Beyond ICA: Identifiability by Symmetry Breaking

We prove the identifiability of deep generative models (DGMs) with piecewise-affine (PWA) decoders and Gaussian mixture model (GMM) priors, in a purely unsupervised setting. We introduce three algebraic contrast principles for symmetry breaking: domain contrast, which trivializes the mixture symmetry group; mechanism contrast, which ensures every decoder branch is witnessed by a unique boundary; and interaction contrast, which forbids parameter conspiracies between latent components and decoder branches. Together they exploit the interplay between the discrete combinatorics of the PWA map and the continuous symmetry structure of the latent GMM. Continuity is replaced by algebraic symmetry conditions; injectivity is decoupled from structural identification and required only for pointwise inversion. Our results form a hierarchy: from law identifiability (LID; latent distribution up to a global affine map) through map identifiability (MID; decoder up to the same map) to posterior and pointwise identifiability. The ICA-form ambiguity emerges under conditions on diagonal component covariances. Assumptions are only on the data-generating process, not on learning methods, except for the interaction contrast. To our knowledge this is the first to make algebraic symmetry-breaking the engine of nonlinear identifiability, the first to admit discontinuous decoders, and the first to handle fully non-injective decoders, where every observation admits multiple latent codes.