Second-order difference subspace
<|reference_start|>Second-order difference subspace: 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.<|reference_end|>
@article{fukui2024second-order,
    title={Second-order difference subspace},
    author={Kazuhiro Fukui, Pedro H.V. Valois, Lincon Souza, Takumi Kobayashi},
    journal={arXiv preprint arXiv:2409.08563},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.08563},
    primaryClass={cs.LG cs.CV}
}