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In variational autoencoders (VAEs), the variational posterior often collapsesto the prior, known as posterior collapse, which leads to poor representationlearning quality. An adjustable hyperparameter beta has been introduced in VAEsto address this issue. This study sharply evaluates the conditions under whichthe posterior collapse occurs with respect to beta and dataset size byanalyzing a minimal VAE in a high-dimensional limit. Additionally, this settingenables the evaluation of the rate-distortion curve of the VAE. Our resultsshow that, unlike typical regularization parameters, VAEs face "inevitableposterior collapse" beyond a certain beta threshold, regardless of datasetsize. Moreover, the dataset-size dependence of the derived rate-distortioncurve suggests that relatively large datasets are required to achieve arate-distortion curve with high rates. These findings robustly explaingeneralization behavior observed in various real datasets with highlynon-linear VAEs.

High-dimensional Asymptotics of VAEs: Threshold of Posterior Collapse and Dataset-Size Dependence of Rate-Distortion Curve