Related papers: Spectral Norm and Nuclear Norm of a Third Order Te…

For a $3$-tensor of dimensions $I_1\times I_2\times I_3$, we show that the nuclear norm of its every matrix flattening is a lower bound of the tensor nuclear norm, and which in turn is upper bounded by $\sqrt{\min\{I_i : i\neq j\}}$ times…

We establish two inequalities for the nuclear norm and the spectral norm of tensor products. The first inequality indicates that the nuclear norm of the square matrix is a matrix norm. We extend the concept of matrix norm to tensor norm. We…

We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field --- the value of the nuclear norm of a…

Biquadratic tensors play a central role in many areas of science. Examples include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann curvature tensor in relativity theory. The singular values and spectral norm of a…

The recent decade has witnessed a surge of research in modelling and computing from two-way data (matrices) to multiway data (tensors). However, there is a drastic phase transition for most tensor optimization problems when the order of a…

This paper studies nuclear norms of symmetric tensors. As recently shown by Friedland and Lim, the nuclear norm of a symmetric tensor can be achieved at a symmetric decomposition. We discuss how to compute symmetric tensor nuclear norms,…

Finding the rank of a tensor is a problem that has many applications. Unfortunately it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of…

The matrix spectral and nuclear norms appear in enormous applications. The generalizations of these norms to higher-order tensors is becoming increasingly important but unfortunately they are NP-hard to compute or even approximate. Although…

We show that the spectral norm of a random $n_1\times n_2\times \cdots \times n_K$ tensor (or higher-order array) scales as $O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right)$ under some sub-Gaussian assumption on the entries. The proof is…

One of the fundamental problems in multilinear algebra, the minimum ratio between the spectral and Frobenius norms of tensors, has received considerable attention in recent years. While most values are unknown for real and complex tensors,…

In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational…

Controlling the spectral norm of the Jacobian matrix, which is related to the convolution operation, has been shown to improve generalization, training stability and robustness in CNNs. Existing methods for computing the norm either tend to…

We study the decomposability and the subdifferential of the tensor nuclear norm. Both concepts are well understood and widely applied in matrices but remain unclear for higher-order tensors. We show that the tensor nuclear norm admits a…

We introduce a new tensor norm, the average spectrum norm, to study sample complexity of tensor completion problems based on the canonical polyadic decomposition (CPD). Properties of the average spectrum norm and its dual norm are…

In this paper, we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion. We introduce a class of tensor norms by allowing for different levels of coherence, which…

We show that the spectral norm of a $d$-mode real or complex symmetric tensor in $n$ variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed…

The maximum of the absolute value of a real homogeneous polynomial of degree $d\ge 3$ on the unit sphere corresponds to the spectral norm of the induced real $d$-symmetric tensor $\mathcal{S}$. We give two sequences of upper bounds on the…

Currently, low-rank tensor completion has gained cumulative attention in recovering incomplete visual data whose partial elements are missing. By taking a color image or video as a three-dimensional (3D) tensor, previous studies have…

The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares…

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…