Yuanxin Li

Off-the-Grid Line Spectrum Denoising and Estimation With Multiple Measurement Vectors

Compressed Sensing suggests that the required number of samples for reconstructing a signal can be greatly reduced if it is sparse in a known discrete basis, yet many real-world signals are sparse in a continuous dictionary. One example is the spectrally-sparse signal, which is composed of a small number of spectral atoms with arbitrary frequencies on the unit interval. In this paper we study the problem of line spectrum denoising and estimation with an ensemble of spectrally-sparse signals composed of the same set of continuous-valued frequencies from their partial and noisy observations. Two approaches are developed based on atomic norm minimization and structured covariance estimation, both of which can be solved efficiently via semidefinite programming. The first approach aims to estimate and denoise the set of signals from their partial and noisy observations via atomic norm minimization, and recover the frequencies via examining the dual polynomial of the convex program. We characterize the optimality condition of the proposed algorithm and derive the expected error rate for denoising, demonstrating the benefit of including multiple measurement vectors. The second approach aims to recover the population covariance matrix from the partially observed sample covariance matrix by motivating its low-rank Toeplitz structure without recovering the signal ensemble. Performance guarantee is derived with a finite number of measurement vectors. The frequencies can be recovered via conventional spectrum estimation methods such as MUSIC from the estimated covariance matrix. Finally, numerical examples are provided to validate the favorable performance of the proposed algorithms, with comparisons against several existing approaches.

Low-Rank Positive Semidefinite Matrix Recovery From Corrupted Rank-One Measurements

We study the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of rank-one measurements using sensing vectors composed of i.i.d. standard Gaussian entries, which are possibly corrupted by arbitrary outliers. This problem arises from applications, such as phase retrieval, covariance sketching, quantum space tomography, and power spectrum estimation. We first propose a convex optimization algorithm that seeks the PSD matrix with the minimum l1-norm of the observation residual. The advantage of our algorithm is that it is free of parameters, therefore, eliminating the need for tuning parameters and allowing easy implementations. We establish that with high probability, a low-rank PSD matrix can be exactly recovered as soon as the number of measurements is large enough, even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes. Moreover, the recovery is also stable against bounded noise. With the additional information of an upper bound of the rank of the PSD matrix, we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy.

Non-convex low-rank matrix recovery from corrupted random linear measurements

Recent work has demonstrated the effectiveness of gradient descent for recovering low-rank matrices from random linear measurements in a globally convergent manner. However, their performance is highly sensitive in the presence of outliers that may take arbitrary values, which is common in practice. In this paper, we propose a truncated gradient descent algorithm to improve the robustness against outliers, where the truncation is performed to rule out the contributions from samples that deviate significantly from the sample median. A restricted isometry property regarding the sample median is introduced to provide a theoretical footing of the proposed algorithm for the Gaussian orthogonal ensemble. Extensive numerical experiments are provided to validate the superior performance of the proposed algorithm.

Super-resolution of mutually interfering signals

We consider simultaneously identifying the membership and locations of point sources that are convolved with different low-pass point spread functions, from the observation of their superpositions. This problem arises in three-dimensional super-resolution single-molecule imaging, neural spike sorting, multi-user channel identification, among others. We propose a novel algorithm, based on convex programming, and establish its near-optimal performance guarantee for exact recovery by exploiting the sparsity of the point source model as well as incoherence between the point spread functions. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Blind calibration of multi-channel samplers using sparse recovery

We propose an algorithm for blind calibration of multi-channel samplers in the presence of unknown gains and offsets, which is useful in many applications such as multi-channel analog-to-digital converters, image super-resolution, and sensor networks. Using a subspace-based rank condition developed by Vandewalle et al., we obtain a set of linear equations with respect to complex harmonics whose frequencies are determined by the offsets, and the coefficients of each harmonic are determined by the discrete-time Fourier transforms of outputs of each of the channels. By discretizing the offsets over a fine grid, this becomes a sparse recovery problem where the signal of interest is sparse with an additional structure, that in each block there is only one nonzero entry. We propose a modified CoSaMP algorithm that takes this structure into account to estimate the offsets. Our algorithm is scalable to large numbers of channels and can also be extended to multi-dimensional signals. Numerical experiments demonstrate the effectiveness of the proposed algorithm.

Performance bounds for modal analysis using sparse linear arrays

We study the performance of modal analysis using sparse linear arrays (SLAs) such as nested and co-prime arrays, in both first-order and second-order measurement models. We treat SLAs as constructed from a subset of sensors in a dense uniform linear array (ULA), and characterize the performance loss of SLAs with respect to the ULA due to using much fewer sensors. In particular, we claim that, provided the same aperture, in order to achieve comparable performance in terms of Cramér-Rao bound (CRB) for modal analysis, SLAs require more snapshots, of which the number is about the number of snapshots used by ULA times the compression ratio in the number of sensors. This is shown analytically for the case with one undamped mode, as well as empirically via extensive numerical experiments for more complex scenarios. Moreover, the misspecified CRB proposed by Richmond and Horowitz is also studied, where SLAs suffer more performance loss than their ULA counterpart.

Outlier-robust recovery of low-rank positive semidefinite matrices from magnitude measurements

We address the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of magnitude measurements that are quadratic in the sensing vectors in the presence of arbitrary outliers. We propose a parameter-free algorithm that seeks the PSD matrix that minimizes the ℓ1-norm of the measurement residual. It is shown that the algorithm can exactly recover a rank-r PSD matrix of size-n from O (nr2) measurements with high probability, even when a fraction of the measurements is corrupted by arbitrary outliers. Furthermore, the recovery is also robust to bounded noise. When an upper bound of the rank of the PSD matrix is known a priori, we further propose a non-convex algorithm based on subgradient descent that demonstrates superior empirical performance.

Parameter estimation for mixture models via convex optimization

Many applications encounter signals that are a linear combination of multiple components, where each component represents a low-resolution observation of a point source model captured through a low-pass point spread function. This paper proposes a convex optimization algorithm to simultaneously separate and identify the point source models of each component from a noisy observation corrupted by possibly adversarial noise, by leveraging the recently proposed atomic norm framework. The proposed algorithm can be solved efficiently via semidefinite programming, where locations of the point sources can be identified via the constructed dual polynomials without estimating the model orders a priori. Stability of the proposed algorithm is established under certain conditions of the point source models and the point spread functions in the presence of bounded noise. Furthermore, numerical examples are provided to corroborate the theoretical analysis, with comparisons against the Cramèr-Rao bound for parameter estimation.