This paper proves the global convergence of a triangularized orthogonalization-free method (TriOFM). TriOFM, in general, applies a triangularization idea to the gradient of an objective function and removes the rotation invariance in minimizers. More precisely, in this paper, the TriOFM works as an eigensolver for sizeable sparse matrices and obtains eigenvectors without any orthogonalization step. Due to the triangularization, the iteration is a discrete-time flow in a non-conservative vector field. The global convergence relies on the stable manifold theorem, whereas the convergence to stationary points is proved in detail in this paper. We provide two proofs inspired by the noisy power method and the noisy optimization method, respectively.

翻译：本文证明了三边化或四边化方法(TriOFM)的全球趋同。 一般来说,三边化方法(TriOFM)将三角化概念应用于客观函数的梯度,并消除最小化器中的旋转变量。 更确切地说,在本文件中,TriOFM作为大量稀少的基质的易分解器,在没有任何正分化步骤的情况下获得分解源。 由于三角化,迭代是非保守矢量字段中的一种离散时间流。 全球趋同依赖于稳定的多管,而与固定点的趋同在本文中得到了详细证明。 我们分别提供了由噪音力法和噪音优化法所启发的两种证据。