# On the Global-Local Dichotomy in Sparsity Modeling

arXiv: https://arxiv.org/abs/1702.03446 · HTML: https://mlsanity.com/paper/1702.03446/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2017.

## Built on by (most cited first, 1 of 1)

- [Patch Craft: Video Denoising by Deep Modeling and Patch Matching](https://mlsanity.com/paper/2103.13767/) (2021)

## Abstract (arXiv)

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.