Approximating Functions on Boxes

TL;DR

This paper explores polynomial function approximation on multi-dimensional boxes, highlighting how vertex-based methods can efficiently approximate functions for biological applications like drug effect prediction and fitness landscape analysis.

Contribution

It introduces a novel approach to approximate functions on boxes using vertices, leveraging polynomial degree properties for biological data modeling.

Findings

Vertex-based polynomial approximation is effective for biological data.

Higher-degree polynomial approximations improve accuracy on box vertices.

Method facilitates predictions in drug combination and fitness landscapes.

Abstract

The vector space of all polynomial functions of degree $k$ on a box of dimension $n$ is of dimension ${n \choose k}$. A consequence of this fact is that a function can be approximated on vertices of the box using other vertices to higher degrees than expected. This approximation is useful for various biological applications such as predicting the effect of a treatment with drug combinations and computing values of fitness landscape.