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We introduce tensor Interpolative Decomposition (tensor ID) for the reduction of the separation rank of Canonical Tensor Decompositions (CTDs). Tensor ID selects, for a user-defined accuracy \epsilon, a near optimal subset of terms of a CTD…

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank…

The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of…

The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor completion. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based…

Dimension reduction techniques are often used when the high-dimensional tensor has relatively low intrinsic rank compared to the ambient dimension of the tensor. The CANDECOMP/PARAFAC (CP) tensor completion is a widely used approach to find…

Based on sketching techniques, we propose two randomized algorithms for tensor ring (TR) decomposition. Specifically, by defining new tensor products and investigating their properties, we apply the Kronecker sub-sampled randomized Fourier…

We present an iterative algorithm, called the symmetric tensor eigen-rank-one iterative decomposition (STEROID), for decomposing a symmetric tensor into a real linear combination of symmetric rank-1 unit-norm outer factors using only…

Numerical methods based on tensor networks have been extensively explored in the research of quantum many-body systems in recent years. It has been recognized that the ability of tensor networks to describe a quantum many-body state…

This paper presents a numerical framework for the low-rank approximation of the solution to three-dimensional parabolic problems. The key contribution of this work is the tensorization process based on a tensor-train reformulation of the…

Tensor train decomposition is a powerful tool for dealing with high-dimensional, large-scale tensor data, which is not suffering from the curse of dimensionality. To accelerate the calculation of the auxiliary unfolding matrix, some…

The canonical polyadic decomposition (CPD) of a low rank tensor plays a major role in data analysis and signal processing by allowing for unique recovery of underlying factors. However, it is well known that the low rank CPD approximation…

From linear classifiers to neural networks, image classification has been a widely explored topic in mathematics, and many algorithms have proven to be effective classifiers. However, the most accurate classifiers typically have…

We introduce the tensor numerical method for solution of the $d$-dimensional optimal control problems with fractional Laplacian type operators in constraints discretized on large $n^{\otimes d}$ tensor-product Cartesian grids. The approach…

In this paper, we study the estimation of a rank-one spiked tensor in the presence of heavy tailed noise. Our results highlight some of the fundamental similarities and differences in the tradeoff between statistical and computational…

This paper proposes a novel method for learning highly nonlinear, multivariate functions from examples. Our method takes advantage of the property that continuous functions can be approximated by polynomials, which in turn are representable…

This paper studies the rank-1 tensor completion problem for cubic tensors when there are noises for observed tensor entries. First, we propose a robust biquadratic optimization model for obtaining rank-1 completing tensors. When the…

In this paper, we consider the estimation of a low Tucker rank tensor from a number of noisy linear measurements. The general problem covers many specific examples arising from applications, including tensor regression, tensor completion,…

Tensor completion aimes at recovering missing data, and it is one of the popular concerns in deep learning and signal processing. Among the higher-order tensor decomposition algorithms, the recently proposed fully-connected tensor network…

Low-rank approximation in data streams is a fundamental and significant task in computing science, machine learning and statistics. Multiple streaming algorithms have emerged over years and most of them are inspired by randomized…

Low-rank tensor completion problem aims to recover a tensor from limited observations, which has many real-world applications. Due to the easy optimization, the convex overlapping nuclear norm has been popularly used for tensor completion.…