High resolution sparse estimation of exponentially decaying two-dimensional signals

In this work, we consider the problem of high-resolution estimation of the parameters detailing a two-dimensional (2-D) signal consisting of an unknown number of exponentially decaying sinusoidal components. Interpreting the estimation problem as a block (or group) sparse representation problem allows the decoupling of the 2-D data structure into a sum of outer-products of 1-D damped sinusoidal signals with unknown damping and frequency. The resulting non-zero blocks will represent each of the 1-D damped sinusoids, which may then be used as non-parametric estimates of the corresponding 1-D signals; this implies that the sought 2-D modes may be estimated using a sequence of 1-D optimization problems. The resulting sparse representation problem is solved using an iterative ADMM-based algorithm, after which the damping and frequency parameter can be estimated by a sequence of simple 1-D optimization problems

Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing

This paper focuses on the estimation of low-complexity signals when they are observed through $M$ uniformly quantized compressive observations. Among such signals, we consider 1-D sparse vectors, low-rank matrices, or compressible signals that are well approximated by one of these two models. In this context, we prove the estimation efficiency of a variant of Basis Pursuit Denoise, called Consistent Basis Pursuit (CoBP), enforcing consistency between the observations and the re-observed estimate, while promoting its low-complexity nature. We show that the reconstruction error of CoBP decays like $M^{-1/4}$ when all parameters but $M$ are fixed. Our proof is connected to recent bounds on the proximity of vectors or matrices when (i) those belong to a set of small intrinsic "dimension", as measured by the Gaussian mean width, and (ii) they share the same quantized (dithered) random projections. By solving CoBP with a proximal algorithm, we provide some extensive numerical observations that confirm the theoretical bound as $M$ is increased, displaying even faster error decay than predicted. The same phenomenon is observed in the special, yet important case of 1-bit CS.Comment: Keywords: Quantized compressed sensing, quantization, consistency, error decay, low-rank, sparsity. 10 pages, 3 figures. Note abbout this version: title change, typo corrections, clarification of the context, adding a comparison with BPD

Deep Signal Recovery with One-Bit Quantization

Machine learning, and more specifically deep learning, have shown remarkable performance in sensing, communications, and inference. In this paper, we consider the application of the deep unfolding technique in the problem of signal reconstruction from its one-bit noisy measurements. Namely, we propose a model-based machine learning method and unfold the iterations of an inference optimization algorithm into the layers of a deep neural network for one-bit signal recovery. The resulting network, which we refer to as DeepRec, can efficiently handle the recovery of high-dimensional signals from acquired one-bit noisy measurements. The proposed method results in an improvement in accuracy and computational efficiency with respect to the original framework as shown through numerical analysis.Comment: This paper has been submitted to the 44th International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2019

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.Comment: ISIT 201

A simultaneous sparse approximation method for multidimensional harmonic retrieval

International audienceIn this paper, a new method for the estimation of the parameters of multidimensional (R-D) harmonic and damped complex signals in noise is presented. The problem is formulated as R simultaneous sparse approximations of multiple 1-D signals. To get a method able to handle large size signals while maintaining a sufficient resolution, a multigrid dictionary refinement technique is associated to the simultaneous sparse approximation. The refinement procedure is proved to converge in the single R-D mode case. Then, for the general multiple modes case, the signal tensor model is decomposed in order to handle each mode separately in an iterative scheme. The proposed method does not require an association step since the estimated modes are automatically "paired". We also derive the Cramér-Rao lower bounds of the parameters of modal R-D signals. The expressions are given in compact form in the single tone case. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed method

Estimating Sparse Signals Using Integrated Wideband Dictionaries

In this paper, we introduce a wideband dictionary framework for estimating sparse signals. By formulating integrated dictionary elements spanning bands of the considered parameter space, one may efficiently find and discard large parts of the parameter space not active in the signal. After each iteration, the zero-valued parts of the dictionary may be discarded to allow a refined dictionary to be formed around the active elements, resulting in a zoomed dictionary to be used in the following iterations. Implementing this scheme allows for more accurate estimates, at a much lower computational cost, as compared to directly forming a larger dictionary spanning the whole parameter space or performing a zooming procedure using standard dictionary elements. Different from traditional dictionaries, the wideband dictionary allows for the use of dictionaries with fewer elements than the number of available samples without loss of resolution. The technique may be used on both one- and multi-dimensional signals, and may be exploited to refine several traditional sparse estimators, here illustrated with the LASSO and the SPICE estimators. Numerical examples illustrate the improved performance