Differentiable Set Operations for Algebraic Expressions

TL;DR

This paper introduces a novel differentiable set operation framework for inequalities, enabling smooth boundary representations useful in machine learning, graphics, and education, with adjustable parameters for accuracy.

Contribution

It formulates a new method to apply set operations directly on inequalities with differentiable boundaries, bridging set theory and differentiable functions.

Findings

Produces inequalities with differentiable boundaries

Offers adjustable parameters for accuracy and curvature

Applicable in machine learning, graphics, and education

Abstract

Basic principles of set theory have been applied in the context of probability and binary computation. Applying the same principles on inequalities is less common but can be extremely beneficial in a variety of fields. This paper formulates a novel approach to directly apply set operations on inequalities to produce resultant inequalities with differentiable boundaries. The suggested approach uses inequalities of the form Ei: fi(x1,x2,..,xn) and an expression of set operations in terms of Ei like, (E1 and E2) or E3, or can be in any standard form like the Conjunctive Normal Form (CNF) to produce an inequality F(x1,x2,..,xn)<=1 which represents the resulting bounded region from the expressions and has a differentiable boundary. To ensure differentiability of the solution, a trade-off between representation accuracy and curvature at borders (especially corners) is made. A set of parameters is introduced which can be fine-tuned to improve the accuracy of this approach. The various applications of the suggested approach have also been discussed which range from computer graphics to modern machine learning systems to fascinating demonstrations for educational purposes (current use). A python script to parse such expressions is also provided.