Ellipsoidal Time Series Forecasting

We argue that long-horizon forecasting requires learning local Jacobians with explicit spectral structure, not only matching conditional means. Our method, Fern (Forecasting with Ellipsoidal RepresentatioNs), 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 avoids post-hoc eigendecomposition of dense Jacobians, enables efficient Householder-based orthogonal factors, and exposes interpretable diagnostics such as local stretching, spectral radius, and volume change. To rigorously evaluate robustness, we introduce controlled synthetic stress tests with nonstationary shocks, together with Wasserstein-based shape metrics and Effective Prediction Time. Fern demonstrates exceptional stability, outperforming baselines like DLinear and Koopa by over two orders of magnitude (up to $790\times$) on nonstationary settings where standard benchmarks fail to expose model brittleness.