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

Abstract: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.

5 pages, 2 figures, 2 tables. Submitted to IEEE ICASSP 2027

Machine Learning (cs.LG)

arXiv:2609.33984 [cs.LG]

(or arXiv:2609.33984v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.33984 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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