Analytic classification of the full-LDA free-energy landscape

Characterize the stationary-point structure of the free energy associated with full Latent Dirichlet Allocation, including the number and nature of its local minima, in order to determine the corresponding mixing behavior of the projected and full collapsed Gibbs samplers.

Background

For full LDA, both the document-topic proportions and topic-word distributions are unknown and are integrated out in the collapsed Gibbs sampler. The resulting mixing analysis is reduced by the mean-field Gibbs–Ornstein–Uhlenbeck framework to the stationary-point structure of an associated free energy.

The paper establishes conditional mixing conclusions when this landscape has one or multiple nondegenerate local minima, but does not provide a complete analytic classification of the landscape. Such a classification is needed to decide when slow mixing reflects genuine metastability and when it is caused only by the symmetry that permits global topic-label switching.

References

For full LDA, it reduces the mixing-time behavior of both the projected and the full chains to the stationary-point structure of an associated free energy. A complete analytic classification of that landscape remains open, but the framework distinguishes genuine metastability, as in , from slow transitions caused solely by topic-label switching, and the proposed landscape picture is supported by numerical examples.

— The mixing time of Mean-Field Gibbs-Ornstein-Uhlenbeck processes  (2610.08103 - Jonasson, 6 Oct 2026) in Introduction; Section 3.4, paragraph "Full LDA"

When the Dirichlet hyperparameters are symmetric, we conjecture, subject to suitable identifiability and nondegeneracy conditions, that the following are equivalent for sufficiently small hyperparameters: the only local minima are two minima related by topic-label switching, and the observed word-frequency matrix admits a genuine underlying LDA factorization in terms of document-specific topic weights and two topic-word distributions.

— The mixing time of Mean-Field Gibbs-Ornstein-Uhlenbeck processes  (2610.08103 - Jonasson, 6 Oct 2026) in Section 6, Discussion and open problems