An Efficient Randomized Fixed-Precision Algorithm for Tensor Singular Value Decomposition

The existing randomized algorithms need an initial estimation of the tubal rank to compute a tensor singular value decomposition. This paper proposes a new randomized fixedprecision algorithm which for a given third-order tensor and a prescribed approximation error bound, automatically finds an optimal tubal rank and the corresponding low tubal rank approximation. The algorithm is based on the random projection technique and equipped with the power iteration method for achieving a better accuracy. We conduct simulations on synthetic and real-world datasets to show the efficiency and performance of the proposed algorithm

Cross Tensor Approximation Methods for Compression and Dimensionality Reduction

Cross Tensor Approximation (CTA) is a generalization of Cross/skeleton matrix and CUR Matrix Approximation (CMA) and is a suitable tool for fast low-rank tensor approximation. It facilitates interpreting the underlying data tensors and decomposing/compressing tensors so that their structures, such as nonnegativity, smoothness, or sparsity, can be potentially preserved. This paper reviews and extends stateof-the-art deterministic and randomized algorithms for CTA with intuitive graphical illustrations.We discuss several possible generalizations of the CMA to tensors, including CTAs: based on  ber selection, slice-tube selection, and lateral-horizontal slice selection. The main focus is on the CTA algorithms using Tucker and tubal SVD (t-SVD) models while we provide references to other decompositions such as Tensor Train (TT), Hierarchical Tucker (HT), and Canonical Polyadic (CP) decompositions. We evaluate the performance of the CTA algorithms by extensive computer simulations to compress color and medical images and compare their performance.Fil: Ahmadi Asl, Salman. Skoltech - Skolkovo Institute Of Science And Technology; RusiaFil: Caiafa, César Federico. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Instituto Argentino de Radioastronomía. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto Argentino de Radioastronomía; ArgentinaFil: Cichocki, Andrzej. Skolkovo Institute of Science and Technology; RusiaFil: Phan, Anh Huy. Skolkovo Institute of Science and Technology; RusiaFil: Tanaka, Toshihisa. Agricultural University Of Tokyo; JapónFil: Oseledets, Ivan. Skolkovo Institute of Science and Technology; RusiaFil: Wang, Jun. Skolkovo Institute of Science and Technology; Rusi

Cross Tensor Approximation Methods for Compression and Dimensionality Reduction

Cross Tensor Approximation (CTA) is a generalization of Cross/skeleton matrix and CUR Matrix Approximation (CMA) and is a suitable tool for fast low-rank tensor approximation. It facilitates interpreting the underlying data tensors and decomposing/compressing tensors so that their structures, such as nonnegativity, smoothness, or sparsity, can be potentially preserved. This paper reviews and extends state-of-the-art deterministic and randomized algorithms for CTA with intuitive graphical illustrations. We discuss several possible generalizations of the CMA to tensors, including CTAs: based on fiber selection, slice-tube selection, and lateral-horizontal slice selection. The main focus is on the CTA algorithms using Tucker and tubal SVD (t-SVD) models while we provide references to other decompositions such as Tensor Train (TT), Hierarchical Tucker (HT), and Canonical Polyadic (CP) decompositions. We evaluate the performance of the CTA algorithms by extensive computer simulations to compress color and medical images and compare their performance.Instituto Argentino de Radioastronomí

Solving bilinear tensor least squares problems and application to Hammerstein identification

Bilinear tensor least squares problems occur in applications such as Hammerstein system identification and social network analysis. A linearly constrained problem of medium size is considered, and nonlinear least squares solvers of Gauss-Newton-type are applied to numerically solve it. The problem is separable, and the variable projection method can be used. Perturbation theory is presented and used to motivate the choice of constraint. Numerical experiments with Hammerstein models and random tensors are performed, comparing the different methods and showing that a variable projection method performs best.Funding Agencies: Mega Grant, Grant/Award Number:14.756.31.0001</p

Image Reconstruction using Superpixel Clustering and Tensor Completion

This paper presents a pixel selection method for compact image representation based on superpixel segmentation and tensor completion. Our method divides the image into several regions that capture important textures or semantics and selects a representative pixel from each region to store. We experiment with different criteria for choosing the representative pixel and find that the centroid pixel performs the best. We also propose two smooth tensor completion algorithms that can effectively reconstruct different types of images from the selected pixels. Our experiments show that our superpixel-based method achieves better results than uniform sampling for various missing ratios

Adaptive Cross Tubal Tensor Approximation

In this paper, we propose a new adaptive cross algorithm for computing a low tubal rank approximation of third-order tensors, with less memory and lower computational complexity than the truncated tensor SVD (t-SVD). This makes it applicable for decomposing large-scale tensors. We conduct numerical experiments on synthetic and real-world datasets to confirm the efficiency and feasibility of the proposed algorithm. The simulation results show more than one order of magnitude acceleration in the computation of low tubal rank (t-SVD) for large-scale tensors. An application to pedestrian attribute recognition is also presented