{"id":"43087e66-7843-4c00-ba43-1ed8fb78bb5a","arxiv_id":"2505.07124","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Sharpened Fenchel-Young losses turn inverse problems over probability measures into convex problems with sample-complexity guarantees, instantiated for inverse UOT and JKO gradient flows.","lead":"This paper introduces a family of convex loss functions, called sharpened Fenchel-Young losses, for recovering unknown potentials or cost functions from samples of probability measures generated by an optimization problem. It proves sample-complexity bounds for inverse unbalanced optimal transport and for learning gradient-flow potentials from snapshot data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The iJKO reduction requires r > η/τ, not just r > 0; if r' ≤ 0, Proposition 4's dual formula fails and Corollary 2 is unsupported.","rationale":"The reader's weakest assumption points to exactly the same load-bearing issue: Proposition 4's reduction to iUOT uses r' = r − η/τ as a divergence coefficient, but positivity is only guaranteed when r > η/τ, while the paper only states r > 0. This matters because Corollary 2, the paper's iJKO sample-complexity claim, is obtained solely by inheriting Theorem 3 through this reduction. If r' ≤ 0, the reduced primal is not a standard convex UOT problem, the dual formula in Proposition 4 is not justified, and the inherited guarantee has no foundation. The concern is substantive but localized: the iUOT analysis in Section 3 appears internally coherent, the forward sample-complexity result is nontrivial, and the numerics happen to use r > η/τ. A secondary and smaller issue is that Theorem 4's proof condition λ_n sqrt(n/log n) → ∞ appears insufficient to make the t/√n term from Proposition 8 vanish; the stronger condition λ_n sqrt(n)/log n → ∞ would be needed. This does not change the conditional verdict, but it reinforces that the paper needs explicit quantitative assumptions in the iJKO and model-consistency sections.","tokens_in":50358,"tokens_out":36229,"duration_ms":333911,"concrete_test":"Run a one-dimensional discrete check of Proposition 4 with α_{k+1} uniform on {0,1}, α_k = δ_0, V = 0, τ = 1, η = 2, r = 1 (so r' = −1). Compute the original iJKO infimum from (FY-iJKO) by direct minimization over α, and compute the claimed dual formula (Kanto-iJKO). If the dual is unbounded above or differs from the primal, the reduction fails for r' ≤ 0. Repeat with r = 3 (r' > 0) and verify that the two values match. This isolates the sign condition without invoking any other assumptions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.2 defines the sharpening term as r KL(α|α_{k+1}) with only r > 0. Proposition 4 sets r' = r − η/τ and rewrites the inner problem (eq. 17) with coefficient r' on KL(α|α_{k+1}). The identity W_{2,η}(α,α_k|α⊗α_k) = W_{2,η}(α,α_k|α_{k+1}⊗α_k) − ηKL(α|α_{k+1}) subtracts (η/τ)KL, so the effective divergence coefficient after the reduction is r − η/τ, not r. If r' ≤ 0, the reduced problem is not a standard UOT problem with a nonnegative marginal divergence; the objective π ↦ (η/τ)KL(π|α_{k+1}⊗α_k) + r'KL(π_1|α_{k+1}) is not of the form required by Assumptions 2–3, and the conjugate/dual step in Proposition 4 (the sup over f,g involving r' exp(−f/r')) is invalid because the conjugate of a negative multiple of KL is not defined. A finite counterexample with α_{k+1} uniform on two points, α_k = δ_0, η/τ = 2, r = 1 makes the claimed dual unbounded above while the original iJKO infimum is finite. Thus the identification with iUOT and the inherited sample bound of Corollary 2 hold only when r > η/τ. The paper never states this constraint; it only fixes r = 1 with η/τ = 10^{-3} in the numerics. This is a missing assumption, not a contradiction of the iUOT theory, but it restricts the claimed iJKO scope and should be stated and proven, or the iJKO analysis redone directly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for inverse problems over probability measures using 'sharpened Fenchel-Young losses', in which a data-dependent discrepancy is inserted into the forward variational problem. An abstract stability theorem is proved and then instantiated in two settings: inverse entropic unbalanced optimal transport (iUOT) and inverse Jordan-Kinderlehrer-Otto (iJKO) flow. For iUOT, the authors establish a forward sample-complexity bound and a parameter-recovery bound with rates of order O(sqrt(m_alpha m_beta (log n + t)/n)) + O(lambda). For iJKO, they show that the sharpened loss reduces to an iUOT problem and inherit a O(sqrt(t/n)) recovery rate from independent snapshots. They also prove model consistency for l1- and nuclear-norm regularization under a nondegeneracy condition on a minimal-norm certificate, and support the theory with numerical experiments on Gaussian data.","tokens_in":50741,"tokens_out":40158,"duration_ms":371441,"significance":"If the results hold as stated, the paper makes a useful contribution: it unifies Fenchel-Young and Fitzpatrick losses into a single sharpened gap-loss construction, extends forward sample-complexity analysis of entropic UOT beyond smooth divergences, and draws an interesting connection between inverse JKO learning and inverse UOT. The appendices are substantial and the main proofs are presented in detail. The numerical study is also valuable, including a check of the certificate condition in two concrete settings. However, several load-bearing statements have missing hypotheses or internal inconsistencies, most notably in the iJKO-to-iUOT reduction and in the abstract stability theorem as stated. These are fixable, but they must be corrected before the central iJKO claims can be accepted.","major_comments":[{"comment":"The iJKO reduction requires r > eta/tau, but the paper only states r > 0. In Proposition 4, r' = r - eta/tau is introduced, and the Kantorovich formulation (Kanto-iJKO) and the subsequent dual step use the function r' exp(-f/r'). If r' <= 0, this is not the convex conjugate of a nonnegative KL divergence, and the displayed dual formula is not the Legendre-Fenchel conjugate of the reduced inner problem. The identification of the iJKO loss with LiUOT(c_V; pi, alpha_k, alpha_{k+1}) with marginal divergence r'KL(pi_1|alpha_{k+1}) is therefore only valid when r' > 0. Corollary 2, which claims the inherited O(sqrt(t/n)) sample bound, omits this condition entirely. The numerical experiments use r = 1 and eta/tau = 10^{-3}, so the demonstrated regime satisfies the missing hypothesis, but the theorem statements must either add r > eta/tau or provide a direct analysis that does not rely on the UOT identification whenever r' <= 0.","section":"§4.2, Proposition 4, Corollary 2"},{"comment":"The sufficient condition in Theorem 1 is stated as 'epsilon + 2 lambda ||p*|| <= alpha^{-1}(K - ||theta*||)', but the proof invokes Lemma 3(iii), whose condition is 'epsilon + 2 lambda ||p*|| <= alpha(K - ||theta*||)'. The displayed inequality in Theorem 1 is dimensionally inconsistent with the concluding bound ||theta_hat - theta*|| <= alpha^{-1}(epsilon + 2 lambda ||p*||), since the left side has units of a gradient norm while alpha^{-1}(K - ||theta*||) does not. This is a straightforward typo, but it affects the main abstract stability result and should be corrected.","section":"Theorem 1"},{"comment":"Theorem 2 states the forward UOT sample-complexity bound under Assumptions 1, 2, and 3 only, without assuming that the cost c is Lipschitz. The proof, however, relies on uniform sup-norm bounds on the Kantorovich potentials that are obtained in Proposition 10 and Proposition 12 using the Lipschitz constant of c, and Lemma D.1 also assumes h is Lipschitz. Compactness of X and Y alone does not provide the required uniform control of f and g. Since all parameterized costs c_theta considered later in the paper are Lipschitz, this omission is fixable, but Theorem 2 as stated is missing an explicit hypothesis (for example, 'c is Lipschitz') and the implicit constant claim should be reconciled with that hypothesis.","section":"Theorem 2"},{"comment":"The reduction of iJKO to iUOT is not written in a verifiable way. In Proposition 4, the statement identifies the iJKO loss with LiUOT(c_V; pi, alpha_k, alpha_{k+1}) and writes 'D_phi1(.|alpha_k) = iota_{.=alpha_k}' and 'D_phi2(.|alpha_{k+1}) = r' KL(pi_1|alpha_{k+1})', but these assignments are inconsistent with the problem in (15), whose hard constraint is on pi_2 = alpha_k and whose KL penalty is on pi_1 relative to alpha_{k+1}. Additionally, the text in §4.3 says 'due to the KL discrepancy for pi_1, phi*_1 is strongly convex', while the relevant iUOT condition in Assumption 4(ii) requires phi*_2 to be locally strongly convex in the orientation that makes the centered matrix E[bar_phi bar_phi^T] the correct identifiability condition. Once the orientation and the phi_1/phi_2 roles are corrected, the claimed curvature condition can be obtained, but as written the proof of Corollary 2 is not checkable. This must be clarified and made consistent with the iUOT assumptions.","section":"§4.3 and Proposition 4"}],"minor_comments":[{"comment":"The stated condition 'm_alpha m_beta * t + log(S) rho_min <= sqrt(n)' has the dependence on the eigenvalue rho_min in the wrong direction; based on Proposition 16 and the proof of Corollary 1, the intended condition should be roughly (m_alpha m_beta (t + log S)) / rho_min <= sqrt(n), or equivalently m_alpha m_beta (t + log S) <= rho_min sqrt(n).","section":"Corollary 1"},{"comment":"The sentence following (FY-iJKO) says that the constants '1/tau W2^2(alpha_{k+1}, alpha_k | alpha_{k+1} tensor alpha_k) + r KL(alpha_k | alpha_{k+1})' are dropped, but the KL term should be r KL(alpha_{k+1} | alpha_{k+1}) (i.e., zero) if the loss is built as in Definition 2. Please correct the notation.","section":"Equation (FY-iJKO)"},{"comment":"The proof uses Hoeffding's inequality with probability at least 1 - exp(-t^2) for the term |A_1|, while the theorem statement is written for probability at least 1 - exp(-t). The parameters should be aligned so that the claimed probability follows from the stated proof.","section":"Theorem 3 proof"},{"comment":"In Proposition 4, the parameter s in equation (15) is declared to be any positive number, but the subsequent identification sets s = r'. If r' is allowed to be nonpositive, the phrase 'for any s > 0' conflicts with the use of r'. A consistent convention for s and r' (together with the condition r' > 0) should be stated explicitly. Similarly, the semi-dual formula in Section 6.1 uses 's' before defining it; it should be identified with r' or another positive parameter.","section":"Proposition 4 and Section 6.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically rich and the main framework is plausible, but the iJKO section contains a missing hypothesis that is load-bearing for Corollary 2. The abstract stability theorem and forward UOT theorem also have statement-level errors. These are all fixable in revision; I do not see a fatal flaw in the overall approach. The notation in Proposition 4 and Section 4.3 needs careful editorial cleanup before the claims can be verified by a reader."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The iUOT half is solid: the sharpened Fenchel-Young loss over measures genuinely generalizes the FY and Fitzpatrick losses, and the paper delivers real contributions — a forward UOT sample-complexity bound without smoothness assumptions (Theorem 2), an iUOT recovery bound (Theorem 3), and a model-consistency result for l1 and nuclear regularization (Theorem 4). The iJKO half has a missing assumption. Proposition 4 reduces iJKO to iUOT with coefficient r' = r − η/τ on the KL marginal penalty, but the paper only states r > 0 when defining the sharpened loss. The reduction, and the inherited O(√(t/n)) bound in Corollary 2, require r' > 0, i.e. r > η/τ. If r' ≤ 0, the inner problem is not a UOT problem: the conjugate of a negative multiple of KL is not defined, the dual can be unbounded while the original infimum is finite, and a finite counterexample is easy to construct. The numerics use r = 1 with η/τ = 10^{-3}, so the experiments are not affected. This is a statement issue, not a contradiction of the iUOT theory, and it is fixable.\n\nWhat the paper does well: Theorem 1 cleanly separates measurement error, forward perturbation, and local curvature; Propositions 2 and 3 verify the curvature conditions in the iUOT setting; and the connection between the sharpened iJKO loss and the quadratic iJKO* surrogate of [63] is elegant, with a useful Gaussian comparison. The paper ships code, which helps for reproducibility.\n\nSofter spots, in proportion: Corollary 2 is asserted by analogy rather than proved in full — it should follow from Theorem 3 once the reduction is valid, so this is minor. The paper assumes the masses are known, which is acknowledged. Assumption 4's invertibility conditions are standard but not always easy to verify in practice.\n\nBottom line: the audience is researchers in inverse optimal transport, trajectory inference, and learning costs or potentials from distributional samples. This deserves a serious referee. Send it to review, and ask the authors to state the r > η/τ condition, prove Proposition 4 under it, and make the hypotheses of Corollary 2 explicit. The iUOT theory should survive; the iJKO claims need the condition stated or a direct analysis.","headline":"Solid iUOT theory and a genuinely new loss framework; the iJKO reduction needs the missing positivity condition r > η/τ (not just r > 0) stated and proven before Corollary 2 is claimed as a theorem.","tokens_in":51280,"tokens_out":8820,"would_cite":true,"duration_ms":74148,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sharpened Fenchel–Young losses turn inverse problems over probability measures into convex estimators with explicit sample-complexity bounds.","keywords":["sharpened Fenchel–Young loss","inverse optimal transport","unbalanced optimal transport","inverse Jordan-Kinderlehrer-Otto flow","sample complexity","model consistency","mirror-stratifiable regularization","entropic optimal transport"],"falsifier":"Take a Gaussian snapshot pair generated by a quadratic potential, set $r=\\eta/(2\\tau)$, and solve the sharpened iJKO objective with a large sample size: if the objective is unbounded below or the parameter error fails to decay at the claimed $\\sqrt{t/n}$ rate, the reduction to iUOT requires the stricter $r>\\eta/\\tau$; if recovery still works, the bound tolerates the weaker condition stated in the paper.","tokens_in":50149,"feed_emoji":"🎯","tokens_out":8680,"duration_ms":82731,"temperature":0.7,"pith_summary":"The paper's aim is to make inverse problems over probability measures tractable: when an observed distribution is the solution of a variational problem that depends on an unknown cost or potential, the paper shows the unknown parameter can be estimated by minimizing a convex loss derived from the forward problem's optimality gap. The proposed sharpened Fenchel–Young loss adds a data-dependent discrepancy inside the forward problem, which preserves calibration while improving the local curvature of the objective. Stability is proved for two optimal-transport settings: inverse entropic unbalanced optimal transport (iUOT), where the estimator recovers costs at rates $O(\\sqrt{m_\\alpha m_\\beta(\\log n+t)/n})+O(\\lambda)$, and inverse one-step Jordan–Kinderlehrer–Otto (JKO) gradient flows, which inherit a $\\sqrt{t/n}$ rate from the iUOT analysis under additional assumptions. The paper also proves support/rank model consistency for $\\ell^1$- and nuclear-norm regularized estimators under a non-degenerate minimal-norm certificate, and illustrates the conditioning and recovery behavior on Gaussian examples.","feed_headline":"Samples reveal hidden costs behind optimal transport at root-n rates","feed_subtitle":"Sharpened Fenchel–Young losses turn nonlinear inverse problems over measures into convex, stable estimation problems.","key_machinery":"The mechanism is the Fenchel–Young gap function $G_\\Omega(v;u)=\\Omega(u)+\\Omega^*(v)-\\langle v,u\\rangle$, sharpened by replacing $\\Omega$ with $\\Lambda_{\\hat\\mu}(\\cdot)=\\Omega(\\cdot)+D(\\cdot\\mid\\hat\\mu)$. The data-dependent discrepancy $D$ is what gives the loss its curvature: it makes the inner minimization over measures push the prediction toward the observed measure, while the outer problem stays convex in the parameter. The abstract stability theorem (Theorem 1) converts local strong convexity, Lipschitz smoothness, measurement stability, and forward-map stability into the parameter bound $\\|\\hat\\theta-\\theta^\\star\\|\\le\\alpha^{-1}(\\varepsilon+2\\lambda\\|p^\\star\\|)$, and the rest of the paper is largely the verification of these four ingredients for entropic unbalanced optimal transport and JKO steps.","core_discovery":"At the core is the observation that the gap function of a convex minimization over measures is itself a well-behaved loss. For a forward map $\\mu=P_\\Omega(f)=\\arg\\min_{\\tilde\\mu}\\langle f,\\tilde\\mu\\rangle+\\Omega(\\tilde\\mu)$, the sharpened loss $L(f;\\hat\\mu,\\Omega,D)=G_{\\Lambda_{\\hat\\mu}}(-f;\\hat\\mu)$ with $\\Lambda_{\\hat\\mu}(\\cdot)=\\Omega(\\cdot)+D(\\cdot\\mid\\hat\\mu)$ is convex, nonnegative, vanishes exactly at the true $f$ under strict convexity, and has gradient $\\hat\\mu-P_{\\Lambda_{\\hat\\mu}}(f)$. The paper's central stability theorem separates the estimation error into measurement error, forward-perturbation error, and the curvature constants of the loss, and this template is instantiated for iUOT and for iJKO. In iUOT, under compactness and local strong convexity of the dual divergences, the regularized empirical estimator satisfies $\\|\\hat\\theta_n^\\lambda-\\theta^\\star\\|_2=O(\\sqrt{m_\\alpha m_\\beta(\\log n+t)/n})+O(\\lambda)$ with probability at least $1-e^{-t}$. In iJKO, the sharpening term $r\\,KL(\\cdot\\mid\\alpha_{k+1})$ converts the loss into an iUOT loss, so the same sample-complexity machinery applies, giving $O(\\sqrt{t/n})+O(\\lambda)$ for independent snapshots.","pith_inferences":["An extension left implicit is that the sharpening device is not tied to optimal transport: any convex forward problem over measures with a strictly convex regularizer and a stable dual potential should admit the same gap-loss treatment, so the sample-complexity template may carry over to other mean-field or generative models.","For multi-snapshot JKO data, the per-step condition $r>\\eta/\\tau$ must hold for every sharpened step; a practitioner who tunes one global sharpening parameter should check it against the largest time step, otherwise later steps may fall outside the theory.","The Gaussian experiments suggest that the degeneracy of the minimal-norm certificate is visible in the covariance geometry of the initial measure, so pre-registering an experiment could include a certificate computation to say in advance whether support recovery is feasible at a given $n$ and $\\lambda$."],"forward_implications":["In link completion and matching applications, the iUOT bound tells practitioners the sample size $n$ needed to identify a ground cost up to a given parameter error, with error decaying like $1/\\sqrt n$ up to a $\\log n$ factor.","For two-snapshot cell-population or particle-system data, the iJKO result makes learning the driving potential from independent samples a convex problem whose sample complexity is explicit, so the same data can be used to validate the inferred drift.","With $\\ell^1$ or nuclear-norm regularization, the minimal-norm certificate condition guarantees that the estimated support or rank eventually matches the truth, not merely that the parameter is close.","In the large-sharpening limit the iJKO loss converges, after rescaling, to a variance-type loss in the Kantorovich potential, which connects the gap-loss framework to quadratic surrogate losses."],"supporting_citations":[{"why":"Defines Fenchel–Young losses, the framework from which the sharpened variant is built.","marker":"[8]"},{"why":"Establishes the convex maximum-likelihood formulation of inverse optimal transport that the loss generalizes.","marker":"[24]"},{"why":"Supplies the proof template and dual strong-convexity argument for the entropic OT sample-complexity bound.","marker":"[53]"},{"why":"Prior sample-complexity and algorithmic results for unbalanced entropic optimal transport that Theorem 2 extends.","marker":"[57]"},{"why":"Introduces the quadratic iJKO* surrogate compared with the sharpened loss in the high-sharpening limit.","marker":"[63]"},{"why":"Provides the manifold-identification theorem used to prove model consistency for l1 and nuclear regularization.","marker":"[26]"},{"why":"Earlier sparsistency analysis for inverse optimal transport that this paper extends to unbalanced and JKO settings.","marker":"[2]"},{"why":"Convex sparsity-constrained inverse optimal transport formulation and solver that the iUOT treatment builds on.","marker":"[16]"}],"fun_headline_variants":["Optimality gaps as convex losses for inverse problems over measures","Sharpened Fenchel-Young losses make inverse measure problems convex","Convex gap losses recover hidden potentials from measure samples","Sharpening makes inverse OT and JKO learning convex at root-n rates","From samples to costs: stable inverse problems via sharpened losses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The iJKO recovery guarantee depends on the sharpening parameter satisfying $r>\\eta/\\tau$, because the reduction to iUOT produces a coefficient $r'=r-\\eta/\\tau$ multiplying the KL term; the manuscript only states $r>0$ where the loss is introduced, and if $r'\\le0$ the inner minimization over measures is unbounded below and the claimed $\\sqrt{t/n}$ rate does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Optimality gaps as convex losses for inverse problems over measures","Sharpened Fenchel-Young losses make inverse measure problems convex","Convex gap losses recover hidden potentials from measure samples","Sharpening makes inverse OT and JKO learning convex at root-n rates","From samples to costs: stable inverse problems via sharpened losses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000751,"raw_usage":{"total_tokens":3406,"prompt_tokens":1071,"completion_tokens":2335,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":687,"completion_tokens_details":{"reasoning_tokens":2249}},"tokens_in":687,"tokens_out":2335,"duration_ms":17485,"temperature":1.0,"reasoning_tokens":2249,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:24:19.128356+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a Gaussian snapshot pair generated by a quadratic potential, set $r=\\eta/(2\\tau)$, and solve the sharpened iJKO objective with a large sample size: if the objective is unbounded below or the parameter error fails to decay at the claimed $\\sqrt{t/n}$ rate, the reduction to iUOT requires the stricter $r>\\eta/\\tau$; if recovery still works, the bound tolerates the weaker condition stated in the paper.","supporting_citations":[{"cited_title":"Estimating matching affinity matrices under low-rank constraints","cited_arxiv_id":null,"evidence_quote":"Establishes the convex maximum-likelihood formulation of inverse optimal transport that the loss generalizes."},{"cited_title":"Learning with fenchel-young losses","cited_arxiv_id":null,"evidence_quote":"Defines Fenchel–Young losses, the framework from which the sharpened variant is built."},{"cited_title":"Unbalanced optimal transport, from theory to numerics","cited_arxiv_id":null,"evidence_quote":"Prior sample-complexity and algorithmic results for unbalanced entropic optimal transport that Theorem 2 extends."},{"cited_title":"Sensitivity analysis for mirror-stratifiable convex functions","cited_arxiv_id":null,"evidence_quote":"Provides the manifold-identification theorem used to prove model consistency for l1 and nuclear regularization."},{"cited_title":"Sparsistency for inverse optimal transport","cited_arxiv_id":null,"evidence_quote":"Earlier sparsistency analysis for inverse optimal transport that this paper extends to unbalanced and JKO settings."},{"cited_title":"Sista: learning op- timal transport costs under sparsity constraints","cited_arxiv_id":null,"evidence_quote":"Convex sparsity-constrained inverse optimal transport formulation and solver that the iUOT treatment builds on."}],"review_version":1}