Matrix Product State for Higher-Order Tensor Compression and Classification

This paper introduces matrix product state (MPS) decomposition as a new and systematic method to compress multidimensional data represented by higher-order tensors. It solves two major bottlenecks in tensor compression: computation and compression quality. Regardless of tensor order, MPS compresses tensors to matrices of moderate dimension which can be used for classification. Mainly based on a successive sequence of singular value decompositions (SVD), MPS is quite simple to implement and arrives at the global optimal matrix, bypassing local alternating optimization, which is not only computationally expensive but cannot yield the global solution. Benchmark results show that MPS can achieve better classification performance with favorable computation cost compared to other tensor compression methods.

Concatenated image completion via tensor augmentation and completion

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.

Matrix Product State for Feature Extraction of Higher-Order Tensors

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.