Randomized and Inner-product Free Krylov Methods for Large-scale Inverse Problems

Ariana N. Brown; James G. Nagy; Julianne Chung; Malena Sabat\'e Landman

REVIEW 3 major objections 5 minor 28 references

Randomized and Inner-product Free Krylov Methods for Large-scale Inverse Problems

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Two new inner-product-free Krylov methods track the residual norms of GMRES and LSQR without computing inner products.

desk verdict The core idea is sound and worth pursuing, but Algorithm 1 as printed cannot implement the claimed method, so the experimental results are not independently verifiable. read the letter →

arxiv 2502.02721 v1 pith:QLOPH2RJ submitted 2025-02-04 math.NA cs.NA

Malena Sabaté Landman , Ariana N. Brown , Julianne Chung , James G. Nagy This is my paper · ORCID

classification math.NAcs.NA MSC 65F1065F2265F2068W20

keywords Krylovsubspacemethodsinner-product-freealgorithmsrandomizedsketchingsketch-and-solveinverseproblemsleastsquaresTikhonovregularizationHessenbergmethod

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces two Krylov subspace methods, sCMRH and sLSLU, for large-scale linear inverse problems that never compute inner products. Existing inner-product-free Krylov methods such as CMRH and LSLU only minimize a quasi-residual, and the gap between the quasi-norm and the true residual norm is uncontrolled; for inverse problems this gap matters because the residual norm is the fit-to-data part of the likelihood. The new methods build a nonorthogonal basis with the Hessenberg method and solve the projected least-squares problem with a randomized sketch, which the paper shows to be unbiased and close to the minimal-residual solution. Numerical experiments indicate that the sketched methods track the residual-norm curves of GMRES and LSQR while producing smaller residual norms than CMRH and LSLU.

What carries the argument

The load-bearing object is the sketched projected least-squares problem $\min_y \|S(AL_k y - b)\|$, in which $L_k$ is the nonorthogonal Krylov basis produced by the (generalized) Hessenberg method with partial pivoting and $S$ is a Gaussian sketching matrix. The Hessenberg construction removes inner products, and the sketch turns the tall-skinny projected problem into a small least-squares problem. The bound in Eq. (14) is the engine: it converts the subspace embedding property into multiplicative control of the residual norm, showing that the sketched solution stays within a factor $(1+\epsilon)/(1-\epsilon)$ of the minimal-residual solution in that subspace.

What would settle it

Recompute the deblurring example with the paper's sketch size and plot per-iteration residual norms of sCMRH against GMRES and CMRH; the central claim fails if the sCMRH curve rises above CMRH, or if its gap above GMRES ever exceeds the sketch condition number $\kappa_2(SQ_{[A b]})$ that the paper identifies with $(1+\epsilon)/(1-\epsilon)$.

Watch

Extended reading notes

Core claim

At iteration $k$, the proposed methods generate a nonorthogonal basis $L_k$ for a Krylov subspace using the Hessenberg method (for sCMRH) or the generalized Hessenberg method (for sLSLU), with no inner products. Instead of the quasi-minimal-residual reduction used by CMRH and LSLU, they solve the sketched projected least-squares problem $\min_{y} \|S(AL_k y - b)\|$, where $S$ is a Gaussian sketch. By the subspace embedding property, the resulting residual norm satisfies $\|Ax_{\mathrm{LS}}-b\| \le \|Ax_k^{(S)}-b\| \le \frac{1+\epsilon}{1-\epsilon}\|Ax_{\mathrm{LS}}-b\|$, where $x_{\mathrm{LS}}$ is the true minimal-residual solution in the same Krylov subspace; and the sketch-and-solve estimate is unbiased, with expected squared residual norm inflated only by a factor $1 + k/(\ell-k-1)$. The paper's numerical experiments show sCMRH residual norms tracking GMRES and sLSLU tracking LSQR, while CMRH and LSLU residual norms drift larger.

Load-bearing premise

The method's practical success rests on the assumption that a small random projection of dimension ten times the iteration count preserves residual norms well enough, but the paper's own theoretical guarantee needs a much larger projection that the experiments do not use.

Editorial extensions

If this is right

If the central claim holds, sCMRH and sLSLU can stand in for GMRES and LSQR in inverse-problem solves where inner products are the bottleneck, because their residual norms follow the same curves.
Inner-product-free operation removes a common source of breakdown in low-precision arithmetic, since no orthogonal projection is computed.
The same sketch-and-solve reduction applies to the Tikhonov-regularized problem with two sketches, giving residual-norm behavior close to LSQR for a fixed regularization parameter.
On the tested problems, the sketched methods achieve smaller residual norms than their non-sketched inner-product-free predecessors CMRH and LSLU.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extending the paper's idea, a natural test is to vary the sketch dimension $\ell$ with the iteration count or problem size: the fixed choice $\ell = 10(\text{maxiter}+1)$ used everywhere is far smaller than the theory requires, so residual tracking might eventually break as the problem grows.
The same sketched-projected-problem idea could be applied to other nonorthogonal Krylov bases, such as truncated Arnoldi or incomplete Golub-Kahan, replacing a reorthogonalization-depth hyperparameter with a sketch.
Because the sketch-and-solve solution is unbiased, averaging several independent sLSLU runs could reduce the variance of the residual-norm estimates; the paper does not explore this averaging.
One further consequence the authors do not draw is that, for hybrid approaches that estimate a regularization parameter from the projected problem, the closer residual norm should make parameter-selection heuristics more reliable than with CMRH and LSLU.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new algorithmic family in this paper is legitimate: take the inner-product-free generalized Hessenberg basis construction from CMRH/LSLU and add a randomized sketch-and-solve step for the projected least-squares problem. Section 3.2 is mostly standard and correct: the unbiasedness argument follows from textbook randomized NLA, and Eq. (14) is the usual subspace embedding bound applied to residuals. The numerical results on deblurring, neutron tomography, and two real x-ray datasets suggest the sketched variants track GMRES/LSQR much more closely than CMRH/LSLU, which is what you would hope for if the method works.

But there is a load-bearing problem in the pseudocode. Algorithm 1 (sLSLU) computes z_k = S2 * (A^T d_k) in line 11 and never uses it, then line 31 asks for the minimizer of ||S2 r0 - Z y|| with Z undefined. To implement Eq. (13) via the generalized Hessenberg relationship A L_k = D_{k+1} H_{k+1,k}, you would need to sketch the D columns (or maintain S2 D_{k+1}) and multiply by H. As printed, the algorithm is non-executable, so the Section 4 results could well come from a different implementation. This is more fundamental than the sketch-size gap.

The sketch-size gap itself is also worth flagging: the experiments use ell = 10*(maxiter+1), far below the theory's ell ~ m log m / eps^2, and the paper honestly admits the guarantee cannot always be met in practice. Without error bars or multiple trials, it is unclear whether the close match to GMRES/LSQR is robust or partly lucky. Minor issues: a few typos in the loops (e.g., line 27 uses u instead of q) and a duplicated Sarlos reference.

Who is this for? Researchers working on communication-avoiding or low-precision Krylov methods for inverse problems. The idea deserves referee time, but the current manuscript is not reproducible. I would send it to peer review with a strong request for a corrected algorithm and code, not desk-reject it.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes two inner-product-free Krylov subspace methods, sCMRH and sLSLU, that combine Hessenberg/generalized-Hessenberg basis construction with randomized sketch-and-solve for the projected least-squares and Tikhonov problems. Section 3.2 derives a residual-norm approximation bound (Eq. 14), notes the unbiasedness of the sketched estimator, and gives an expected squared residual formula. Algorithms 1 and 2 provide pseudocode, and Section 4 reports experiments on deblurring, simulated neutron tomography, and two real X-ray datasets, with the conclusion that the sketched methods track GMRES/LSQR residual norms more closely than CMRH/LSLU while remaining inner-product free.

Significance. If the methods perform as claimed, the contribution is useful: it offers inner-product-free alternatives to GMRES and LSQR whose residual norm is closer to the true minimal residual than that of quasi-minimal residual methods, with potential benefits for parallel and low-precision computation. The theoretical derivation in Section 3.2 is standard and largely correct, and the paper honestly acknowledges that the subspace-embedding guarantee cannot always be guaranteed for the sketch sizes used in practice. However, the pseudocode as printed does not define the sketched projected problem it claims to solve, and the experiments use sketch dimensions far below the theory with no variance reporting. These issues currently prevent the numerical evidence from being verified.

major comments (3)

[Algorithm 1, line 31; Algorithm 2, line 32] The solve step is not executable from the pseudocode. Line 31 states that y_k is the minimizer of ||S2 r0 - S2 A L_k y||^2 = ||S2 r0 - Z y||^2, but no matrix Z is constructed anywhere. The only sketch computed in the loop, z_k = S2 * A^T d_k (line 11), is never used and is not a column of S2 D_{k+1} H_{k+1,k} or of S2 A L_k; similarly, Algorithm 2 computes f1 and f_{k+1} but never assembles S1 L_k. Consequently, the numerical results in Section 4 may come from an implementation different from the method defined by Eq. (13). Please specify explicitly how S2 A L_k (or an equivalent factored form) is formed, for example by maintaining sketches of the D columns and multiplying by H, and correct both algorithms accordingly.
[Algorithm 1, lines 17, 21, and 27] The basis update contains inconsistent variable reuse that prevents verification of the generalized Hessenberg recurrence (7). Line 17 overwrites d_k with the newly computed basis vector, so d_k no longer denotes the kth column of D when it is used in line 21 and in the subsequent W update; the new vector should be stored as d_{k+1}. In addition, line 27 sets W(k+1,k+1) using u(g(i2)), although the quantity available for the L-basis update at that point is q(g(i2)). These are not mere notation slips: they make the algorithm ambiguous about which vectors enter the H and W relations.
[Section 4 and Eq. (14)] The experiments set ell = 10*(maxiter+1), which equals at most a few hundred in the reported runs, while the theory in Section 3.2 requires ell ~ m log(m)/eps^2 for Eq. (14) to hold with high probability; for the tested image and tomography problems, m is orders of magnitude larger. The paper acknowledges that the guarantee cannot always be ensured in practice, but every figure shows a single run with no error bars or repeated trials, and the random pivoting sample size is not systematically studied. Please report the distribution of residual norms over repeated sketches and pivot samples, or otherwise provide empirical justification for the small sketch dimension.

minor comments (5)

[Throughout] There are several typos and grammatical slips, including "mulitplications" (Section 1), "are are" (Section 4.1), and "perforamcne" (Section 5). A careful proofread is needed.
[References] References [15] and [23] are the same paper (Sarlos, FOCS 2006) and should be consolidated.
[Section 3.3, Eq. (17)] The two terms in Eq. (17) are missing the superscript 2 and the subscript 2: the expression should read ||S2(ALky - b)||_2^2 + lambda^2 ||S1(Lky)||_2^2. The same omission appears in Algorithm 2, line 32.
[Figure 2 caption and Section 4.1] The caption states that sCMRH uses a sample size of 5 for the random pivot search, but Section 4.2 describes a sample size of 25 and the text never defines how the sample size for sCMRH was chosen or whether the results are sensitive to it.
[Figures 4, 6, 8, and 11] Several residual-norm panels appear to lack curve labels or legends, making it difficult to verify which curve corresponds to sLSLU, LSLU, and LSQR; please add legends or otherwise identify each curve.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the sketch-and-solve residual-norm claim is a standard embedding argument, not a restatement of the method's inputs.

full rationale

The central derivation—that solving the sketched projected problem (Eq. 13) yields residual norms bounded relative to the minimal-residual solution by (1+ε)/(1−ε) (Eq. 14)—is a direct application of the subspace-embedding property (Eq. 4), not an equivalence-by-construction. The Gaussian expectation identity for sketch-and-solve is cited to [24,25] and is independent of the paper's fitted values. The generalized Hessenberg relations (Eqs. 6–7) are inherited from the authors' prior work [9], but they are used as a basis-construction tool rather than as the source of the residual-norm claim; the residual-norm claim stands on the embedding property. The comparisons with CMRH, LSLU, GMRES, and LSQR are benchmarks, not circular premises. The paper explicitly concedes that the small sketch dimension used in experiments does not always satisfy the theory ('this cannot, however, always be guaranteed in practice'), which is a validity gap rather than a circular reduction. One non-circular but serious issue: Algorithm 1 line 31 refers to an undefined Z (and line 11's z_k = S2 A^T d_k is unused), so the printed pseudocode may not implement Eq. (13); this is an algorithmic correctness concern, not a circularity.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The new methods rest on standard randomized NLA embedding theory and the authors' earlier generalized Hessenberg basis construction. The main user-chosen inputs are the sketch dimension and sampling size, which are set by hand rather than derived from the theory for the reported experiments. No new physical or mathematical entities are introduced.

free parameters (3)

sketch dimension ell = 10 * (maxiter + 1)
Chosen by hand for all experiments; the theoretical guarantee in Eq. (14) requires a much larger dimension (ell ~ m log(m)/eps^2), so the practical choice is an ad hoc compromise between cost and accuracy.
sampling size for partial pivoting = 25 entries
The paper replaces the exact infinity norm with a maximum over 25 randomly sampled entries to reduce communication; sample size is fixed without sensitivity analysis beyond a note that it does not drastically change results.
regularization parameter lambda = 26 (neutron tomography), 1 (cheese and walnut)
Fixed by hand for each Tikhonov example; the paper notes that adaptive selection is future work, so the numerical demonstration depends on a user-supplied lambda.

assumptions (3)

standard math The sketching matrix S satisfies the subspace embedding property (4) for the vectors in the projected Krylov subspace, with a given probability and distortion 0 < eps < 1.
This is a standard result for Gaussian embeddings from the randomized NLA literature (e.g., [14,23]); the paper uses it to derive the residual norm bound (14).
domain assumption The generalized Hessenberg method with partial pivoting generates linearly independent basis vectors for K_k(A^T A, A^T b) and K_k(A A^T, b), with manageable condition numbers.
The basis construction and its stability are taken from the authors' prior work [9]; the current paper uses this construction without re-proving it.
domain assumption The matrix A L_k has full column rank, so the least-squares and sketched least-squares problems have unique solutions and the unbiasedness result E[x(S)_k] = x(LS)_k holds.
This is assumed in Section 3.2 where the projection and sketch-and-solve are applied; breakdown is avoided by partial pivoting but full rank is not proven for all cases.

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Cite this review

Pith. "Pith review of Randomized and Inner-product Free Krylov Methods for Large-scale Inverse Problems." pith.science (2026). https://pith.science/paper/QLOPH2RJ

@misc{pith2026250202721,
  author       = {Pith},
  title        = {Pith review of: Randomized and Inner-product Free Krylov Methods for Large-scale Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLOPH2RJ}},
  note         = {Machine review of arXiv:2502.02721}
}

read the original abstract

Iterative Krylov projection methods have become widely used for solving large-scale linear inverse problems. However, methods based on orthogonality include the computation of inner-products, which become costly when the number of iterations is high; are a bottleneck for parallelization; and can cause the algorithms to break down in low precision due to information loss in the projections. Recent works on inner-product free Krylov iterative algorithms alleviate these concerns, but they are quasi-minimal residual rather than minimal residual methods. This is a potential concern for inverse problems where the residual norm provides critical information from the observations via the likelihood function, and we do not have any way of controlling how close the quasi-norm is from the norm we want to minimize. In this work, we introduce a new Krylov method that is both inner-product-free and minimizes a functional that is theoretically closer to the residual norm. The proposed scheme combines an inner-product free Hessenberg projection approach for generating a solution subspace with a randomized sketch-and-solve approach for solving the resulting strongly overdetermined projected least-squares problem. Numerical results show that the proposed algorithm can solve large-scale inverse problems efficiently and without requiring inner-products.

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Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

In: Scherzer, O

Chung, J., Knepper, S., Nagy, J.G.: Large-scale inverse problems in imaging. In: Scherzer, O. (ed.) Handbook of Mathematical Methods in Imaging, pp. 47–90. Springer, New York, NY (2015). https://doi.org/10.1007/978-1-4939-0790-8 2

work page doi:10.1007/978-1-4939-0790-8 2015

[2] [2]

SIAM, Philadel- phia (2010)

Hansen, P.C.: Discrete Inverse Problems: Insight and Algorithms . SIAM, Philadel- phia (2010)

work page 2010

[3] [3]

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Vogel, C.R.: Computational Methods for Inverse Problems. SIAM , Philadelphia (2002)

work page 2002

[4] [4]

Zhdanov, M.: Geophysical Inverse Theory and Regularization Pr oblems vol. 36. Elsevier, New York (2002)

work page 2002

[5] [5]

Freeman, San Francisco and London (1963)

Faddeev, D.K., Faddeeva, V.N.: Computational Methods of Linear Algebra. Freeman, San Francisco and London (1963). Translated by R. C. W illiams

work page 1963

[6] [6]

Sadok, H.: CMRH: A new method for solving nonsymmetric linear sys tems based on the Hessenberg reduction algorithm. Numer. Algor. 20, 303–321 (1999)

work page 1999

[7] [7]

B IT Numerical Mathematics 52, 485–501 (2012) https://doi.org/10.1007/s10543-011- 0365-x

Sadok, H., Szyld, D.B.: A new look at CMRH and its relation to GMRES. B IT Numerical Mathematics 52, 485–501 (2012) https://doi.org/10.1007/s10543-011- 0365-x

work page doi:10.1007/s10543-011- 2012

[8] [8]

SIAM J ournal on Matrix Analysis and Applications 46(1), 232–255 (2025) https://doi.org/10.1137/ 24M1634874

Brown, A.N., Sabat´ e Landman, M., Nagy, J.G.: H-CMRH: An inner pr oduct free hybrid Krylov method for large-scale inverse problems. SIAM J ournal on Matrix Analysis and Applications 46(1), 232–255 (2025) https://doi.org/10.1137/ 24M1634874

work page 2025