Yonghan Sun

Yonghan Sun, Hou-Duo Qi, Deren Han et al.

In this paper, we further investigate and refine the subspace-constrained preconditioning technique to enhance the theoretical and numerical convergence properties of randomized iterative methods for solving linear systems. In particular, we design a QR-like factorization that transforms the original linear system into an equivalent block-orthogonal form, thus avoiding the full-rank assumptions adopted in existing work. Moreover, this reformulation reduces the problem to solving a smaller linear system with a favorable singular value distribution, provided an appropriate initial point is employed. The proposed framework can be implemented implicitly within the iteration and does not require explicitly constructing either a preconditioner matrix or a preconditioned linear system, which eliminates the prohibitive cost of forming a fully preconditioned system. Furthermore, we construct orthogonalized search directions from stochastic gradients and develop accelerated variants of the framework. We prove that the proposed algorithmic framework converges linearly in expectation. Numerical experiments demonstrate the benefits of the proposed preconditioning strategy.

Yijie Wang, Yonghan Sun, Deren Han et al.

The generalized Gearhart-Koshy acceleration is a recent exact affine search technique designed for the method of cyclic projections onto hyperplanes, i.e., the Kaczmarz method. However, its convergence properties, particularly the linear convergence rate, have not been thoroughly established. In this paper, we systematically establish the linear convergence of the generalized Gearhart-Koshy accelerated Kaczmarz method for tensor linear systems, proving that it converges linearly to the unique least-norm solution. Our analysis is general and applies to several popular Kaczmarz variants, including incremental, shuffle-once, and random-reshuffling schemes, and demonstrates that this acceleration approach yields a better convergence upper bound compared to the plain Kaczmarz method. We also propose an efficient Gram-Schmidt-based implementation that computes the next iterate in linear time. Building on this implementation, we establish a connection between this acceleration framework and Arnoldi-type Krylov subspace methods, further highlighting its efficiency and potential. Our theoretical results are supported by numerical experiments.