Second-order difference subspace

TL;DR

This paper introduces the second-order difference subspace, a novel geometric tool for analyzing the dynamics of subspaces over time, extending previous methods to capture acceleration in subspace changes.

Contribution

It proposes the second-order difference subspace, extending first-order differences to analyze subspace acceleration and intersection, with applications in shape and biometric time series analysis.

Findings

Validates the second-order difference subspace through numerical experiments.

Demonstrates improved analysis of subspace dynamics in shape and biometric data.

Shows the method captures velocity and acceleration of subspace changes.

Abstract

Subspace representation is a fundamental technique in various fields of machine learning. Analyzing a geometrical relationship among multiple subspaces is essential for understanding subspace series' temporal and/or spatial dynamics. This paper proposes the second-order difference subspace, a higher-order extension of the first-order difference subspace between two subspaces that can analyze the geometrical difference between them. As a preliminary for that, we extend the definition of the first-order difference subspace to the more general setting that two subspaces with different dimensions have an intersection. We then define the second-order difference subspace by combining the concept of first-order difference subspace and principal component subspace (Karcher mean) between two subspaces, motivated by the second-order central difference method. We can understand that the first/second-order difference subspaces correspond to the velocity and acceleration of subspace dynamics from the viewpoint of a geodesic on a Grassmann manifold. We demonstrate the validity and naturalness of our second-order difference subspace by showing numerical results on two applications: temporal shape analysis of a 3D object and time series analysis of a biometric signal.