论文标题
算术电路张量网络，多变量函数表示和高维积分
Arithmetic circuit tensor networks, multivariable function representation, and high-dimensional integration
论文作者
论文摘要
许多计算问题可以根据高维功能提出。此类功能的简单表示和与之相关的计算通常会遭受“维度的诅咒”，这是对维度的指数成本依赖性。张量网络提供了一种用多项式内存来表示某些高维函数类别的方法。这导致计算指数成本得到改善或在某些情况下删除，如果可以获得张量网络表示。在这里，我们将函数算术电路的直接映射引入到算术电路张量网络，以避免执行任何优化或功能拟合。我们证明了电路构建的功率，以多达50个维度的单位超数据集对单位超数据集进行多变量集成的示例，其中可以从电路结构中理解积分的复杂性。与这些情况下的准蒙特 - 卡洛集成相比，我们发现成本缩放非常有利，并进一步举例说明，在理论上不了解潜在的张量网络电路结构的情况下，理论上不能在理论上执行高效的准蒙特 - 卡洛。
Many computational problems can be formulated in terms of high-dimensional functions. Simple representations of such functions and resulting computations with them typically suffer from the "curse of dimensionality", an exponential cost dependence on dimension. Tensor networks provide a way to represent certain classes of high-dimensional functions with polynomial memory. This results in computations where the exponential cost is ameliorated or in some cases, removed, if the tensor network representation can be obtained. Here, we introduce a direct mapping from the arithmetic circuit of a function to arithmetic circuit tensor networks, avoiding the need to perform any optimization or functional fit. We demonstrate the power of the circuit construction in examples of multivariable integration on the unit hypercube in up to 50 dimensions, where the complexity of integration can be understood from the circuit structure. We find very favorable cost scaling compared to quasi-Monte-Carlo integration for these cases, and further give an example where efficient quasi-Monte-Carlo cannot be theoretically performed without knowledge of the underlying tensor network circuit structure.
