Introduction

BodyWalk is a python library implementing popular random walk techniques for sampling uniformly over convex bodies. More precisely, this package offers the following functionalities:

1. Efficient implementation of several random walk algorithms, including Ball Walk, Hit-and-Run, and Billiard Walk;
2. Support for general convex bodies, including rectangles, balls, and polytopes;
3. Implementation of rounding techniques for improved mixing time.

Example

from bodywalk.sampling import ball_walk, billiard_walk, hit_and_run
from bodywalk.geometry import Polytope

convex_body = Polytope([[1, 0], [-1, 0], [0, 1], [0, -1]], [0.5]*4)  # Square of side 1 centered at (0, 0)
initial_sample = [0, 0]  # Initial point to start the Markov Chain
random_state = 42  # RNG seed

chain = ball_walk(convex_body, initial_sample, delta=0.5)
# chain = billiard_walk(convex_body, initial_sample, tau=0.5)
# chain = hit_and_run(convex_body, initial_sample)

for sample in chain.generate(random_state):
  # process the samples one-by-one

# generate a batch of 10 samples, ignoring the first 100
samples = chain.sample(n=10, warmup=100, random_state=random_state)

Dependencies

This module was tested using the following dependencies:

• Python >= 3.9
• Numpy >= 1.21.4
• Pytest >= 6.2.5

License

This project is released under the BSD-3 Clause license.

References

[1] L. Lovász. An Algorithmic Theory of Numbers, Graphs and Convexity, volume 50 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, 1986. ISBN 9781611970203.

[2] L. Lovász. Hit-and-run mixes fast. Mathematical Programming, 86(3):443–461, 1999. ISSN 0025-5610. doi: 10.1007/s101070050099.

[3] B.T. Polyak and E.N. Gryazina. Billiard walk-a new sampling algorithm for control and optimization. IFAC Proceedings Volumes, 47(3):6123-6128, 2014.