The Poisson tensor completion parametric estimator

TL;DR

This paper introduces the Poisson tensor completion estimator, which leverages inter-sample relationships and Poisson process modeling to improve histogram-based density estimation for multivariate distributions.

Contribution

It presents a novel Poisson tensor decomposition approach that enhances histogram completion by modeling count data as a non-homogeneous Poisson process, ensuring non-negativity and improved accuracy.

Findings

Outperforms standard histogram estimators for sub-Gaussian distributions.

Guarantees non-negative estimates without additional constraints.

Utilizes inter-sample relationships for better density approximation.

Abstract

We introduce the Poisson tensor completion (PTC) estimator that exploits inter-sample relationships to compute a low-rank Poisson tensor decomposition of the frequency histogram for samples of a multivariate distribution. Our crucial observation is that the histogram bins are an instance of a space partitioning of counts and thus can be identified with a spatial non-homogeneous Poisson process. The Poisson tensor decomposition leads to a completion of the mean measure over all bins -- including those containing few to no samples -- and leads to our proposed estimator. A Poisson tensor decomposition models the underlying distribution of the count data and guarantees non-negative estimated values obviating the need for additional constraints to ensure non-negativity. Furthermore, we demonstrate that our PTC estimator is a substantial improvement over standard histogram-based estimators for sub-Gaussian probability distributions because of the concentration of norm phenomenon.