Gongguo Tang

Compressed Sensing Off the Grid

This paper investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. An atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies, which is then reformulated as an exact semidefinite program. Even with this continuous dictionary, it is shown that O(slog s log n) random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are well separated. Extensive numerical experiments are performed to illustrate the effectiveness of the proposed method.

A game-theoretic approach for optimal time-of-use electricity pricing

Demand for electricity varies throughout the day, increasing the average cost of power supply. Time-of-use (TOU) pricing has been proposed as a demand-side management (DSM) method to influence user demands. In this paper, we describe a game-theoretic approach to optimize TOU pricing strategies (GT-TOU). We propose models of costs to utility companies arising from user demand fluctuations, and models of user satisfaction with the difference between the nominal demand and the actual consumption. We design utility functions for the company and the users, and obtain a Nash equilibrium using backward induction. In addition to a single-user-type scenario, we also consider a scenario with multiple types of users, each of whom responds differently to time-dependent prices. Numerical examples show that our method is effective in leveling the user demand by setting optimal TOU prices, potentially decreasing costs for the utility companies, and increasing user benefits. An increase in social welfare measure indicates improved market efficiency through TOU pricing.

Atomic Norm Denoising With Applications to Line Spectral Estimation

Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzows, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.

Performance Analysis for Sparse Support Recovery

The performance of estimating the common support for jointly sparse signals based on their projections onto lower-dimensional space is analyzed. Support recovery is formulated as a multiple-hypothesis testing problem. Both upper and lower bounds on the probability of error are derived for general measurement matrices, by using the Chernoff bound and Fanos inequality, respectively. The upper bound shows that the performance is determined by a quantity measuring the measurement matrix incoherence, while the lower bound reveals the importance of the total measurement gain. The lower bound is applied to derive the minimal number of samples needed for accurate direction-of-arrival (DOA) estimation for a sparse representation based algorithm. When applied to Gaussian measurement ensembles, these bounds give necessary and sufficient conditions for a vanishing probability of error for majority realizations of the measurement matrix. Our results offer surprising insights into sparse signal recovery. For example, as far as support recovery is concerned, the well-known bound in Compressive Sensing with the Gaussian measurement matrix is generally not sufficient unless the noise level is low. Our study provides an alternative performance measure, one that is natural and important in practice, for signal recovery in Compressive Sensing and other application areas exploiting signal sparsity.

Compressive sensing off the grid

We consider the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressive sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. We propose an atomic norm minimization approach to exactly recover the unobserved samples, which is then followed by any linear prediction method to identify the frequency components. We reformulate the atomic norm minimization as an exact semidefinite program. By constructing a dual certificate polynomial using random kernels, we show that roughly s log s log n random samples are sufficient to guarantee the exact frequency estimation with high probability, provided the frequencies are well separated. Extensive numerical experiments are performed to illustrate the effectiveness of the proposed method. Our approach avoids the basis mismatch issue arising from discretization by working directly on the continuous parameter space. Potential impact on both compressive sensing and line spectral estimation, in particular implications in sub-Nyquist sampling and superresolution, are discussed.

Multiobjective Optimization of OFDM Radar Waveform for Target Detection

We propose a multiobjective optimization (MOO) technique to design an orthogonal-frequency-division multiplexing (OFDM) radar signal for detecting a moving target in the presence of multipath reflections. We employ an OFDM signal to increase the frequency diversity of the system, as different scattering centers of a target resonate variably at different frequencies. Moreover, the multipath propagation increases the spatial diversity by providing extra “looks” at the target. First, we develop a parametric OFDM radar model by reformulating the target-detection problem as the task of sparse-signal spectrum estimation. At a particular range cell, we exploit the sparsity of multiple paths and the knowledge of the environment to estimate the paths along which the target responses are received. Then, to estimate the sparse vector, we employ a collection of multiple small Dantzig selectors (DS) that utilizes more prior structures of the sparse vector. We use the ℓ1-constrained minimal singular value (ℓ1-CMSV) of the measurement matrix to analytically evaluate the reconstruction performance and demonstrate that our decomposed DS performs better than the standard DS. In addition, we propose a constrained MOO-based algorithm to optimally design the spectral parameters of the OFDM waveform for the next coherent processing interval by simultaneously optimizing two objective functions: minimizing the upper bound on the estimation error to improve the efficiency of sparse-recovery and maximizing the squared Mahalanobis-distance to increase the performance of the underlying detection problem. We provide a few numerical examples to illustrate the performance characteristics of the sparse recovery and demonstrate the achieved performance improvement due to adaptive OFDM waveform design.

Experimental robustness of Fourier ptychography phase retrieval algorithms

Fourier ptychography is a new computational microscopy technique that provides gigapixel-scale intensity and phase images with both wide field-of-view and high resolution. By capturing a stack of low-resolution images under different illumination angles, an inverse algorithm can be used to computationally reconstruct the high-resolution complex field. Here, we compare and classify multiple proposed inverse algorithms in terms of experimental robustness. We find that the main sources of error are noise, aberrations and mis-calibration (i.e. model mis-match). Using simulations and experiments, we demonstrate that the choice of cost function plays a critical role, with amplitude-based cost functions performing better than intensity-based ones. The reason for this is that Fourier ptychography datasets consist of images from both brightfield and darkfield illumination, representing a large range of measured intensities. Both noise (e.g. Poisson noise) and model mis-match errors are shown to scale with intensity. Hence, algorithms that use an appropriate cost function will be more tolerant to both noise and model mis-match. Given these insights, we propose a global Newtons method algorithm which is robust and accurate. Finally, we discuss the impact of procedures for algorithmic correction of aberrations and mis-calibration.

Convergence of a Class of Multi-Agent Systems In Probabilistic Framework

Multi-agent systems arise from diverse fields in natural and artificial systems, and a basic problem is to understand how locally interacting agents lead to collective behaviors (e.g., synchronization) of the overall system. In this paper, we will consider a basic class of multi-agent systems that are described by a simplification of the well-known Vicsek model. This model looks simple, but the rigorous theoretical analysis is quite complicated, because there are strong nonlinear interactions among the agents in the model. In fact, most of the existing results on synchronization need to impose a certain connectivity condition on the global behaviors of the agents’ trajectories (or on the closed-loop dynamic neighborhood graphs), which are quite hard to verify in general. In this paper, by introducing a probabilistic framework to this problem, we will provide a complete and rigorous proof for the fact that the overall multi-agent system will synchronize with large probability as long as the number of agents is large enough. The proof is based on a detailed analysis of both the dynamical properties of the nonlinear system evolution and the asymptotic properties of the spectrum of random geometric graphs.

Linear system identification via atomic norm regularization

This paper proposes a new algorithm for linear system identification from noisy measurements. The proposed algorithm balances a data fidelity term with a norm induced by the set of single pole filters. We pose a convex optimization problem that approximately solves the atomic norm minimization problem and identifies the unknown system from noisy linear measurements. This problem can be solved efficiently with standard, free software. We provide rigorous statistical guarantees that explicitly bound the estimation error (in the ℌ2-norm) in terms of the stability radius, the Hankel singular values of the true system and the number of measurements. These results in turn yield complexity bounds and asymptotic consistency. We provide numerical experiments demonstrating the efficacy of our method for estimating linear systems from a variety of linear measurements.

Sparse recovery over continuous dictionaries-just discretize

In many applications of sparse recovery, the signal has a sparse representation only with respect to a continuously parameterized dictionary. Although atomic norm minimization provides a general framework to handle sparse recovery over continuous dictionaries, the computational aspects largely remain unclear. By establishing various convergence results as the discretization gets finer, we promote discretization as a universal and effective way to approximately solve the atomic norm minimization problem, especially when the dimension of the parameter space is low.