We present a subspace method to solve large-scale trace ratio problems. This method is matrix-free, only needing the action of the two matrices in the trace ratio. At each iteration, a smaller trace ratio problem is addressed in the search subspace. Additionally, our algorithm is endowed with a restarting strategy, that ensures the monotonicity of the trace ratio value throughout the iterations. We also investigate the behavior of the approximate solution from a theoretical viewpoint, extending existing results on Ritz values and vectors, as the angle between the search subspace and the exact solution to the trace ratio approaches zero. In the context of multigroup classification, numerical experiments show that the new subspace method tends to be more efficient than iterative approaches that need a (partial) eigenvalue decomposition in every step.

翻译：我们提出了一种用于求解大规模迹比问题的子空间方法。该方法无需显式存储矩阵，仅需利用迹比中两个矩阵的作用。在每次迭代中，我们在搜索子空间中求解一个规模较小的迹比问题。此外，我们的算法配备了一种重启策略，确保迹比值在整个迭代过程中具有单调性。我们还从理论角度研究了近似解的行为，当搜索子空间与迹比精确解之间的夹角趋近于零时，扩展了关于Ritz值和Ritz向量的现有结论。在多组分类的背景下，数值实验表明，新的子空间方法通常比每步需要（部分）特征值分解的迭代方法更高效。