Beyond Mean-Field: Tree-Copula Variational Autoencoders for Structured Latent Dependencies

Variational Autoencoders (VAEs) typically assume a factorized {Gaussian} posterior and isotropic normal prior, implicitly imposing mean-field independence in the latent space. Rewriting the {ELBO} in copula form, we isolate a dependence-mismatch term that contributes to approximation and amortization gaps. We propose a Tree-Copula VAE that replaces the mean-field posterior with a tree-structured copula while retaining flexible learned marginals. The tree is inferred per datapoint via a Chow–Liu maximum-weight spanning tree, exploiting the monotonic relationship between mutual information and copula parameters. We further study an Average-of-Trees (AoT) copula prior for prior–posterior alignment and a rank-1 {Gaussian}-copula likelihood for low-rank observation correlations. Experiments on correlated dSprites and Fashion-{MNIST} show tighter importance-weighted bounds than mean-field VAEs when latent dependence is present, and highlight the role of prior alignment and likelihood modeling beyond mean-field inference.