where y l and x l-1 represent the activations of layer l and the preceding layer l -1, respectively. The function f l is the activation function, W l is the weight matrix defining the connections between layers, and b l is the bias vector for layer l. The choice of activation function is crucial for introducing nonlinearity into the network, allowing it to effectively capture the nonlinear features of the one-dimensional manifold. The weights and biases serve as trainable parameters optimized during training to minimize the loss function. Now, consider the derivative of the additional skewnessrelated loss function term, L skew , with respect to the NN parameters {θ} n := {{w ij l }, {b i l }} n , which include all trainable variables: the weight elements {w ij l } from matrix W l and the bias terms {b i l } from vector b l across all layers l in the Skewencoder. The subscript n denotes that these parameters are optimized based on the n-th batch. The gradient of L skew , as shown in Equation ( ), appears in any gradient-based optimization method, ensuring the integration of skewness considerations into the training process.