## 3 Methodology
### Data Complexity Measurement  
As mentioned in Section 1, the specification of easy samples in curriculum learning is a task-specific issue which may vary due to different tasks. The aim of this study was to design an approach that could rank the training images from easy to complex in a relatively unsupervised manner. This approach is inspired by the K-means algorithm, where the complexity of the training samples can be measured through the data distribution density. We assumed that by training a neural network, each class of samples can be mapped into a deep feature space. Higher local densities indicate more likely clean samples with correct labels, because they are often clustered together with a similar appearance. Conversely, samples with noisy labels have a small density, i.e., poor aggregation. In this case, we conduct the following procedures in each category:  
Step 1: Set the training sample set as \(T\), the number of sample categories as \(n\), and the training sample dataset of the \(i\)th class as \(T_{i}\), and \(T_{i}\) contains \(m_{i}\) samples, \(i=1,\ 2,\ldots n\). In \(T_{i}\), we calculate the Euclidean distance between the \(j\)th sample feature vector and the 1 st sample feature vector, and construct an \(m_{i}\times m_{i}\) intra-class Euclidean distance matrix:  
\[D_{i}= \begin{bmatrix}d_{i,\ 1.1}&\cdots&d_{i,\ 1,\ 1}&\cdots&d_{i,\ 1,\ m_{i}}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ d_{i,\ j,\ 1}&\cdots&d_{i,\ j,\ l}&\cdots&d_{i,\ j,\ m_{i}}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ d_{i,m_{i},\ 1}&\cdots&d_{i,\ m_{i},\ l}&\cdots&d_{i,\ m_{i},\ m_{i}}\end{bmatrix}  \tag{1}\]  
where \(i=1,\ 2,\ldots n,\ j=1,\ 2,\ldots m_{i}\), \(l=1,\ 2,\ldots m_{i}\). Then, sorting the element \(d_{i,\ j,\ 1}\) in the intra-class Euclidean distance matrix \(D_{i}\) to determine the median value \(d_{i,\ th}\). Step 2: For the \(j\)th sample in the \(i\)th class, calculate its intra-class local density, \(p_{i,\ j}\):  
\[p_{i,\ j}=\sum_{l=1}^{m_{i}}Y\Big{(}d_{i,\ j,\ l}-d_{i,\ th}\Big{)} \tag{2}\]  
where \(Y(x)= \begin{cases}1,&x<0\\ 0,&x\geq 0^{{}^{\prime}}\end{cases} \) and \(p_{i,\ j}\) is the number of samples whose Euclidean distance between the feature vector in \(T_{i}\) and the \(j\)th sample feature vector is less than \(d_{i,\ th}\). The larger \(p_{i,\ j}\) indicates that, in the feature space, more samples are close to each other with the feature vector of the \(j\)th sample. Step 3: For the \(j\)th sample in the \(i\)th class, calculating its intra-class distance is performed as follows:  
\[\delta_{i,\ j}= \begin{cases}max_{l\in\{1,\ 2,\ \ldots,\ m_{i}\}}(d_{i,\ j,\ l}),& \forall l,\ s,\ t.\ p_{i,\ j}\geq p_{i,\ l}\\ \quad min_{l:\ p_{i,\ l}>p_{i,\ j}}(d_{i,\ j,\ l}),&\exists l,\ s.\ t.\ p_{i,\ j}<p_{i,\ l}\end{cases}  \tag{3}\]  
when the \(j\)th sample of the \(i\)th class has the largest intra-class local density of the \(i\)th class, \(\delta_{i,\ j}\) represents the largest value of the \(j\)th row of the distance matrix, \(D_{i}\); otherwise, \(\delta_{i,\ j}\) represents the distance between the sample point with the smallest distance from the \(j\)th sample point and the \(j\)th sample point among all the sample points in the \(i\)th class, whose intra-class local density is greater than the \(j\)th sample. Step 4: Since the cluster center should have the largest \(p_{i,\ j}\) value and the largest \(\delta_{i,\ j}\) value at the same time [30], determining the cluster center involves calculating the product of the intra-class local density, \(p_{i,\ j}\), and the intra-class distance, \(\delta_{i,\ j}\):  
\[\theta_{i,\ j}=p_{i,\ j}\times\delta_{i,\ j} \tag{4}\]Step 5: Sorting the \(\theta_{i,\ j}\) value of the \(i\)th samples (\(j=1,\ 2,\ldots m_{i}\)) in descending order, the feature vectors of the samples corresponding to the first three \(\theta_{i,\ j}\) values are selected as the initial cluster center.