Badri Narayan Bhaskar

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.

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.

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.

Atomic norm denoising with applications to line spectral estimation

The sub-Nyquist estimation of line spectra is a classical problem in signal processing, but currently popular subspace-based techniques have few guarantees in the presence of noise and rely on a priori knowledge about system model order. Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectrum estimation that provides theoretical guarantees for the mean-square-error performance in the presence of noise and without advance knowledge of the model order. We propose an abstract theory of denoising with atomic norms which is specialized to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials with guaranteed bounds on the mean-squared-error. In general, our proposed optimization problem has no known polynomial time solution, but we provide an efficient algorithm, called DAST, based on the Fast Fourier Transform that achieves nearly the same error rate. We compare DAST with Cadzows canonical alternating projection algorithm, which performs marginally better under high signal-to-noise ratios when the model order is known exactly, and demonstrate experimentally that DAST outperforms other denoising techniques, including Cadzows, over a wide range of signal-to-noise ratios.

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.

Sketching Sparse Matrices, Covariances, and Graphs via Tensor Products

This paper considers the problem of recovering an unknown sparse p×p matrix X from an m×m matrix Y=AXBT, where A and B are known m×p matrices with m≪p. The main result shows that there exist constructions of the sketching matrices A and B so that even if X has O(p) nonzeros, it can be recovered exactly and efficiently using a convex program as long as these nonzeros are not concentrated in any single row/column of X. Furthermore, it suffices for the size of Y (the sketch dimension) to scale as m = O(√(# nonzeros in X) × log p). The results also show that the recovery is robust and stable in the sense that if X is equal to a sparse matrix plus a perturbation, then the convex program we propose produces an approximation with accuracy proportional to the size of the perturbation. Unlike traditional results on sparse recovery, where the sensing matrix produces independent measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require nonstandard techniques. Indeed, our approach relies on a novel result concerning tensor products of bipartite graphs, which may be of independent interest. This problem is motivated by the following application, among others. Consider a p×n data matrix D, consisting of n observations of p variables. Assume that the correlation matrix X:=DDT is (approximately) sparse in the sense that each of the p variables is significantly correlated with only a few others. Our results show that these significant correlations can be detected even if we have access to only a sketch of the data S=AD with A ∈ Rm×p .

Robust line spectral estimation

Line spectral estimation is a classical signal processing problem that finds numerous applications in array signal processing and speech analysis. We propose a robust approach for line spectral estimation based on atomic norm minimization that is able to recover the spectrum exactly even when the observations are corrupted by arbitrary but sparse outliers. The resulting optimization problem is reformulated as a semidefinite program. Our work extends previous work on robust uncertainty principles by allowing the frequencies to assume values in a continuum rather than a discrete set.

Near minimax line spectral estimation

Line spectral estimation is a classical signal processing problem involving estimation of frequencies and amplitudes from noisy equispaced samples of a sparse combination of complex sinusoids. We view this as a sparse recovery problem with a continuous, infinite dictionary, and employ tools from convex optimization for estimation. In this paper, we establish that using atomic norm soft thresholding (AST), we can achieve near minimax optimal prediction error rate for line spectral estimation, in spite of having a highly coherent dictionary corresponding to arbitrarily close frequencies. We also derive guarantees on the frequency localization performance of AST.

Multipath-cluster channel models

In multipath channels, paths often arrive in clusters. An early model that exhibits clustering was proposed by Saleh and Valenzuela in 1987, and since then, several extensions have been developed. Although these models are straightforward to simulate, little analysis of them has been carried out. Here, a generalized multipath-cluster channel model is proposed that subsumes the Saleh-Valenzuela model and its extensions as special cases. The proposed model provides a unified framework for the understanding and analysis of multipath-cluster channel models. The framework is used to derive formulas for the power-delay profile, excess-delay moments, and similar channel characteristics.