张量标度、量子边缘和矩多面体的有效算法
Efficient algorithms for tensor scaling, quantum marginals and moment polytopes
摘要：
我们提出了一个多项式时间算法，近似缩放张量的任何格式，以任意规定的边缘（只要可能）。这统一和推广了过去的工作序列矩阵，运营商和张量标度。我们的算法提供了一个有效的弱成员预言机相关联的时刻多面体，一个重要的家庭的隐式定义的凸多面体指数许多方面和广泛的应用。这些包括量子信息论的纠缠多面体（特别是，我们获得了一个有效的解决方案，臭名昭著的单体量子边缘问题）和克罗内克多面体从表示论（捕捉克罗内克系数的渐近支持）。我们的算法可以适用于简洁的描述的输入张量的边际可以有效地计算，在重要的情况下，矩阵产品的状态或张量列车分解，广泛应用于计算物理和数值数学。我们加强和推广的交替最小化方法，以前的文件，通过引入理论的最高权向量表示理论的数值优化框架。我们表明，最高权重向量的缩放算法是自然的潜在功能，并证明了新的界限，他们的评价，以获得多项式时间收敛。我们的技术是一般的，我们相信，他们将有助于获得有效的算法时刻多面体超出这里考虑的，更广泛地说，其他优化问题具有自然对称性。
英文摘要：
We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our algorithm provides an efficient weak membership oracle for the associated moment polytopes, an important family of implicitly-defined convex polytopes with exponentially many facets and a wide range of applications. These include the entanglement polytopes from quantum information theory (in particular, we obtain an efficient solution to the notorious one-body quantum marginal problem) and the Kronecker polytopes from representation theory (which capture the asymptotic support of Kronecker coefficients). Our algorithm can be applied to succinct descriptions of the input tensor whenever the marginals can be efficiently computed, as in the important case of matrix product states or tensor-train decompositions, widely used in computational physics and numerical mathematics. We strengthen and generalize the alternating minimization approach of previous papers by introducing the theory of highest weight vectors from representation theory into the numerical optimization framework. We show that highest weight vectors are natural potential functions for scaling algorithms and prove new bounds on their evaluations to obtain polynomial-time convergence. Our techniques are general and we believe that they will be instrumental to obtain efficient algorithms for moment polytopes beyond the ones consider here, and more broadly, for other optimization problems possessing natural symmetries.
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