A Simple Algorithm for Approximation by Nomographic Functions

TL;DR

This paper presents a new algorithm for approximating multivariate functions with nomographic functions using cone-constrained optimization, combining ANOVA decomposition and monotone polynomial optimization, applicable to distributed computation.

Contribution

It introduces a novel algorithm that approximates multivariate functions by nomographic functions through cone-constrained Rayleigh-Quotient optimization, integrating ANOVA and monotone polynomial methods.

Findings

Effective approximation of multivariate functions by nomographic functions.

Application to distributed function computation over multiple-access channels.

Demonstrated feasibility through an illustrative example.

Abstract

This paper introduces a novel algorithmic solution for the approximation of a given multivariate function by a nomographic function that is composed of a one-dimensional continuous and monotone outer function and a sum of univariate continuous inner functions. We show that a suitable approximation can be obtained by solving a cone-constrained Rayleigh-Quotient optimization problem. The proposed approach is based on a combination of a dimensionwise function decomposition known as Analysis of Variance (ANOVA) and optimization over a class of monotone polynomials. An example is given to show that the proposed algorithm can be applied to solve problems in distributed function computation over multiple-access channels.