Discrete Differential Principle for Continuous Smooth Function Representation

TL;DR

This paper introduces a novel discrete differential operator based on Vandermonde matrices that accurately estimates derivatives and represents smooth functions, overcoming limitations of traditional methods in high-dimensional and discrete settings.

Contribution

The paper proposes a new discrete differential operator using Vandermonde matrices that improves derivative estimation and function representation, reducing error propagation and curse of dimensionality.

Findings

Achieves high-order accuracy with equidistant sampling.

Provides tighter error bounds for lower-order derivatives.

Outperforms finite difference and spline methods in experiments.

Abstract

Taylor's formula holds significant importance in function representation, such as solving differential difference equations, ordinary differential equations, partial differential equations, and further promotes applications in visual perception, complex control, fluid mechanics, weather forecasting and thermodynamics. However, the Taylor's formula suffers from the curse of dimensionality and error propagation during derivative computation in discrete situations. In this paper, we propose a new discrete differential operator to estimate derivatives and to represent continuous smooth function locally using the Vandermonde coefficient matrix derived from truncated Taylor series. Our method simultaneously computes all derivatives of orders less than the number of sample points, inherently mitigating error propagation. Utilizing equidistant uniform sampling, it achieves high-order accuracy while alleviating the curse of dimensionality. We mathematically establish rigorous error bounds for both derivative estimation and function representation, demonstrating tighter bounds for lower-order derivatives. We extend our method to the two-dimensional case, enabling its use for multivariate derivative calculations. Experiments demonstrate the effectiveness and superiority of the proposed method compared to the finite forward difference method for derivative estimation and cubic spline and linear interpolation for function representation. Consequently, our technique offers broad applicability across domains such as vision representation, feature extraction, fluid mechanics, and cross-media imaging.