Discrete Tree Flows via Tree-Structured Permutations

TL;DR

This paper introduces a novel discrete flow model based on decision trees, called Tree-Structured Permutations, which efficiently encodes permutations of discrete data and overcomes gradient-based optimization challenges.

Contribution

It proposes a decision tree-based discrete flow model with a tree-structured permutation that enables efficient density estimation and sampling without pseudo-gradient approximations.

Findings

Successfully applied to multiple datasets.

Efficient computation of density and sampling.

Overcomes limitations of previous discrete flow models.

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. Previous works approximate pseudo-gradients of the discrete functions but do not solve the problem on a fundamental level. In addition to that, backpropagation can be computationally burdensome compared to alternative discrete algorithms such as decision tree algorithms. 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 build TSPs that learns the tree structure and permutations at each node via novel criteria. We empirically demonstrate the feasibility of our method on multiple datasets.