Ho N. Phien

Ho N. Phien, Hoang D. Tuan, Johann A. Bengua et al.

This paper proposes a novel formulation of the tensor completion problem to impute missing entries of data represented by tensors. The formulation is introduced in terms of tensor train (TT) rank which can effectively capture global information of tensors thanks to its construction by a well-balanced matricization scheme. Two algorithms are proposed to solve the corresponding tensor completion problem. The first one called simple low-rank tensor completion via tensor train (SiLRTC-TT) is intimately related to minimizing the TT nuclear norm. The second one is based on a multilinear matrix factorization model to approximate the TT rank of the tensor and called tensor completion by parallel matrix factorization via tensor train (TMac-TT). These algorithms are applied to complete both synthetic and real world data tensors. Simulation results of synthetic data show that the proposed algorithms are efficient in estimating missing entries for tensors with either low Tucker rank or TT rank while Tucker-based algorithms are only comparable in the case of low Tucker rank tensors. When applied to recover color images represented by ninth-order tensors augmented from third-order ones, the proposed algorithms outperforms the Tucker-based algorithms.

Johann A. Bengua, Hoang D. Tuan, Ho N. Phien et al.

This paper proposes a novel framework called concatenated image completion via tensor augmentation and completion (ICTAC), which recovers missing entries of color images with high accuracy. Typical images are second- or third-order tensors (2D/3D) depending if they are grayscale or color, hence tensor completion algorithms are ideal for their recovery. The proposed framework performs image completion by concatenating copies of a single image that has missing entries into a third-order tensor, applying a dimensionality augmentation technique to the tensor, utilizing a tensor completion algorithm for recovering its missing entries, and finally extracting the recovered image from the tensor. The solution relies on two key components that have been recently proposed to take advantage of the tensor train (TT) rank: A tensor augmentation tool called ket augmentation (KA) that represents a low-order tensor by a higher-order tensor, and the algorithm tensor completion by parallel matrix factorization via tensor train (TMac-TT), which has been demonstrated to outperform state-of-the-art tensor completion algorithms. Simulation results for color image recovery show the clear advantage of our framework against current state-of-the-art tensor completion algorithms.

Johann A. Bengua, Ho N. Phien, Hoang D. Tuan et al.

This paper introduces matrix product state (MPS) decomposition as a computational tool for extracting features of multidimensional data represented by higher-order tensors. Regardless of tensor order, MPS extracts its relevant features to the so-called core tensor of maximum order three which can be used for classification. Mainly based on a successive sequence of singular value decompositions (SVD), MPS is quite simple to implement without any recursive procedure needed for optimizing local tensors. Thus, it leads to substantial computational savings compared to other tensor feature extraction methods such as higher-order orthogonal iteration (HOOI) underlying the Tucker decomposition (TD). Benchmark results show that MPS can reduce significantly the feature space of data while achieving better classification performance compared to HOOI.