On the separability of multivariate functions

TL;DR

This paper introduces a Monte Carlo-based method to determine the separability of multivariate functions, which is crucial for simplifying optimization problems, especially when the functions are black-box and high-dimensional.

Contribution

It presents a novel Monte Carlo algorithm for estimating function separability and extends it to identify the number and composition of disjoint variable subsets.

Findings

The algorithm can estimate separability with varying computational complexity.

It can identify the number of disjoint subsets where the function is separable.

The method is applicable to functions over arbitrary domains, including the unit cube.

Abstract

Separability of multivariate functions alleviates the difficulty in finding a minimum or maximum value of a function such that an optimal solution can be searched by solving several disjoint problems with lower dimensionalities. In most of practical problems, however, a function to be optimized is black-box and we hardly grasp its separability. In this study, we first describe a general separability condition which a function defined over an arbitrary domain satisfies if and only if the function is separable with respect to given disjoint subsets of variables. By introducing an alternative separability condition, we propose a Monte Carlo-based algorithm to estimate the separability of a function defined over unit cube with respect to given disjoint subsets of variables. Moreover, we extend our algorithm to estimate the number of disjoint subsets and the disjoint subsets such that a function is separable with respect to them. Computational complexity of our extended algorithm is function-dependent and varies from linear to exponential in the dimension.