# Feature Transformation-Based Few-Shot Class-Incremental Learning
## 4 Simulation Experiments
### Ablation Experiments
#### 4.3.4 Comparison with Other Feature Transformation Methods  
There exist various commonly used feature transformation methods, such as Principal Component Analysis (PCA), widely adopted for dimensionality reduction, and Transfer Component Analysis (TCA), frequently used in transfer learning. To validate that the proposed transformation method performs better under the constraint of a fixed feature extractor, we compare it with these alternative transformation methods in Figure 11. All methods use the same feature extractor with identical parameters for fair comparison. PCA aims to reduce the dimensionality of the data while ensuring that the sample points remain well-separated in the new low-dimensional feature space. A common measure of separation is the covariance between vectors--when the covariance between two vectors in the transformed space is zero, they are considered completely uncorrelated, achieving the objective of separation. It is important to note that PCA involves centering the data by subtracting the mean. Therefore, if the test samples and class prototypes are jointly transformed via PCA, the distribution of test classes can significantly influence the  
Figure 11: Performance of different feature transformation methods. Figure 10: Performance comparison of base and novel classes before and after feature transformation. resulting feature space. To avoid this issue, we apply PCA separately for each test sample and the class prototypes before performing classification. The original feature dimension is 512, and after PCA, the dimensionality is reduced to match the number of class prototypes available in the current session. TCA is a classical method in transfer learning. It aims to minimize the Maximum Mean Discrepancy (MMD) between the marginal distributions of the source domain dataset \(X_{S}\) and the target domain dataset \(X_{T}\) through feature transformation:  
\[MMD(P_{s}(x),P_{t}(x))=\|\frac{1}{|X_{S}|}\sum_{i=1}^{|X_{S}|}\phi(x_{i})-\frac {1}{|X_{T}|}\sum_{j=1}^{|X_{T}|}\phi(x_{j})\|^{2}_{\mathcal{H}_{k}} \tag{4}\]  
where \(\mathcal{H}_{k}\) denotes a Reproducing Kernel Hilbert Space (RKHS) defined by a kernel function \(k\). The mapping \(\phi(\cdot)\) defines a transformation from the original data space to \(\mathcal{H}_{k}\), where the kernel function corresponds to the inner product in the mapped space. In our experiments, we use the Gaussian (RBF) kernel with a kernel width of 1. When both the source and target domain data are specified, the TCA problem becomes an optimization problem that can be solved using the Lagrange dual method, without the need for gradient-based training. However, in incremental learning, there is no explicit target domain for domain adaptation. Upon analysis, we consider the maintained set of class prototypes to serve as a relatively good approximation of the target domain. The prototypes of the base classes are derived from pre-training and are thus highly representative. The prototypes of the novel classes are estimated as the mean of the support samples, representing the feature distribution of new classes after incremental updates. Therefore, the matrix formed by class prototypes after the \(i\)-th incremental session can be regarded as a good approximation of the target domain features. Since TCA is also a dimensionality reduction method, the dimension of the target feature space is set to the number of class prototypes available at the current stage. In addition to PCA and TCA, we also compare with a recently proposed method named Graph Random Neural Networks (GRAND) [26]. From a structural perspective, GRAND combines a multilayer perceptron (MLP) with label propagation techniques. It includes additional mechanisms such as random feature dropout and propagation depth control, which contribute to its strong performance in graph-based tasks. In our implementation, the node features are extracted by the fixed feature extractor, edge weights are computed using cosine similarity between node pairs, and the propagation depth is set to 1.