  Jointly identifying a mixture of discrete and continuous factors of
variability without supervision is a key problem in unraveling complex
phenomena. Variational inference has emerged as a promising method to learn
interpretable mixture representations. However, posterior approximation in
high-dimensional latent spaces, particularly for discrete factors remains
challenging. Here, we propose an unsupervised variational framework using
multiple interacting networks called cpl-mixVAE that scales well to
high-dimensional discrete settings. In this framework, the mixture
representation of each network is regularized by imposing a consensus
constraint on the discrete factor. We justify the use of this framework by
providing both theoretical and experimental results. Finally, we use the
proposed method to jointly uncover discrete and continuous factors of
variability describing gene expression in a single-cell transcriptomic dataset
profiling more than a hundred cortical neuron types.
