# On the Global-Local Dichotomy in Sparsity Modeling

## TL;DR

This paper addresses the gap between local sparsity models and global signal modeling by proposing a method to construct a global model from local assumptions, ensuring unique and stable recovery.

## Contribution

It introduces a framework to derive global signal representations from local sparsity assumptions using constrained linear systems and modern optimization techniques.

## Key findings

- Global models can be constructed from local sparsity assumptions.
- Conditions for unique and stable recovery are established.
- Numerical experiments support the theoretical results.

## Abstract

The traditional sparse modeling approach, when applied to inverse problems with large data such as images, essentially assumes a sparse model for small overlapping data patches. While producing state-of-the-art results, this methodology is suboptimal, as it does not attempt to model the entire global signal in any meaningful way - a nontrivial task by itself. In this paper we propose a way to bridge this theoretical gap by constructing a global model from the bottom up. Given local sparsity assumptions in a dictionary, we show that the global signal representation must satisfy a constrained underdetermined system of linear equations, which can be solved efficiently by modern optimization methods such as Alternating Direction Method of Multipliers (ADMM). We investigate conditions for unique and stable recovery, and provide numerical evidence corroborating the theory.