Identifiable Feature Learning for Spatial Data with Nonlinear ICA

TL;DR

This paper introduces a nonlinear ICA framework using t-process latent components suitable for spatial and spatio-temporal data, with new algorithms and theoretical guarantees for identifiability in complex dependency structures.

Contribution

It develops a novel nonlinear ICA method employing t-process priors for higher-dimensional dependencies and provides theoretical identifiability results for these models.

Findings

Proposed a t-process nonlinear ICA framework for spatial data.

Proved identifiability of t-process components under general conditions.

Validated the approach on simulated and real spatio-temporal data.

Abstract

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.