The best rank-one approximation ratio of a tensor space

Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

47 Citations (Scopus)


In this paper we define the best rank-one approximation ratio of a tensor space. It turns out that in the finite dimensional case this provides an upper bound for the quotient of the residual of the best rankone approximation of any tensor in that tensor space and the norm of that tensor. This upper bound is strictly less than one, and it gives a convergence rate for the greedy rank-one update algorithm. For finite dimensional general tensor spaces, third order finite dimensional symmetric tensor spaces, and finite biquadratic tensor spaces, we give positive lower bounds for the best rank-one approximation ratio. For finite symmetric tensor spaces and finite dimensional biquadratic tensor spaces, we give upper bounds for this ratio.
Original languageEnglish
Pages (from-to)430-442
Number of pages13
JournalSIAM Journal on Matrix Analysis and Applications
Issue number2
Publication statusPublished - 15 Aug 2011


  • Best rank-one approximation ratio
  • Bounds
  • Tensors

ASJC Scopus subject areas

  • Analysis

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