool for studying molecular structures, interactions and dynamics in bio-engineering. The data acquisition time, however, is relatively long and non-uniform sampling can be applied to reduce this time. To obtain the full spectrum,a reconstruction method with Vandermonde factorization is proposed.This method explores the general signal property in magnetic resonance spectroscopy: Its time domain signal is approximated by a sum of a few exponentials. Results on synthetic and realistic data show that the new approach can achieve faithful spectrum reconstruction and outperforms state-of-the-art low rank Hankel matrix method.

physics.med-ph↗

arXiv2016

Fast and Provable Algorithms for Spectrally Sparse Signal Reconstruction via Low-Rank Hankel Matrix Completion

A spectrally sparse signal of order $r$ is a mixture of $r$ damped or undamped complex sinusoids. This paper investigates the problem of reconstructing spectrally sparse signals from a random subset of $n$ regular time domain samples, which can be reformulated as a low rank Hankel matrix completion problem. We introduce an iterative hard thresholding (IHT) algorithm and a fast iterative hard thresholding (FIHT) algorithm for efficient reconstruction of spectrally sparse signals via low rank Hankel matrix completion. Theoretical recovery guarantees have been established for FIHT, showing that $O(r^2\log^2(n))$ number of samples are sufficient for exact recovery with high probability. Empirical performance comparisons establish significant computational advantages for IHT and FIHT. In particular, numerical simulations on $3$D arrays demonstrate the capability of FIHT on handling large and high-dimensional real data.

cs.IT↗

arXiv2016

Guarantees of Riemannian Optimization for Low Rank Matrix Recovery

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an $m\times n$ rank $r$ matrix from $p < mn$ number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant $R_{3r}$ of the sensing operator is less than $C_κ/\sqrt{r}$, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank $r$ matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.

math.NA↗

arXiv2016

Guarantees of Riemannian Optimization for Low Rank Matrix Completion

We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume $m$ entries of an $n\times n$ rank $r$ matrix are sampled independently and uniformly with replacement. We first prove that with high probability the Riemannian gradient descent and conjugate gradient descent algorithms initialized by one step hard thresholding are guaranteed to converge linearly to the measured matrix provided \begin{align*} m\geq C_κn^{1.5}r\log^{1.5}(n), \end{align*} where $C_κ$ is a numerical constant depending on the condition number of the underlying matrix. The sampling complexity has been further improved to \begin{align*} m\geq C_κnr^2\log^{2}(n) \end{align*} via the resampled Riemannian gradient descent initialization. The analysis of the new initialization procedure relies on an asymmetric restricted isometry property of the sampling operator and the curvature of the low rank matrix manifold. Numerical simulation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements.

math.NA↗

arXiv2015

Fast Multi-class Dictionaries Learning with Geometrical Directions in MRI Reconstruction

Objective: Improve the reconstructed image with fast and multi-class dictionaries learning when magnetic resonance imaging is accelerated by undersampling the k-space data. Methods: A fast orthogonal dictionary learning method is introduced into magnetic resonance image reconstruction to providing adaptive sparse representation of images. To enhance the sparsity, image is divided into classified patches according to the same geometrical direction and dictionary is trained within each class. A new sparse reconstruction model with the multi-class dictionaries is proposed and solved using a fast alternating direction method of multipliers. Results: Experiments on phantom and brain imaging data with acceleration factor up to 10 and various undersampling patterns are conducted. The proposed method is compared with state-of-the-art magnetic resonance image reconstruction methods. Conclusion: Artifacts are better suppressed and image edges are better preserved than the compared methods. Besides, the computation of the proposed approach is much faster than the typical K-SVD dictionary learning method in magnetic resonance image reconstruction. Significance: The proposed method can be exploited in undersapmled magnetic resonance imaging to reduce data acquisition time and reconstruct images with better image quality.

cs.CV↗

arXiv2015

Projected Iterative Soft-thresholding Algorithm for Tight Frames in Compressed Sensing Magnetic Resonance Imaging

Compressed sensing has shown great potentials in accelerating magnetic resonance imaging. Fast image reconstruction and high image quality are two main issues faced by this new technology. It has been shown that, redundant image representations, e.g. tight frames, can significantly improve the image quality. But how to efficiently solve the reconstruction problem with these redundant representation systems is still challenging. This paper attempts to address the problem of applying iterative soft-thresholding algorithm (ISTA) to tight frames based magnetic resonance image reconstruction. By introducing the canonical dual frame to construct the orthogonal projection operator on the range of the analysis sparsity operator, we propose a projected iterative soft-thresholding algorithm (pISTA) and further accelerate it by incorporating the strategy proposed by Beck and Teboulle in 2009. We theoretically prove that pISTA converges to the minimum of a function with a balanced tight frame sparsity. Experimental results demonstrate that the proposed algorithm achieves better reconstruction than the widely used synthesis sparse model and the accelerated pISTA converges faster or comparable to the state-of-art smoothing FISTA. One major advantage of pISTA is that only one extra parameter, the step size, is introduced and the numerical solution is stable to it in terms of image reconstruction errors, thus allowing easily setting in many fast magnetic resonance imaging applications.

physics.med-ph↗

arXiv2015

Precise Phase Transition of Total Variation Minimization

Characterizing the phase transitions of convex optimizations in recovering structured signals or data is of central importance in compressed sensing, machine learning and statistics. The phase transitions of many convex optimization signal recovery methods such as $\ell_1$ minimization and nuclear norm minimization are well understood through recent years' research. However, rigorously characterizing the phase transition of total variation (TV) minimization in recovering sparse-gradient signal is still open. In this paper, we fully characterize the phase transition curve of the TV minimization. Our proof builds on Donoho, Johnstone and Montanari's conjectured phase transition curve for the TV approximate message passing algorithm (AMP), together with the linkage between the minmax Mean Square Error of a denoising problem and the high-dimensional convex geometry for TV minimization.

cs.IT↗

arXiv2015

Block Iterative Reweighted Algorithms for Super-Resolution of Spectrally Sparse Signals

We propose novel algorithms that enhance the performance of recovering unknown continuous-valued frequencies from undersampled signals. Our iterative reweighted frequency recovery algorithms employ the support knowledge gained from earlier steps of our algorithms as block prior information to enhance frequency recovery. Our methods improve the performance of the atomic norm minimization which is a useful heuristic in recovering continuous-valued frequency contents. Numerical results demonstrate that our block iterative reweighted methods provide both better recovery performance and faster speed than other known methods.

cs.IT↗

arXiv2015

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.

cs.IT↗

arXiv2015

Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction

This paper explores robust recovery of a superposition of $R$ distinct complex exponential functions from a few random Gaussian projections. We assume that the signal of interest is of $2N-1$ dimensional and $R<<2N-1$. This framework covers a large class of signals arising from real applications in biology, automation, imaging science, etc. To reconstruct such a signal, our algorithm is to seek a low-rank Hankel matrix of the signal by minimizing its nuclear norm subject to the consistency on the sampled data. Our theoretical results show that a robust recovery is possible as long as the number of projections exceeds $O(R\ln^2N)$. No incoherence or separation condition is required in our proof. Our method can be applied to spectral compressed sensing where the signal of interest is a superposition of $R$ complex sinusoids. Compared to existing results, our result here does not need any separation condition on the frequencies, while achieving better or comparable bounds on the number of measurements. Furthermore, our method provides theoretical guidance on how many samples are required in the state-of-the-art non-uniform sampling in NMR spectroscopy. The performance of our algorithm is further demonstrated by numerical experiments.

cs.IT↗

arXiv2013

Precise Semidefinite Programming Formulation of Atomic Norm Minimization for Recovering d-Dimensional ($d\geq 2$) Off-the-Grid Frequencies

Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In particular, atomic norm minimization was proposed in \cite{tang2012csotg} to recover $1$-dimensional spectrally sparse signal. However, in spite of existing research efforts \cite{chi2013compressive}, it was still an open problem how to formulate an equivalent positive semidefinite program for atomic norm minimization in recovering signals with $d$-dimensional ($d\geq 2$) off-the-grid frequencies. In this paper, we settle this problem by proposing equivalent semidefinite programming formulations of atomic norm minimization to recover signals with $d$-dimensional ($d\geq 2$) off-the-grid frequencies.

cs.IT↗