**3. Minimum Distribution Support Vector Clustering**

In this section, we briefly delineate the process of MDSVC, including three subsections, the formula of MDSVC, which minimizes both the mean and the variance, the optimization algorithms based on dual coordinate descent method, and the statistical property of MDSVC that shows the upper bound of the expectation of error. In this research, as mentioned before, we take the Gaussian kernel as a nonlinear transformation approach to map data points to the feature space, and then we derive *k*(*x*, *x*) = 1, which is critical for us to simplify the variance and solve the objective function. In addition, we define the mean and variance based on the Euclidean distance. The reason we employ the Euclidean distance is that we can take the objective function as the convex quadratic programming function and the Euclidean norm represents the actual distance between two points rather than the distance on the surface.

We delineate the idea of our algorithm in the feature space in Figure 1 roughly, and more detailed descriptions are given in Sections 3.1.1 and 3.1.2. First, the hyperplanes

of MDSVC, SVC, and the unit ball are shown in Figure 1a. By characterizing and minimizing our mean and variance, we can, thus, have the hypersphere of MDSVC as an inclined curved surface in the feature space, as indicated in red in Figure 1a. The intersection of the SVC's hypersphere and the unit sphere is a cap-like area. We further illustrate the main difference between MDSVC and SVC through a lateral view and top view, which are shown in Figure 1b,c, respectively. Figure 1b is the schematic diagram of the MDSVC's Cap and the SVC's Cap. We can find that the center *a* of MDSVC's hypersphere moves away from the center of the ball and inclines to the distribution of the overall data because of the mean and variance. In Figure 1c, we use *Soft-Rsvc* to represent the soft boundary of SVC. The centers of the three spheres, namely the unit ball, SVC's hypersphere, and MDSVC's hypersphere, are denoted by *o*, *asvc*, and *a*, respectively. We also use red points to indicate the SVs of MDSVC. As shown in Figure 1c, we can see how the boundary of MDSVC *R* is determined. Finally, we use Figure 1d to show the distribution of data points and the details of the Cap formed by SVC.

**Figure 1.** (**a**) Hyperplanes of SVC and MDSVC. (**b**) Two caps formed by SVC and MDSVC with the unit-ball respect tively. (**c**) Top view of Figure 1a. (**d**) Data distribution in the cap.
