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

Additive Decoders for Latent Variables Identification and Cartesian-Product Extrapolation