# Hyperbolic Time Series Forecasting with Learnable Decomposition and Manifold Dynamics

arXiv: https://arxiv.org/abs/2610.15760 · HTML: https://mlsanity.com/paper/2610.15760/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 42 references that are in the map.
- First posted on arXiv 11 Oct 2026.

## Builds on (most cited first, 29 of 29)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting](https://mlsanity.com/paper/2201.12740/) (2022)
- [TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis](https://mlsanity.com/paper/2210.02186/) (2022)
- [iTransformer: Inverted Transformers Are Effective for Time Series Forecasting](https://mlsanity.com/paper/2310.06625/) (2023)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting](https://mlsanity.com/paper/2405.14616/) (2024)
- [FITS: Modeling Time Series with 10k Parameters](https://mlsanity.com/paper/2307.03756/) (2023)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [SparseTSF: Modeling Long-term Time Series Forecasting with 1k Parameters](https://mlsanity.com/paper/2405.00946/) (2024)
- [Generative Time Series Forecasting with Diffusion, Denoise, and Disentanglement](https://mlsanity.com/paper/2301.03028/) (2023)
- [PM2.5-GNN: A Domain Knowledge Enhanced Graph Neural Network For PM2.5 Forecasting](https://mlsanity.com/paper/2002.12898/) (2020)
- [Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Space](https://mlsanity.com/paper/2407.01290/) (2024)
- [Language Models as Hierarchy Encoders](https://mlsanity.com/paper/2401.11374/) (2024)
- [Hyperbolic Fine-Tuning for Large Language Models](https://mlsanity.com/paper/2410.04010/) (2024)
- [HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Experts](https://mlsanity.com/paper/2505.24722/) (2025)
- [Poincar'e ResNet](https://mlsanity.com/paper/2303.14027/) (2023)
- [HMamba: Hyperbolic Mamba for Sequential Recommendation](https://mlsanity.com/paper/2505.09205/) (2025)
- [PhaseFormer: From Patches to Phases for Efficient and Effective Time Series Forecasting](https://mlsanity.com/paper/2510.04134/) (2025)
- [Hierarchical Mamba Meets Hyperbolic Geometry: A New Paradigm for Structured Language Embeddings](https://mlsanity.com/paper/2505.18973/) (2025)
- [LETS Forecast: Learning Embedology for Time Series Forecasting](https://mlsanity.com/paper/2506.06454/) (2025)
- [Hyperbolic Large Language Models](https://mlsanity.com/paper/2509.05757/) (2025)

## Abstract (arXiv)

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 Poincar'e 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 Poincar'e 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.