New theory proves deep generative models identifiable via symmetry breaking

A new theoretical paper by Pengzhou Wu, titled 'Beyond ICA: Identifiability by Symmetry Breaking,' tackles the fundamental challenge of identifiability in deep generative models (DGMs). While linear independent component analysis (ICA) is well-understood, nonlinear extensions have struggled to guarantee that learned representations match the true latent factors. Wu proves that DGMs with piecewise-affine (PWA) decoders and Gaussian mixture model (GMM) priors are identifiable in a purely unsupervised setting, without requiring injective decoders or continuity.

The proof relies on three algebraic contrast principles that exploit the interplay between the discrete combinatorics of the PWA map and the continuous symmetries of the latent GMM. Domain contrast trivializes the mixture symmetry group; mechanism contrast ensures every decoder branch has a unique boundary; and interaction contrast prevents parameter conspiracies. The results form a hierarchy: law identifiability (latent distribution up to a global affine map), map identifiability (decoder up to the same map), and posterior/pointwise identifiability. ICA-form ambiguity arises only under diagonal component covariances. This work is the first to make algebraic symmetry-breaking the engine of nonlinear identifiability, and the first to accommodate both discontinuous and fully non-injective decoders.