Extrinsic Principal Component Analysis

TL;DR

This paper introduces a fast extrinsic PCA method for manifold data analysis by embedding the object space into a numerical space, enabling efficient dimension reduction especially for high-dimensional or infinite-dimensional data.

Contribution

It proposes a novel extrinsic PCA approach on manifolds using embedding and chord distances, facilitating analysis of complex data structures.

Findings

Effective dimension reduction for shape data from imaging.

Applicable to high and infinite-dimensional data.

Provides a computationally efficient PCA methodology.

Abstract

One develops a fast computational methodology for principal component analysis on manifolds. Instead of estimating intrinsic principal components on an object space with a Riemannian structure, one embeds the object space in a numerical space, and the resulting chord distance is used. This method helps us analyzing high, theoretically even infinite dimensional data, from a new perspective. We define the extrinsic principal sub-manifolds of a random object on a Hilbert manifold embedded in a Hilbert space, and the sample counterparts. The resulting extrinsic principal components are useful for dimension data reduction. For application, one retains a very small number of such extrinsic principal components for a shape of contour data sample, extracted from imaging data.