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Entropy-Controlled Flow Matching

T0 review · 4 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A per-path entropy budget on generative flows provably rules out mode collapse and connects the method to optimal transport in the small-budget limit.

desk verdict New formulation, real gap: the Γ-limit claim fails for generic endpoints because the constraint does not vanish, so the abstract overstates what is proven. read the letter →

arxiv 2602.22265 v3 pith:BQP4UGST submitted 2026-02-25 cs.LG cs.CV

Chika Maduabuchi This is my paper · ORCID
classification cs.LGcs.CV MSC 49Q2249J4560J6068T07
keywords entropy-ratecontrolflowmatchingmodecollapseoptimaltransportSchrödingerbridgeGamma-convergencegenerativemodelingdensityfloors
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

This paper tries to show that low-entropy bottlenecks along the trajectory of a flow-based generative model are a structural cause of mode collapse, and that a single global constraint—the entropy rate must never fall below a user-chosen budget—eliminates that failure mode with mathematical guarantees. The author proposes Entropy-Controlled Flow Matching (ECFM), a constrained variational problem over continuity-equation paths, and proves that its minimizers exist, are unique in natural special cases, satisfy explicit mode-coverage and density-floor certificates, and remain stable under perturbations. In the pure transport regime, the paper claims ECFM trajectories coincide with entropic optimal-transport geodesics and, as the budget tends to zero, Gamma-converge to classical Benamou–Brenier optimal transport. It also constructs explicit near-optimal collapsing trajectories for unconstrained flow matching, arguing that the constraint is necessary for any certificate-level non-collapse claim.

What carries the argument

The central object is the entropy-rate budget constraint dH(μt)/dt ≥ −λ, equivalently an expected-divergence constraint E_{μt}[∇·v] ≥ −λ. This single inequality imposes a global ceiling on how fast the transport can concentrate mass, and the KKT multiplier η(t) acts as an adaptive anti-collapse pressure: it is zero when the reference flow is feasible and activates exactly when entropy dissipation exceeds the budget, injecting a score-direction correction into the optimal velocity field. The λ→0 limit is studied via Gamma-convergence of the constrained kinetic action to the Benamou–Brenier functional, under the vanishing-entropic-correction assumption that the entropy and Fisher-information c

What would settle it

Take μ0 = N(0, I) and μT = σ N(0, I) with σ < 1 on R^d, and compute the Benamou–Brenier geodesic: its entropy rate is dH/dt = (1−σ)/t < 0 for small t. If the claimed Γ-convergence holds, one can find λ_n→0 and feasible ECFM paths converging to this geodesic; computing the minimal kinetic energy under the constraint Ḣ ≥ −λ_n for this endpoint pair and showing that its limit exceeds the classical OT value would falsify Theorem 6. A concrete numerical check is to solve the constrained problem for σ = 0.5, T = 1, and compare the limiting cost to W_2^2(μ0, μT)/2.

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Extended reading notes

Core claim

The central claim is that enforcing the pathwise inequality dH(μt)/dt ≥ −λ, where H is differential entropy, turns trajectory compressibility into a controllable constraint. Under the continuity equation this constraint is literally an expected-divergence budget E[∇·v] ≥ −λ, so it forbids transient divergence spikes that deplete semantic modes. The paper proves that the induced primal–dual system has a KKT/Pontryagin form with a nonnegative time-dependent multiplier that acts only when the path tries to dissipate entropy too fast, adding a score-like correction. It then establishes that, in the transport specialization with zero reference drift, ECFM recovers entropic OT geodesics and, under

Load-bearing premise

The Γ-convergence to classical optimal transport rests on a vanishing-entropic-correction regime (Assumption 74) that requires either the diffusion coefficient to shrink to zero or the entropy and Fisher-information corrections to vanish along the minimizing sequence; for generic endpoint pairs the classical OT geodesic has an initially negative entropy rate, making the recovery sequence infeasible for small λ.

Editorial extensions

If this is right

  • If the claims hold, flow-matching and diffusion generators can be trained with an explicit, checkable anti-collapse constraint rather than relying on ad hoc regularization or architectural tricks.
  • The equivalence to Schrödinger bridges implies that ECFM inherits strict convexity and path-law uniqueness in the bridge specialization, giving trajectory-level uniqueness where unconstrained flow matching has none.
  • The mode-coverage and density-floor certificates are SOTA-free: they hold for any minimizer satisfying the regularity assumptions, so practitioners could certify non-collapse from minibatch entropy-rate diagnostics alone.
  • The Gamma-convergence result, if valid in the stated regime, would place entropy-controlled transport in a common mathematical framework with entropic OT and classical OT, allowing existing OT algorithms and stability theory to be imported.
  • The near-optimal collapse counterexamples for unconstrained FM show that any flow-matching method without an explicit entropy-like constraint lacks a uniform mode-coverage guarantee under the same assumptions.

Reading between the lines

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

  • The entropy-rate constraint could be applied not only to the marginal path but to any statistic of the trajectory (e.g., mode-specific mass or moment evolution), yielding a family of certifiable anti-collapse constraints; the paper only explores the global entropy version.
  • A testable extension is to replace the scalar budget λ by a time-dependent schedule λ(t) or by per-mode budgets λ_k(t); the proof machinery appears to generalize, but the paper does not develop these variants.
  • The Gamma-convergence claim likely requires the endpoint pair to satisfy H(μ_T) ≥ H(μ_0) − λT for small λ and the classical OT geodesic to have nonnegative entropy rate; for endpoint pairs where the OT geodesic initially compresses (e.g., μ0 = N(0,I), μT = σN(0,I) with σ<1), the recovery-sequence construction seems to break down, suggesting the true λ→0 limit may be a constrained OT problem rather
  • If the vanishing-entropic-correction assumption fails, the method still yields mode-coverage guarantees, so the practical anti-collapse value of ECFM is largely independent of the OT-limit claim; the two contributions should be evaluated separately.
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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 proposes Entropy-Controlled Flow Matching (ECFM), a constrained variational principle for continuity-equation transports with the pathwise entropy-rate budget dH(μ_t)/dt ≥ −λ, and a flow-matching-style objective against a reference drift u⋆. The main claims are: (i) existence of minimizers under standing assumptions; (ii) a KKT/Pontryagin optimality system with an adaptive nonnegative multiplier; (iii) identification of the pure-transport specialization with Schrödinger bridges / entropic OT geodesics through an 'endogenous' effective regularization ε(λ); (iv) Γ-convergence to classical Benamou–Brenier optimal transport as λ→0; (v) mode-coverage and density-floor certificates with Lipschitz perturbation stability; and (vi) a counterexample showing unconstrained flow matching admits near-optimal collapsing trajectories. The bulk of the paper is an appendix developing measure-theoretic preliminaries, entropy identities, convex duality, the Schrödinger-bridge equivalence, and the Γ-limits. The central theoretical claims fail on internal-consistency grounds: the core entropy-rate identity has a sign error; the Γ-limit functional retains the entropy-rate constraint in the limit and is not the Benamou–Brenier action; and the entropic-OT identification is conditional on a 'matched gauge' assumption that is not satisfied in general.

Significance. If correct, the results would deliver a principled, certificate-level anti-collapse mechanism for flow-based generative models and a novel interpolation between entropic and classical optimal transport. The paper's architecture is ambitious, and several standalone pieces of the scaffolding—metric absolute continuity, convex duality in flux variables, weak compactness of CE trajectories—are standard and appear internally sound. The paper ships no code, so the abstract's assertions rest entirely on the proofs, and those proofs fail at load-bearing points: the entropy identity (Eq. 3) contradicts the paper's own Appendix B.3; the Γ-convergence to classical OT is false for an open family of endpoints; the Theorem 5 identification with entropic OT depends on an unconstructed ε(λ) and an unverified gauge condition; and the mode-coverage argument in App. G uses a reversed monotonicity inequality. The counterexample construction (Theorem 10) is a falsifiable prediction, but its mechanism (25) is driven by the same sign error. The attempted contributions are thus not established as stated.

major comments (4)
  1. [§2.1 Eq. (3); §2.11 (27); Thm 1 (8); Thm 10; App. B.3 Thm 19; App. B.4 Prop. 11] Sign error in the central entropy-rate identity. For H(μ)=∫ρlogρ (Def. 10), the correct identity along CE paths is dH/dt = −∫∇·v dμ. Eq. (3) has the opposite sign, and the paper's own App. B.3 Thm 19 derives dH/dt = −∫ρ_t Δφ_t dx, a direct contradiction: for the Gaussian geodesic N(0,σ_t²), Eq. (3) integrates to +d log σ while the endpoint difference is −(d/2)log σ². The sign of the score correction also alternates across sections (Thm 1: −η∇logρ; Prop. 20/Cor. 13: +η∇logρ). Under the paper's convention, constraint (5) limits de-concentration, not the 'low-entropy bottlenecks' motivating the paper; Thm 10's claim that negative divergence drives H→−∞ is inverted; and App. G.2 Lemma 26's monotonicity bound is reversed (b(m) is decreasing on (0,1/2), so depleting a small mode increases b).
  2. [§2.7 Thm 6; App. F.1 Def. 52; App. F.5 Thm 75 / Assumption 74] The Γ-limit functional F0 defined in App. F.1 is the kinetic action restricted to E0 = {Ḣ ≥ 0}—a constrained problem, not the Benamou–Brenier action on all CE paths. Theorem 75's identification F0 = BB is conditioned on Assumption 74 ('vanishing-entropic-correction regime'), an extra postulate that neither follows from the ECFM functional nor removes the pointwise constraint. More decisively, for endpoints with H(μ_T)<H(μ_0), e.g., μ0=N(0,I), μT=N(0,σ²I) with σ>1 in the paper's sign convention, integrating (5) forces H(μ_T)−H(μ_0) ≥ −λT; for λ<[H(μ0)−H(μT)]/T the feasible set A_λ is empty, so Fλ≡+∞ although the BB value is finite. The recovery sequence (Thm 69) is the constant sequence and only covers limits already in E0; classical OT geodesics are generically not in E0. Hence the abstract's Γ-convergence claim fails for an open set of endpoints, and standing assumption S2 is silently v
  3. [§2.6 Thm 5; App. D.4 Lemma 16, Def. 40] The equivalence to entropic OT / Schrödinger bridges is conditional on 'matched gauge' (Def. 40), i.e., constancy of the correction C(μ;u⋆,ε)=−ε∫Ḣ dt+ε∫∇logρ·u⋆ dμdt−(ε²/2)∫I(μ)dt on the admissible class. For u⋆≡0, C=−ε(H(μ_T)−H(μ_0))−(ε²/2)∫₀^T I(μ_t)dt; the Fisher-information integral is path-dependent, so the gauge fails generically. Theorem 5 postulates ε=ε(λ)>0 with no construction, and Sec. 1 itself frames the SB link as 'an analytical lens'. Without gauge constancy, the ECFM objective is not proportional to the KL/Schrödinger objective, so the claimed identification of ECFM trajectories with entropic OT geodesics is not established by the argument given.
  4. [§2.8 Thms 7–8; App. G.2 Lemmas 24–26] The mode-coverage argument inherits the sign error. Lemma 26 claims b((1−η)m)−b(m) ≤ −ηm log((1−m)/m) < 0 for b(m)=mlogm+(1−m)log(1−m), but b′(m)=log(m/(1−m))<0 on (0,1/2), so depleting a small mode increases b; the correct bound runs in the opposite direction. Lemma 24's 'entropy drop under mode depletion' is therefore an entropy gain in the paper's convention, and constraint (5) does not restrict the mode-depletion channel it is meant to rule out. The proofs of Theorem 7 (and Theorem 8's Harnack step) are built on this reversed estimate, so the asserted mode floors (17) and density floors (18) are not established as stated.
minor comments (4)
  1. [Throughout; bibliography] Cross-references are unstable: App. B.1 cites 'Sec. 3', Remark 11 cites 'App. 3', and the vision protocol is referred to as 'Sec. 1'. Bibliography entries are duplicated or corrupted: refs [10] and [11] are the same DiPerna–Lions paper; [26] and [32] are the same Léonard survey with different titles; 'Kynkääniemi' and 'Léonard' are mangled; ref. [21] (Katanaev) appears unrelated to any cited claim.
  2. [Algorithm 1; §2.2] The residual g^n_k = −b˙H_n(θ^k) − λ_n and the 'Explanation' paragraph are formally consistent with Eq. (3), but would need to change under a corrected entropy identity. The time-grid budgets {λ_n} are introduced in the caption without reconciling them with the global budget λ of (5).
  3. [App. H.2, Prop. 49] The final displayed inequality in Proposition 49 appears incomplete ('∥v⋆ −u⋆∥L' with no closing norm or argument); the statement should be completed.
  4. [§2.3, App. C.2, C.6] The KKT stationarity formula changes sign across sections: Theorem 1 has v = u⋆ − ∇ϕ − η∇logρ, Prop. 20/Cor. 13 have +η∇logρ, and Prop. 24 has −η∇logρ with φ redefined. A single consistent convention (and a sign table for H, Ḣ, and the score correction) would materially improve verifiability.

Circularity Check

3 steps flagged · score 7.0 of 10

Headline OT/entropic-OT claims are secured by postulating epsilon=lambda and by assuming the entropic/Fisher correction is constant or vanishes; the Gamma-limit is not a derived consequence of ECFM.

  1. fitted input called prediction [Sec. 2.6, Theorem 5]
    "Assume (S1)–(S2) and u⋆ ≡ 0. Let (µλ, vλ) minimize (13). Then: 1. (Identification). There exists ε = ε(λ) > 0 and a Schrödinger bridge Pε between µ0 and µT such that µλ_t = Pε_t for a.e. t. Equivalently, (µλ, vλ) is the entropic OT geodesic (current-velocity form) with regularization ε(λ)."

    The entropic OT geodesic is defined as the Schrödinger bridge with regularization parameter ε. The theorem asserts that an ECFM minimizer is such a bridge for some ε(λ), but provides no construction or formula for ε(λ); the proof sketch only says the entropy-rate constraint 'induces an effective regularization level ε(λ) through the KKT multiplier'. Since ε(λ) is chosen after the fact to make the identity hold, the claimed 'recovery of entropic OT geodesics' is a fitted parameter renamed as a prediction rather than a derived consequence of the ECFM functional.

  2. self definitional [App. D.4, Definition 40 (matched gauge)]
    "Definition 40 (Matched gauge). We say (u⋆, ε, λ) is in matched gauge if, on the admissible class, C(µ; u⋆, ε) = C0 for a constant C0 independent of (µ, v) (e.g., via calibrated reference drift, fixed entropy endpoints, and absorbed Fisher/score terms in baseline energy)."

    Lemma 16 defines the correction C(µ;u⋆,ε) = −ε∫Ḣ dt + ε∫∇logρ·u⋆ dµdt − (ε²/2)∫I(µ)dt. This correction is precisely the Fisher-information and score-energy part that distinguishes ECFM from a plain Schrödinger-bridge KL objective. Theorem 50 obtains the exact equivalence Vλ = 2εKε + C0 only in this 'matched gauge'. Thus the paper's central identification of ECFM with Schrödinger bridges/entropic OT is made true by definitionally assuming the distinguishing entropic terms are constant, rather than by deriving the equivalence from the ECFM objective.

1 more flagged steps
  1. fitted input called prediction [App. F.5, Assumption 74 and Theorem 75]
    "Assumption 74 (Vanishing-entropic-correction regime). Along the λ↓0 minimizer sequence (ρλ, vλ), assume: 1. either ε = ε(λ) ↓ 0, or 2. ε > 0 fixed but ε |H(µλ_T)−H(µλ_0)| + ε² ∫ I(µλ_t) dt → 0."

    The claimed Γ-limit to classical Benamou–Brenier (Theorem 6, Sec. 2.7) is completed only by identifying the limit functional F0 with BB under Assumption 74. But F0 itself is defined (App. F.1, Def. 52) as the zero-budget entropy-constrained action on E0 = {Ḣ ≥ 0}, not the unconstrained BB action. The identification with BB is thus an unverified assumption that the entropic/Fisher correction vanishes. Moreover, from the paper's own integrated constraint H(µt)−H(µs) ≥ −λ(t−s), endpoints with H(µ_T) < H(µ_0) make A_λ empty for λ < (H(µ0)−H(µT))/T, so Fλ = +∞ while the BB value is finite; no recovery sequence can exist. The 'vanishing-entropic-correction regime' is therefore a fitted input introduced to force the Γ-limit, not a consequence of ECFM.

full rationale

The paper's abstract and main-text contributions claim: (i) ECFM recovers entropic OT geodesics, and (ii) Γ-converges to classical OT as λ→0. Both of these central theoretical claims reduce, by the paper's own definitions and assumptions, to postulating the very entropic structure being derived. Theorem 5 asserts existence of ε=ε(λ) with no construction; the proof sketch appeals to the KKT multiplier, but the parameter is never tied to λ or to the data beyond the assertion. In App. D.4 the equivalence to Schrödinger bridges is obtained only in the 'matched gauge' of Definition 40, where the correction functional C(µ;u⋆,ε)—containing ∫I(µ)dt and ∫∇logρ·u⋆ dµdt—is assumed constant. That is not a derivation; it is an assumption that the entropic/Fisher terms are irrelevant. The Γ-limit to classical OT is similarly conditional: App. F.5 needs Assumption 74 (vanishing entropic correction), and the proof sketch of Theorem 6 uses a recovery sequence obtained by smoothing an OT geodesic, which is feasible only if the OT geodesic already satisfies Ḣ≥0. For generic endpoints with H(µ_T)<H(µ_0), the admissible set A_λ is empty for small λ, so Fλ≡+∞ despite finite BB cost; the paper's standing assumption S2 is silently violated exactly in the λ→0 regime where the headline claim is made. These are not self-citations or ad hominem issues; the paper also contains independent content, notably the mode-coverage/density-floor arguments (Theorems 7–8) and the explicit unconstrained-collapse counterexample (Theorem 10), which are not circular in the same way. But the two flagship 'geometric' identifications are secured by postulating ε(λ) and by assuming the entropic correction is constant/vanishing. Hence the central derivation is substantially circular: the predicted entropic-OT and classical-OT limits are assumed into existence rather than implied by the ECFM objective. Score 7 reflects that the mode-coverage and counterexample components retain independent mathematical content, while the core OT/entropic-OT claims reduce to fitted inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claim rests on user-chosen λ, an asserted effective entropic level ε(λ), a matched-gauge assumption containing the Fisher/score terms, and a vanishing-correction assumption that removes the very constraint the method is built on. The axiom count is high because the headline equivalences are imported rather than derived.

free parameters (3)
  • λ (entropy-rate budget) = user-specified (λ ≥ λ̂^LCB_eff)
    The constraint level d/dt H ≥ −λ is chosen by hand or by feasibility heuristic (App J.2, J.4); all guarantees scale with λ.
  • ε (effective entropic level) = ε = ε(λ) asserted to exist (Theorem 5)
    The diffusivity of the Schrödinger-bridge reference is introduced after the fact to represent the ECFM trajectory; no construction is given, so it functions as a fitted number.
  • ρ, penalty, step sizes (Algorithm 1) = ρ>0, α_k, β_k, grid {t_n}, budgets {λ_n}
    Numerical hyperparameters of the proposed primal–dual algorithm; not central to the theorems but required to implement the method.
assumptions (5)
  • domain assumption Continuity equation / absolute continuity of µ_t with H(µ_t)∈AC and entropy-rate identity dH/dt = E[∇·v] holds for minimizers (App. B).
    The entropy-rate constraint is meaningful only when the path is absolutely continuous in time and regular enough; cannot be verified for the neural-network learned flows in practice.
  • domain assumption Feasibility (S2): A_λ(µ0, µT) nonempty for the chosen λ.
    For small λ this forces H(µ_T) ≥ H(µ_0); typical generative settings (Gaussian base → data) have entropy decrease, so small λ is infeasible, undermining the λ→0 results.
  • ad hoc to paper Matched gauge (Definition 40): correction C(µ; u*, ε) is constant on the admissible class.
    Used in D.4/E.2 to claim exact SB equivalence; includes ∫∇logρ·u* and ∫I(µ) terms, which are generally path-dependent, so the assumption is essentially the desired conclusion.
  • ad hoc to paper Vanishing-entropic-correction regime (Assumption 74) to identify F0 with Benamou–Brenier in F.5.
    The λ→0 functional is F0 = A_BB + constraint Ḣ≥0; assuming a correction vanishes cannot remove the constraint; this identification is not justified for generic endpoints.
  • domain assumption Slater condition (Assumption 29) for KKT and strong duality.
    Strict feasibility δ>0 of the entropy-rate constraint is assumed; may fail near the λ→0 boundary.
invented entities (1)
  • ε(λ) — effective entropic regularization / reference diffusivity
    purpose: Identifies ECFM minimizers with entropic OT geodesics and Schrödinger bridges (Theorems 4–5); gives the KL representation a diffusion coefficient.
    No construction or measurement is given for ε(λ); its existence is asserted to make the equivalence hold. There is no falsifiable prediction attached.

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Pith. "Pith review of Entropy-Controlled Flow Matching." pith.science (2026). https://pith.science/paper/BQP4UGST

@misc{pith2026260222265,
  author       = {Pith},
  title        = {Pith review of: Entropy-Controlled Flow Matching},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BQP4UGST}},
  note         = {Machine review of arXiv:2602.22265}
}
read the original abstract

Modern vision generators transport a base distribution to data through time-indexed measures, implemented as deterministic flows (ODEs) or stochastic diffusions (SDEs). Despite strong empirical performance, standard flow-matching objectives do not directly control the information geometry of the trajectory, allowing low-entropy bottlenecks that can transiently deplete semantic modes. We propose Entropy-Controlled Flow Matching (ECFM): a constrained variational principle over continuity-equation paths enforcing a global entropy-rate budget d/dt H(mu_t) >= -lambda. ECFM is a convex optimization in Wasserstein space with a KKT/Pontryagin system, and admits a stochastic-control representation equivalent to a Schrodinger bridge with an explicit entropy multiplier. In the pure transport regime, ECFM recovers entropic OT geodesics and Gamma-converges to classical OT as lambda -> 0. We further obtain certificate-style mode-coverage and density-floor guarantees with Lipschitz stability, and construct near-optimal collapse counterexamples for unconstrained flow matching.

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Reference graph

Works this paper leans on

190 extracted references · 6 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bakry, D., Gentil, I., Ledoux, M.: Logarithmic Sobolev Inequalities, pp. 235–

  2. [2]

    Numerische Mathematik84(3), 375–393 (2000).https://doi.org/10.1007/s002110050002,https://doi.org/ 10.1007/s002110050002

    Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the monge–kantorovich mass transfer problem. Numerische Mathematik84(3), 375–393 (2000).https://doi.org/10.1007/s002110050002,https://doi.org/ 10.1007/s002110050002

    doi:10.1007/s002110050002 2000

  3. [3]

    Communications on Pure and Applied Mathematics44, 375–417 (1991), https://api.semanticscholar.org/CorpusID:123428953

    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued func- tions. Communications on Pure and Applied Mathematics44, 375–417 (1991), https://api.semanticscholar.org/CorpusID:123428953

    1991

  4. [4]

    In: Bengio, S., Wallach, H., Larochelle, H., Grauman, K., Cesa-Bianchi, N., Garnett, R

    Chen, R.T.Q., Rubanova, Y., Bettencourt, J., Duvenaud, D.K.: Neural or- dinary differential equations. In: Bengio, S., Wallach, H., Larochelle, H., Grauman, K., Cesa-Bianchi, N., Garnett, R. (eds.) Advances in Neu- ral Information Processing Systems. vol. 31. Curran Associates, Inc. (2018),https://proceedings.neurips.cc/paper_files/paper/2018/file/ 69386f...

    2018

  5. [5]

    IEEE Transactions on Automatic Control62, 2137–2152 (2015),https: //api.semanticscholar.org/CorpusID:15303335

    Chen, Y., Georgiou, T.T., Pavon, M.: Optimal transport over a linear dynamical system. IEEE Transactions on Automatic Control62, 2137–2152 (2015),https: //api.semanticscholar.org/CorpusID:15303335

    2015

  6. [6]

    Journal of Opti- mization Theory and Applications169(2), 671–691 (2016).https://doi.org/10

    Chen, Y., Georgiou, T.T., Pavon, M.: On the relation between optimal trans- port and schr¨ odinger bridges: A stochastic control viewpoint. Journal of Opti- mization Theory and Applications169(2), 671–691 (2016).https://doi.org/10. 1007/s10957-015-0803-z,https://doi.org/10.1007/s10957-015-0803-z

    doi:10.1007/s10957-015-0803-z 2016

  7. [7]

    In: Burges, C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K

    Cuturi, M.: Sinkhorn distances: Lightspeed computation of optimal transport. In: Burges, C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K. (eds.) Advances in Neural Information Processing Systems. vol. 26. Curran Associates, Inc. (2013),https://proceedings.neurips.cc/paper_files/paper/2013/file/ af21d0c97db2e27e13572cbf59eb343d-Paper.pdf

    2013

  8. [8]

    In: Proceedings of the 35th International Conference on Neural Information Processing Systems

    De Bortoli, V., Thornton, J., Heng, J., Doucet, A.: Diffusion schr¨ odinger bridge with applications to score-based generative modeling. In: Proceedings of the 35th International Conference on Neural Information Processing Systems. NIPS ’21, Curran Associates Inc., Red Hook, NY, USA (2021)

    2021

Show all 190 references
  1. [9]

    In: Beygelzimer, A., Dauphin, Y., Liang, P., Vaughan, J.W

    Dhariwal, P., Nichol, A.Q.: Diffusion models beat GANs on image synthesis. In: Beygelzimer, A., Dauphin, Y., Liang, P., Vaughan, J.W. (eds.) Advances in Neu- ral Information Processing Systems (2021),https://openreview.net/forum?id= AAWuCvzaVt

    2021

  2. [11]

    Inventiones Mathematicae98(3), 511–548 (1989).https://doi

    DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and sobolev spaces. Inventiones Mathematicae98(3), 511–548 (1989).https://doi. org/10.1007/BF01393835

    1989 doi

  3. [12]

    Proceedings of the National Academy of Sciences72(3), 780–783 (1975).https://doi.org/10.1073/pnas.72.3.780, https://www.pnas.org/doi/abs/10.1073/pnas.72.3.780

    Donsker, M.D., Varadhan, S.R.S.: On a variational formula for the principal eigen- value for operators with maximum principle. Proceedings of the National Academy of Sciences72(3), 780–783 (1975).https://doi.org/10.1073/pnas.72.3.780, https://www.pnas.org/doi/abs/10.1073/pnas.72.3.780

    1975 doi

  4. [13]

    In: International Conference on Artificial In- 16 C

    Goldfeld, Z., Greenewald, K.H.: Gaussian-smoothed optimal transport: Metric structure and statistical efficiency. In: International Conference on Artificial In- 16 C. Maduabuchi et al. telligence and Statistics (2020),https://api.semanticscholar.org/CorpusID: 210168147

    2020

  5. [14]

    In: Ghahra- mani, Z., Welling, M., Cortes, C., Lawrence, N., Weinberger, K

    Goodfellow, I.J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial nets. In: Ghahra- mani, Z., Welling, M., Cortes, C., Lawrence, N., Weinberger, K. (eds.) Ad- vances in Neural Information Processing Systems. ...

    2014

  6. [15]

    In: International Confer- ence on Learning Representations (2019),https://openreview.net/forum?id= rJxgknCcK7

    Grathwohl, W., Chen, R.T.Q., Bettencourt, J., Duvenaud, D.: Scalable reversible generative models with free-form continuous dynamics. In: International Confer- ence on Learning Representations (2019),https://openreview.net/forum?id= rJxgknCcK7

    2019

  7. [16]

    Annals of Mathematics20(4), 292–296 (1919), http://www.jstor.org/stable/1967124

    Gronwall, T.H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Annals of Mathematics20(4), 292–296 (1919), http://www.jstor.org/stable/1967124

    1919

  8. [17]

    In: Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., Lin, H

    Ho, J., Jain, A., Abbeel, P.: Denoising diffusion probabilistic models. In: Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., Lin, H. (eds.) Advances in Neu- ral Information Processing Systems. vol. 33, pp. 6840–6851. Curran Associates, Inc. (2020),https://proceedings.neur...

    2020

  9. [18]

    Journal of the American Statistical Association58(301), 13–30 (1963).https://doi.org/10.1080/01621459.1963.10500830,https://doi.org/ 10.1080/01621459.1963.10500830

    Hoeffding, W.: Probability inequalities for sums of bounded random vari- ables. Journal of the American Statistical Association58(301), 13–30 (1963).https://doi.org/10.1080/01621459.1963.10500830,https://doi.org/ 10.1080/01621459.1963.10500830

    1963

  10. [19]

    Kallianpur, G., Karandikar, R.L.: Girsanov’s Theorem, pp. 95–101. Birkh¨ auser Boston, Boston, MA (2000).https://doi.org/10.1007/978-1-4612-0511-1_5, https://doi.org/10.1007/978-1-4612-0511-1_5

    2000 doi

  11. [20]

    In: Koyejo, S., Mohamed, S., Agar- wal, A., Belgrave, D., Cho, K., Oh, A

    Karras, T., Aittala, M., Aila, T., Laine, S.: Elucidating the design space of diffusion-based generative models. In: Koyejo, S., Mohamed, S., Agar- wal, A., Belgrave, D., Cho, K., Oh, A. (eds.) Advances in Neural Infor- mation Processing Systems. vol. 35, pp. 26565–26577. Curr...

    2022

  12. [21]

    Physica Scripta98(10), 104001 (sep 2023).https: //doi.org/10.1088/1402-4896/acf251,https://doi.org/10.1088/1402-4896/ acf251

    Katanaev, M.O.: Complete separation of variables in the geodesic hamilton-jacobi equation in four dimensions. Physica Scripta98(10), 104001 (sep 2023).https: //doi.org/10.1088/1402-4896/acf251,https://doi.org/10.1088/1402-4896/ acf251

    2023 doi

  13. [22]

    Universitext, Springer London, London, 1 edn

    Klenke, A.: Probability Theory: A Comprehensive Course. Universitext, Springer London, London, 1 edn. (2008).https://doi.org/10.1007/978-1-84800-048-3

    2008 doi

  14. [23]

    In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950

    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950. pp. 481–

    1950

  15. [24]

    In: Proceedings of the 33rd In- ternational Conference on Neural Information Processing Systems

    Kynk¨ a¨ anniemi, T., Karras, T., Laine, S., Lehtinen, J., Aila, T.: Improved precision and recall metric for assessing generative models. In: Proceedings of the 33rd In- ternational Conference on Neural Information Processing Systems. pp. 3927–3936. Curran Associates, Inc. (2019)

    2019

  16. [25]

    arXiv: Optimization and Control (2010),https://api.semanticscholar.org/ CorpusID:119683072 Entropy-Controlled Flow Matching 17

    L’eonard, C.: From the schr¨ odinger problem to the monge-kantorovich problem. arXiv: Optimization and Control (2010),https://api.semanticscholar.org/ CorpusID:119683072 Entropy-Controlled Flow Matching 17

    2010

  17. [27]

    In: Meila, M., Zhang, T

    Lin, C.H., Azabou, M., Dyer, E.: Making transport more robust and interpretable by moving data through a small number of anchor points. In: Meila, M., Zhang, T. (eds.) Proceedings of the 38th International Conference on Machine Learning. Proceedings of Machine Learning Researc...

    2021

  18. [28]

    In: The Eleventh International Conference on Learning Representations (2023),https://openreview.net/forum?id=PqvMRDCJT9t

    Lipman, Y., Chen, R.T.Q., Ben-Hamu, H., Nickel, M., Le, M.: Flow matching for generative modeling. In: The Eleventh International Conference on Learning Representations (2023),https://openreview.net/forum?id=PqvMRDCJT9t

    2023

  19. [29]

    SIAM Journal on Numerical Analysis60(3), 1385–1449 (2022).https://doi.org/ 10.1137/20M1344986,https://doi.org/10.1137/20M1344986

    Liu, S., Li, W., Zha, H., Zhou, H.: Neural parametric fokker–planck equation. SIAM Journal on Numerical Analysis60(3), 1385–1449 (2022).https://doi.org/ 10.1137/20M1344986,https://doi.org/10.1137/20M1344986

    2022 doi

  20. [30]

    Liu, S., Ma, S., Chen, Y., Zha, H., Zhou, H.: Learning high dimensional wasserstein geodesics (2021),https://arxiv.org/abs/2102.02992

    2021 arXiv

  21. [31]

    In: The Eleventh International Conference on Learning Representations (2023),https://openreview.net/forum?id=XVjTT1nw5z

    Liu, X., Gong, C., qiang liu: Flow straight and fast: Learning to generate and trans- fer data with rectified flow. In: The Eleventh International Conference on Learning Representations (2023),https://openreview.net/forum?id=XVjTT1nw5z

    2023

  22. [32]

    Discrete and Continuous Dynamical Systems34(4), 1533–1574 (2014).https://doi.org/10.3934/dcds.2014.34.1533,https://www

    L´ eonard, C.: A survey of the schr¨ odinger problem and some of its connec- tions with optimal transport. Discrete and Continuous Dynamical Systems34(4), 1533–1574 (2014).https://doi.org/10.3934/dcds.2014.34.1533,https://www. aimsciences.org/article/id/d5bcf817-901d-4104-b7da...

    2014 doi

  23. [33]

    Advances in Mathematics128(1), 153–179 (1997).https://doi.org/https://doi.org/ 10.1006/aima.1997.1634,https://www.sciencedirect.com/science/article/ pii/S0001870897916340

    McCann, R.J.: A convexity principle for interacting gases. Advances in Mathematics128(1), 153–179 (1997).https://doi.org/https://doi.org/ 10.1006/aima.1997.1634,https://www.sciencedirect.com/science/article/ pii/S0001870897916340

    1997

  24. [34]

    Mikami, T., Thieullen, M.: Duality theorem for the stochastic optimal con- trol problem. Stochastic Processes and their Applications116(12), 1815– 1835 (2006).https://doi.org/https://doi.org/10.1016/j.spa.2006.04.014, https://www.sciencedirect.com/science/article/pii/S0304414906000627

    2006 doi

  25. [35]

    Nikodym, O.: Sur une g´ en´ eralisation des int´ egrales de M. J. Radon. Errata et remarques. Fundam. Math.15, 358 (1930),https://eudml.org/doc/212339

    1930

  26. [36]

    Journal of Machine Learning Research24(374), 1–51 (2023),http://jmlr.org/papers/v24/23-0527.html

    Peluchetti, S.: Diffusion bridge mixture transports, schr ˜A¶dinger bridge problems and generative modeling. Journal of Machine Learning Research24(374), 1–51 (2023),http://jmlr.org/papers/v24/23-0527.html

    2023

  27. [37]

    American Journal of Mathematics12(3), 211–294 (1890),http://www.jstor.org/ stable/2369620

    Poincar´ e, H.: Sur les equations aux d´ eriv´ ees partielles de la physique math´ ematique. American Journal of Mathematics12(3), 211–294 (1890),http://www.jstor.org/ stable/2369620

  28. [38]

    Siberian Mathematical Journal8(5), 801–816 (Sep 1967).https: //doi.org/10.1007/BF01040656,https://doi.org/10.1007/BF01040656

    Reshetnyak, Y.G.: General theorems on semicontinuity and on convergence with a functional. Siberian Mathematical Journal8(5), 801–816 (Sep 1967).https: //doi.org/10.1007/BF01040656,https://doi.org/10.1007/BF01040656

    1967 doi

  29. [39]

    In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)

    Rombach, R., Blattmann, A., Lorenz, D., Esser, P., Ommer, B.: High-resolution image synthesis with latent diffusion models. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 10684– 10695 (June 2022)

    2022

  30. [40]

    In: Oh, A., Naumann, T., Globerson, A., Saenko, K., Hardt, M., Levine, S

    Shi, Y., De Bortoli, V., Campbell, A., Doucet, A.: Diffusion schr¨ odinger bridge matching. In: Oh, A., Naumann, T., Globerson, A., Saenko, K., Hardt, M., Levine, S. (eds.) Advances in Neural Information Pro- cessing Systems. vol. 36, pp. 62183–62223. Curran Associates, Inc. 1...

    2023

  31. [41]

    In: Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., Scarlett, J

    Song, Y., Dhariwal, P., Chen, M., Sutskever, I.: Consistency models. In: Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., Scarlett, J. (eds.) Proceedings of the 40th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 202...

    2023

  32. [42]

    In: Interna- tional Conference on Learning Representations (2021),https://openreview.net/ forum?id=PxTIG12RRHS

    Song, Y., Sohl-Dickstein, J., Kingma, D.P., Kumar, A., Ermon, S., Poole, B.: Score- based generative modeling through stochastic differential equations. In: Interna- tional Conference on Learning Representations (2021),https://openreview.net/ forum?id=PxTIG12RRHS

    2021

  33. [43]

    Probability Theory and Related Fields97(4), 515–542 (Dec 1993).https://doi.org/10.1007/BF01192962,https://doi.org/10.1007/ BF01192962

    Stummer, W.: The Novikov and entropy conditions of multidimensional diffusion processes with singular drift. Probability Theory and Related Fields97(4), 515–542 (Dec 1993).https://doi.org/10.1007/BF01192962,https://doi.org/10.1007/ BF01192962

    1993 doi

  34. [44]

    In: Dasgupta, S., Mandt, S., Li, Y

    Tong, A.Y., Malkin, N., Fatras, K., Atanackovic, L., Zhang, Y., Huguet, G., Wolf, G., Bengio, Y.: Simulation-free Schr¨ odinger bridges via score and flow match- ing. In: Dasgupta, S., Mandt, S., Li, Y. (eds.) Proceedings of The 27th Inter- national Conference on Artificial In...

    2024

  35. [45]

    Villani, C.: Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften, vol. 338. Springer-Verlag Berlin Heidelberg (2009).https://doi. org/10.1007/978-3-540-71050-9

    2009 doi

  36. [46]

    Pro- ceedings of the Royal Society of London Series A87(594), 225–229 (Aug 1912)

    Young, W.H.: On Classes of Summable Functions and their Fourier Series. Pro- ceedings of the Royal Society of London Series A87(594), 225–229 (Aug 1912). https://doi.org/10.1098/rspa.1912.0076 Entropy-Controlled Flow Matching 19 Appendix T able of Contents Abbreviations . . . ...

    1912

  37. [49]

    Moreover, among all such v, there exists a unique (a.e

    There exists a Borel field vt ∈L 2(µt) such that ∂tµt +∇ ·(µtvt) = 0inD ′((0, T)×R d), and Z T 0 Z Rd |vt(x)|2 dµt(x)dt <∞. Moreover, among all such v, there exists a unique (a.e. in t, µt-a.e. in x) minimal-norm representative vt ∈T µt P2 satisfying |˙µt|=∥v t∥L2(µt) = min n ...

  38. [50]

    t∈(0, T)

    ˙h(t)≥ −λfor a.e. t∈(0, T)

  39. [51]

    For all 0≤s≤t≤T , h(t) +λt≥h(s) +λs

  40. [52]

    Proof.(1 ⇒2): Integrate ˙h+λ≥0 over [s, t]

    For all nonnegative η∈C ∞ c ((0, T)), − Z T 0 h(t)η′(t)dt≥ −λ Z T 0 η(t)dt. Proof.(1 ⇒2): Integrate ˙h+λ≥0 over [s, t]. (2 ⇒1): Monotonicity of h+λt implies distributional derivative nonnegative, hence ˙h+λ≥0 a.e. (1 ⇔3): dis- tributional characterization of a.e. lower bound o...

  41. [53]

    µ0, µ1 ∈ Pac 2 (Rd), µ0 ̸=µ 1

  42. [54]

    µ0 ≪ Ld, hence the Brenier map T=∇ψ from µ0 to µ1 exists and is unique

  43. [55]

    along interpolation: det (1−t)I+tDT(x) >0, µ 0-a.e.,∀t∈(0,1)

    Jacobian is non-singular µ0-a.e. along interpolation: det (1−t)I+tDT(x) >0, µ 0-a.e.,∀t∈(0,1). Theorem 16 (Displacement convexity of Boltzmann entropy).Let (µt)t∈[0,1] be the W2-geodesic between µ0, µ1 ∈ P2(Rd). Then H(µt)≤(1−t)H(µ 0) +tH(µ 1), t∈[0,1]. Hence H is 0-displaceme...

  44. [56]

    Hence µ⋆ 0 =µ ⋆ 1

    = 0. Hence µ⋆ 0 =µ ⋆ 1. Proposition 8 (Second variation along smooth geodesics (formal Hes- sian lower bound)).Let (µt, vt) be a smooth geodesic with µt =ρ tdx, and assume µt solves CE with potential velocity vt =∇ϕ t, where ∂tϕt + 1 2 |∇ϕt|2 = 0. 38 C. Maduabuchi et al. Then ...

  45. [57]

    convexity of entropy along transport paths, crucial for existence/uniqueness in Section C

  46. [58]

    strict convexity mechanisms under nondegeneracy, used in uniqueness proofs

  47. [59]

    κ-convex structure for relative-entropy penalized formulations in Sections D– F. B.3. Entropy along W asserstein geodesicsWe derive quantitative formulas for entropy along displacement interpolations: first derivative, second derivative (under smoothness), and integral inequal...

  48. [60]

    (µt)t∈[0,1] ⊂ Pac 2 (Rd) is the unique W2-geodesic between µ0, µ1 ∈ Pac 2

  49. [61]

    µt =ρ t dx, with ρt ∈C 1 t C 2 x, ρt >0

  50. [62]

    Entropy-Controlled Flow Matching 39 Lemma 7 (Pointwise Jacobian representation along geodesic).Let T= ∇ψ be Brenier map µ0 7→µ 1, and Xt(x) := (1−t)x+tT(x), µ t = (Xt)#µ0

    there exists ϕt ∈C 1 t C 3 x such that vt =∇ϕ t, ∂ tρt +∇·(ρ t∇ϕt) = 0, ∂ tϕt + 1 2 |∇ϕt|2 = 0. Entropy-Controlled Flow Matching 39 Lemma 7 (Pointwise Jacobian representation along geodesic).Let T= ∇ψ be Brenier map µ0 7→µ 1, and Xt(x) := (1−t)x+tT(x), µ t = (Xt)#µ0. Then for ...

  51. [63]

    ρt ∈C 1 t C 2 x, ρt >0 , R ρt dx= 1

  52. [64]

    b∈L 1 loc(0, T;W1,1 loc (Rd;R d))

  53. [65]

    Set µt =ρ t dx

    all boundary terms at infinity vanish in integrations by parts (e.g., via finite second moment and suitable decay). Set µt =ρ t dx. Definition 15 (Fisher information).For µ=ρ dx with ρ >0 and √ρ∈ H 1(Rd), define I(µ) := Z Rd |∇ρ|2 ρ dx= 4 Z Rd |∇√ρ|2 dx. If the integral diverg...

  54. [66]

    u⋆ has at most linear growth: |u⋆ t (x)| ≤at +b t|x|, a, b∈L 2(0, T), b≥0

  55. [67]

    admissible curves satisfy uniform second-moment bound (from A.2/A.3): sup t∈[0,T] m2(µt)<∞

  56. [68]

    endpoint entropies are finite: H(µ0),H(µ T )<∞. Lemma 10 (Nonemptiness criterion for entropy-feasible set).If there exists (¯µ,¯v)∈A(µ0, µT ) with H(¯µt)∈AC([0, T]) and ess inf t∈(0,T) d dt H(¯µt)≥ −λ, then Aλ(µ0, µT )̸=∅ , hence Vλ <+∞ . Proof.Immediate from Definition 18: (¯...

  57. [69]

    Then (ECFMλ) admits at least one minimizer

    sequential closedness of entropy-rate constraint: if (µn, vn)∈A λ, µn t ⇀ µt for each t, vnµn ⇀ vµweakly as vector measures, and supn R |vn|2dµndt < ∞, then (µ, v)∈A λ. Then (ECFMλ) admits at least one minimizer. Proof.Take minimizing sequence (µn, vn)⊂A λ with JFM(µn, vn)↓V λ...

  58. [70]

    (Slater-type condition) there exists (¯µ,¯v)∈A(µ0, µT ) with ˙H(¯µt) +λ≥δ >0a.e.t

  59. [71]

    in (µ, m=vµ)

    the map (µ, v)7→ JFM(µ, v)is convex in v, l.s.c. in (µ, m=vµ)

  60. [72]

    Definition 25 (F easible cone and critical directions).Let (µ⋆, v⋆)∈A λ

    entropy-rate mapping (µ, v)7→ ˙H(µ) is weakly closed on admissible sequences (as used in C.1 existence theorem). Definition 25 (F easible cone and critical directions).Let (µ⋆, v⋆)∈A λ. A perturbation (δµ, δv)is an admissible first-order direction if:

  61. [73]

    linearized CE holds: ∂tδµ+∇·(δµ v ⋆ +µ ⋆δv) = 0, δµ |t=0 =δµ |t=T = 0

  62. [74]

    on{t: ˙H(µ⋆ t ) +λ= 0}, where δHt = Z 1 + logρ ⋆ t d(δµt) (µ ⋆ t =ρ ⋆ t dx)

    linearized entropy-rate is feasible on active set: d dt h δHt i ≥0a.e. on{t: ˙H(µ⋆ t ) +λ= 0}, where δHt = Z 1 + logρ ⋆ t d(δµt) (µ ⋆ t =ρ ⋆ t dx). Theorem 30 (KKT conditions in measure space).Under Assumption 29, if (µ⋆, v⋆) solves (ECFMλ), then there exist multipliers η⋆ ∈L ...

  63. [75]

    µ⋆ t =ρ ⋆ t dx, ρ⋆ t >0 , ρ⋆ ∈C 1 t C 2 x

  64. [76]

    v⋆, u⋆,∇logρ ⋆,∇φ ⋆ ∈L 2(dt dµ⋆ t ); Entropy-Controlled Flow Matching 59

  65. [77]

    entropy derivative identity holds: ˙H(µ⋆ t ) = Z ∇logρ ⋆ t ·v ⋆ t dµ⋆ t a.e.t

  66. [78]

    endpoint constraints µ⋆ |0 =µ 0, µ⋆ |T =µ T . Definition 26 (Hamiltonian density).For (t, x, ρ, v, p, η)with ρ >0, define h(t, x, ρ, v, p, η) :=1 2 |v−u ⋆ t (x)|2 ρ+p∇·(ρv)−η∇·(ρ∇logρ v-linearized form), where the entropy term is interpreted via first variation: −η ˙H=−η Z ∇lo...

  67. [79]

    restricting η to W 1,∞ with η(0) =η(T) = 0 , removing endpoint entropy terms

  68. [80]

    In this appendix we use the first route for clean closed dual constraints

    or augmenting dual with a scalar state s(t) =H(µ t) and its own adjoint. In this appendix we use the first route for clean closed dual constraints. Definition 29 (Reduced dual admissible set).Define Kλ :=   (φ, η) : φ∈W 1,∞ loc ((0, T)×R d), η∈W 1,∞(0, T), η≥0, η(0) =η(T) =...

  69. [81]

    Slater condition from Assumption 29

  70. [82]

    of primal integrand in (ρ, m)

    convexity and l.s.c. of primal integrand in (ρ, m)

  71. [83]

    Then Dred λ =V λ

    closedness of CE and entropy-rate constraints in the topology used for C.1 existence. Then Dred λ =V λ. Moreover, any primal-dual optimal pair (ρ⋆, m⋆;φ ⋆, η⋆) satisfies saddle-point conditions and KKT system from C.3–C.4. Entropy-Controlled Flow Matching 65 Proof.Apply Fenche...

  72. [84]

    KKT/EL system from C.3–C.4

  73. [85]

    Therefore C.3–C.6 provide equivalent primal-dual characterizations of entropy- controlled flow matching

    PMP system from Theorem 38. Therefore C.3–C.6 provide equivalent primal-dual characterizations of entropy- controlled flow matching. Proof. (1)⇒(2) : KKT stationarity in v yields explicit optimizer of strictly concave Hamiltonian, hence maximization condition. Path stationarit...

  74. [86]

    R0T has strictly positive density r0T (x, y)w.r.t. dx dy

  75. [87]

    there exists at least one π∈Π(µ 0, µT ) with KL(π∥R0T )<∞ . Entropy-Controlled Flow Matching 69 Definition 32 (Static Schr¨ odinger problem).The static Schr¨ odinger prob- lem is [26] Sstat(µ0, µT ;R 0T ) := inf π∈Π(µ 0,µT ) KL(π∥R0T ), where KL(π∥R0T ) =    Z log dπ dR0T d...

  76. [88]

    there exists P with P0 =µ 0, PT =µ T ,KL(P∥R)<∞

  77. [89]

    R0, RT are finite

    endpoint relative entropies w.r.t. R0, RT are finite. Theorem 44 (Static–dynamic equivalence).Under Assumption 43, Sdyn(µ0, µT ;R) =S stat(µ0, µT ;R 0T ), and the dynamic minimizer P ⋆ is uniquely characterized by dP ⋆ dR = dπ⋆ dR0T (X0, XT ), where π⋆ is the static minimizer ...

  78. [90]

    u⋆ ∈L 2 loc(dt dµt) with at most linear growth in x

  79. [91]

    admissible (µ, v)satisfy µt =ρ tdx, ρt >0 , ρ∈C 1 t C 2 x, R T 0 I(µ t)dt <∞

  80. [92]

    CE holds: ∂tρ+∇·(ρv) = 0 , ρ|0 =ρ 0, ρ|T =ρ T

  81. [93]

    entropy-rate constraint: ˙H(µt)≥ −λa.e

  82. [94]

    Definition 39 (Primal values).Define ECFM value Vλ := inf (µ,v)∈Aλ 1 2 Z T 0 Z |v−u ⋆|2 dµ dt, and KL-control value Kε := inf w:P w 0 =µ0, Pw T =µT KL(P w∥R⋆)

    KL-control is with reference diffusion dXt =u ⋆ t (Xt)dt+ √ 2ε dWt, X 0 ∼µ 0, and controlled drift correction w such that dXt = (u⋆ t +w t)(Xt)dt+ √ 2ε dWt, X T ∼µ T . Definition 39 (Primal values).Define ECFM value Vλ := inf (µ,v)∈Aλ 1 2 Z T 0 Z |v−u ⋆|2 dµ dt, and KL-control...

  83. [95]

    smooth CE paths (ρ, v)with fixed endpoints

  84. [96]

    Proof.Given (ρ, v), define w=v−u ⋆ +ε∇logρ

    controlled FP paths (ρ, w)solving ∂tρ+∇· ρ(u⋆ +w) =ε∆ρ with same endpoints. Proof.Given (ρ, v), define w=v−u ⋆ +ε∇logρ . Then ρ(u⋆ +w) =ρv+ε∇ρ. Hence ∂tρ+∇·(ρ(u ⋆ +w)) =∂ tρ+∇·(ρv) +ε∆ρ=ε∆ρ, Entropy-Controlled Flow Matching 79 using CE. Conversely, given (ρ, w), define v=u ⋆ +...

  85. [97]

    CE for (ρ, v)⇐ ⇒FP for (ρ, u⋆ +w)

  86. [98]

    Hamiltonian stationarity in v ⇐ ⇒quadratic minimization in w

  87. [99]

    equivalence to Schr¨ odinger bridge under entropy control

    entropy multiplier η enforces active entropy-rate boundary, matching the KL regularization pressure through the score term. Proof.(1) is Lemma 15. (2) follows from strict convex quadratic relationship between v−u ⋆ and w−ε∇logρ . (3) from complementary slackness in C and ident...

  88. [100]

    The reference path law is Brownian-with-drift R⋆: dXt =u ⋆ t (Xt)dt+ √ 2ε dWt, X 0 ∼µ 0, with u⋆ satisfying linear growth and integrability from C.1/D.4. 82 C. Maduabuchi et al

  89. [101]

    The ECFM problem (ECFMλ) admits minimizers (µλ, vλ), and Section D equivalence holds

  90. [102]

    The associated KL/SB problem has unique minimizer P ε,λ (with marginals µ0, µT ), and µε,λ t := (Xt)#P ε,λ has density ρε,λ t ∈C 1 t C 2 x, >0

  91. [103]

    Finite Fisher action: Z T 0 I(µ ε,λ t )dt <∞. Definition 41 (Entropic OT geodesic (Schr¨ odinger interpolation)).Given (µ0, µT ) and reference R⋆, the entropic OT geodesic is (¯µε t )t∈[0,T] := (Xt)# ¯P ε t∈[0,T] , where ¯P ε is the unique dynamic SB minimizer: ¯P ε ∈arg min{K...

  92. [104]

    any ECFM minimizer induces a KL/SB minimizer

  93. [105]

    by uniqueness of dynamic SB minimizer ¯P ε, induced law equals ¯P ε

  94. [106]

    therefore marginals coincide: µλ t = ¯µε t ∀t

  95. [107]

    via flux uniqueness: mλ t =ρ λ t vλ t = ¯ρε t ¯vε t

    velocities coincide dt dµ-a.e. via flux uniqueness: mλ t =ρ λ t vλ t = ¯ρε t ¯vε t

  96. [108]

    action identity follows from AFM = 2εSdyn +C 0. 86 C. Maduabuchi et al. Hence all claims of Theorem 53 hold. Proof.(1) from Proposition 33. (2) uniqueness of SB minimizer (D.1/D.2 strict convexity). (3) equal path laws imply equal one-time marginals. (4) strict con- vexity of ...

  97. [109]

    (LSI route) Along admissible µt, a uniform log-Sobolev constant CLSI exists: H(µt |γ)≤ CLSI 2 I(µ t |γ) for suitable reference γ

  98. [110]

    (displacement-convex route) Entropy functional is displacement convex along relevant geodesics, ensuring lower semicontinuity and convex interpolation bounds. 88 C. Maduabuchi et al. Lemma 19 (W ell-definedness of entropy-rate term).Under Definition 45(3)- (4), Z T 0 Z |∇logρ ...

  99. [111]

    ECFM minimizer exists

  100. [112]

    induced KL/SB minimizer is unique

  101. [113]

    ECFM marginal path (µt) is unique

  102. [114]

    Proof.Existence: direct method using coercivity + CE stability (Lemma 20) + entropy closedness from Theorem 27(3)

    velocity is unique dt dµ-a.e. Proof.Existence: direct method using coercivity + CE stability (Lemma 20) + entropy closedness from Theorem 27(3). SB uniqueness: strict convexity of KL on path space with fixed endpoints. Path uniqueness: from D/E identification of ECFM minimizer...

  103. [115]

    +W 2(µ1 T , µ2 T ), drift discrepancy ∆u :=∥u ⋆ 1 −u ⋆ 2∥L2([0,T]×B R) for radius R capturing 1−η mass uniformly (tail handled below), and budget discrepancy ∆λ :=|λ 1 −λ 2|. Proposition 37 (V alue sensitivity to drift perturbation).Under Assump- tion 60, for same endpoints an...

  104. [116]

    Proof.Use dynamic plan coupling along two optimal current-velocity fields

    +d T (µ1 T , µ2 T ) +∆ u +∆ λ , for some L=L(T, M 1, M2, M3, C⋆, δ). Proof.Use dynamic plan coupling along two optimal current-velocity fields. Dif- ferentiate squared distance along coupled flow: d dt W 2 2 (µ1 t , µ2 t )≤2W 2(µ1 t , µ2 t )∥v 1 t −v 2 t ∥L2(πt), with πt optim...

  105. [117]

    Apply Gr¨ onwall

    +ε∇log(β 1/α1)−ε∇log(β 2/α2) + budget-response term, using endpoint/diffusion potential stability and multiplier bound from Proposi- tion 38. Apply Gr¨ onwall. Remark 20 (Consequence for training robustness).These bounds imply that small perturbations in marginals, drift param...

  106. [118]

    minimizers of Fλ converge to minimizers of F0

  107. [119]

    F.Γ-Convergence asλ→0 F.1

    under matched gauge and vanishing entropic correction, F0 identifies classi- cal OT action. F.Γ-Convergence asλ→0 F.1. Definition ofΓ-convergence and functional settingWe fix the am- bient topological space, define the λ-indexed functionals, and state the precise notion of Γ -...

  108. [120]

    (liminf inequality) for every zλ τ − →z, F0(z)≤lim inf λ↓0 Fλ(zλ)

  109. [121]

    Definition 52 (Limit candidate functional).Define F0(ρ, m) := ( A(ρ, m),(ρ, m)∈ E0, +∞,otherwise, where E0 = n (ρ, m)∈ CE(µ0, µT ) : ˙H(ρt)≥0a.e

    (recovery sequence) there exists zλ τ − →zsuch that F0(z)≥lim sup λ↓0 Fλ(zλ). Definition 52 (Limit candidate functional).Define F0(ρ, m) := ( A(ρ, m),(ρ, m)∈ E0, +∞,otherwise, where E0 = n (ρ, m)∈ CE(µ0, µT ) : ˙H(ρt)≥0a.e. o . Remark 23 (Interpretation of λ↓0 ).As λ→0 , admis...

  110. [122]

    uniform integrability of ˙H(ρn) in L1(0, T),

  111. [123]

    weak convergence ˙H(ρn)⇀ g in L1,

  112. [124]

    Then g≥0 a.e

    identification g= ˙H(ρ) via entropy-chain-rule stability. Then g≥0 a.e. since gn := ˙H(ρn)≥ −λn →0 . Theorem 67 ( Γ -compactness principle for minimizers).Assume:

  113. [125]

    Γ -convergence: Γ-lim λ↓0 Fλ =F 0

  114. [126]

    equicoercivity (Assumption 65)

  115. [127]

    Then every cluster point z⋆ of zλ is a minimizer of F0, and lim λ↓0 minF λ = minF 0

    each Fλ attains a minimizer zλ. Then every cluster point z⋆ of zλ is a minimizer of F0, and lim λ↓0 minF λ = minF 0. If F0 has unique minimizer, zλ →z ⋆ in τ . Proof.Standard fundamental theorem of Γ -convergence: equicoercivity gives compactness of minimizers; liminf + recove...

  116. [128]

    (ρn, mn)∈ CE(µ0, µT ),

  117. [129]

    Step 1: CE and endpoints pass to the limit.By Lemma 22, (ρ, m)∈ CE(µ 0, µT )

    A(ρn, mn)≤C . Step 1: CE and endpoints pass to the limit.By Lemma 22, (ρ, m)∈ CE(µ 0, µT ). Step 2: entropy-rate inequality passes to the limit.By Assumption 66, from (ρn, mn)∈ Eλn , λn ↓0 , and τ -convergence, we infer (ρ, m)∈ E0,i.e. ˙H(ρt)≥0 a.e. Step 3: l.s.c. of action.By...

  118. [130]

    Γ -convergence from Theorem 70,

  119. [131]

    equicoercivity (Assumption 65),

  120. [132]

    Then lim λ↓0 Vλ =V 0

    Mλ ̸=∅ for small λ, and M0 ̸=∅ . Then lim λ↓0 Vλ =V 0. Proof.By Γ -convergence + equicoercivity (fundamental theorem), minima con- verge: lim λ↓0 minF λ = minF 0. By Definition 53, this is exactly Vλ →V 0. Theorem 72 (Compactness and convergence of minimizers).Let λn ↓0 , and ...

  121. [133]

    (zn) is relatively compact in (Y, τ)

  122. [134]

    Fλn (zn)→ F0(z⋆) =V 0 along any convergent subsequence

    every cluster point z⋆ = (ρ⋆, m⋆) belongs to M0; 3. Fλn (zn)→ F0(z⋆) =V 0 along any convergent subsequence. Proof.Equicoercivity yields compactness of minimizing sequence (zn). Γ -compactness theorem implies cluster points are minimizers of F0. Value convergence follows from T...

  123. [135]

    either ε=ε(λ)↓0 , or

  124. [136]

    Remark 28 (Why Assumption 74 is natural).In entropic OT, classical OT is re- covered as diffusion/entropy regularization vanishes

    ε >0 fixed but ε H(µλ T )− H(µλ 0 ) +ε 2 Z T 0 I(µ λ t )dt→0. Remark 28 (Why Assumption 74 is natural).In entropic OT, classical OT is re- covered as diffusion/entropy regularization vanishes. Assumption 74 is the pre- cise dynamic counterpart: KL-control energy dominates whil...

  125. [137]

    Γ -convergence Fλ → F0 as λ↓0

  126. [138]

    convergence of minima and minimizers

  127. [139]

    identification of F0 with classical BB action under vanishing entropic cor- rection

  128. [140]

    recovery of classical OT/Wasserstein geodesics from entropy-controlled flow matching. Transition to Section G.With the asymptotic limit resolved, Section G proves the mode-coverage theorem: entropy-rate control prevents singular concentra- tion and supplies quantitative lower-...

  129. [141]

    each Ak has Lipschitz boundary and finite diameter

  130. [142]

    there exist disjoint open neighborhoods Uk ⊃A k with separation dist(Ui, Uj)≥ ∆sep >0 for i̸=j

  131. [143]

    Assumption 78 (Entropy-controlled admissibility)Trajectory (µt, vt) is ECFM-admissible: ∂tµt +∇·(µ tvt) = 0, ˙H(µt)≥ −λa.e

    along admissible trajectories, µt =ρ tdx with ρt >0 a.e., H(µt)∈AC , and finite Fisher action. Assumption 78 (Entropy-controlled admissibility)Trajectory (µt, vt) is ECFM-admissible: ∂tµt +∇·(µ tvt) = 0, ˙H(µt)≥ −λa.e. for some λ≥0 , with fixed endpoints (µ0, µT ). Definition ...

  132. [144]

    Let ρ, ρ′ be corresponding optimal densities

    +W 2(µT , µ′ T ) +∥u ⋆ −u ⋆′∥U +|λ−λ ′|. Let ρ, ρ′ be corresponding optimal densities. Theorem 87 (Stability of density floors under perturbations).Assume hypotheses of E.4 stability and local regularity hold uniformly for θ, θ′. Then for each core Kk ⋐A k, there exists Ck >0 ...

  133. [145]

    set-mass floor: µ′ t(Ak)≥(c k − ˜Ck∆θ)πk,

  134. [146]

    Thus mode coverage is perturbation-robust

    density floor on cores: ρ′ t|Kk ≥ρ k −C k∆θ. Thus mode coverage is perturbation-robust. Proof.Set-mass floor is E.4/G.3 stability; density floor is Theorem 87. Remark 35 (Interpretation for generative vision).For semantic modes Ak (class/attribute regions in representation spa...

  135. [147]

    finite uniform action/Fisher/moment bounds: sup Θ∈N Z T 0 Z |vΘ t |2 dµΘ t dt+ Z T 0 I(µ Θ t )dt+ sup t m2(µΘ t ) ! ≤M

  136. [148]

    unique minimizers and strict-convexity regime (E.4)

  137. [149]

    local FP coefficient bounds ensuring Harnack/regularity constants are uni- form. Definition 64 (T rajectory distance).For two trajectories µ1, µ2, define d∞(µ1, µ2) := sup t∈[0,T] W2(µ1 t , µ2 t ), and energy-weighted distance d2(µ1, µ2) := Z T 0 W2(µ1 t , µ2 t )2 dt !1/2 . Le...

  138. [150]

    L∈L 1(0, T)controlling one-sided Lipschitz growth of optimal velocity fields

  139. [151]

    κ≥0 controlling endpoint-conditioning amplification from Schr¨ odinger sys- tem

  140. [152]

    Lemma 34 (F orward perturbation inequality).Let D(t) :=W 2(µt,˜µt)

    finite uniform moment/Fisher/action bounds. Lemma 34 (F orward perturbation inequality).Let D(t) :=W 2(µt,˜µt). Then for a.e. t, ˙D(t)≤L(t)D(t) +κ ∆ init. Hence D(t)≤e R t 0 L(s)ds ∆init +κ Z t 0 e R t s L(r)dr ds ∆init. Proof.Use H.1 differential inequality: ˙D(t)≤ ∥vt −˜vt∥L...

  141. [153]

    Total perturbation size: ∆tot :=∆ par +∆ noi

    +W 2(µT , µ′ T ) +∥u ⋆ −u ⋆′∥U +|λ−λ ′|, and noise magnitude ∆noi :=∥ξ∥ N = Z T 0 Z |ξt|2 d¯µtdt !1/2 . Total perturbation size: ∆tot :=∆ par +∆ noi. Entropy-Controlled Flow Matching 127 Assumption 97 (Unified envelope)All instances in a neighborhood of Θ satisfy:

  142. [154]

    uniform regularity/coercivity and uniqueness assumptions from E.3–E.4

  143. [155]

    trajectory Lipschitz constants from H.1 bounded by Lpar

  144. [156]

    velocity-noise response constants from H.2 bounded by Lnoi

  145. [157]

    Lemma 35 (Two-step decomposition).For any t∈[0, T] , W2(˜µt, µ⋆ t )≤W 2(˜µt,¯µt) +W 2(¯µt, µ⋆ t )

    local parabolic/Harnack constants from G.4 uniformly bounded. Lemma 35 (Two-step decomposition).For any t∈[0, T] , W2(˜µt, µ⋆ t )≤W 2(˜µt,¯µt) +W 2(¯µt, µ⋆ t ). Consequently, sup t W2(˜µt, µ⋆ t )≤sup t W2(˜µt,¯µt) + sup t W2(¯µt, µ⋆ t ). Proof.Triangle inequality in (P2, W2), ...

  146. [158]

    match endpoint marginals (or approximate them arbitrarily well),

  147. [159]

    keep FM regression objective finite (even small),

  148. [160]

    This proves anti-collapse guarantees in Sections G–H are not inherited by clas- sical unconstrained FM

    collapse intermediate-time mass onto low-dimensional/single-mode regions. This proves anti-collapse guarantees in Sections G–H are not inherited by clas- sical unconstrained FM. Definition 65 (Unconstrained FM functional).Given supervision field v†(x, t), define LFM(v) := Z T ...

  149. [161]

    µ0, µT are matched exactly

  150. [162]

    FM objective is arbitrarily close to a non-collapsing reference optimum

  151. [163]

    Therefore unconstrained FM does not guarantee mode preservation/coverage

    intermediate measures collapse to near-singular unimodal concentration. Therefore unconstrained FM does not guarantee mode preservation/coverage. Proof.Combine Lemma 36, Proposition 53, and Lemma 37. Corollary 49 (Entropy-rate violation along collapsing paths).For the collapsi...

  152. [164]

    exact endpoint matching: µ(n) 0 =µ 0, µ(n) T =µ T

  153. [165]

    t∈(0, T)

    interior-time singular limit: µ(n) t ⇀ δ0 for a.e. t∈(0, T)

  154. [166]

    failure theorem package

    FM risk remains asymptotically near reference: LFM(v(n))≤infL FM +o(1) for suitable teacher/reference v† from I.1. Hence unconstrained FM admits asymptotically near-optimal yet singular/collapsing trajectories. Proof.(1)–(2): Lemma 38, Proposition 55. (3): same short-window ar...

  155. [167]

    CE/FP regularity from Sections C–E,

  156. [168]

    entropy-rate lower bound ˙H(µt)≥ −λa.e.,

  157. [169]

    Then all Section G/H guarantees apply verbatim:

    endpoint and coercivity assumptions of Sections F–H. Then all Section G/H guarantees apply verbatim:

  158. [170]

    quantitative mode-mass floors,

  159. [171]

    modal-core density floors,

  160. [172]

    Proof.Sections G/H depend on trajectory-level properties (CE/regularity + entropy-rate budget), not on specific parameterization of vθ

    perturbation robustness to endpoint/drift/noise/initialization shifts. Proof.Sections G/H depend on trajectory-level properties (CE/regularity + entropy-rate budget), not on specific parameterization of vθ. Hence any model class satisfying these assumptions inherits the same c...

  161. [173]

    ∇θLAL is L-Lipschitz on bounded iterates

  162. [174]

    unbiased stochastic gradients with bounded variance

  163. [175]

    step sizes satisfy Robbins–Monro: P k αk =∞, P k α2 k <∞ , P k βk = ∞, P k β2 k <∞

  164. [176]

    Then every limit point of iterates generated by Algorithm 1 is a first-order KKT point of Definition 75 (in expectation / a.s

    Slater-type feasibility holds for the discrete constraints. Then every limit point of iterates generated by Algorithm 1 is a first-order KKT point of Definition 75 (in expectation / a.s. subsequential sense). Proof.Apply stochastic primal–dual convergence for nonconvex constra...

  165. [177]

    discrete entropy-budget constraints hold

  166. [178]

    certified modal floors hold at sampled times; Entropy-Controlled Flow Matching 147

  167. [179]

    Proof.(1) from Proposition 66

    by Section H stability, small deployment perturbations preserve floors up to O(∆tot). Proof.(1) from Proposition 66. (2) from Theorem 111. (3) from Theorem 98. Remark 46 (Computational cost).Additional overhead is dominated by diver- gence/score diagnostics: –FM/rectified flow...

  168. [180]

    time grid {tn}N n=0 (uniform or curvature-adaptive)

  169. [181]

    confidence level 1−α

  170. [182]

    entropy budget range [λmin, λmax]

  171. [183]

    modal sets {Ak} and optional cores {Kk ⋐A k}

  172. [184]

    minibatch sizes Bn, Hutchinson probes Rn

  173. [185]

    Proposition 67 (Time-grid adequacy criterion).Let ˙H be locally Lipschitz with modulus LH on intervals

    robustness radius target ∆max tot (deployment shift envelope). Proposition 67 (Time-grid adequacy criterion).Let ˙H be locally Lipschitz with modulus LH on intervals. If grid satisfies max n ∆tn ≤ ϵH LH , then discretization error in peak negative entropy-rate obeys ess sup t ...

  174. [186]

    training follows Algorithm 1 with robust residuals

  175. [187]

    diagnostics satisfy J.2 concentration conditions

  176. [188]

    grid adequacy/refinement condition in Proposition 67

  177. [189]

    Then with probability 1−α , the deployed model enjoys:

    stability envelope constants are estimated conservatively. Then with probability 1−α , the deployed model enjoys:

  178. [190]

    entropy-budget feasibility up to discretization slack ϵH

  179. [191]

    certified modal floors at sampled times and interpolated times under refine- ment

  180. [192]

    Proof.(1) from Proposition 68

    perturbation-robust mode/density floors for all shifts ∆tot ≤∆ max tot : mdeploy k ≥m cert k − bCM ∆max tot , ρdeploy k ≥ρ cert k − bCρ∆max tot . Proof.(1) from Proposition 68. (2) from Theorem 111 plus refinement/interpolation control. (3) from unified perturbation theorem H....

  181. [275]

    Springer International Publishing, Cham (2014).https://doi.org/10.1007/ 978-3-319-00227-9_5,https://doi.org/10.1007/978-3-319-00227-9_5

    2014 doi

  182. [492]

    California Press, Berkeley-Los Angeles, Calif

    Univ. California Press, Berkeley-Los Angeles, Calif. (1951)

    1951

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.