Deep Manifold Learning of Symmetric Positive Definite Matrices with Application to Face Recognition

In this paper, we aim to construct a deep neural network which embeds high dimensional symmetric positive definite (SPD) matrices into a more discriminative low dimensional SPD manifold. To this end, we develop two types of basic layers: a 2D fully connected layer which reduces the dimensionality of the SPD matrices, and a symmetrically clean layer which achieves non-linear mapping. Specifically, we extend the classical fully connected layer such that it is suitable for SPD matrices, and we further show that SPD matrices with symmetric pair elements setting zero operations are still symmetric positive definite. Finally, we complete the construction of the deep neural network for SPD manifold learning by stacking the two layers. Experiments on several face datasets demonstrate the effectiveness of the proposed method. Introduction Symmetric positive definite (SPD) matrices have shown powerful representation abilities of encoding image and video information. In computer vision community, the SPD matrix representation has been widely employed in many applications, such as face recognition (Pang, Yuan, and Li 2008; Huang et al. 2015; Wu et al. 2015; Li et al. 2015), object recognition (Tuzel, Porikli, and Meer 2006; Jayasumana et al. 2013; Harandi, Salzmann, and Hartley 2014; Yin et al. 2016), action recognition (Harandi et al. 2016), and visual tracking (Wu et al. 2015). The SPD matrices form a Riemannian manifold, where the Euclidean distance is no longer a suitable metric. Previous works on analyzing the SPD manifold mainly fall into two categories: the local approximation method and the kernel method, as shown in Figure 1(a). The local approximation method (Tuzel, Porikli, and Meer 2006; Sivalingam et al. 2009; Tosato et al. 2010; Carreira et al. 2012; Vemulapalli and Jacobs 2015) locally flattens the manifold and approximates the SPD matrix by a point of the tangent space. The kernel method (Harandi et al. 2012; Wang et al. 2012; Jayasumana et al. 2013; Li et al. 2013; Quang, San Biagio, and Murino 2014; Yin et al. 2016) embeds the manifold into a higher dimensional Reproducing Kernel Hilbert Space (RKHS) via kernel functions. On new ⇤corresponding author Copyright c 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. SPD Manifold Tangent Space