Abstract

Tensorizations of functions are studied, that is, tensors with elements \(A({{i}_{1}}, \ldots ,{{i}_{d}}) = f(x({{i}_{1}}, \ldots ,{{i}_{d}}))\) \(A({{i}_{1}}, \ldots ,{{i}_{d}}) = f(x({{i}_{1}}, \ldots ,{{i}_{d}}))\), where \(f(x)\) is some function defined on an interval and \(\{ x({{i}_{1}}, \ldots ,{{i}_{d}})\} \) is a grid on this interval. For tensors of this type, the problem of approximation by tensors admitting a tensor train (ТТ) decomposition with low ТТ ranks is posed. For the class of functions that are traces of analytic functions of a complex variable in some ellipses on the complex plane, upper and lower bounds for ТТ ranks of optimal approximations are deduced. These estimates are applied to tensorizations of polynomial functions. In particular, the well-known upper bound for ТТ ranks of approximations of such functions is improved to \(O(\log n)\), where \(n\) is the degree of the polynomial.

中文翻译：

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一些平滑函数的近似张量化的 TT 秩

摘要

研究了函数的张量化，即具有元素的张量\(A({{i}_{1}}, \ldots ,{{i}_{d}}) = f(x({{i}_{ 1}}, \ldots ,{{i}_{d}}))\) \(A({{i}_{1}}, \ldots ,{{i}_{d}}) = f( x({{i}_{1}}, \ldots ,{{i}_{d}}))\)，其中\(f(x)\)是定义在区间上的某个函数，而\(\{ x({{i}_{1}}, \ldots ,{{i}_{d}})\} \)是这个区间的网格。对于这种类型的张量，提出了通过张量进行逼近的问题，该问题允许具有低 ТТ 等级的张量序列 (ТТ) 分解。对于复平面上某些椭圆中复变量解析函数迹的函数类，推导出最优逼近的ТТ秩的上限和下限。这些估计应用于多项式函数的张量化。特别是，众所周知的此类函数的 ТТ 秩的上限改进为\(O(\log n)\)，其中\(n\)是多项式的次数。