Discrete Differential Principle for Continuous Smooth Function Representation
This work addresses a foundational mathematical bottleneck in fields like visual perception and fluid mechanics, offering a novel method for derivative estimation and function representation, though it appears incremental as an improvement over existing discrete approaches.
The authors tackled the problem of error propagation and the curse of dimensionality in Taylor's formula for derivative computation and function representation by proposing a new discrete differential operator based on Vandermonde matrices, achieving high-order accuracy with rigorous error bounds and demonstrating superiority over methods like finite forward differences and interpolation in experiments.
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.