pith. sign in

arxiv: 2507.17036 · v2 · submitted 2025-07-22 · 💻 cs.IT · cs.DS· cs.NA· math.IT· math.NA

Fast One-Pass Sparse Approximation of the Top Eigenvectors of Huge Approximately Low-Rank Matrices? Yes, MAM^*!

Edem Boahen , Simone Brugiapaglia , Hung-Hsu Chou , Mark Iwen , Felix Krahmer This is my paper

Pith reviewed 2026-05-19 02:54 UTC · model grok-4.3

classification 💻 cs.IT cs.DScs.NAmath.ITmath.NA
keywords one-pass algorithmssparse eigenvector approximationcompressive sensinglow-rank matriceslinear sketchessparse PCAsublinear runtime
0
0 comments X

The pith

A single linear sketch called MAM* enables one-pass sparse approximation of the top eigenvectors for huge approximately low-rank matrices.

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

The paper introduces provably accurate algorithms that approximate the leading eigenvectors of extremely large matrices after a single pass over their entries. It achieves this with a compact linear sketch of the form MAM* whose size is on the order of the target sparse approximations, followed by compressive sensing recovery. The total memory stays proportional to the sparsity of the output vectors rather than the full matrix, and the recovery step runs in time governed mainly by the approximation size. This matters for settings like sparse PCA where matrices with 10^16 entries cannot be stored or revisited.

Core claim

For approximately low-rank matrices whose top eigenvectors admit good sparse approximations, a single compact linear sketch MAM* suffices to recover those sparse vectors accurately via compressive sensing, with the recovery algorithm running in time sublinear in the matrix dimensions and with total storage scaling only with the size of the sought sparse approximations.

What carries the argument

The MAM* linear sketch, a compact compressive measurement operator that preserves enough information for accurate recovery of sparse top eigenvectors through standard compressive sensing techniques.

Load-bearing premise

The input matrices are approximately low-rank so their top eigenvectors admit accurate sparse approximations that the chosen sketch can recover.

What would settle it

Measuring large reconstruction error in the recovered sparse vectors when the matrix rank is increased beyond the regime where sparse approximations are known to exist.

Figures

Figures reproduced from arXiv: 2507.17036 by Edem Boahen, Felix Krahmer, Hung-Hsu Chou, Mark Iwen, Simone Brugiapaglia.

Figure 1
Figure 1. Figure 1: Measurement and CS approximation errors for fixed rank [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Measurement and CS approximation errors for rank [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Measurement and CS approximation errors for rank [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plot of average βm values associated with Theorem 1.2 across the experiments reported in Section 5. Note that all the reported values are less than 8. Behavior of β [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
read the original abstract

Motivated by applications such as sparse PCA, in this paper we present provably-accurate one-pass algorithms for the sparse approximation of the top eigenvectors of extremely massive matrices based on a single compact linear sketch. The resulting compressive-sensing-based approaches can approximate the leading eigenvectors of huge approximately low-rank matrices that are too large to store in memory based on a single pass over its entries while utilizing a total memory footprint on the order of the much smaller desired sparse eigenvector approximations. Finally, the compressive sensing recovery algorithm itself (which takes the gathered compressive matrix measurements as input, and then outputs sparse approximations of its top eigenvectors) can also be formulated to run in a time which principally depends on the size of the sought sparse approximations, making its runtime sublinear in the size of the large matrix whose eigenvectors one aims to approximate. Preliminary experiments on huge matrices having $\sim 10^{16}$ entries illustrate the developed theory and demonstrate the practical potential of the proposed approach.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims to present provably-accurate one-pass algorithms for the sparse approximation of the top eigenvectors of extremely massive approximately low-rank matrices. These algorithms rely on a single compact linear sketch of the form MAM*, achieve memory usage on the order of the desired sparse approximations, and support sublinear runtime via a compressive-sensing recovery step that takes the sketch as input.

Significance. If the central claims hold with rigorous guarantees, the work would enable practical sparse PCA and eigenvector computations on matrices too large to store in memory, with costs scaling primarily with the sparsity level rather than matrix dimensions. This would be valuable for streaming and distributed big-data settings.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (Recovery Procedure): The abstract asserts provable accuracy and sublinear runtime, yet the manuscript must explicitly derive how the matrix sketch S = MAM* produces effective linear measurements Φv of the unknown sparse eigenvector v. Standard CS recovery requires that the derived operator satisfies RIP or incoherence; the approximate-low-rank assumption alone does not automatically supply this, and the current presentation leaves the mapping from S to the recovery optimization unclear.
  2. [Theorem statements (likely §4)] Theorem statements (likely §4): Any error bounds for the recovered sparse eigenvectors must be stated with explicit dependence on the sketch parameters (dimensions of M, number of rows), the sparsity level, and the low-rank approximation error. Without these, the sublinear-runtime claim cannot be verified against the cost of the CS recovery step.
minor comments (2)
  1. [Title] The title uses informal punctuation; a more conventional title would improve clarity for the journal audience.
  2. [Experiments] Preliminary experiments on ~10^16-entry matrices should report quantitative error metrics (e.g., eigenvector approximation error) and wall-clock timings relative to the sketch size to support the sublinear claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (Recovery Procedure): The abstract asserts provable accuracy and sublinear runtime, yet the manuscript must explicitly derive how the matrix sketch S = MAM* produces effective linear measurements Φv of the unknown sparse eigenvector v. Standard CS recovery requires that the derived operator satisfies RIP or incoherence; the approximate-low-rank assumption alone does not automatically supply this, and the current presentation leaves the mapping from S to the recovery optimization unclear.

    Authors: We thank the referee for highlighting this point. In Section 3 we outline the recovery procedure that takes S = MAM* as input and applies a compressive sensing solver to obtain the sparse eigenvector approximation. To make the mapping explicit, we will add a dedicated derivation in the revised manuscript showing that, for a matrix A that is approximately low-rank, the sketch S yields effective linear measurements of the form Φv (where v is the sparse top eigenvector) and that the operator Φ satisfies the RIP with high probability for random M with appropriate row dimension. The approximate low-rank assumption is used to control the deviation from the exact eigenvector equation, ensuring the recovery guarantees hold. This clarification will be inserted before the recovery optimization is presented. revision: yes

  2. Referee: [Theorem statements (likely §4)] Theorem statements (likely §4): Any error bounds for the recovered sparse eigenvectors must be stated with explicit dependence on the sketch parameters (dimensions of M, number of rows), the sparsity level, and the low-rank approximation error. Without these, the sublinear-runtime claim cannot be verified against the cost of the CS recovery step.

    Authors: We agree that explicit dependence improves verifiability. While the proofs in Section 4 already track these quantities (sketch row count r, sparsity s, and low-rank error ε), the theorem statements themselves present the bounds in a more compact form. In the revision we will restate the main theorems to display the explicit dependence on r, s, and ε. This will also make transparent that the CS recovery step runs in time O(s log n + poly(r)) which is sublinear in the ambient matrix dimensions, confirming the claimed runtime scaling. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on independent primitives

full rationale

The paper proposes one-pass algorithms using the MAM* linear sketch plus compressive sensing recovery to approximate sparse top eigenvectors of approximately low-rank matrices. The central claims rely on standard concentration properties of random sketching matrices and established CS recovery guarantees (RIP, incoherence) applied to the derived measurements, without redefining the eigenvectors in terms of the sketch outputs or fitting parameters to the target data in a self-referential loop. No load-bearing step reduces by construction to a self-citation chain or an ansatz imported from the authors' prior work; the approximate-low-rank assumption and sublinear runtime follow from external matrix sketching theory rather than being forced by the paper's own equations. This is the typical non-circular case for sketching papers that compose known primitives.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on the domain assumption that the matrices are approximately low-rank and admit sparse eigenvector approximations recoverable via the MAM* sketch; no free parameters, invented entities, or additional axioms are stated in the abstract.

axioms (1)
  • domain assumption The input matrix is approximately low-rank.
    Explicitly stated in the abstract as the setting for which the algorithms apply.

pith-pipeline@v0.9.0 · 5730 in / 1368 out tokens · 81408 ms · 2026-05-19T02:54:18.583443+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Ailon and B

    N. Ailon and B. Chazelle. The fast Johnson-Lindenstrauss transform and approximate nearest neighbors. SIAM Journal on Computing , 39(1):302–322, 2009

    work page 2009

  2. [2]

    Ailon and E

    N. Ailon and E. Liberty. Fast dimension reduction using Rademacher series on dual BCH codes. Discrete & Computational Geometry , 42(4):615–630, Sept. 2008

    work page 2008

  3. [3]

    Andoni and H

    A. Andoni and H. L. Nguyen. Eigenvalues of a matrix in the streaming model. In Pro- ceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 1729–1737. SIAM, 2013

    work page 2013

  4. [4]

    Bailey, M

    J. Bailey, M. A. Iwen, and C. V. Spencer. On the design of deterministic matrices for fast recovery of Fourier compressible functions. SIAM Journal on Matrix Analysis and Applications, 33(1):263–289, 2012

    work page 2012

  5. [5]

    Baraniuk, M

    R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin. A simple proof of the restricted isometry property for random matrices. Constructive Approximation, 28:253–263, 2008

    work page 2008

  6. [6]

    Bourgain, S

    J. Bourgain, S. Dirksen, and J. Nelson. Toward a unified theory of sparse dimension- ality reduction in euclidean space. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing , pages 499–508, 2015

    work page 2015

  7. [7]

    K. L. Clarkson and D. P. Woodruff. Numerical linear algebra in the streaming model. In Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing , pages 205–214, 2009

    work page 2009

  8. [8]

    Cormode and S

    G. Cormode and S. Muthukrishnan. Combinatorial algorithms for compressed sensing. In P. Flocchini and L. Gasieniec, editors, Structural Information and Communication Complexity, pages 280–294, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg

    work page 2006

  9. [9]

    R. A. DeVore. Deterministic constructions of compressed sensing matrices. Journal of Complexity, 23(4):918 – 925, 2007

    work page 2007

  10. [10]

    Foucart and H

    S. Foucart and H. Rauhut. A Mathematical Introduction to Compressive Sensing . Ap- plied and Numerical Harmonic Analysis. Birkh¨ auser/Springer, New York, 2013

    work page 2013

  11. [11]

    G. H. Golub and C. F. Van Loan. Matrix Computations . Johns Hopkins University Press, Baltimore, MD, USA, 3rd edition, 1996

    work page 1996

  12. [12]

    Halko, P.-G

    N. Halko, P.-G. Martinsson, and J. A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53(2):217–288, 2011

    work page 2011

  13. [13]

    R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Presss, Cambridge, 1991. 37

    work page 1991

  14. [14]

    M. Iwen. Compressed sensing with sparse binary matrices: Instance optimal error guarantees in near-optimal time. Journal of Complexity , 30(1):1 – 15, 2014

    work page 2014

  15. [15]

    M. Iwen. A Mathematical Introduction to Fast and Memory Efficient Algo- rithms for Big Data. https: // math. msu. edu/~ iwenmark/ Notes_ Fall2024_ Iwen_ Classes. pdf, 2024

    work page 2024

  16. [16]

    M. A. Iwen. A deterministic sub-linear time sparse Fourier algorithm via non-adaptive compressed sensing methods. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms , SODA 08, pages 20–29, USA, 2008. Society for Industrial and Applied Mathematics

    work page 2008

  17. [17]

    M. A. Iwen. Simple deterministically constructible RIP matrices with sublinear Fourier sampling requirements. In 2009 43rd Annual Conference on Information Sciences and Systems, pages 870–875, March 2009

    work page 2009

  18. [18]

    M. A. Iwen, D. Needell, E. Rebrova, and A. Zare. Lower memory oblivious (tensor) subspace embeddings with fewer random bits: modewise methods for least squares. SIAM Journal on Matrix Analysis and Applications , 42(1):376–416, 2021

    work page 2021

  19. [19]

    M. A. Iwen, B. Schmidt, and A. Tavakoli. On Fast Johnson-Lindenstrauss Embeddings of Compact Submanifolds of RN with Boundary. Discrete Comput. Geom., 71(2):498– 555, 2024

    work page 2024

  20. [20]

    Kacham and D

    P. Kacham and D. Woodruff. Approximating the top eigenvector in random order streams. Advances in Neural Information Processing Systems, 37:113461–113483, 2024

    work page 2024

  21. [21]

    B. S. Kashin. The diameters of octahedra. Uspekhi Matematicheskikh Nauk , 30(4(184)):251–252, 1975

    work page 1975

  22. [22]

    Krahmer and R

    F. Krahmer and R. Ward. New and improved Johnson–Lindenstrauss embeddings via the restricted isometry property. SIAM Journal on Mathematical Analysis , 43(3):1269– 1281, 2011

    work page 2011

  23. [23]

    Lau and J

    I. Lau and J. Jedwab. Construction of binary matrices for near-optimal compressed sensing. In 2021 IEEE International Symposium on Information Theory (ISIT) , pages 1612–1617. IEEE, 2021

    work page 2021

  24. [24]

    Martinsson and J

    P.-G. Martinsson and J. A. Tropp. Randomized numerical linear algebra: Foundations and algorithms. Acta Numerica, 29:403–572, 2020

    work page 2020

  25. [25]

    Muthukrishnan

    S. Muthukrishnan. Data streams: Algorithms and applications. Foundations and Trends® in Theoretical Computer Science, 1(2):117–236, 2005

    work page 2005

  26. [26]

    Nakatsukasa and J

    Y. Nakatsukasa and J. A. Tropp. Fast and accurate randomized algorithms for linear systems and eigenvalue problems. SIAM Journal on Matrix Analysis and Applications , 45(2):1183–1214, 2024. 38

    work page 2024

  27. [27]

    Needell, W

    D. Needell, W. Swartworth, and D. P. Woodruff. Testing positive semidefiniteness using linear measurements. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 87–97. IEEE, 2022

    work page 2022

  28. [28]

    Needell and J

    D. Needell and J. A. Tropp. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis , 26(3):301–321, 2009

    work page 2009

  29. [29]

    Price and Z

    E. Price and Z. Xun. Spectral guarantees for adversarial streaming PCA. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) , pages 1768–1785. IEEE, 2024

    work page 2024

  30. [30]

    Y. Saad. Numerical Methods for Large Eigenvalue Problems: Revised Edition . SIAM, 2011

    work page 2011

  31. [31]

    Swartworth and D

    W. Swartworth and D. P. Woodruff. Optimal eigenvalue approximation via sketching. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing , pages 145–155, 2023

    work page 2023

  32. [32]

    K. M. Tan, Z. Wang, H. Liu, and T. Zhang. Sparse generalized eigenvalue problem: Optimal statistical rates via truncated rayleigh flow. Journal of the Royal Statistical Society Series B: Statistical Methodology , 80(5):1057–1086, 2018

    work page 2018

  33. [33]

    Vershynin

    R. Vershynin. High-Dimensional Probability: An Introduction with Applications in Data Science, volume 47. Cambridge University Press, 2018

    work page 2018

  34. [34]

    M. J. Wainwright. High-dimensional statistics: A non-asymptotic viewpoint , volume 48. Cambridge University Press, 2019

    work page 2019

  35. [35]

    D. P. Woodruff. Sketching as a tool for numerical linear algebra. Foundations and Trends® in Theoretical Computer Science, 10(1–2):1–157, 2014

    work page 2014

  36. [36]

    Y. Yu, T. Wang, and R. J. Samworth. A useful variant of the Davis–Kahan theorem for statisticians. Biometrika, 102(2):315—-323, 2015

    work page 2015

  37. [37]

    Yuan and T

    X.-T. Yuan and T. Zhang. Truncated power method for sparse eigenvalue problems. Journal of Machine Learning Research , 14(4), 2013

    work page 2013

  38. [38]

    H. Zou, T. Hastie, and R. Tibshirani. Sparse principal component analysis. Journal of Computational and Graphical Statistics , 15(2):265–286, 2006. 39

    work page 2006