Fourier Latent framework for 3D Point Cloud Generation

Generating 3D point clouds poses unique challenges due to their unordered nature and variable cardinality. Existing generative approaches often couple point cloud shape representations to the number of points, limiting their flexibility and scalability. We propose FourierDiT, a Fourier latent framework that decouples shape representation from point sampling. Our key idea is to represent a point cloud through its underlying shape distribution using empirical characteristic-function evaluations at a set of learnable spatial frequencies. The resulting Fourier latent representation is permutation-invariant by construction and has a fixed dimensionality independent of point-cloud cardinality. Building on this representation, we train a shape-conditioned vector-field decoder via flow matching, allowing an arbitrary number of points to be sampled from a single latent code. A frequency-conditioned Diffusion Transformer further learns the distribution of Fourier latent codes, enabling the synthesis of novel shapes. Extensive experiments demonstrate competitive generation quality and improved training efficiency, while reconstruction and interpolation results highlight the effectiveness of the learned Fourier latent space. Our framework provides a flexible and efficient approach to 3D point cloud generation without tying the point cloud representation to a fixed number of points.