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Learning from samples: inverse problems over measures

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Pith's one-line read Sharpened Fenchel–Young losses turn inverse problems over probability measures into convex estimators with explicit sample-complexity bounds.

desk verdict 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. read the letter →

arxiv 2505.07124 v3 pith:XNWGXVB6 submitted 2025-05-11 cs.LG math.STstat.MLstat.TH

Francisco Andrade , Gabriel Peyré , Clarice Poon This is my paper · ORCID
classification cs.LGmath.STstat.MLstat.TH MSC 49Q2290C25
keywords sharpenedFenchel–YounglossinverseoptimaltransportunbalancedJordan-Kinderlehrer-Ottoflowsamplecomplexitymodelconsistencymirror-stratifiableregularizationentropic
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'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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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 $\lambda$.
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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 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.

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 (4)
  1. [§4.2, Proposition 4, Corollary 2] 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.
  2. [Theorem 1] 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.
  3. [Theorem 2] 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.
  4. [§4.3 and Proposition 4] 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.
minor comments (4)
  1. [Corollary 1] 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).
  2. [Equation (FY-iJKO)] 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.
  3. [Theorem 3 proof] 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.
  4. [Proposition 4 and Section 6.1] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main iUOT and iJKO guarantees are derived self-containedly; self-citations are non-load-bearing.

full rationale

The paper's core claims are not derived by assuming their conclusions. Theorem 1 is a standalone convex-analysis stability statement whose proof uses Lemma 3; Theorem 2 extends the Rigollet–Stromme entropic-OT sample-complexity argument to unbalanced OT with nonsmooth divergences, and its proof in Appendix D is self-contained. Theorem 3 checks the hypotheses of Theorem 1 using Corollary 1 and Theorem 2, and Theorem 4 invokes the published manifold-identification theorem [26] together with Proposition 6 for the minimal-norm certificate; these are external mathematical facts with stated assumptions, not conclusions of this paper. The iJKO-to-iUOT reduction in Proposition 4 is explicit algebra, writing the sharpened loss as an infimum over couplings and computing the dual, and the inherited Corollary 2 is therefore a genuine transfer of independently proven bounds; no quantity is fitted and then renamed a prediction. Self-citations to the authors' earlier iOT paper [2] appear when the balanced-OT case is set aside in Remark 2 and when semi-dual solvers are recalled in Section 6; these citations are disclosed and are not load-bearing for the new UOT/JKO results. Two non-circular gaps should be noted as correctness risks rather than circularity: Proposition 4 and Corollary 2 require the sharpening coefficient r to exceed eta/tau so that r' = r - eta/tau is a valid divergence coefficient, which the paper only states as r > 0; and Corollary 2 assumes coupling samples or a valid proxy even though only independent marginals are observed. Neither gap makes a claimed result equal to its input by construction.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claims rest on compactness, strict-convexity-type assumptions on divergences, identifiability of the parameterization, known masses, and for iJKO a positivity condition on the sharpening coefficient. No new physical entities or data-fitted constants are introduced.

free parameters (2)
  • Sharpening parameter r = not fitted; set to r=1 in numerical experiments (Section 6.4)
    Controls the KL sharpening term in the iJKO loss; the theory in Proposition 5 studies the r to infinity limit, and positivity r > eta/tau is needed for the iUOT reduction.
  • Regularization strength lambda = not fitted; grid-searched over lambda in the numerical experiments
    Regularizes the parameterized loss in (iUOTn) and appears linearly in the error bounds of Theorem 3 and Theorem 4.
assumptions (7)
  • domain assumption Compactness of X and Y (Assumption 1)
    Used throughout for bounded Kantorovich potentials, strong duality, and coercivity of the UOT dual problem (Section 3.1).
  • domain assumption Cross-mass inequalities for the divergences (Assumption 2)
    Ensures coercivity of K and restriction of dual optimization to compact sets; satisfied by KL, Jensen-Shannon, chi2, Hellinger, and alpha-divergences (Remark 1).
  • domain assumption Local strong convexity of one conjugate divergence (Assumption 3)
    Needed for the PL inequality and for local strong convexity of the dual function in the forward sample complexity proof (Appendix D.3).
  • domain assumption Identifiability and invertibility of the basis Gram matrix (Assumption 4)
    Gives local strong convexity of the parameterized loss and excludes unidentifiable directions such as skew-symmetric quadratic potentials in Example 4.
  • domain assumption Known masses m(pi_star), m(alpha_star), m(beta_star) in the sampling model
    Stated in Section 3.4; the theory is for known masses rather than estimated ones.
  • domain assumption Sharpening parameter satisfies r > eta/tau for iJKO
    Needed for r' = r - eta/tau to multiply KL with a positive sign in eq. (17); this regime condition is not stated explicitly in Section 4.2.
  • standard math Standard convex analysis toolkit (strong duality, Fenchel-Moreau, PL inequality, McDiarmid, matrix Bernstein, implicit function theorem, Gamma convergence)
    Used without proof in Sections 3 through 5 and the appendices; these are accepted background results.

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Cite this review

Pith. "Pith review of Learning from samples: inverse problems over measures." pith.science (2026). https://pith.science/paper/XNWGXVB6

@misc{pith2026250507124,
  author       = {Pith},
  title        = {Pith review of: Learning from samples: inverse problems over measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XNWGXVB6}},
  note         = {Machine review of arXiv:2505.07124}
}
abstract

We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward problem. This keeps the estimator calibrated while improving the local geometry of the loss. Our main stability theorem separates the inverse error analysis into measurement error, forward perturbation, and empirical curvature. We instantiate this principle for inverse entropic unbalanced optimal transport and for inverse Jordan--Kinderlehrer--Otto (JKO) learning from independent snapshot samples, obtaining high-probability parameter recovery bounds. JKO schemes discretize Wasserstein gradient flows through a sequence of variational problems over measures, making them a natural language for population dynamics observed through snapshots. In this JKO case, the sharpened objective reduces to an unbalanced transport problem, which also clarifies the connection between variational gap losses and quadratic iJKO\(^\star\) surrogates. Numerical experiments illustrate the conditioning effect of sharpening and its benefits for sparse inverse-gradient-flow recovery.

Figures

Figures reproduced from arXiv: 2505.07124 by the authors.

Figure 1
Figure 1. Left: Plots of zmax := ∥P ⊥ T z∥2 where z is the precertificate, a problem setting is nonde￾generate when zmax is small than 1 (i.e. below the red line). FY denotes Fenchel-Young. Here, d = 6, m⋆ = 0. We plot zmax for different values of ω, where ω denotes the angle between Σ⋆ := Σω and θ ⋆ . Both certificates become degenerate as the angle ω increases. Right: Plots of zmax := maxi∈I c zi and a problem setting is no… view at source ↗
Figure 2
Figure 2. Visualization of trajectories. In this case, θ is a rank-1 matrix, and the red line shows the eigenvector of θ. Darkest blue is at t = 0 and lightest blue is at t = τT. Here, τ = 0.1 and T = 10. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_2.png] view at source ↗
Figure 3
Figure 3. Here, both θ ⋆ = u ⋆ (u ⋆ ) ⊤ is a rank 1 matrix and we consider Σ⋆ = vv⊤, varying the angle ω between u ⋆ and v. The x-axis shows in log-scale the parameter λ and the y-axis shows ∥U˜ λU˜ ⊤ λ − u ⋆ (u ⋆ ) ⊤∥2 + |Errrank| where U˜ λ are the reconstructed eigenvectors. Observe that the eigenvectors are better recovered for small ω, which corroborates the fact that the certificate is nondegenerate when ω is small. Not… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Evolution across 6 time points with τ = 0.1, m⋆ = 2 · 1 and Σ⋆ = σ 2 Id. becomes degenerate [PITH_FULL_IMAGE:figures/full_fig_p045_4.png]
Figure 5
Figure 5. Figure 5: This figure shows the fraction of incorrectly estimated positions (y-axis) against log(λ) (x-axis) for different numbers of samples N. A position is considered incorrectly estimated if it takes an absolute value greater than 10−5 . The initial distribution α 0 = N (2 ·…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.