# Greedy Signal Space Recovery Algorithm with Overcomplete Dictionaries in Compressive Sensing

## TL;DR

This paper introduces a greedy signal space recovery algorithm called SSSP for signals sparse in overcomplete dictionaries, providing theoretical guarantees and demonstrating superior performance over existing methods.

## Contribution

It develops a generalized iterative recovery algorithm for overcomplete dictionaries using D-RIP, with proven convergence and improved accuracy in noisy and noise-free settings.

## Key findings

- Algorithm outperforms existing recovery methods
- Provides theoretical bounds on measurement requirements
- Demonstrates high accuracy and low complexity

## Abstract

Compressive Sensing (CS) is a new paradigm for the efficient acquisition of signals that have sparse representation in a certain domain. Traditionally, CS has provided numerous methods for signal recovery over an orthonormal basis. However, modern applications have sparked the emergence of related methods for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary, particularly due to their potential to generate different kinds of sparse representation of signals. To this end, we apply a signal space greedy method, which relies on the ability to optimally project a signal onto a small number of dictionary atoms, to address signal recovery in this setting. We describe a generalized variant of the iterative recovery algorithm called Signal space Subspace Pursuit (SSSP) for this more challenging setting. Here, using the Dictionary-Restricted Isometry Property (D-RIP) rather than classical RIP, we derive a low bound on the number of measurements required and then provide the proof of convergence for the algorithm. The algorithm in noisy and noise-free measurements has low computational complexity and provides high recovery accuracy. Simulation results show that the algorithm outperforms best compared with the existing recovery algorithms.