Nystr\"{o}m Regularization for Time Series Forecasting

This paper focuses on learning rate analysis of Nystr\"{o}m regularization with sequential sub-sampling for $\tau$-mixing time series. Using a recently developed Banach-valued Bernstein inequality for $\tau$-mixing sequences and an integral operator approach based on second-order decomposition, we succeed in deriving almost optimal learning rates of Nystr\"{o}m regularization with sequential sub-sampling for $\tau$-mixing time series. A series of numerical experiments are carried out to verify our theoretical results, showing the excellent learning performance of Nystr\"{o}m regularization with sequential sub-sampling in learning massive time series data. All these results extend the applicable range of Nystr\"{o}m regularization from i.i.d. samples to non-i.i.d. sequences.