On the Nuclear Norm and the Singular Value Decomposition of Tensors

2013/08/18 by Harm Derksen, Derksen, Harm · 8 citations
Computer Science · Engineering · Mathematics · #15A18 #15A69 #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Spectral Theory (math.SP) #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.1308.3860

openalex publication_date 2013/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 rank of a tensor. We determine the nuclear norm of various tensors of interest. Along the way, we also do a systematic study various measures of orthogonality in tensor product spaces and we give a new generalization of the Singular Value Decomposition to higher order tensors.

Citations

• Analysis of Individual Differences in Multidimensional Scaling Via an N-way Generalization of “Eckart-Young” Decomposition

Cited by

• Elementary Symmetric Polynomial Inequalities for Centered Vectors and Matrices
• Waring decompositions of special binomials
• Causal Inference on Sequential Treatments via Tensor Completion
• The Real Vanishing Ideals of Nuclear p-Norm Balls
• The border rank of the 4 × 4 determinant tensor is twelve
• Tensors, entanglement, separability, and their complexity
• Slice rank and partition rank of the determinant
• Bounds for Geometric rank in Terms of Subrank

Related

• Nuclear Norm of Higher-Order Tensors
• Symmetric Tensor Nuclear Norms
• On decomposability and subdifferential of the tensor nuclear norm
• Biquadratic Tensors, Biquadratic Decomposition and Norms of Biquadratic Tensors
• Relations of the Nuclear Norms of a Tensor and its Matrix Flattenings