Recently, nonlinear ICA has surfaced as a popular alternative to the many
heuristic models used in deep representation learning and disentanglement. An
advantage of nonlinear ICA is that a sophisticated identifiability theory has
been developed; in particular, it has been proven that the original components
can be recovered under sufficiently strong latent dependencies. Despite this
general theory, practical nonlinear ICA algorithms have so far been mainly
limited to data with one-dimensional latent dependencies, especially
time-series data. In this paper, we introduce a new nonlinear ICA framework
that employs $t$-process (TP) latent components which apply naturally to data
with higher-dimensional dependency structures, such as spatial and
spatio-temporal data. In particular, we develop a new learning and inference
algorithm that extends variational inference methods to handle the combination
of a deep neural network mixing function with the TP prior, and employs the
method of inducing points for computational efficacy. On the theoretical side,
we show that such TP independent components are identifiable under very general
conditions. Further, Gaussian Process (GP) nonlinear ICA is established as a
limit of the TP Nonlinear ICA model, and we prove that the identifiability of
the latent components at this GP limit is more restricted. Namely, those
components are identifiable if and only if they have distinctly different
covariance kernels. Our algorithm and identifiability theorems are explored on
simulated spatial data and real world spatio-temporal data.

最近，非线性独立成分分析（nonlinear ICA）已经成为深度表示学习和特征解缠中许多启发式模型的热门替代方法之一。本文介绍了一种新的非线性 ICA 框架，采用适用于具有高维度依赖结构的数据的 $t$-process (TP) 潜在成分。我们发展了一种新的学习和推理算法，将变分推断方法扩展到将深度神经网络混合函数与 TP 先验结合起来，并采用诱导点的方法以提高计算效率。在理论方面，我们证明了这些 TP 独立成分在非常普遍的条件下是可识别的。此外，高斯过程（GP）非线性 ICA 被建立为 TP 非线性 ICA 模型的极限，并且我们证明了该 GP 极限下潜在成分的可识别性更受限制。也就是说，只有当这些成分具有不同的协方差核时，它们才是可识别的。我们的算法和可识别性定理在模拟空间数据和真实的时空数据上进行了探索。