  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.
