A subspace method is introduced to solve large-scale trace ratio problems. This approach is matrix-free, requiring only the action of the two matrices involved in the trace ratio. At each iteration, a smaller trace ratio problem is addressed in the search subspace. Additionally, the algorithm is endowed with a restarting strategy, that ensures the monotonicity of the trace ratio value throughout the iterations. The behavior of the approximate solution is investigated from a theoretical viewpoint, extending existing results on Ritz values and vectors, as the angle between the search subspace and the exact solution approaches zero. Numerical experiments in multigroup classification show that this new subspace method tends to be more efficient than iterative approaches relying on (partial) eigenvalue decompositions at each step.

翻译：本文提出一种子空间方法用于求解大规模迹比问题。该方法无需显式矩阵存储，仅需利用迹比问题中两个矩阵的矩阵-向量乘积运算。在每次迭代中，算法在搜索子空间内求解一个较小规模的迹比问题。此外，算法配备了重启策略，确保迹比值在迭代过程中保持单调性。当搜索子空间与精确解之间的夹角趋近于零时，我们从理论角度分析了近似解的性质，扩展了关于Ritz值和Ritz向量的现有结论。在多组分类问题的数值实验中表明，相较于每一步依赖（部分）特征值分解的迭代方法，该新型子空间方法通常具有更高的计算效率。