README.md

Safe screening rules for the identification of zeros in the solution of the Slope problem

This repository contains numerical procedure to evaluate the solution of the SLOPE problem with / without safe screening [1]. More precisely, the SLOPE problem is defined as

\text{Find }
\mathbf{x}^\star\in\underset{\mathbf{x}\in\mathbb{R}^n}{\arg\min}\;
\frac{1}{2}\left\Vert 
	\mathbf{y} - \mathbf{A}\mathbf{x}
\right\Vert_2^2
+
\lambda\sum_{k=1}^n \gamma_k\vert\mathbf{x}\vert_{[k]}

where

• \mathbf{x}\in\mathbb{R}^m is the observation vector
• \mathbf{A}\in\mathbb{R}^{m\times n} is the so-called dictionary
• \lambda>0 a (positive) scalar
• \{\gamma_k\}_{k=1}^n a sequence of non-increasing nonnegative scalars such that \gamma_1=1,
• \vert\mathbf{x}\vert_{[k]} denotes the kth largest entry of \mathbf{x} in absolute value

[1] Clément Elvira, Cédric Herzet: “Safe rules for the identification of zeros in the solution of the Slope problem”, arXiv, septembre 2021; arXiv:1911.07508

The above paper contains theoretical results and several applications that can be reproduced with this toolbox.

This python toolbox is currently under development and is hosted on Gitlab. If you encounter a bug or something unexpected please let me know by raising an issue on the project page or by contacting me by mail.

Requirements

This toolbox works with python 3.5+.

Dependencies:

• NumPy
• SciPy
• Matplotlib

Install from sources

1. Clone the repository

git clone https://gitlab-research.centralesupelec.fr/2020elvirac/slope-screening

2. Enter the folder

cd slope-squeezing

3. (Optional) Create a virtual environment and activate it

virtualenv venv -p python3
source venv/bin/activate

4. Install the dependencies