Fast Derivatives for Multilinear Polynomials

TL;DR

This paper introduces a fast algorithm for computing derivatives of multilinear polynomials, significantly reducing computational costs and resembling the structure of FFT, with potential applications in high-dimensional polynomial analysis.

Contribution

The paper presents a novel minimal-operation algorithm for derivatives of multilinear polynomials, improving efficiency over traditional methods.

Findings

Derivative evaluation cost approaches 1/8 of polynomial evaluation as variables grow

Algorithm minimizes floating point operations for non-sparse polynomials

Structural similarity to FFT enables efficient computation

Abstract

The article considers linear functions of many (n) variables - multilinear polynomials (MP). The three-steps evaluation is presented that uses the minimal possible number of floating point operations for non-sparse MP at each step. The minimal number of additions is achieved in the algorithm for fast MP derivatives (FMPD) calculation. The cost of evaluating all first derivatives approaches to only 1/8 of MP evaluation with a growing number of variables. The FMPD algorithm structure exhibits similarity to the Fast Fourier Transformation (FFT) algorithm.