{"id":"6d1451af-c2ce-4183-a720-bb464666494b","arxiv_id":"2507.14940","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Optimal Frobenius-norm perturbation bounds for subunitary and positive polar factors are derived from convex analysis, along with strengthened Lee, AM-GM, and Cauchy-Schwarz inequalities.","lead":"This paper proves sharp upper and lower bounds on how much the polar factors of a matrix change under a small perturbation. The bounds use the singular values of the matrices and are shown to be optimal, refining several classical inequalities.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.9 is undefined on a singular face of C1 and its minimum formula fails in the r=s, σ=σ~=c case; the sharp lower-bound claim and the proof of Theorems 1.3/1.9 rest on this.","rationale":"The reader's weakest assumption correctly identifies Lemma 2.9 as the load-bearing step: continuity on C1 fails because the denominator vanishes, and the discrete monotonicity argument uses strict inequalities that can fail with equality. My independent check sharpens this: in the degenerate case r=s, σ_j=σ~_j=c, the denominator vanishes on the entire Birkhoff-type face of C1, the k=0 term of the stated minimum is 0/0, and the true ratio for every nonzero perturbation is 1/c², not 0. This directly affects the claimed optimality of the lower bound in Theorem 1.9. The upper bound in Theorem 1.3 may still be correct, and the lower bound inequality remains true (since it becomes 0), but the paper's central claim of a sharp, completely resolved perturbation problem is not established until Lemma 2.9 is repaired or its domain of validity is stated precisely. This does not change the reader's conditional verdict: the manuscript is promising but should not be accepted as a definitive resolution without fixing this gap.","tokens_in":18502,"tokens_out":46098,"duration_ms":549348,"concrete_test":"","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.9 is the engine for both The upper-bound Theorem 1.3 and the lower-bound Theorem 1.9, but its proof is not valid as written. The function f is defined only where D(X)=F_{r,s}-2Σσ~_iσ_j x_ij > 0, while Theorem 2.6 is invoked for a function continuous on all of C1. D actually vanishes on a whole face of C1: for r=s and σ_j=σ~_j=c, every X with row and column sums equal to 1 (e.g. any permutation matrix) gives D=0. On the domain D>0 in this case, f(X)=1/c² identically. The claimed minimum formula has a 0/0 term at k=0, while every k≥1 term equals 1/c². If 0/0 is interpreted as 0, Lemma 2.9 is false and the claimed optimal lower bound in Theorem 1.9 is not sharp: taking A=cQ and Ã=cQ~ yields ∥Q−Q~∥²_F/∥E∥²_F = 1/c² exactly. Separately, the step from the strict inequalities (2.2)-(2.5) to 'it is always possible to increase by moving right or upward' is not justified: when NΔ_right ≤ 2D ≤ NΔ_up, neither move improves, and equality cases are not discussed. Until this lemma is repaired, the sharp constants asserted in the main theorems are unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18806,"tokens_out":14741,"duration_ms":649666,"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":[{"comment":"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":"Section 2, Lemma 2.9"},{"comment":"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":"Section 2, proof of Lemma 2.9, conditions (2.2)-(2.5)"},{"comment":"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":"Section 3, proof of Theorems 1.3 and 1.9"},{"comment":"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.","section":"Section 3, proof of Theorem 1.4"}],"minor_comments":[{"comment":"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":"Section 3, proof of Theorems 1.3 and 1.9"},{"comment":"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":"Section 3, proof of Theorem 1.2"},{"comment":"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.","section":"Section 2, Lemma 2.9"},{"comment":"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.","section":"Table 1"},{"comment":"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.","section":"Theorems 1.24 and 1.25"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nSeparately, 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.\n\nWhat 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.\n\nMy 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.","headline":"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.","tokens_in":19308,"tokens_out":20044,"would_cite":false,"duration_ms":224343,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A45","15A60","47A30","47A50","65F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["perturbation bound","Frobenius norm","polar decomposition","subunitary polar factor","positive polar factor","optimal constant","convex analysis","singular values"],"falsifier":"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.","tokens_in":18287,"feed_emoji":"📐","tokens_out":18453,"duration_ms":177446,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sharp bounds pin polar-factor perturbation","feed_subtitle":"Optimal constants refine classical inequalities by using singular-value data.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["The rank-changing case $r