LSMR: An iterative algorithm for sparse least-squares problems (1006.0758v2)

Abstract: An iterative method LSMR is presented for solving linear systems $Ax=b$ and least-squares problem $\min \norm{Ax-b}_2$, with $A$ being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation $A\T Ax = A\T b$, so that the quantities $\norm{A\T r_k}$ are monotonically decreasing (where $r_k = b - Ax_k$ is the residual for the current iterate $x_k$). In practice we observe that $\norm{r_k}$ also decreases monotonically. Compared to LSQR, for which only $\norm{r_k}$ is monotonic, it is safer to terminate LSMR early. Improvements for the new iterative method in the presence of extra available memory are also explored.

• The paper introduces LSMR, an iterative algorithm that ensures a monotonic decrease in the residual norm Aᵀr for sparse least-squares problems.
• It employs backward error estimates and selective reorthogonalization, enabling faster convergence and earlier termination compared to LSQR.
• Extensive experiments on varied system configurations validate LSMR's reliability and efficiency in handling large-scale, sparse computations.

An Analytical Assessment of "LSMR: An Iterative Algorithm for Sparse Least-Squares Problems"

The paper presented by Fong and Saunders introduces LSMR, a novel iterative algorithm designed to solve linear systems and least-squares problems effectively, particularly where the matrix $A$ is sparse or serves as a fast linear operator. Central to LSMR's innovation is its foundation on the Golub-Kahan bidiagonalization process, establishing an analytical equivalence to the MINRES method applied to the normal equation $A^TAx = A^Tb$. This equivalence ensures that the norm of the residual $A^Tr$ decreases monotonically, enhancing the effectiveness of early termination versus the traditional LSQR method, where only $r$ is guaranteed to be monotonically decreasing.

Key Findings and Methodology

One compelling aspect of LSMR is its advantageous stopping criteria. The algorithm maintains a cheaply computable backward error that remains close to optimal (i.e., the smallest possible backward error), an attribute that is both unexpected and beneficial. This feature significantly improves reliability when prematurely terminating iterations, which is often practical in computational environments where early results are sufficient or desired due to limited computational resources.

The paper delineates the operational principles of LSMR through a series of derived subproblems and recurrent formulae. Notably:

• Backward Errors: The paper provides a detailed analysis of backward error estimates, both practical and theoretical, held pivotal for establishing stopping conditions. Particularly, LSMR showcases superiority over LSQR in terms of backward error, notably terminating sooner in most instances without compromising solution integrity.
• Reorthogonalization and Stability: Experimental evidence suggests that only the vectors associated with one of the sets—either $V_k$ or $U_k$ from the Golub-Kahan process—need to be reorthogonalized to maintain high computational accuracy. Such insights present a significant optimization over full reorthogonalization requirements common in similar algorithms.
• Regularized Least Squares: The paper extends the methodology to tackle regularized least squares problems by incorporating Tikhonov regularization $\alpha I$. The extension ensures that LSMR remains versatile, applicable to an increased variety of practical problems where regularization becomes necessary to combat ill-conditioning.

Experimental Results

The researchers carried out extensive empirical tests utilizing problems sourced from the University of Florida Sparse Matrix Collection. Results include experiments on both overdetermined and square system configurations, showcasing LSMR's robust performance under varying conditions. Notably:

• LSMR often converged with fewer iterations than LSQR for inconsistent problems, partly due to its better handling of backward error criteria.
• Its performance remained stable even in scenarios necessitating iterative stopping preemptively due to computational constraints.
• In square systems, LSMR's application proved advantageous even under limited storage and reorthogonalization scenarios, indicating its effective handling of resource-restricted environments.

Practical Implications and Future Outlook

The algorithm is meticulously detailed with ample considerations for its trade-offs between computational demand and convergence speed, making it suitable for large-scale, high-dimensional, and sparse problems commonly encountered in scientific computing and data analysis. Its backward error proximity to the optimal solution without requiring rigorous stopping parameters is particularly noteworthy.

Looking forward, further exploration into partial reorthogonalization techniques, as suggested by Larsen and detailed by the authors, may yield additional efficiency gains. Moreover, adapting LSMR for parallel and distributed computing environments is a potential area for advancement, amplifying its utility in handling modern computational workloads.

In conclusion, the LSMR algorithm addresses significant challenges associated with sparse least-squares problems, providing a reliable, efficient tool with proven performance enhancements over traditional methods like LSQR. The algorithm stands as a recommended technique in the field, offering substantial practical and theoretical advantages for sparse matrix computations.