Monte Carlo computation of \(L_p\) distance between two densities on the unit hypersphere

notes
geometric statistics
computation
Author

Kisung You

Published

August 9, 2022

Problem Statement

A \(d\)-dimensional unit hypersphere \(\mathbb{S}^d = \lbrace x \in \mathbb{R}^{d+1}~|~ \|x\|_2^2 = \sum_{i=1}^{d+1} x_i^2 = 1\rbrace\) is one of the standard mathematical spaces in the field of objected-oriented data analysis (Marron and Dryden 2021). Let \(\mathcal{P}(\mathbb{S}^d)\) denote a space of probability densities on \(\mathbb{S}^d\). For two densities \(f,g\in\mathcal{P}(\mathbb{S}^d)\), it is frequently needed to measure dissimilarity between the two. Unfortunately, even for the most well-known distributions on the hypersphere, analytic formula for any discrepancy measure is rarely available, leading to require numerical schemes for approximation. Here we focus on \(L_p\) distance between the two densities,

\[ L_p (f,g) = \left( \int_{\mathbb{S}^d} |f(x) - g(x)|^p \right)^{1/p} \tag{1}\]

and we show how to combine Monte Carlo way of integration by means of importance sampling to approximate Equation 1.

Computation

Importance sampling requires a proposal density. The easiest choice is to use uniform density \(u(x)\) as an importance proposal since sampling from \(u(x)\) is trivial. First, take a random sample from standard normal distribution \(x \sim \mathcal{N}(0,I)\) in \(\mathbb{R}^{d+1}\). Then, the rest is to take \(L_2\) normalization, i.e., \(x \leftarrow x / \|x\|_2\), which makes a sampled vector to have a unit norm. Given the sample generation process, we have the following

\[\begin{aligned} L_p (f,g)^p &= \int_{\mathbb{S}^d} |f(x)-g(x)|^p dx \\ &= \int_{\mathbb{S}^d} \frac{|f(x)-g(x)|^p}{u(x)} u(x) dx \\ &= \mathbb{E}_{u(x)} \left\lbrack \frac{|f(x)-g(x)|^p}{u(x)} \right\rbrack\\ &\approx \frac{1}{N} \sum_{n=1}^N \frac{|f(x)-g(x)|^p}{u(x)} \,\,\textrm{for}\,\, x_n \overset{iid}{\sim} u(x), \end{aligned} \] where the last term gets better approximation as \(N\rightarrow \infty\).

References

Marron, James Stephen, and I. L. Dryden. 2021. Object Oriented Data Analysis. Boca Raton: Taylor & Francis Group, LLC.

Citation

BibTeX citation:
@online{you2022,
  author = {Kisung You},
  title = {Monte {Carlo} Computation of {\$L\_p\$} Distance Between Two
    Densities on the Unit Hypersphere},
  date = {2022-08-09},
  url = {https://kisungyou.com/posts/note001-spherical-distance},
  langid = {en}
}
For attribution, please cite this work as:
Kisung You. 2022. “Monte Carlo Computation of $L_p$ Distance Between Two Densities on the Unit Hypersphere.” August 9, 2022. https://kisungyou.com/posts/note001-spherical-distance.