Multiple penalized principal curves: analysis and computation

TL;DR

This paper introduces a new functional for approximating data with multiple principal curves, proves the existence of minimizers, and develops an efficient algorithm capable of handling noisy data.

Contribution

It proposes a novel functional for multiple principal curves, proves theoretical properties, and provides an efficient algorithm for practical computation.

Findings

Enlarging the configuration space simplifies the energy landscape.

The method effectively captures one-dimensional structures in noisy data.

The approach outperforms single-curve models in complex data scenarios.

Abstract

We study the problem of finding the one-dimensional structure in a given data set. In other words we consider ways to approximate a given measure (data) by curves. We consider an objective functional whose minimizers are a regularization of principal curves and introduce a new functional which allows for multiple curves. We prove the existence of minimizers and establish their basic properties. We develop an efficient algorithm for obtaining (near) minimizers of the functional. While both of the functionals used are nonconvex, we argue that enlarging the configuration space to allow for multiple curves leads to a simpler energy landscape with fewer undesirable (high-energy) local minima. Furthermore we note that the approach proposed is able to find the one-dimensional structure even for data with considerable amount of noise.