Numerically Stable Cholesky-QR on GPU via Mixed-Precision Randomized Preconditioning

Abstract

Cholesky-QR is among the fastest algorithms for computing the thin QR factorization of tall-and-skinny matrices on GPUs, relying entirely on BLAS-3 operations. However, it is numerically unstable: forming the Gram matrix squares the condition number, causing breakdown when $κ_2(\boldsymbol{A}) \gtrsim 10^8$. We present MRCQR (Mixed-Precision Randomized Cholesky-QR), a stable GPU algorithm that addresses this limitation. MRCQR uses a subsampled randomized trigonometric transform to construct a preconditioner $\boldsymbol{R}_s$ that reduces $κ_2(\boldsymbol{A}\boldsymbol{R}_s^{-1})$ to near unity with high probability, then applies Cholesky-QR in double precision to the preconditioned matrix. The key insight -- supported by perturbation analysis -- is that the preconditioner requires far less accuracy than the final result: single (FP32) precision suffices when $κ_2(\boldsymbol{A}) \lesssim 10^8$, and half (FP16) when $κ_2(\boldsymbol{A}) \lesssim 10^4$. MRCQR produces an explicit orthogonal factor $\widehat{\boldsymbol{Q}}$ satisfying $\|\boldsymbol{I} - \widehat{\boldsymbol{Q}}^\top\widehat{\boldsymbol{Q}}\|_2 = \cal O(\mathbf{u})$ ($\mathbf{u} \approx 10^{-16}$, double-precision unit roundoff) for condition numbers up to $10^{16}$, far beyond the $10^8$ limit of CholQR2. Experiments on an NVIDIA H100 GPU show that MRCQR (FP16) outperforms rand-cholQR by $1.4$--$1.8\times$ across all tested column counts and is $1.8$--$13.5\times$ faster than cuSOLVER geqrf, while the FP16 sketch (used when $κ_2(\boldsymbol{A}) \lesssim 10^4$) is $2\times$ cheaper than FP64 at no accuracy cost.