Global Convergence of Triangularized Orthogonalization-free Method

TL;DR

This paper establishes the global convergence of TriOFM, a novel orthogonalization-free eigensolver that uses triangularization to efficiently find eigenvectors of large sparse matrices.

Contribution

It introduces TriOFM, a new method that removes the need for orthogonalization in eigensolvers, and proves its global convergence using advanced mathematical tools.

Findings

TriOFM converges globally for large sparse matrices.

The method avoids orthogonalization steps, improving efficiency.

Convergence proofs are based on the stable manifold theorem and two different approaches.

Abstract

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.