论文标题
张量列车格式给出的功能的优化
Optimization of Functions Given in the Tensor Train Format
论文作者
论文摘要
张量列车（TT）格式是用于计算高效工作的一种常见方法，该方法具有多维阵列，向量，矩阵和离散功能，包括广泛的应用，包括计算数学和机器学习。在这项工作中，我们提出了一种用于TT量优化的新算法，这导致了最小和最大张量元件的非常准确的近似值。该方法由TT核的顺序张量乘法组成，最佳选择了候选者。我们提出了该方法的概率解释，并对其复杂性和收敛性进行估计。我们使用随机张量和各种多变量基准函数进行广泛的数值实验，其输入尺寸数量最高为$ 100 $。我们的方法为所有模型问题提供了接近确切最佳的解决方案，而在常规笔记本电脑上的运行时间不超过50美元。
Tensor train (TT) format is a common approach for computationally efficient work with multidimensional arrays, vectors, matrices, and discretized functions in a wide range of applications, including computational mathematics and machine learning. In this work, we propose a new algorithm for TT-tensor optimization, which leads to very accurate approximations for the minimum and maximum tensor element. The method consists in sequential tensor multiplications of the TT-cores with an intelligent selection of candidates for the optimum. We propose the probabilistic interpretation of the method, and make estimates on its complexity and convergence. We perform extensive numerical experiments with random tensors and various multivariable benchmark functions with the number of input dimensions up to $100$. Our approach generates a solution close to the exact optimum for all model problems, while the running time is no more than $50$ seconds on a regular laptop.
