Maximum relative distance between real rank-two and rank-one tensors

2021/11/24 by Henrik Eisenmann, Eisenmann, Henrik, André Uschmajew +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Blind Source Separation Techniques #FOS: Mathematics #Optimization and Control (math.OC) #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2111.12611

openalex publication_date 2021/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that the relative distance in Frobenius norm of a real symmetric order-d tensor of rank two to its best rank-one approximation is upper bounded by √1-(1-1/d)^d-1. This is achieved by determining the minimal possible ratio between spectral and Frobenius norm for symmetric tensors of border rank two, which equals (1-1/d)^(d-1)/2. These bounds are also verified for arbitrary real rank-two tensors by reducing to the symmetric case.

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