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 L\'evy 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++.

(1) Problem: Artificial intelligence models learn to generate new images or forecast future events by studying examples. However, they often struggle to grasp the "big picture" from limited data. Furthermore, generating this new content usually requires a slow, step-by-step process that consumes massive amounts of computing power. (2) Solution: To solve this, we translated the data into a continuous, wave-like mathematical format. This global view allowed us to develop a new AI system that treats data generation as a direct, solvable equation. Instead of taking dozens of small steps to produce an image or a forecast, our method calculates the final high-quality result in a single, rapid step. (3) Impact: This approach proved highly effective. It significantly reduced errors when predicting complex trends like solar energy output or stock market fluctuations. When generating images, it matched the quality of leading methods while using up to 170 times less computing power. Ultimately, our work makes advanced AI generation much faster, more accurate, and far more energy-efficient.