## 2 Grassmann Manifold
### A Brief Review of the Grassmann Manifold  
A Grassmann manifold \(G_{r,s}\), which shows a nonlinear structure, can be defined as a set of \(r\)-dimensional linear subspaces embedded in an \(s\)-dimensional Euclidean space, where \(r\leq s\). A linear subspace can be perceived as a point on a Grassmann manifold, which may be specified by an \(s\times s\) orthogonal matrix \(\mathbf{X}=\mathbf{X}^{T}\) of rank \(r\). Theoretically, the Grassmann manifold is defined as a particular subset of the symmetric matrices \(Sym_{s}\)[48] as follows:  
\[G_{r,s}=\{\mathbf{X}\in Sym_{s}|\mathbf{X}^{T}=\mathbf{X},rank(\mathbf{X})=r\}. \tag{3}\]  
A linear dynamical system (LDS) can thus be regarded as a point on the Grassmann manifold corresponding to the column space of the extended observability matrix [20]. Geodesic distance is used for the measurement between two MTS points on the manifold. This geodesic distance is defined as the length of the shortest curve between the two points [23], as shown in Figure 2.  
In this context, to represent the subspace spanned by the columns of extended observability matrices \(\mathbf{O}_{m}^{T}\) defined in (2), an orthonormal basis can be computed by using thin singular value decomposition (thinSVD) [49] (see Algorithm 1). In other words, a linear subspace corresponding to (2) is operationally stored as a thin-tall orthonormal matrix \(\mathbf{U}\in\mathbb{R}^{q\times d}\) for computation. Algorithm 1 details the process of representing an MTS as a linear subspace.  
```
1:Input: Given a set of \(n\) MTSs, \(\{\mathbf{Y}_{i}\}_{i=1}^{n}\); \(\forall\mathbf{Y}_{i}\in\mathbb{R}^{p\times L}\), state-space dimension \(d\), size of truncated observability matrix, \(m=d\)
2:for\(i\gets 1\) to \(n\)do
3: Compute \(\mathbf{T}_{i}\in\mathbb{R}^{d\times d}\) and \(\mathbf{M}_{i}\in\mathbb{R}^{p\times d}\) of LDS model using (1) for each \(\mathbf{Y}_{i}\).
4: Compute \(\mathbf{O}_{i}^{T}=\left[\mathbf{M}_{i}^{T},(\mathbf{M}_{i}\mathbf{T}_{i})^{ T},(\mathbf{M}_{i}\mathbf{T}_{i}^{2})^{T},...,(\mathbf{M}_{i}\mathbf{T}_{i}^{m-1})^{T} \right]\in\mathbb{R}^{mp\times d}\) using the \(\mathbf{M}_{i}\) and \(\mathbf{T}_{i}\).
5: Compute \(\mathbf{U}_{i}\mathbf{\Sigma}_{i}\mathbf{V}_{i}^{T}=thinSVD(\mathbf{O}_{i}^{T})\)
6:endfor
7:Output: Subspaces \(\{\mathbf{U}_{i}\}_{i=1}^{n}\); \(\forall\mathbf{U}_{i}\in\mathbb{R}^{q\times d}\), where \(q=mp\).
```  
**Algorithm 1** Subspace generation for MTS  
Figure 2: (a) State-space dynamical models can be described in \(\mathbb{R}^{d}\) by linear subspaces. To compare two linear subspaces, the principal angles between them can be used conventionally. (b) Linear subspaces in \(\mathbb{R}^{d}\) can be represented as points on the Grassmann manifold.  
The canonical distance between two subspaces is called a principal angle [50]. Let \(\mathbf{U}\) and \(\mathbf{V}\) be two orthonormal matrices; the principal angles \(0\leq\theta_{1}\leq...\leq\theta_{d}\leq\pi/2\) between two subspaces \(span(\mathbf{U})\) and \(span(\mathbf{V})\) are defined recursively by  
\[\begin{split} cos\theta_{k}&=\max_{u_{k}\in span( \mathbf{U})}\ \max_{v_{k}\in span(\mathbf{V})}u_{k}^{T}v_{k}\\ \text{subject to}& u_{k}^{T}u_{k}=1,v_{k}^{T}v_{k}=1,u_{k}^{T}u_{j}=0\text{ and }v_{k}^{T}v_{j}=0,\end{split} \tag{4}\]  
where \(j=1,...,k-1\). The cosine of the principal angle is the first canonical correlation [47].