Additive Decoders for Latent Variables Identification and Cartesian-Product Extrapolation

TL;DR

This paper demonstrates that additive decoders can uniquely identify latent variables and enable novel image generation through Cartesian-product extrapolation, even with complex, dependent, and arbitrarily supported latent factors.

Contribution

It introduces additive decoders as a means to achieve latent variable identification and out-of-support image generation under weak assumptions, expanding theoretical understanding of OCRL and nonlinear ICA.

Findings

Additive decoders guarantee latent variable identification under weak assumptions.

Additive decoders enable Cartesian-product extrapolation for novel image generation.

Empirical results confirm the importance of additivity for identifiability and extrapolation.

Abstract

We tackle the problems of latent variables identification and ``out-of-support'' image generation in representation learning. We show that both are possible for a class of decoders that we call additive, which are reminiscent of decoders used for object-centric representation learning (OCRL) and well suited for images that can be decomposed as a sum of object-specific images. We provide conditions under which exactly solving the reconstruction problem using an additive decoder is guaranteed to identify the blocks of latent variables up to permutation and block-wise invertible transformations. This guarantee relies only on very weak assumptions about the distribution of the latent factors, which might present statistical dependencies and have an almost arbitrarily shaped support. Our result provides a new setting where nonlinear independent component analysis (ICA) is possible and adds to our theoretical understanding of OCRL methods. We also show theoretically that additive decoders can generate novel images by recombining observed factors of variations in novel ways, an ability we refer to as Cartesian-product extrapolation. We show empirically that additivity is crucial for both identifiability and extrapolation on simulated data.