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Inexact subspace projection methods for low-rank tensor eigenvalue problems.

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Title: Inexact subspace projection methods for low-rank tensor eigenvalue problems.
Authors: Dektor, Alec1 (AUTHOR) adektor@lbl.gov, DelMastro, Peter2 (AUTHOR), Ye, Erika1 (AUTHOR), Van Beeumen, Roel1 (AUTHOR), Yang, Chao1 (AUTHOR)
Source: Linear Algebra & its Applications. Aug2026, Vol. 743, p286-321. 36p.
Subjects: Lanczos method, Krylov subspace, Eigenvalues, Iterative methods (Mathematics)
Abstract: We propose inexact subspace iteration for solving high-dimensional eigenvalue problems with low-rank structure. Inexactness stems from low-rank compression, enabling efficient representation of high-dimensional vectors in a low-rank tensor format. A primary challenge in these methods is that standard operations, such as matrix-vector products and linear combinations, increase tensor rank, necessitating rank truncation and hence approximation. We compare the proposed methods with an existing inexact Lanczos method with low-rank compression. This method constructs an approximate orthonormal Krylov basis, which is often difficult to represent accurately in low-rank tensor formats, even when the eigenvectors themselves exhibit low-rank structure. In contrast, inexact subspace iteration uses approximate eigenvectors (Ritz vectors) directly as a subspace basis, bypassing the need for an orthonormal Krylov basis. Our analysis and numerical experiments demonstrate that inexact subspace iteration is much more robust to rank-truncation errors compared to the inexact Lanczos method. We also demonstrate that rank-truncated subspace iteration can converge for problems where the DMRG method stagnates. Furthermore, the proposed subspace iteration methods do not require a Hermitian matrix, in contrast to DMRG, which is designed specifically for Hermitian matrices. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:We propose inexact subspace iteration for solving high-dimensional eigenvalue problems with low-rank structure. Inexactness stems from low-rank compression, enabling efficient representation of high-dimensional vectors in a low-rank tensor format. A primary challenge in these methods is that standard operations, such as matrix-vector products and linear combinations, increase tensor rank, necessitating rank truncation and hence approximation. We compare the proposed methods with an existing inexact Lanczos method with low-rank compression. This method constructs an approximate orthonormal Krylov basis, which is often difficult to represent accurately in low-rank tensor formats, even when the eigenvectors themselves exhibit low-rank structure. In contrast, inexact subspace iteration uses approximate eigenvectors (Ritz vectors) directly as a subspace basis, bypassing the need for an orthonormal Krylov basis. Our analysis and numerical experiments demonstrate that inexact subspace iteration is much more robust to rank-truncation errors compared to the inexact Lanczos method. We also demonstrate that rank-truncated subspace iteration can converge for problems where the DMRG method stagnates. Furthermore, the proposed subspace iteration methods do not require a Hermitian matrix, in contrast to DMRG, which is designed specifically for Hermitian matrices. [ABSTRACT FROM AUTHOR]
ISSN:00243795
DOI:10.1016/j.laa.2026.04.021