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.

本研究通过低秩泊松张量模型和神经网络学习算法，在理论上支持了未完全统计的多方面数据完成，提出了可以证明从部分观测中恢复张量的完全统计条目及其每个条目的欠计数概率的 UC-TC 公式，并进行了模拟和实际数据实验。

神经属性融合的低计数张量完成

Under-Counted Tensor Completion with Neural Incorporation of Attributes

We show that the standard stochastic gradient decent (SGD) algorithm is
guaranteed to learn, in polynomial time, a function that is competitive with
the best function in the conjugate kernel space of the network, as defined in
Daniely, Frostig and Singer. The result holds for log-depth networks from a
rich family of architectures. To the best of our knowledge, it is the first
polynomial-time guarantee for the standard neural network learning algorithm
for networks of depth more that two.
As corollaries, it follows that for neural networks of any depth between $2$
and $\log(n)$, SGD is guaranteed to learn, in polynomial time, constant degree
polynomials with polynomially bounded coefficients. Likewise, it follows that
SGD on large enough networks can learn any continuous function (not in
polynomial time), complementing classical expressivity results.

本文证明标准随机梯度下降算法保证在多项式时间内学习到与神经网络的共轭核空间最优函数相竞争的函数； 对于深度在 2 到 log (n) 之间的神经网络，SGD 保证多项式时间内学习到多项式系数多项式边界的常微商； 大型网络上的 SGD 可以学习连续函数。