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Abstract:
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Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this paper, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions $p$ is comparable to, or even larger than the sample size $n$. Aided by the Hilbert--Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding of the sample covariance function capturing the variability of individual functional entries. We investigate the convergence and support recovery properties of our proposed estimator under a high-dimensional regime where $p$ can grow exponentially with $n$. Our simulations demonstrate that the adaptive functional thresholding estimators significantly outperform the competing estimators. Finally, we illustrate the proposed method by the analysis of brain functional connectivity using two neuroimaging datasets.
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