# Linearly Constrained Smoothing Group Sparsity Solvers in Off-grid Model

## TL;DR

This paper develops efficient algorithms for off-grid DoA estimation in compressed sensing, addressing matrix perturbations with various optimization formulations and convergence analyses.

## Contribution

It introduces novel group-sparsity solvers using ADMM, Nesterov smoothing, and primal-dual methods tailored for off-grid model perturbations.

## Key findings

- Algorithms demonstrate high accuracy in numerical simulations.
- Proposed methods converge efficiently with reduced computational cost.
- Effective handling of off-grid effects in compressed sensing scenarios.

## Abstract

In compressed sensing, the sensing matrix is assumed perfectly known. However, there exists perturbation in the sensing matrix in reality due to sensor offsets or noise disturbance. Directions-of-arrival (DoA) estimation with off-grid effect satisfies this situation, and can be formulated into a (non)convex optimization problem with linear inequalities constraints, which can be solved by the interior point method (using the CVX tools), but at a large computational cost. In this work, in order to design efficient algorithms, we consider various alternative formulations, such as unconstrained formulation, primal-dual formulation, or conic formulation to develop group-sparsity promoted solvers. First, the consensus alternating direction method of multipliers (C-ADMM) is applied. Then, iterative algorithms for the BPDN formulation is proposed by combining the Nesterov smoothing technique with accelerated proximal gradient method, and the convergence analysis of the method is conducted as well. We also developed a variant of EGT (Excessive Gap Technique)-based primal-dual method to systematically reduce the smoothing parameter sequentially. Finally, we propose algorithms for quadratically constrained L2-L1 mixed norm minimization problem by using the smoothed dual conic optimization (SDCO) and continuation technique. The performance of accuracy and convergence for all the proposed methods are demonstrated in the numerical simulations.