{"id":"b71492f0-5a53-46e5-a024-b7f801b7d852","arxiv_id":"2607.16782","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A modified Benamou–Brenier action with Euclidean invariance equals the static Procrustes–Wasserstein distance, and for Gaussians this distance is the distance between square-root spectra.","lead":"This paper builds rotation- and translation-invariance directly into the optimal-transport action, so moving a distribution by a rigid motion costs zero. It proves that this dynamic problem equals a static alignment distance, gives a closed formula for Gaussians, and adds a local convergence analysis for a numerical solver.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central mathematical claims are sound, but the computational theorem is not instantiated by the experiments: condition (48) and the local-regime assumption (57) are unverified, and Section 8 shows the relevant stability constant degenerates near symmetric inputs.","rationale":"I read the paper first as a claim about a dynamic–static equivalence and a Gaussian closed form. Theorem 1 is proved by a rigid-flow pullback: any admissible MBB flow is gauged by the rigid flow generated by (E,C), reducing it to a Benamou–Brenier curve between μ0 and a rotated/translated copy of μ1; conversely, any static isometry can be realized dynamically by reversing the same change of variables. The endpoint and energy-bookkeeping arguments in Lemmas 5–6 are consistent, so the equality ¯d² = d_S² appears correct. Theorem 4 also survives close reading: the singular-value inequality gives the upper bound ⟨√a0,√a1⟩, and equality is attained by aligning ordered eigenbases; the only oddity is the convention that the optimal orthogonal transformation is reported as P1P0^T rather than P0P1^T, but the distance value is unaffected. The existence arguments and quotient-metric statements are standard and do not change the assessment. The genuine soft spot is in the computational half. The paper is honest that Theorem 7 is local and conditional, but the numerical section does not actually instantiate the theorem's hypotheses: practical stopping uses inner caps and adaptive step sizes, while the theorem requires a residual-based rule and fixed steps. Moreover, Section 8 demonstrates a mechanism by which the stability constant in Proposition 7 can blow up: the linearized E-operator has a kernel at symmetric densities and becomes arbitrarily ill-conditioned under arbitrarily small perturbations. The pure-rotation benchmark's recovered angle (33° rather than 45°) is exactly the kind of symptom one would expect from running outside the local regularity regime. This is a reproducibility/verification gap rather than a contradiction, so the reader's CONDITIONAL verdict is appropriate and I would not change it.","tokens_in":41888,"tokens_out":20381,"duration_ms":187677,"concrete_test":"Take the final discrete solution (ρ*, m*, E*) of the pure-rotation experiment on the 64×64 grid and evaluate the left-hand side of (48), i.e. C_h ∥m^*/ρ^* − Π0g_{E^*}∥_{L∞}, together with the smallest singular value of D_z A_{E^*} (or equivalently its inverse norm). If this ratio is ≥ c_h or the smallest singular value is small (say <10^{-2} relative to the largest), the reference solution does not meet Proposition 7's sufficient condition, so Assumption 2 and hence Theorem 7 are not verified for the benchmark. Completing the check requires running the same experiment with the theorem-conforming residual stopping rule (56) and fixed stepsizes satisfying Corollary 3, and monitoring (57) at every outer iteration; if (57) is ever violated or (48) fails at the converged solution, the concern lands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing assumption sits in the computational part, exactly where the reader places it. Theorem 7 is a conditional result whose hypotheses are (i) local regularity of the reference solution via Proposition 7, in particular the smallness condition C_h ∥m^*/ρ^* − Π0g_{E^*}∥_{L∞} < c_h in (48), and (ii) the local-regime assumption (57) that all iterates remain in U_{E^*}×U_{z^*}. The numerical experiments do not verify either. Section 7.5 states that the inner loop is capped at five PDHG iterations and the outer loop at 500, and that adaptive residual-balancing stepsizes are used; Theorem 7 instead requires the residual-based acceptance rule (56) and fixed stepsizes satisfying Corollary 3 and Proposition 5. More importantly, Section 8 shows that the linearized E-operator has a kernel for radial densities and a stability constant growing like 1/ε for arbitrarily small non-symmetric perturbations (example 2). The pure-rotation benchmark recovers only ≈33° instead of the nominal 45°, which is unexplained and is consistent with the iterates operating outside the local regime where the theorem applies. This does not refute the theorem—it is explicitly conditional—but it means the paper's 'computation' half is not connected to the theory; the reported results are numerical illustrations only.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a Modified Benamou–Brenier (MBB) formulation of optimal transport in which only the non-rigid part of the velocity field is penalized, so that the dynamics is invariant under the special Euclidean group SE(d). The central theoretical result, Theorem 1, proves that the MBB value equals the static orientation-preserving Procrustes–Wasserstein distance d_S, with an analogous statement for the full orthogonal group; Theorem 2 gives existence of static and dynamic minimizers. For Gaussian measures, Theorem 4 derives the closed form d̄² = ||√a₀ − √a₁||², where a₀, a₁ are the ordered eigenvalues of the covariance matrices. The second half of the paper develops an abstract conditional local convergence framework for parameter-dependent saddle-point problems (Section 5) and applies it to a fixed-grid discretization of the MBB problem, yielding the conditional subsequential convergence result Theorem 7. Numerical experiments on 2D densities illustrate the scheme. The paper is explicit about the conditional nature of the computational result in Remarks 11–12 and in Section 8.","tokens_in":42252,"tokens_out":12989,"duration_ms":124444,"significance":"If the main theoretical results stand, they are a clean and useful contribution: the dynamic–static equivalence in Theorem 1 is proved by an elegant rigid-flow change of variables, and the Gaussian closed form in Theorem 4 is a compact, usable formula. The paper also re-derives the Gaussian result independently rather than importing it. The abstract saddle-point analysis in Section 5 is a reasonable, carefully stated local convergence framework. However, the computational half is currently not connected to the theory: the numerical experiments do not verify the hypotheses of Theorem 7, and a central benchmark (pure rotation) recovers only about 33° instead of the nominal 45°, which contradicts the qualitative claim that the scheme 'correctly captures a purely rotational motion.' The paper's honest caveats are a strength, but they imply that the reported computations are numerical illustrations rather than a validated instantiation of the convergence theorem.","major_comments":[{"comment":"The experiments do not instantiate Theorem 7. The theorem requires the residual-based acceptance rule (56) with a summable tolerance sequence, fixed step sizes satisfying Corollary 3 and Proposition 5, and iterates remaining in the local neighborhoods (57). Section 7.5 instead uses a fixed inner iteration cap of five PDHG steps, an outer horizon of 500, adaptive residual-balancing step sizes, and a stopping/diagnostic criterion based on outer iterate differences. Thus the convergence guarantee of Theorem 7 does not apply to the reported numerical results, and the paper's 'From Theory to Computation' claim is not supported on the computational side.","section":"§7.5, Theorem 7, Eq. (56)–(57)"},{"comment":"The discretization used in the experiments violates an explicit hypothesis of the theory. Section 7.1 states n_t = 64 grid points in time with h_t = 1/(n_t−1), which gives K = 63 time intervals, an odd number. Lemma 11, Lemma 18, and Proposition 7 all explicitly assume that K is even. Consequently, the discrete compatibility, coercivity, and Assumption 2 results in Section 6 do not apply to the grid used in the numerical section, independently of the inner/outer loop choices.","section":"§7.1, Lemmas 11 and 18, Proposition 7"},{"comment":"The pure-rotation benchmark recovers an angle of approximately 33° instead of the nominal 45°. This is a large error for a configuration that should be absorbed entirely by the orthogonal component at zero transport cost. The text attributes the discrepancy to numerical effects, but does not provide any quantitative error analysis or explain why the optimizer stops at 33°. This undermines the claim in the Discussion that the MBB formulation 'correctly captures a purely rotational motion' and also calls into question the meaning of the reported kinetic-energy comparison, since a suboptimal E would require a compensating V with non-negligible energy.","section":"§7.2, Figure 5 and Discussion"},{"comment":"The paper's own stability analysis shows that the linearized E-operator has a kernel at radial densities (Example 1) and that arbitrarily small non-symmetric perturbations produce a stability constant growing like 1/ε (Example 2). Hence the local neighborhoods U_{E*} × U_{z*} required by condition (57) may be arbitrarily small near symmetric inputs, and the smallness condition (48) in Proposition 7 is not verified anywhere. The convergence theorem is therefore not only conditional but also potentially inapplicable precisely in regimes close to the type of data used in the experiments. The section honestly explains why local analysis is needed, but it also underscores that the computational claims are not validated.","section":"§8, Examples 1–2"}],"minor_comments":[{"comment":"The proof would benefit from stating explicitly that the eigenvector matrices P₀, P₁ are chosen with matching orientation (det P₀ = det P₁) so that the optimal transformation P₁P₀ᵀ lies in SO(d). As written, the statement 'optimal orthogonal transformation' is ambiguous when arbitrary eigendecompositions are used.","section":"Theorem 4, proof"},{"comment":"The quantity plotted as 'fixed-point residual' is not defined in the experimental section. The paper should specify the exact residual norm being computed in the figures, since this is not the residual Res in (29) used in the theory.","section":"§7, Figures 3, 8, 13"},{"comment":"Remark 6 says 'As shown in Lemma 9, the variables split into three blocks'; Lemma 9 concerns convex-concave structure, not the block splitting. The reference should be to the algorithmic presentation in Section 4.2.","section":"Remark 6"},{"comment":"The text states that the adaptive step-size strategy is used 'only in the implementation' and that the theory requires fixed step sizes. This is an important caveat, but it is placed late and should be emphasized more prominently in the numerical section to avoid any impression that the experiments verify the convergence theorem.","section":"§7.5"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core (Theorems 1–4 and the abstract saddle framework) is sound and publishable. The major problem is the disconnect between the computational section and the convergence theory: the experiments use an odd-K grid, fixed inner caps, adaptive step sizes, and a different acceptance rule than Theorem 7 requires, and the pure-rotation benchmark misses the true angle by 12°. I would encourage the editor to seek a revision in which the computational claims are either made compatible with the theorem (by verifying its hypotheses on the test problems) or explicitly downgraded to illustrations without convergence claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague:\n\nThe core mathematics is in good shape. Theorem 1 — the modified Benamou-Brenier value equals the static orientation-preserving Procrustes-Wasserstein distance — is new and the proof is clean and self-contained. The existence theorem and the Gaussian closed form (Euclidean distance between square roots of ordered eigenvalues) follow naturally. The Gaussian formula is credited to their own preprint and re-derived here, so the self-citation seems legitimate rather than circular. Section 8 is also honest: radial densities create a kernel, and non-symmetric perturbations cause the stability constant to blow up. That is real information, not a handwave.\n\nThe computational part is the soft spot. Theorem 7 is explicitly local and conditional, requiring the smallness condition (48) and the local-regime assumption (57), plus fixed step sizes and a residual-based inner acceptance rule. The experiments use adaptive step sizes, a five-iteration inner cap, and a 500-iteration outer horizon, and they never verify those conditions. The pure-rotation benchmark recovers ~33 degrees of a nominal 45, and that gap is left unexplained. Given Section 8, that is consistent with the iterates being outside the local regularity region. Compounding this, no code or data is provided, so the numbers are not independently checkable. The paper's own Remark 12 is explicit that the convergence analysis is fixed-grid and the experiments are illustrations, so the authors know the limitation; it still means the numerical half does not actually instantiate the theorem.\n\nThe abstract saddle framework is a standard implicit-function-plus-inexact-gradient setup, and it is handled competently. Not original in outline, but fit for purpose.\n\nThis deserves a serious referee. The main theorems should be checked carefully, and the authors should be asked to either numerically verify the local conditions or present the experiments as heuristic only. The dynamic-static equivalence is worth knowing for anyone in invariant OT or shape alignment.","headline":"Core math is sound and the dynamic-static equivalence is a genuine contribution; the numerical half is not yet connected to the theory, but the paper deserves a serious referee.","tokens_in":42708,"tokens_out":2476,"would_cite":true,"duration_ms":40892,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","90C47","65K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A transport metric that makes rigid motions free equals the static Procrustes–Wasserstein distance.","keywords":["modified Benamou–Brenier","Procrustes–Wasserstein distance","Euclidean invariance","quotient Wasserstein space","Gaussian closed form","skew-symmetric drift","saddle-point problem","conditional convergence"],"falsifier":"Take two identical radially symmetric densities on a fixed grid and run the alternating scheme: the action is zero for any skew-symmetric field E, so the inner residual can vanish while E varies. If the algorithm produces unbounded or initialization-dependent E sequences, or if the stability constant in Section 8 blows up as shown for perturbed radial densities, the local branch assumption behind the convergence theorem is violated.","tokens_in":1466,"feed_emoji":"📐","tokens_out":3911,"duration_ms":80843,"temperature":0.7,"pith_summary":"The paper introduces a modified Benamou–Brenier formulation of optimal transport in which velocity fields are orthogonally projected away from infinitesimal rigid motions, so that global rotations and translations incur zero cost. It proves that this dynamical value equals the minimum over orientation-preserving rigid motions of the classical Wasserstein distance to the target—the static Procrustes–Wasserstein distance. For Gaussian measures the distance admits a closed form: it is the Euclidean distance between the vectors of square roots of the ordered eigenvalues of the two covariance matrices. The paper also develops a discretized primal–dual algorithm and proves a local, conditional convergence result for it. A reader should care because the equivalence turns invariant shape comparison into a geodesic-based, PDE-amenable problem rather than a heuristic two-step alignment.","feed_headline":"Zero-cost rigid motions define a transport metric that matches Procrustes–Wasserstein","feed_subtitle":"For Gaussians the distance reduces to the Euclidean distance between square roots of sorted eigenvalues.","key_machinery":"The key object is the MBB seminorm on the Wasserstein tangent space. For each measure μ it projects a velocity field onto the orthogonal complement of the finite-dimensional space of rigid fields x ↦ Ex + C, with E skew-symmetric. The dynamical action is the integral of this projected norm squared. The dynamic–static equivalence is carried by the rigid flow g_t(x)=Q_t x + b_t generated by (E,C); pulling back the measure curve by g_t converts the MBB action into classical Benamou–Brenier action between aligned endpoints, which then realizes the static Procrustes–Wasserstein distance.","core_discovery":"The central claim is Theorem 1: for any two probability measures with finite second moments, the modified Benamou–Brenier value—built by assigning zero energy to the rigid component of the velocity field—equals the orientation-preserving Procrustes–Wasserstein distance, i.e. the minimum of the squared Wasserstein distance after applying a special Euclidean isometry to one measure. The proof uses a rigid flow that 'undoes' the rotation and translation, pulling back the dynamics to a classical Benamou–Brenier problem and then optimizing over the final isometry. Theorem 4 gives the Gaussian closed form: the distance equals the Euclidean norm of the difference between the square roots of the ord","pith_inferences":["The dynamic–static equivalence suggests that the MBB action is the natural kinetic energy on the quotient of Wasserstein space by Euclidean isometries, so gradient flows in this quotient could serve as principled shape interpolations.","Because the recovered rotation generator suffers from a gauge-type nonuniqueness near symmetric densities, any practical solver that produces a rotation path will need an explicit gauge or regularization if that path is to be meaningful.","A testable extension beyond Gaussians: for elliptically symmetric or log-concave families, the eigenvalue-square-root distance might emerge as an approximation; numerical experiments could check whether alignment reduces to eigendecomposition for those measures."],"forward_implications":["The equivalence reduces computation of the MBB distance to a two-step problem: first align optimally, then compute classical Wasserstein distance.","For Gaussians, Procrustes–Wasserstein distance can be computed in closed form by sorting and square-rooting eigenvalues, enabling fast alignment and comparison of normal distributions.","The quotient space of centered measures modulo rotations is a complete, path-connected, geodesic metric space; geodesics in the quotient can be realized by projecting Wasserstein geodesics after optimal alignment.","Under the stated local regularity and smallness conditions, the alternating primal–dual algorithm yields accumulation points that satisfy the discrete optimality conditions.","The full Procrustes–Wasserstein distance is obtained by taking the minimum of two orientation-preserving MBB values, one with a reflected target, so the computational scheme extends to orientation-reversing comparisons."],"fun_headline_variants":["Zero-cost rigid motions yield Procrustes–Wasserstein metric","Rotation-invariant transport: Gaussian closed form via eigenvalue square roots","New metric: MBB transport equals Procrustes–Wasserstein","Procrustes–Wasserstein from dynamical optimal transport","Euclidean invariance reshapes transport into Procrustes distance"],"cache_read_input_tokens":44032,"weakest_assumption_plain":"The computational convergence theorem rests on the unverified premise that the iterates remain in a local regularity neighborhood and that at the reference solution the effective non-rigid velocity satisfies a smallness condition (C_h ||m*/ρ* − Π0 g_{E*}||_∞ < c_h); if either fails, the scheme may not converge to a stationary point.","fun_headline_variants_meta":{"raw":{"variants":["Zero-cost rigid motions yield Procrustes–Wasserstein metric","Rotation-invariant transport: Gaussian closed form via eigenvalue square roots","New metric: MBB transport equals Procrustes–Wasserstein","Procrustes–Wasserstein from dynamical optimal transport","Euclidean invariance reshapes transport into Procrustes distance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000191,"raw_usage":{"total_tokens":1134,"prompt_tokens":655,"completion_tokens":479,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":399,"completion_tokens_details":{"reasoning_tokens":387}},"tokens_in":399,"tokens_out":479,"duration_ms":5225,"temperature":1.0,"reasoning_tokens":387,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T19:57:03.007877+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two identical radially symmetric densities on a fixed grid and run the alternating scheme: the action is zero for any skew-symmetric field E, so the inner residual can vanish while E varies. If the algorithm produces unbounded or initialization-dependent E sequences, or if the stability constant in Section 8 blows up as shown for perturbed radial densities, the local branch assumption behind the convergence theorem is violated.","supporting_citations":[],"review_version":1}