# Second-order difference subspace

arXiv: https://arxiv.org/abs/2409.08563 · HTML: https://mlsanity.com/paper/2409.08563/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 28 references that are in the map.
- First posted on arXiv 13 Sep 2024.

## Builds on (most cited first, 2 of 2)

- [Text Classification based on Word Subspace with Term-Frequency](https://mlsanity.com/paper/1806.03125/) (2018)
- [Time-series Anomaly Detection based on Difference Subspace between Signal Subspaces](https://mlsanity.com/paper/2303.17802/) (2023)

## Abstract (arXiv)

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.