约束张量网络：离散线性约束上的嵌入与优化
Cons-training Tensor Networks: Embedding and Optimization Over Discrete Linear Constraints
AI总结：
本文提出约束矩阵乘积态张量网络，通过量子区域嵌入离散线性约束，并开发规范形式与无监督训练，在二次背包问题上优于非线性整数规划求解器。
摘要：
在本研究中，我们引入了一类新型张量网络，称为约束矩阵乘积态（MPS），旨在将任意离散线性约束（包括不等式）精确地纳入稀疏块结构中。这些张量网络特别适用于建模支持严格位于可行空间内的分布，其优势包括缩小优化问题中的搜索空间、减轻过拟合、提高训练效率以及减小模型规模。我们方法的核心是量子区域的概念，这是对传统U(1)对称张量网络中使用的量子数的扩展，适用于捕获任何线性约束，包括无约束场景。我们进一步为这些新的MPS开发了一种新颖的规范形式，该形式允许根据量子区域融合规则对张量块进行合并和分解，并支持最优截断方案。利用这种规范形式，我们应用了一种无监督训练策略来优化受离散线性约束的任意目标函数。我们方法的有效性通过求解二次背包问题得到证明，与领先的非线性整数规划求解器相比，取得了更优的性能。此外，我们分析了我们方法的复杂性和可扩展性，展示了其在解决复杂约束组合优化问题方面的潜力。
英文摘要：
In this study, we introduce a novel family of tensor networks, termed constrained matrix product states (MPS), designed to incorporate exactly arbitrary discrete linear constraints, including inequalities, into sparse block structures. These tensor networks are particularly tailored for modeling distributions with support strictly over the feasible space, offering benefits such as reducing the search space in optimization problems, alleviating overfitting, improving training efficiency, and decreasing model size. Central to our approach is the concept of a quantum region, an extension of quantum numbers traditionally used in U(1) symmetric tensor networks, adapted to capture any linear constraint, including the unconstrained scenario. We further develop a novel canonical form for these new MPS, which allow for the merging and factorization of tensor blocks according to quantum region fusion rules and permit optimal truncation schemes. Utilizing this canonical form, we apply an unsupervised training strategy to optimize arbitrary objective functions subject to discrete linear constraints. Our method's efficacy is demonstrated by solving the quadratic knapsack problem, achieving superior performance compared to a leading nonlinear integer programming solver. Additionally, we analyze the complexity and scalability of our approach, demonstrating its potential in addressing complex constrained combinatorial optimization problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
