Sharp perturbation bounds on the Frobenius norm of subunitary and positive polar factor

Teng Zhang

REVIEW 4 major objections 5 minor 1 cited by

Sharp perturbation bounds on the Frobenius norm of subunitary and positive polar factor

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves sharp optimal-constant Frobenius-norm perturbation bounds for the subunitary and positive polar factors, using one extremal ratio to refine several classical inequalities.

desk verdict The results are probably right and genuinely valuable, but the paper's central lemma is not proven as written and needs a real repair before the sharpness claims can be trusted. read the letter →

arxiv 2507.14940 v1 pith:4VLEY7JD submitted 2025-07-20 math.FA

Teng Zhang This is my paper

classification math.FA MSC 15A4515A6047A3047A5065F10

keywords perturbationboundFrobeniusnormpolardecompositionsubunitaryfactorpositiveoptimalconstantconvexanalysissingularvalues

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

The paper tries to resolve how much the two factors in a generalized polar decomposition $A=QH$ can move, measured in Frobenius norm, when $A$ is changed to $\widetilde A=A+E$. It gives upper and lower bounds whose only inputs, besides $\|E\|_F$, are the singular values of $A$ and of $\widetilde A$, and it proves the multiplicative constants in those bounds are optimal. In the equal-rank case the subunitary bound refines the best previous bound; for the positive factor it refines the classical constant-$\sqrt2$ bound and is strictly smaller unless the singular values coincide. The same extremal device also yields sharp versions of the conjecture about $\|A+\widetilde A\|_F$ versus $\|H+\widetilde H\|_F$, of the matrix AM-GM and Cauchy-Schwarz inequalities, and a matching lower bound for a known positive-factor inequality.

What carries the argument

The central object is the rational function $f(X)=(r+s-2\sum_{i,j}x_{ij})/(\sum_j\sigma_j^2+\sum_i\widetilde\sigma_i^2-2\sum_{i,j}\widetilde\sigma_i\sigma_jx_{ij})$ on the convex set $C_1=\{X\in\mathbb{R}^{s\times r}:\sum_i|x_{ij}|\le1,\ \sum_j|x_{ij}|\le1\}$, whose entry-wise absolute values form doubly substochastic constraints. The paper shows $f$ is quasi-convex (every sublevel set is convex) and quasi-concave, so its extremes are attained at the extreme points of $C_1$, which are exactly sign-permutation matrices with at most one nonzero entry per row and column. The rearrangement inequality then selects the optimal placement of those entries, reducing the extremal problem to a maximum over $k=0,\dots,r$. In the SVD proof, $x_{ij}$ is identified with $\Re(s_{ij}t_{ij})$ from two unitary matrices $S=\widetilde U^*U$ and $T=\widetilde V^*V$, and this identification makes $f$ control both the subunitary and the positive polar factor ratios.

What would settle it

Take $r=s=2$ with $\sigma=(2,1)$ and $\widetilde\sigma=(2,1)$, and let $X$ be the $2\times2$ identity, an extreme point of $C_1$. The denominator of $f$ becomes $(4+1+4+1)-2(2\cdot2+1\cdot1)=0$, so $f$ is undefined there; testing whether $f$ has a continuous extension on that face of $C_1$, and whether the extreme-point argument survives that extension, would settle Lemma 2.9 and hence the claimed optimal constants. A direct check is also possible: numerical search for $A,E$ that violate Theorem 1.3's bound would falsify it.

Watch

Extended reading notes

Core claim

Let $A\in\mathbb{C}^{m\times n}_r$ and $\widetilde A=A+E\in\mathbb{C}^{m\times n}_s$ have generalized polar decompositions $A=QH$, $\widetilde A=\widetilde Q\widetilde H$, with singular values $\sigma_1\ge\cdots\ge\sigma_r>0$ and $\widetilde\sigma_1\ge\cdots\ge\widetilde\sigma_s>0$. The paper proves that for $r\le s$, $$\|Q-\widetilde Q\|_F \le \sqrt{\max_{0\le k\le r} \frac{s-r+4k}{\sum_{j=1}^{r-k}(\sigma_j-\widetilde\sigma_j)^2 + \sum_{j=1}^k(\sigma_{r+1-j}+\widetilde\sigma_{s-k+j})^2 + \sum_{j=r-k+1}^{s-k}\widetilde\$sigma_j^{2}$}}\,\|E\|_F,$$ and that the coefficient is optimal; a matching lower bound appears as Theorem 1.9. For the positive factors it proves $$\|H-\widetilde H\|_F \le \sqrt{\frac{F_{r,s}-\sqrt{F_{r,s}^2-2G_{r,r}F_{r,s}}}{G_{r,r}}}\,\|E\|_F,$$ with $F_{r,s}=\sum_{j=1}^r\sigma_j^2+\sum_{j=1}^s\widetilde\sigma_j^2$ and $G_{r,r}=\sum_{j=1}^r\sigma_j\widetilde\sigma_j$, again with an optimal coefficient and a lower bound in Theorem 1.10. The bounds are derived by controlling the ratio $\|Q-\widetilde Q\|_F^2/\|E\|_F^2$ through one rational function of a doubly substochastic matrix. The paper also derives optimal-constant versions of the conjecture on $\|A+\widetilde A\|_F$ versus $\|H+\widetilde H\|_F$, of the matrix AM-GM and Cauchy-Schwarz inequalities, and a sharp lower bound for a known normal-matrix inequality.

Load-bearing premise

The argument's load-bearing premise is that the extremal ratio reaches its largest and smallest values at the corner (sign-permutation) choices of a doubly substochastic matrix, and the ratio is defined at every point of the convex set; the denominator of that ratio can vanish when the two matrices share singular values and a corner matrix aligns them perfectly, which is exactly where the premise needs checking.

Editorial extensions

If this is right

The rank-changing case $r<s$, for which the paper notes no significant previous bounds existed, now has sharp upper and lower Frobenius-norm bounds for the subunitary factor.
In the equal-rank case the subunitary bound refines the classical bound, reducing to it only when the trailing singular values align; the optimal constant can be attained.
The positive-factor coefficient is at most $\sqrt2$ and is strictly smaller than $\sqrt2$ whenever $r\ne s$ or the singular values are not paired equal, so the old constant is achieved only in a degenerate alignment.
The sharp version of the conjecture on $\|A+\widetilde A\|_F$ gives a two-sided bound in terms of $\|H+\widetilde H\|_F$, with strict inequality in generic cases.
The strengthened AM-GM and Cauchy-Schwarz inequalities are strict whenever the ranks differ or the singular values are not proportional, which is the generic situation in applications.

Reading between the lines

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

One implication the paper leaves implicit is that the extremizers' sign-permutation structure gives a concrete recipe for constructing equality cases in numerical tests.
A natural extension would be to test whether the same quasi-convex ratio method transfers to other unitarily invariant norms; the paper states results only for the Frobenius norm.
The strictness results suggest an applied consequence: in generic matrix algorithms the old constants overestimate the true error, and the gap is quantified by how far the singular-value ratio $F_{r,s}/G_{r,r}$ is from its minimum.

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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 central results are probably true and genuinely valuable, but the paper as posted does not prove them. Lemma 2.9 is the engine for Theorems 1.3 and 1.9, and its proof mishandles the domain of f. The function is defined only where the denominator is positive, while Theorem 2.6 is applied as if f were continuous on all of C1. There are whole faces of C1 on which the denominator vanishes—e.g. r=s, sigma=sigma~, and X a permutation—and the statement includes k=0 terms that are 0/0 there. The scalar equal-singular-value case shows the formula survives if those undefined terms are excluded (k=1 gives 1/c^2), but the lemma as written is not a valid statement and the claimed max/min over extreme points is not justified. This is not a cosmetic issue: the optimality assertions in the main theorems rest on it.

The monotonicity step in the proof of Lemma 2.9 also needs tightening. The inequalities (2.2)-(2.5) only give strict increase when strict inequalities hold; in the equality case you get plateau moves, not increases. The conclusion that some maximum is reached on the boundary k1+k2=r is likely still reachable by a non-decreasing path, but that argument has to be written.

Separately, the norm identities in Section 3 have a notational slip: the real part should be Re(s_ij * conjugate(t_ij)), not Re(s_ij t_ij). Since both uses satisfy the same row/column estimates, this is fixable, but it should be corrected before refereeing.

What is genuinely good: the idea of optimizing the ratio over the convex set C1 with singular-value weights is new for this class, and the sharp constants, if correct, refine Li-Sun, Araki-Yamagami, and Lee's conjecture in a real way. The paper engages honestly with the prior literature; the self-citation is auxiliary and not load-bearing. No circularity.

My read: this deserves a serious referee, so I'd send it out. But I would not accept it in current form. The author needs to repair Lemma 2.9—restrict the domain, handle the D=0 faces by a limiting argument or by excluding undefined k, and justify the boundary maximum. If that repair works, the paper is a solid contribution to matrix perturbation theory.

Referee Report

4 major / 5 minor

Summary. The paper studies perturbation of the subunitary and positive polar factors in the Frobenius norm when the rank changes from r to s >= r. The main results are sharp upper and lower bounds for the subunitary factor (Theorems 1.3 and 1.9), sharp upper and lower bounds for the positive factor (Theorems 1.4 and 1.10), a strengthened Lee conjecture (Theorems 1.12 and 1.13), and refinements of the AM-GM, Cauchy-Schwarz, and Kittaneh inequalities. The proofs reduce the matrix ratio ||Q - Qtilde||_F^2 / ||E||_F^2 to a linear-fractional function on a doubly substochastic set C1, whose extrema are computed in Lemma 2.9.

Significance. If the main theorems are correct, they would resolve sharp constants in a natural rank-changing generalization of the Li-Sun bound, refine the Araki-Yamagami inequality, and strengthen Lee's conjecture, with explicit extremal constructions and numerical examples. However, the central lemma has substantive gaps in both its statement and its proof, and the matrix identities feeding into it contain notational errors. The validity of the claimed sharp constants is therefore not yet established.

major comments (4)

[Section 2, Lemma 2.9] The function f is defined only where D(X)=F_{r,s}-2*sum sigmaTilde_i sigma_j x_ij > 0, but Theorem 2.6 is applied on all of C1. The denominator vanishes on a substantial face: for r=s and sigma_j=sigmaTilde_j=c, any X with all row and column sums equal to 1, for example a permutation matrix, gives D=0. In this case f(X)=1/c^2 identically on the domain D>0, while the stated minimum formula contains a 0/0 term at k=0. If that term is interpreted as 0, the lemma is false; if it is left undefined, the lemma does not cover the cases asserted in Theorems 1.3 and 1.9. This is load-bearing because Lemma 2.9 is the engine for the sharp-coefficient claims in both theorems.
[Section 2, proof of Lemma 2.9, conditions (2.2)-(2.5)] The conclusion that 'it is always possible to increase the function value by moving right or upward' does not follow from Delta_right <= Delta_up. The right move increases only when 2D > N*Delta_right, and the up move increases only when 2D < N*Delta_up; when N*Delta_right <= 2D <= N*Delta_up, neither move improves. The proof does not rule out this interval, and the same gap appears in the minimization argument. The reduction to k1+k2=r is therefore not established.
[Section 3, proof of Theorems 1.3 and 1.9] The displayed identities for ||Q-Qtilde||_F^2 and ||E||_F^2 are algebraically incorrect for complex S,T. With S=Utilde*U and T=Vtilde*V, the cross term is Re(Tr((S I(r))*(I(s) T))) = sum_{i=1}^s sum_{j=1}^r Re(overline{s_ij} t_ij), not sum Re(s_ij t_ij). The subsequent definition x_ij=Re(s_ij t_ij) and all estimates that feed into Lemma 2.9 therefore concern a different quantity. This is correctable by inserting conjugates, but as written it breaks the central reduction.
[Section 3, proof of Theorem 1.4] The bound |M| <= G_{r,r}^{1/2} N^{1/2} requires the inequality sum_{i,j} sigmaTilde_i sigma_j |s_ij|^2 <= G_{r,r}. This is true by the same extreme-point and rearrangement argument used for N, but it is not stated or proved. Since Theorems 1.4, 1.10, 1.12, and 1.13 all rely on this bound, the missing justification should be supplied explicitly.

minor comments (5)

[Section 3, proof of Theorems 1.3 and 1.9] The Cauchy-Schwarz display for the row constraint has the indices interchanged: for each fixed j one needs sum_{i=1}^s |t_ij|^2 <= 1, and for each fixed i one needs sum_{j=1}^r |s_ij|^2 <= 1; the text as written mixes the two.
[Section 3, proof of Theorem 1.2] The phrase 'cos 2 alpha <= cos beta' should read 'cos^2 alpha <= cos beta', since the subsequent inequality uses cos beta >= cos^2 alpha.
[Section 2, Lemma 2.9] The block-matrix descriptions of the maximizer and minimizer are not fully specified; the dimensions of the zero blocks and the meaning of the reversal matrices S_k should be stated explicitly so that the claimed row and column supports are unambiguous.
[Table 1] The table lists four values per row under the heading f(k), but the correspondence between these values and k=0,...,3 is not spelled out; a brief explanation would improve readability.
[Theorems 1.24 and 1.25] The denominators in these theorems can vanish for particular eigenvalue configurations; the statements should either exclude those cases or specify a limiting interpretation, as in the degenerate cases of Lemma 2.9.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the sharp Frobenius bounds are obtained by optimizing a ratio over C1, not by assuming the target inequalities.

full rationale

The main derivation chain starts by rewriting ||Q - Qtilde||_F^2 / ||E||_F^2 as the ratio f(X) with X = (Re(s_ij t_ij)) lying in C1, and then proves Lemma 2.9 by optimizing this ratio over the extreme points of C1 using the Birkhoff-type structure, quasi-convexity (Lemma 2.7), and the rearrangement inequality (Lemma 2.8). The bound coefficients are outputs of that optimization, not inputs. Theorems 1.3 and 1.9 then follow by the same reduction, and optimality is checked by explicitly constructed S and T, so the claims are not fitted to the data. The only self-citation, [29], appears in the introduction as background about an alternative proof of Lee's conjecture and is not used in any proof of the new theorems; the cited Lin-Zhang result [24] is external. The skeptical observation that f is only defined where the denominator is positive and that the proof of Lemma 2.9 may fail at exceptional singular-value coincidences is a correctness and domain gap, not a circularity: a failed proof of an extremal lemma does not make the theorem equivalent to its assumptions. Thus no step in the derivation reduces, by construction or by self-citation, to the claimed result.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard convex analysis, the Birkhoff theorem for doubly substochastic matrices, and the rearrangement inequality. No free parameters are fitted; the singular values are inputs to the bounds. The paper introduces no new entities. The main unproved external input is the Lin-Zhang lemma used in the auxiliary proof of Araki-Yamagami.

assumptions (5)

standard math Birkhoff-type theorem: every doubly substochastic matrix is a convex combination of subpermutation matrices (Lemma 2.3).
Used in Lemma 2.4 to characterize the extreme points of the set C1.
standard math Continuous quasi-convex functions attain maxima at extreme points of a compact convex set (Lemma 2.6).
Used in Lemma 2.9 to reduce the optimization to extreme points; the paper does not verify continuity on the whole set.
standard math Rearrangement inequality for real sequences (Lemma 2.8).
Used to identify the optimal pairings of singular values at extreme points in Lemma 2.9.
domain assumption Lemma 3 from Lin-Zhang [24], that the angle inequality cos 2α ≤ cos β holds for matrices.
Invoked in the new proof of Araki-Yamagami in Section 3; the lemma is cited but not proved or stated in this paper.
domain assumption Without loss of generality assume m ≥ n ≥ r in the main theorems.
The paper states this reduction in Section 3 but does not give the explicit adjoint argument for the case m < n.

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Cite this review

Pith. "Pith review of Sharp perturbation bounds on the Frobenius norm of subunitary and positive polar factor." pith.science (2026). https://pith.science/paper/4VLEY7JD

@misc{pith2026250714940,
  author       = {Pith},
  title        = {Pith review of: Sharp perturbation bounds on the Frobenius norm of subunitary and positive polar factor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4VLEY7JD}},
  note         = {Machine review of arXiv:2507.14940}
}

read the original abstract

Leveraging tools from convex analysis and incorporating additional singular value information of matrices, we completely resolve the problem of establishing perturbation bounds for the Frobenius norm of subunitary and positive polar factors. We derive corresponding sharp upper and lower bounds. As corollaries, we refine the results of Li and Sun [SIAM J. Matrix Anal. Appl., 23 (2002), pp. 1183--1193] and strengthen the classical Araki-Yamagami inequality [Comm. Math. Phys., 81 (1981), no. 1, pp. 89--96]. The versatility of our method also allows us to strengthen Lee's conjecture, providing a sharper version along with a matching sharp lower bound. Furthermore, we generalize the classical matrix arithmetic-geometric mean inequality and Cauchy-Schwarz inequality into tighter and more robust forms. Finally, we establish a sharp lower bound for a result by Kittaneh [Comm. Math. Phys., 104 (1986), no. 2, pp. 307--310].

Figures

Figures reproduced from arXiv: 2507.14940 by the authors.

**Figure 2.1.** r = 6, s = 7, the objective function f(k1, k2) is analyzed at the initial grid point P(1, 1) (black circle). The dashed line represent the linear constraint k1 + k2 = 6. Arrows show stepwise changes from (1,1) to adjacent grid points, with conditions for f to increase labeled in italic. where Ir−k⋆ is a (r − k ⋆ ) × (r − k ⋆ ) identity matrix and Sk⋆ is a k ⋆ × k ⋆ reversal matrix. Minimizing f on the set ext(C1) … view at source ↗

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Araki, S

H. Araki, S. Yamagami, An inequality for Hilbert-Schmidt norm, Comm. Math. Phys. 81(1) (1981) 89–96

work page 1981

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Barrlund, Perturbation bounds on the polar decomposition, BIT 30 (1989) 101–113

A. Barrlund, Perturbation bounds on the polar decomposition, BIT 30 (1989) 101–113

work page 1989

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R. Bhatia, Matrix Analysis, Springer, New York, 1997

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S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2009

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work page 2005

[6] [6]

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work page 2006

[7] [7]

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