  Systematic under-counting effects are observed in data collected across many
disciplines, e.g., epidemiology and ecology. Under-counted tensor completion
(UC-TC) is well-motivated for many data analytics tasks, e.g., inferring the
case numbers of infectious diseases at unobserved locations from under-counted
case numbers in neighboring regions. However, existing methods for similar
problems often lack supports in theory, making it hard to understand the
underlying principles and conditions beyond empirical successes. In this work,
a low-rank Poisson tensor model with an expressive unknown nonlinear side
information extractor is proposed for under-counted multi-aspect data. A joint
low-rank tensor completion and neural network learning algorithm is designed to
recover the model. Moreover, the UC-TC formulation is supported by theoretical
analysis showing that the fully counted entries of the tensor and each entry's
under-counting probability can be provably recovered from partial observations
-- under reasonable conditions. To our best knowledge, the result is the first
to offer theoretical supports for under-counted multi-aspect data completion.
Simulations and real-data experiments corroborate the theoretical claims.
