We examine the possibility of using a reinforcement learning (RL) algorithm to solve large-scale eigenvalue problems in which the desired the eigenvector can be approximated by a sparse vector with at most $k$ nonzero elements, where $k$ is relatively small compare to the dimension of the matrix to be partially diagonalized. This type of problem arises in applications in which the desired eigenvector exhibits localization properties and in large-scale eigenvalue computations in which the amount of computational resource is limited. When the positions of these nonzero elements can be determined, we can obtain the $k$-sparse approximation to the original problem by computing eigenvalues of a $k\times k$ submatrix extracted from $k$ rows and columns of the original matrix. We review a previously developed greedy algorithm for incrementally probing the positions of the nonzero elements in a $k$-sparse approximate eigenvector and show that the greedy algorithm can be improved by using an RL method to refine the selection of $k$ rows and columns of the original matrix. We describe how to represent states, actions, rewards and policies in an RL algorithm designed to solve the $k$-sparse eigenvalue problem and demonstrate the effectiveness of the RL algorithm on two examples originating from quantum many-body physics.

中文翻译：

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用强化学习解决 k 稀疏特征值问题

我们研究了使用强化学习 (RL) 算法解决大规模特征值问题的可能性，其中所需的特征向量可以由至多具有 $k$ 非零元素的稀疏向量近似，其中 $k$ 相对较小比较到要部分对角化的矩阵的维数。这种类型的问题出现在所需特征向量表现出定位特性的应用程序和计算资源量有限的大规模特征值计算中。当可以确定这些非零元素的位置时，我们可以通过计算从原始矩阵的 $k$ 行和列中提取的 $k\times k$ 子矩阵的特征值来获得原始问题的 $k$-sparse 近似。我们回顾了先前开发的用于增量探测 $k$-sparse 近似特征向量中非零元素位置的贪婪算法，并表明可以通过使用 RL 方法改进 $k$ 行和列的选择来改进贪婪算法的原始矩阵。我们描述了如何在旨在解决 $k$-稀疏特征值问题的 RL 算法中表示状态、动作、奖励和策略，并在源自量子多体物理学的两个示例上展示 RL 算法的有效性。