汇报标题 (Title)：Can Tensor Product Functions Represent High-Dimensional Problems with Antisymmetry Constraints in Polynomial Complexity? （张量积函数能否暗示多项式复杂度中拥有否决称约束的高维问题？）
汇报人 (Speaker)：刘歆 教授（中国科学院数学与系统科学钻研院）
汇报功夫 (Time)：2025年6月5日(周四) 14:30
汇报地址 (Place)：校本部F309
约请人(Inviter)：余长君 教授
主办部门：亿万先生MR理学院数学系、亿万先生MR运筹与优化盛开尝试室、上海市运筹学会
汇报提要：Tensor product function (TPF) approximations are widely used to solve high-dimensional problems, such as partial differential equations and eigenvalue problems, achieving remarkable accuracy with computational costs that scale linearly with problem dimensions. However, recent studies have highlighted the prohibitively high computational cost of TPFs in quantum many-body problems, even for systems with as few as three particles. A key factor contributing to this challenge is the antisymmetry requirement imposed on the unknown functions. In this work, we rigorously demonstrate that the minimum number of terms required for a class of TPFs to satisfy exact antisymmetry grows exponentially with the problem dimension. This class includes both traditionally discretized TPFs and those parameterized by neural networks. By establishing a connection between antisymmetric TPFs and their corresponding antisymmetric tensors, we analyze the Canonical Polyadic rank of the latter to derive our results.Our findings reveal a fundamental incompatibility between antisymmetry and low-rank TPFs in high-dimensional settings. This work provides new insights into the limitations of TPFs and offers guidance for future developments in this area.
