Posterior collapse as a phase transition in variational autoencoders

We investigate the phenomenon of posterior collapse in variational autoencoders (VAEs) from the perspective of statistical physics, and reveal that it constitutes a phase transition governed jointly by data structure and model hyper-parameters. By analyzing the stability of the trivial solution associated with posterior collapse, we identify a critical hyper-parameter threshold. In particular, we derive an explicit criterion for the onset of collapse: posterior collapse occurs when the decoder variance exceeds the largest eigenvalue of the data covariance matrix. This critical boundary, separating meaningful latent inference from collapse, is characterized by a discontinuity in the KL-divergence between the approximate posterior and the prior distribution, where the KL-divergence and its derivatives exhibit clear non-analytic behavior. We validate this critical behavior on both synthetic and real-world datasets, confirming the existence of a phase transition. The experimental results align well with our theoretical predictions, demonstrating the robustness of our collapse criterion across various VAE architectures. Our stability-based analysis demonstrate that posterior collapse is not merely an optimization failure, but rather an emerging phase transition arising from the interplay between data structure and variational constraints. This perspective offers new insights into the trainability and representational capacity of deep generative models.