Principal manifold estimation via model complexity selection

TL;DR

This paper introduces a flexible framework for principal manifolds using Sobolev spaces, incorporating model complexity selection to enhance data modeling, outlier robustness, and computational efficiency, with applications in medical imaging.

Contribution

It presents a novel principal manifold estimation method with model complexity selection, extending PCA and principal curves, and introduces techniques for identifying complex geometric structures.

Findings

Effective in modeling high-dimensional data

Reduces overfitting and outlier effects

Improves computational speed

Abstract

We propose a framework of principal manifolds to model high-dimensional data. This framework is based on Sobolev spaces and designed to model data of any intrinsic dimension. It includes principal component analysis and principal curve algorithm as special cases. We propose a novel method for model complexity selection to avoid overfitting, eliminate the effects of outliers, and improve the computation speed. Additionally, we propose a method for identifying the interiors of circle-like curves and cylinder/ball-like surfaces. The proposed approach is compared to existing methods by simulations and applied to estimate tumor surfaces and interiors in a lung cancer study.