Hyperbolic Time Series Forecasting with Learnable Decomposition and Manifold Dynamics

Sarang Rajendra Patil; Vishal Fenn; Mengjia Xu

cs.LGOct 11, 2026

Hyperbolic Time Series Forecasting with Learnable Decomposition and Manifold Dynamics

Authors: Sarang Rajendra Patil, Vishal Fenn, Mengjia Xu

Organizations: Department of Data Science New Jersey Institute of Technology Newark, NJ 07102, USA

Abstract

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.

Figures & tables

Figure 1: Multiscale temporal decomposition and geometric embedding in HyperForecast. (a) A time series is decomposed into trend, coarse seasonal, fine seasonal, and residual components with increasing frequency. (b) Euclidean space provides no natural separation of these components, whereas the Poincaré ball naturally separates them radially through its exponential volume growth.

Figure 2: Overview of HyperForecast pipeline. Input $\mathbf{X}\!\in\!\mathbb{R}^{L\times C}$ is decomposed into trend $\mathbf{T}$ , coarse seasonal $\mathbf{S}_{c}$ , fine seasonal $\mathbf{S}_{f}$ , and residual $\mathbf{R}$ components and normalized by a shared RevIN using statistics from the reconstructed signal $\mathbf{X}\!=\!\mathbf{T}\!+\!\mathbf{S}_{c}\!+\!\mathbf{S}_{f}\!+\!\mathbf{R}$ . Each component is then segment-encoded, projected onto the Poincaré ball $\mathbb{P}^{d}_{c}$ , forecast via geodesic dynamics on the manifold, and Möbius-fused before Euclidean decoding.

Dataset	HF NAR		HF AR		PhaseFormer		TimeBase		SparseTSF		TimeMixer		FITS		iTransformer		DLinear		PatchTST		TimesNet		FEDformer		Autoformer		Informer
	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Weather	0.218	0.273	0.221	0.270	0.223	0.261	0.219	0.263	0.244	0.285	0.238	0.281	0.244	0.286	0.233	0.273	0.242	0.299	0.223	0.264	0.254	0.293	0.355	0.391	0.446	0.457	0.567	0.513
AQShunyi	0.657	0.503	0.665	0.504	0.722	0.523	0.674	0.506	0.760	0.546	0.736	0.536	0.763	0.548	0.722	0.520	0.757	0.572	0.690	0.503	0.788	0.559	1.130	0.757	1.393	0.878	1.688	0.966
AQWan	0.751	0.486	0.762	0.495	0.837	0.517	0.779	0.499	0.826	0.526	0.822	0.521	0.813	0.519	0.843	0.538	0.883	0.560	0.810	0.513	0.857	0.543	0.931	0.565	1.001	0.654	1.123	0.744
CzeLan	0.221	0.286	0.224	0.286	0.225	0.273	0.225	0.262	0.240	0.292	0.234	0.282	0.250	0.304	0.245	0.296	0.241	0.290	0.229	0.275	0.258	0.312	0.266	0.314	0.289	0.368	0.347	0.426
ZafNoo	0.485	0.443	0.489	0.448	0.533	0.456	0.495	0.445	0.531	0.490	0.505	0.464	0.520	0.480	0.540	0.500	0.540	0.495	0.510	0.469	0.545	0.502	0.601	0.539	0.667	0.635	0.726	0.697

Table 1: Average long-term forecasting results (MSE/MAE) across horizons

H\in\{96,192,336,720\}

on 11 datasets (input length

L=720

for all models). Best values are red-bold ; second-best blue-bold .

Figure 3: Per-feature Welch-PSD spectral centroid of each decomposed component on Solar (137 channels,

H=96

). Black dashed: original (undecomposed) signal. Legend values are the per-component average across all 137 features. The ordering

T\!<\!S_{c}\!<\!S_{f}\!<\!R

holds on the dataset average and for the large majority of individual features.

Dataset	Var	Length	Freq	$\rho$	$\delta_{D}$	Dataset	Var	Length	Freq	$\rho$	$\delta_{D}$
Weather	21	52,696	10m	0.737	0.0139	METR-LA	207	34,272	5m	1.023	0.0143
AQShunyi	11	35,064	1h	1.127	0.0095	PM2.5	184	11,688	3h	0.997	0.0194
AQWan	11	35,064	1h	0.848	0.0200	Solar	137	52,560	10m	0.897	0.0150
CzeLan	11	19,934	30m	0.749	0.0025	Temp	184	11,688	3h	1.020	0.0227
ZafNoo	11	19,225	30m	0.515	0.0021	Wind	7	48,673	15m	0.935	0.0153
Exchange	8	7,588	1d	0.886	0.0097

Table 2: Dataset statistics and frequency-hierarchy geometric diagnostics for the 11 LTSF benchmarks. Var : number of variables. Length : total series length. Freq : sampling frequency.

\rho

: paired norm ratio between consecutive components’ embedding radii (

\rho\!<\!1

indicates good hierarchical separation).

\delta_{D}

: Gromov

\delta

-hyperbolicity of the trained embeddings, both averaged across

H\in\{96,192,336,720\}

Figure 4: Each point is a (Scale, TimeSegment, Feature) tuple. The radial hierarchy shows our model learns frequency-based decomposition (Trend $<$ Coarse $<$ Fine $<$ Residual). Labels follow t{segment}_{feature} (e.g., t0_Feature_1 = segment 0, feature 1).

Panel of Figure 5 — Figure 5: Ablation studies: (a) Hyperbolic vs. Euclidean geometry and (b) lookback window sensitivity, shown for both forecasting variants.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: Efficiency comparison across all fourteen models on Weather ( $\text{seq\_len}=\text{pred\_len}=720$ ). (a) Parameter count vs. MACs for a single sample. (b) Peak memory at batch size 12, sorted ascending.

Figure 7: Efficiency of the two HyperForecast variants on Weather, seq_len/pred_len

=720/720

. NAR is shown in green and AR in blue.

Figure 8: Weather (21 channels,

H=96

). The full ordering holds on average centroids:

\Psi_{T}\!=\!0.0051<\Psi_{S_{c}}\!=\!0.0062<\Psi_{S_{f}}\!=\!0.0180<\Psi_{R}\!=\!0.0224

Figure 9: AQWan (11 channels,

H=96

). The full ordering holds on average centroids:

\Psi_{T}\!=\!0.0249<\Psi_{S_{c}}\!=\!0.0371<\Psi_{S_{f}}\!=\!0.0481<\Psi_{R}\!=\!0.1186

Figure 10: METR-LA (207 channels,

H=96

). The full ordering holds on average centroids:

\Psi_{T}\!=\!0.0036<\Psi_{S_{c}}\!=\!0.0100<\Psi_{S_{f}}\!=\!0.0286<\Psi_{R}\!=\!0.0368

Figure 11: PM2.5 (184 channels,

H=96

). The full ordering holds on average centroids:

\Psi_{T}\!=\!0.0100<\Psi_{S_{c}}\!=\!0.0232<\Psi_{S_{f}}\!=\!0.0255<\Psi_{R}\!=\!0.0588

Dataset	Variant	96		192		336		720
		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
AQShunyi	NAR w/o Decomp.	0.638	0.490	0.679	0.507	0.713	0.527	0.776	0.554
	NAR w/ Decomp.	0.607	0.475	0.640	0.498	0.670	0.510	0.712	0.528
	AR w/o Decomp.	0.631	0.483	0.690	0.512	0.713	0.524	0.773	0.551
	AR w/ Decomp.	0.606	0.473	0.642	0.493	0.669	0.508	0.742	0.542
AQWan	NAR w/o Decomp.	0.717	0.471	0.779	0.508	0.802	0.514	0.876	0.547

Appendix

Table 3: Impact of learnable decomposition on forecasting accuracy, NAR and AR. Models with decomposition (trend, coarse/fine seasonal, residual) vs. without decomposition across prediction horizons

H\in\{96,192,336,720\}

Dataset	Variant	96	192	336	720
Solar	Channel-Dependent	0.207	0.234	0.239	0.223
Solar	Channel-Independent	0.177	0.204	0.221	0.217
AQShunyi	Channel-Dependent	0.667	0.678	0.693	0.689
AQShunyi	Channel-Independent	0.607	0.646	0.678	0.717
AQWan	Channel-Dependent	0.774	0.754	0.770	0.829
AQWan	Channel-Independent	0.727	0.741	0.773	0.822

Appendix

Table 4: Channel-independent vs. channel-dependent processing (Poincaré, NAR; MSE).

Dataset	Variant	96	192	336	720
Solar	Channel-Dependent	0.206	0.738	0.754	0.757
Solar	Channel-Independent	0.170	0.206	0.227	0.243
AQShunyi	Channel-Dependent	0.658	0.684	0.692	0.713
AQShunyi	Channel-Independent	0.615	0.658	0.690	0.756
AQWan	Channel-Dependent	0.757	0.765	0.787	0.839
AQWan	Channel-Independent	0.704	0.754	0.781	0.867

Appendix

Table 5: Channel-independent vs. channel-dependent processing (Poincaré, AR; MSE).

Dataset	$K$	96	192	336	720
AQShunyi	2	0.615	0.658	0.690	0.756
	4	0.617	0.662	0.689	0.758
	5	0.613	0.661	0.691	0.758
	6	0.616	0.662	0.689	0.737
Temp	2	0.126	0.125	0.136	0.637
	4	0.124	0.126	0.138	0.293

Appendix

Table 6: AR window size

K

ablation (MSE). Best per dataset/horizon in bold .

Dataset	Variant	96	192	336	720
CzeLan	Main	0.171	0.205	0.237	0.270
	w/o TC	0.183	0.203	0.239	0.331
	w/o Hier	0.190	0.223	0.240	0.279
	w/o both	0.200	0.211	0.238	0.285
AQWan	Main	0.692	0.733	0.760	0.819
	w/o TC	0.699	0.735	0.764	0.815

Appendix

Table 7: Effect of removing the temporal-consistency loss

\mathcal{L}_{\text{tc}}

and hierarchy loss

\mathcal{L}_{\text{hier}}

on HyperForecast (NAR), per horizon. Best per dataset/horizon in bold .

	$\lambda_{\text{hier}}$
Dataset	0	0.0001	0.001	0.005	0.01	0.05	0.1
ZafNoo	0.439	0.429	0.429	0.425	0.423	0.419	0.422
Temp	0.128	0.128	0.135	0.124	0.124	0.128	0.138
AQShunyi	0.610	0.605	0.617	0.619	0.619	0.615	0.609

Appendix

Table 8: Hierarchy-loss weight sensitivity at

H=96

(NAR), with

\lambda_{\text{tc}}=0.1

fixed. MSE reported; best per dataset in bold .

	$\lambda_{\text{tc}}$
Dataset	0	0.05	0.1	0.2
ZafNoo	0.425	0.425	0.423	0.430
Temp	0.125	0.124	0.124	0.124
AQShunyi	0.619	0.618	0.619	0.618

Appendix

Table 9: Temporal-consistency loss weight sensitivity at

H=96

(NAR), with

\lambda_{\text{hier}}=0.01

fixed. MSE reported; best per dataset in bold .

Dataset	$H$	MSE (mean $\pm$ std)	MAE (mean $\pm$ std)
Weather	96	0.146 $\pm$ 0.002	0.210 $\pm$ 0.003
	192	0.188 $\pm$ 0.001	0.253 $\pm$ 0.002
	336	0.245 $\pm$ 0.010	0.301 $\pm$ 0.012
	720	0.309 $\pm$ 0.005	0.349 $\pm$ 0.006
AQShunyi	96	0.609 $\pm$ 0.003	0.474 $\pm$ 0.002
	192	0.646 $\pm$ 0.002	0.498 $\pm$ 0.005

Appendix

Table 10: Mean

\pm

std MSE/MAE over 5 seeds for HyperForecast NAR.

Figure 12: Each point represents a (Scale_TimeSegment_Feature) tuple at a specific frequency scale. Trend (innermost) captures low-frequency patterns closest to the origin, while Residual (outermost) encodes high-frequency noise farther from the origin. The radial hierarchy demonstrates that our model successfully learns frequency-based decomposition (Trend $<$ Coarse $<$ Fine $<$ Residual). Node labels follow the format t{segment}_{feature} (e.g., t0_Feature_1 = time segment 0, feature 1).

Figure 13: Segment embeddings learned on AQShunyi. Left: Poincaré disk from HyperForecast. Trend embeddings cluster near the origin, while coarse, fine, and residual components sit at increasing radii. Right: Euclidean PCA from the same architecture. All four frequency components collapse to coincident 2D positions, eliminating the radial hierarchy.

Figure 14: Segment embeddings learned on AQWan; conventions follow Figure 13 . Similar radial ordering emerges in the Poincaré panel and is absent in the Euclidean PCA, where the four components stack at identical coordinates.

Figure 15: Poincaré-disk embeddings learned on AQWan with and without the hierarchical radial loss $\mathcal{L}_{\mathrm{hier}}$ .

Methods		HF NAR		HF AR		PhaseFormer		TimeBase		SparseTSF		TimeMixer		FITS		iTransformer		DLinear		PatchTST		TimesNet		FEDformer		Autoformer		Informer
Metrics	H	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Weather	96	0.145	0.206	0.143	0.202	0.148	0.195	0.146	0.198	0.174	0.231	0.163	0.223	0.176	0.232	0.159	0.212	0.174	0.242	0.149	0.199	0.176	0.234	0.289	0.342	0.323	0.389	0.283	0.361
	192	0.188	0.252	0.190	0.248	0.193	0.237	0.185	0.241	0.216	0.267	0.201	0.254	0.216	0.268	0.203	0.252	0.215	0.277	0.193	0.243	0.219	0.270	0.340	0.394	0.389	0.423	0.445	0.461
	336	0.237	0.292	0.238	0.289	0.242	0.278	0.236	0.281	0.260	0.299	0.258	0.300	0.261	0.299	0.253	0.291	0.262	0.319	0.240	0.281	0.277	0.311	0.370	0.408	0.497	0.495	0.587	0.526
	720	0.303	0.344	0.315	0.341	0.309	0.332	0.309	0.331	0.325	0.345	0.329	0.348	0.325	0.346	0.317	0.337	0.319	0.359	0.312	0.334	0.344	0.356	0.420	0.421	0.573	0.520	0.953	0.703
AQShunyi	96	0.607	0.475	0.606	0.473	0.650	0.493	0.629	0.489	0.751	0.553	0.703	0.534	0.759	0.555	0.686	0.508	0.751	0.579	0.643	0.476	0.759	0.560	1.247	0.819	1.394	0.931	1.221	0.864

Appendix

Table 11: Long-term forecasting results across 11 datasets and 4 horizons (

H\in\{96,192,336,720\}

). HF: HyperForecast (ours). The input length L is set to 720 across all models. Best results are marked in red-bold ; second-best are blue-bold .

Long-Term Time Series Forecasting

Long-term time series forecasting predicts distant future values from historical sequences in domains such as energy, weather, traffic, and health.

See all 115 papers

HypLTSF: A Hyperbolic Geometric View of Multi-Scale Hierarchies for Long-Term Time Series Forecasting
Sep 8, 2026Namwoo Kim, Hyungryul Baik, Yoonjin YoonLong-Term Time Series Forecasting Hierarchical Representation Learning
Urban AI Institute, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea · Department of Mathematical Science, KAIST, Daejeon, Republic of Korea · Department of Civil and Environmental Engineering, KAIST, Daejeon, Republic of Korea
Learning to Forecast on Multiple Spatiotemporal Hierarchies
Oct 15, 2026Valentina Moretti, Cesare Alippi, Andrea CiniTime Series Forecasting Spatiotemporal Forecasting
IDSIA USI-SUPSI, Universit`a della Svizzera italiana · Politecnico di Milano · EPFL
LeapTS: Rethinking Time Series Forecasting as Adaptive Multi-Horizon Scheduling
May 11, 2026Sheng Pan, Ming Jin, Bo Du +1Neural Controlled Differential Equations Time Series Forecasting
Griffith University

Sep 8, 2026J/K move · Enter open · S save

HypLTSF: A Hyperbolic Geometric View of Multi-Scale Hierarchies for Long-Term Time Series Forecasting

Namwoo Kim, Hyungryul Baik, Yoonjin Yoon

Urban AI Institute, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea · Department of Mathematical Science, KAIST, Daejeon, Republic of Korea · Department of Civil and Environmental Engineering, KAIST, Daejeon, Republic of Korea

Long-Term Time Series Forecasting Hierarchical Representation Learning Multiscale Time-Series Modeling Hyperbolic Representation Learning

Open paper AskPDF arXiv

Multi-scale modeling has become an effective approach for long-term time series forecasting, capturing temporal patterns that range from fine-grained local dynamics to coarse global trends. Representations across these temporal scales are inherently hierarchical, with coarser scales abstracting and aggregating information from finer ones. While existing approaches readily exchange information across these scales, the hierarchy itself is typically left as an emergent byproduct of such interactions rather than captured as a geometric structure in its own right. In this paper, we introduce HypLTSF, a framework that endows the multi-scale hierarchy with a concrete geometric form by embedding scale-wise representations into the Poincaré ball, whose exponentially expanding volume naturally accommodates hierarchical structures. To align this geometry with the temporal hierarchy, HypLTSF imposes two constraints: (1) a radial constraint that orders embeddings by their level of abstraction, and (2) an angular constraint that groups fine-scale patterns sharing a common coarser-scale ancestor. Extensive experiments on long-term time series forecasting benchmarks show that HypLTSF achieves state-of-the-art performance, suggesting that explicitly modeling the multi-scale hierarchy as a geometric structure is effective for forecasting.

Abstract

Long-Term Time Series Forecasting

Explore similar work

HypLTSF: A Hyperbolic Geometric View of Multi-Scale Hierarchies for Long-Term Time Series Forecasting

Learning to Forecast on Multiple Spatiotemporal Hierarchies

LeapTS: Rethinking Time Series Forecasting as Adaptive Multi-Horizon Scheduling