  Many scientific and economic applications involve the statistical learning of
high-dimensional functional time series, where the number of functional
variables is comparable to, or even greater than, the number of serially
dependent functional observations. In this paper, we model observed functional
time series, which are subject to errors in the sense that each functional
datum arises as the sum of two uncorrelated components, one dynamic and one
white noise. Motivated from the fact that the autocovariance function of
observed functional time series automatically filters out the noise term, we
propose a three-step procedure by first performing autocovariance-based
dimension reduction, then formulating a novel autocovariance-based block
regularized minimum distance estimation framework to produce block sparse
estimates, and based on which obtaining the final functional sparse estimates.
We investigate theoretical properties of the proposed estimators, and
illustrate the proposed estimation procedure via three sparse high-dimensional
functional time series models. We demonstrate via both simulated and real
datasets that our proposed estimators significantly outperform the competitors.
