Deterministic Sampling of Multivariate Densities based on Projected Cumulative Distributions

TL;DR

This paper introduces a novel deterministic sampling method for multivariate densities using projected cumulative distributions, enabling efficient approximation by leveraging Radon transforms and one-dimensional sorting.

Contribution

It proposes a new approach employing projections and cumulative distributions to approximate multivariate densities deterministically, improving efficiency and simplicity over existing methods.

Findings

Efficient computation mainly involves one-dimensional sorting.

The method provides a tractable distance measure for multivariate densities.

The approach is simple to implement and effective for approximation.

Abstract

We want to approximate general multivariate probability density functions by deterministic sample sets. For optimal sampling, the closeness to the given continuous density has to be assessed. This is a difficult challenge in multivariate settings. Simple solutions are restricted to the one-dimensional case. In this paper, we propose to employ one-dimensional density projections. These are the Radon transforms of the densities. For every projection, we compute their cumulative distribution function. These Projected Cumulative Distributions (PCDs) are compared for all possible projections (or a discrete set thereof). This leads to a tractable distance measure in multivariate space. The proposed approximation method is efficient as calculating the distance measure mainly entails sorting in one dimension. It is also surprisingly simple to implement.