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.

翻译：在一系列广泛的领域,包括高分辨率视频、MRI和FMRI扫描等多式成像、商业网络、信号处理等工程(如信号处理等)和其他地方,都普遍存在着高调高压强数据。塔克分解可能是高调高压高压多功能分解组中最普遍的低级近似方法,具有强大的压缩能力,而诱发的塔克拉索尔要素模型(TTTTTFaM)的统计特性仍然是巨大的挑战,但在它为机器学习及以后的应用提供理由之前,仍然至关重要。现有的理论发展主要侧重于时间序列领域,假设按时间订购的观测之间有强大的自动-反应,这对于独立和依赖弱的喇叭观测是无效的。在相当温和的假设下,这一文章将图TOTFM的组合中原始的弱度相关高压观测的分母化为最普遍,并提出了两套基于五氯苯的估算程序,即MOPCA及其精细的IPMOPA的估算程序,后者的趋同率有所提高。我们发展了它们的时间序列行为,主要包括统一率和低度的软质实验性实验性观测方法中的现有沙压结构中,以及目前基于沙压性沙压结构中的现有数质变变压方法中的现有数质结构中的现有数质压方法。