SCIENCE CHINA Information Sciences, Volume 61 , Issue 9 : 092104(2018) https://doi.org/10.1007/s11432-017-9233-4

## Locality preserving projection on SPD matrix Lie group: algorithm and analysis

Yangyang LI 1,2,*,
• AcceptedAug 16, 2017
• PublishedApr 27, 2018
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### Abstract

Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data.For high-dimensional SPD matrices, directly using manifold learning algorithms to reduce the dimension of matrix-form data is impossible. The SPD matrix must first be transformed into a long vector, and then the dimension of this vector must be reduced. However, this approach breaks the spatial structure of the SPD matrix space. To overcome this limitation, we propose a new dimension reduction algorithm on SPD matrix space to transform high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size has a Lie group structure, and we aim to transform the manifold learning to the SPD matrix Lie group. We use the basic idea of the manifold learning algorithm called locality preserving projection (LPP) to construct the corresponding Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach Lie-LPP to emphasize its Lie group character. We present a detailed algorithm analysis and show through experiments that Lie-LPP achieves effective results on human action recognition and human face recognition.

### Acknowledgment

This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB1000902), National Natural Science Foundation of China (Grant Nos. 61232015, 61472412, 61621003), Beijing Science and Technology Project on Machine Learning-based Stomatology, and Tsinghua-Tencent-AMSS-Joint Project on WWW Knowledge Structure and its Application.

Appendix A.

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• Table 1   Time comparison, accuracy rates of HDM05 database under different reduced dimensions and different Riemannian metrics, i.e., EM and LEM
 Method Dimension $d\times~d$ Time (s) Accuracy rate Lie-LPP-EM $10~\times~10$ $16.1961$ 0.5905 Lie-LPP-EM $15~\times~15$ $21.0645$ 0.6095 Lie-LPP-EM $20~\times~20$ $30.2843$ 0.6095 Lie-LPP-LEM $10~\times~10$ $275.156$ 0.7619 Lie-LPP-LEM $15~\times~15$ $822.289$ 0.7857 Lie-LPP-LEM $20~\times~20$ $1003.13$ 0.7905 Lie-LPP-EM $93~\times~93$ $438.777$ 0.5857 Lie-LPP-LEM $93~\times~93$ $21133.9$ 0.6000
• Table 2   Classification performance of CMU Motion Graph database together with the comparison results for Lie-LPP and traditional manifold learning algorithms
 Method CMU Dimension $d$ Lie-LPP 0.9750 $3~\times~3$ LPP [8] 0.8500 9 LEP [21] 0.9000 9 PCA [27] 0.2500 9
• Table 3   Classification performance of YFB DB together with the comparison results for Lie-LPP and traditional manifold learning algorithms PCA, LPP, LEML, SPD-ML-Stain, and SPD-ML-Airm$^{\rm~a)}$
 YFB DB YFD-trn20/tst40 YFD-trn30/tst30 YFD-trn40/tst20 YFD-trn50/tst10 PCA [27] $46.4~\pm~1.8$ $50.2~\pm~2.2$ $61.8~\pm~1.7$ $63.7~\pm~1.7$ LPP [8] $48.2~\pm~1.5$ $57.2~\pm~2.4$ $62.8~\pm~1.8$ $69.4~\pm~1.4$ LEML [15] $46.5~\pm~1.6$ $53.6~\pm~2.1$ $57.3~\pm~1.7$ $72.9~\pm~1.6$ SPD-Stain [17] $44.6~\pm~1.9$ $52.4~\pm~2.1$ $56.7~\pm~1.3$ $69.4~\pm~1.7$ SPD-Airm [17] $45.2~\pm~1.8$ $53.1~\pm~1.7$ $58.3~\pm~1.6$ $68.5~\pm~1.9$ Lie-LPP $\textbf{50.8}~\pm~\textbf{2.4}$ $\textbf{66.3}\pm~\textbf{1.2}$ $\textbf{70.2}~\pm~\textbf{1.7}$ $\textbf{73.9}~\pm~\textbf{2.1}$

a) “trn" is the abbreviation of training; “tst" is the abbreviation of testing.

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