TY - JOUR
T1 - Solving the $k$-Sparse Eigenvalue Problem with Reinforcement Learning
AU - Zhou , Li
AU - Lihao Yan , 
AU - Mark A. Caprio , 
AU - Weiguo Gao , 
AU - Yang , Chao
JO - CSIAM Transactions on Applied Mathematics
VL - 4
SP - 697
EP - 723
PY - 2021
DA - 2021/11
SN - 2
DO - http://doi.org/10.4208/csiam-am.2020-0220
UR - https://global-sci.org/intro/article_detail/csiam-am/19989.html
KW - Large-scale eigenvalue problem, quantum many-body problem, eigenvector localization, reinforcement learning, approximate $Q$-learning, stochastic sampling, high performance computing.
AB - <p style="text-align: justify;">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 the eigenvalue of a $k×k$ 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.</p>