A randomized preconditioned Cholesky-QR algorithm

TL;DR

This work addresses stable, efficient thin QR factorization for tall matrices by introducing rpCholesky-QR, a randomized preconditioned two-stage algorithm. It combines a small randomized preconditioner built from row-sampled transforms with a Cholesky-QR pass, achieving a machine-precision residual and a two-norm deviation from orthonormality that scales with the preconditioned matrix conditioning rather than the square of the original conditioning. The authors provide rigorous two-norm perturbation bounds under minimal assumptions and demonstrate, through extensive numerics, that the method remains robust on numerically singular inputs using as few as $3n$ preconditioner rows; it also shows competitive accuracy with Cholesky-QR2 for moderately conditioned matrices. The results indicate strong practical impact for high-performance linear algebra tasks requiring reliable orthogonalization, particularly in Krylov methods and GPU-accelerated environments where Cholesky-based approaches are favorable.

Abstract

We a present and analyze rpCholesky-QR, a randomized preconditioned Cholesky-QR algorithm for computing the thin QR factorization of real mxn matrices with rank n. rpCholesky-QR has a low orthogonalization error, a residual on the order of machine precision, and does not break down for highly singular matrices. We derive rigorous and interpretable two-norm perturbation bounds for rpCholesky-QR that require a minimum of assumptions. Numerical experiments corroborate the accuracy of rpCholesky-QR for preconditioners sampled from as few as 3n rows, and illustrate that the two-norm deviation from orthonormality increases with only the condition number of the preconditioned matrix, rather than its square -- even if the original matrix is numerically singular.

Figures (8)

• Figure 1: Logarithm of errors and condition number versus sampling amount $c$ for $\boldsymbol{A}\in\mathbb{R}^{6,000\times 100}$ in (\ref{['e_ErrorRange']}). Upper panel: The red region delineates the smallest and largest deviation from orthonormality $\|\widehat{\boldsymbol{Q}}^T\widehat{\boldsymbol{Q}}-\boldsymbol{I}\|_2$ over 10 runs for each sampling amount $c$, and the red circles represent the mean. The thin blue region delineates the smallest and largest residual $\|\boldsymbol{A}-\widehat{\boldsymbol{Q}}\widehat{\boldsymbol{R}}\|_2/\|\boldsymbol{A}\|_2$ over 10 runs for each sampling amount $c$, and the blue crosses represent the mean. Lower panel: The red region delineates the smallest and largest condition number $\kappa_2(\boldsymbol{A}_1)$ of the preconditioned matrix $\boldsymbol{A}_1$ over 10 trials for each sampling amount $c$, and the red squares represent the mean.
• Figure 2: Logarithm of errors and condition number versus sampling amount $c$ for $\boldsymbol{A}\in\mathbb{R}^{6,000\times 1,000}$ in (\ref{['e_ErrorRange']}). Upper panel: The red region delineates the smallest and largest deviation from orthonormality $\|\widehat{\boldsymbol{Q}}^T\widehat{\boldsymbol{Q}}-\boldsymbol{I}\|_2$ over 10 runs for each sampling amount $c$, and the red circles represent the mean. The thin blue region delineates the smallest and largest residual $\|\boldsymbol{A}-\widehat{\boldsymbol{Q}}\widehat{\boldsymbol{R}}\|_2/\|\boldsymbol{A}\|_2$ over 10 runs for each sampling amount $c$, and the blue crosses represent the mean. Lower panel: The red region delineates the smallest and largest condition number $\kappa_2(\boldsymbol{A}_1)$ of the preconditioned matrix $\boldsymbol{A}_1$ over 10 trials for each sampling amount $c$, and the red squares represent the mean.
• Figure 3: Logarithm of deviation from orthonormality and residual versus number of columns $n$ for rpCholesky-QR with $c=3n$ samples, applied to matrices $\boldsymbol{A}\in\mathbb{R}^{6,000\times n}$ in (\ref{['e_ErrorRange']}) with $n=100, \ldots, 2,000$ columns. The red stars represent the deviation from orthonormality and the blue diamonds the residuals.
• Figure 4: Logarithm of deviation from orthonormality and estimate (\ref{['e_ex1']}) versus sampling amount $c$ for $\boldsymbol{A}\in\mathbb{R}^{6,000\times 100}$ in (\ref{['e_ErrorRange']}). The red region delineates the smallest and largest deviation from orthonormality $\|\widehat{\boldsymbol{Q}}^T\widehat{\boldsymbol{Q}}-\boldsymbol{I}\|_2$ over 10 runs for each sampling amount $c$, and the red circles represent the mean. The blue region delineates the smallest and largest estimate (\ref{['e_ex1']}) over 10 runs for each sampling amount $c$, and the blue crosses represent the mean.
• Figure 5: Logarithm of deviation from orthonormality and estimate (\ref{['e_ex1']}) versus sampling amount $c$ for $\boldsymbol{A}\in\mathbb{R}^{6,000\times 1,000}$ in (\ref{['e_ErrorRange']}) with $\kappa(\boldsymbol{A})=10^{15}$. The red region delineates the smallest and largest deviation from orthonormality $\|\widehat{\boldsymbol{Q}}^T\widehat{\boldsymbol{Q}}-\boldsymbol{I}\|_2$ over 10 runs for each sampling amount $c$, and the red circles represent the mean. The blue region delineates the smallest and largest estimate (\ref{['e_ex1']}) over 10 runs for each sampling amount $c$, and the blue crosses represent the mean.

Theorems & Definitions (13)

• Lemma 1
• Theorem 2
• Proof 1
• Remark 3.1
• Theorem 1
• Proof 2
• Corollary 2: First-order version of Theorem \ref{['t_perturb2']}
• Lemma 1
• Proof 3
• Theorem 2