Solving the $k$-Sparse Eigenvalue Problem with Reinforcement Learning (CSIAM Transactions on Applied Mathematics, 2021)
We examine the possibility of using a reinforcement learning (RL) algorithmto solve large-scale eigenvalue problems in which the desired the eigenvector can beapproximated by a sparse vector with at most $k$ nonzero elements, where $k$ is relativelysmall compare to the dimension of the matrix to be partially diagonalized. This typeof problem arises in applications in which the desired eigenvector exhibits localizationproperties and in large-scale eigenvalue computations in which the amount of computational resource is limited. When the positions of these nonzero elements can bedetermined, we can obtain the $k$-sparse approximation to the original problem by computing the eigenvalue of a $k×k$ submatrix extracted from $k$ rows and columns of theoriginal matrix. We review a previously developed greedy algorithm for incrementallyprobing the positions of the nonzero elements in a $k$-sparse approximate eigenvectorand show that the greedy algorithm can be improved by using an RL method to refinethe 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 ontwo examples originating from quantum many-body physics.