# Global and Local Topology-Aware Attention with Persistent Homology and Euler Biases for Time-Series Forecasting

arXiv: https://arxiv.org/abs/2605.03163 · HTML: https://mlsanity.com/paper/2605.03163/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 32 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context](https://mlsanity.com/paper/1901.02860/) (2019)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [Topological Attention for Time Series Forecasting](https://mlsanity.com/paper/2107.09031/) (2021)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [Topology-Informed Graph Transformer](https://mlsanity.com/paper/2402.02005/) (2024)
- [Attending to Topological Spaces: The Cellular Transformer](https://mlsanity.com/paper/2405.14094/) (2024)
- [Linear Transformer Topological Masking with Graph Random Features](https://mlsanity.com/paper/2410.03462/) (2024)
- [Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms](https://mlsanity.com/paper/2410.02622/) (2024)
- [Topological Machine Learning for Multivariate Time Series](https://mlsanity.com/paper/1911.12082/) (2019)

## Abstract (arXiv)

Scientific time series often encode predictive geometric structure, including connectivity, cycles, shell-like geometry, directional changes, and nonlinear neighborhoods, that standard dot-product attention does not explicitly represent. We introduce a topology-aware attention framework that adds such structure to attention logits using persistent homology (H0-H2), anchored Euler characteristic transforms, and kernel-Hilbert channels. A validation-gated local residual captures local topological signals, including a Zeng-style local H0 component, only when held-out validation data support the correction. Exact Vietoris-Rips computations and smooth topological surrogates are evaluated under a no-leakage protocol with train-only calibration, validation-only selection, and test-only reporting. We evaluate guarded topology-aware variants across three architecture families: lightweight attention/Ridge, PatchTSTForRegression, and TimeSeriesTransformerForPrediction. Experiments include synthetic benchmarks isolating higher-order topology and real datasets covering CO2, S&P 500 return-window geometry, and NASA IMS bearing degradation. The audit uses matched paired comparisons across seven dataset units, three random seeds, and three chronological splits, giving 63 paired units per architecture and 189 paired units overall. Topology-aware models show positive paired effects when geometry is predictive, with heterogeneous magnitude across datasets and architectures. Lightweight attention/Ridge improves in 46 of 63 units, with mean relative RMSE reduction of 12.5% and paired randomization p=7.2e-4; PatchTST improves in 33 units and retains the baseline in 20 units, with 23.5% reduction and p=3.5e-5; and TimeSeriesTransformer improves in 47 units, with 47.8% reduction and p<1e-4. The results support topology as a validation-selected, architecture-compatible inductive bias.