# Atomic decomposition via polar alignment: the geometry of structured optimization

Foundations and Trends in Optimization, 3(4):280–366, 2020

Foundations and Trends in Optimization, 3(4):280–366, 2020

Structured optimization uses a prescribed set of atoms to assemble a solution that fits a model to data. Polarity, which extends the familiar notion of orthogonality from linear sets to general convex sets, plays a special role in a simple and geometric form of convex duality. This duality correspondence yields a general notion of alignment that leads to an intuitive and complete description of how atoms participate in the final decomposition of the solution. The resulting geometric perspective leads to variations of existing algorithms effective for large-scale problems. We illustrate these ideas with many examples, including applications in matrix completion and morphological component analysis for the separation of mixtures of signals.

```
@article{2019PolarAlignment,
url = {http://dx.doi.org/10.1561/2400000028},
year = {2020},
volume = {3},
journal = {Foundations and Trends in Optimization},
title = {Atomic Decomposition via Polar Alignment: The Geometry of Structured Optimization},
doi = {10.1561/2400000028},
issn = {2167-3888},
number = {4},
pages = {280-366},
author = {Zhenan Fan and Halyun Jeong and Yifan Sun and Michael P. Friedlander}
}
```