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REVIEW 4 major objections 4 minor 43 references

Stabilization of the Gradient Method for Solving Linear Algebraic Systems -- A Method Related to the Normal Equation

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proposes a residual-stabilized gradient iteration that claims convergence for every nonsingular matrix and every real step size, at the price of one large stabilization parameter.

desk verdict The stabilized gradient method has a correct error bound, but the advertised step-size-independent global convergence fails against the paper's own condition (28). read the letter →

arxiv 2506.00702 v1 pith:IITNCGHX submitted 2025-05-31 math.NA cs.NA

Ibrahima Dione This is my paper
classification math.NAcs.NA MSC 65F1065F22
keywords stabilizedgradientmethodlinearsystemsnormalequationsingularvaluedecompositionfilterfactorsill-conditionedproblemsstep-sizeindependenceglobalconvergence
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 proposes replacing the gradient iteration $x^{k+1}=x^k-\alpha_k(Ax^k-b)$ by the stabilized recurrence $(I+\gamma A^T A)x^{k+1}_\gamma=(I-\alpha_k A)x^k_\gamma+\alpha_k b+\gamma A^T b$, where $\gamma>0$ is a user-set stabilization parameter. The author's claim is that, for any nonsingular matrix $A$, with no symmetry, positive definiteness, or eigenvalue-structure assumption, and for any real step size $\alpha_k$, taking $\gamma$ large enough makes the iterates converge to $A^{-1}b$, with an error bound that contracts geometrically. The practical payoff is that step-size tuning and matrix-structure conditions, the usual obstacles of the plain gradient method, are replaced by one parameter that also controls the number of iterations. Numerical tests on nonsymmetric, increasingly ill-conditioned, inverse, and finite-element problems show the method reaching the tolerance in one or a few iterations when $\gamma$ is large, while very large $\gamma$ degrades accuracy on severely ill-conditioned problems.

What carries the argument

The load-bearing object is the regularized normal-equation operator $I+\gamma A^T A$, whose inverse damps the small singular values of $A$: the bound $\|(I+\gamma A^T A)^{-1}\|\le \kappa(V)/(1+\gamma\sigma_n^2)$ from the SVD is what turns the possibly non-contracting factor $\|I-\alpha_k A\|$ into a product that can be made contracting. The second mechanism is the filtered SVD expansion of the iterate, with filter factors $\phi^{(k)}_{(\gamma,i)}$ showing how increasing $\gamma$ progressively includes more SVD components, up to the naive solution in the $\gamma\to\infty$ limit.

What would settle it

Set $A=1$, $b=1$, $x_0=0$, $\gamma=1$, and $\alpha=100$. The stabilized iteration gives $x_1=((1-100)\cdot 0+100+1)/(1+1)=101/2$, so the error grows from $1$ to $49.5$, and the scheme diverges. This shows that convergence cannot hold for every real $\alpha$ at a fixed $\gamma$; if the claim is that $\gamma$ may be chosen after $\alpha$, the same calculation marks the regime where the proof's contraction condition must be checked before convergence is asserted.

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Extended reading notes

Core claim

The central object is the stabilized iteration above, which adds the normal-equation terms $\gamma A^T A$ and $\gamma A^T b$ to the gradient method. Subtracting the exact solution $x_\star$ gives the error recurrence $e^{k+1}_\gamma=(I+\gamma A^T A)^{-1}(I-\alpha_k A)e^k_\gamma$. Using the SVD $A=U\Sigma V^T$, the paper bounds $\|(I+\gamma A^T A)^{-1}\|\le \kappa(V)/(1+\gamma\sigma_n^2)$, where $\sigma_n$ is the smallest singular value of $A$, and concludes that the full contraction factor is $\kappa(V)\|I-\alpha_k A\|/(1+\gamma\sigma_n^2)$. Because $\gamma$ can be made arbitrarily large, the factor can be brought below $1$ for any real $\alpha_k$ and any nonsingular $A$, which is the basis for the claimed global convergence. The SVD/filter-factor analysis expresses $x^k_\gamma$ as a filtered version of the naive solution $\sum_i (v_i^T b/\sigma_i)v_i$, with filter factors approaching $1$ as $\gamma\to\infty$; the paper notes this limit explains both the one-iteration direct behavior for large $\gamma$ and the instability on severely ill-conditioned problems.

Load-bearing premise

The argument assumes that a stabilization parameter $\gamma$ can be chosen large enough to make $\kappa(V)\|I-\alpha_k A\|/(1+\gamma\sigma_n^2)$ smaller than $1$ while the inner system $(I+\gamma A^T A)x = \cdots$ is still solved accurately; for a fixed $\gamma$, large step sizes can violate the bound, and for severely ill-conditioned $A$, very large $\gamma$ makes the inner system itself hard to solve.

Editorial extensions

If this is right

  • If the convergence claim holds, a single algorithm solves $Ax=b$ for any nonsingular $A$ without symmetry or definiteness assumptions, and $\alpha$ can be fixed once and for all to a neutral value such as $1$.
  • Large $\gamma$ makes the iteration behave like a direct method, often reaching the target accuracy in one step on well-conditioned systems.
  • The convergence rate is governed by $\gamma$ and the smallest singular value rather than by the eigenvalue distribution of $A$, so the method sidesteps the classic slow-convergence behavior of steepest descent.
  • For severely ill-conditioned matrices the user must balance $\gamma$: too small means slow convergence, while too large drives the solution toward the naive SVD solution and can amplify noise or quadrature error.

Reading between the lines

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

  • In the Euclidean norm used throughout, the SVD matrix $V$ is orthogonal, so $\kappa(V)=1$ and the paper's stated error factor simplifies to $\|I-\alpha A\|/(1+\gamma\sigma_n^2)$; the symbolic $\kappa(V)$ does not add a real penalty.
  • Because the SVD/filter-factor calculation does not actually use squareness or nonsingularity, running the scheme from $x_0=0$ with very large $\gamma$ formally reaches the minimum-norm least-squares solution $A^{+}b$; extending the proof to rectangular and rank-deficient systems is a natural next step the paper leaves open.
  • An adaptive strategy that increases $\gamma$ until the residual is controlled, rather than fixing a huge $\gamma$ a priori, would address the documented trade-off between fast convergence and naive-solution instability; this is testable on the same shaw, heat, and gravity examples.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a stabilized gradient method (Eq. (9)) for solving nonsingular linear systems Ax=b, obtained by adding the regularization term γA^tA to the standard gradient iteration. The main advertised results are that the method converges globally for any nonsingular matrix A, without assumptions of symmetry or positive definiteness, and that the choice of the stepsize α_k is irrelevant because convergence holds for every α_k ∈ R. The analysis derives an error recursion (Theorem 1), a product error bound (Corollary 1, Eq. (26)–(27)), and a contraction condition (Remark 3, Eq. (28)). Numerical experiments on well-conditioned, increasingly ill-conditioned, and severely ill-conditioned problems are reported, including large-scale finite element systems.

Significance. If the advertised claims were correct, the method would be a substantial contribution: a genuinely step-size-independent convergent iterative solver for general nonsingular systems. The paper does contain a correct error recursion and a valid sufficient contraction bound, and the numerical experiments are fairly extensive. However, the central claim of convergence for every real stepsize is false as stated; the contraction condition itself depends on the stepsize, and for fixed γ the iteration diverges for sufficiently large |α|. In addition, the singular value expansion in Section 2.2 is only valid for symmetric matrices, undermining the stated mechanism for non-symmetric cases. The contribution, after correcting these points, would be much weaker than advertised.

major comments (4)
  1. [Abstract and Remark 3 (Eq. (28))] The claim that the method converges for every α_k ∈ R is false. The contraction factor in Eq. (28) is (κ(V)/(1+γσ_n^2))·||I−αA||, which depends on α. For any fixed γ > 0 and any nonsingular A, ||I−αA|| tends to infinity as |α| tends to infinity, so condition (28) is violated for sufficiently large |α|. A concrete scalar counterexample is A=1, γ=1, α=100: the error recursion e^{k+1} = (1−α)/(1+γ) e^k gives e^{k+1} = −49.5 e^k, so the error grows without bound. Thus the advertised property in the abstract and the statement in Remark 3 that convergence is 'no longer based on the choice of the stepsize' are internally inconsistent with the paper's own contraction condition.
  2. [Corollary 1 and Remark 3 (Eqs. (26)–(29))] The 'if and only if' in Eq. (28) is not a valid characterization of convergence. Equation (27) provides a sufficient condition for the decay of the error bound, not a necessary condition for the convergence of the iterates; the actual spectral radius condition may be satisfied even when the norm bound exceeds one, and the norm bound can indicate divergence when the iteration still converges. Consequently, the subsequent discussion in Eqs. (29)–(30) and the conclusion that the stepsize has only a limited influence are not justified by the analysis.
  3. [Section 2.2, Eq. (12)] The decomposition A = VΣV^t in Eq. (12) is only valid for symmetric matrices. For a general nonsingular matrix the singular value decomposition is A = UΣV^t with U ≠ V. As a result, the filtered SVD expansions in Eqs. (13)–(17), the filter factor plots in Figure 1, and the interpretation in Remark 2 do not apply to the non-symmetric matrices that the paper claims to handle. This technical error undermines the explanatory mechanism for the stabilization, even though the error bound in Corollary 1 does not rely on this expansion.
  4. [Sections 4 and 5 (Numerical experiments and conclusion)] The numerical experiments do not test the claimed 'any α_k ∈ R' property. All reported runs use moderate values of α (mostly α=1 or powers of ten in a bounded range) and always choose γ large enough to drive the tested iteration matrix to a contraction. The conclusion in Section 5 that 'the value of the stepsize α has a limit influence on the convergence' is not supported by the analysis, which shows an unbounded dependence of the error bound on α. A test with a large α (e.g., α=100 for a scalar or diagonal system) would immediately demonstrate the failure of the claimed step-size independence.
minor comments (4)
  1. [Throughout] There are numerous typographical errors, including 'marixA' for 'matrix A', 'ect.' for 'etc.', 'choix' for 'choice', and inconsistent use of subscripts in norms (||·||_n vs. ||·||_2); the manuscript would benefit from a careful proofreading.
  2. [Figure 1] The reference to subfigures as superscripts '1', '2', '3', etc., is unconventional and confusing; standard subfigure labels (a), (b), ... or panel numbers would improve readability.
  3. [Algorithm 1] The algorithm does not discuss how the matrix M = I + γA^tA is formed or solved; for large sparse systems, forming A^tA explicitly may be prohibitive, and the condition number of M grows like γ σ_1^2, which can make the inner solve inaccurate or expensive for very large γ. A brief complexity or implementation discussion would be helpful.
  4. [References] Some references are cited in the text with incomplete details, and the preprint contains a footnote statement about originality that is more appropriate for a cover letter than the main text; this should be removed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the convergence bound follows in-line from the error recursion and SVD estimates with no fitted parameters; the only self-citation ([14]) is an incidental future-work remark. The stepsize-independence overclaim contradicts the paper's own Eq. (28), which is a correctness defect, not a circular reduction.

full rationale

Walking the derivation chain: the method (9) is shown consistent with (1) in Eq. (25), and subtracting (9) from (25) yields the exact error recursion e^{k+1} = (I+γA^TA)^{-1}(I−α_kA)e^k. Theorem 1 applies only submultiplicativity, Lemma 1 bounds the first factor by the SVD estimate κ(V)/(1+γσ_n^2), and Corollary 1 iterates this to the product bound (26)-(27); condition (28) is simply the literal contraction statement of that bound. No quantity in this chain is defined in terms of the target conclusion, and no parameter is fitted to data: γ and α are user-chosen inputs, and the contraction factor (κ(V)/(1+γσ_n^2))||I−αA|| is computed, not selected, to produce the advertised result. The paper also honestly records its own limitations — Remark 2 warns that as γ→∞ the iterate converges to the inaccurate 'naive' solution, and the conclusion states that residual control alone 'may not provide accurate approximations under severely ill-conditioning context,' consistent with the degraded results in Tables 6-8. The single self-citation [14] appears only in the closing paragraph as a suggestion for future regularization and carries no argumentative weight, so it is at most the minor non-load-bearing self-citation of the rubric. The genuine defects are soundness issues, not circularity: Remark 3's claim that convergence is 'no longer based on the choice of the stepsize α' is internally inconsistent with Eq. (28), which contains ||I−αA|| and hence fails for large α at fixed γ (scalar example A=1, γ=1, α=100 gives factor 49.5); and the SVD expansion (12) writes A = VΣV^t, valid only for symmetric A, though Lemma 1 states the general-rank SVD. Per the hard rules these are correctness/rigor concerns ('not standard consensus' and false-as-stated theorems are not circularity), so the honest finding is minor or no circularity, scored 1.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central bound relies only on standard SVD theory, norm inequalities, and the nonsingularity of A. The parameters γ and α are user-chosen algorithmic constants, not fitted to data. No new entities are postulated. The main fragility is not in the axioms but in the interpretation of condition (28).

free parameters (2)
  • γ (stabilization parameter)
    User-chosen weight; the contraction factor (28) depends on it, and the paper suggests taking it as large as possible, which can make the inner system ill-conditioned.
  • α (stepsize)
    User-chosen; the paper claims the method converges independently of α, but condition (28) depends on α, so α remains a tuning parameter in practice.
assumptions (4)
  • domain assumption A is nonsingular
    Stated as the only structural requirement; ensures a unique solution and σ_n > 0.
  • standard math Singular value decomposition of A
    Used in Lemma 1 to bound (I+γA^TA)^{-1}; existence of SVD is standard.
  • standard math Submultiplicativity and compatibility of the matrix norm
    Used in proof of Theorem 1 to get the error bound.
  • standard math V is orthogonal so κ(V)=1 (though the paper keeps κ(V) in the bound)
    For a square nonsingular A, V from the SVD is orthogonal; the paper's retention of κ(V) is redundant but harmless.

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Pith. "Pith review of Stabilization of the Gradient Method for Solving Linear Algebraic Systems -- A Method Related to the Normal Equation." pith.science (2026). https://pith.science/paper/IITNCGHX

@misc{pith2026250600702,
  author       = {Pith},
  title        = {Pith review of: Stabilization of the Gradient Method for Solving Linear Algebraic Systems -- A Method Related to the Normal Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IITNCGHX}},
  note         = {Machine review of arXiv:2506.00702}
}
read the original abstract

Although it is relatively easy to apply, the gradient method often displays a disappointingly slow rate of convergence. Its convergence is specially based on the structure of the matrix of the algebraic linear system, and on the choice of the stepsize defining the new iteration. We propose here a simple and robust stabilization of the gradient method, which no longer assumes a structure on the matrix (neither symmetric, nor positive definite) to converge, and which no longer requires an approach on the choice of the stepsize. We establish the global convergence of the proposed stabilized algorithm under the only assumption of nonsingular matrix. Several numerical examples illustrating its performances are presented, where we have tested small and large scale linear systems, with and not structured matrices, and with well and ill conditioned matrices.

Figures

Figures reproduced from arXiv: 2506.00702 by the authors.

Figure 1
Figure 1. The stabilized filter factors ϕ [k] (γ,i) represented as functions of the singular values σi ∈ [10−6 , 100 ] (in 1 , 2 , 3 ), for each of the stepsize α = 10i , i = −3, 0, 3 and with respect to the parameter γ = 10i , i = 3, 6, 8, 10. The same filter factors are represented in 4 , 5 , 6 , 7 , 8 , 9 , for each α = 10i , i = −3, 0, 3 and with respect to the iterations k = 10, 20, 40, 80. The figure is displayed on a l… view at source ↗
Figure 2
Figure 2. The spectral radii ϱ(M−1 α Nα) and ϱ(M−1 γ Nk) of iterative matrices M−1 α Nα and M−1 γ Nk from the gradient method (3) and the stabilized gradient method (9), respectively. The former is represented with respect to the stepsize α ∈ [10−5 , 105 ], and the latter with respect to the stabilization parameter γ ∈ [100 , 1015] for the different values of the fixed stepsize αk = α = 10i , i = −5, −2, 0, 2, 5. In the solvi… view at source ↗
Figure 3
Figure 3. The spectral radius ϱ(M−1 α Nα) of the iteration matrix M−1 α Nα from the gradient method (3) applied to the problem (35) is represented in 1 and 2 , with respect to the parameters a ∈ [10−5 , 105 ] (for α = 10i , i = −5, −2, 0, 2, 5) and α ∈ [10−5 , 105 ] (for a = 10i , i = −5, −2, 0, 2, 5), respectively. The spectral radius ϱ(M−1 γ Nk) of the iteration matrix M−1 γ Nk from the stabilized gradient method (9) applie… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The spectral radius ϱ(M−1 γ Nk) of the stabilized gradient method (9) applied to the discret linear system (41), for the Shaw (in 1 ), the Heat (in 2 ), and the Gravity (in 3 ) test problems. These are represented with respect to the stabilization parameter γ, for the …
Figure 5
Figure 5. Figure 5: The exact solution x ⋆ (in green solid line) computed from (43), and the stabilized solution x [k] γ (in black dashed line) obtained from (9), are presented with respect to the index i = 1, · · · , 1 000, for different values of γ = 105 , 1010 , 1012 , 1014. Data A and…
Figure 6
Figure 6. Figure 6: The exact solution x ⋆ (in green solid line) computed from (46), and the stabilized solution x [k] γ (in black dashed line) obtained from (9), are presented with respect to the index i = 1, · · · , 1 000, for different values of γ = 105 , 1010 , 1012 , 1016. Data A and…
Figure 7
Figure 7. Figure 7: The exact solution x ⋆ (in green solid line) computed from (47), and the stabilized solution x [k] γ (in black dashed line) obtained from (9), are presented with respect to the index i = 1, · · · , 1 000, for different values of γ = 105 , 1010 , 1012 , 1015. Data A and…
Figure 8
Figure 8. Figure 8: The spectral radius ϱ(M−1 γ Nk) of the stabilized method (9) (or (10)) applied to the linear system derived from the finite element approximation of the problem (48)-(49), represented with respect to the stabilization parameter γ for the different values of the fixed s…

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