Discrete Tree Flows via Tree-Structured Permutations

Abstract

While normalizing flows for continuous data have been extensively researched, flows for discrete data have only recently been explored. These prior models, however, suffer from limitations that are distinct from those of continuous flows. Most notably, discrete flow-based models cannot be straightforwardly optimized with conventional deep learning methods because gradients of discrete functions are undefined or zero, and backpropagation can be computationally burdensome compared to alternative discrete algorithms such as decision tree algorithms. Previous works approximate pseudo-gradients of the discrete functions but do not solve the problem on a fundamental level. Our approach seeks to reduce computational burden and remove the need for pseudo-gradients by developing a discrete flow based on decision trees---building upon the success of efficient tree-based methods for classification and regression for discrete data. We first define a tree-structured permutation (TSP) that compactly encodes a permutation of discrete data where the inverse is easy to compute; thus, we can efficiently compute the density value and sample new data. We then propose a decision tree algorithm to learn TSPs that estimates the tree structure and simple permutations at each node via a novel criteria. We empirically demonstrate the feasibility of our method on multiple datasets.