# From HL to H+L-1 Parameters: A Hankel-Toeplitz Forecaster for Long-Term Time Series Forecasting

arXiv: https://arxiv.org/abs/2609.33984 · HTML: https://mlsanity.com/paper/2609.33984/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 18 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 11 of 11)

- [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)
- [FITS: Modeling Time Series with 10k Parameters](https://mlsanity.com/paper/2307.03756/) (2023)
- [SparseTSF: Modeling Long-term Time Series Forecasting with 1k Parameters](https://mlsanity.com/paper/2405.00946/) (2024)
- [An Analysis of Linear Time Series Forecasting Models](https://mlsanity.com/paper/2403.14587/) (2024)
- [Revisiting Long-term Time Series Forecasting: An Investigation on Linear Mapping](https://mlsanity.com/paper/2305.10721/) (2023)
- [MixLinear: Extreme Low Resource Multivariate Time Series Forecasting with 0.1K Parameters](https://mlsanity.com/paper/2410.02081/) (2024)
- [Disentangled Parameter-Efficient Linear Model for Long-Term Time Series Forecasting](https://mlsanity.com/paper/2411.17257/) (2024)
- [PULSE: Generative Phase Evolution for Non-Stationary Time Series Forecasting](https://mlsanity.com/paper/2605.16793/) (2026)
- [A prediction perspective on the Wiener-Hopf equations for time series](https://mlsanity.com/paper/2107.04994/) (2021)

## Abstract (arXiv)

Linear forecasters have shown competitive accuracy against Transformer-based models in long-term time series forecasting. We study how classical stationary prediction theory can guide parameter sharing for more compact linear forecasters. For centered second-order stationary processes with nonsingular history covariance, the minimum-MSE finite-window linear predictor factors into a Hankel cross-covariance matrix and an inverse Toeplitz covariance matrix. Shared lags and scale cancellation specify this predictor using H+L-1 autocorrelations for lookback L and horizon H. Building on the innovations representation, our Hankel-Toeplitz Forecaster (HTF) learns one impulse response that defines both an inverse filter and a forecast map. We characterize the finite-history correction and, under summability assumptions, bound the excess risk of truncating the true filters. HTF uses H+L-1 trainable coefficients while allowing a full-rank forecasting matrix. Across seven benchmarks at L=336, its horizon-averaged MSE is within 1.2% of Dense Linear on each dataset with 75-229 times fewer trainable parameters.