基于可学习分解与流形动力学的双曲时间序列预测
Hyperbolic Time Series Forecasting with Learnable Decomposition and Manifold Dynamics
AI总结：
提出首个双曲空间长期时间序列预测框架HyperForecast，通过可学习分解与流形动力学建模，在11个基准的9个上优于现有方法，长 horizon 下均方误差最高降58%。
摘要：
多尺度结构是时间序列预测的核心：时间信号包含缓慢变化的趋势、中间的季节模式以及高频残差。传统预测器在欧氏空间中对这种多尺度结构进行建模，未表示其在分量间诱导的有序关系。我们发现级联分解按特征频率对分量排序，形成自然的频率层级，该层级与双曲空间的指数体积增长相契合，双曲空间随半径增大而提升表示能力，且天然适配嵌套层级结构。我们提出HyperForecast，首个在双曲空间中运行的长期时间序列预测（LTSF）框架。HyperForecast将分解后的时间片段嵌入庞加莱球，其中径向距离捕捉该嵌套结构，从靠近原点的低频分量到靠近边界的高频分量。我们通过两种流形动力学变体对测地速度进行建模：（a）基于庞加莱残差动力学的自回归预测；（b）并行预测所有未来片段的非自回归预测。HyperForecast在11个长期预测基准中的9个上始终优于现有及最先进的基线，在长预测 horizon 下均方误差降低最多达58%。进一步分析表明，学习到的径向嵌入可恢复频率层级，且双曲几何对预测性能有显著贡献。
英文摘要：
Multiscale structure is central to time series forecasting: temporal signals contain slowly varying trends, intermediate seasonal patterns, and high-frequency residuals. Traditional forecasters model this multiscale structure in Euclidean space, without representing the ordering it induces among components. We identify that cascaded decomposition orders components by their characteristic frequencies, forming a natural frequency hierarchy. This hierarchy aligns with the exponential volume growth of hyperbolic space, which provides increasing representational capacity with radius and naturally accommodates nested hierarchies. We propose HyperForecast, the first long-term time series forecasting (LTSF) framework operating in hyperbolic space. HyperForecast embeds decomposed temporal segments on the Poincaré ball where radial distance captures this nested structure, from low-frequency components near the origin to high-frequency components near the boundary. We model geodesic velocity through two manifold-dynamics variants: (a) autoregressive forecasting via Poincaré residual dynamics, and (b) non-autoregressive forecasting that predicts all future segments in parallel. HyperForecast consistently outperforms existing and state-of-the-art baselines on 9 of 11 long-term forecasting benchmarks, achieving up to 58% lower mean squared error at long horizons. Further analysis reveals that the learned radial embeddings recover the frequency hierarchy and that hyperbolic geometry contributes substantially to forecasting performance.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
