Zero-Truncated Poisson Regression for Sparse Multiway Count Data Corrupted by False Zeros

TL;DR

This paper introduces a zero-truncated Poisson regression method for multiway count data with false zeros, leveraging tensor completion and low-rank structure to achieve accurate inference despite data corruption.

Contribution

It proposes a novel zero-truncated Poisson approach combined with tensor completion for sparse, corrupted multiway count data, providing theoretical guarantees.

Findings

Accurately estimates Poisson tensor parameters with fewer non-zero counts.

Quantifies error introduced by zero-truncation under bounded parameters.

Demonstrates effectiveness through numerical experiments.

Abstract

We propose a novel statistical inference methodology for multiway count data that is corrupted by false zeros that are indistinguishable from true zero counts. Our approach consists of zero-truncating the Poisson distribution to neglect all zero values. This simple truncated approach dispenses with the need to distinguish between true and false zero counts and reduces the amount of data to be processed. Inference is accomplished via tensor completion that imposes low-rank tensor structure on the Poisson parameter space. Our main result shows that an $N$-way rank-$R$ parametric tensor $\boldsymbol{\mathscr{M}}\in(0,\infty)^{I\times \cdots\times I}$ generating Poisson observations can be accurately estimated by zero-truncated Poisson regression from approximately $IR^2\log_2^2(I)$ non-zero counts under the nonnegative canonical polyadic decomposition. Our result also quantifies the error made by zero-truncating the Poisson distribution when the parameter is uniformly bounded from below. Therefore, under a low-rank multiparameter model, we propose an implementable approach guaranteed to achieve accurate regression in under-determined scenarios with substantial corruption by false zeros. Several numerical experiments are presented to explore the theoretical results.