Primal-Spectral Generative Modeling: Fast Analytical Generation via Pseudoinverse Lévy Inversion

A probability distribution $\mathbb{P}$ is a measure on a $\sigma$-algebra, assigning mass to sets rather than points. This poses a challenge for the training of neural networks, which often struggle to reconstruct the global topology of continuous manifolds from sparse samples. We mitigate this issue by innovatively transforming $\mathbb{P}$ into a continuous function via spectral methods, providing theoretical guarantees for the convergence of the learned distribution to the true distribution. Specifically, we introduce a network, PriSpecNet, with a single-function evaluation (1-NFE) Pseudoinverse Lévy Inversion (PiLI) solver that regards generation as a fast analytical problem, eliminating the need for iterative numerical integration. By reformulating the generation in the spectral domain, we bypass the computationally expensive sampling trajectory while maintaining full compatibility with the stochastic interpolants. We test our PriSpecNet in two applications. For time series, it unifies generation and forecasting, outperforming state-of-the-art (SOTA) baselines with Context-FID reductions of 50.0\%, 41.5\%, 80.6\%, and 63.1\% on Sines, Solar, ETTh, and Stock benchmarks, respectively, also decreasing forecasting MSE by 29.8\% on Solar and 23.8\% on Stock. For vision on ImageNet $256 \times 256$, 1-NFE PiLI achieves a competitive FID of **1.66** using only **26** Gflops, representing a **170** $\times$ reduction in total Gflops compared to the 4,436 Gflops required by the advanced 25-NFE DPM-Solver++.