arXiv subjects

Xunmeng Wu

Publications and source records attributed to Xunmeng Wu.

10 recordsLinked to original sources

arXiv2026

Rank-Optimal Sample Complexity of Riemannian Gauss-Newton for Low-Rank Matrix Recovery

In this paper, we study the recovery of an unknown rank-$r$ matrix $\boldsymbol{X}_\star \in \mathbb{R}^{n_1\times n_2}$ from Gaussian linear measurements using the Riemannian Gauss-Newton (RGN) algorithm. Each iteration of RGN solves the RGN equation followed by a fixed-rank retraction. We prove that, with high probability, RGN with spectral initialization converges quadratically to $\boldsymbol{X}_\star$ using $\mathcal{O} ( r(n_1+n_2)κ^2 )$ measurements, where $κ$ is the condition number of $\boldsymbol{X}_\star$. This is the first result to simultaneously achieve rank-optimal sample complexity and quadratic convergence for Gaussian low-rank matrix sensing. The key to the proof is a sharp spectral-norm analysis of the iterates, which requires controlling the inverse operator arising from the RGN equation. Our analysis develops an exact finite Neumann expansion of this inverse operator and combines it with a virtual-sequence decoupling argument and the restricted isometry property. Extensive numerical experiments support our theoretical findings and demonstrate the advantages of RGN.

math.NA↗

arXiv2026

Superfast Spectral Compressed Sensing with Quadratic Convergence

In this paper, we study the spectral compressed sensing problem of recovering a spectrally sparse signal from linear compressive measurements. Exploiting the low-rank Hankel structure associated with the signal, we formulate a nonconvex fixed-rank optimization problem with a Hankel regularization. We propose a Superfast Iterative Hard Thresholding (SIHT) algorithm, where each iteration consists of a tangent-space least-squares update followed by a fixed-rank hard thresholding projection. SIHT is the first nonconvex algorithm with provable quadratic convergence for spectral compressed sensing. For random Gaussian measurements, it achieves exact recovery in the noiseless case and stable recovery in the noisy case with near-optimal sample complexity. To efficiently implement SIHT, we develop a parameterized conjugate gradient method for solving the least-squares subproblem, endowing SIHT with state-of-the-art computational and memory efficiency. Numerical experiments are provided to validate our results.

cs.IT↗

arXiv2026

Lifted Restricted Isometry Property and Fast Algorithm for Spectral Compressed Sensing With Near-Optimal Sample Complexity

This paper studies the spectral compressed sensing problem, which aims to recover an $r$-spectrally-sparse signal of length $N$ from linear compressive measurements. We reformulate the problem as the recovery of a low-rank Hankel matrix. Motivated by this formulation, we propose a Lifted Restricted Isometry Property (LRIP) for the measurement matrix and show that Gaussian measurement matrices satisfy LRIP under certain conditions. Unlike classical RIP, LRIP imposes a rank constraint in the lifted matrix domain while taking measurements in the signal domain through the adjoint Hankel operator. To solve the resulting nonconvex problem efficiently, we introduce a Fast Iterative Hard Thresholding (FIHT) algorithm. Based on LRIP, we show that FIHT converges linearly to the true signal with high probability using $\mathcal{O}(r \ln r \ln^2 N)$ scaled Gaussian measurements, which is near-optimal up to log factors and improves the state-of-the-art order of $r^2$ for nonconvex Hankel methods. Extensive numerical results demonstrate the superiority of FIHT as compared to state-of-the-art convex methods in terms of phase transitions, computational time, and robustness to noise.

math.OC↗

arXiv2024

Low-Complexity Algorithms for Multichannel Spectral Super-Resolution

This paper studies the problem of multichannel spectral super-resolution with either constant amplitude (CA) or not. We propose two optimization problems based on low-rank Hankel-Toeplitz matrix factorization. The two problems effectively leverage the multichannel and CA structures, while also enabling the design of low-complexity gradient descent algorithms for their solutions. Extensive simulations show the superior performance of the proposed algorithms.

math.OC↗

arXiv2024

Multichannel Frequency Estimation with Constant Amplitude via Convex Structured Low-Rank Approximation

We study the problem of estimating the frequencies of several complex sinusoids with constant amplitude (CA) (also called constant modulus) from multichannel signals of their superposition. To exploit the CA property for frequency estimation in the framework of atomic norm minimization (ANM), we introduce multiple positive-semidefinite block matrices composed of Hankel and Toeplitz submatrices and formulate the ANM problem as a convex structured low-rank approximation (SLRA) problem. The proposed SLRA is a semidefinite programming and has substantial differences from existing such formulations without using the CA property. The proposed approach is termed as SLRA-based ANM for CA frequency estimation (SACA). We provide theoretical guarantees and extensive simulations that validate the advantages of SACA.

math.OC↗

arXiv2023

Multichannel Frequency Estimation in Challenging Scenarios via Structured Matrix Embedding and Recovery (StruMER)

Multichannel frequency estimation with incomplete data and miscellaneous noises arises in array signal processing, modal analysis, wireless communications, and so on. In this paper, we consider maximum-likelihood(-like) optimization methods for frequency estimation in which proper objective functions are adopted subject to observed data patterns and noise types. We propose a universal signal-domain approach to solve the optimization problems by embedding the noiseless multichannel signal of interest into a series of low-rank positive-semidefinite block matrices of Hankel and Toeplitz submatrices and formulating the original parameter-domain optimization problems as equivalent structured matrix recovery problems. The alternating direction method of multipliers (ADMM) is applied to solve the resulting matrix recovery problems in which both subproblems of ADMM are solved in (nearly) closed form. The proposed approach is termed as structured matrix embedding and recovery (StruMER). Extensive numerical simulations are provided to demonstrate that StruMER has improved threshold performances in various challenging scenarios, e.g., limited data, low signal-to-noise ratio, impulsive noise, and closely spaced frequencies, as compared with state-of-the-art methods.

eess.SP↗

arXiv2023

Direction-of-Arrival Estimation for Constant Modulus Signals Using a Structured Matrix Recovery Technique

This paper addresses the problem of direction-of-arrival (DOA) estimation for constant modulus (CM) source signals using a uniform or sparse linear array. Existing methods typically exploit either the Vandermonde structure of the steering matrix or the CM structure of source signals only. In this paper, we propose a structured matrix recovery technique (SMART) for CM DOA estimation via fully exploiting the two structures. In particular, we reformulate the highly nonconvex CM DOA estimation problems in the noiseless and noisy cases as equivalent rank-constrained Hankel-Toeplitz matrix recovery problems, in which the Vandermonde structure is captured by a series of Hankel-Toeplitz block matrices, of which the number equals the number of snapshots, and the CM structure is guaranteed by letting the block matrices share a same Toeplitz submatrix. The alternating direction method of multipliers (ADMM) is applied to solve the resulting rank-constrained problems and the DOAs are uniquely retrieved from the numerical solution. Extensive simulations are carried out to corroborate our analysis and confirm that the proposed SMART outperforms state-of-the-art algorithms in terms of the maximum number of locatable sources and statistical efficiency.

eess.SP↗

arXiv2022

A Robust and Statistically Efficient Maximum-Likelihood Method for DOA Estimation Using Sparse Linear Arrays

A recent trend of research on direction-of-arrival (DOA) estimation is to localize more uncorrelated sources than sensors by using a proper sparse linear array (SLA) and the Toeplitz covariance structure, at a cost of robustness to source correlations. In this paper, we make an attempt to achieve the two goals simultaneously by using a single algorithm. In order to statistically efficiently localize a maximal number of uncorrelated sources, we propose an effective algorithm for the stochastic maximum likelihood (SML) method based on elegant problem reformulations and the alternating direction method of multipliers (ADMM). We prove that the SML is robust to source correlations though it is derived under the assumption of uncorrelated sources. The proposed algorithm is usable for arbitrary SLAs (e.g., minimum redundancy arrays, nested arrays and coprime arrays) and is named as {\em m}aximum-likelihood {\em e}stimation via {\em s}equential {\em A}DMM (MESA). Extensive numerical results are provided that collaborate our analysis and demonstrate the statistical efficiency and robustness of MESA among state-of-the-art algorithms.

eess.SP↗

arXiv2022

Robust spectral compressive sensing via vanilla gradient descent

This paper investigates the recovery of a spectrally sparse signal from its partially revealed noisy entries within the framework of spectral compressive sensing. Nonconvex optimization approaches have recently been proposed based on low-rank Hankel matrix completion and projected gradient descent (PGD). The PGD however involves unknown tuning parameters and its theoretical analysis is available only in the absence of noise. In this paper, we propose a hyperparameter-free, vanilla gradient descent (VGD) algorithm and prove that the VGD enables robust recovery of an $N$-dimensional $K$-spectrally-sparse signal from order $K^2 log^2N$ number of noisy samples under coherence and other mild conditions. The above sample complexity increases by factor $logN$ as compared with PGD without noise. Numerical simulations are provided that corroborate our analysis and show advantageous performances of VGD.

cs.IT↗

arXiv2021

New Low Rank Optimization Model and Convex Approach for Robust Spectral Compressed Sensing

This paper investigates recovery of an undamped spectrally sparse signal and its spectral components from a set of regularly spaced samples within the framework of spectral compressed sensing and super-resolution. We show that the existing Hankel-based optimization methods suffer from the fundamental limitation that the prior of undampedness cannot be exploited. We propose a new low rank optimization model partially inspired by forward-backward processing for line spectral estimation and show its capability in restricting the spectral poles on the unit circle. We present convex relaxation approaches with the model and show their provable accuracy and robustness to bounded and sparse noise. All our results are generalized from the 1-D to arbitrary-dimensional spectral compressed sensing. Numerical simulations are provided that corroborate our analysis and show efficiency of our model and advantageous performance of our approach in improved accuracy and resolution as compared to the state-of-the-art Hankel and atomic norm methods.

cs.IT↗