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Randomized algorithms for low-rank matrix approximation: Design, analysis, and applications

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arxiv 2306.12418 v3 pith:V2PH7PIC submitted 2023-06-21 math.NA cs.NA

Joel A. Tropp , Robert J. Webber This is my paper · ORCID
classification math.NAcs.NA
keywords randomizedanalysiscomputationaliterationapplicationsblockdatakrylov
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This survey explores modern approaches for computing low-rank approximations of high-dimensional matrices by means of the randomized SVD, randomized subspace iteration, and randomized block Krylov iteration. The paper compares the procedures via theoretical analyses and numerical studies to highlight how the best choice of algorithm depends on spectral properties of the matrix and the computational resources available. Despite superior performance for many problems, randomized block Krylov iteration has not been widely adopted in computational science. The paper strengthens the case for this method in three ways. First, it presents new pseudocode that can significantly reduce computational costs. Second, it provides a new analysis that yields simple, precise, and informative error bounds. Last, it showcases applications to challenging scientific problems, including principal component analysis for genetic data and spectral clustering for molecular dynamics data.

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Cited by 11 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Faster Linear Algebra Algorithms with Structured Random Matrices

    cs.DS 2025-08 accept novelty 8.0 of 10

    Randomized sketching needs only the new OSI property, not the full subspace embedding, and multiple structured matrices satisfy it with near-optimal cost.

  2. Quasi-optimal hierarchically semi-separable matrix approximation

    math.NA 2025-05 conditional novelty 8.0 of 10

    A randomized algorithm produces an HSS approximation with expected error at most O(log(N/k)) times optimal, using O(k log(N/k)) matrix-vector products.

  3. A structural bound for cluster robustness of randomized small-block Lanczos

    math.NA 2025-07 conditional novelty 7.0 of 10

    A matrix-polynomial analysis shows that the subspace error of randomized small-block Lanczos is controlled by the b-th order relative gap times random factors, with the crucial random-factor bound conjectured and tested.

  4. Approximate full conformal prediction in an RKHS

    stat.ML 2026-01 conditional novelty 6.0 of 10

    For RKHS/Tikhonov predictors, computable approximations to the full conformal region contain it (hence cover at level 1−α) with explicit thickness rates, improved from O(1/(λn)) to O(1/(λ³n²)) via influence functions.

  5. Accelerating Large-Scale Regularized High-Order Tensor Recovery

    cs.LG 2025-06 conditional novelty 6.0 of 10

    Randomized Tucker compression plus nonconvex gradient-based regularization gives faster, more accurate large-scale tensor recovery.

  6. What is a Sketch-and-Precondition Derivation for Low-Rank Approximation? Inverse Power Error or Inverse Power Estimation?

    math.NA 2025-02 conditional novelty 6.0 of 10

    Sketched inverse iteration applied to the sketching error gives a top-k eigensolver whose convergence rate is proportional to the quality of a Nyström preconditioner and depends only on the final spectral gap.

  7. Reduced Basis Method for Simulating Thermal Transients in Electric Machines

    math.NA 2026-08 conditional novelty 5.0 of 10

    A 5-dimensional reduced basis model reproduces the calibrated finite element thermal response of an induction motor at LPTN-like cost, based on one measured operating cycle.

  8. Intrinsic Low-Tucker-Rank Theory and Unified Tensor CUR Decomposition for High-Dimensional Hyperinterpolation

    math.NA 2026-07 reject novelty 5.0 of 10

    The paper asserts low-epsilon-Tucker-rank compressibility of hyperinterpolation coefficient tensors and gives unified TCUR error bounds, but the central rank formula exceeds the tensor dimensions and the advertised pr...

  9. How many integrals should be evaluated at least in two-dimensional hyperinterpolation?

    math.NA 2025-10 reject novelty 5.0 of 10

    The paper presents adaptive CUR-based algorithms that evaluate only a subset of Fourier integrals for 2D hyperinterpolation, but the central error bound is invalid because it confuses max-norm and Frobenius-norm of th...

  10. Tomography-assisted noisy quantum circuit simulator using matrix product density operators

    quant-ph 2025-08 conditional novelty 4.0 of 10

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  11. A Comparative Analysis of Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) as Dimensionality Reduction Techniques

    cs.CV 2025-06 conditional

    PCA and SVD are derived and compared on interpretability, numerical stability, and matrix shape suitability, yielding qualitative selection guidelines without empirical testing.

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