Pith. sign in

REVIEW 2 major objections 2 minor 13 references

The paper claims that when a low-dimensional summary R(X) of the covariates is sufficient for a response Y, the optimal conditional transport map from a reference response to the conditional law of Y given X factors through R—and so does th

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 14:07 UTC pith:XD7FSFEM

load-bearing objection The factorization theorem is a real addition to SDR–OT; the Hilbert consistency theorems, however, are gated by an interpolation-compression assumption that is false for the paper's own FF4 design, and the abstract overpromises what the theorems deliver. the 2 major comments →

arxiv 2607.18861 v1 pith:XD7FSFEM submitted 2026-07-21 math.ST stat.MEstat.TH

Learning sufficient low-dimensional structures through conditional optimal transport

Kaiqiang Alan Zeng , Efstathia Bura This is my paper
classification math.ST stat.MEstat.TH MSC 49Q2262G0862G2062R10
keywords sufficient dimension reductionconditional optimal transporttriangular mapsflow matchingcentral subspaceHilbert-space statisticsGaussian continuity equationfunctional data
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tries to show that the classical sufficiency condition in dimension reduction—Y independent of X given R(X)—is not merely a probabilistic statement but is mirrored in the geometry of optimal transport: the optimal triangular transport map and its associated velocity depend on the covariate only through the low-dimensional reduction R(x). It establishes this factorization on separable Hilbert spaces, and uses it to build a flow-matching objective that learns the reduction by making the transported displacement predictable from (R(X), current response). For linear reductions, it proves consistency: under regularity conditions, minimizing the empirical objective recovers the true central subspace. The proofs avoid global injectivity of the terminal transport map, and work even when the response is infinite-dimensional. The functional-consistency result, however, rests on an interpolation-compression assumption that the paper concedes is not verified for one of its own simulation designs (FF4), and nonlinear consistency is explicitly left open.

Core claim

The central claim is that sufficiency descends from the sigma-algebra level to the transport geometry itself. Under Assumptions 4.1–4.3, the response component of the optimal triangular conditional optimal transport map satisfies T*_Y(x,y) = G*_Y(R(x), y) for µ0-almost every (x,y), and the Monge-induced current-state velocity can be chosen so that v*_t(x,y) = (0, u*_Y(t, R(x), y)) for almost every t and µ*_t-almost every point. This means the whole conditional law of Y given X is generated from a fixed reference law by a map that reads the covariate only through its sufficient low-dimensional form. The paper also proves that, for quadratic cost, the displacement interpolation admits a Borel

What carries the argument

The central object is the conditional optimal transport (COT) problem from a product source η⊗ρ to the joint law of (X,Y), solved by a triangular map (x,y) ↦ (x, T_Y(x,y)) that is, for each x, the optimal transport map from the reference response law ρ to the conditional law P(Y|X=x). The factorization theorem uses uniqueness of these slicewise optimal maps together with a measurability/Doob–Dynkin-type lemma: because the conditional law is, up to null sets, a function of R(x), the unique optimal map inherits the same dependence. The velocity factorization uses the displacement interpolation and a Borel current-state velocity constructed on the full-measure images of the interpolation maps,

Load-bearing premise

The load-bearing premise is that, along the truncated interpolation, each conditional Monge interpolation admits a density bounded in L^r with respect to a fixed Gaussian reference measure (Assumption 6.4); if this 'interpolation compression' fails, the uniqueness argument that identifies the central subspace from zero loss collapses, and the paper itself concedes it is not verified for its FF4 design.

What would settle it

Run the linear SDR-COT estimator on a model where the conditional law of Y given X is a Gaussian whose covariance is rescaled by a function of the true index in infinitely many directions (the paper's FF4 design), and check whether the estimated subspace converges to the truth as the sample size grows. The paper's theory does not cover this case because the interpolation-compression assumption is not established, so convergence would strengthen the result, while divergence or a systematically biased subspace would confirm the current scope limitation.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the factorization theorem holds, then sufficiency can be tested through transport: a representation R that fails to make the optimal COT map factor through R cannot be sufficient.
  • For linear reductions, the population objective is exhaustive—any representation containing the central σ-algebra attains the full-information value—and, under the regularity and uniform-convergence conditions, empirical minimization recovers the central subspace consistently.
  • The existence of a Borel current-state velocity without global injectivity extends the dynamic theory to infinite-dimensional response spaces, covering functional responses such as curves on a bounded interval.
  • The method captures distributional information beyond the conditional mean (heteroscedasticity, tails, multimodality), which classical moment-based SDR can miss.
  • The functional consistency theory requires an interpolation-compression condition (Assumption 6.4) that the paper does not verify for its own covariance-rescaled design FF4; the FF4 simulation is presented as empirical evidence beyond the stated theorem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The factorization suggests a direct diagnostic: fit an unrestricted COT map and measure how much of the dependence on x remains after conditioning on R(x); a data-driven test of sufficiency could be built by thresholding this residual dependence.
  • Because the population objective is constant across all representations containing the central σ-algebra, the learned representation is generically non-unique; parsimony must come from penalties, output-dimension constraints, or a topology on σ-algebras, and a nonlinear consistency theory will need a minimality-inducing mechanism.
  • The interpolation-compression assumption (6.4) is the main analytical bottleneck for Hilbert-valued responses; if one could prove compression for singular Gaussian translations (as in the FF4 design), the consistency theorem would extend to covariance-rescaled models, which are a natural stress-test for the method.
  • The paper's diagonal-tuning argument for the relaxation parameter gives no explicit rate; a quantitative stability bound for conditional optimal couplings would turn the qualitative consistency result into a convergence rate and is a concrete open problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper proposes SDR-COT, a sufficient-dimension-reduction framework built from conditional optimal transport. It proves that, under a finite-generator sufficiency assumption, the response component of the optimal triangular COT map and a suitable Borel current-state velocity representative factor through the sufficient reduction (Propositions 4.3–4.4, Theorem 3.1). It introduces a conditional-flow-matching objective and analyses its population exhaustiveness (Proposition 5.1). For linear reductions it develops consistency theory: Euclidean responses via slicewise Caffarelli contraction, Hilbert-valued responses via Gaussian–Sobolev admissibility and interpolation compression. The theory is complemented by numerical experiments, including function-on-function designs and a Bikeshare data analysis.

Significance. The factorisation theorems are conceptually valuable and technically demanding, and the construction of a Borel current-state velocity without global injectivity of the terminal map is a genuine improvement over earlier Hilbert-space dynamic COT results. The proofs are generally careful about regular conditional measures, measurable selections, and null sets; the central structural claims appear sound. The statistical theory is honest about its high-level assumptions, but the functional branch is narrower than the narrative suggests, and the paper's own FF4 example lies outside the stated functional consistency theorem.

major comments (2)
  1. [§6.4, Assumption 6.4, Example 6.1, Table 4] Assumption 6.4 is not merely unverified for FF4; it is false. In FF4, conditionally on X, the response law is N(m_x, (8<beta2,X>)^2 C_rho), so the quadratic COT map is T_x(y)=m_x+8<beta2,X> y and the interpolation is mu^x_t = N(t m_x, (1-t+8t<beta2,X>)^2 C_rho). For a.e. t, the scale factor differs from 1, and Feldman–Hajek (Prop. B.3) gives mu^x_t perpendicular to rho, so no density f^x_t exists. Thus Lemma 6.2 and Theorems 6.2–6.3 do not apply to FF4. The text's concession that 'current tools do not verify' understates a genuine falsity. Since FF4 is reported as a main empirical success, the abstract and introduction should be revised to state explicitly that the functional consistency theory excludes heteroscedastic covariance rescaling, or the uniqueness argument must be extended to cover such singular Gaussian mixtures.
  2. [§6, Eqs. (6.21), (6.32), (6.35)] The consistency theorems are conditional on a high-level uniform-convergence assumption on the empirical criterion. Proposition 6.1 establishes W2-consistency of the relaxed empirical coupling, but, as the paper itself notes, coupling convergence does not imply uniform convergence of the fitted criterion. The conditions (6.21)/(6.32)/(6.35) are imposed rather than derived, and no concrete velocity class is given for which they are verified. This is acceptable if the theorems are explicitly framed as sufficient conditions, but the abstract's phrase 'we prove consistency' overstates what is established. Please qualify the consistency claims and, if possible, identify a class of velocity models and an empirical-process bound under which (6.21)/(6.35) can be checked.
minor comments (2)
  1. [§6.3] The theorem is attributed to 'Caffarelli's contraction theorem (Alexander V Kolesnikov 2011)'. Please cite Caffarelli's original result as the primary source, with Kolesnikov's paper as a secondary reference.
  2. [§5.2/§7.2] Algorithm 3 specifies a Gaussian process source law, while the Bikeshare experiment uses an empirical response covariance with 10% shrinkage. This discrepancy should be noted explicitly so that the algorithm and data analysis are understood as using different reference laws.

Circularity Check

0 steps flagged

No significant circularity: the factorization and consistency claims are derived from stated sufficiency/regularity inputs rather than assumed as outputs.

full rationale

Walking the derivation chain, Proposition 4.3 takes Assumptions 4.1–4.3 as genuine inputs: conditional independence Y1⊥⊥X|G, a finite generator R of G, and COT source regularity. Lemma 4.2 derives a Borel kernel κ with PY1|X(·|X)=κ(R(X)); the slicewise Monge uniqueness theorem (Hosseini–Hsu–Taghvaei, Prop. 2.2) then forces the optimal triangular map to factor through R. The factorization is a theorem, not an assumption smuggled in as a conclusion. Proposition 4.4 transfers the factorization to the current-state velocity by constructing the reduced COT problem and pulling back the Borel velocity representative; this is a derived equality of representatives, not a redefinition of the predicted object. In the Euclidean linear theory, Lemma 6.1 does not fit the subspace and then call it predicted: zero population loss plus slicewise Lipschitz regularity is used to reconstruct Y1 as a Borel function of (B*X,Y0), which proves conditional independence and hence identifies the central subspace. In the Hilbert case, Lemma 6.2 invokes the Ambrosio–Figalli uniqueness criterion as an external theorem, with Assumption 6.4 as an explicit measure-side hypothesis. The paper openly states that current tools do not verify Assumption 6.4 for the FF4 design, and later calls the FF4 result empirical evidence beyond the scope of the functional theorem. Whether Assumption 6.4 actually fails for FF4 is a correctness/scope concern, not a circularity: the theorem is conditional on that assumption and the paper does not claim the assumption follows from earlier structure. No load-bearing self-citation appears: the only author self-citation (Fertl and Bura 2022) is background context in the introduction. No fitted parameter is renamed as a prediction, and no known result is merely relabelled as a new derivation. The core factorization and identification results are therefore self-contained given their stated assumptions.

Axiom & Free-Parameter Ledger

6 free parameters · 11 axioms · 0 invented entities

The factorization programme rests on the SDR conditional-independence assumption, finite-generator existence, and external uniqueness theorems for conditional Monge maps; the consistency theory adds substantially stronger, partially unverifiable regularity: slicewise log-concavity, correct specification, imposed uniform convergence, Gaussian–Sobolev admissibility, and L^r compression. The paper is transparent about most of these; the fragility is that the advertised 'consistency' is conditioned on an empirical-process condition that is imposed rather than proved, and on a compression condition that fails for one of the paper's own simulated designs (FF4). No new physical or mathematical entities are postulated.

free parameters (6)
  • structural dimension d = assumed known and set equal to true d in all experiments
    Identification in Section 6 fixes the output dimension at d; misspecification breaks the zero-loss ⇒ central-subspace argument. In practice d is unknown and must be selected; the paper treats it as given.
  • anisotropic cost weight ε = not specified; 'small' in practice
    The empirical coupling uses cost ||x−x′||² + ε||y−y′||² (Section 5.2). Prop 6.1 only guarantees existence of a non-constructive diagonal sequence ε_n ↓ 0 with no rate; the practical value is hand-set.
  • time truncation τ = not specified in the simulations
    All losses integrate over [0, 1−τ] (Section 5.1). τ enters the theoretical Lipschitz constants ℓ0(s) = (1+L_T(s))/τ (Prop 6.2) and the Gaussian uniqueness threshold c > rT_τ.
  • reference response law ρ = Gaussian N(mρ, Σρ) (Euclidean); GP law; Law(σW) in FF simulations; 10%-shrunk empirical covariance in Bikeshare
    ρ is user-chosen. The factorization holds for any regular ρ, but the Euclidean Caffarelli argument requires Gaussian ρ, and the functional compression condition (Assumption 6.4) and admissibility (Definition 6.1) are ρ-specific and unverified for some designs (FF4).
  • basis truncation K for functional covariates = K = 8, 20, 24 in simulations; 8 FPCA scores in Bikeshare
    Functional covariates are projected to K coefficients (Section 5.2); the projection error is acknowledged but not part of the consistency statements.
  • velocity/reduction network architectures = SiLU MLPs; FNO width 20, 10 Fourier modes, 3 spectral layers; Adam 800 steps, lr 0.002 (Bikeshare)
    Implementation hyperparameters chosen by hand; the uniformity conditions (6.21)/(6.35) would require complexity control for these classes, which the paper does not quantify.
axioms (11)
  • domain assumption Assumption 4.1: Y1 ⊥⊥ X | G (SDR conditional independence)
    The object of study; enters Lemma 4.2 and Props 4.3–4.4. If no sufficient reduction exists, the factorization target is vacuous.
  • domain assumption Assumption 4.3: the SDR σ-algebra admits a bounded finite-dimensional Borel generator R
    Needed for the Doob–Dynkin step (Lemma 4.2, Prop 4.3). Weak in practice since any countably generated σ-algebra over a standard Borel space admits a real Borel generator (the paper notes this).
  • domain assumption Source regularity: ρ ∈ P²_r (zero mass on Gaussian null sets), η ∈ P², µ1 ∈ P^{2,η}
    Uniqueness of the conditional Monge map (Prop 2.2, from Hosseini–Hsu–Taghvaei) requires source regularity; the whole programme rests on this external theorem.
  • standard math Prop 2.2 (Hosseini, Hsu, Taghvaei 2025, Prop 3.8): existence and uniqueness of the triangular conditional Monge map in separable Hilbert spaces
    Used in Thm 3.1, Prop 4.3, and Prop 6.1. External result, not re-derived.
  • standard math Standard measure/OT facts: cyclic monotonicity of optimal supports, measurable selection (Villani Cor 5.22), Lusin–Souslin, Doob–Dynkin, Borel support graphs of kernels (Lemma A.1)
    Backbone of Thm 3.1 and the measurable-version arguments; standard but non-elementary.
  • domain assumption Assumption 6.2: slicewise strongly log-concave conditional laws and Gaussian source; Caffarelli contraction (Kolesnikov 2011, Thm 2.2)
    Gives the Lipschitz velocity bound (Prop 6.2) needed for Euclidean zero-loss identification. Excludes multimodal and heavy-tailed conditional laws — the paper says so explicitly.
  • ad hoc to paper Assumption 6.3: correct specification (true reduced velocity in the fitted class), slicewise Lipschitz velocity fields, lower semicontinuous population criterion
    Theorem 6.1 needs V = u^{θ0}(t, B*0 X, Yt) and the Lipschitz envelope; not derivable from the transport structure.
  • ad hoc to paper Uniform convergence of the empirical criterion (6.21)/(6.32)/(6.35)
    Imposed, not proved; the paper explicitly notes W2 coupling convergence does not imply it. Without it, Theorems 6.1–6.3 do not deliver consistency.
  • ad hoc to paper Assumption 6.4: L^r Gaussian interpolation compression (µ_x_t = f_x_t ρ, f_x ∈ L^∞((0, T_τ); L^r(ρ)))
    Load-bearing for functional identification (Lemma 6.2); ρ-dependent; unverified for the paper's own FF4 design.
  • domain assumption Definition 6.1: Gaussian–Sobolev admissibility of the fitted velocity + uniqueness of the Gaussian continuity equation (Ambrosio–Figalli Thm 3.1, restated as Prop B.6)
    The functional theory replaces Caffarelli with Malliavin-divergence control and one-sided exponential integrability; a strong, hard-to-check condition.
  • domain assumption Assumption 6.1: existence of a unique d-dimensional linear central subspace
    Standard SDR identifiability; used to convert containment into equality in Lemmas 6.1–6.2 and Theorems 6.1–6.3.

pith-pipeline@v1.3.0-alltime-deepseek · 53809 in / 28042 out tokens · 228956 ms · 2026-08-01T14:07:54.437419+00:00 · methodology

0 comments
read the original abstract

Sufficient dimension reduction seeks a low-dimensional covariate representation that preserves the conditional law of a response. We introduce SDR-COT, which represents that law by conditional optimal transport from an independent reference response. On separable Hilbert spaces, sufficiency forces the response component of the optimal triangular map to factor through the reduction. For quadratic cost, the induced interpolation has a Borel current-state velocity on every truncated time interval, without global injectivity of the terminal map, and this velocity has the same factorisation. These results motivate a conditional-flow-matching criterion. For linear reductions, we prove consistency using a suitably tuned relaxed empirical coupling. Euclidean responses are treated through slicewise Caffarelli bounds; Hilbert-valued responses are treated through Gaussian Sobolev regularity, interpolation compression and uniqueness of a Gaussian continuity equation. Numerical studies with Euclidean and functional data show competitive performance, especially when sufficient information is not solely contained in the conditional mean.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

13 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    Since Φ(P(Y)) is Borel and W is G-measurable, the map fW is G-measurable

    Choose some fixedν 0 ∈ P(Y), and define fW(ω) := ( W(ω), ω∈D, Φ(ν0),otherwise. Since Φ(P(Y)) is Borel and W is G-measurable, the map fW is G-measurable. Moreover, fW (ω) ∈ Φ(P(Y)), for allω∈Ω. Now define eZ(ω) := Φ−1(fW(ω)), then the map eZ: Ω→ P(Y) isG-measurable. Finally, forω /∈N, fW(ω) =W(ω) = Φ(Z(ω)) =⇒ eZ(ω) = Φ −1(Φ(Z(ω))) =Z(ω). Thus eZ=ZP-a.s., s...

    1995

  2. [2]

    cylindrical

    ii)m 1 −m 2 ∈ H1. iii) Let L := C −1/2 1 C 1/2 2 be understood as its bounded extension under condition i), the operator LL∗ −I belongs toS 2(Y). 18 The Feldman–H´ ajek theorem implies that, if Y is infinite-dimensional, a > 0, and a̸ = 1, then N(0, a2C1) andN(0, C1) are mutually singular. Specifically,LL ∗ −I= (a 2 −1)I, and ∥(a2 −1)I∥ 2 S2(Y) = ∞X j=1 ∥...

    2014

  3. [10]

    Since x7→K (x, Un) is measurable for every open Un, each set {x : K(x, Un) = 0 } is measurable

    Hence Graph(supp(K)) c = [ n∈N ({x:K(x, U n) = 0} ×Un). Since x7→K (x, Un) is measurable for every open Un, each set {x : K(x, Un) = 0 } is measurable. Therefore the right-hand side is Borel, so the graph itself is Borel. Lemma A.2.Let Y be Polish, and let Z : Ω → P(Y)be a random probability measure. Let G ⊆ Fbe a sub-σ-algebra. Suppose that there exists ...

    2012

  4. [12]

    stochastic dot product

    16The kernel is defined as ker(C ρ) :={y∈ Y:C ρy= 0}. 48 Definition B.3(Abstract Wiener space).An abstract Wiener space is a triple( E,H, µ)in which E is a separable Banach space, µ is a centred nondegenerate Gaussian measure on E, and H is its Cameron–Martin space, continuously and densely embedded in E. Thus( Y,H ρ, ρ)is an abstract Wiener space. In the...

    2014

  5. [338]

    Springer, 2009.isbn: 978-3-540-71050-9.doi:10.1007/978-3-540-71050-9

    Grundlehren der mathematischen Wis- senschaften. Springer, 2009.isbn: 978-3-540-71050-9.doi:10.1007/978-3-540-71050-9. Wald, Christian and Gabriele Steidl. ‘Flow Matching: Markov kernels, stochastic processes and transport plans’. In:Variational and Information Flows in Machine Learning and Optimal Transport(2025), pp. 185–254. Xia, Yingcun and Weiqiang H...

    arXiv 2009

  6. [904]

    New York: Chapman and Hall/CRC, 2018.isbn: 9781315119427.doi:10.1201/9781315119427

    Li, Bing.Sufficient Dimension Reduction: Methods and Applications with R. New York: Chapman and Hall/CRC, 2018.isbn: 9781315119427.doi:10.1201/9781315119427. Li, Bing, Andreas Artemiou and Lexin Li. ‘Principal support vector machines for linear and nonlinear sufficient dimension reduction’. In: (2011). Li, Bing and Jun Song. ‘Dimension reduction for funct...

    Pith/arXiv arXiv 2018

  7. [1999]

    ‘Measuring and testing dependence by correlation of distances’

    Sz´ ekely, G´ abor J, Maria L Rizzo and Nail K Bakirov. ‘Measuring and testing dependence by correlation of distances’. In: (2007). Tong, Alexander et al. ‘Improving and generalizing flow-based generative models with minibatch optimal transport’. In:arXiv preprint arXiv:2302.00482(2023). Varadarajan, V. S. ‘On the Convergence of Sample Probability Distrib...

    Pith/arXiv arXiv 2007

  8. [2005]

    ‘Well-Posedness of Lagrangian Flows and Continuity Equations in Metric Measure Spaces’

    Ambrosio, Luigi and Dario Trevisan. ‘Well-Posedness of Lagrangian Flows and Continuity Equations in Metric Measure Spaces’. In:Analysis & PDE7.5 (2014), pp. 1179–1234.doi: 10.2140/apde.2014. 7.1179. Baptista, Ricardo et al. ‘Conditional sampling with monotone GANs: from generative models to likelihood- free inference’. In:arXiv preprint arXiv:2006.06755(2...

    Pith/arXiv arXiv 2014

  9. [2007]

    Invariant Feature Extraction Through Conditional Independence and the Optimal Transport Barycenter Problem: the Gaussian case

    Bogachev, Vladimir I. and Alexander V. Kolesnikov. ‘Sobolev Regularity for the Monge–Amp` ere Equation in the Wiener Space’. In:Kyoto Journal of Mathematics53.4 (2013), pp. 713–738.doi: 10.1215/ 21562261-2366078. Bounos, Ian et al.Invariant Feature Extraction Through Conditional Independence and the Optimal Transport Barycenter Problem: the Gaussian case....

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2512.20914 2013

  10. [2012]

    ‘Dynamic conditional optimal transport through simulation-free flows’

    Kerrigan, Gavin, Giosue Migliorini and Padhraic Smyth. ‘Dynamic conditional optimal transport through simulation-free flows’. In:Advances in Neural Information Processing Systems37 (2024), pp. 93602– 93642. — ‘Functional Flow Matching’. In:arXiv preprint arXiv:2305.17209(2023).doi: 10.48550/arXiv. 2305.17209.url:https://arxiv.org/abs/2305.17209. Kolesniko...

    Pith/arXiv arXiv doi:10.48550/arxiv.2305.17209 2024

  11. [2015]

    ‘An RKHS formulation of the inverse regression dimension-reduction problem’

    Hsing, Tailen and Haobo Ren. ‘An RKHS formulation of the inverse regression dimension-reduction problem’. In: (2009). Huang, Jian et al. ‘Deep Dimension Reduction for Supervised Representation Learning’. In:arXiv preprint arXiv:2006.05865(2020).url:https://arxiv.org/abs/2006.05865. Kallenberg, Olav.Foundations of modern probability. Springer,

    Pith/arXiv arXiv 2009

  12. [2021]

    ‘On Flows Associated to Sobolev Vector Fields in Wiener Spaces: An Approach ` a la DiPerna–Lions’

    Ambrosio, Luigi and Alessio Figalli. ‘On Flows Associated to Sobolev Vector Fields in Wiener Spaces: An Approach ` a la DiPerna–Lions’. In:Journal of Functional Analysis256.1 (2009), pp. 179–214.doi: 10.1016/j.jfa.2008.05.007. Ambrosio, Luigi, Nicola Gigli and Giuseppe Savar´ e.Gradient flows: in metric spaces and in the space of probability measures. Springer,

    work page doi:10.1016/j.jfa.2008.05.007 2009

  13. [2025]

    Lisini, Stefano

    arXiv: 2507.03669 [math.OC].url:https://arxiv.org/abs/2507.03669. Lisini, Stefano. ‘Characterization of absolutely continuous curves in Wasserstein spaces’. In:Calculus of variations and partial differential equations28.1 (2007), pp. 85–120. Shao, Jun.Mathematical statistics. Springer,

    Pith/arXiv arXiv 2007