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Injecting structure into neural networks enables learning functions thatsatisfy invariances with respect to subsets of inputs. For instance, whenlearning generative models using neural networks, it is advantageous to encodethe conditional independence structure of observed variables, often in the formof Bayesian networks. We propose the Structured Neural Network (StrNN), whichinjects structure through masking pathways in a neural network. The masks aredesigned via a novel relationship we explore between neural networkarchitectures and binary matrix factorization, to ensure that the desiredindependencies are respected. We devise and study practical algorithms for thisotherwise NP-hard design problem based on novel objectives that control themodel architecture. We demonstrate the utility of StrNN in three applications:(1) binary and Gaussian density estimation with StrNN, (2) real-valued densityestimation with Structured Autoregressive Flows (StrAFs) and StructuredContinuous Normalizing Flows (StrCNF), and (3) interventional andcounterfactual analysis with StrAFs for causal inference. Our work opens up newavenues for learning neural networks that enable data-efficient generativemodeling and the use of normalizing flows for causal effect estimation.

Structured Neural Networks for Density Estimation and Causal Inference