The Incomplete Rosetta Stone Problem: Identifiability Results for Multi-View Nonlinear ICA

TL;DR

This paper proves that multiple noisy nonlinear views of independent sources can be jointly used to recover the original sources, overcoming previous impossibility results for nonlinear ICA.

Contribution

It provides novel identifiability proofs showing that combining multiple diverse views enables source recovery in nonlinear ICA, which was previously impossible.

Findings

Identifiability is achievable with multiple views in nonlinear ICA.

Deep neural networks can theoretically undo complex nonlinear mixing.

Recovery of arbitrary sources is possible with sufficiently different noisy views.

Abstract

We consider the problem of recovering a common latent source with independent components from multiple views. This applies to settings in which a variable is measured with multiple experimental modalities, and where the goal is to synthesize the disparate measurements into a single unified representation. We consider the case that the observed views are a nonlinear mixing of component-wise corruptions of the sources. When the views are considered separately, this reduces to nonlinear Independent Component Analysis (ICA) for which it is provably impossible to undo the mixing. We present novel identifiability proofs that this is possible when the multiple views are considered jointly, showing that the mixing can theoretically be undone using function approximators such as deep neural networks. In contrast to known identifiability results for nonlinear ICA, we prove that independent latent sources with arbitrary mixing can be recovered as long as multiple, sufficiently different noisy views are available.