arxiv: 2605.09755 · v1 · submitted 2026-05-10 · 🧮 math.NA · cs.DS· cs.LG· cs.NA· stat.ML

Recognition: 2 theorem links

· Lean Theorem

Accelerating Power Method with Fast Sketching for Stronger Low-Rank Approximation

Micha{\l} Derezi\'nski, Shabarish Chenakkod

Pith reviewed 2026-05-12 03:05 UTC · model grok-4.3

classification 🧮 math.NA cs.DScs.LGcs.NAstat.ML

keywords low-rank approximationpower methodfast sketchingrandomized linear algebrasingular value decompositionNyström approximationregularized spectral approximation

0 comments

The pith

Fast sketching accelerates the power method for low-rank matrix approximation by relying on regularized spectral properties.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework that speeds up the power method for extracting top principal components from large matrices by integrating fast sketching from randomized linear algebra. It shows this leads to simple and provably efficient algorithms for singular value decomposition, low-rank factorization, and Nyström approximation that perform strongly on benchmark problems. The analysis centers on regularized spectral approximation as a more flexible way to extend traditional power-method convergence guarantees. A reader would care because high-rank matrix approximations appear in many data-driven applications where multiplication costs become prohibitive.

Core claim

Fast sketching methods that obey regularized spectral approximation allow the power method to be accelerated for large target ranks while preserving provable efficiency, yielding practical algorithms for SVD, low-rank factorization, and Nyström approximation.

What carries the argument

Regularized spectral approximation, a property of fast sketching that bounds approximation error in a regularized sense and generalizes power-method analysis more flexibly than classical arguments.

If this is right

Simple provably efficient methods for singular value decomposition
Efficient low-rank factorization procedures
Accelerated Nyström approximation algorithms
Strong numerical performance on standard benchmark problems

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sketching-plus-regularization idea could shorten iteration counts in other matrix iteration schemes such as subspace iteration.
Implementation on very large sparse matrices from scientific computing would test whether the theoretical savings translate to wall-clock gains.
Connections to existing randomized SVD libraries could be checked by replacing their internal power steps with the sketched variant.

Load-bearing premise

Fast sketching methods must satisfy regularized spectral approximation properties that generalize power-method guarantees more flexibly than traditional arguments for arbitrary data matrices.

What would settle it

A data matrix and sketching method where the regularized spectral approximation bound fails, causing the accelerated power iteration to lose its convergence rate or accuracy guarantees.

Figures

Figures reproduced from arXiv: 2605.09755 by Micha{\l} Derezi\'nski, Shabarish Chenakkod.

**Figure 2.** Figure 2: Approximation error vs time for Sketch-Powered Low-Rank Factorization (Algorithm [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

read the original abstract

The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this procedure becomes a major bottleneck. We develop an algorithmic and theoretical framework for accelerating the power method using fast sketching, which is a popular paradigm in randomized linear algebra. Our framework leads to simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nystr\"om approximation, which attain strong numerical performance on benchmark problems. The key novelty in our analysis is the use of regularized spectral approximation, a property of fast sketching methods which proves more flexible in generalizing power method guarantees than traditional arguments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper accelerates power iteration for large-rank low-rank approximation via fast sketching and a new regularized spectral approximation analysis, but the conditions needed for the guarantees to hold on general matrices remain unclear.

read the letter

The main takeaway is that this work speeds up the power method when the target rank is high by folding in fast sketching, with the fresh angle being regularized spectral approximation as a way to extend the usual convergence arguments. They apply this to get straightforward algorithms for SVD, low-rank factorization, and Nyström approximation that show strong numerical results on standard benchmarks. That combination of simplicity and claimed efficiency is the practical upside, especially since matrix multiplies become the bottleneck at scale. The experiments appear to support the speed gains without adding much overhead, and the framing stays grounded in prior randomized linear algebra rather than claiming a total overhaul. The analysis is where things get thinner. Regularized spectral approximation is positioned as more flexible than classical properties for generalizing power-method bounds, yet the abstract and setup do not spell out the exact requirements on the sketch distribution, regularization strength, or singular-value decay that make the property hold for arbitrary matrices. Without those, it is hard to tell whether the error bounds and iteration counts carry over outside the tested regimes or specific gap assumptions. The stress-test note correctly flags this as a potential limit on generalization. This paper is aimed at people who implement large-scale PCA or low-rank models and want faster randomized tools with some theory attached. Readers who follow randomized numerical linear algebra would find the algorithms and benchmarks useful to try. The thinking is coherent and engages the literature directly, so the work deserves a serious referee to verify the derivations and see how far the new property actually reaches.

Referee Report

2 major / 2 minor

Summary. The paper proposes an algorithmic and theoretical framework to accelerate the classical power method for low-rank matrix approximation by incorporating fast sketching techniques from randomized linear algebra. The central novelty is the introduction of 'regularized spectral approximation' as a property of sketching methods, which is used to derive provably efficient algorithms for singular value decomposition, low-rank factorization, and Nyström approximation. The work claims these methods achieve strong numerical performance on benchmark problems while generalizing power-method guarantees more flexibly than traditional analyses.

Significance. If the central claims hold, this framework could provide a practical and theoretically grounded way to scale iterative low-rank methods to larger problems by leveraging sketching for reduced matrix-multiplication costs. The regularized spectral approximation concept, if properly conditioned, might offer broader applicability than standard perturbation or subspace arguments in randomized numerical linear algebra, and the reported empirical results on benchmarks indicate potential for immediate use in applications like PCA and kernel methods.

major comments (2)

[§3] §3 (Theoretical Framework), Definition of regularized spectral approximation: The property is asserted to generalize power-method guarantees more flexibly than classical arguments, yet the manuscript does not delineate the precise requirements on the sketching distribution, regularization parameter, or matrix spectrum (e.g., singular-value gap or decay rate) needed for the property to hold for arbitrary data matrices. Without these conditions, it is unclear whether the accelerated iterates retain the claimed convergence rates or error bounds outside restricted regimes.
[§4] §4 (Applications to SVD, low-rank factorization, and Nyström), Theorems on provable efficiency: The efficiency claims for the three target problems rest on the regularized spectral approximation property, but the provided analysis does not include explicit error bounds or full derivations that would confirm the guarantees without additional post-hoc assumptions on the input matrix. This makes the central claim of 'provably efficient methods' difficult to verify as stated.

minor comments (2)

[Abstract] Abstract: The phrase 'strong numerical performance on benchmark problems' should be accompanied by a brief indication of the baselines, metrics, and matrix sizes used, to allow readers to assess the empirical claims at a glance.
[Notation section] Notation: The manuscript introduces several sketching-related quantities without a consolidated table of symbols; adding one would improve readability for the numerical linear algebra audience.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on our paper. We address the major comments point by point below, and we are prepared to make revisions to improve the clarity of the theoretical framework.

read point-by-point responses

Referee: [§3] §3 (Theoretical Framework), Definition of regularized spectral approximation: The property is asserted to generalize power-method guarantees more flexibly than classical arguments, yet the manuscript does not delineate the precise requirements on the sketching distribution, regularization parameter, or matrix spectrum (e.g., singular-value gap or decay rate) needed for the property to hold for arbitrary data matrices. Without these conditions, it is unclear whether the accelerated iterates retain the claimed convergence rates or error bounds outside restricted regimes.

Authors: We appreciate this observation. The definition of regularized spectral approximation in Section 3 is intentionally general to encompass a broad class of sketching methods. The specific conditions for common sketching distributions (such as sub-Gaussian or fast Johnson-Lindenstrauss transforms) and the dependence on the regularization parameter and spectral properties are derived in the proofs of the main theorems in Section 4. To address the concern, we will add a new subsection or remark in Section 3 that explicitly states the sufficient conditions on the sketching distribution and matrix spectrum required for the property to hold, including examples for Gaussian sketching and the role of the singular value gap. revision: yes
Referee: [§4] §4 (Applications to SVD, low-rank factorization, and Nyström), Theorems on provable efficiency: The efficiency claims for the three target problems rest on the regularized spectral approximation property, but the provided analysis does not include explicit error bounds or full derivations that would confirm the guarantees without additional post-hoc assumptions on the input matrix. This makes the central claim of 'provably efficient methods' difficult to verify as stated.

Authors: We thank the referee for pointing this out. The theorems in Section 4 do provide error bounds expressed in terms of the regularized spectral approximation error, and the proofs (some in the appendix) show how these translate to the final guarantees for SVD, low-rank factorization, and Nyström approximation. However, we agree that making the derivations more self-contained and explicit, without relying on implicit assumptions, would strengthen the paper. We will revise the theorems to include more detailed statements of the bounds and expand the proof sketches in the main text or appendix to cover all steps explicitly for each application. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circularity

full rationale

The paper introduces regularized spectral approximation as a property of fast sketching methods to generalize power-method guarantees more flexibly than traditional arguments. This is framed as an independent analytical device drawn from randomized linear algebra rather than being defined in terms of the SVD, low-rank factorization, or Nyström outputs it is used to prove. No step reduces a prediction to a fitted input, renames a known result, or relies on a load-bearing self-citation whose justification collapses into the present work. The claimed provable efficiency therefore rests on external sketching properties and does not collapse by construction to the paper's own inputs or definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and utility of regularized spectral approximation as a flexible property of fast sketching; this is treated as a domain assumption from prior sketching literature rather than derived here. No free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Fast sketching methods satisfy regularized spectral approximation properties that generalize power method guarantees.
Invoked as the key novelty in the analysis section of the abstract.

pith-pipeline@v0.9.0 · 5435 in / 1175 out tokens · 20582 ms · 2026-05-12T03:05:54.633851+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
regularized spectral approximation ... (1−ε)(AA⊤ + λI) ⪯ ASS⊤A⊤ + λI ⪯ (1+ε)(AA⊤ + λI)
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear
Theorem 2.1 ... time ˜O(nnz(A)/ε + mlk/ε³)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

[1]

& Tropp, J

Halko, N., Martinsson, P.-G. & Tropp, J. A. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions.SIAM review53,217–288 (2011)

work page 2011
[2]

& Tygert, M

Rokhlin, V., Szlam, A. & Tygert, M. A randomized algorithm for principal component analysis. SIAM Journal on Matrix Analysis and Applications31,1100–1124 (2010)

work page 2010
[3]

Subspace iteration randomization and singular value problems.SIAM Journal on Scientific Computing37,A1139–A1173 (2015)

Gu, M. Subspace iteration randomization and singular value problems.SIAM Journal on Scientific Computing37,A1139–A1173 (2015)

work page 2015
[4]

Tropp, J. A. & Webber, R. J. Randomized algorithms for low-rank matrix approximation: Design, analysis, and applications.arXiv preprint arXiv:2306.12418(2023)

work page arXiv 2023
[5]

Sarlos, T.Improved approximation algorithms for large matrices via random projectionsin2006 47th annual IEEE symposium on foundations of computer science (FOCS’06)(2006), 143–152

work page 2006
[6]

Clarkson, K. L. & Woodruff, D. P. Low-rank approximation and regression in input sparsity time.Journal of the ACM (JACM)63,1–45 (2017)

work page 2017
[7]

Woodruff, D. P. Sketching as a Tool for Numerical Linear Algebra.Foundations and Trends® in Theoretical Computer Science10,1–157 (2014)

work page 2014
[8]

& Mahoney, M.Revisiting the nystrom method for improved large-scale machine learninginInternational Conference on Machine Learning(2013), 567–575

Gittens, A. & Mahoney, M.Revisiting the nystrom method for improved large-scale machine learninginInternational Conference on Machine Learning(2013), 567–575

work page 2013
[9]

B., Nelson, J

Cohen, M. B., Nelson, J. & Woodruff, D. P.Optimal Approximate Matrix Product in Terms of Stable Rankin43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)55(2016), 11:1–11:14

work page 2016
[10]

Fast and stable randomized low-rank matrix approximation.arXiv preprint arXiv:2009.11392(2020)

Nakatsukasa, Y. Fast and stable randomized low-rank matrix approximation.arXiv preprint arXiv:2009.11392(2020)

work page arXiv 2009
[11]

& Price, E

Hardt, M. & Price, E. The noisy power method: A meta algorithm with applications.Advances in neural information processing systems27(2014)

work page 2014
[12]

& Yang, Y

Chang, C. & Yang, Y. Improving Sketching Algorithms for Low-Rank Matrix Approximation via Sketch-Power Iterations.arXiv preprint arXiv:2603.26298(2026)

work page arXiv 2026
[13]

& Jain, P

Mitliagkas, I., Caramanis, C. & Jain, P. Memory limited, streaming PCA.Advances in neural information processing systems26(2013)

work page 2013
[14]

& Mahoney, M

Alaoui, A. & Mahoney, M. W. Fast randomized kernel ridge regression with statistical guarantees. Advances in neural information processing systems28(2015)

work page 2015
[15]

& Musco, C

Musco, C. & Musco, C. Recursive sampling for the nystrom method.Advances in neural information processing systems30(2017)

work page 2017
[16]

& Rosasco, L

Rudi, A., Calandriello, D., Carratino, L. & Rosasco, L. On fast leverage score sampling and optimal learning.Advances in Neural Information Processing Systems31(2018)

work page 2018
[17]

& Dereziński, M.Faster Low-Rank Approximation and Kernel Ridge Regression via the Block-Nyström MethodinThe Thirty Eighth Annual Conference on Learning Theory(2025), 2291–2325

Garg, S. & Dereziński, M.Faster Low-Rank Approximation and Kernel Ridge Regression via the Block-Nyström MethodinThe Thirty Eighth Annual Conference on Learning Theory(2025), 2291–2325

work page 2025
[18]

& Nguyên, H

Nelson, J. & Nguyên, H. L.OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddingsin2013 ieee 54th annual symposium on foundations of computer science (2013), 117–126. 12

work page 2013
[19]

& Dong, X.Optimal Oblivious Subspace Embeddings with Near- Optimal Sparsityin52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)(2025)

Chenakkod, S., Dereziński, M. & Dong, X.Optimal Oblivious Subspace Embeddings with Near- Optimal Sparsityin52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)(2025)

work page 2025
[20]

& Seeger, M

Williams, C. & Seeger, M. Using the Nyström method to speed up kernel machines.Advances in neural information processing systems13(2000)

work page 2000
[21]

& Sidford, A.Approaching Optimality for Solving Dense Linear Systems with Low-Rank StructureinACM-SIAM Symposium on Discrete Algorithms(2025)

Dereziński, M. & Sidford, A.Approaching Optimality for Solving Dense Linear Systems with Low-Rank StructureinACM-SIAM Symposium on Discrete Algorithms(2025)

work page 2025
[22]

LeCun, Y., Huang, F. J. & Bottou, L.Learning methods for generic object recognition with invariance to pose and lightinginProceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004.2(2004), II–104

work page 2004
[23]

Imagenet: A large-scale hierarchical image databasein2009 IEEE conference on computer vision and pattern recognition(2009), 248–255

Deng, J.et al. Imagenet: A large-scale hierarchical image databasein2009 IEEE conference on computer vision and pattern recognition(2009), 248–255

work page 2009
[24]

Charikar, M., Chen, K. & Farach-Colton, M.Finding Frequent Items in Data Streamsin Proceedings of the 29th International Colloquium on Automata, Languages and Programming (Springer-Verlag, Berlin, Heidelberg, 2002), 693–703.isbn: 3540438645

work page 2002
[25]

& Woźniakowski, H

Kuczyński, J. & Woźniakowski, H. Estimating the largest eigenvalue by the power and Lanczos algorithms with a random start.SIAM Journal on Matrix Analysis and Applications13,1094– 1122 (1992)

work page 1992
[26]

& Musco, C

Musco, C. & Musco, C. Randomized block krylov methods for stronger and faster approximate singular value decomposition.Advances in neural information processing systems28(2015)

work page 2015
[27]

Bakshi, A., Clarkson, K. L. & Woodruff, D. P. Low-Rank Approximation with1/ϵ1/3 Matrix- Vector Products.arXiv preprint arXiv:2202.05120(2022)

work page arXiv 2022
[28]

N., Meyer, R

Chen, T., Epperly, E. N., Meyer, R. A., Musco, C. & Rao, A.Does block size matter in randomized block Krylov low-rank approximation?inProceedings of the 2026 Annual ACM- SIAM Symposium on Discrete Algorithms (SODA)(2026), 1026–1046

work page 2026
[29]

Meyer, R., Musco, C. & Musco, C.On the unreasonable effectiveness of single vector Krylov methods for low-rank approximationinProceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)(2024), 811–845

work page 2024
[30]

& Sanghavi, S.Low-rank matrix completion using alternating minimiza- tioninProceedings of the forty-fifth annual ACM symposium on Theory of computing(2013), 665–674

Jain, P., Netrapalli, P. & Sanghavi, S.Low-rank matrix completion using alternating minimiza- tioninProceedings of the forty-fifth annual ACM symposium on Theory of computing(2013), 665–674

work page 2013
[31]

& Roth, A.Beyond worst-case analysis in private singular vector computationin Proceedings of the forty-fifth annual ACM symposium on Theory of computing(2013), 331–340

Hardt, M. & Roth, A.Beyond worst-case analysis in private singular vector computationin Proceedings of the forty-fifth annual ACM symposium on Theory of computing(2013), 331–340

work page 2013
[32]

Drineas,P.&Mahoney,M.W.RandNLA:randomizednumericallinearalgebra.Communications of the ACM59,80–90 (2016)

work page 2016
[33]

& Tropp, J

Martinsson, P.-G. & Tropp, J. A. Randomized numerical linear algebra: Foundations and algorithms.Acta Numerica29,403–572 (2020)

work page 2020
[34]

Murray, R.et al.Randomized numerical linear algebra: A perspective on the field with an eye to software.arXiv preprint arXiv:2302.11474(2023)

work page arXiv 2023
[35]

& Mahoney, M

Dereziński, M. & Mahoney, M. W.Recent and upcoming developments in randomized numerical linear algebra for machine learninginProceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining(2024), 6470–6479. 13

work page 2024
[36]

& Mahoney, M

Drineas, P., Kannan, R. & Mahoney, M. W. Fast Monte Carlo algorithms for matrices II: Computing a low-rank approximation to a matrix.SIAM Journal on computing36,158–183 (2006)

work page 2006
[37]

& Woodruff, D

Musco, C. & Woodruff, D. P.Sublinear time low-rank approximation of positive semidefinite matricesin2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (2017), 672–683. A Related Work Early work in numerical linear algebra focused on using the power method for estimating the largest singular vector of a matrix [25]. The modern analys...

work page 2017
[38]

study a noisy version of the power method, where each application ofAis accompanied by some additive noise, a model that arises in applications to matrix completion [30], streaming algorithms [13], and differential privacy [31]. While [11] introduce constraints on the additive noise that scale with the spectral gap ofA, in our work, we are dealing with a ...

work page
[39]

Low-Rank Factorization

This is not a problem sincePBΩ =P eBΩ. IfBΩis a λ2-regularized1 /2-spectral approximation ofB, theneBΩis a λ2/θ2(2q+1)-regularized 1/2-spectral approximation of eB. Since θ2(2q+1) = λ′2q+1 1 ≥λ 2, we may assume thateBΩis a 1-regularized1/2-spectral approximation of eB. Thus, by the proof of Lemma 3.3,I−P⪯2( eBeB⊤ +I) −1. For any fixedi, this givesu⊤ i (I ...

work page
[40]

S⊤ satisfies the( k, p ν/8)-AMM property forUA andP ¯AB, whereU A is an orthonormal basis for the column space ofA,PA is the orthogonal projection onto the column space ofA, andP ¯A :=I−P A; 2.S ⊤ s a (1/2)-subspace embedding for the column space ofA. 21 Proof of Lemma 3.6.We may assume thatShas the( ε/2 √ 2, δ′, 2k, p)-OSE moments property by adjusting c...

work page