Triangularized Orthogonalization-Free Method for Solving Extreme Eigenvalue Problems

A novel orthogonalization-free method together with two specific algorithms is proposed to address extreme eigenvalue problems. On top of gradient-based algorithms, the proposed algorithms modify the multicolumn gradient such that earlier columns are decoupled from later ones. Locally, both algorithms converge linearly with convergence rates depending on eigengaps. Momentum acceleration, exact linesearch, and column locking are incorporated to accelerate algorithms and reduce their computational costs. We demonstrate the efficiency of both algorithms on random matrices with different spectrum distributions and matrices from computational chemistry.

Automated summary

A novel orthogonalization-free method, along with two specific algorithms, is proposed to address extreme eigenvalue problems. Both algorithms converge linearly at the local level, with convergence rates depending on eigengaps. Additional measures, such as momentum acceleration, exact linesearch, and column locking, are incorporated to accelerate algorithms and reduce computational costs.