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Dimensionality reduction of SPD data based on Riemannian manifold tangent spaces and local affinity

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Abstract

Non-Euclidean data is increasingly used in practical applications. As a typical representative, Symmetric Positive Definite (SPD) matrices can form a Riemannian manifold rather than a flat linear space. Hence, constructing a dimensionality reduction (DR) algorithm for SPD data directly on manifolds will encounter difficulties in modeling and solving. This paper proposes a novel DR technique based on Riemannian Manifold Tangent Spaces and Local Affinity for SPD data (RMTSLA-SPDDR). Our primary contributions are listed below: (1) We transfer the data from SPD manifolds to tangent spaces, which are all Euclidean spaces. Besides, the tangent spaces are symmetric matrix spaces, which indicates the proposed method greatly preserves the data form and properties; (2) Under the Affine-Invariant Riemannian Metric (AIRM), we incorporate tangent spaces and log transformation, so as to keep the geodesic distance between SPD data and the identity matrix equal to the corresponding Euclidean distance between the transformed data and the origin of the tangent space; (3) The method adopts the local affinity criterion to determine the bilinear transformation between tangent spaces, and take it as the transformation for SPD data consequently. There are no similar reports as the technique presented in this paper before, which means it is a new attempt. On five benchmark datasets, abundant experimental results indicate RMTSLA-SPDDR outperforms the other five state-of-the-art DR methods.

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Authors and Affiliations

  1. School of Electronics and Information Technology, Sun-Yat-Sen University, Guangzhou, China

    Wenxu Gao, Zhengming Ma, Chenkui Xiong & Ting Gao

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Gao, W., Ma, Z., Xiong, C. et al. Dimensionality reduction of SPD data based on Riemannian manifold tangent spaces and local affinity. Appl Intell 53, 1887–1911 (2023). https://doi.org/10.1007/s10489-022-03177-0

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