论文标题
近似真实的对称张量等级
Approximate Real Symmetric Tensor Rank
论文作者
论文摘要
我们研究了$ \ varepsilon $ - 扰动耐受性对对称张量分解的效果。更确切地说，假设一个真实的对称$ d $ -tensor $ f $，norm $ ||。在$ \ varepsilon $ -F $的$ \ varepsilon $中，最小的对称张量排名是多少？换句话说，在巧妙的$ \ varepsilon $ perterbation之后，$ f $的对称张量排名是什么？我们证明了两个定理，并开发了三种相应的算法，这些算法为这个问题提供了建设性的上限。考虑说明性目标；我们在结果背后介绍了概率和凸数学思想，重现了一些已知的结果，并指出了开放问题。
We investigate the effect of an $\varepsilon$-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric $d$-tensor $f$, a norm $||.||$ on the space of symmetric $d$-tensors, and $\varepsilon >0$ are given. What is the smallest symmetric tensor rank in the $\varepsilon$-neighborhood of $f$? In other words, what is the symmetric tensor rank of $f$ after a clever $\varepsilon$-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind; we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.
