Ellipsoidal Time Series Forecasting

TL;DR

Fern is a novel time series forecasting method that uses spectral-structured Jacobians and optimal transport to improve robustness and interpretability, especially in non-stationary environments.

Contribution

The paper introduces Fern, a spectral Jacobian-based forecasting approach leveraging Brenier's theorem, reducing computational costs and enhancing robustness against non-stationary shocks.

Findings

Fern outperforms baselines like DLinear and Koopa by over 790x in nonstationary settings.

The method provides interpretable, geometry-aware projections.

Fern demonstrates exceptional stability in synthetic benchmarks with controlled shocks.

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.