The decomposition or approximation of a linear operator on a matrix space as a sum of Kronecker products plays an important role in matrix equations and low-rank modeling. The approximation problem in Frobenius norm admits a well-known solution via the singular value decomposition. However, the approximation problem in spectral norm, which is more natural for linear operators, is much more challenging. In particular, the Frobenius norm solution can be far from optimal in spectral norm. We describe an alternating optimization method based on semidefinite programming to obtain high-quality approximations in spectral norm, and we present computational experiments to illustrate the advantages of our approach.

翻译：矩阵空间上线性算子分解或近似为Kronecker积之和，在矩阵方程与低秩建模中具有重要作用。Frobenius范数下的近似问题可通过奇异值分解获得经典解。然而，对于线性算子更为自然的谱范数下的近似问题则更具挑战性。特别是，在谱范数下Frobenius范数解可能远非最优。本文提出一种基于半定规划的交替优化方法，可在谱范数下获得高质量近似，并通过计算实验验证了该方法的优势。