Discriminative Analysis for Symmetric Positive Definite Matrices on Lie Groups

Xu, Chunyan; Lu, Canyi; Gao, Junbin; Zheng, Wei; Wang, Tianjiang; Yan, Shuicheng

doi:10.1109/tcsvt.2015.2392472

Cited by 22 publications

(14 citation statements)

References 28 publications

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“…More recently, and probably inspired by our preliminary study Harandi et al (2014), this bilinear form was employed to perform DR on the SPD manifold and on the Grassmannian by exploiting notions of Riemannian geometry Huang et al (2015b,a); Yger and Sugiyama (2015). We also acknowledge that the work of Xu et al Xu et al (2015) is somehow relevant to the log-Euclidean development done in §4.1. However, in contrast to our proposal, in Xu et al (2015) authors did not impose an orthogonality constraint on W .…”

Section: Related Workmentioning

confidence: 91%

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Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

Harandi

Salzmann

Hartley

2018

IEEE Trans. Pattern Anal. Mach. Intell.

168

162

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Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones-comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods.

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Section: Related Workmentioning

confidence: 91%

“…We also acknowledge that the work of Xu et al Xu et al (2015) is somehow relevant to the log-Euclidean development done in §4.1. However, in contrast to our proposal, in Xu et al (2015) authors did not impose an orthogonality constraint on W .…”

Section: Related Workmentioning

confidence: 91%

Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

Harandi

Salzmann

Hartley

2018

IEEE Trans. Pattern Anal. Mach. Intell.

168

162

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show abstract

“…The first Riemannian metric learning scheme [34], [35], [36], [37], [38], [39] typically first flattens the underlying Riemannian manifold via tangent space approximation, and then learns a discriminant metric in the resulting tangent (Euclidean) space by employing traditional metric learning methods. However, the map between the manifold and the tangent space is locally diffeomorphic, which inevitably distorts the original Riemannian geometry.…”

Section: Riemannian Metric Learningmentioning

confidence: 99%

Cross Euclidean-to-Riemannian Metric Learning with Application to Face Recognition from Video

Huang

Wang

Shan

et al. 2018

IEEE Trans. Pattern Anal. Mach. Intell.

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Abstract-Riemannian manifolds have been widely employed for video representations in visual classification tasks including videobased face recognition. The success mainly derives from learning a discriminant Riemannian metric which encodes the non-linear geometry of the underlying Riemannian manifolds. In this paper, we propose a novel metric learning framework to learn a distance metric across a Euclidean space and a Riemannian manifold to fuse the average appearance and pattern variation of faces within one video. The proposed metric learning framework can handle three typical tasks of video-based face recognition: Video-to-Still, Still-to-Video and Video-to-Video settings. To accomplish this new framework, by exploiting typical Riemannian geometries for kernel embedding, we map the source Euclidean space and Riemannian manifold into a common Euclidean subspace, each through a corresponding high-dimensional Reproducing Kernel Hilbert Space (RKHS). With this mapping, the problem of learning a cross-view metric between the two source heterogeneous spaces can be expressed as learning a single-view Euclidean distance metric in the target common Euclidean space. By learning information on heterogeneous data with the shared label, the discriminant metric in the common space improves face recognition from videos. Extensive experiments on four challenging video face databases demonstrate that the proposed framework has a clear advantage over the state-of-the-art methods in the three classical video-based face recognition tasks.Index Terms-Riemannian manifold, video-based face recognition, cross Euclidean-to-Riemannian metric learning.

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“…And Riemannian data has been proven to be more robust as the feature descriptors for images and videos than traditional Euclidean feature vectors. Hence, Riemannian machine learning algorithms obtain extensive researches in the recent years and are successful in kernel learning [9], [10], [11], metric learning [12], [13], discriminant analysis [14], dimensionality reduction [15], and so on. We study the DSLC algorithm on SPD manifold in this paper.…”

Section: Introductionmentioning

confidence: 99%

Robust Dictionary Learning and Sparse Coding With Riemannian Geometry Preserving Method in Symmetric Matrices Inner Product Space

Zhang

Zhu

2020

IEEE Access

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Existing Dictionary Learning and Sparse Coding (DLSC) algorithms for Symmetric Positive Definite (SPD) matrices usually adopt Reproducing Kernel Hilbert Space as workspace to perform necessary linear operations. But those methods heavily rely on ideal kernel maps and they are lack of robustness when facing different SPD data, especially in the case of high-sparsity coding. Different from existing methods, we explore a new workspace called Symmetric Matrices Inner Product Space (SMIPS) for modeling robust DLSC of SPD matrices. SMIPS, which can be treated as the minimal linear expansion space of SPD manifold, is a linear space equipped with a pre-defined inner product so it supports linear operations. Modelling DLSC in SMIPS is more intuitive, and SPD data can preserve matrix form in SMIPS. Then, in this paper, a Riemannian Geometry Preserving (RGP) method based on graph-regularized is proposed to include the Riemannian geometric information of original SPD data, so that improves the discriminability of sparse codes, and a sparse coding algorithm is developed to acquire sparse codes of SPD matrices in SMIPS efficiently, which extend the feature-sign search to such matrix space. For dictionary learning, an overall learning strategy utilizing the closed form solution of RGP codes is proposed to speed up the iterative dictionary learning process. Experiments on several computer vision tasks show that our algorithm is more robust even with high-sparsity coding across different datasets and outperforms other comparative algorithms in terms of recognition accuracy and learning efficiency, which also demonstrates the validity of SMIPS as a workspace.

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Discriminative Analysis for Symmetric Positive Definite Matrices on Lie Groups

Cited by 22 publications

References 28 publications

Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

Cross Euclidean-to-Riemannian Metric Learning with Application to Face Recognition from Video

Robust Dictionary Learning and Sparse Coding With Riemannian Geometry Preserving Method in Symmetric Matrices Inner Product Space

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