Volume of Hypercubes Clipped by Hyperplanes
Project description
hyperclip
This Python 3.5+ package implements volume computation of hypercubes clipped by hyperplanes. All methods implemented here have been proposed by Yunhi Cho and Seonhwa Kim (2020) in the article Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities. An arxiv paper is available here. The documentation is available on Read the Doc.
Installation
Hyperclip is available through PyPI, and may be installed using pip
:
pip install hyperclip
Introduction
The package is essentially composed by two classes : hyperclip.Hyperplane
and hyperclip.Hyperclip
.
hyperclip.Hyperplane
allows users to create a n-dimensional hyperplane defined asa.x + r \geq 0
. It is possible to directly seta
andr
or to providen
distinct points which belongs to the hyperplane, i.ea.x + r = 0
.hyperclip.Hyperclip
allows users to create an hyperclip object. It aims to compute the volume ofA.X+R \leq 0
forX
inside the uniform hypercube[0,1]^n
. It is possible to directly setA
andR
or to set a list ofhyperclip.Hyperplane
objects.
Example code
Here's an example showing the usage of hyperclip.Hyperclip
for a 2-dimensional case.
The result provided by Hyperclip is compared to a MonteCarlo volume estimation.
import numpy as np
import hyperclip
from matplotlib import pyplot as plt
n = 2
m = 3
np.random.seed(29)
hyperplanes = [hyperclip.Hyperplane().set_by_points(np.random.random((n,n))) for i_m in range(m)]
np.random.seed(None)
X = np.random.random((10**6,n))
id_pos_side = np.ones(X.shape[0])
for hyp in hyperplanes:
id_pos_side = np.all((id_pos_side, hyp.side(X)), axis=0)
fig, axs = plt.subplots()
axs.set_aspect('equal', 'box')
plt.scatter(X[id_pos_side, 0], X[id_pos_side, 1], s=2, color='gray')
for hyp in hyperplanes:
sol = hyp.compute_n_solutions()
x_a, y_a, x_b, y_b = sol.flat
a = (y_b-y_a)/(x_b-x_a)
b = y_a - x_a * a
y_0 = b
y_1 = a * 1 + b
plt.plot([0, 1], [y_0, y_1])
hc = hyperclip.Hyperclip().set_hyperplanes(hyperplanes)
vol = hc.volume()
plt.text(0.25,0.2, "10**6 MonteCarlo : "+str(round(id_pos_side.mean(),4)))
plt.text(0.25,0.1, "Hyperclip : "+str(round(vol,4)))
plt.xlim([0,1])
plt.ylim([0,1])
plt.show()
Contact
Please, send me an email at francois-remi.mazy@inria.fr.
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