## 2 Materials and Methods
### Classification
#### 2.6.4 CK-Svm  
The Gaussian kernel embeds the original space \(R^{N}\) into an infinite dimensional space \(F\) as a Riemannian manifold lying on a unit ball centered at the origin of the space \(F\) (see the author's previous work [34] for an illustration), and the kernel-induced Riemannian metric is given by the following:  
\[g_{ij}(\mathbf{x})=\left.\frac{\partial k(\mathbf{x},\mathbf{x}^{\prime})}{ \partial x_{i}\partial x_{j}^{\prime}}\right|_{\mathbf{x}^{\prime}=\mathbf{x}} \tag{7}\]where \(x_{i}\) denotes the \(i\)th element of \(\mathbf{x}\). The relationship between the Riemannian distance \(ds\) and a small displacement \(d\mathbf{x}\), as follows,  
\[ds^{2}=\sum_{i,j}g_{ij}(\mathbf{x})dx_{i}dx_{j} \tag{8}\]  
indicates how a local volume in \(R^{N}\) is magnified or contracted in \(F\) under the mapping of \(\varphi\). A conformal transformation of \(\varphi\) is defined by \(\widetilde{\varphi}(\mathbf{x})=Q(\mathbf{x})\varphi(\mathbf{x})\), where \(Q(\mathbf{x})\) is a real-valued conformal function. The function \(Q(\mathbf{x})\) can be chosen in a way in which its value is the largest at the vicinity of \(\varphi(\mathbf{x})\) and decreases with the distance from the position of \(\varphi(\mathbf{x})\). Designing such a transformation yields the conformal transformation of a primary kernel \(k\) as follows:  
\[\widetilde{k}(\mathbf{x},\mathbf{x}^{\prime})=Q(\mathbf{x})Q(\mathbf{x}^{ \prime})k(\mathbf{x},\mathbf{x}^{\prime}) \tag{9}\]  
where the transformed kernel \(\widetilde{k}\) is called a conformal kernel. Let a set \(T\) contain a set of data points \(\mathbf{x}_{i}\)s. To magnify the spatial resolution around the images of the data points \(\mathbf{x}_{i}\)s in the space \(F\), a conformal function consisting of a set of Gaussian functions can be defined as follows [21]:  
\[Q(\mathbf{x})=\sum_{\mathbf{x}_{i}\in T}exp\Bigg{(}\frac{\left\|\varphi( \mathbf{x})-\varphi(\mathbf{x}_{i})\right\|^{2}}{\frac{1}{n}\sum_{j}\left\| \varphi(\mathbf{x}_{j})-\varphi(\mathbf{x}_{i})\right\|^{2}}\Bigg{)} \tag{10}\]  
where the denominator term \(\frac{1}{n}\sum_{j}\left\|\varphi(\mathbf{x}_{j})-\varphi(\mathbf{x}_{i}) \right\|^{2}\) computes the mean squared distance from the image of \(\mathbf{x}_{i}\) to its \(n\) nearest neighbors \(\varphi(\mathbf{x}_{j})\). According to Equation (9), the conformal function \(Q(\mathbf{x})\) has the largest value at the position of the image of the set \(T\) and decays with the distance from the image of \(T\) in the feature space \(F\). [PERSON] and [PERSON] [21] proposed that the set \(T\) should collect the training data points whose \(\alpha_{i}>0\) (i.e., the support vectors) based on the fact that most support vectors lie within the margin of separation. However, some support vectors may be far from the margin. Accordingly, [PERSON] et al. [22] defined the set \(T\) as follows:  
\[T=\big{\{}\mathbf{x}_{i}|\mathbf{x}_{i}\in SV\text{ }and\left|D_{SVM}(\mathbf{x}_{i}) \right|\leq 1\big{\}}. \tag{11}\]  
Because the training data points in the set \(T\) are the support vectors falling inside of the separation margin, enlarging the spatial resolution around the image of the set \(T\), defined as Equation (11), is equivalent to increasing the spatial resolution of the separation margin in the feature space \(F\). In this paper, we adopted the conformal function and the set \(T\) suggested in [21; 22], respectively. Because the Gaussian function is adopted as the kernel, \(k(\mathbf{x},\mathbf{y})=1,\ \forall\mathbf{x}=\mathbf{y}\). Thus, the conformal function expressed in Equation (10) can be simplified as follows:  
\[Q(\mathbf{x})=\sum_{\mathbf{x}_{i}\in T}exp\Bigg{(}\frac{2-2k(\mathbf{x}, \mathbf{x}_{i})}{\frac{1}{n}\sum_{j}\left(2-2k\big{(}\mathbf{x}_{j},\mathbf{x} _{i}\big{)}\right)}\Bigg{)}. \tag{12}\]  
In this study, we set \(n=3\). The training of CK-SVM consists of two steps: firstly, train an SVM with a Gaussian kernel; and secondly, retrain the SVM with the conformal kernel defined in Equation (9).