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Context: Smooth deformations are key to generate realistic and locally transformed samples.A naïve application of a random local transformation within its finite neighborhood may result in a discontinuous shape and an overlap of different parts. It has a high chance to lose discriminative structures.Instead, we employ the Nadaraya-Watson kernel regression [27, 28] to smoothly interpolate the local transformations in the 3D space.Given M local transformations \{T_{j}\}_{j=1}^{M}, our smoothly varying transformation at an arbitrary point \mathbf{p}_{i} is given as:\small\hat{T}(\mathbf{p}_{i})=\frac{\sum_{j=1}^{M}{K_{h}}(\mathbf{p}_{i},\mathbf{p}^{\mathcal{A}}_{j})T_{j}}{\sum_{k=1}^{M}{K_{h}}(\mathbf{p}_{i},\mathbf{p}^{\mathcal{A}}_{k})},(3)where K_{h}(\cdot,\cdot) is a kernel function with bandwidth h, and T_{j} is the local transformation in (2) centered at \mathbf{p}^{\mathcal{A}}_{j}.To define \hat{T}(\mathbf{p}_{i}) at any point in the 3D space, we use a kernel function that has a strictly positive value for any pair of points,i.e., K_{h}(\mathbf{p}_{i},\mathbf{p}_{j})>0 for \forall\mathbf{p}_{i},\forall\mathbf{p}_{j}.The following proposition theoretically guarantees that our augmentation is a smooth transformation under mild conditions. The proof is in the supplement. Thus, CDA is simply a similarity transformation with small jittering that cannot simulate diverse shapes and deformable objects.Unlike synthetic datasets like ModelNet [14] and ShapeNet [26], a real-world dataset like ScanObjectNN [1] further necessitates the generation of sophisticated deformations such as a mixture of local transformations.These are exemplified in Figure 1: airplanes with varying lengths and directions of wings and body, guitars in varying sizes and aspect ratios, and people with different heights and postures (e.g., crossing legs).

Question: What does it mean the realistic sample?
A realistic sample in this context refers to a 3D object that has undergone a smooth deformation, meaning that the shape of the object changes gradually rather than abruptly. The realistic samples that the authors aim to generate are those that resemble real-world objects with diverse shapes and deformations, such as airplanes with varying wing lengths and directions, guitars with different sizes and aspect ratios, and people with different heights and postures.