A Critical $\beta$-Scale for Posterior Collapse in Dirichlet $\beta$-VAEs

Posterior collapse is a common failure mode in variational autoencoders, but existing theory has mostly focused on Gaussian latent variables. Dirichlet VAEs are comparatively less understood, despite their use in settings where latent variables represent proportions, abundances, memberships, or convex mixture weights. We derive a closed-form local stability threshold $\beta_c$ for collapse in symmetric Dirichlet $\beta$-VAEs with linear Gaussian decoders. This threshold shows that collapse in those models depends not only on the data spectrum, but also on the simplex dimension and prior concentration through a trigamma curvature factor induced by the Dirichlet KL. Although $\beta_c$ is derived from a local analysis, we find that it accurately tracks the empirical onset of collapse beyond the local regime. We show its practical relevance in two real-world linear-mixture settings: hyperspectral unmixing and convex archetypal analysis. Across experiments, $\beta/\beta_c$ consistently organizes collapse-related behavior, providing an interpretable coordinate and a practical way to narrow the search over the $\beta$ hyperparameter before training. Taken together, these empirical findings suggest that $\beta_c$ can serve, beyond its local derivation, as a practical reference point for anticipating collapse and guiding $\beta$ selection in Dirichlet $\beta$-VAEs.