Python library implementing proximal operators to allow solving non-smooth, constrained convex problems with proximal algorithms.

# PyProximal

:vertical_traffic_light: :vertical_traffic_light: This library is under early development. Expect things to constantly change until version v1.0.0. :vertical_traffic_light: :vertical_traffic_light:

## Objective

This Python library provides all the needed building blocks for solving non-smooth convex optimization problems using the so-called proximal algorithms.

Whereas gradient based methods are first-order iterative optimization algorithms for solving unconstrained, smooth optimization problems, proximal algorithms can be viewed as an analogous tool for non-smooth and possibly constrained versions of these problems. Such algorithms sit at a higher level of abstraction than classical algorithms like Steepest descent or Newton’s method and require a basic operation to be performed at each iteration: the evaluation of the so-called proximal operator of the functional to be optimized.

Whilst evaluating a proximal operator does itself require solving a convex optimization problem, these subproblems often admit closed form solutions or can be solved very quickly with ad-hoc specialized methods. Several of such proximal operators are therefore implemented in this library.

Here is a simple example showing how to compute the proximal operator of the L1 norm of a vector:

```import numpy as np
from pyproximal import L1

l1 = L1(sigma=1.)
x = np.arange(-5, 5, 0.1)
xp = l1.prox(x, 1)
```

and how this can be used to solve a basic denoising problem of the form: `min ||x - y||_2^2 + ||Dx||_1`:

```import numpy as np
from pylops import FirstDerivative
from pyproximal import L1, L2

np.random.seed(1)

# Create noisy data
nx = 101
x = np.zeros(nx)
x[:nx//2] = 10
x[nx//2:3*nx//4] = -5
n = np.random.normal(0, 2, nx)
y = x + n

# Define functionals
l2 = L2(b=y)
l1 = L1(sigma=5.)
Dop = FirstDerivative(nx, edge=True, kind='backward')

L = np.real((Dop.H * Dop).eigs(neigs=1, which='LM'))
tau = 1.
mu = 0.99 * tau / L
x0=np.zeros_like(x), niter=200)
```

## Why another library for proximal algorithms?

Several other projects in the Python ecosystem provide implementations of proximal operators and/or algorithms, which present some clear overlap with this project.

A (possibly not exahustive) list of other projects is:

All of these projects are self-contained, meaning that they implement both proximal and linear operators as needed to solve a variety of problems in different areas of science.

The main difference with PyProximal lies in the fact that we decide not to intertangle linear and proximal operators within the same library. We leverage the extensive set of linear operators provided by the PyLops project and focus only on the proximal part of the problem. This makes the codebase more concise, and easier to understand and extend. Moreover many of the problems that are solved in PyLops can now be also solved by means of proximal algorithms!

## Project structure

This repository is organized as follows:

• pyproximal: python library containing various orthogonal projections, proximial operators, and solvers
• pytests: set of pytests
• testdata: sample datasets used in pytests and documentation
• docs: sphinx documentation
• examples: set of python script examples for each proximal operator to be embedded in documentation using sphinx-gallery
• tutorials: set of python script tutorials to be embedded in documentation using sphinx-gallery

## Getting started

You need Python 3.6 or greater.

#### From PyPi

If you want to use PyProximal within your codes, install it in your Python environment by typing the following command in your terminal:

``````pip install pyproximal
``````

Open a python terminal and type:

``````import pyproximal
``````

#### From Github

You can also directly install from the master node (although this is not reccomended)

``````pip install git+https://git@github.com/PyLops/pyproximal.git@master
``````

## Contributing

Feel like contributing to the project? Adding new operators or tutorial?

We advise using the Anaconda Python distribution to ensure that all the dependencies are installed via the `Conda` package manager. Follow the following instructions and read carefully the CONTRIBUTING file before getting started.

### 1. Fork and clone the repository

Execute the following command in your terminal:

``````git clone https://github.com/your_name_here/pyproximal.git
``````

### 2. Install PyLops in a new Conda environment

To ensure that further development of PyLops is performed within the same environment (i.e., same dependencies) as that defined by `requirements-dev.txt` or `environment-dev.yml` files, we suggest to work off a new Conda enviroment.

The first time you clone the repository run the following command:

``````make dev-install_conda
``````

To ensure that everything has been setup correctly, run tests:

``````make tests
``````

Make sure no tests fail, this guarantees that the installation has been successfull.

Remember to always activate the conda environment every time you open a new terminal by typing:

``````source activate pyproximal
``````

## Documentation

The official documentation of PyProximal is available here.

Moreover, if you have installed PyProximal using the developer environment you can also build the documentation locally by typing the following command:

``````make doc
``````

Once the documentation is created, you can make any change to the source code and rebuild the documentation by simply typing

``````make docupdate
``````

Note that if a new example or tutorial is created (and if any change is made to a previously available example or tutorial) you are required to rebuild the entire documentation before your changes will be visible.

## Contributors

• Matteo Ravasi, mrava87
• Nick Luiken, NickLuiken

## Project details

This version 0.1.0 0.0.0