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Recently, nonlinear ICA has surfaced as a popular alternative to the manyheuristic models used in deep representation learning and disentanglement. Anadvantage of nonlinear ICA is that a sophisticated identifiability theory hasbeen developed; in particular, it has been proven that the original componentscan be recovered under sufficiently strong latent dependencies. Despite thisgeneral theory, practical nonlinear ICA algorithms have so far been mainlylimited to data with one-dimensional latent dependencies, especiallytime-series data. In this paper, we introduce a new nonlinear ICA frameworkthat employs $t$-process (TP) latent components which apply naturally to datawith higher-dimensional dependency structures, such as spatial andspatio-temporal data. In particular, we develop a new learning and inferencealgorithm that extends variational inference methods to handle the combinationof a deep neural network mixing function with the TP prior, and employs themethod of inducing points for computational efficacy. On the theoretical side,we show that such TP independent components are identifiable under very generalconditions. Further, Gaussian Process (GP) nonlinear ICA is established as alimit of the TP Nonlinear ICA model, and we prove that the identifiability ofthe latent components at this GP limit is more restricted. Namely, thosecomponents are identifiable if and only if they have distinctly differentcovariance kernels. Our algorithm and identifiability theorems are explored onsimulated spatial data and real world spatio-temporal data.

Identifiable Feature Learning for Spatial Data with Nonlinear ICA