Fitting scattered data with optional monotonicity constraints on GPU: LipFit package

2026-06-03

AI Summary

This work addresses the problem of continuous interpolation and approximation of multivariate scattered data under optional monotonicity constraints. The authors propose a training-free, Lipschitz-continuous approximation method that constructs tight upper and lower bounds, integrating local interpolation with smoothing strategies to ensure strict adherence to prescribed monotonicity while avoiding discontinuities. The key contribution lies in achieving shape-preserving, continuous, and computationally efficient instance-wise approximation. By leveraging GPU-based parallelization, the method achieves substantial performance gains. The proposed algorithms have been implemented in the open-source Python package LipFit, making them readily applicable to large-scale constrained scattered data fitting tasks.

Abstract

This paper presents a method of multivariate scattered data interpolation and approximation that produces optimal Lipschitz-continuous approximation, subject to the desired monotonicity constraints. This method relies on tight upper and lower approximations to the data, and is similar in its spirit to the nearest-neighbour approximation but does not suffer from discontinuities. Local Lipschitz interpolation and Lipschitz smoothing are also presented. This approach falls under the umbrella of instance-based approximation with no training phase, and it is suitable for GPU-based parallelisation. A Python GPU-friendly package LipFit which implements the methods discussed is discussed.

Problem

• scattered data
• monotonicity constraints
• Lipschitz continuity
• interpolation
• approximation

Innovation

• Lipschitz interpolation
• monotonicity constraints
• GPU parallelization
• instance-based approximation
• scattered data fitting