Manifold Data Analysis with Applications to High-Frequency 3D Imaging

TL;DR

This paper introduces Manifold Data Analysis (MDA), a new statistical framework for analyzing complex data where variables are manifolds, demonstrated through 3D facial imaging to understand influences like age and genetics.

Contribution

The paper develops a novel framework converting manifolds into functional objects, along with a 2-step functional PCA and manifold-on-scalar regression, specifically tailored for high-dimensional manifold data.

Findings

Effective analysis of 3D facial shape data

Identification of age and genetic influences on facial morphology

Demonstration of the framework's utility in anthropological studies

Abstract

Many scientific areas are faced with the challenge of extracting information from large, complex, and highly structured data sets. A great deal of modern statistical work focuses on developing tools for handling such data. This paper presents a new subfield of functional data analysis, FDA, which we call Manifold Data Analysis, or MDA. MDA is concerned with the statistical analysis of samples where one or more variables measured on each unit is a manifold, thus resulting in as many manifolds as we have units. We propose a framework that converts manifolds into functional objects, an efficient 2-step functional principal component method, and a manifold-on-scalar regression model. This work is motivated by an anthropological application involving 3D facial imaging data, which is discussed extensively throughout the paper. The proposed framework is used to understand how individual characteristics, such as age and genetic ancestry, influence the shape of the human face.