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, which poses a challenge for the training of neural networks that often struggle to reconstruct the global topology of continuous manifolds from sparse samples. We mitigate this issue by 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 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 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 25-NFE DPM-Solver++.