Monotone Cubic B-Splines with a Neural-Network Generator

TL;DR

This paper introduces a novel approach for fitting monotone cubic B-splines using neural networks, which improves efficiency and accuracy, especially under high noise, and is applicable to scientific phenomena like star formation.

Contribution

It proposes a neural network-based generator for monotone spline fitting, offering a faster alternative to traditional optimization methods with theoretical and empirical validation.

Findings

The generator approach accelerates spline fitting in repeated computations.

The method outperforms existing techniques in high-noise conditions.

Application to astrophysics demonstrates practical utility.

Abstract

We present a method for fitting monotone curves using cubic B-splines, which is equivalent to putting a monotonicity constraint on the coefficients. We explore different ways of enforcing this constraint and analyze their theoretical and empirical properties. We propose two algorithms for solving the spline fitting problem: one that uses standard optimization techniques and one that trains a Multi-Layer Perceptrons (MLP) generator to approximate the solutions under various settings and perturbations. The generator approach can speed up the fitting process when we need to solve the problem repeatedly, such as when constructing confidence bands using bootstrap. We evaluate our method against several existing methods, some of which do not use the monotonicity constraint, on some monotone curves with varying noise levels. We demonstrate that our method outperforms the other methods, especially in high-noise scenarios. We also apply our method to analyze the polarization-hole phenomenon during star formation in astrophysics. The source code is accessible at \texttt{\url{https://github.com/szcf-weiya/MonotoneSplines.jl}}.