Yet Another Representation of Binary Decision Trees: A Mathematical Demonstration

TL;DR

This paper presents a novel mathematical framework for representing binary decision trees using computational graphs and numerical operations, unifying various decision tree types.

Contribution

It introduces a numerical representation of decision trees with bitvector matrices, converting logical operations into arithmetic, and unifies different decision tree concepts.

Findings

Decision trees can be expressed as shallow binary networks.

Bitvector matrices enable numerical traversal of trees.

Logical AND operations are converted to arithmetic operations.

Abstract

A decision tree looks like a simple directed acyclic computational graph, where only the leaf nodes specify the output values and the non-terminals specify their tests or split conditions. From the numerical perspective, we express decision trees in the language of computational graph. We explicitly parameterize the test phase, traversal phase and prediction phase of decision trees based on the bitvectors of non-terminal nodes. As shown, the decision tree is a shallow binary network in some sense. Especially, we introduce the bitvector matrix to implement the tree traversal in numerical approach, where the core is to convert the logical `AND' operation to arithmetic operations. And we apply this numerical representation to extend and unify diverse decision trees in concept.