**5. Discussion**

It has been proved that trade-off parameters, *q* and *C*, have a significant impact on the results of SVC [5,7]. Obviously, we may spend more time in finding the optimal parameters that characterize a better boundary of clusters for SVC. This will result in a large number of SVs during the tuning process, which may affect the partition of clusters and is unreasonable, obviously. We know that it is feasible to adjust parameter *C* to obtain better performance, but it comes at the cost of increasing outliers. To solve these problems and inspired by the margin theory, we reconstruct a new hypersphere to identify the clusters to make denser sets more easily divided by employing the margin distribution, and then we establish the corresponding theory. We circumvent the high complexity resulting from the variance by demonstrating Theorem 1 and employing the Gaussian kernel, and then we derive the convex optimization problem.

As for the MDSVC algorithm, we design the customized DCD method to solve the convex optimization problem [25]. MDSVC has two other trade-off parameters compared to SVC, namely, *λ*1, *λ*2. Furthermore, we demonstrate that both of them play an important role in MDSVC through experiments shown in Figure 2 and equations about hypersphere we derive in Section 2. In Figure 4, we can obtain some useful instructive insights as an avenue for adjusting the number of SVs. Therefore, we can obtain better performance by increasing the *λ*<sup>1</sup> value while there are few SVs. Moreover, we can increase *λ*<sup>2</sup> value to reduce SVs. If one focuses on forming better outlines of clusters, the recommendation is to control the ratio of *λ*<sup>1</sup> and *λ*<sup>2</sup> to between 10−<sup>2</sup> and 102. Once the number of SVs changes drastically, there is no need for us to increase the value of *λ*<sup>1</sup> and *λ*2. Meanwhile, what we should be aware of is that *λ*<sup>1</sup> should not be zero. We further theoretically prove that the error has an upper bound in Section 3. Due to the lack of prior knowledge (true labels) of clustering algorithms, it is difficult for us to achieve our error bound in a manner similar to the approach used in LDM. We make it by taking the advantage of the error proposed in SVDD [6] and the lemma derived in CCL [9]. According to Figure 1b,c and Figure 4c–e, minimizing the mean and variance can make datasets properly outlined with a proper amount of SVs from a practical and theoretical perspective, while the outlines of SVC are inappropriate. However, we found that our method performed generally when the edge points of datasets are separated relatively densely, where edge points are a collection of relatively sparsely distributed points in the data space. Based on the experiments and formulas obtained; thus, we think that our method performs better on the datasets with edge points dispersing sparsely.

In short, the novel contribution of our work is that we redefine the hyperplane and the center in feature space considering the distribution of data to form better boundaries with a proper amount of SVs. Furthermore, experimental results in most datasets indicate that MDSVC achieves better performance, which further demonstrates the superiority of our method. In the future, we will design a corresponding method to improve the performance, which redefines the clustering partition.
