Transportation of Subspaces

Abstract

This talk introduces a distance metric for subspaces using transportation between the Stiefel manifolds. Principal component analysis (PCA) is a fundamental methodology for data processing in pattern recognition, computer vision, physiology and many natural and social sciences.

Organs, cells and microstructures in cells dealt with in biomedical image analysis are volumetric data. Sampled values of volumetric data are expressed as three-way array data. Therefore, PCA of multi-way data is an essential technique for subspace-based pattern recognition, data retrievals and data compression of volumetric data.

For dealing with sampled organs as three-way data from the viewpoint of object oriented data analysis, we introduce PCA for three-way data arrays employing the Tucker-3 tensor decomposition. The tensor PCA for volumetric data allows us to simultaneously extract both outline shapes of volumetric objects and statistical properties of interior textures of volumetric data from data projected onto a low-dimensional linear subspace spanned by tensors. 

For the quantitative discrimination of multi-way forms from the view point of PCA-based pattern recognition, a distance metric for subspaces of multi-way data arrays is requested. The canonical angle between two subspaces has been used to define a metric between subspaces. Furthermore, a distance between two subspaces is computed by the Grassmann distance based on a set of canonical angles between two subspaces. However, the computation of canonical angles between tensor subspaces is computationally expensive from the view point of numerical computation since we need to compute the eigendecomposition of a large projection matrix of size mn by mn for images of size m by n. Moreover, the distance between Stiefel manifolds has also been proposed.