## 1 Introduction
### Three Approaches to Grassmann Clustering  
These linear subspaces representing MTS can be analyzed within a special topological space known as the Grassmann manifold--a high-dimensional differentiable manifold defined by a set of linear subspaces [23]. According to this definition, each linear subspace (i.e., MTS sample) is an element of the Grassmann manifold, and can thus be regarded as a point on the manifold, as shown in Figure 1. In this manifold, traditional Euclidean distance-based dissimilarity measures are not applicable due to its non-Euclidean geometry. Consequently, several studies have investigated clustering methods that account for the geometric properties of the Grassmann manifold where the data resides [24]. As shown in Figure 1, we categorize the existing works on Grassmann clustering into three approaches: extrinsic, intrinsic, and semi-intrinsic. Among these, the extrinsic approach, which performs clustering in an embedded low-dimensional space (i.e., local planes) derived from the Grassmann manifold [18, 22, 25], has been a popular choice due to its computational efficiency. Techniques such as reproducing kernel Hilbert space (RKHS) embeddings [15] and Gaussian radial basis function kernels [26] are commonly used to define the embedded space. By enabling the calculation of similarities in a local Euclidean space, extrinsic approach methods have been successfully applied in various studies, including dictionary learning [27] and neighborhood-preserving embedding [22]. However,  
Figure 1: An overview of the three Grassmann MTS clustering approaches: (1) extrinsic approach, which embeds MTS into an Euclidean space (e.g., a tangent space) where the conventional clustering methods can be adopted, (2) intrinsic approach, which uses a geodesic distance for computing the similarity between MTS samples on the Grassmann manifold, and (3) semi-intrinsic approach, in which the sample cluster mean is computed using the Karcher mean. selecting appropriate embedding techniques for different datasets remains a significant challenge [15]. This difficulty arises because the clustering process is performed in the embedded space, which may not fully capture the manifold's global structure. Different from the extrinsic approach, methods belonging to the intrinsic approach [20; 24; 28] cluster MTS samples directly on the surface of the Grassmann manifold, as shown in the middle panel of Figure 1. In this approach, geodesic distance--the shortest path between two points on the manifold--is used as the measure for clustering. By using the geodesic distance, the intrinsic approach takes advantages of preserving the natural distribution of MTS data on the Grassmann manifold. For example, the authors of [20] effectively analyzed image data acquired under variations of lighting conditions and camera perspectives using the intrinsic approach. Moreover, applying this approach to multi-view clustering problems across datasets has been observed to improve clustering accuracy [28]. However, the intrinsic approach imposes certain limitations on the available clustering methods. For instance, since the concept of a mean is not well defined on the manifold, traditional mean-based clustering methods cannot be applied. Both approaches have their respective pros and cons. The last semi-intrinsic approach was designed to take advantages of the two approaches [29]. Specifically, this approach clusters MTS data directly on the manifold using a special type of mean called the Karcher mean [30]. The Karcher mean, which is also known as a Riemannian center of mass, is defined as a local minimizer of the sum of squared distances to a finite set of points on the Grassmann manifold, thereby providing a natural notion of an \"average\" subspace in a non-Euclidean setting. Although it calculates approximated mean, conventional mean-based clustering methods such as \(k\)-means clustering and mean shift can be applied [29]. According to [21; 31], even better MTS clustering accuracy was obtained by using the Karcher mean.