Fast randomized PVR-PCA for NLA-oriented leading subspace computation

Computing the leading k-dimensional subspace of large-scale matrices is a foundational task in numerical linear algebra (NLA), but repeated full-data products and matrix factorizations can be costly when both matrix dimensions are large. To address these issues, we propose Projected Variance-Reduced PCA (PVR-PCA), a fast randomized algorithm tailored for NLA-oriented subspace computation. PVR-PCA integrates two NLA-optimized innovations: (1) projection-enhanced variance reduction, which replaces VR-PCA's SVD-based orthogonal alignment by an unconstrained least-squares projection; and (2) QR-based orthogonalization, which replaces polar/SVD orthogonalization by thin QR factorization. Together, these changes reduce the dense-kernel per-inner-iteration cost by [Formula Presented] FLOPs compared with VR-PCA. Theoretical analysis establishes high-probability convergence to the leading subspace of the sample covariance matrix. It also gives a conditional warm-start guarantee for a Gaussian-plus-one-power-iteration initializer whenever its realized subspace error satisfies the stated threshold. Experiments on CCAT and MNIST show lower wall-clock cost than VR-PCA with comparable subspace accuracy in the reported comparisons. When covariance products are streamed or blocked, PVR-PCA requires O(dk) additional working memory, excluding storage of the input data.