Description:

simplexers 1.0.0

Euclidean Projections onto Positive and Capped Simplices















About |
Installation |
Dependencies |
Attribution |
Contributions |
Issues |
License |
Acknowledgements

About
The s-capped simplex is defined as:
\Delta_{s}^{=} := \{\mathbf{x} \in \mathbb{R}^{n} \mid \mathbf{x}^T\mathbf{1} = s,
\mathbf{0} \leq \mathbf{x} \leq \mathbf{1} \}




Geometrically, the simplex is a slice at xT1=s of
a hypercube 0≤x≤1 shown by the blue
region in the image.
Projecting a vector onto a simplex is an important subproblem that appears in
imaging, statistics, and machine learning applications 1 2 3.
The projection of vector y onto the simplex amounts to finding a
vector x∗ that lives in the blue (feasible) region that is closest to y. This vector is
the shadow of y. Formally, this projection is
written as:
\mathbf{x}^* = proj_{\Delta_{s}^{=}}\left(\mathbf{y}\right) = \underset{x}{\mathrm{argmin}}\{
\frac{1}{2}\|\mathbf{x} - \mathbf{y}\|^2 \mid \mathbf{x} \in
\Delta_{s}^{=}\}


This software computes the projection of vectors onto s-capped and positive
simplices (simplices where x∗'s components can be > 1) using sorting methods and fast root
finding methods of the Lagrangian's critical points.
2
4.

Example
import numpy as np
from simplexers import capped, positive

rng = np.random.default_rng()
x = rng.uniform(0, 3, size=(4, 100))

# construct capped and positive projections
capped_projection = capped.capped_simplexer(x, s=1, axis=-1)
positive_projection = positive.positive_simplexer(x, s=1, axis=-1)

# validate the sum of each of the 4 vectors is 1
print(np.allclose(np.sum(capped_projection, axis=-1), 1))
print(np.allclose(np.sum(positive_projection, axis=-1), 1))

Installation
Simplexers is available on pypi for easy intallation into virtual environments.
Python Virtual Environment

Create your virtual environment, Here we name it my_venv.

$ python3 -m venv my_venv


Activate your 'my_venv' environment

$ source my_venv/bin/activate


Install openseize into your virtual environment

(my_venv)$ pip install simplexers

Conda Virtual Environment


Download the simplexers environment configuration yaml


Create a conda simplexers environment.


$ conda env create --file environment.yml


Activate the simplexers environment.

$ conda activate simplexers

From Source
To get the development version:

Create a virtual environment with latest pip version.

$ python3 -m venv env
$ source env/bin/activate
$ pip install --upgrade pip


Get the source code

$ git clone https://github.com/mscaudill/simplexers.git


CD into the directory containing the pyproject.toml and create an
editable install with pip using the development dependencies

$ pip install -e .[dev]

Dependencies
Simplexers requires Python ≥ 3.10 and has the
following dependencies:


package
pypi
conda


numpy
https://pypi.org/project/numpy/
✓


scipy
https://pypi.org/project/scipy/
✓


ipython
https://pypi.org/project/ipython/
✓


notebook
https://pypi.org/project/jupyter/
✓


Attribution
Please see the Cite this repository under the About section or the citation
file.
Contributions
Contributions are what makes open-source fun and we would love for you to
contribute. Please check out our contribution guide
to get started.
Issues
Simplexers provides custom issue templates for filing bugs, requesting
feature enhancements, suggesting documentation changes, or just asking
questions. You can file an issue here.
License
Simplexers is licensed under the terms of the 3-Clause BSD License.
Acknowledgements
This work is generously supported through the Ting Tsung and Wei Fong Chao
Foundation

License

For personal and professional use. You cannot resell or redistribute these repositories in their original state.

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