The Best Rank-One Approximation Ratio of a Tensor Space
The Best Rank-One Approximation Ratio of a Tensor Space
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DOI：
10.1137/100795802
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发表时间：
2011-06
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影响因子：
--
通讯作者：
L. Qi
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文献类型：
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作者：
L. Qi
本文定义了张量空间的最佳一阶逼近比。结果表明，在有限维的情况下，这提供了该张量空间中任何张量的最佳一阶逼近的残余商和该张量的范数的上界。这个上界严格小于1，它给出了贪婪的一阶更新算法的收敛速度。对于有限维一般张量空间、三阶有限维对称张量空间和有限双二次张量空间，我们给出了最佳一阶逼近比的正下界。对于有限对称张量空间和有限维双二次张量空间，我们给出了这一比例的上界。
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 rank-one 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.