Convexity in source separation: Models, geometry, and algorithms

Source separation or demixing is the process of extracting multiple components entangled within a signal. Contemporary signal processing presents a host of difficult source separation problems, from interference cancellation to background subtraction, blind deconvolution, and even dictionary learning. Despite the recent progress in each of these applications, advances in high-throughput sensor technology place demixing algorithms under pressure to accommodate extremely high-dimensional signals, separate an ever larger number of sources, and cope with more sophisticated signal and mixing models. These difficulties are exacerbated by the need for real-time action in automated decision-making systems. Recent advances in convex optimization provide a simple framework for efficiently solving numerous difficult demixing problems. This article provides an overview of the emerging field, explains the theory that governs the underlying procedures, and surveys algorithms that solve them efficiently. We aim to equip practitioners with a toolkit for constructing their own demixing algorithms that work, as well as concrete intuition for why they work

Settling-time improvement in global convergence lagrangian networks

In this brief, a modification of Lagrangian networks given in (Xia Y., 2003) is presented. This modification improves the settling time of the convergence of Lagrangian networks to a stationary point; which is the optimal solution to the nonlinear convex programming problem with linear equality constraints. This is important because, in many real-time applications where Lagrangian networks are used to find an optimal solution, such as in signal and image processing, this settling time is interpreted as the processing time. Simulation results applied to a quadratic optimization problem show that settling time is improved from about to 2000 to 20 seconds. Lyapunov theory was used to obtain our main result.Postprint (published version

1 A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal

Abstract—Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured “noises”. As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by wave-field bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for time-varying adaptive filtering of seismic signals, using sparse representations, based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimony-promoting wavelet frames. The designed primal-dual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters. The approach demonstrates significantly good performance in low signal-to-noise ratio conditions, both for simulated and real field seismic data. Index Terms—Convex optimization, Parallel algorithms, Wavelet transforms, Adaptive filters, Geophysical signal processing, Signal restoration, Sparsity, Signal separation

Joint Optimization of Waveform Covariance Matrix and Antenna Selection for MIMO Radar

In this paper, we investigate the problem of jointly optimizing the waveform covariance matrix and the antenna position vector for multiple-input-multiple-output (MIMO) radar systems to approximate a desired transmit beampattern as well as to minimize the cross-correlation of the received signals reflected back from the targets. We formulate the problem as a non-convex program and then propose a cyclic optimization approach to efficiently tackle the problem. We further propose a novel local optimization framework in order to efficiently design the corresponding antenna positions. Our numerical investigations demonstrate a good performance both in terms of accuracy and computational complexity, making the proposed framework a good candidate for real-time radar signal processing applications.Comment: This paper is accepted for publication in the 2019 IEEE Asilomar Conference on Signals, Systems, and Computers (Asilomar 2019

Designing Gabor windows using convex optimization

Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal l2-norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good time-frequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler-Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found