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.

翻译：我们提出了一种不精确子空间迭代法，用于求解具有低秩结构的高维特征值问题。不精确性源于低秩压缩，使得高维向量能够以低秩张量格式高效表示。这类方法面临的一个主要挑战是，标准运算（如矩阵-向量乘积和线性组合）会增加张量秩，从而需要进行秩截断并引入近似。我们将所提出的方法与现有的低秩压缩不精确Lanczos方法进行了比较。后者构建近似的正交Krylov基，即使特征向量本身具有低秩结构，该基通常也难以用低秩张量格式精确表示。相比之下，不精确子空间迭代直接使用近似特征向量（Ritz向量）作为子空间基，从而避免了构建正交Krylov基的需要。我们的分析和数值实验表明，与不精确Lanczos方法相比，不精确子空间迭代对秩截断误差具有更强的鲁棒性。我们还证明了，对于DMRG方法停滞的问题，秩截断子空间迭代仍可收敛。此外，与专门针对厄米矩阵设计的Lanczos和DMRG方法不同，所提出的子空间迭代方法不要求矩阵为厄米矩阵。