A New Riemannian Structure in SPD(n)

Zhang, Shiqiang; Cao, Yueqi; Li, Wenyu; Yan, Fang-Jia; Luo, Yihao; Sun, Hongwei

doi:10.1109/icsidp47821.2019.9173017

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…The general linear group with Euclidean metric (GL(n), g E ) and projection σ is a trivial principal bundles on SPD(n) with orthogonal group O(n) as the structure group. This bundle structure establishes two facts [10]: SPD(n) ∼ = GL(n)/O(n), and g E remains invariant under the group action of O(n).…”

Section: Proposition 1 the Projectionmentioning

confidence: 52%

“…This expression only depends on the eigenvalue decomposition. More details can be found in [10]. Algorithm 1 will be used frequently in the following passage, and it helps us to comprehend the geometry of SPD(n).…”

Section: Preliminary 21 Notationmentioning

confidence: 99%

“…Recently, by constructing a principle bundle, Y. Li, M. Wong et al [9] and S. Zhang et al [10] defined a new Riemannian metric on SPD(n) whose geodesic distance is equivalent to the Wasserstein-2 [11,12] distance, the so-called Wasserstein metric. This distance rather than the metric has been widely used in artificial intelligence [13].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, equation (10) shows that the dimension of Wasserstein isometric group is higher than the dimension of O(n). Therefore, by dim(SPD(n)) = n 2 +n 2 = 4 and dim(O(n)) = n 2 −n 2 , we obtain the desired result.…”

mentioning

confidence: 99%

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Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

Luo

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Cao

et al. 2021

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The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.

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Section: Proposition 1 the Projectionmentioning

confidence: 52%

Section: Preliminary 21 Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

Luo

Zhang

Cao

et al. 2021

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“…Since each n multivariate normal distribution density function can be determined by an n -dimensional vector (mean) and an n -order symmetric positive definite matrix (covariance matrix), the manifold that consists of the family of normal distributions is closely related to manifold of the symmetric positive definite matrices [ 12 ].…”

Section: Preliminarymentioning

confidence: 99%

Application of Statistical K-Means Algorithm for University Academic Evaluation

Zhou

Pan

et al. 2022

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With the globalization of higher education, academic evaluation is increasingly valued by the scientific and educational circles. Although the number of published papers of academic evaluation methods is increasing, previous research mainly focused on the method of assigning different weights for various indicators, which can be subjective and limited. This paper investigates the evaluation of academic performance by using the statistical K-means (SKM) algorithm to produce clusters. The core idea is mapping the evaluation data from Euclidean space to Riemannian space in which the geometric structure can be used to obtain accurate clustering results. The method can adapt to different indicators and make full use of big data. By using the K-means algorithm based on statistical manifolds, the academic evaluation results of universities can be obtained. Furthermore, through simulation experiments on the top 20 universities of China with the traditional K-means, GMM and SKM algorithms, respectively, we analyze the advantages and disadvantages of different methods. We also test the three algorithms on a UCI ML dataset. The simulation results show the advantages of the SKM algorithm.

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AWCD: An Efficient Point Cloud Processing Approach via Wasserstein Curvature

Luo

Yang

Sun

et al. 2021

2021 IEEE International Conference on Artificial Intelligence and Computer Applications (ICAICA)

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In this paper, we introduce the adaptive Wasserstein curvature denoising (AWCD), an original processing approach for point cloud data. By collecting curvatures information from Wasserstein distance, AWCD consider more precise structures of data and preserves stability and effectiveness even for data with noise in high density. This paper contains some theoretical analysis about the Wasserstein curvature and the complete algorithm of AWCD. In addition, we design digital experiments to show the denoising effect of AWCD. According to comparison results, we present the advantages of AWCD against traditional algorithms.

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A New Riemannian Structure in SPD(n)

Cited by 5 publications

References 20 publications

Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

Application of Statistical K-Means Algorithm for University Academic Evaluation

AWCD: An Efficient Point Cloud Processing Approach via Wasserstein Curvature

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