Learning the Structure of Triangular Transport Maps

Abstract

Triangular transport maps provide a flexible approach to sampling-based probabilistic modeling, including density estimation, generative modeling, and Bayesian inference. They transform an unknown target distribution into a simpler reference through a monotone triangular map. The map structure is defined by a variable ordering and sparsity pattern, which together encode a directed acyclic graph. Map quality can depend strongly on this structure, yet finding a good structure is computationally expensive because each candidate generally requires fitting a different map. A central challenge is therefore to learn density and structure jointly, while keeping computation manageable as dimension grows. We introduce Self-Structuring Transport Maps (SSTM), which learn the map, ordering, and sparsity jointly. We use SoftSort to learn the variable ordering and $L_0$ gates to learn the sparsity, while preserving a triangular structure. To keep the map scalable, we use a monotone BatchEnsemble that shares one weight matrix across all map components through rank-one adapters. Across synthetic and real data, jointly learning the structure and map gives better density estimates than estimating the structure first. When the structure is identifiable from the density, SSTM matches the density performance of a map fitted with the true structure and outperforms autoregressive flows. On large datasets, SSTM is competitive with autoregressive flows.