Inexact subspace projection methods for low-rank tensor eigenvalue problems

TL;DR

The paper tackles the challenge of computing multiple extreme eigenpairs of high-dimensional operators represented in low-rank tensor formats. It introduces inexact polynomial-filtered TT subspace iteration, which directly uses Ritz vectors as a subspace basis, avoiding the problematic orthonormal Krylov basis in low-rank settings. The authors derive explicit truncation-error bounds that guarantee progress and demonstrate, through extensive numerical experiments on Heisenberg, Laplacian, and Henon–Heiles systems, that TT subspace iteration is significantly more robust to rank truncation than low-rank Lanczos and can outperform DMRG in several scenarios. The approach offers a practical, non-Hermitian-capable framework for computing multiple eigenpairs with substantial efficiency gains, and the results suggest promising extensions to related methods such as FEAST and randomized truncation strategies.

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 Lanczos and DMRG, which are designed specifically for Hermitian matrices.

Figures (7)

• Figure 1: (a) Illustration of one iteration of truncated power iteration. The radius of the dashed ball is the upper bound on the size of absolute truncation error $\|e\|$ that guarantees progress is made towards convergence in a single iteration. (b) Upper bound (re-scaled by eigenvalue ratio) on truncation error sufficient for truncated power iteration convergence in \ref{['thm:pwr_convergence']}.
• Figure 2: Comparison of convergence of the rank-truncated subspace iteration and Lanczos method for each test problem \ref{['Heis']}-\ref{['henon_heiles']}. Top row: Heisenberg Hamiltonian \ref{['Heis']} with $L=10$. The errors of the first five Ritz values are shown in column (a) for subspace iteration and column (b) for Lanczos method, both using a fixed truncation rank of 6. For the subspace iteration, subspace dimension $m=5$ and polynomial filter of degree $k=2$ were used. Shown in the top row of column (c) are the ranks needed to represent the exact eigenvectors $\psi_j$ (blue) and the orthogonal Krylov basis vectors $v_j$ with accuracy $10^{-10}$ in the Frobenius norm (red). Also shown are the condition numbers of the approximate low-rank Lanczos basis at each iteration. Middle row: Laplacian with $d=3$, $n=16$, truncation rank $1$. The second row of column (c) shows the condition number of the low-rank approximate Krylov basis at each iteration. Bottom row: Hamiltonian with Hénon Heiles potential with $d=3$, $n=16$ and truncation rank 10. Column (c) shows the condition number of the low-rank approximate Krylov basis at each iteration.
• Figure 3: Ground state (blue) and first excited state (red) energies of spin-1/2 (left) and spin-1 (right) Heisenberg Hamiltonians \ref{['Heis']} with $L=100$, $J=1$, $h=0$ and periodic boundary conditions. We computed the energies using rank-truncated subspace iteration (dashed line) and DMRG method (solid line) and plot the results versus CPU-time. On the right we show the reference ground state energy -140.14840390392 White2005.
• Figure 4: Numerical demonstration of the sufficient condition for convergence of truncated power iteration provided in \ref{['eq:err_bnd1']} for Laplacian \ref{['eq:laplacian-operator']} with polynomial acceleration. We set dimension $d=2$, mode sizes $n=32$, and polynomial filter degree $k=8$ with appropriately chosen linear map \ref{['eq:linear_trans']}, which yields an eigenvalue ratio $|p_k(\lambda_2)/p_k(\lambda_1)|\approx 0.6807$. (a) Upper bound \ref{['eq:err_bnd1']} used as error tolerance in TT-SVD truncation and the resulting truncation error. (b) Truncation rank adaptively selected by TT-SVD truncation to achieve truncation error smaller than the tolerance. (c) Angle between current iteration and dominant eigenvector.
• Figure 5: Numerical demonstration of the truncation error upper bounds in \ref{['eq:err_bnd1']} and \ref{['thm:pwr_convergence']} during the computation of the dominant eigenvector of the Laplacian \ref{['eq:laplacian-operator']} with dimension $d=2$ and $n=32$ points per dimension. A Chebyshev polynomial filter of degree $8$ was used, yielding an eigenvalue ratio $|p_k(\lambda_2)/p_k(\lambda_1)|\approx 0.6807$. The figure shows the projections of iterates onto $\mathbb{R}^2$, as defined in \ref{['eq:R2_proj']}, at various iterations both before and after truncation.