用几何结构建模时间序列，让长期预测更稳定可靠。
Ellipsoidal Time Series Forecasting
- 基于最优传输理论，将预测建模为从高斯分布到数据椭球的变换
- 在非平稳场景下性能超越基线790倍，有效预测时长提升显著
- 适合需要高鲁棒性的长期预测任务，如金融或气候建模
我们认为，长期预测需要学习具有显式谱结构的局部雅可比矩阵，而不仅仅是条件均值匹配。我们的方法Fern利用Brenier定理，直接参数化雅可比矩阵为对称半正定（SPD）分解形式，将预测视为从固定高斯源到数据相关椭球的概率质量最优传输。该框架将特征分解的计算成本从立方级降低至线性级，同时提供可解释、几何感知的投影。为严格评估鲁棒性，我们引入一个带有可控非平稳冲击的合成基准，并提出新指标有效预测时间（EPT）。Fern在非平稳设置下表现出极强稳定性，其性能优于DLinear和Koopa等基线模型达两个数量级以上（最高达790倍），而标准基准无法暴露这些模型的脆弱性。
原文摘要 · Abstract (English)
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.
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