Computing eigenspaces with low rank constraints
Christian Krumnow, Max Pfeffer, and André Uschmajew
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Submission date: 17. Dec. 2019 (revised version: February 2021)
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In this work, the task to find a simultaneous low rank approximation of several lowest eigenpairs of a matrix-valued symmetric operator is considered. This problem arises for instance in the density matrix renormalization group algorithm (DMRG) for the accurate simulation of quantum chains. The usual approach is to compute the desired set of eigenvectors and then to identify a low rank approximation by truncating the singular value decomposition (SVD). Since SVD truncation is a norm projection, this yields only sub-optimal results for the eigenvalues. As an alternative, this article explores a direct trace minimization on the intersection of the Stiefel and a low rank manifold using a Riemannian optimization method, which gives better approximations to the eigenspace. A second algorithm based on alternating optimization is also considered but it is less stable. Compared to SVD truncation, the proposed Riemannian method can be seen as a more natural choice for a sub-solver in the DMRG algorithm, and appears to yield better results when applied to spin chains for which the singular values of the exact eigenvectors decay only moderately.