# Generalizing CoSaMP to Signals from a Union of Low Dimensional Linear Subspaces

## TL;DR

This paper extends the CoSaMP algorithm to recover signals from a union of low-dimensional subspaces, providing theoretical guarantees and a new greedy reconstruction method, supported by experimental validation.

## Contribution

It generalizes CoSaMP for union of subspaces using Gaussian mean width and introduces a new greedy algorithm inspired by this theory.

## Key findings

- The generalized CoSaMP guarantees recovery under Gaussian measurements.
- The new greedy method performs well in combined sparse-synthesis and cosparse-analysis models.
- Experimental results validate the effectiveness of the proposed approach.

## Abstract

The idea that signals reside in a union of low dimensional subspaces subsumes many low dimensional models that have been used extensively in the recent decade in many fields and applications. Until recently, the vast majority of works have studied each one of these models on its own. However, a recent approach suggests providing general theory for low dimensional models using their Gaussian mean width, which serves as a measure for the intrinsic low dimensionality of the data. In this work we use this novel approach to study a generalized version of the popular compressive sampling matching pursuit (CoSaMP) algorithm, and to provide general recovery guarantees for signals from a union of low dimensional linear subspaces, under the assumption that the measurement matrix is Gaussian. We discuss the implications of our results for specific models, and use the generalized algorithm as an inspiration for a new greedy method for signal reconstruction in a combined sparse-synthesis and cosparse-analysis model. We perform experiments that demonstrate the usefulness of the proposed strategy.