Tail Annealing for Heavy-Tailed Flow Matching

摘要

Standard generative models struggle with heavy-tailed data: Lipschitz architectures cannot produce power-law tails from Gaussian noise, and interpolating between heavy-tailed data and Gaussians is ill-posed. We propose a simple fix: apply the soft-log transform $\phi(x) = \mathrm{sign}(x) \cdot \log(1 + |x|)$ to data before training, then exponentiate samples after generation. This compresses heavy tails into a range where standard flow matching succeeds. The approach requires no tail parameter estimation, no heavy-tailed base distributions, and no architectural modifications. We provide theoretical intuition for why this works: the log-transform maps Pareto tails to exponentials, and the induced dynamics implement a form of tail annealing via power transformations. Experiments on synthetic benchmarks and real financial data show that this simple trick achieves competitive sample quality for heavy-tailed distributions ($\nu \leq 5$), with improved stability over specialized methods in moderate dimensions ($d = 50$).