On the Relation between Rectified Flows and Optimal Transport
Authors: Johannes Hertrich
Antonin Chambolle
Julie Delon
Venue: NeurIPS 2025 poster ()
Keywords: rectified flows
optimal transport
flow matching
Primary area: theory

Abstract: This paper investigates the connections between rectified flows, flow matching, and optimal transport. Flow matching is a recent approach to learning generative models by estimating velocity fields that guide transformations from a source to a target distribution. Rectified flow matching aims to straighten the learned transport paths, yielding more direct flows between distributions. Our first contribution is a set of invariance properties of rectified flows and explicit velocity fields. In addition, we also provide explicit constructions and analysis in the Gaussian (not necessarily independent) and Gaussian mixture settings and study the relation to optimal transport. Our second contribution addresses recent claims suggesting that rectified flows, when constrained such that the learned velocity field is a gradient, can yield (asymptotically) solutions to optimal transport problems. We study the existence of solutions for this problem and demonstrate that they only relate to optimal transport under assumptions that are significantly stronger than those previously acknowledged. In particular, we present several counterexamples that invalidate earlier equivalence results in the literature, and we argue that enforcing a gradient constraint on rectified flows is, in general, not a reliable method for computing optimal transport maps.

On the Relation between Rectified Flows and Optimal Transport
Johannes Hertrich Université Paris Dauphine- PSL & Inria Mokaplan, Paris johannes www.heertrrtuuuueee hertrichhdauphine. wwwplaaupppieepps......ssmss.
Antonin Chambolle Université Paris Dauphine PSL & Inria Mokaplan, Paris chambolle@ceremadee dauphine. fr
Julie Delon
ENS Paris
julie wew.nnens...... frr
Abstract
This paper investigates the connections between rectified flows, flow matching, and optimal transport. Flow matching is recent approach to learning generative models by estimating velocity fields that guide transformations from source to a target distribution. Rectified flow matching aims straighten the learned transport paths, yielding more direct flows between distributions. Our first contribution is a set of invariance properties of rectified flows and explicit velocity fields. In addition, we also provide explicit constructions and analysis in the Gaussian (not necessarily independent) and Gaussian mixture settings and study the relation to optimal transport. Our second contribution addresses recent claims suggesting that rectified flows, when constrained such that the learned velocity field gradient, yield (asymptotically) solutions optimal transport problems. We study the existence of solutions for this problem and demonstrate that they only relate optimal transport under assumptions that are significantly stronger than those previously acknowledged. In particular, we present several counterexamples that invalidate earlier equivalence results in the literature, and argue that enforcing a gradient constraint on rectified flows is, in general, not a reliable method for computing optimal transport maps.
Introduction
Optimal transport is the problem of transporting probability distribution No to another distribution J4 such that a given cost function has minimal expected value. More precisely, we aim to find a coupling (Xu, X1) of random variables Xo Nu and X1 μ4 such that E[c(Xo, X2)] is minimal,
where is a cost function, most commonly c(z, y) = ||x | (1-Wasserstein or earth-mover distance) c(x,y) y) | y| (squared 2- Wassersttein distance). The problem was originally formulated by Monge (39 in terms of optimal transport maps and later generalized by Kantorovich [25) using 51 for overview. Nowadays, it is heavily used in
more general couplings. We refer 42
45
machine learning for clustering 23
38 domain adaptation [12 generative modeling [5] 27 or
model selection
[6], to cite just few references.
Computing solutions to the optimal transport problem can very challenging in practice. In the discrete case, it amounts solving linear program and can be efficiently accelerated using entropic regularization and the Sinkhorn algorithm [3] 47]. A dynamic formulation of optimal transport, introduced by Benamou and Brenier 77]. characterizes optimal solutions those induced by a velocity field v, of minimal L norm (for the 2- Wasserstein distance) among fields transporting μl to μ4 via
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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the ordinary differential equation (x) uy (z)()())). From an analytic viewpoint this corresponds to finding a curve + f4 interpolating between pn to μi in the space of probability measures, such that the continuity equation 0,44 + div(ve))) = 0 holds and J, [| is minimal.
This dynamic perspective transporting probability measures is also used in
generative models, such as continuous normalizing flows I 19 or denoising diffusion models [22 48) However, the transport maps learned by these models are in general not optimal.
Recently, Liu et al. [35] proposed to learn such a velocity field by starting from an arbitrary coupling and averaging up all linear interpolations between Xo and X1- Similar ideas were introduced the
same time in 2 32 under the name flow matching and stochastic interpolants. Building on this framework, rectified flow matching seeks to straighten the learned transport paths, leading to more direct flows between distributions.
Outline and Contributions In this paper, we study the relation between rectified flows and optimal transport. In particular, we show that the iterative rectification proposed in (34) is not, in general, a suitable tool for computing optimal transport maps. In Section 2 we start with revisiting the backgrounds on rectified flows. Additionally, we show some invariance properties under affine transformations and derive the optimal velocity fields for the Gaussian case. Then, we show in Section how these properties change if we constrain the velocity fields to be gradient. In partieular, we prove that a solution of the constrained problem always exists in weak form. Afterwards, in Section 4 we construct the following two counterexamples. where the relation between optimal transport couplings and fixed points of the rectification procedure is false
First, in Section 4.1 we consider an example where the support of the interpolated distri- butions μ4 is disconnected. In this case, subdomains. Then, we can construct a transport which is optimal on each subdomain, but not globally optimal. This example shows in particular, that the claim from 34 Theorem 5.6] is not true without additionally assuming that X, has connected support. Since datasets in applications are often disconnected this massively restricts the applicability of rectified flows for computing the optimal transport.
divide the space in several disconnected
Second, we pay attention to so-called non-rectifiable couplings in Section 4.2 couplings such that the velocity field v, learned by the rectification does not lead to a unique solution of the ODE 2g = Ve (.). ). We show an example that such couplings exist and can have zero loss even though they are not optimal.
These are
We prove a sufficient criterion for coupling to be rectifiable based on the smoothness of the conditional probability of Xo given Xj. This criterion is particular fulfilled for the independent coupling with smooth initial distribution oo or when small amount of noise is injected into Xg- In Section 5 we consider the special case where Mo is Gaussian distribution. Then, we modify the iterative rectification by injecting noise in each step to ensure that the iterates remain rectifiable. We show that the arising procedure still marginal preserving and has the same decay of the loss function as without noise injection. We discuss the implications of our results and draw conclusions in Section 6
and further investi- for an overview. Error of the generated distribution by the training error in the velocity field were proven by 8 44 The basic idea of defining an interpolation path of latent and target measure was previously used in diffusion models and several other models [20 [
Related Work Rectified flows or flow matching was introduced in 2 32 35
gated in [1] see 333 52
Applications and improvements of the training procedure of rectified flows were considered in
110 29 30
36 37].
In order to compute the optimal transport using a flow-based generative model, a regularization term of the velocity field into the loss function of a continuous noraalizin flow However, this approach alters the marginals of the solution. Other papers [24 26) propose to compute optimal transport maps by using Brenier's ' theorem 9 and representing the convex potential of the transport map by an input convex neural network. The authors (28) use the dual formulation of optimal transport and solve the arising saddle point problem with neural networks. The authors of 2 relate flow matching with (entropic) optimal transport by considering non-linear interpolants between the marginals.
include

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To initialize rectified flows close to the optimal coupling, 43 49
propose minibatch OT. That is, they draw minibatches from both marginals Ao and μl and pair the data points within these batches by computing the discrete optimal transport between them. However, the coupling generated by minibatch OT is again not optimal in general.
In [1 the authors propose a noisy version of flow matching. Iterating this procedure is related to the Schrödinger bridge problem under the name Diffusion Schrödinger Bridge Matching (DSBM) [4 46]. This corresponds to computing the entropically regularized optimal transport plan, see 31 for an overview. While convergence guaranteed for DSBM under mild assumptions, our counterexamples suggest that the convergence rate becomes arbitrary slow when the entropic regularization parameter tends zero. We include some numerical tests in this direction in Appendix
2 Rectified Flows
In this section, we provide an overview of the backgrounds of rectified flows. Afterwards, we derive invariance properties of the rectification procedure and study the case where all involved measures (mixtures of) Gaussians. We will see that these invariance properties already have some similarities to optimal transport. Additionally, they will be needed in the proofs later in the paper.
2.1 Backgrounds
Wasserstein Distance We define coupling between two probability measures do and μn on Rd as a pair (Xo, X1) of random variables with Xo μá and X1 ~ μl. The Wasserstein-2 distance is given by
W (oo 11)
inf B| X1113..
In addition we say that a coupling (Xo, X2) between AA and μl is optimal if it fulfills W? (Ho. p1) = B|I|X, X2 [12]. The Wasserstein distance W2 defines metric the space of probability measures with finite second moment. Throughout the paper, we assume that all considered probability measures belong to this space. The distance W2 belongs to the family of optimal transport discrepancies, which are defined in the same way by replacing the squared Fuclidean norm by some more general cost function c.
Rectified Flows In order build generative model for some target measure μl based on some latent measure μag the authors of [4 35) propose rectified flows, which are also known by the name flow matching 332 33 or stochastic interpolants 2 Given coupling (Xo, X,) between Ho and we consider the interpolations X, = (1 X, + tX, and denote by the distribution of X,- Then, construct velocity field (14))ε(o.1] for fe by minimizing the loss function
Vt arg min [(wr|Xo, X1), [(we|Xo, we w (()e))
EX,,,,(,,) Xi Xoll?ddt.
The authors of 2 32 35
show that the minimizer of this problem exists and is unique. Using the optimal prediction property of conditional expectations, the solution in of ( can be formulated as the conditional expectation
1
1 E X, x] - (E| = x] -)) (2)
va (xr) = E[X, - Xo X; = x]
Additionally, they show that the solution o, fulfills the continuity equation with respect to plcs i.c., that
0,44 + div(u.p))) =
in distributional sense. In particular, under the assumption that e, is smooth enough, es defines transport from μi to ini in the sense that μi
z1 # No. where 2x(x)) is the solution of the ODE
(x)) = (=())))), with initial condition za <xx = x.
the ODE 4 has a unique solution, we can sample from μ4 by sampling from 140 and solving the ODE 4 In the literature, the arising generative model produces tfae-of-the-aa art results 2 32 35).
3

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Iterative Rectification In order to obtain simpler velocity fields, Liu et al. [35 propose to construct a new "rectified" coupling as follows.
Definition 1. Let (Xo, X1) be coupling between M4o and μi and denote by i, the minimizer of the loss function M. Then, we call (Xo, XI) "rectifiable" the ODE 4 has a unique solution and define (Z., Z R(xo X1) with Zo Mo and Z1 P4 by setting Zo Xo and Z1 to the solution of Z, 24 (Za) at time 1.
The authors of 35 prove that this procedure always reduces the transport distance of the coupling,
that is, it holds Z Zi [1 Xo X1 11. Morcover, we can iterate this procedure by generating a sequence (xo ( x((+t) = R(Xx x(x)). Then, the minimal loss function over
the first K iteration converges to zero, i... it holds mink=....... =O....K {(mit,,, L(w, [x x
0 as
K + op. Additionally, any coupling (Xo: with velocity field ii such that C( X1) = is fixed point of R. Intuitively, these plans can be characterized by the property that the paths of the ODE (4) are straight. That is, for po-almost every x, the solution path + 24 (x) of the ODE 0y (x)) has constant velocity in time 2s (x))..
2
2.2 A ffine Invariance and Gaussian Case
Next, we consider some equivariances of the rectification step R with respect to translations and scalings of one or both marginals of the argument The proofs are given in Appendix (A
Theorem 2 (Affine Transformations). Let (Xo, X1), be a coupling between μi and μt, let Vt = arg min,,, [(w
be the minimizer of the loss function (T] and let Bddd be invertible,
b E IRd and c E RRso Then, the following holds true.
(i) The velocity vg
arg min, w. [(welAXo + b. AX1 + b) is given by of (x) =
Ave(A-1(x (ol--b)) b))-
(ii) The velocity uf = arg min... [(w/ X1 + b) is given by (f(()) v2(x tb) +
(iil) The velocity of = arg min,,, [(w, Xoo, cX1) is given by
o 1-t+a Ur (1-2 (1-t-tc)
c-
tc
tc
+ 1-t tc
x.
with r 1- + te
If (Xo: X1) is in addition rectifiable with (Zo, 21) = R(Xo. X1). then it holds that R(AX, AX; b) (AZo + b, AZ, + b), X1 b) (Zo, Z1 +) and R(Xu, cX2) (Zo, cZ1)
The invariances (ii) and (iii) hold also true for the optimal transport and the corresponding velocity fields from the Bernamou-Brenne Remark
theorem. Part (i) is false for optimal transport and we will see in 7 that it no longer true if we use the loss function 7 instead of (I. In the specific case that oi and 11 are Gaussian and the joint distribution of (Xo, X1) is a Gaussian as well, we can write down analytically the velocity field v, which solves M. Additionally, for the independent coupling of two Gaussians sharing the same eigenvectors, can show that already the first rectification step leads to the optimal coupling. The proof is given in Appendix A
Σo Σni
Σιo
Theorem 3 (Gaussian Case). Assume that (1) NOO with Σ
.oo Σ④
positive definite En and Σx. Then, the following holds true.
(i) The minimizer vi = arg min w. C(urlXoo of the loss function T is given by
Uy (x( = 1-t ((1- (t) + +tEn)
- Id x.
(5)
where E, = Cov(X,) = (1- (1
(1- -
+ Σ1o) + 12
(ii) Let Eo1 = Σ10 = O and assume that Eo and Σ₁ can be jointly diagonalized. Then,
(Zu 71) R(Xo, X1) is the unique optimal coupling between pi and p1.
In the special case where E. = Id, part (ii) was already proven Prop 4.12]. Note that as direct consequence of Theorem 2 (i) and the explicit representation of the optimal transport for

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Gaussian measures, part (ii) of the previous theorem is no longer true if we skip the assumption that Eg and Σ, can be jointly diagonalized. The authors of 44 also emphasize the one-dimensional case. However, it is straightforward to see that in the one-dimensional case any rectifiable coupling leads to the optimal transport after one step. For completeness, we formalize the result in the following proposition and include proof in Appendix A
Proposition 4. Consider the one-dimensional case and let (Xo, X1) be rectifiable coupling between No and f1. Then (7o. 71) = RR(o, X1) is the optimal coupling between PAo and j41.
The explicit representation of v, in the case where the coupling (X,; X1) follows Gaussian distribution can be generalized to Gaussian mixture models by averaging the vector fields induced by the components, as outlined in the following theorem. The proof included Appendix
Theorem 5 (Gaussian Mixture Case). Assume that (No. Σt= TTN(mm 52 with m mg
(
Σ
and E*
210 Σ/
for positive definite Σg and Σ/ Write 2tt the velocity field (5)
mi
Σ
for the covariance matrix h* and write w = of (x - tm (1 (1--tyn t)mmi) + m m Then, the Xo,
minimizer arg min.,, [(w)lXo, X1) ()))e))))) of the loss function T given by
K
vx(x)
(z)
(6)
Jmn
where a (x) = Σ (x)) with p1 the Gaussian density of N Σ/) with m = tm/ +(1-t)m) m)mb and Σ. = PEY + (1 t)20// - t(1 - (Eio t)
(x)
+ Eo1)
Note that the same result holds for degenerated GMMs where E and Σ are only positive semi- definite. In the case of generative models, is classical to assume that Xo ~ (O, Td) and X1 ~ Σ R Om.. If Xo and Xı are independent, then X1) follows (degenerated) GMM and the velocity field U2 solution is explicit and given by (6) with wx(x) I m tm and Σ (1 Eg In these cases, it is evident that our goal is not compute the velocity field exactly, but rather to rely on its approximation by neural network, a key element which gives the model its generalization properties.
3 Rectified Flows and Optimal Transport
Next, we are interested how rectified flows are related to optimal transport. To this end, we first provide overview over (34] which relates the optimality of the velocity in rectified flows with the condition that they are gradient field. Afterwards, we study how this condition effects the solutions of I
3.1 Backgrounds on Rectified Flows with Gradient Fields
As already observed does not necessarily define an optimal transport. Based on the observation that the optimal velocity field from the Sitaamou-Breniee theorem (7] always admits a potential
35 a coupling (Xo: with velocity field v2 such that L(v,) Xo, X1)
Thm. 8.3. I], one of the authors of (35] suggests in (34) to impose the additional constraint that the velocity field v, from has a potential. More precisely, the loss function ! is altered to
Ut E arg min L(w, x1) (.E))((p))
subject to w, = Vyp; for some op : IRd + R
(7)
This leads to a rectification step analogously to Definition 1 where the loss ① is replaced by (7). Definition 6. Let (Xo, be coupling between μo and μ and denote by v the minimizer of the loss function 77. We denote by (Zo, Zi) = (X,, X1) the rectification step with potential, where Zo = Xo and where Z₁ is the solution of 2, = 02 (Zz) at time 1.
Note that it is unclear whether the minimum in 7 really exists and that this question is not addressed in 34 Proposition & we will show that such solutions always exist in a weak form and relate them to the minimal-now norm solution of the continuity equation defined in [4] Thm. 8.3.
5

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Now, the author of [3 claims that
(8) We will see later Section 4.2 that this result requires several assumptions. In particular, we have to assume that X has full support for all t E (0, 1), that the minimizer of (7) is sufficiently smooth and that (Xo, X1) is rectifiable. While the last two assumptions are stated in [34], the first one is missing and we show that without this assumption the claim (8) indeed false.
(Xo, .x1) (Xo,
3u, Vip (o, X1) (1) is optimal.
Let us stress that Liu 34 also considers rectified flows for more general cost functions c(x; y) than just the quadratic cost. Considering that our examples already appear for the "simple" case of the quadratic cost, we only consider this case.
3.2 Velocity Fields with Potential
In the following, we are interested in the effects of imposing that the velocity field admits a gradient. More precisely, we study, how the solutions problems (I) and 7 differ. To this end, first consider how the affine invariances from Theorem 2 change in this case. Afterwards, we prove existence solutions
in a weak form.
Remark 7 (Affine Invariances with Potential). It straightforward to show that parts (ii) and (iii) of Theorem 2 are also true if we replace R by Ry- To this end, we just have to show that e = Vy implies that there exist spt and fo such that uf = Vi"" and os Vpp.. This is fulfilled for
2 = pe(x (b) + (b,2). and of = cspr (-
2(1 + tc) |=|
However, item (i) is not for Ry. Indeed, velocity field IRd has a potential if and only if the Jacobian is symmetric, see [18] Thm 6.6.3]. For the specific choice of vAi,, this is the case if and only if A (z), Ju, (x),)). A A for all
In general, it is not clear whether solutions 7 exist. However, in the following we prove existence in weak form. To this end, we consider the space T,. == {Vip: : yp E (RR)) where the closure
taken in (Hee. Now, we weaken the constraint in by only assuming that i E Tp. More precisely, consider the problem
Vt E arg min [(wi|Xo, w)EL)(pp)
Xo, X2) subject to w4 T
(9)
The next proposition shows that the solution of (9) is the limit of minimizing sequence in the optimization problem from (7). In particular, both solutions coincide whenever the minimizer in (7) exists. To this end, we first show that the solution of ( is the orthogonal projection onto T of the
solution of ( The proof is given in Appendix A
Proposition 8. Let Ut and of be the solutions of and (9). Then, the the following holds true.
(i) For any t E [0. we have that v. = arg min wT l|ve well
(ii) There exist 4tth C,, 1] x Rdd) such that Vip" + vp in Ld @ pe) and [(Vipt) X1) + inf in XowelXo, X1).
(ii)) The vector field of is the minimal-norm solution of the continuity equation depe div(u? = O. That is, it minimizes the norm fo [lu,l[dduddt among all solutions of the continuity equation wrt. P4.
(iv) If the minimizer in 7 exists, then it coincides with uf.
The minimal- -nrimaml--ommm norm velocity field from part (iii) was defined 44 Thm. 8.3.1], see also Ex. 8.5 in [51]. It minimizes the same objective as the Benamou-Brenin theorem. However, stress that Benamou-Bret Bernnoru--reiereeee also optimizes over the path μ4, which is here fixed as the distribution of X, such that ei does not directly lead the optimal transport. Following the proposition, say that uf is solution of (7). if is the limit in (dt @ 14) minimizing sequence of gradients potentials E C,0((0 IRd). Given the universal approximation theorem, this particularly implies that the solution of can be approximated by the gradient of sufficiently large neural network.

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(a) Optimal Coupling
10
(b) Non-optimal Fixed Point
(c) Non-optimal Fixed Point for Gaussian μi from Remark 12
Rp for (T0)
Figure 1: Construction of non-optimal couplings which are fixed points of Rp.
If Xo ~ (V Id) and if Xo and X are independent, rectified flows are equivalent to denoising diffusion models outlined in (35] Section 3.5]. In particular, the solutions of ( and (7) coincide by the next corollary. The proof is a direct consequence of (2) and Tweedie's formula [16 and can be found, e.g.. in [52] Proposition 4.11].
po & Mn with pa (N( Id). Then, it holds that the velocity field is from ( is given by (z(x) = - ((()) + 1 x. where s((x) = -v log(p) is the Stein score for the density Pt of Xt- In particular, va admits the potential o, Vip for (p(x)) L-t - log(p:(zz) 22 | so that the solutions of and coincide.
Corollary 9. Let X1)
Counterexamples
In the following, we study specific cases in which the equivalence in is not true. Here, first consider an example where μl0 and have a disconnected support leading to a fixed point of Rd which is not the optimal transport plan. Afterwards, construct non-rectifiable coupling which has zero loss in but is again not optimal.
4.1 Disconnected Supports
We construct simple example of non-optimal velocity field which admits straight paths and a potential contradicting (8) and [34] Theorem 5.6]. To this end, let 17 E P(R?) an arbitrary
probability measure with support(no) (x E R.2 ||^||| s 0.3) and denote by r7" = +++ b) shifted versions of n, Then, we define
(1/ (22)) + 111 and 11 (n1 (11-2-11+7(2.1)) 1222.1).
No 2
+
(10)
see Figure la and lb for an illustration. Now let No = Xo ~ and define
X, (0,2)
if (Xx)1 <-
Xo (4,0) if (Xx)2 < -05
X1
and X1
(1I)
Xo + (0,2) 2)
(Xo) >
Xo + (4,0) iX)) > 0.5
see Figure Ta and 1b for an illustration. Then the following proposition shows that the coupling (Xo, X2) is counterexample to (8). The proof is straightforward calculation. For completeness, we include it in Appendix
Proposition 10. Let X1) and (Xo. be defined as above. Then, both couplings (Xo, X1)
and (Xo, X1) are fixed points of RR and have zero loss in
7 Le..
min Vi
[(w,)oo, =
min C(welKo. X1) = 0.
10 V4p
Moreover, it holds B[[[X 8o11 > B|J|X, - - such that the coupling (Xo, X1) is not optimal.
The proof velocity field v, with Vyp and |Xo
34 Theorem 5.6] fails, since in the direction ii)-iii) the author only shows that any 0 has straight paths X--almost everywhere. However, they then use (34 Lemma 5.9] which requires that the velocity field has straight paths everywhere. Our example shows that this assumption cannot be neglected. The statement of [34]

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Theorem 5.6] is correct if we assume that supp(X;) = R is connected in addition to the other assumptions of the theorem, although we conjecture that the smoothness assumptions on yp (assumed
in C (TRe x (0, 1]) in (34 Theorem 5.6]) can probably be lowered. The corrected statement of [34
Theorem 5.6] reads as follows. The proof is the same as [34]
Theorem 11. Assume that (Xo; X1) is rectifiable and let ii Vip E
X1) 1I). Moreover, suppose that (upp(X.) = R for X; = (1 - X
fulfill that yr: CR2.( x [0,
Then it holds
R (oo, X1) = (Xo, No,Xy X1)
( Xo, X1) = 00 +
(Xo, X1) is an optimal coupling.
Section 6], the author raises the question, whether for (7) (((+1) Rx(Z0' _ the optimality gap
In 34
El|l 1--
inf 7ampa.24 101M
E Zo ||
E
converges to zero for i - oc. The above example gives a negative answer to this question, since leads to constant but strictly positive optimality gap.
Remark 12. Many applications consider the case where f40 is a standard normal distribution. However, we note that we can construct a similar counterexample for this case. To this end let Xo ~ (0, Id) and define
Xo + (-2,2) if (Xo) < 0, (-2.2) if (Xo)1 0,
X1
(12)
see Figure Ic for an illustration. With similar arguments as for the previous example, we can observe that (Xu, X is fixed point of R. but is not optimal. The full statement and a corresponding proof are included as Proposition 19]in in Appendix B Note that also in this example, the support of X, is disconnected for any / 0. is also possible to make more complicated counterexample where the X, have full support but with very irregular transport and potential, see Example 20 in Appendix B
We verify numerically that the couplings discussed in this subsection are indeed fixed points of Ry in Appendix D
4.2 Non-Rectifiable Couplings
In this subsection, consider the of non-rectifiable couplings in more detail. More precisely, we give an example of a non-rectifiable coupling X1) such that the minimizer e, Vy of 7 fulfills [(v= Xo, X1) 0, showing that (8) is again false in this case. Afterwards, provide some sufficient condition for coupling y to be rectifiable.
We consider No = |/1 (0, Id) and define the coupling (Xo: X2) by X1 = -Xo which is illustrated in Figure 2 Then, the next proposition shows that the loss from 77) is indced zero and that (Xo, X1) is nonoptimal. The proof is given in Appendix B
Proposition 13. Let X1) be defined as above. Then, the following holds true.
(i) The minimizer v arg min w, L(velXo. is given by ve(x) (x)e(z) for (r(x)) T-
1-21"
T +| for t # and by vr(x) = 0 0. I z.
(ii) It holds C(v, X1) even though (Xo, X1) is not optimal.
(iii) The coupling (Xo, X1) is not rectifiable, i.e., the ODE i, = vz((e) does not admit a unique solution.
The next theorem provides a sufficient condition which ensures that given coupling is rectifiable. The proof is given in Appendix B
Theorem 14. Let (Xu, X1) be a coupling between pa and μn and denote by PxalX, =xı the condi- tional distribution of Xo given X, = I1. If PxolX is absolutely continuous with a smooth and positive density X (fa), then (Xo, X1) is rectifiable and the solutions v, and of of ( and 77 are smooth.
8

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We highlight two examples, where the assumptions of the theorem are fulfilled.
Example 15. Let be absolutely continuous with smooth density and consider the independent coupling XI) - Po @ fli. Then, the conditional distribution PxolX = Po has smooth density. In particular, the assumptions of Theorem 14 are fulfilled and (Xo, X1) is rectifiable.
Nevertheless, since it is not clear that R( X1) is rectifiable when (1) is rectifiable it is still open whether any sequence generated by (Xx ( xx(+1) E
Figure 2: Illustration of the non- rectifiable coupling XI) with Xo - Id)
R(X))) x remains rectifiable if the initial coupling (x0%) x x()) is rectifiable. The next example shows that any coupling can be made rectifiable by injecting an arbi- trary small amount of noise.
Example 16. For any coupling (Xo. between 140
and Y1 and an independent noise variable W ~ (O, Id), we can define a smoothed coupling (Xf,-xf) == (Xg- + cW, X1) between M == μ4 * *0, e'Id) and μr. By Theorem 14 this coupling is guaranteed to be rectifiable. Note that this procedure alters the left marginal No by the convolution with Gaussian. But if e becomes small, becomes arbitrarily close to /Ao.
we obtain that small loss value [(vr| X1) for V arg min L(w,] [[w,(XXo_5 [x,, X1) does not imply that the velocity field of 2 and coupling (Xo, X1) are close to be optimal. We formalize this finding in the following corollary. The proof is given Appendix B
As a consequence of Proposition 13 and Example 16
Corollary 17. Let (Xo, X1) with Xo = -X1 ~ N(0, Id) be a coupling between po =
(O, Denote by (X6, X() the smoothed coupling from Example [or e 0 and by of arg min = L(w,l X6; Xf) the corresponding velocity field. Then, for any > 0, there exists
> 0 small enough such that [(u)) X6 XG) < and W2 (16. p1) but
L BlIbs
and [f X611 > 4-
Smoothed Rectification
The rectification process proposed by Liu in (34] never ensures that the iterates remain rectifiable, although we understand from the previous section that this property can be crucial. In the follow- ing, we propose smoothing procedure of rectified couplings in order to ensure that the iterates always remain rectifiable. To this end, we require that μı ~ (N Id). More precisely, starting
with some initial coupling (2) 2(0)) we define sequence of couplings (Z)) 2 by defining (Z((tt) z (((+1) R(x(") x()),, where x(() = -I- c,2 Ve,W with W ~ N/ Id)
independent of (Z)"), ( and x z(6) and some G; (0, 1). The following theorem proves that the iteration still optimizes the loss function up to some error which depends on the injected
noise levels ei. The proof is given in Appendix C
Theorem 18. Let (Z zz(1) and (x(") xx(") be defined as above. Denote by ( =
inf C(wel (x(() x(()) ) the loss values in the rectification steps and by Vi ||= dur(a) the second moment of μt. Then, the following holds true.
(i) We have that (x(") x(²)) rectifiable and that (x(" x(()) and (2 z((0) are couplings between pio and μr for all i, i.e., that x(() z(1 14 and x(11 z(60 μn;
(ii) For ēk = KK-1 C₁ it holds
min ()) E E ( + tk).
For constant noise levels e = e, Theorem states that the minimal loss value of the iterates tends to zero up to error which depends linearly on the variance of the injected noise. However, as soon

---

as the noise levels tend to zero, also the averages čK converge to zero. In particular, for summable
noise levels with Di= G C < oo, we have that min. K-I L E (L) which is the same
rate as in (3 without noise injection. By the same proof, Theorem T8 a8 holds true if we replace the step
) = R x x by (7) (1(11) 2 ) Rx(x0' x
x
Additionally, we note that we always have (X,) Rd within the smoothed rectification. Therefore, also the counterexamples from Section 4.1 not apply in this case. However, remains still unclear whether the smoothed rectification converges to optimal transport. Numerically, we test the approach in Appendix E on the example from Remark 12, where convergence to the optimal transport indeed seems to be fulfilled.
Conclusions
Even though rectified flows have shown to define efficient generative models in the literature, have seen in this paper that they are not a suitable tool for computing optimal transport maps between two distributions. In particular, we have identified the following two main reasons for that:
Non-optimal fixed points of R,; In the case where the distributions μt have disconnected support, we showed in Section T.T that there exist fixed points of the rectification step, where the resulting velocity field has a potential but does not lead to the optimal coupling. In particular, 34 Theorem 5.6] is not true without additional assumptions. Given that datasets in applications are often disconnected this limits the applicability of rectified flows for computing optimal couplings significantly.
state that the loss function (1) or becomes arbitrary small. However, already in the simple case that (Xo, X1) follows Gaussian distribution, we showed in Corollary 17 that there exist couplings with an arbitrary small loss function which are arbitrary far away from the optimal coupling.
Vanishing loss does not imply optimality: All convergence guarantees in 34 35
Moreover, we studied the assumption that a coupling is rectifiable. While we can indeed give a simple example where this assumption is violated, they are heavily based on symmetry such that they probably will not appear in practice. Indeed, we show that injecting small amount of noise in each step ensures that couplings remain rectifiable and do not alter the thcoretical guarantees.
Limitations and Open Questions While we have shown the existence of non-optimal fixed points of R.p it is unclear to us, to which fixed point the iterative rectification converges. This question will heavily depend on the initial coupling. Here, interesting cases to consider would be the independent coupling and couplings defined by minibatch optimal transport 43 49] Additionally, noisy version of rectified flows was considered in [ [4, 46) in connection with Schrödinger bridges. While in this case convergence to the entropic optimal transport plan is guaranteed, we show in Appendix F numerically that this convergence can become arbitrary slow for small regularization parameters. Similar for rectified flowsm it could be interesting consider the convergence behavior locally around the (regularized) optimal transport plan.
The same limitation applies to non-rectifiable couplings. Even though we showed the existence of non-rectifiable couplings, we do not know so far whether they appear during the iterative rectification when we start with a "smooth" coupling like the independent one. Moreover, the noise injection from Section guaranteeiig the rectifiability throughout the itcrations is so far limited to the case where Mo is Gaussian. While we can show the same convergence result for the rectification with and without noise-injection, these result only states that the loss converges to zero. However, we have seen in Section 4. and Corollary 17 that this is not sufficient to show convergence to the optimal coupling even to a fixed point of Rp-
Finally, we restricted our considerations to the case of the quadratic cost function, while 34 considers more general choices. However, the counterexamples from Section 4.2 are independent of the cost function as the issue arises from the fact that the interpolation X, degenerates. Also the counterexamples from Section 4.1 can be transferred to more general cost functions with very similar arguments.
10

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Acknowledgements
JH is funded by the Deutsche Forschungsgemmiinccaatt (DFG, German Research Foundation) within project no 530824055. AC and JD acknowledge the support of the "France 2030" funding ANR-23- PEIA-0004 ("PDE-AI"). AC also acknowledges the support of the "France 2030" funding ANR-23- IACL-0008 ("PRIAIJRIE-PSAT),
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13

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A Proofs from Section And 3
Proof of Theorem from If (Xo, Xi) is rectifiable, we use the notation Ze for the random variable defined as Zo Xo and Zi (Z)).
2 For the proof, we use the representation of o via the conditional expectation
(i) By definition it holds that
U/ (x) E[AX, b- - (AX, bX, bb= x] B[A(X) - Xo)IAX, +b= z]
A XolX, A (x b)] (A (A-1xx (x b)).
If (Xo, X2) is rectifiable, we observe for Z,4. = AZ, + b that Zooi AZo + = AXo b = Zn and
(z) = AZ: + = Au;(Z,) bb= (n, (Z - b]) = es (2,4.0)
Thus, we have (AZo + b, AZI b) R(AX. + AX, +).
iii We have by definition that
2 E(x, + - X, + tb 21
= = - Xx = x - tb] + = -v(( (bb] + b.
If (Xu, is rectifiable, we observe for Z. = -2 + tb that Z3 = Zo = Xu and
(7(x) - 2. + b - Ave (2,) + = (z (b) + v
Thus, we have (Zo, 21 + b) = R(Xo, X1 + +))
iii) Denote (Xp. Xf) (Xo, cx1). Then, it holds
X = (1 x)x + +tx (1 - Xo + tcX1
1 1-6
le
= (1 - + te)
Xo
X1 = 10--4 -1+
1-t+ + tc
1 - t - tc
with r= = II-+-c0 Thus, we have
1
o((x)
(E[x) Xi- x]] -))
(Blex [[- - =x] =))
1-1 1
1
1
cR X1
x
t
1 1- t + tc
c-1
E X,
t
1- t
1 t + tc
1- 1-t+b tc
x
C- 1
1 tc Ur
(1-t+ tc
+ 1 1- + tc
If (Xo. X1) is rectifiable, we observe for 2i = (1 - 1 + (c) 2r that
2 = (1 - + tc) + (c- 1)2, = (1 - + (2,) + (c 1)2,
C
(I-(+le) + +1
c-1 Zi = =f((2))
Ur
1- 1-ttle tc
where we used (1-1+16)= Thus, we have cZ,) = R(xo,
14

---

Proof of Theorem (0, ) with
3
We obtain by the calculation rules for Gaussian distributions that (Xe, X1) ~
Σg
E1t
= ( (1 (1-t)ld rId
Eo
E10 Σ1
)
(1
t
0
Σ1
Id
Evi
fld
Id
(1 - 2) + []) + Σ10) + 1
(1- E10 + 131
(1 E) + +111
Σ
Thus, we have that X T-E (E[xi x x) proves the first part.
x] En Eo x. Inserting this formula in vx(x) =
For part (ii), note that we will see later in Example 15 that (Xo, X1) is rectifiable. Then, the claim
was proven for En Id in 44 Proposition 4.1 The general case follows from Theorem 2 (i). L
Proof of Proposition Let 02 = arg min... [(w and define by <x the solution of the
ODE (xx)
with initial condition 20 =0(x) x. By Brenier's theorem [9) it suffices to show
that 21 R + I. has a convex potential, which is equivalent to 21 being monotone increasing. Assume in contrary that there exist 2 such that 21 (ā) 21 (z). Since the mapping f(t) (x) 24 (z)
is continuous and f(0) = I - x < 0 < =1( 21 (x) = f(1), there exists some E (0. 1) such that (t) = 0. This implies that 24 (8) = 24 (z) which means that the ODE (x))) = ey (2 (x)) admits
crossing paths and contradicts the uniqueness of the solution.
Writing Z the latent variable for the mixture, we can always write |[ = x] = | 2, X] x = Σ piz = k|X, = x]B[[x|XX I, Z = k). Using
Proof of Theurem 5
ey = - ( X = x] 2)) follows that
K
(x()) = Σ
plz = x|x, = xx) = Σ (z)
Arm
Proof of Proposition 8
For part (i), let W. T pr and be the solution of ① Then have that
Cw,,XN, X1) =
E[l (/www(X3 (Xi) X1 - Xoll²] dt
EE| (X))) (xx)) + (x())) - X1 + xo
E[|l (X (x))) dt S Ex)le(())) Xx Xull 11
+ 2 [ (X(, 0. (X,,, u (X,) X1 Xa)} dt
It remains to show that the last term is zero. To this end, note that
(X,, - 21 (x.,, - X1 + Xoy]
E [Ble (x) - (x..). ue (X,)) - X1 + [x.] E (Xx, (X.), E(v. -X, Xx)xx,))
E [Xx) - vx.),,, (.,))--xxX))))
(X,) E[X, XoIX 0,
where the first step comes from the properties of conditional expectations, the second and third step uses that we (Xt) and vy (X,) are mm-meauuablee and the last step uses that (X,,) = E[X] - Xo X.].
For part (ii) and (iii), we prove here that one can define, in weak form, solutions to We introduce the measure dt & μs, defined by
f (t(t x: & n4 S.' (el,, (((1)
15

---

for any test function & E Ce( 1] x )"), and the space (dt & (2) (4)) Consider
now minimizing sequence for (7). of the form (Vp")n21 where each function yo C ([0, 1] x R Thanks to part (i), we know that (Vp")221 also a minimizing sequence for
inf {%/ J [V - 1/ u/ldd up E C 1] x Rd))
(13)
where Us is the solution of (T), which is such that (3) holds (in the distributional sense). Up subsequence, Vup" has a weak limit w in (dl @ pe). which is such that
Ilu, [/// Rd
"ou.--wl' dp < lim inf ( JR
IV
vell
Now, thanks to Mazur's 's lemma (see, e.g., the text book [3. Lem 8. 8.14]), w is also the strong limit of convex combinations of the Vy": there exist fn.m E [0, 1] such that for each n. Em 6n.m = 1 and all f nm. vanish but finite number, and defining 6" : Σn, fn,m 4 one has that V Vi" + w strongly in (dt & pt). Obviously, (V m²) is also minimizing sequence for D. and by construction, one has:
S/
/1/ ||w, vyl1 dp = lim / /
|V - -v:ll vell2 (value of 13)
Let now & E C2 ([0, 1] x I and s RR ||||| < one has
1. Rd ww + sVb. - v. lim J /
[V + sti), - velld 2 (value of (33
Rdd
Differentiating
0 we deduce that for any test function 4.
L.' Vib - (we - ut) = 0
which precisely means that in the distributional sense,
div(wape) div(p)
showing that the continuity equation 3 holds with the speed replaced with wy. In addition, for a.e. E [0, 1]. by construction, wy belongs to Ty, so that by uniqueness the vector field with minimal norm defined in 4 Thm. 8.3.1] (see also equation (8.0.1)). Observe that without any knowledge on the regularity of w. it is unclear how to associate a unique transport map X (1,,z) satisfying X = (X), (0, = x for all x.
Part (iv) follows from the facts that any wi = Vi E (px) belongs to Ty, and that of is the limit of minimizing sequence
B Proofs from Section
Proof of Proposition 10 Define the potentials ypr and c.. such that
(x) = (z. (0, -)))
if < -1, and pe(x) o > 1,
(z, (-4,0), (z, (4,0)),
if < -0.5, if > 0.5,
(x. (0,2)) 2)),
and extend them smoothly to the full I Then, the velocity fields = V and ūN = V are given by
(x) (0. 2)), if < 11, and (x(x) (0. if 1,
(4,0), (-4,0).
if < -0.5
if if > 0.5,
Plugging in the definitions of (Xo, and (Xo, X1) this yields that [(v, X1) [6, X2) =
and that X, = (X,) and Xt = (x,)) such that (Xo, X1) and (Xo, X1) are fixed points of R On the other side, we directly obtain that
E||li; xol1 = 16 4 E||| Xol1
16

---

Proposition 19 (Formal Statement of Remark 12 Let d = Xu ~ (O Id) and define
X1 = Xo + (-2,2)
(X)) <0, Xo - (-2,2) (X)) >
Then it holds (Xo, X1) (x X1), but (Xo, X1) is not optimal coupling.
Proof. For 1 >> 0 we define the potentials
(z, (-2,2)
if -4, 21 1
(p(())
(ar, (2, -2))
and extend them smoothly the full Rd. Then, the velocity fields Uy Vip are given by
(-2,2)
if < -t,
(x))
(2, -2) if
Noting that (Xu, Xx, X1) is given by plugging in the definition of X1) this yields [(vl Xo. X1) = and X, vy (X+) such that (Xo, Xi) = ox,, X1).
Next, we show that (Xo, X1) is not optimal by contradiction. To this end, note that X₁ TXx)) where T(x) is defined for (21, 22) as
x + (-2,2) x (-2,2)
if < 0, if x1 >
T(x)
Assuming that (Xp, X2) is optimal, there exists by Brenier's theorem 9 some convex function ub: IR E such that Vis almost everywhere.
Then, we should have (²xx) = (x) == /z/12/2- + ((-2,2),x) + c- for # < 0 and vxxx) = vi (x) == |2 (( 2), ar) c for 2n 0, for constants ct , E R. Yet if we want s to be continuous across the interface {=1 = 0}. we need that (z) = (x) there, which boils
down to ((-2,)), x) constant: this means that the interface should be normal to (-2, which clearly is not the case.
Example 20 (Counterexample regularity/support. We show here that it is also possible to find path (o,,1) of measures with full support at all time, which fixed point for the rectification. However, the corresponding potential are not regular (or rather, do not really exist) and we do not expect the existence of transport map in this case. Let (20) be such that Q² {zn E N). and such that both {In n E 2N} and (In n E 2N + 1} are dense in R2, Let then (an)n20 a sequence of positive numbers with Σn On co; C1 two vectors (specified later), and define:
a, a P1
Σ nE2N
an den +
Σ an
WE2N+1
By slight abuse of notation we denote n[2] the remainder mod 2, and see that the straight trajectory between (Ag and j4 is simply
+00
Ne
Σ
an +langj
E [0, 1),
which satisfies the continuity equation with the speed (x))) Cn(2) if x En + ten[2]. 2 0. Actually, this is true only if the paths do not cross, that is,
Vn 2N, Vm = 2N + 1, Vt E (0, 1), Im - In t(co e1).
Choosing eo, G1 such that ég - e (1, with & & Q clearly ensures this property, since I' Fr, =(cp- e1), the first coordinate imposes that t @ and then the second that & Q contradiction.
Since for all t E [0. 1], (z, + tenpp))n2 is dense in IR, the support (defined classically as the complement of the largest open negligible set) of pr is Ig Then, solving ! will return the same
17

---

speed. In addition, we can check that Us E THe at all time, so that solving will also return the same speed and He a fixed point of the rectification process.
To see that Up E Tp, (for a given time & E [0, 1)), we observe that for fixed N, one can find small radii Ph n 0,.... N, such that the balls (a + tenp)), Pw) are disjoint for n I N. Then, y E C (B, 1); [0, II) is a smooth cut-off function with compact support, with v = neighborhood of 0, the function
of (x)
Σn'
Zn,t Pi
en|2] - (x-) CC (IR*)
(where we denoted Tntt == Un + 1en/2] for n
0) is such that vz(x) = Vipi" (x) for = Untts
n O,...,NN. N. One has, for x E IR?
Visi (2) = Cn/2)")
x - In.t (=
+ en(2]
In.t
Vn
I In,1
PN where Vn((x= xn,1)//)) is not vanishing,
PN
P.
and since || Fn < py
VV (x)1 S
max([leill- |eell] |V oo)) Hence, one has:
(x)) ex) f |V N
(x) (x)
E an m 1122)) (V))l +0 a>N
as N - +oo. Notice that possibly reducing the radii N. the same construction will produce a function ip E C. (10, 1] x (R with Vip + Ut in (dt @ pe) (in both time and space).
Proof of Proposition 13 Part (i) follows from Theorem 3 with Eo E₁ Id and Σoi Eio Idd. Moreover, holds that
[(ve Xo, X1) B[I [[[ 2t) + 2Xx Elll 2X, + +2Xxo111 0,
+ + X1 Xo
which shows (ii). For (iii) assume that the ODE (z) (=))((z)) with 2x x admits a unique solution. Since de is locally Lipschitz continuous on t E [0, 2 this solution is determined by
(x) = (1 - 2 fort t E (0,5) which implies that 21 (x) = 0 for all x. However, since 2 solves the ODE, we have for all x E RRd that
21 (z) 21 =x_((x))
2x
lim c-
= =2(()) (2 = my (0) = 0.
c
This is contradiction.
Proof of Theorem 14
For the solution of (T) suffices by to show that E[X]|X, x is smooth in for all t < 1. Using the transformation
I)ld ld (x,Ix==((x) (1 - tja PxpPxi/X==
Xx
(1
Xo
X1
Id
X1
we obtain by the transformation formula the conditional distribution Px, X1=y
has the density
(1-
x xx=ty ty
(1-t)
Thus, the distribution μ of X, is absolutely continuous with density
(x) PX.IXI=V
(T-7)
x ty dup (v).
Consequently, we obtain by Bayes' theorem that for x within the support of X, it holds (=px)xx==,(()dx0000
E|X = 2] = [ (z)dp1(1)
(x)
[UXXoXX
dun
x-
dur(v)
18

---

which admits the same smoothness as pxp Xx (x)().)
For the solution of 77. note that by the first part Ut is smooth for 1 (by symmetry, if Px (X smooth with full support for Ao-a.ee zoo U is smooth for > 0). In this case, using the projection property (i) from Proposition 8 we easily derive (1) that for a.e. Pi Hae (RR); (2) that Wy Vyp for such E; (3) that Pp:, as a solution of
Ap: V 1n (vt Vip() + div 12
C in RRd by standard bootstrap argument (since for any h 2 stt Hie implies yr: e
Hr ) In particular, we can integrate = wy (X) for any X(, x) = x and build unique
corresponding transport map. This map itself is smooth if both Px,l X mm and Px smooth and positive.
Σo Eg
Σιo Σi
Proof of Corollary 17 Note that (Xp. Xf) O), for Σ
with Eg (1
Id, 2 ld and Noi Σio -dd. Hence, have by Theorem
(i) that the minimizer of in
① is given by
of = 21 - c 1)2
1)²
2 x
(1 () t) (2t 1)2)
for + Since of a multiple of the identity. it has potential such that of is the minimizer in 7 Moreover we have that for c -> 0 it holds
1
f.
2
dt (' E |1xx01]] dt 4.
X1
21
Thus for small enough we have that fo (X:) > 4 e. Similarly, we have for e - 0 that
Ef X,5 E Xo cW] Elll 2X. cW| 2 | 4
such that for small enough it holds Xf - X=11 > - On the other side it holds that MÚ = -N (1 + Idd and μ1 =AV Id) such that by the explicit form of the Wasserstein distance
of two Gaussians we have that (:2 p1) + 0 for - O. Finally, the loss function fulfills
[(1 Xi) / ([(lex()))) Xi dd
2
Ell
Xx
X1
Xo Pldt
1-2t
E1|
2Xo
2Xool
Thus for e small enough the loss is smaller than
C Proof of Theorem 18
(x) is rectifiable by Theorem 14 and Example 16 Moreover, have by the definition, that x is the sum of two independent Gaussian random variables with zero mean and covariances (1 - c)ld and cld. Thus, x(( ~ Id) Since R preserves the marginals, this yields the claim.
Proof of part (i). We have that (X) x
For the proof of the second part, we need the following lemma.
Lemma 21. We have [ x E[|Z 2 +
Proof. Since W is independent of (z(() (z())) it holds
E x E1|lz(0) - vI 2. 2(1) - -V@W*(12)
E||1Z(0 vi 7()(1) [01 + + (
=] - /1 - 2,2( 1 +
19

---

We bound the remaining term as
BillZ - -1- -
- / - (1 - v1 -11-v--)22 |3
= =E|IZ z 112 2 +(1 - v1 11
(1 - v1 - 4) (E((Z
Ell 2 (P)
Using Hölder's 's inequality and the estimate 1 v1 G S c₁ this is smaller or equal than
E 5 - 1 + (1 - vI - 2 21 0 11
+ (1 - -I - ci) (V |111
17 | 2 111 1P)
- E - 1| + c T 11
Ell 2 ( [EE|l e0 ||| Z 10 191
Since by definition it holds that / n and 2 ,h, we have 11 V? and E||l 7 [0 ||
= Vi2 = such that the above formula is is equal to
EII - +
Now, we can proof the second part of Theorem
Proof of Theorem 18
(ii).
By
34 eqt (28)1. we have that
E X|| 0 |s] --||
(6+1) (6+1)
119] 11(t (0)
Thus, we get by Lemma 21 that
(
s [|] |||| 1 E1|22| (11(((+)))
+ c + ot
z| - 11 | i+11 2 1 (2 + Vide.
Summing this equation up for i 0, ..., K we obtain
K-1
Σ
L
<
B1|1Z(0)
0 | Elll((k) - z 11 + K(2 + Vi)ex
1=0
B -
2 0 ز + K(2 + V1)xEK-
Noting that the minimum of the (8) is always smaller or equal than the mean, this yields
1 z | + (2 (2+K/)) E cO ( + +εx) +xx)
min
K
K
D Numerical Verification of the Counterexamples
In this appendix, we numerically verify the findings from Section 4.1 To this end, we consider the iteration (6+1) xxxxxx+++ x((+1)
((x x(0)), which corresponds to minimizing
0
D(2 1 X x(()
Ellly ((1-))x + +xx(1)
x((" + x
For the implementation, parameterize the velocity fields ( v, ( (x) - (() (1, x)), where (8) is a fully connected ReLU neural network with three hidden layers and 512 neurons per hidden layer. We minimize the loss functions [(v:()) (x(() xx()) with the Adam optimizer for 40000 steps with batch size 256 and initial step size 10 which is reduced by a factor 0.995 every 40 steps. For the
initial coupling (Xx() _x(()) we consider three cases. These are
20

---

(a) Step 1
(b) Step 2
(c) Step
Figure Iterative rectification for the example from initialized with the optimal coupling (Xo, X1).
Step
Step
Step
Figure 4: Iterative rectification for the example from
initialized with the non-optimal fixed point
(Xo. of Rp.
(a) Step
Step
Step
Figure 5: Iterative rectification for the example from Section 4.1 initialized with the independent coupling.
the optimal coupling (Xo, X1) from
the non-optimal coupling (Xo, X1) from and
the independent coupling, i.e., we choose x=(0) and x to be independent.
The results are given Figure (optimal coupling). Figure (non-optimal fixed point R2,) and Figure (independent coupling), where we always plot samples from x-()) the trajectory (1) of (() (Z) and the final samples x We plot these results for the first, second and fifth step, i... for 1, 2 and i == 5. The results verify that both couplings from (TT) are indeed fixed points of Rp. we start with the independent coupling, it does not converge to either of them within the first couple of steps. Instead, it splits the mass of both modes from (lD and transports it to either of the modes from (41. We observe that approximating this non-smooth velocity is hard for neural networks such that numerical errors appear. Nevertheless, it seems to be close fixed point of Rp. since the coupling does not change much throughout the iterations.
We redo the experiment for the example from Remark 12] Note that here, we do not have access to the analytical form the optimal coupling. As a remedy, we generate two datasets of 20000 points from nn and bn and compute the discrete optimal coupling between them using the Python Optimal Transport (POT) package [17]. The results are given Figure (optimal coupling), Figure (non-optimal fixed point of R,,) and Figure g (independent coupling). The black line indicates
21

---

(a) Step
(b) Step
(c) Step
Figure 6: Iterative rectification for the example in Remark 12 initialized with the optimal coupling.
(a) Step 1
(b) Step 2
(c) Step
Figure 7: Iterative rectification for the example in Remark 12 initialized with the non-optimal fixed
point of R2 from 12.
(a) Step 1
(b) Step 2
(c) Step
Figure Iterative rectification for the example in Remark 12 initialized with the independent coupling.
22

---

Table 1:
Caption
step 1
step 2
step 3
step 4
step 5
0
2.84
2.81
2.78
2.80
2.79
0.01
2.79
2.75
2.73
2.70
2.68
0.05
2.73
2.68
2.69
2.68
2.67
0.1
2.70
2.69
2.70
2.70
2.71
0.2
2.68
2.64
2.65
2.67
2.66
the separation between the left and right half-plane. We can see that the non-optimal coupling from Remark T2 is indeed fixed point of R2 and transports the mass the left and right half- plane separately. In contrast, the optimal coupling splits the mass differently. we start with the independent coupling, the process seems to converge the optimal coupling, even though the lines are not fully straight even after five iterations.
E Numerical Experiments for the Smoothed Rectification
We consider the example of Remark 12 and apply the smoothed rectification with parameter = for co E O, 0.01, 0.05. 0.1, 0.2, where cp = resembles the case of the standard rectification Ry. The
initial coupling (X) x1 ) is set to the non-optimal fixed point of Ry from Remark [2 (defined
eqt (T2). visualization
Figure [c.. We report the transport distance [R _x1 |
versus the number of steps of the smoothed rectification procedure in Table [ A lower transport
distance indicates that the coupling is closer to optimal transport. The optimal coupling admits a transport distance of 2.66 (evaluated based on 20000 samples using the POT package (17). Since all experiments are initialized with the same coupling, the first step coincides over all runs, independent of the noise level. Due numerical errors in the marginals, the transport distance is sometimes slightly smaller than the analytical optimum. Overall, we observe that the smoothed rectification indeed escapes the non-optimal fixed points from Section 4.1 However, the convergence becomes slower if the noise level approaches zero. The generated couplings a visualized in for the different choices of co in the Figures 9 to
Relation to Diffusion Schrödinger Bridge Matching
In this appendix, we first briefly describe the relation of rectified flows to Schrödinger bridge matching in Section F.II While for this method, in contrast to rectified flows, the convergence to the regularized optimal transport plan is guaranteed, our counterexamples suggest that the convergence speed becomes arbitrary slow for these examples when the regularization strength tends to zero. We justify this hypothesis numerically in Subsection F.2
F1 Diffusion Schrödinger Bridge Matching
In 1 46), the authors introduce a noisy version of rectified flows, where the interpolation variables X₃ are replaced by noisy versions given as
X, = - ()Xo LX1 + V 1)Z, Z ~~,((011)
By denoting the law of & by ft and choosing the drift term v2(xx) E[X, -XX--x/XX== can be shown that the Pakker-Planck equation
3,44 + div(r.p.) VeDp,
fulfilled, see 46 for details. In particular, samples from X, can be generated by sampling from the stochastic differential equation
dY, u. (Yi)dt + VedW,
(14)
where W, denotes Brownian motion. Denoting by (Zz)<e(o.1) solution of this SDE, we obtain the noisy rectification R. X,) (Yo, Y1), where 0 recovers the standard rectification R.
23

---

(a) Step
(b) Step
(c) Step
(d) Step
(e) Step
Figure 9: Smoothed rectification for the example in Remark 12 with co
0.01.
(a) Step 1
(b) Step 2
(c) Step 5
(d) Step
(e) Step
Figure 10: Smoothed rectification for the example in Remark 12 with Co
0.05.
(a) Step 1
(b) Step 2
(c) Step 5
(d) Step
(e) Step
Figure 11: Smoothed rectification for the example in Remark 12 with Co 0.1.
(a) Step 1
(b) Step 2
(c) Step
(d) Step
Step
Figure 12: Smoothed rectification for the example in Remark 12 with Co 0.2.
24

---

Optimal Coupling
Non-optimal Coupling
Independent Coupling
2.04
322
2.03
3.9-
2.02
tarspont dsunse
3.5
transport dpannce
2.8-
2.01
2.5
2.30
2.4
1:40-
1.35
22-
1.87
2.0
2.0
20
20
55
70
10
Figure 13: Transport distance for DSBM for the example from Section 4.1 initialized with the
coupling (Xo, X2) from (IT) (left), the coupling (Xo, X1) from (1]) (middle) and the independent coupling (right).
Optimal Coupling
Non-optimal Coupling
Independent Coupling
0.16
mug cene Jenery drannd
margrai entr denegy dennce)
0.14
0.32
0.25
9.4
0.50
0.20
0.58
0.23
0.3
0.36-
AA
0:30
82-
0.34
9.3
0.02
0.00
0.00
10
20
20
terations
number Firatioos
Figure 14: Error introduced in the marginals introduced by errors of DSBM measured in the energy distance for the example from Section 4.1 initialized with the coupling (Xo, X,) from 11 (left), the coupling (Xo, X1) from [1] (middle) and the independent coupling (right).
Now, the authors of 44) showed that the iterative noisy rectification given by (x(++t),{(**)) (xx(") xx(") converges to a solution of the Schrödinger bridge problem, which equivalent to entropically regularized optimal transport, see [31] for overview. Based this observation, they propose a variation of the rectified flows algorithm, called Diffusion Schrödinger Bride Matching (DSBM), where the velocity field iy is approximated by neural network which is then trained by the loss function
[(w X1) = L.¹ IE wi)()))
Uz E arg min w, (.E))(())
dt. (15)
1-t
Again, for e this loss function coincides with the loss function
I for rectified flows.
In practice, for the sake of numerical stability, the authors of (46) propose to train not only the drift of
the SDE (14), but also the time- linerreversal which is has the drift (z) B|X,- (oK--x,,, -x which leads
an analogous loss function as (5). Note that related generative models for computing Schrödinger
bridges were proposed in 15 41 50
F22 Numerical Examples with Diffusion Schrödinger Bridge Matching
Next, we numerically investigate the convergence speed of DSBM for our example IT.
To this end,
we run DSBM for 50 iterations where the initial coupling is given (as in Appendix D by
the optimal coupling (Xo, X1) from
the non-optimal coupling (Xo, Xx)) from and
the independent coupling, i.e., we choose x- (0)
and x to be independent.
We run this test for different regularization parameters e E (0.05, 0.1, 1}. For evaluating the results, plot two error measures versus the number of iterations:
25

---

(a) Step 1
(b) Step
(c) Step 20
(d) Step 50
Figure 15: Iterative rectification with the DSBM algorithm [46
0.1) for the example from
Section 4.1 initialized with the optimal coupling.
(a) Step 1
(b) Step
(c) Step 20
(d) Step 50
Figure 16: Iterative rectification with the DSBM algorithm 46 (c 0.1) for the example from
Section 4.1
initialized with the coupling (Xo, X1) from
(a) Step
(b) Step
Step 20
(d) Step
Figure 17: Iterative rectification with the DSBM algorithm 146 (c 0.1) for the example from Section initialized with the independent coupling.
First, we consider the transport cost E||
1 of the coupling (x)² xx(")))))))) .x()).
This quantity measures how close the coupling is to the optimal transport plan.
Second, we consider the distance the distributions ( of x and μ of x to the
original distributions ni and μu. To this end, we evaluate the energy distance
E(P,V) [ |2- v\(d(=- - v)
1/2
between do and ( and between μl and 14( This quantity measures the error which is introduced by the neural network approximation of the drift terms, the optimization error in the loss function and the sampling error in the SDE simulation.
For both evaluation metrics, we discretize the expectation by 50000 samples. The results are given in Figure 13 and [4 Additionally, we plot the corresponding coupling and trajectories for 0.1 and iteration E I 5, 20, 50} in Figure 15 (optimal coupling from (IIb). Figure 16 (non-optimal and Figure 17 (independent coupling). We observe that for our counterexample from Section 4.1 a very large regularization parameter is required in order to converge to the entropic optimal transport plan. However, when initialized with the independent coupling DSBM seems to converge in a reasonable time even for moderate e. However, for larger also the error introduced in the distributions of x and ( increases. In summary, the examples show that cannot hope for reasonable global convergence rates the DSBM algorithm. Whether such rates can be derived locally, remains an open question.
coupling from
26

---

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Reviews:
[{"review_id": "on8yn69CPG", "reviewer": "Reviewer_3EWA", "summary": "The authors clarify the relationship between couplings obtained from rectified flows and optimal transport in the setting of vector fields induced by potentials. In particular, they demonstrate via counter-examples, that fixed points and zero-minimisers of the rectified flow objective do not necessarily obtain optimal couplings. This corrects a previous result in the literature which claims the opposite. Moreover, the authors clarify conditions when solutions to the rectified flow objective are indeed rectifiable, and propose a procedure for ensuring rectifiability.", "questions": "To what extend can the counter-examples/proofs/intuition from the paper be generalised to rectified flows under the setting of a generic cost function, as presented in [1]?\n\n[1] Qiang Liu (2022) Rectified Flow: A Marginal Preserving Approach to Optimal Transport", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n- While the main contribution presented in the paper is probably already known to theorists in this area, due to the simplicity of the counter-examples presented in the paper, the paper is still a nice addition to the literature for providing a single citable reference, for newcomers to the area, in clarifying the extent to which rectified flows can recover optimal transport (as well as correcting a previous mistake in the literature). \n\nWeaknesses:\n\n- The structure of the paper is a bit confusing. For instance, what is the purpose of Section 3.1? This section appears abruptly from the previous discussion of rectified flows with the gradient potential conditions, and appears to have little connection with the rest of the presentation. From inspection, it appears that Section 3.1 is only used in the proofs of later results. The layout disrupts from the presentation of the main argument of the paper and makes the paper harder to follow.  \n- Moreover, the paper is a bit overstuffed with off-hand remarks of results which have no accompanying justification. For example:\n    - Line 193: \"$v^{A,b}_t$ has a potential if and only if the Jacobian is symmetric.\" - what is the justification of this? I'm not sure how this immediately follows?\n    - Line 244: \"The statement of...\". The full statement of the corrected proposition and proof would be valuable to include in this paper to resolve any ambiguity, instead of leaving this offhand remark halfway in the paper  \n- For Theorem 2 and Theorem 4, covariance matrices are referred to as both $\\Sigma_0$ and $\\Sigma_{00}$. Why is this inconsistent? \n- In line 184, I do not understand how the independent coupling of $(X_0, X_1)$ follows a GMM distribution as $X_1$ follows a discrete measure. Hence, I do not see how Theorem 4 can be applied to this case.\n- The notation in Remark 5 is confusing. I assume that here $\\varphi_t^a$ is referring to the transformation by adding the constant $a$. However, in Theorem 1, the notation for this transformation uses the letter $b$ in $v_t^b$. I don't see any reason for changing up the notation here, apart from to reduce clarity.", "was_revised": true, "final_justification": "The authors have been able to address my concerns from the review making me increase my score to 5 for acceptance. I believe this will be a useful addition to the literature to help clarify misconception about rectified flows.", "rebuttal": "Thank you for having read accurately our paper and for your comments! We reply to all of these separately.\n\n> While the main contribution presented in the paper is probably already known to theorists in this area [...]\n\nWe are not aware of any comparable results from the literature. In addition, the slightly incorrect results which we discuss are cited in plenty of papers, without any discussion of their validity. We really feel that pointing this out is very useful to the community.\n\n> The structure of the paper is a bit confusing. For instance, what is the purpose of Section 3.1? This section appears abruptly from the previous discussion of rectified flows with the gradient potential conditions, and appears to have little connection with the rest of the presentation. From inspection, it appears that Section 3.1 is only used in the proofs of later results. The layout disrupts from the presentation of the main argument of the paper and makes the paper harder to follow. \n\nSome relations between rectified flows and optimal transport are already true for the case without the gradient potential condition. Section 3.1 gathers these foundational results, including\n\n- Rectified flows and optimal transport share some invariance properties (Theorem 1 (ii) and (iii))\n- The observation that, in both the Gaussian setting (Theorem 2 (ii)) and the one-dimensional case (Proposition 3), the optimal coupling is achieved after a single rectification step.\n\nAs a side effect, these results are quite useful to understand the basic behavior of rectified flows and are consequently used in the proofs of subsequent results.\n\nThe current structure aims to provide a general overview of the literature in Section 2, followed by our contributions in the subsequent sections. However, we recognize that a revised layout could enhance accessibility from a pedagogical perspective. In the next revision, we consider to restructure Sections 2 and 3 as follows:\n\n- Section 2 will present the general background, excluding the discussion on the gradient potential condition (currently lines 88–119), and incorporate the results presently found in Section 3.1.\n- Section 3 will begin with the background on the gradient potential condition (lines 120–137), followed by the results currently contained in Section 3.2.\n\n> Line 193: \"$v^{A,b}_t$ has a potential if and only if the Jacobian is symmetric.\" - what is the justification of this? I'm not sure how this immediately follows?\n\nHere we refer to the fact that a vector field admits a potential (locally or in a \"simple\" topology) if and only if the Jacobian is symmetric (such vector fields are often referred as \"conservative\"). This is a non-trivial, yet well-known result, contained in many text books: One direction immediately follows from the fact that the Jacobian of the vector field is the Hessian of the potential and consequently symmetric (known as Schwarz's theorem). The other direction is known as Poincaré's lemma (and equivalent to the property that the circulation of such field along a closed loop vanishes). In the next revision, we will add an additional sentence that we use the above equivalence and add a reference to some text book.\n\n> Line 244: \"The statement of...\". The full statement of the corrected proposition and proof would be valuable to include in this paper to resolve any ambiguity, instead of leaving this offhand remark halfway in the paper\n\nWe agree that presenting the corrected statement could be beneficial. In the next revision, we will include the following version of Thm 5.6 from [1] for $c(x-y)=\\\\|x-y\\\\|^2$ (where the additonal assumption is marked italic):\n\nAssume that $(X_0,X_1)$ is rectifiable and let $v_t=\\nabla \\varphi_t\\in\\mathrm{argmin}_{w_t=\\nabla \\psi_t}\\mathcal L(w_t|X_0,X_1)$ fulfill that $\\varphi_t\\in C^{2,1}(\\mathbb R^d\\times[0,1])$. _Moreover, suppose that $\\mathrm{supp}(X_t)=\\mathbb R^d$ for $X_t=(1-t)X_0+tX_1$_. Then the following are equivalent:\n- $\\mathcal R_p(X_0,X_1)=(X_0,X_1)$,\n- $\\mathcal L(v_t|X_0,X_1)=0$,\n- $(X_0,X_1)$ is an optimal coupling.\n\nThe proof is mainly the same as in [1]. But we will add an appendix, which provides a more detailed explanation on this point.\n\n> For Theorem 2 and Theorem 4, covariance matrices are referred to as both $X_0$ and $X_{00}$. Why is this inconsistent? \n\nThis is a typo, thank you for pointing this out. We will correct it within the next revision.\n\n> In line 184, I do not understand how the independent coupling of $(X_0,X_1)$ follows a GMM distribution as $X_1$ follows a discrete measure. Hence, I do not see how Theorem 4 can be applied to this case.\n\nHere we use the definition of the multivariate normal distribution which includes singular covariance matrices. More precisely, we say that a $d$-dimensional random variable $X$ follows normal distribution if and only if all one-dimensional projections $a^T X$ follow a normal distribution with the notation that $\\delta_x=\\mathcal N(x,0)$ for $x\\in\\mathbb{R}$. It can be shown that using this definition, Gaussian distributions can still be identified by their mean and covariance matrix (which is still positivie semi-definite but not necessarily invertible) and that it preserves all computation rules of Gaussian distributions which are used within the statement and proofs of the Theorem 2 and 4. Then, the joint distribution of independent $X_0\\sim\\mathcal N(0,\\mathrm{Id})$ and $X_1\\sim\\sum_{k=1}^K\\frac1K\\delta_{m^k}$ follows the GMM \n$$\\sum_{k=1}^K\\frac1K\\mathcal N\\left(\\left(\\begin{array}{c}0\\\\\\\\m^k\\end{array}\\right),\\left(\\begin{array}{cc}\\mathrm{Id}&0\\\\\\\\0&0\\end{array}\\right)\\right)$$\nIn the next revision, we will add a remark about this and add a reference to some text book introducing Gaussian distributions and their computation rule for singular covariance matrices.\n\n> The notation in Remark 5 is confusing. I assume that here $\\varphi^a_t$ is referring to the transformation by adding the constant $a$. However, in Theorem 1, the notation for this transformation uses the letter $b$ in $v_t^b$. I don't see any reason for changing up the notation here, apart from to reduce clarity.\n\nThis is again a typo, which we will correct in the next revision. We are sorry for the confusion!\n\n> To what extend can the counter-examples/proofs/intuition from the paper be generalised to rectified flows under the setting of a generic cost function, as presented in [1]?\n\nThe counterexample regarding the non-rectifiable transport plan from Section 4.2 works for any cost function fulfilling the assumptions from [1]. The main argument here is that the distribution $\\mu_t$ of $X_t=(1-t)X_0+tX_1$ becomes singular for $t=\\frac12$. Therefore, for $v_t$ fulfilling the continuity equation $\\partial_t \\mu_t+\\mathrm{div}(v_t\\mu_t)=0$ (which is true for the minimizer of the loss function) and for any solution of $\\dot Z_t=v_t(Z_t)$ with $Z_0=X_0$, it holds that $Z_{1/2}=0$ almost surely. In other words, almost-every ODE path collapses to a single point at time $\\frac12$ which contradicts the uniqueness of solutions.\n\nThe counterexample from Section 4.1 might not work \"as-is\" for other cost funcitions (already because it might be possible to construct cost functions such that both couplings from eqt (11) admit the same cost). However, it should be straight-forward to construct analogous examples for other cost functions (which are convex and have a $C^1$ convex conjugate) by the same idea: Construct two locally optimal transport plans which are sepatated by some area where the coupling has zero-mass. Then, glue them together such that the global plan is not optimal.\n\nWithin the next revision, we will add a remark about other cost functions in the Conclusions section.\n\n\nReference:  \n[1] Qiang Liu (2022) Rectified Flow: A Marginal Preserving Approach to Optimal Transport"}, {"review_id": "MV9gchBnqB", "reviewer": "Reviewer_pPGx", "summary": "The authors of this paper focus on further study of rectified flows, and in particular their relation to optimal transport. Rectified flows straighten the paths of a coupling (transport plan in the language of optimal transport) between two distributions. Given that the paths generated by the velocity field of an optimal transport (in the euclidean case) are straight, the goal is to learn an optimal transport through this rectification. The authors show by providing counterexamples that the assumptions for this to hold are stronger than in a previous paper. They also prove invariances for rectified flows under affine transformations, find formulas for rectified flows (which are optimal transports) in the gaussian case, and propose to smooth the starting distribution at each rectification step to ensure that the couplings stay rectifiable.", "questions": "1. Isn't there a mistake in the definition (12) of $X_1$ in Remark 9? Shouldn't the second case also contain $X_0$ instead of $X_1$?\n2. To make sure that I understand the smoothed rectification in Theorem 15, you assume $\\mu_0$ is gaussian, and since rectification preserves the marginals, $X_0$ will remain gaussian, but it is the conditional distribution of $X_0$ (given $X_1$) that must be absolutely continuous with a smooth and positive density, which is why you add an independent Gaussian noise?", "limitations": "1. Counterexamples are provided which mean that stronger assumptions are needed for rectification to converge to an optimal transport, but these stronger assumptions are not investigated. This is of course minor as I think the paper is rich enough in results.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. Many theoretical contributions to the study of rectified flows\n\nWeaknesses:\n1. To improve clarity, Definitions should be included. For example, the rectified coupling is defined on line 110 with a reference to equation (1). I think the paper would have been easier to follow if such important definitions were in Definition sections such as theorems and propositions.", "was_revised": true, "final_justification": "This paper proves useful results about the relation between rectified flows and optimal transport. In my opinion this outweighs the lack of experimentation on large datasets. Furthermore, the rebutal and proposed changes do indeed improve the paper. Therefore I recommend to accept this paper.", "rebuttal": "We would like to thank the reviewer for having read carefuly our paper and their efforts in evaluating it. We provide answers to the specific critiques separately.\n\n> To improve clarity, Definitions should be included. For example, the rectified coupling is defined on line 110 with a reference to equation (1). I think the paper would have been easier to follow if such important definitions were in Definition sections such as theorems and propositions.\n\nThank you for the suggestion! We will revise the paper and put important definitions (like the rectification) into a definition environment.\n\n > Isn't there a mistake in the definition (12) of $X_1$ in Remark 9? Shouldn't the second case also contain instead $X_0$ of $X_1$?\n \nYou are absolutely right! Thank you for pointing out this typo.\n \n> To make sure that I understand the smoothed rectification in Theorem 15, you assume $\\mu_0$ is gaussian, and since rectification preserves the marginals, $X_0$ will remain gaussian, but it is the conditional distribution of $X_0$ (given $X_1$) that must be absolutely continuous with a smooth and positive density, which is why you add an independent Gaussian noise?\n \nCorrect! Adding independent Gaussian noise (and rescaling $Z_0^{(i)}$) in the definition to $X_0^{(i)}$ ensures that the conditional distribution $P_{X_0|X_1=x_1}$ has a smooth positive density such that we can apply Theorem 11. At the same time $X_0$ remains a Gaussian distribution since the rectification preserves the marginals and a sum of two Gaussians remains Gaussian.\n \n> Counterexamples are provided which mean that stronger assumptions are needed for rectification to converge to an optimal transport, but these stronger assumptions are not investigated. This is of course minor as I think the paper is rich enough in results.\n \nWell, we can at least state sufficient assumptions such that Theorem 5.6 from [1] is correct (where the additional assumption is marked italic):\n\nAssume that $(X_0,X_1)$ is rectifiable and let $v_t=\\nabla \\varphi_t\\in\\mathrm{argmin}_{w_t=\\nabla \\psi_t}\\mathcal L(w_t|X_0,X_1)$ fulfill that $\\varphi_t\\in C^{2,1}(\\mathbb R^d\\times[0,1])$. _Moreover, suppose that $\\mathrm{supp}(X_t)=\\mathbb R^d$ for $X_t=(1-t)X_0+tX_1$_. Then the following are equivalent:\n- $\\mathcal R_p(X_0,X_1)=(X_0,X_1)$,\n- $\\mathcal L(v_t|X_0,X_1)=0$,\n- $(X_0,X_1)$ is an optimal coupling.\n\nHowever, we stress that within the iterative rectification $(X_0^{(i+1)},X_1^{(i+1)})=\\mathcal R(X_0^{(i)},X_1^{(i)})$ it is usually intractable to check, whether these assumptions remain fulfilled for all $i$ based on the initial coupling $(X_0^{(0)},X_1^{(0)})$. Moreover, we showed in Corollary 14 that the fact that the loss decays does not necessary imply convergence of the (couplings) to a fixed point of $\\mathcal R$. How (or if at all) the decay of the loss relates to convergence of the couplings to a fixed point remains an open question. In the next revision, we will include the statement of Theorem 5.6 from [1] with the corrected assumptions (also considering the comments of reviewer 3EWA) and extend the explanations on this point in the conclusions."}, {"review_id": "C5fLiCTVq5", "reviewer": "Reviewer_u4yX", "summary": "This paper re-examines the relationship between rectified flows and optimal transport (OT), arguing decisively that rectified flows, even with gradient constraints, are not a reliable method for computing OT solutions. The authors substantiate this claim through counterexamples, including a key scenario where distributions on disconnected supports cause a non-optimal transport plan to become a fixed point of the iterative rectification process. They further show that a \"non-rectifiable\" coupling can achieve zero loss without being optimal, demonstrating that a vanishing training objective does not guarantee optimality. Beyond these refutations, the paper provides a foundational analysis of rectified flows—deriving invariance properties, giving explicit solutions for Gaussian cases, and establishing conditions for rectifiability. Finally, to address the identified issues with ill-posed dynamics, the authors propose a \"smoothed rectification\" procedure that injects noise at each iteration to ensure the transport problem remains well-defined.", "questions": "Does noise injection also help the iterative process escape these non-optimal fixed points? For instance, in your disconnected support counterexample (Fig. 1), if one were to run the smoothed rectification procedure starting from the independent coupling, would it be more likely to converge to the globally optimal transport plan (Fig. 1a) compared to the standard procedure? My evaluation of the paper's practical contribution would be further strengthened if the authors could provide either (a) an empirical study, perhaps by extending the experiments in Appendix B, to show that smoothing helps find the correct OT plan, or (b) a theoretical argument or conjecture on how noise might provide a regularizing effect that favors the true optimum.", "limitations": "Yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. The core contribution—the systematic refutation of the equivalence between rectified flows and OT—is original. The idea of linking failure modes to disconnected supports and non-rectifiable couplings is a novel perspective. \n\n2. The paper has good technical quality. The arguments are backed by mathematical proofs and precisely constructed counterexamples.\n\n3. The paper has good contribution. By demonstrating the limitations of using rectified flows for optimal transport, it may clarify misconceptions and prevents the community from pursuing a flawed direction.\n\nWeaknesses\n1. While the proposed \"smoothed rectification\" method is a valuable idea, its analysis feels preliminary. The theoretical guarantees are shown for the specific case where the source distribution is Gaussian, which may not cover all use cases. More importantly, while the authors prove that their method maintains a good convergence rate for the *loss*, they do not investigate if it actually helps in converging to the *true optimal transport plan*, especially in the challenging cases (like disconnected supports) that they identify.\n\n2. While the counterexample in Section 4.1 based on disconnected supports is technically sound and arguably the paper's most critical contribution, its practical significance is not immediately obvious from the text. The authors should explicitly state why this scenario is not just a theoretical curiosity but a common and fundamental challenge in practice.\n\n3.No experiments demonstrate the proposed smoothed rectification procedure.", "was_revised": true, "final_justification": "This paper studies some basic properties of reflow and reveals its relations to optimal transport. While theoretically sound, it's hard to see the real-world value without using standard benchmarks. Instead of just using examples like Gaussian noise, the authors should use common datasets like MNIST and ImageNet to justify their theory. \n\nSo I stick to my original score.", "rebuttal": "Many thanks for your reading of our paper, and your review, comments and suggestions! We reply to all of your critiques separately.\n\n> While the proposed \"smoothed rectification\" method is a valuable idea, its analysis feels preliminary. The theoretical guarantees are shown for the specific case where the source distribution is Gaussian, which may not cover all use cases. More importantly, while the authors prove that their method maintains a good convergence rate for the loss, they do not investigate if it actually helps in converging to the true optimal transport plan, especially in the challenging cases (like disconnected supports) that they identify.\n\nPlease find our answer to this comment together with the answer to your last comment at the end of our reply.\n\n> While the counterexample in Section 4.1 based on disconnected supports is technically sound and arguably the paper's most critical contribution, its practical significance is not immediately obvious from the text. The authors should explicitly state why this scenario is not just a theoretical curiosity but a common and fundamental challenge in practice.\n\nOne key motivation and application of rectified flows is distribution translation, i.e., constructing a mapping that transports one probability distribution to another. In this setting, connecting rectified flows to **optimal transport** is particularly appealing, as it yields a mapping that is **unique** (under absolute continuity), **stable**, and **interpretable** (minimizing the expected squared distance between inputs and outputs).\n\nHowever, our counterexample demonstrates that this connection **breaks down when the distributions are disconnected**. Since disconnected datasets are common in practice, this indicates that **rectified flows cannot reliably compute optimal transport mappings** in general.\n\nAnother major motivation for rectified flows is the fact that the flow paths tend to become \"straighter\", improving computational efficiency during sampling by simplifying the ODE trajectories. This aspect is largely independent of optimal transport, and our counterexample does not affect this advantage.\n\nIn the next revision, we will extend the comments in introduction, Section 4.1 and Conclusions based on this discussion.\n\n> No experiments demonstrate the proposed smoothed rectification procedure.\n\nIn the next revision, we will extend Appendix B by some numerical experiments in the setting of Remark 9 (see our reply to the next comment, a brief overview over these experiments is given at the end of the rebuttal).\n\n> Does noise injection also help the iterative process escape these non-optimal fixed points? For instance, in your disconnected support counterexample (Fig. 1), if one were to run the smoothed rectification procedure starting from the independent coupling, would it be more likely to converge to the globally optimal transport plan (Fig. 1a) compared to the standard procedure? My evaluation of the paper's practical contribution would be further strengthened if the authors could provide either (a) an empirical study, perhaps by extending the experiments in Appendix B, to show that smoothing helps find the correct OT plan, or (b) a theoretical argument or conjecture on how noise might provide a regularizing effect that favors the true optimum.\n\nFrom a theoretical side, the definition of the smoothed rectification ($X_0^{(i)}=\\sqrt{1-c_i}Z_0^{(i)}+\\sqrt{c_i}W^{(i)}$ for standard Gaussian noise $W^{(i)}$) ensures not only rectifiability, but also that $\\mathrm{supp}(X_t^{(i)})=\\mathbb R^{d}$ (for $t<1$). Consequently, the smoothed rectification indeed overcomes the problem with disconnected supports which is the main idea for the counterexamples from Section 4.1. However, while the smoothed rectification helps for the counterexamples from Section 4.1, we are currently unable to provide a proof that it generally converges to the optimal coupling (for appropriate $c_i$).\n\nWe have now also tested the smoothed rectification for the counterexample from Remark 9 (we do not use the other example from Section 4.1, because the smoothed rectification assumes that $\\mu_0$ is Gaussian). As initialization, we use the non-optimal fixed point from eqt (12). The results (see below) suggest that smoothed rectification does converge to the optimal coupling. Furthermore, the convergence speed significantly decreases as the noise level approaches zero. This is expected since a small change in the coupling results only in the small change in the rectification.\n\nWe will present these findings and the corresponding discussion in Appendix B of the next revision. A brief overview of the experiments and results is given below (for the paper we will also generate figures in the style of Figure 6/7/8 visualizing the resulting couplings).\n\n---\n\n### Additional Experiment with Smoothed Rectification\n\nWe consider the example of Remark 9 and apply the smoothed rectification with parameter $c_k=\\frac{c_0}{k}$ for $c_0\\in\\{0,0.01,0.05,0.1,0.2\\}$, where $c_0=0$ resembles the case of the standard rectification $\\mathcal R_p$. The initial coupling $(X_0^{(0)},X_1^{(0)})$ is set to the non-optimal fixed point of $\\mathcal R_p$ from Remark 9 (defined in eqt (12), visualization in Figure 1c). We report the transport distance $(\\mathbb E[\\\\|X_0-X_1\\\\|^2])^{1/2}$ versus the number of steps of the smoothed rectification procedure in the table below. A lower transport distance indicates that the coupling is closer to otpimal transport. The optimal coupling admits a transport distance of $2.66$ (evaluated based on $20000$ samples using the POT package). Since all experiments are initialized with the same coupling, the first step coincides over all runs, independent of the noise level. Due to numerical errors in the marginals, the transport distance is sometimes slightly smaller than the anlytical optimum. Overall, we observe that the smoothed rectification indeed escapes the non-optimal fixed points from Section 4.1. However, the convergence becomes slower if the noise level approaches zero. \n\nIn the paper, we will also include figures (similar to Fig 6/7/8) for this experiments.\n\n$c_0$ | step 1 | step 2 | step 3 | step 4 | step 5\n------|--------|--------|--------|--------|-------\n$0$   | $2.84$ | $2.81$ | $2.78$ | $2.80$ | $2.79$ \n$0.01$| $2.79$ | $2.75$ | $2.73$ | $2.70$ | $2.68$ \n$0.05$| $2.73$ | $2.68$ | $2.69$ | $2.68$ | $2.67$ \n$0.1$ | $2.70$ | $2.69$ | $2.70$ | $2.70$ | $2.71$ \n$0.2$ | $2.68$ | $2.64$ | $2.65$ | $2.67$ | $2.66$"}]

Author rebuttal:

Meta-review: During the rebuttal the authors have addressed several concerns raised by the reviewers and all the reviewers agree that the paper has merits especially on the theoretical side. I have also assigned an additional reviewer (see the comments below) who also provided a positive evaluation. I recommend an acceptance. 

## Additional review
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Summary 

The broad context of this work is flow models, i.e., for a given source $\mu_0$ and target $\mu_1$ probability measures, the process of learning a time-continuous vector field $v_t: \mathbb R^d \rightarrow  \mathbb  R^d$ which satisfies an ordinary differential equation (ODE) and transforms a sample $X_0 \sim \mu_0$ such that the target $X_1 = v_1 (X_0)$ at the final time $t=1$ is distributed according to $\mu_1$. The pushforward of $\mu_0$ by $v_t$ generates a time-dependent probability density $\mu_t$ (probability path).  Initially, flow models were trained by solving the ODE (simulation) and maximizing the likelihood of the target training examples. Subsequently, training by so-called flow matching was introduced - it relies on parametric models of $v_t$ generating the probability path $\mu_t$ learned by regression This approach avoids simulation and makes training more computationally tractable, and lead to state-of-the art diffusion and other generative models. 

More specifically, this work focuses on so-called rectified flows defined as follows. Given a joint probability distribution, called coupling, $\pi$ of $X_0$ and $X_1$ with marginals $X_0 \sim \mu_0$ and $X_1 \sim \mu_1$, we define the interpolation $X_t = (1-t) X_0 + X_1$ and let $\mu_t$ be the law of $X_t$.  A rectified flow is a minimizer of 

$$(1) ~\mathcal L (v_t|X_0, X_1) = \int_0^1 \mathbb E \| v_t (X_t)-X_1 +X_0\|^2 dt.$$

where $v_t$ belongs to the space of $L^2(\mu_t)$ functions.  Previous work showed that such minimizer exists, is unique and satisfies the continuity equation 
$$(2) ~\partial_t \mu_t + div (v_t \mu_t) = 0 $$
in the distributional sense. Based on that $v_t$ is a feasible vector-valued measure in the minimization problem defining the hydrodynamic formulation of the classic Wasserstein-2 ($W_2$) distance (Brenier-Benamou). If $v_t$ is smooth enough, $v_t$ defines a transport map from $\mu_0$ to $\mu_1$ given by $z_1$ where $z_t$ solves the ODE $\dot z_t(x) = v_t(z_t(x))$ with $z_0 = x$. If this ODE has a unique solution, then the underlying coupling $\pi$  is called "rectifiable" and one can sample $\mu_1$ by sampling $\mu_0$ and solving this ODE. Moreover, it is possible to construct a series of coupling $( X^k_0, X_1^k) $ such that $\min_{k \in [K]} \mathcal L (v_t|X^k_0, X^k_1) \rightarrow 0$ as $K$ get large. The time-dependent velocity $v_t$ corresponding to the optimal coupling being the "rectified" flow.

Previous work showed that even if $\mathcal L (v_t|X_0, X_1) =0$, the corresponding $v_t$ is not necessarily the optimal vector-valued measure in the Brenier-Benamou formulation of $W_2$.  However, since an optimal measure is a potential, previous work proposed minimizing  (1) subject to the additional constraint $v_t = \nabla \phi$ for some $\phi: \mathbb R^d \rightarrow \mathbb R$. 

The main contributions of this work are

1. It shows existence of a minimizer of (1) with the additional potential constraint, closing a gap in the literature. 

2. It provides a counterexample that a fixed point of the iterative rectification with this constraint does not lead to an optimal coupling in the $W_2$ minimization problem, in particular this may not happen when $\mu_0$ and $\mu_1$ have disconnected support, revealing a gap in the literature.

3. It provides affine invariances and other properties of minimizers of (1) with and without the potential constraint, and compares them with the corresponding properties of the optimal vector field in the Brenier-Benamou formulation of $W_2$. The manuscript also derives explicit solutions  for the minimizers of (1) without the potential constraint in the Gaussian measure case.

Strengths and weaknesses

The classic $W_p$ metric possesses several properties that make it attractive as a loss function in machine learning and other data-driven problems, as an alternative to the standard $L^p$ norms and other mismatch functionals.  Computing $W^p_p$ is a convex optimization problem that in general does not have a known closed form solution, and this problem is challenging in high dimensional settings.  One important exception is when $\mu_0$ and $\nu_0$ are Gaussian, the $W_2$ distance has a closed form solution  (we also have a closed form expression for $W_p$ when $\mu$ and $nu$ are measures over 1D domains).

On fundamental level,  this work improves our understanding of when iterative rectification may lead to optimal $W_2$ transport couplings and optimal velocity fields, revealing a gap in the existing literature on this subject. I believe the paper provides a collection of interesting and original results at the intersection of generative modeling and optimal transport. However, if I understood the paper correctly, it does not affirmatively prove any results when  iterative rectification may lead to optimal $W_2$ transport couplings, it only provides counterexamples.  

The present paper is a theoretical paper with limited numerical experiments involving toy data. From application oriented perspective, my main question is how relevant these results given that when the distributions are Gaussian, which is typical for diffusion models, the $W_2$ distance, the corresponding optimal transport map, and the optimal density and optimal vector field in the Brenier-Benamou formulation of $W_2$ have  explicit closed form solutions.  More generally are rectified flows, relevant in the context of any problems involving real data and non-Gaussian models?  Are there any references / compelling applications of rectified flows with models that are not Gaussian (and therefore optimal couplings/flows can be computed in closed form) to real world problems/ datasets?

Questions

Main questions

You seems to be suggesting that the claim of (6) has not been proven in [30] or there were gap in their proof, but if I understood the manuscript correctl you only provide a counterexample when (6) does not hold . Are you able to prove some version of (6) when $X_t$ has full support, i.e. when iterative rectification leads to optimal coupling. Do you need to lower bound $\mu_0$ or $\mu_1$ by some constant strictly above zero uniformly on the domain to achieve this? 

Minor comments

* Were any new conceptual ideas/proof techniques developed in the proofs in the present manuscript?

* To contextualize this work better, why was $W_2$ used instead of $W_p$ with some other value of $p$? 

* page 3 paragraph entitled "Iterative Rectification" - does the beginning of this paragraph define $\mathcal R(X_0, X_1)$? If so, it should be made clear how $\mathcal R$ is defined. Similarly, the next paragraph should be a little clearer  how $\mathcal R_p$ is defined.

Limitations

Yes, except as noted above.

Paper formatting concerns

None.
Decision: Accept (poster)