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REVIEW 3 major objections 4 minor 26 references

Forward Reverse Kernel Regression for the Schr\"{o}dinger bridge problem

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that the Schr\"odinger bridge between two endpoint distributions can be computed by a forward–reverse kernel regression iteration that needs only samples of the reference process, converges after about $\log N$…

desk verdict Promising kernel Monte Carlo method for Schrödinger bridge potentials with a solid lower bound, but the central upper-bound rate is not proven as written due to a conditioning gap. read the letter →

arxiv 2507.00640 v1 pith:NGFPFH55 submitted 2025-07-01 stat.ML cs.LGcs.NAmath.NA

Denis Belomestny , John. Schoenmakers This is my paper · ORCID
classification stat.MLcs.LGcs.NAmath.NA MSC 90C4065C0562G08
keywords SchrödingerbridgeproblementropicoptimaltransportMonteCarlokernelregressionforward-reversesimulationHilbertprojectivemetricminimaxconvergenceratenonparametricestimationconditionaldiffusion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The Schr\"odinger bridge problem asks for the most likely evolution between two fixed endpoint distributions, measured by relative entropy against a reference process. This paper proposes to solve it by an iterative Monte Carlo method that never needs the reference transition density: at each step, kernel (Nadaraya–Watson) regression estimates conditional expectations from forward samples and from samples of an associated reverse process. The iteration is shown to stay inside a positive cone where the fixed-point map is a contraction in Hilbert's projective metric, giving a provable convergence rate. With a bandwidth $\delta_N = N^{-2/(2(1+\alpha)+d)}$ and about $\log N$ iterations, the sup-norm error decays as $N^{-(1+\alpha)/(2(1+\alpha)+d)}$, and a lower bound shows no estimator based on $N$ forward samples can do better uniformly over the smoothness class.

What carries the argument

The central object is the fixed-point operator $C = E_0 \circ D_0 \circ E_T \circ D_T$ on the cone $L^\infty_+(S_T)$ of positive functions, where $D_0,D_T$ take reciprocals, $E_T$ integrates forward transition densities against $\rho_T$, and $E_0$ integrates against the reverse process via the representation $\int q(0,x;T,z)g(x)\,dx = \mathbb{E}[g(Y_T^z)Y_T^z]$. The Hilbert projective metric $d_H(f,g)=\log(\sup(f/g)/\inf(f/g))$ is the natural yardstick because it ignores positive scalar multiples, and Birkhoff's contraction theorem makes $C$ a strict contraction with coefficient at most $\tanh^2(\tfrac12\log(q_{\max}/q_{\min}))$. The algorithm replaces $E_T$ and $E_0$ by kernel regression estimates, adds an $L^1$ normalization and a truncation step, and inherits the contraction, which is what turns a crude Monte Carlo update into a provably convergent iteration.

What would settle it

Take a reference Markov process whose transition density is not generated by a smooth diffusion, simulate the proposed forward and reverse samples, and compare the Monte Carlo estimate $E^N_0[f]$ against the true integral $\int \rho_0(y) f(y)\, q(0,y;T,\cdot)\,dy$ as $N$ grows; if the bias does not vanish, the reverse-process representation (36) fails and the algorithm's core estimates are invalid.

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Extended reading notes

Core claim

The paper's central claim is that the Schr\"odinger potentials $\nu_0,\nu_T$, equivalently the fixed point $g^\star$ of the operator $C = E_0 \circ D_0 \circ E_T \circ D_T$, can be approximated by a forward-reverse kernel regression iteration with rigorous convergence guarantees. Each application of the estimated operator $C_N$ replaces the forward expectation $E_T[f](x)=\mathbb{E}[\rho_T(X_T^x)f(X_T^x)]$ and the reverse expectation $E_0[f](z)=\mathbb{E}[\rho_0(Y_T^z)f(Y_T^z)Y_T^z]$ by Nadaraya–Watson estimates built from simulated forward and reverse trajectories. Because the random approximation preserves positivity and the underlying deterministic operator is a strict contraction in the Hilbert metric on $L^\infty_+(S_T)$, the iterates $\hat g_k$ converge in $L^\infty$ at the rate stated in Corollary 7, and Theorem 8 proves this rate is minimax optimal. The same potentials feed a non-nested Monte Carlo estimator for the finite-dimensional distributions of the Schr\"odinger bridge process.

Load-bearing premise

The whole Monte Carlo scheme depends on the existence of a reverse process $(Y,\mathcal{Y})$ satisfying $\mathbb{E}[g(Y_T^z)\mathcal{Y}_T^z] = \int q(0,x;T,z)g(x)\,dx$; the paper constructs such a process only for diffusions with smooth coefficients, so for a general Markov reference process this load-bearing ingredient is asserted but not supplied.

Editorial extensions

If this is right

  • After $k \ge \frac{1+\alpha}{2(1+\alpha)+d}\log(N)/\log(1/\kappa(C))$ iterations with bandwidth $\delta_N = N^{-2/(2(1+\alpha)+d)}$, the $L^\infty$ error $\mathbb{E}\|\hat g_k - g^\star\|_\infty$ is of order $N^{-(1+\alpha)/(2(1+\alpha)+d)}$.
  • No estimator built from $N$ independent forward samples can achieve smaller Hilbert-metric error uniformly over the H\"older class of transition densities, so the rate is minimax optimal.
  • The method needs no closed-form transition density of the reference process, only the ability to simulate forward paths and the associated reverse process, which opens the door to high-dimensional reference models.
  • Given the estimated potentials, finite-dimensional distributions of the Schr\"odinger bridge can be estimated by a non-nested Monte Carlo estimator based on forward-reverse simulation, without simulating the bridge SDE itself.

Reading between the lines

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

  • If a reverse-process representation were constructed for non-diffusion references such as jump processes or discrete-time Markov chains, the same forward-reverse kernel iteration would yield a sample-based Sinkhorn-type solver for those models; the paper leaves that extension open.
  • The contraction-plus-regression proof pattern is not tied to the Schr\"odinger problem: positive fixed-point iterations with kernel integral operators, such as Sinkhorn's algorithm, may inherit the same Hilbert-metric contraction and similar rates when the integrals are replaced by nonparametric regressions.
  • The KL bound in Section 5 suggests that pathwise simulation of the bridge from an estimated potential becomes fragile near the terminal time; the paper's non-nested estimator avoids that sensitivity when only expectations of the finite-dimensional distributions are needed.
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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

3 major / 4 minor

Summary. The paper proposes an iterative Monte Carlo kernel-regression scheme for estimating the Schrödinger bridge boundary potentials from samples of a forward reference process and an associated reverse process, building on the Hilbert-metric contraction framework of Chen, Georgiou, and Pavon. The central theoretical claims are convergence of the iterates to the fixed point g* in Hilbert metric and in sup norm at rate N^{-(1+α)/(2(1+α)+d)} after k ≍ log N iterations, and a matching minimax lower bound. The paper further develops a non-nested forward-reverse Monte Carlo estimator for finite-dimensional distributions of the Schrödinger bridge and a perturbation analysis for the drift approximation.

Significance. If the rate result is established, the paper is a genuinely useful contribution: it provides a nonparametric estimation procedure for Schrödinger potentials under a generative-model-only assumption, with convergence rates and a minimax lower bound, and the non-nested estimator in Section 6 is a practical advance over nested simulation. The lower-bound proof (Theorem 8) is a plausible Assouad-style argument, and the paper is transparent about importing the Hilbert-metric machinery from [5] and the forward-reverse representations from [19] and [2]. However, the upper-bound proof currently has a load-bearing gap in the conditional-expectation step, and several scope and bandwidth issues need correction.

major comments (3)
  1. [Section 7.1, proof of Proposition 6] The tower-property step is not justified. The proof defines F_ell = σ(bg_1,...,bg_ell) and bounds E[||E_N^0(g_{0,ell}) - E_0(g_{0,ell})||_∞ | F_ell] by the iid uniform regression bound of Corollary 19. But bg_ell is constructed in (17) as T[ C_N(bg_{ell-1})/||C_N(bg_{ell-1})|| ] with C_N = E_N^0 ∘ D_0 ∘ E_N^T ∘ D_T, so bg_ell is a non-injective function of the same forward and reverse samples that enter E_N^T and E_N^0 at iteration ell. Conditioning on bg_ell therefore conditions on a nonlinear function of those samples, and the conditional law of the samples is no longer the product law with densities φ_0 and φ_T. Moreover g_{0,ell} and g_{1,ell} are themselves random functions of those samples, so neither the fixed-function bias bound (45) nor the VC-type uniform bound of Corollary 19 applies to them. Consequently the 'with probability 1' bounds following (45) are not established, and the rate (18) in Corollary 7 does not follow. A fix would require using fresh samples per iteration and conditioning only on F_{ell-1} (so that the regression functions are fixed), or a genuine decoupling/empirical-process argument covering the data-dependent functions inf f / f uniformly.
  2. [Abstract/Introduction vs Appendix B] The paper claims to handle 'general reference processes', but the reverse process (Y,Y) satisfying (36) is constructed in Appendix B only for diffusions whose coefficients are C^∞ with bounded derivatives of all orders. The Monte Carlo estimates (13)-(14) presuppose the existence of such a reverse process; for a general Markov reference process with transition density q, no existence or approximation result is supplied. The scope should be restricted to diffusion reference processes of the form (34) with the stated regularity, or the reverse-process construction must be extended to the claimed generality.
  3. [Proposition 6 and Section 7.1, bandwidth choice] The stated bandwidth δ_N = N^{-2/(2(1+α)+d)} appears inconsistent with the bias-variance balance in the proof. With this choice, the stochastic bound of Corollary 19 is of order N^{-1/2} δ^{-d} = N^{-1/2 + 2d/(2(1+α)+d)}, whereas the bias is of order δ^{1+α} = N^{-2(1+α)/(2(1+α)+d)}; for d>0 these are not of the claimed common order N^{-(1+α)/(2(1+α)+d)}. The bandwidth exponent should be checked against the actual stochastic error of the ratio estimators E_N^0 and E_N^T, and the proof adjusted accordingly (the exponent is likely a typo for N^{-1/(2(1+α)+d)}).
minor comments (4)
  1. [Section 4.1, Corollary 7] The sup-norm bound (18) is stated without derivation; since bg_k is not explicitly normalized to unit L1 mass, the passage from dH(bg_k,g*) to ||bg_k-g*||_∞ requires an argument (e.g., via Lemma 13 and the bounded ratio) and should be included.
  2. [Section 7.1, after (45)] The notation 'κ(δ/2)^{1+α}' after (45) is confusing: κ has already been used for the Hilbert-metric contraction coefficient, while the bias bound (45) involves (δ/2)^{α+1}. Please use distinct notations.
  3. [References] References [17] and [18] refer to the same survey by Léonard; one entry should be removed or merged.
  4. [Section 3, Eqs. (13)-(14)] The definitions of E_N^T and E_N^0 use the same symbol N for the sample size in both the forward and reverse samples; in the lower-bound Theorem 8 the symbol N has a different meaning (the number of iid forward pairs). Please clarify or use separate symbols (e.g., N and M).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the paper's fixed-point contraction theory and reverse-process representations are cited from published prior work, while the kernel-regression rate analysis is a new statistical derivation; the only self-citations are supporting tools, not the target claim.

full rationale

The paper's central derivation is the statistical rate analysis of a Monte Carlo kernel regression iteration for the Schrödinger fixed-point operator (Proposition 6, Corollary 7). It does not define the target potentials in terms of the estimator, nor fit parameters that are later called predictions. The contraction theory is cited from Chen-Georgiou-Pavon [5], and the reverse-process representation is cited from Milstein-Schoenmakers-Spokoiny [19] and Bayer-Schoenmakers [2]; although the latter two involve a present author, they are published, peer-reviewed, parameter-free results with stated assumptions that do not include the present paper's target claim, so they count as independent support. The minimax lower bound (Theorem 8) is derived via an independent Assouad-lemma argument. The proof of Proposition 6 uses a conditioning step (F_ell = sigma(bg_1,...,bg_ell)) that is potentially invalid because the kernel estimators use the same sample as bg_ell, but this is a proof-technical gap, not a circular reduction of the claimed rate to its inputs. There is no self-definition, no fitted-input-called-prediction, and no load-bearing self-citation chain. The paper is therefore essentially non-circular, with only minor self-citation of background tools.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new postulated entities. It relies on published existence theory, an existing reverse-process construction, known Hilbert-metric contraction results, and standard kernel-regression inequalities. The main free hyperparameter is the kernel bandwidth, and the reverse-process existence is a strong domain assumption not satisfied by the claimed 'general reference processes'.

free parameters (2)
  • kernel bandwidth delta_N = N^{-2/(2(1+alpha)+d)}
    The bandwidth is chosen by hand as a function of the smoothness exponent alpha, which is an unknown problem parameter. In practice this would be a tuning parameter.
  • iteration count k = O(log N / log(1/kappa(C)))
    The number of Picard iterations is selected to balance the contraction term with the Monte Carlo error; this is a user-chosen stopping rule, not a fitted constant.
assumptions (6)
  • standard math Existence and uniqueness of Schrödinger potentials from Chen-Georgiou-Pavon [5, Prop. 1], requiring continuous strictly positive transition density and compact supports of the marginals.
    Theorem 1 in the paper is quoted from [5] and is the foundation of the fixed-point formulation.
  • domain assumption The reverse process (Y,Y) from Milstein-Schoenmakers-Spokoiny [19] satisfies the stochastic representation (36); this construction requires the reference diffusion to have C^infinity coefficients with bounded derivatives.
    Section 3 and Appendix B rely on this representation for the Monte Carlo estimate of E0. The paper does not provide a reverse process for general Markov reference processes, despite the abstract's wording.
  • standard math The operator C is a contraction in Hilbert metric with coefficient kappa(C) = tanh^2(0.5 log(qmax/qmin)) < 1.
    Quoted from [5] and used throughout Section 4 as the basis of the convergence proof.
  • standard math Uniform kernel regression bounds from Dony-Einmahl (Appendix F, Theorem 18) apply to the Nadaraya-Watson estimators; this requires a VC-type kernel class and bounded Hölder smoothness of the regression functions.
    Used in the proof of Proposition 6 to control the sup-norm error of the kernel estimates.
  • domain assumption Known positive bounds qmin, qmax, Qmin, Qmax, rho_min, rho_max on the transition kernel, its integrals, and the marginal densities (Assumption 4).
    These bounds define the truncation interval [g*_min, g*_max] in (11) and appear in all constants and in the algorithm's truncation operator.
  • domain assumption The normalization Assumption 5 that g* integrates to 1 on S_T.
    Used to remove the scaling ambiguity of the potentials and to derive the bound (11).

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Cite this review

Pith. "Pith review of Forward Reverse Kernel Regression for the Schr\"{o}dinger bridge problem." pith.science (2026). https://pith.science/paper/NGFPFH55

@misc{pith2026250700640,
  author       = {Pith},
  title        = {Pith review of: Forward Reverse Kernel Regression for the Schr\"odinger bridge problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGFPFH55}},
  note         = {Machine review of arXiv:2507.00640}
}
read the original abstract

In this paper, we study the Schr\"odinger Bridge Problem (SBP), which is central to entropic optimal transport. For general reference processes and begin--endpoint distributions, we propose a forward-reverse iterative Monte Carlo procedure to approximate the Schr\"odinger potentials in a nonparametric way. In particular, we use kernel based Monte Carlo regression in the context of Picard iteration of a corresponding fixed point problem. By preserving in the iteration positivity and contractivity in a Hilbert metric sense, we develop a provably convergent algorithm. Furthermore, we provide convergence rates for the potential estimates and prove their optimality. Finally, as an application, we propose a non-nested Monte Carlo procedure for the final dimensional distributions of the Schr\"odinger Bridge process, based on the constructed potentials and the forward-reverse simulation method for conditional diffusions.

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Reference graph

Works this paper leans on

26 extracted references · 24 canonical work pages

  1. [5]

    Entropic and displacement interpolation: a computa- tional approach using the Hilbert metric.SIAM J

    Yongxin Chen, Tryphon Georgiou, and Michele Pavon. Entropic and displacement interpolation: a computa- tional approach using the Hilbert metric.SIAM J. Appl. Math., 76(6):2375–2396, 2016

    work page 2016

  2. [19]

    Milstein, John G

    Grigori N. Milstein, John G. M. Schoenmakers, and Vladimir Spokoiny. Transition density estimation for stochastic differential equations via forward-reverse representations.Bernoulli, 10(2):281–312, 2004

    work page 2004

  3. [2]

    Simulation of forward-reverse stochastic representations for condi- tional diffusions.Ann

    Christian Bayer and John Schoenmakers. Simulation of forward-reverse stochastic representations for condi- tional diffusions.Ann. Appl. Probab., 24(5):1994–2032, 2014

    work page 1994

  4. [1]

    Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional bre- nier maps.arXiv preprint arXiv:2411.07154, 2024

    Ricardo Baptista, Aram-Alexandre Pooladian, Michael Brennan, Youssef Marzouk, and Jonathan Niles-Weed. Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional bre- nier maps.arXiv preprint arXiv:2411.07154, 2024

    arXiv 2024

  5. [3]

    Iterative bregman projections for regularized transportation problems

    Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Lorenzo Nenna, and Gabriel Peyré. Iterative bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2):A1111– A1138, 2015

    work page 2015

  6. [4]

    An automorphism of product measures.Ann

    Arne Beurling. An automorphism of product measures.Ann. Math. (2), 72:189–200, 1960

    work page 1960

  7. [6]

    Georgiou, and Michele Pavon

    Yongxin Chen, Tryphon T. Georgiou, and Michele Pavon. On the relation between optimal transport and schrödinger bridges: A stochastic control viewpoint. Journal of Optimization Theory and Applications, 169(2):671–691, 2016

    work page 2016

  8. [7]

    On the convergence rate of sinkhorn’s algorithm.SIAM Journal on Optimization, 31(2):1478–1498, 2021

    Roberto Cominetti, José Soto, and Cristóbal Ríos. On the convergence rate of sinkhorn’s algorithm.SIAM Journal on Optimization, 31(2):1478–1498, 2021

    work page 2021

Show all 26 references
  1. [8]

    P. Dai Pra. A stochastic control approach to reciprocal diffusion processes.Applied Mathematics and Opti- mization, 23(1):313–329, 1991

    1991

  2. [9]

    Diffusion schrödinger bridge with applications to score-based generative modeling

    Valentin De Bortoli, James Thornton, Jeremy Heng, Benjamin Graham, and Yee Whye Teh. Diffusion schrödinger bridge with applications to score-based generative modeling. InAdvances in Neural Information Processing Systems (NeurIPS), volume 34, pages 24667–24681, 2021

    2021

  3. [10]

    Uniform in bandwidth consistency of kernel regression estimators at a fixed point

    Julia Dony and Uwe Einmahl. Uniform in bandwidth consistency of kernel regression estimators at a fixed point. InHigh dimensional probability V: The Luminy volume, volume 5, pages 308–326. Institute of Mathe- matical Statistics, 2009

    2009

  4. [11]

    Hilbert’s projective metric for functions of bounded growth and exponential convergence of Sinkhorn’s algorithm.Probability Theory and Related Fields, pages 1–37, 2025

    Stephan Eckstein. Hilbert’s projective metric for functions of bounded growth and exponential convergence of Sinkhorn’s algorithm.Probability Theory and Related Fields, pages 1–37, 2025

    2025

  5. [12]

    R. Fortet. Résolution d’un système d’équations de M. Schrödinger.J. Math. Pures Appl. (9), 19:83–105, 1940

    1940

  6. [13]

    U. G. Haussmann and E. Pardoux. Time reversal of diffusions.Ann. Probab., 14:1188–1205, 1986

    1986

  7. [14]

    Reciprocal processes.Z

    Benton Jamison. Reciprocal processes.Z. Wahrscheinlichkeitstheor. Verw. Geb., 30:65–86, 1974

    1974

  8. [15]

    Kurbanmuradov, U

    O. Kurbanmuradov, U. Rannik, K. Sabelfeld, and T. Vesala. Direct and adjoint Monte Carlo algorithms for the footprint problem.Monte Carlo Methods Appl., 5(2):85–111, 1999

    1999

  9. [16]

    Birkhoff’s version of hilbert’s metric and applications

    Bas Lemmens and Roger Nussbaum. Birkhoff’s version of hilbert’s metric and applications. In A Papadopoulos and M Troyanov, editors,Handbook of Hilbert Geometry, IRMA Lectures in Mathematics and Theoretical Physics, pages 275–303. European Math. Soc., December 2014

    2014

  10. [17]

    A survey of the schrödinger problem and some of its connections with optimal transport

    Christian Leonard. A survey of the schrödinger problem and some of its connections with optimal transport. Discrete and Continuous Dynamical Systems, 34(4):1533–1574, 2014

    2014

  11. [18]

    A survey of the schrödinger problem and some of its connections with optimal transport

    Christian Léonard. A survey of the schrödinger problem and some of its connections with optimal transport. Discrete and Continuous Dynamical Systems A, 34(4):1533–1574, 2013

    2013

  12. [20]

    Tabak, and Giulio Trigila

    Michele Pavon, Esteban G. Tabak, and Giulio Trigila. The data-driven schrödinger bridge.arXiv preprint arXiv:1806.01364, 2018. 22 D. BELOMESTNY AND J. SCHOENMAKERS

    2018 arXiv

  13. [21]

    Computational optimal transport: With applications to data science.Foun- dations and Trends in Machine Learning, 11(5–6):355–607, 2019

    Gabriel Peyré and Marco Cuturi. Computational optimal transport: With applications to data science.Foun- dations and Trends in Machine Learning, 11(5–6):355–607, 2019

    2019

  14. [22]

    Plug-in estimation of Schrödinger bridges

    Aram-Alexandre Pooladian and Jonathan Niles-Weed. Plug-in estimation of Schrödinger bridges. arXiv preprint arXiv:2408.11686, 2024

    2024 arXiv

  15. [23]

    Guided proposals for simulating multi- dimensional diffusion bridges.Bernoulli, 23(4A):2917–2950, 2017

    Moritz Schauer, Frank van der Meulen, and Harry van Zanten. Guided proposals for simulating multi- dimensional diffusion bridges.Bernoulli, 23(4A):2917–2950, 2017

    2017

  16. [24]

    Über die umkehrung der naturgesetze.Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse, pages 144–153, 1932

    Erwin Schrödinger. Über die umkehrung der naturgesetze.Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse, pages 144–153, 1932

    1932

  17. [25]

    D. J. Thomson. Criteria for the selection of stochastic models of particle trajectories in turbulent flows.Journal of Fluid Mechanics, 180:529–556, 1987

    1987

  18. [26]

    Springer Science & Business Media, 2003

    Alexandre B Tsybakov.Introduction à l’estimation non paramétrique, volume 41. Springer Science & Business Media, 2003. 1F aculty of Mathematics, Duisburg-Essen University, Thea-Leymann-Str. 9, D-45127 Essen, Ger- many Email address: denis.belomestny@uni-due.de 2Weierstrass Ins...

    2003

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