Robust Order Determination for Tensors Using Data Augmentation

• TU Wien, Vienna, Austria [una.radojicic(at)tuwien.ac.at]

• University of Turku, Turku, Finlad [niko.lietzen@utu.fi]

• University of Jyväskylä, Jyväskylä, Finland [klaus.k.nordhausen@jyu.fi]

• University of Turku, Turku, Finlad [joni.virta@utu.fi]

Keywords: Augmentation – HOSVD – Order Determination – Dimension Reduction – Scree Plot – Minimum Matrix Covariance Determinant

1 Abstract

Tensor-valued data benefits significantly from dimension reduction, as the reduction in size grows exponentially with the number of modes. Achieving maximal reduction without information loss requires a systematic approach to selecting optimal reduced dimensionality. In this work, we present an automated and theoretically grounded procedure that combines a novel data augmentation technique, presented in Luo and Li [2021], with the higher-order singular value decomposition (HOSVD) [De Lathauwer et al., 2000] in a tensorially consistent manner.

We provide theoretical guidelines for selecting tuning parameters and investigate their influence through simulation studies. Our primary result demonstrates that the proposed method consistently estimates true latent dimensions under a noisy tensor model, both at the population and sample levels. Additionally, we introduce a bootstrap-based alternative to the augmentation estimator to enhance flexibility.

Since the procedure relies on sample covariance estimators, it is sensitive to outliers. To address this limitation, we incorporate robust Minimum Matrix Covariance Determinant (MMCD) estimators that were recently developed in Mayrhofer et al. [2024].

Comprehensive simulations illustrate the estimation accuracy and robustness of the proposed methods across diverse scenarios, highlighting their practical relevance for dimension reduction in tensor data.