Variational autoencoders (VAEs) have received considerable attention, since they allow us to learn expressive neural density estimators effectively and efficiently. However, learning and inference in VAEs is still problematic due to the sensitive interplay between the generative model and the inference network. Since these problems become generally more severe in high dimensions, we propose a novel hierarchical mixture model over low-dimensional VAE experts. Our model decomposes the overall learning problem into many smaller problems, which are coordinated by the hierarchical mixture, represented by a sum-product network. In experiments we show that our model consistently outperforms classical VAEs on all of our experimental benchmarks. Moreover, we show that our model is highly data efficient and degrades very gracefully in extremely low data regimes.