## 2 Preliminaries
### The Geometry of SPD Manifold  
The \(D\times D\) dimensional matrix \(X\) is symmetric and positively definite if \(X^{T}=X\) and the scalar \(v^{T}Xv>0\) for any non-zero column vector \(v\in R^{D}\). The space of all \(D\times D\) dimensional SPD matrixes is expressed as \(Sym_{++}^{D}\). If \(Sym_{++}^{D}\) is given a Riemannian metric, the space of the SPD matrix becomes a Riemannian manifold. Namely, the SPD manifold is defined as  
\[Sym_{++}^{D}=\Big{\{}X\in R^{D\times D}\Big{|}X^{T}=X,v^{T}Xv>0\;for\;\forall v \in R^{D}\,\Big{\}}. \tag{1}\]  
Figure 1: Overview of the proposed framework. Taking face video as example, the four boxes of red, yellow, blue, and green in (**a**) show the regions of interest (ROIs) on human face and time series of ROIs stringed through motion estimation. The regional covariance descriptors (RCDs) are computed for each region in (**b**). Hence, a family of feature curves on symmetric positive definite (SPD) manifold are built in (**c**). For all \(X\in Sym_{++}^{D}\), its tangent space of symmetric \(D\times D\) matrices with logarithm mapping \(Log_{X}:Sym_{++}^{D}\to T_{X}\big{(}Sym_{++}^{D}\big{)}\) :  
\[T_{X}\big{(}Sym_{++}^{D}\big{)}=\Big{\{}\Phi\Big{|}\Phi\in R^{D\times D},\Phi^{T }=\Phi\Big{\}}. \tag{2}\]  
The geometry of SPD manifolds is usually learned through Riemannian metrics \(\omega\). The Riemannian metric defines the inner product on tangent space. For any \(X\in Sym_{++}^{D}\) and \(\Phi,\Theta\in T_{x}\big{(}Sym_{++}^{D}\big{)}\),  
\[\omega(\Phi,\Theta)=\left\langle\Phi,\Theta\right\rangle_{X}. \tag{3}\]  
And the inner product reflects the length of the curve between corresponding points on SPD manifold \(Sym_{++}^{D}\). The curve with the shortest distance is the geodesic between two elements on \(Sym_{++}^{D}\). The length of the geodesic is called the geodesic distance. The affine invariant Riemannian metric (AIRM) [1] is the most frequently used Riemannian metric. For all \(\Phi,\Theta\in T_{x}\big{(}Sym_{++}^{D}\big{)}\), the AIRM is defined as  
\[\left\langle\Phi,\Theta\right\rangle_{X}=\left\langle X^{-1/2}\Phi X^{-1/2},X ^{-1/2}\Theta X^{-1/2}\right\rangle_{F}, \tag{4}\]  
where \(\left\langle A,B\right\rangle_{F}=tr\big{(}AB^{T}\big{)}\) and \(X\in Sym_{++}^{D}\). \(\left\langle\cdot,\cdot\right\rangle_{X}\) are an inner product and its smoothing over \(Sym_{++}^{D}\), respectively. The logarithm mapping projecting \(X_{2}\in Sym_{++}^{D}\) to tangent space \(T_{X_{1}}\big{(}Sym_{++}^{D}\big{)}\) is defined as  
\[Log_{X_{1}}(X_{2})=X_{1}^{\frac{1}{2}}log\bigg{(}X_{1}^{-\frac{1}{2}}X_{2}X_{ 1}^{-\frac{1}{2}}\bigg{)}X_{1}^{\frac{1}{2}}. \tag{5}\]  
For any given pair \(X_{1},X_{2}\in Sym_{++}^{D}\), the unique geodesic [23] induced from AIRM connecting \(T_{X_{1}}^{X_{2}}(0)=X_{1}\) with \(T_{X_{1}}^{X_{2}}(1)=X_{2}\) is given by  
\[\Gamma_{X_{1}}^{X_{2}}(t)=X_{1}^{\frac{1}{2}}exp\bigg{(}t\ logX_{1}^{-\frac{1 }{2}}X_{2}X_{1}^{-\frac{1}{2}}\bigg{)}X_{1}^{\frac{1}{2}},t>0. \tag{6}\]  
The geodesic distance between \(X_{1},X_{2}\in Sym_{++}^{D}\) induced from AIRM is as follows:  
\[\delta_{AIRM}^{2}(X_{1},X_{2})=\big{\|}log\Big{(}X_{1}^{-1/2}X_{2}X_{1}^{-1/2 }\Big{)}\big{\|}_{F}^{2}, \tag{7}\]  
where \(\big{\|}A\big{\|}_{F}^{2}=\left\langle A,A\right\rangle_{F}\) is the Frobenius norm of a matrix. The AIRM possesses a property of invariance to affine transformations, i.e., \(\delta_{AIRM}^{2}(X_{1},X_{2})=\delta_{AIRM}^{2}\big{(}AX_{1}A^{T},AX_{2}A^{T} \big{)}\) for \(\forall A\in GL(D)\). For all \(\Phi,\Theta\in T_{x}\big{(}Sym_{++}^{D}\big{)}\), the log-Euclidean metric (LEM) [2; 20] is defined as  
\[\left\langle\Phi,\Theta\right\rangle_{X}=\left\langle D_{X}log(\Phi),D_{X}log( \Theta)\right\rangle, \tag{8}\]  
where \(X\in Sym_{++}^{D}\) and \(D_{X}log(\Phi)\) denote the directional derivative of \(log(\Theta)\) at \(X\).