A note on the problem of straight-line interpolation by ridge functions

TL;DR

This paper investigates the limitations of using ridge functions with fixed directions for interpolating data on straight lines, proving impossibility results for certain configurations.

Contribution

It provides a constructive proof that arbitrary data cannot be interpolated on three or more lines using two fixed directions, and reduces the general case to set existence problems.

Findings

Interpolation on three or more lines with two fixed directions is impossible.

The problem reduces to the existence of specific sets in the union of lines.

The paper offers a geometric and linear algebraic approach to the problem.

Abstract

In this paper we discuss the problem of interpolation on straight lines by linear combinations of ridge functions with fixed directions. By using some geometry and/or systems of linear equations, we constructively prove that it is impossible to interpolate arbitrary data on any three or more straight lines by sums of ridge functions with two fixed directions. The general case with more straight lines and more directions is reduced to the problem of existence of certain sets in the union of these lines.