Identifiability of deep generative models without auxiliary information

TL;DR

This paper proves that a broad class of deep generative models, including common variational autoencoders, are identifiable without auxiliary information, advancing theoretical understanding and practical implications for unsupervised learning.

Contribution

It establishes the first general identifiability results for deep latent variable models without supervision, covering models like VaDE and MFC-VAE with mixture priors.

Findings

Identifiability up to affine transformations for certain models.

Broader class of models now proven identifiable without auxiliary data.

Experimental validation on simulated and real datasets.

Abstract

We prove identifiability of a broad class of deep latent variable models that (a) have universal approximation capabilities and (b) are the decoders of variational autoencoders that are commonly used in practice. Unlike existing work, our analysis does not require weak supervision, auxiliary information, or conditioning in the latent space. Specifically, we show that for a broad class of generative (i.e. unsupervised) models with universal approximation capabilities, the side information $u$ is not necessary: We prove identifiability of the entire generative model where we do not observe $u$ and only observe the data $x$. The models we consider match autoencoder architectures used in practice that leverage mixture priors in the latent space and ReLU/leaky-ReLU activations in the encoder, such as VaDE and MFC-VAE. Our main result is an identifiability hierarchy that significantly generalizes previous work and exposes how different assumptions lead to different "strengths" of identifiability, and includes certain "vanilla" VAEs with isotropic Gaussian priors as a special case. For example, our weakest result establishes (unsupervised) identifiability up to an affine transformation, and thus partially resolves an open problem regarding model identifiability raised in prior work. These theoretical results are augmented with experiments on both simulated and real data.