Tucker tensor factor models for high-dimensional higher-order tensor observations
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Higher-order tensor data are prevailing in a wide range of fields including high-resolution videos, multimodality imaging such as MRI and fMRI scans, commercial networks, engineering such as signal processing, and elsewhere. Tucker decomposition may be the most general low-rank approximation method among versatile decompositions of higher-order tensors owning to its strong compression ability, whilst statistical properties of the induced Tucker tensor factor model (TuTFaM) remains a big challenge and yet critical before it provides justification for applications in machine learning and beyond. Existing theoretical developments mainly focus on the field of time series with the assumption of strong auto-correlation among temporally ordered observations, which is ineffective for independent and weakly dependent tensor observations. Under quite mild assumptions, this article kicks off matricization of raw weakly correlated tensor observations within the TuTFaM setting, and proposes two sets of PCA based estimation procedures, moPCA and its refinement IPmoPCA, the latter of which is enhanced in rate of convergence. We develop their asymptotic behaviors, including mainly convergence rates and asymptotic distributions of estimators of loading matrices, latent tensor factors and signal parts. The theoretical results can reduce to those in low-order tensor factor models in existing literature. The proposed approaches outperform existing auto-covariance based methods for tensor time series in terms of effects of estimation and tensor reconstruction, in both simulation experiments and two real data examples.
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