Ellipsoidal Time Series Forecasting

Abstract

We argue that long-term forecasting requires learning local Jacobians with explicit spectral structure, going beyond simple conditional mean matching. Our method, Fern, invokes Brenier's theorem to directly parameterize the Jacobian as a symmetric positive semi-definite (SPD) factorization, treating forecasting as the optimal transport of probability mass from a fixed Gaussian source to data-dependent ellipsoids. This formulation reduces the computational cost of eigendecomposition from cubic to linear time while providing interpretable, geometry-aware projections. To rigorously evaluate robustness, we introduce a synthetic benchmark with controlled non-stationary shocks alongside new metrics like Effective Prediction Time (EPT). Fern demonstrates exceptional stability, outperforming baselines like DLinear and Koopa by over two orders of magnitude (up to 790x) on nonstationary settings where standard benchmarks fail to expose model brittleness.