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Solving the $$k$$-Sparse Eigenvalue Problem with Reinforcement Learning

Journal Article · · CSIAM Transactions on Applied Mathematics
 [1];  [2];  [2];  [3];  [4]
  1. Fudan Univ., Shanghai (China); University of Notre Dame
  2. Univ. of Notre Dame, IN (United States)
  3. Fudan Univ., Shanghai (China)
  4. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
+ Show Author Affiliations

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. Here, 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 × 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.

Research Organization:
Univ. of Notre Dame, IN (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Nuclear Physics (NP); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC); National Science Foundation of China
Grant/Contract Number:
FG02-95ER40934
OSTI ID:
1864862
Journal Information:
CSIAM Transactions on Applied Mathematics, Journal Name: CSIAM Transactions on Applied Mathematics Journal Issue: 4 Vol. 2; ISSN 2708-0560
Publisher:
Global Science PressCopyright Statement
Country of Publication:
United States
Language:
English

References (3)

Intrinsic operators for the translationally-invariant many-body problem journal November 2020
Colloquium : Many-body localization, thermalization, and entanglement journal May 2019
A Modified Principal Component Technique Based on the LASSO journal September 2003

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Related Subjects

97 MATHEMATICS AND COMPUTING
approximate Q-learning
eigenvector localization
high performance computing
large-scale eigenvalue problem
quantum many-body problem
reinforcement learning
stochastic sampling