Learning Arbitrary Sum-Product Network Leaves with   Expectation-Maximization
  Sum-Product Networks with complex probability distribution at the leaves have been shown to be powerful tractable-inference probabilistic models. However, while learning the internal parameters has been amply studied, learning complex leaf distribution is an open problem with only few results available in special cases. In this paper we derive an efficient method to learn a very large class of leaf distributions with Expectation-Maximization. The EM updates have the form of simple weighted maximum likelihood problems, allowing to use any distribution that can be learned with maximum likelihood, even approximately. The algorithm has cost linear in the model size and converges even if only partial optimizations are performed. We demonstrate this approach with experiments on twenty real-life datasets for density estimation, using tree graphical models as leaves. Our model outperforms state-of-the-art methods for parameter learning despite using SPNs with much fewer parameters. 
Mattia Desana and Christoph Schn\"orr
cs.LG