generate random unit vectors

geometrystatistics

How

• Draw from a standard distribution for each dimension.
• Do not draw from a uniform distribution.
  • Or will need to apply Box-Muller transform

Implementation

import numpy as np
import torch
 
# numpy
v = np.random.normal(0, 1, size)
 
# torch
v = torch.normal(torch.tensor([0.0] * size), torch.tensor([1.0] * size))

Why that works

• Considering a 2D space, the probability of a point being at $(x,y)$ is $P(x)P(y)$
• The PDF for the standard distribution is $P(x)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{x}^2}$, which is roughly in the form of $e^{x^2}$
• Thus $P(x)P(y)$ is in the form of $e^{x^2+y^2}$, which is a function of the distance to the origin
• So the resulting distribution is radially symmetric around the origin

An intuitive explanation of why uniform sampling don’t work

• Considering a 2D space, uniformly sampling from both dimension results in a square population
• Whereas a random distribution in 2D space should be a circle
• This is called the corner effect