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原文摘要

We prove the population-level 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, a label-free counterpart of the sufficient-variability assumption of nonlinear ICA; *mechanism contrast*, which ensures every decoder branch gets an observational boundary across which it toggles alone; 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. The interaction contrast is the only one of our contrasts required on the estimator side. To our knowledge this is the first to develop algebraic symmetry-breaking into an identifiability engine for a concrete nonlinear model class, the first to admit discontinuous decoders, and the first to handle fully non-injective decoders, where every observation admits multiple latent codes.