Gradient Variance Reveals Failure Modes in Flow-Based Generative Models
Authors: Teodora Reu
Sixtine Dromigny
Michael M. Bronstein
Francisco Vargas
Venue: NeurIPS 2025 spotlight ()
Keywords: CFM
Rectified Flows
Generative AI
Primary area: deep_learning
TL;DR: The paper shows that Rectified Flows trained with deterministic interpolants memorize training pairings.

Abstract: Rectified Flows learn ODE vector fields whose trajectories are straight between source and target distributions, enabling near one-step inference. We show that this straight-path objective reveals fundamental failure modes: under deterministic training, low gradient variance drives memorization of arbitrary training pairings, even when interpolant lines between training pairs intersect. To analyze this mechanism, we study Gaussian-to-Gaussian transport and use the loss gradient variance across stochastic and deterministic regimes to characterize which vector fields optimization favors in each setting. We then show that, in a setting where all interpolating lines intersect, applying Rectified Flow yields the same specific pairings at inference as during training. More generally, we prove that a memorizing vector field exists even when training interpolants intersect, and that optimizing the straight-path objective converges to this ill-defined field. At inference, deterministic integration reproduces the exact training pairings. We validate our findings empirically on the CelebA dataset, confirming that deterministic interpolants induce memorization, while the injection of small noise restores generalization.

Gradient Variance Reveals Failure Modes in Flow-Based Generative Models
Teodora Reu University of Oxford
Sixtine Dromigny University of Oxford
Michael Bronstein University of Oxford
teodora. reuccs. ox, ac.
Francisco Vargas
Xaira Therapeutics
Abstract
Rectified Flows learn ODE vector fields whose trajectories are straight between source and target distributions, enabling near one-step inference. We show that this straight-path objective reveals fundamental failure modes: under deterministic training, low gradient variance drives memorization of arbitrary training pairings, even when interpolant lines between training pairs intersect. To analyze this mech- anism, study Gaussian-to- toa-ssii transport and use the loss gradient variance across stochastic and deterministic regimes to characterize which vector fields optimization favors in each setting. We then show that, in setting where all interpolating lines intersect, applying Rectified Flow yields the same specific pair- ings inference during training. More generally, prove that memorizing vector field exists even when training interpolants intersect, and that optimizing the straight- path objective converges to this ill-defined field. At inference, deterministic integration reproduces the exact training pairings. We validate our findings em- pirically on the CelebA dataset, confirming that deterministic interpolants induce memorization, while the injection of small noise restores generalization.
Introduction
The current state of the art in generative modeling involves learning dynamics between a known source distribution-such as standardized Gaussian-and target distribution from which many samples are available. When the learned dynamics are based on an ordinary differential equation (ODE), the model effectively Iearns a vector field connecting the two distributions [Lipman et al.) 2022 Tong et al. 2023bla When the learned dynamics are based on a stochastic differential equation (SDE). either the score function (Ho al.) 2020 Song al. 2020 both the and the vector field [llergoo et al. 2023). are learned, depending on the choice of SDE. Ultimately, an SDE can be rewritten as ODE referred to in the literature as the probability flow ODE [Song et al.) 2020
These methods perform very well in practice across wide range of data modalities: images HH
et al. 2022a Balaji
al. 2022 Rombach et al. [022]] video (Ho et al.) 2022b Wang et al. 2024b
Zhou et al. 2022], audio (Huang et al. 2023 Kong et al. 2020 Liu et al. 2023 Ruan al.
2023
and molecular data [Hoogeboom et al. [222 Xu et al. [202]] However, their main limitations are the
high computational cost of the generation samples (as integration over an ODE required) [Song
et al. 2020, Tong et al.) 2023b Albergo et al. 2023), and their inability to learn optimal transport maps, which are often essential for unpaired data translation tasks (Shi et al. 2024
To address computational inefficiency, new models have been proposed. Consistency models accel- erate sampling for score-based approaches [Song et
2023
Salimans and Ho [022]] Kim al.)
2023 while Rectified Flows LLu 2022 Bansal et al. 2024 Lee et al. 2024 aim to learn straight
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Does low Var(Val/ac) translate into learning straight optimal vector
Why not?
Consequences ReFlow
fields?
No, in deterministic regimes (e.g. ReFlow(k=1))
Under a deterministic regime, points where interpolating lines intersect are measure-zero, making (2t, t) - (#o, 21) injective over the training samples.
ReFlow tends to memorize arbitrary deterministic pairings training dala, without improving transport couplings (Proposition 2).
Yes, stochastic regimes
(e.g. CFM, SBM)
Figure 1: Intuition behind the main results.
vector fields that can be integrated in single step. Rectified Flows iteratively update couplings and retrain vector fields to straighten transport paths, but repeated rectifications can accumulate errors, and it remains theoretically unclear whether one or two rectifications suffice under general conditions. Input-Convex Neural Networks (JCNN)-based parameterizations can, in theory, guarantee optimal couplings for noiseless interpolants, but are difficult to optimize in practice [Makkuva et al. 2020 Huang et al. 2020].
Schrödinger Bridge Matching (SBM) (Shi et al. 2024 Peluchetti 2023 De Bortoli et al. 2024
extends these ideas by learning both forward and backward vector fields using noisy interpolants. approximating entropic optimal transport. This bidirectional approach can improve stability and mitigate eтrоr accumulation, but requires training two networks per iteration.
In this paper, we address fundamental limitations of iterative generative models by analyzing how gradient variance reveals suboptimality in learned vector fields. Our theoretical and empirical study uncovers why standard neural architectures struggle to represent even simple transports and how repeated rectifications can lead memorization rather than improvement.
Contributions In Section we investigate the question "When is gradient variance informative?" in the Gaussian-tov to to-Gaussian setting. We show that the answer depends on the training regime (stochastic or deterministic), and that in deterministic regimes, ill-defined vector fields that memorize the training pairs may emerge. Figure bounds for the loss and gradient variance under both regimes and demonstrate that our empirical observations align with the theoretical predictions from Proposition 1
central illustration of this phenomenon. Additionally, we derive
In Section we extend these results from the datasets within the ReFlow paradigm. We prove (Proposition that there exists a minimizer on the finite dataset that reproduces the training pairings. Moreover, due the deterministic nature of the integration method, the model can reproduce these exact pairings at inference time by effectively "jumping over" intersections (Remark[). We validate our theoretical findings empirically on both Mixture of Gaussians and the CelebA dataset in Section 4.2, and Section 4.3
case to general finite training
Background
Notation. Let IRd denote d-dimensional Euclidean space. We use P(Rd) for the set of probability distributions on Rd. The source and target distributions are To and IT with Xo no and X1 mis transport map T Rd >> Rdd, and T To denotes the pushforward of ιa by T. Interpolants are written X, T(Xo- X1, l). with / E [0, 1j. We write B[[] for expectation, War[-] for variance, and Ve for gradients with respect to parameters 0. Bold symbols (c.g., x) denote vectors. All other notation is defined in context. For & E IR, R& denotes d x d rotation matrix corresponding to a counterclockwise rotation by k degrees in the plane of interest. All other notation is defined in context.
Optimal Transport and Entropy-Regularizee OT. Let wo कi be probability measures Rd The Monge problem [Monge] 1781 seeks a transport map T minimizing:
inf / [T(x) x|l d (x) s.t. TạTO 11. T
(1)

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where TH denotes the pushforward of To under T. The Kantorovich Kantorovitch 1958 relax- ation introduces couplings π E [[ Ti) and solves:
inf (m mι) =Ell(
[(xxxx))~<IXX -
(2)
where [1 mi) denotes the of joint distributions with marginals no and ni Entropy- regularized Optimal Transport (eOT) adds a Kullback-Leibler divergence penalty (mT, mT)
inf ( |x Xoll²) + ( @ m)).
When No is absolutely continuous, the Monge and Kantorovich problems admit the same deterministic optimal plan. A dynamic formulation describes OT evolving path (ex) connecting
Xn ~ and X1 ~ P1. For convex costs, the optimal path is given by the straight-line interpolant X, X) + LX1 [McCann] 1997].
Conditional Flow Matching. Given the interpolants X, = I(Xo, X1, = oxxo + B,x1 + +
where I), the CFM objective objentiveLLipmnn al. [2022 is:
LCFM (v) (No ell((l Xo) ((X,,
(3)
The learned vector field generates flows via the ODE:
d X, ((xx:()
dt
Rectified Flows (or ReFlow). Liu et al. [002 propose an iterative procedure to straighten
transport paths. At each iteration k, a vector field ()()) is trained using the CFM loss LCrM
on the current coupling (X/*) (x(x), The updated coupling is then generated via (((+1)
ODE-Solvefu() = 1), and the process repeats until convergence. A coupling (Xg ((()) is considered straight if
E
(M) =x 1x - x x
-x]= = x x (R)
(5)
where x (1 - 1)x + +(x(1) (k) A coupling is considered straight if, for deterministic couplings (Xo: X1), the mapping (X,, + (x) is injective. Although some works claim one or two rectifications suffice to obtain straight couplings (Lee et 20224] and [Bansal et al., [024] (in the case of isotropic Gaussian), there are theoretical results guaranteeing it for any continuos source and target distributions.
Gradient Variance. We begin by rewriting the CFM objective from Equation B
L(c,T,I) Ex [IT(X.) - Xu v u(x,, X, I(Xo, T(Xxx,((Xx)) T(X)).
(6)
where T' is a (possibly random or deterministic) map satisfying TTTO Ti. A Monte Carlo approximation of this loss, using samples ( )- given by:
s
Luc(E, T,I) - Σ T
v (x (x,"))) -x
x(")= =[x==((x'),___)_) I(Xx"),TXX ).)). (7)
The gradient variance will have the following formulation:
T (x6²) x((()) (x!") ))
Var Vor T, Var V
(8)
S
$=1
Memorization and Generalization. Memorization and Generalization have been extensively
explored in the field of generative modeling [Bamberger et al. 2025 Buchanan et al. 2025| Somepalli
jet al., 2023 Ren et al. 2024 Rahman et al.) 2025 Wang et al. [2024a] Stein et al. [023] with privacy
concerns Ghaaebbikssbbi et al. 2020] and copyright infringement [Cui et al. 2023 being key issues
that would result from such frameworks. Our setting differs from most prior memorization studies
by assuming that each training target has deterministic counterpart in the source, as in Rectified
3

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LMc noiseless interpolants for deterministic pairings
Lc on noisy interpolants for deterministic pairings
OT vector field
high Var Var( (VLLMc) region
OT vector field
non-OT vector field
non-OT vector field (but optimal for the deterministic pairings)
Figure 2: Schematic representing a hypothetical loss (Imc) landscape. The gradient variance of the loss acts indicator of solution quality. The schematic illustrates how the variance of the loss gradient reveals information about the optimality of the vector field under different interpolant types.
Flows, rather than sampling source target pairs independently as is standard in the CFM literature. As a consequence, the appropriate notion of memorization changes: instead of asking whether model starting from arbitrary source points reproduces pairs from the training dataset, we consider deterministic training set of pairs (Xo: Xi) and say that memorization occurs if the ODE integration
of the learned vector field from a training source X, returns its paired target X; = X1 at inference, effectively reproducing the training coupling. This reframing highlights the central question in rectified-flor flow training: what value is added by learning a new vector field deterministic integration simply recovers the original pairings, i.e., amounts to memorization rather than improved transport structure or generalization.
3 Gradient Variance of Gaussian-to- transport
Analyzing Var[V T, [)] across different choices of pairings T (e.g. random vs. optimal vs. straight), inter- polants I (noiseless vs. stochastic), and vector-field classes 19 provides insight into which solutions are favored by the loss landscape. For clarity, we distinguish vector fields that are optimal in the OT sense from those that are optimal for specific pairing (minimizing error given that pairing); we refer to the latter as pair-optimal. Although loss may have multiple minima, optimization often prefers those with lower gradient variance, even if they are suboptimal in terms of transport cost. Figure 2 illustrates this effect: for deterministic couplings and noiseless interpolants, the lowest variance and loss minimizer can be non-OT, while introducing stochasticity can shift preference toward more optimal solutions.
[Pa6]
Verancee 97%
120"
182"
Rancor
3.0
Interpolation
Figure 3: Gradient variance over time for vector field rotating by 120", under various pairing types including rotated and random couplings. In the top right, the two Gaussians are shown along with sample trajectories, color-coded by inte- gration time. Optimal pairings exhibit the highest variance, while random pair- ings do not yield the maximum variance.
OT is analytically tractable (Lemma DT. the exact gradient variance can be derived for a broad set of vector-fiell classes, interpolants, and pairing schemes in Proposition [ To broaden the comparison, rotational OT (rOT) field is included that first rotates the source and then maps to the target; these trajectories are straight for all angles except the degenerate 180" case, which is not dell-defined. This controlled setting enables closed-form analysis of variance across configurations and direct alignment with empirical measurements.
Because

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In Lemmal 1 we derive this closed-form expression for the optimal vector field between two Gaussian distributions with different means and covariances, and we show that a multilayer perceptron (MLP) cannot exactly represent it without error, motivating the use of the closed-form parameterization rather than an MLP surrogate.
Lemma 1. Lef Xo ~ NO, Ia) and X1 (iu, Ma). where Ma a positive definite and symmetric matrix. The OT vector field is given by
Bor t, 6 = 0 + (X, -X,- t0),
and the rotating vector field:
to (X 1,
R Por
0) - 0 6 N0T 11 16 .or. [ 10),
where Xt (1 x)X tX R M d θ M Is. θ p., and R is rotation matrix with R + -I. Furthermore, the function èOT cannot be exactly represented by an MLP, CNN, Transformer architecture when given concatenated inputs tX.
An intuitive reason why this vector field cannot be represented with zero error by typical neural network parameterizations is the difficulty in capturing the matrix inverse term Ia + 1 For experiments quantifying the training impact of this approximation error, see App. G.2 for full results. Proposition (Informal). Let To NO, Ia), and Tı (n, Ma), where Ma is positive definite and symmetric matrix. Lel the following push forward maps: Tor(Xo) M Xo + p. To (Xo) M RXu 14, and T Trand(X, parameterisatiin be v (Xe, t) = 0 + I t - -t0), where Xt (1 Xo + tX2. Let:
X2, where R is rotation matrix. Let the following
N = N Σ [T(x - -x6" -
,t, ,t,
(9)
(6.0) with ToT and (θ,OT. 0) with ToT achieve zero loss and zero gradient variance. If the rotation angle is not 180', trajectories remain straight, but any learned field that does not correspond to the true angle exhibits nonzero gradient variance. In addition, for TooT and Trand evaluated with the VOT class, gradient variance does not increase merely because multiple straight interpolants pass near one another (for analytical values check Appendix B
common misconception in flow matching is that gradient variance primarily originates at points (z,, t) where straight-line interpolants intersect [Gagneux et al. 2025 Fjelde al. 2024]; Propo- sition challenges this by showing that variance can be nonzero even for straight couplings with
(1 - tixo tX + oy ((1 tjz
Veriance of [aad]
(o -0)
[o 11)
[o -2)
(o -4)
0.5
698%
25
022
044
100
09
0.2
8.
interpolation time
Interpolation
Interpolation
Interpolation
(1 ()X, + DX; - oZ
Rotation Argles
VT
Varincce of [9a4]
[a -0
(a 1)
(a 2
(7 44)
000
0.50
0.75
0.20
0.
0.2
34
088
1.3
0.0
0.2
04
0.
0.E
1.0
0.0
0.2
0.6
0.8
1.0
0.0
0.2
0.6
0.8
Interpolation time it)
Inte Piterpolaoion time Iti
Interpolation time (t)
Interpoladion time
Figure 4: Gradient variance (with 95% confidence intervals) for Gaussian transport paired (Xo. R3g--Np + μ4) under different noise levels (a of the two interpolants mentioned in the title of the figures). We compare vector fields inducing 0 (OT, blue), 30" (pair-optimal, orange), and 60 (non-OT, pon-pair- green) rotations. Shaded regions show confidence intervals calculated over 100 bootstrap samples. As noise increases (a - 4), the optimal transport (OT) field exhibits significantly reduced variance (p < 0.01, paired t-test) while maintaining lower transport error.
5

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Inocence of Arrst Hamasctror vs Inarssition Time
Vor snce of [Ped]
Veriancee of 1l
0.15
32 644 05
08 30
Intepplation
Figure 5: Gradient variance over time for several pairing types: (Xo, MM R Xo +
with {0, 30, 60, 120, 180), and (Xo, X1) with X1 Ni,. Ma). over the OT vector field. Shaded regions indicate 95% confidence intervals computed via 100 bootstrap samples. Notably, variance does not increase in regions where interpolant trajectories come closer together: for example. with 180 rotation, all interpolants intersect at & = 1/2. yet variance is lowest there. Random pairings exhibit lower variance under the OT field than structured pairings with 120 (straight) or 180 (not- straight) rotations, highlighting that variance is not simply a function of interpolant density.
no intersections, while intersections or regions of high interpolant density do not, by themselves, induce elevated variance. Consequently, gradient variance under L is not governed by geometric proximity of trajectories but by mismatch between the learned vector field and the pairing structure (e.g., misaligned rotation). This also clarifies that low gradient variance does not certify transport optimality: when and T are aligned (e.g., they apply the same rotation), both loss and gradients vanish, whereas under the OT field, even straight pairings can exhibit nonzero gradient variance (see Figures 3 6)
3.1 Empirical validation
This subsection studies empirically what happens with the gradient variance across choices of pairings, interpolants, and vector field classes.
Figure 6 iillustrates the key experiment that motivates Section 4 We consider a deterministic pairing with Xo ~ (0, I2) and X1 = Risu Xo + [5, 5] so that all straight- -ine interpolants meet at the midpoint when t 1 Under noiseless interpolants, the learned vector field memorizes this ill-defined pairing and, at inference, reproduces the same mapping that rotates the source Gaussian by 180 and translates it by [5, 5]T. This occurs because the probability of sampling exactly = during training is zero, so the vector field is never constrained at the intersection point; at inference, deterministic numerical integration effectively bypasses this singular location, yielding the same pairing. This provides concrete failure case in which rectification does not resolve interpolant intersections but instead preserves the training-specific pairing.
Key experiment:
Under noiseless interpolants. We now characterize the variance of the gradients over the noiseless interpolant 2, (1 t)xg + frı, as this is the commonly used type of interpolant in ReFlow architectures. Notably, when the vector field is OT, straight pairings can exhibit higher gradient variance than random pairings (see 5). Conversely, with a non-OT vector field, the loss can display substantial variance for pairings that are themselves OT (see Figure 3). Furthermore, for
Integrating over #1a0 vector
11- + 06 ovtil
=0
J0 -1)
Wabno Portt
InPeppaaiion timee
Intepplattie S134 11!
Figure 6: Integration over a vector field performing 180 rotation. Although all interpolation paths intersect at f = 1/2, and E Xo X; = 1122 * X1 - Xo. we still recover -rtatedd couplings. This is because the numerical integration procedure discretizes time and does not evaluate the vector field precisely at the point of intersection, effectively bypassing it.

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straight non- non-OT pairings, the most stable solution is the corresponding pair-optimal (non-OT) vector field, Figure 6 Variance does not arise from intersections or regions where the interpolating lines come close together, and random pairings exhibit lower variance than structured couplings, such as 120" rotation (see Figure 5)-
Variance under stochastic interpolants. We now consider noisy interpolant I; (1 t) Zo - tæ + f(t,o) 2,
Z (0, Id), where f [0, - 20 modulates the noise level over time = a I)). As shown in Figure 4 and 6 adding noise to the interpolant causes the optimal vector field to exhibit lower gradient variance than the pairing- oopiimal vector field. Furthermore, in the second frame of Figure 6 we can now see that at t = 1/2 the variance of the gradients is very high, meaning that, even though we never sample = 1/2, the loss landscape has very high variance around that area for this vector field solution.
[(1, (e.g.,
4 Rectified Flow Dynamics: Minimizer Memorizes
This section extends the experiment in Figure 6 by formalizing it through Proposition which analyzes stagnation for finite datasets under de- terministic training and clarifies its implications. We then validate these implications empirically on synthetic data and CelebA in Sections 4.2
Rellew on noise interpolants
and
4.1 Theoretical Results
Reflow naisy interpolants
We now formalize two key properties of ReFlow: first, its tendency to stagnate once it reaches straight coupling (Lemma 2}. a known result previously established in Bansal et al. (2024), Liu 2022]; and second, its ability to memorize arbitrary deterministic pairings when trained on a finite dataset (Proposition [}. which represents the main novel contribution of this work.
Lemma 2 (Idempotence of Rectified Flows). Let To, Ti be distributions on RD with densi- ties, and (Zu, Z1) their straight-line coupling via linear interpolant Z = (1 Zu + tZN- Subsequent Rectified Flow iterations with this noiseless interpolant yield identical couplings:
Figure 7: Schematic showing the two types of interpolants. When training with deterministic pairings, sampling at intersection has zero probability mass. As a result, the mapping (Z,,t) - (Zo. T(Zo). becomes injective, mak- ing it straightforward to define a vector field at these points by v v(Z,, Zo -o-T(ZZ)). In contrast, for noisy interpolants, this injectivity no longer holds.
ReFl owl (Zo, Z1) (Zo, Z1) V 1.
This stagnation stems from the bijective rela- tionship between interpolants (Z,,t) t) and initial pairs (Zo, Z1), given by our assumption of the straightness of the deterministic couplings.
Some recent works have suggested that 2-
ReFlow might be sufficient to recover straight pairings Lee et al. [2024), and Bansal et al. [2024] showed that 2 rectifications are enough under some regularity assumptions to straighten paths for isotropic Gaussian distributions. In the spirit of Figure (see where this not possible for non-independent pairings.
(see Appendix A), we provide an example
Counter Example (1-ReFlow May Fail Under Mild Assumptions). Even if the learned transport map T(xu) is injective (e.g., under standard Lipschitz and growth conditions; see Appendix A), the straight-line interpolant [(I0, T(xa), 1) need not be. For instance, a rotation-basel map like T(o)) = Ris0e + 5 (which can be realized by continuous vecior field) leads to overlapping interpolants, breaking injectivity of (xa, 1) +> Fn- As a result, a new ReFlow step cannot reconstruct (0o, 21) and fails to straighten the coupling. See Appendix for details and a visual example.
fails
7

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Relaxing the straightness assumption to consider general deterministic couplings and finite training datasets leads to a nuanced but equally critical limitation:
Proposition 2 (The Minimizer Memorizes). Let no and का be two probubility densities on R2, and let T IRD + R be a deterministic transport mup pushing wo to a (so if Zo ~ No, then
Z1 = (Zo) ~ TT). Suppose we draw a finite dataset of N i..dd samples (2/0 - = T(Z) j= and for each i we also sample m time-points {t(-d) ym C [0, 1] (e.g. uniformly). Let Z, (1d)
)
(1 -1-#te -(h-)) % + gli-5) z Define the empirical loss over this doubly indexed dataset by
N ΣE =1
det (ve) MC
1
ue (z(2) ,0,,))
N
Then there exists (deterministic) vector field v attaining zero loss: (c( (v) 0.
Next, we discuss the implications of Lemma 2 and Proposition 2
Remark 1 (Condition for Recovering the Same Couplings). Defining look-up vector field over the training data-or or approximating it via the Universal Approximation Theorem (UAT))does not guarantee that integration will recover the original pairings. To ensure this, the integration process must traverse the specific time steps (1(6.) J==1 associated with each training pair X(s)- These are the only points along the interpolation = (1 - t)zo + tr where we are guaranteed that v t) T1 To- While this condition might appear restrictive, the proposition remains valid for any natural number m.
This insight arose from empirical observations. For example, when training neural network on pairs (Xo. (X))) with T(Xo) = Risn Xy + μ., we found desi-d all interpolants intersecting at t 1/2 the model successfully learned the rotation and preserved the pairings during integration. See Figure for a visualization.
Remark (Noise Breaks Proof Assumptions). A key advantage of using noisy interpolants that they break the bijection between (æ,, t) and (zo, (zo)), thereby violating the key steps of the proofs of Lemma 2 and Proposition 2 This disruption prevents the model from becoming stuck in suboptimal or deterministic straight pairings. As shown in Figure 4 this can lead to learning vector fields that are closer to the optimal solution.
Why Doesn't 't CFM Memorize Random Pairings? Consider a finite dataset of targets, while at
each training step, the source In independently drawn from continuous distribution. Initially, we hypothesized that CFM could memorize these random couplings, since Proposition 3 (Appendix C shows that interpolants intersect at x, with probability zero. However, as noted in Remark 2 the independent sampling of no from continuous distributions breaks the bijection between (z,, F) and (xo, 21).
While Rectified Flows can help straighten trajectories, they risk collapsing onto memorized pairings
the first iteration, when deterministic couplings are formed. This memorization undermines generalization and highlights a limitation of relying solely on early deterministic transport.
4.2 Memorization for Mixture of Gaussian
To probe CFM's tendency to memorize deterministic pairings, we train two variants-one one using noiseless interpolants and one with added noise-on on an CFM model that maps To = (0, Ia) to a target Gaussian mixture n (more details for the experiments in Appendix(F). We evaluate in both low (d = 3) and high (d = 50) dimensions using three metrics (log-likelihood under the true mixture, MMD, and Sinkhorn distance) and four comparisons: (i) Gen: generated held-out vs. true samples, (ii) Mem: generated vs. training pair integrations, (iii) True: true vs. true reference, and (iv) Data: training pair vs. true samples. This setup isolates whether noiseless CFM simply memorizes its finite dataset, and whether stochastic interpolants improve generalization without sacrificing sample quality.
As shown in Table [ CFM tends to memorize the deterministic pairings it trained on, reflected in very low memorization distances but weaker generalization. In contrast, CFM( = 0.05) demon- strates consistently lower distances to the true distribution across both dimensions, suggesting superior generalization and less reliance on memorized pairings.
8

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4.3 Memorization for CelebA
To empirically validate Proposition 2, we conduct experiments on the CelebA dataset using Condi- tional Flow Matching (CFM), with ground-truth optimal transport (OT) pairings from Korotin et al. [02T]] as reference. Our primary objective is to test the claim that CFM trained with deterministic interpolants memorizes its input pairings, as predicted by theory.
We compare two versions of CFM: CFM 0): trained using deterministic (noiseless) interpolants, CFM(( = 0.05): trained using slightly noisy interpolants.
Both models use identical U-Net architectures. The noise parameter is not chosen for improved performance, but specifically to break the injectivity between data and interpolant trajectories thereby disabling the memorization pathway outlined in Proposition 2.
Adversarial Pairings. To directly probe memorization, we construct an adversarial dataset by shuffling the targets of the Korotin al. (202T) optimal OT pairings. This breaks correct transport structure, producing random pairings with no coherent OT semantics. The model variants are compared on two criteria: generalization (L2 error to true OT targets), and Memorization (L2 error to the shuffled (training) targets). We measure both on held-out subsets (5K and 50K samples). Lower L2 to true OT indicates better generalization, while low L2 shuffled indicates greater memorization.
Simulated 1-ReFlow. We simulate 1-step ReFlow scenario to test if iterative flow models reinforce memorization further- central prediction of Proposition 2. Here, base CFM model is first trained on random (non-OT) pairings, then used to generate endpoints via ODE integration. We then retrain CFM on these generated endpoints.
Memorization effects diminish with larger datasets, as fixed model capacity makes perfect memoriza- tion infeasible for 50K samples compared 5K. However, optimization still favors memorization when possible, as predicted by Proposition 2. Larger datasets impede perfect memorization, but do not alter the underlying objective.
Our experiments provide direct empirical support for Proposition 2: CFM trained on deterministic interpolants reliably memorizes its inputs, even when those pairings random flawed. Introducing noise to interpolants breaks the pathway for memorization, enabling principled generalization toward the true OT map. These findings highlight a key limitation of deterministic training regimes for flow-based generative models on empirical datasets. For completion we mention FID values of these models in Appendix
Conclusion
We amalyzed flow-based models through the lens of gradient variance and showed that deterministic -especiall in Rectified Flows-cca can lead to memorization and stagnation. Noise, whether explicit (stochastic interpolants) implicit (random pairings), breaks this effect, improving both generalization and sample quality. Our theoretical results and experiments on Gaussians,
Table I: Comparison of CFM and CFM with stochastic interpolants (CFM( = 0.05)) across low and high dimensions. CFM( o 0.05) consistently generates samples that better match the target distribution (as indicated by lower MMD and Sinkhorn distances), without resorting to memorization, evidenced by its strong performance on both generated and true data metrics. This highlights CFM(a 0.05)'s superior generalization and sample quality compared to CFM. Experiments done 10 seeds, the mean is presented, for full table check Appendix F
Dimension
50
Gen
Mem
True
Data
Gen
Mem
True
Data
LogProb
4.0150
4,0890
4.1330
4.0155
54.8299
53.6502
52.5094
53.6244
CFM (a
MMD
0.0034
1.758 x 10
0.0014
0.0032
0.0021
9.089 10
0.0020
0.0019
Sinkhor
0.0730
1.411 x 10-5
0.0637
0.0790
15.1000
0.0045
14.3221
15.7400
LogProb
4.1270
4.0960
4.1330
4.0155
54.7220
53,8890
52.5094
53.6244
CFM (a 0.05)
MMD
0.0018
3.105 x 10-5
0.0014
0.0032
0.0020
6.09 x 10-5
0.0020
0.0019
Sinkhom
0.0680
3.557 x 10-4
0.0637
0.0790
15. 15.1689
0.0304
14.3221
15.7400

---

Table Adversarial Pairings Results: These results show that CFM with deterministic interpolants (a 0) strongly memorizes non-optimal shuffled targets, minimizing training loss even on arbitrary pairings. Adding slight noise (a 0.05) disrupts this, enabling the model to resist overfitting and generalize toward the true OT map.
Metric
CFM(a = 0.05)
CFM(a = 0)
5K Generalization (L2 to OT)
34.25 7.54
50.40 16.73
5K Memorization (L2 to Shuffled)
55.02 = 16.51
28.57 = 5.49
50K Generalization (L.2 to OT)
30.05 6.77
46.78 14.87
50K Memorization (12 to Shuffled)
56.48 + 18.55
45.98 + 11.85
Table 3: Simulated I-ReFloow
The findings are consistent: deterministic interpolants 0) facilitate memorization of training pairings, while small amount of noise substantially increases resistance to memorization and restores generalization ability.
Metric
CFM(( = 0.05)
CFM( = 0)
5K Generalization (L22 to OT)
31.35 7.38
43.35 + 14,21
5K Memorization (L2 to Generated)
25.08 + 8.59
8,63 + 1.76
50K Generalization (L2 OT)
32.54 8.86
38.16 11.37
50K Memorization (L2 Generated)
16.29 4.82
12.76 4.01
Gaussian Mixtures and CelebA demonstrate that variance-sensitive optimization prefers suboptimal flows unless stochasticity is introduced. These findings challenge the reliability of rectified flows and underscore the importance of noise in guiding models toward better transport.
Limitations. Our evaluation focuses on: Gaussians, Gaussian mixtures, and CelebA as the sole real-world benchmark; broader modalities and higher resolutions may surface additional behaviors. While critique Rectified Flows' propensity memorize deterministic pairings, we do propose new remedy beyond introducing stochasticity into interpolants. Notably, adding noise is known to steer training toward entropic OT [De Bortoli et al. 2024] over many iterations, which in practice manifests as improved generalization rather than principled cure for deterministic memorization.
6
Acknowledgements
TR and MB are supported by EPSRC Turing AI World-Leading Research Fellowship No. EP/X040062/1 and EPSRC AI Hub on Mathematical Foundations of Intelligence: An "Erlangen Programme" " for No. EP/Y028872/1. SD supported by the Clarendon Scholarship and NERC DTP studentship (NE/S007474/1). FV declares specific funding for this work.
10

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A
Assumptions
Assumption 1. Suppose that e(x,t) satisfies the following conditions:
Lipschitz continuity in z: There exists L 0 such that for all x, RD and t [0, 1].
||((, t) v(y. t)ll S L||x= 10-0 yll.
(10)
Linear growth: There exists such that for all TR. and t [0, 1]. [[v(,, 1)l s C(1 ||=+])..
(11)
With Assumption in place we can conclude via Arnold [1992], that the ODE dr. v(rs, t) admits unique solutions for all initial conditions Tn, and that T(m) z0 So v(se, l)dt is an injective map.
B Proofs Gaussian Setting:
Extra-Lemma Let q(x) 1. with x E Rdd. There is no finite parameterization of Multilayer Perceptron (MLP) that represent this function with zero error.
Proof. Assume, for the sake of contradiction, that there exists a finite MLP that represents g(x) exactly. To avoid issues
0 where ((x) becomes undefined, we restrict the domain of æ to E b). where 0 & [a, b] and a, b E R. This ensures that ((x) is Lipschitz continuous on this domain.
Next, consider the scaling property of ((r): for any scalar e # 0, g(cx) = (x)) This property implies that (zx) scales inversely with
However, standard MLPs do not inherently possess this scaling property. For an MLP to exhibit this behavior, its activation functions would need to satisfy (cz)) (a(x), which is not true for commonly used activation functions such as ReLU or tanh.
Since the desired function g(x) exhibits a scaling property that standard MLPs cannot replicate, we conclude that no finite MLP can represent ((x) exactly.
Lemma I.
Let Xo NO, Ia) and Xı NNu,, Ma). where Ma a positive definite and symmetric matrix. The OT vector field is given by
for (Xe, Ô = - -Xt=))
tt),
and the rotating vector field:
êTT (X,, t, 2 ror, 0) - - O -0 -tt).
where X; = 1)X, + 1X1 -= M Ia, Ô MM RA Ia Ô = ns and Res is rotation matrix with Rge # - R2.. Furthermore, the function for cannot be exactly represented by MLP, CNN, Transformer architecture when given concatenated inputs (X,, 6).
Proof. The optimal transport map from N/0,, Ia) to (n, Ma) is Tor(x) = 1 M2 Thus, we denote the optimal coupling by Xū and Xi. we have Xi T(Xi) = μ + / Xg. The displacement interpolation is defined as X, = (1 t)Xā tXi. Substitute the expression for Xi into the interpolation:
X, (1-t)Xo t (p+ M M2/2xx) (1- -) +al++tMi +M X6 tp.
(12)
Solving for X gives
(1- (1-1)4 1) L2222 -1
(Xe - -4).
14

---

Now, substitute X back into the transport map to get
= M / x - μ + M (11 - t) + +M (X, tp)..
(13)
The optimal displacement (vector field) is defined as v(Xt,1) Xi - Xū- Therefore, we have
(X, = M (1 + -1
(X:- tpp) -t)
(14)
-[u- +1M (Xt 141
(15)
= - (1 t) + +M -1 (X, (X,-tt)
(16)
This is the desired expression for the vector field.
For BrT,, the computation is analogous, with the key difference that the transport map is now given by (1) j++ + M RXp. This defines valid pushforward map, since for any rotation matrix Reo, ~ (0, Ia), then RZ ~ (O,,, Ia) as well, due to the rotational invariance of the standard Gaussian.
Proceeding as before, we solve for Xū. substitute it back into the transport map, and derive the corresponding vector field. This yields the expression for U,OT. The derivation follows the same steps as the case of Bor, with the only difference being that M M//2 is replaced by M/2 R.
Also we can't t have R = -R (180" rotation), because then the inverse [LL - -
Ta - 1L-tMM/// + 2) is not computable when t( 1/2 + I) = I, (for example for Ma = I, and
(M)
1/2). For more complete approach, see Liu 2022
We proceed to prove that this parameterization cannot be represented with zero error by an MLP. Representing the function 1+40 with zero error is equivalent to representing with zero error. By applying Extra-Lemma [ we conclude that such representation is not possible.
L
Proposition Let To ~ N(O,, Ia), and m Níu, Ma), where Ma is positive definite and
symmetric matrix. Let the following push forward maps: Tor (Xo) = M Xo + n. TroT(Xo) = / RX, p. and Trana(Xy) (Xy) X1, where R is rotation matrix. Let the linear transformation be u(x,,1) = 0 + ① + 11) - 10), where X. = =(1 I)Xo + LX, Let:
M.c (0,,,T,))
N [T(x!()) x(() - -((! 0)12
(17)
Then, for optimal, V θ θ VOT, Tor)] Var(V e I θ i VroT, T.or)] and VarlV e ( θ NOT. T.. ] & (Ir((M2/ 2)) Moreover, for the maps Trot and Trand and vor vector field, the variance of the gradients of the optimal vector field does not increase in regions where the interpolant trajectories come closer together.
(Ô,
we have
Var V L. (,, 0 vor, Tror)]
Var(Vo(E) O θ vror, Tror)]
O. and VarlVo O vor T
N ||2
Proof. We break this proof into the computation of variance over time and the value at time zero to simplify computations.
Way I: for 1 = 0 but closed form is nice: We compute gradient variance in two ways, one relies on the fact that when I is deterministic there is a direct bijection between X, and Xo. this of
15

---

course won't give us the values in terms of t, but the formulas are somehow nicer and easier to work with. We are basically looking at the variance for (Xo. 0)
M 10 - x!" - - -010
Var
V N
(18)
Σ T (u+ (M - - 1)x -0) -00
Var
(19)
N
[ TT(( (M (M - -0
Var
(20)
122 - - I (M 12 Q --M
-
(21)
1/2 x(!)) -x,6') 0 - =012
Var Vo,
M
(22)
N
Σ X (++ (M2/2 Ia - 0 0) N
2
= Var
(23)
Gradient variance for Tor and vor For the rotated case:
Note that our vector field optimal when è M2 Ia and 1 jr. Then we have:
Var VolMIc(((0,0.xx0) = 0,
(24)
and
4 N
VarlXo (a 0)] 411 011
Var Ve L OT 0, (1,,,
0.
(25)
N
Gradient variance for Tror and vor
For the rotated case:
12 RXx," -x(")) - - 0 -01²]
Var V
M
(26)
2
Σ T (u+(M) (M R - 1)x((²) - 8)
Var
(27)
N
4
Var
I ((M
1/2 R- - --
(28)
N
At the optimum, where ô M1 Ids we obtain:
4 N
Var
1 (MM2 M 1 [xo]
1 (M 2 (R
11
(29)
(
d 1/2 (R ()) - M 1/2T
(30)
(2Ia 1 T M (21a - -R - R)M 1.
(31)
N
For the variance of the gradient w.r.t. O at the optimum, we get:
Var
roT (6, V Lo
= Var xx MM 22 (R- 1)x1]
4
(32)
N
8
1/2 (R T))²)
(33)
N
The last equality follows from the known variance formula of quadratic form for Gaussian random vectors, specifically when Xo ~N
16

---

Gradient variance for Tror and PoT
For the rotated case:
Var
Ve N
Σ
M 1
RXO (m) -x(0") - - -x -
(n)
(34)
= Var
[N N
2
E
1 (++ (M R Q 1)x0 - 0)
(35)
(36)
At the optimum, where θ = M2/ R/ La- obtain:
Var I 1MM R M R) = 0
(37)
N
For the variance of the gradient w.r.t. O at the optimum, we get:
Var
V L MC (6, 0)
N X
Var
M 1/2 (R- R)Xx
(38)
0
(39)
Way (for any i, closed form is not nice): Gradient variance over time Before analyzing
different kinds of pairings that we want to analyze, first, want true formulation of the gradients:
Var V
MC
type (+,0)
0]]
(40)
1
(Ttype type (X(x) - -x(") (X," .t)1(²]
Var
Vo
(41)
N
Var
2
Σ (eype (X)(²) - -x(0m) - ---0 10) -t0) (-1- + +0
11)
N
(42)
2
Σ (Teype (X) (x(x)) -x(") 0 - 1 (0)) (14a + 10)
Var
N
(43)
For now, let C, [La + = We arrive at more simplified:
4
Var Vo T
type (a, 0)]
Var( (ar[(Ttgpee (Trgpe (Xo) - X, - 0 -0) C..,)
Ci (44)
N
Var (T (Xo) - Xx - -0 C,,11]
(45)
N
(C,,,l)² 4
Varl (XoT)ppp()))))) - Xo O + -,)
(46)
N
Moving forward gradients with respect to we get:
Var Ve
type (θ, 0)]
(47)
MC
1 Σ11(T = Var Ve N
(((TTpee(x(()) ) - x(()) - t v (x.(), (x,)(x((,,,()
type (x
(48)
4 Var (Teype (Xo) Xo -00 (0) ((CC,, 1x),, 10))
N
(49)
(50)
CASE (OT): This case is nice because we just need to prove that:
(Tr( - Xo -
0 - --0 -
-t0) =0,
(51)
17

---

which actually happens once we replace e = (M 1), and = μ.. We will show this for completion:
M - Xo - 10)
(52)
-MM1/2 r)(1a 1MM122 (X 1(M2/1121110-1 1)Xo - 1M - ())
(53)
-M I)( -1)
(54)
Remark our network is not quite at the optima so M Xo T a 10) - = c we can notice that the variance of and è decrease in time because 1+
and 1+frtt decreasing in (1+tz)²
No connection between gradient variance and crossings show the lack of connection between crossings and variance of gradients, it is enough to offer counter-examples. In the rotation case, we will look at the variance when the rotation matrix does 180 degree rotation, which means all interpolation lines should meet at / = 1/2. As we will show, there is no peak in variance at time t = 1/2. In the Gaussian case, we will observe very similar behaviour.
Case 2 (rOT):
For 0 the variance of the gradients will be:
4
Var VorVooL LC
type ( (θ, = CT
N
1
1.6
(55)
4
1 1,6 A
A AC, ,
(56)
where M1/2 12 1)[I + t(M - - (f(M 1) + I), with A A ~ M + - R RD + 2(1 t) - I)" (M I)(R I), for small 1/2 (approximation obtained via taylor expansion).
Counter-example For Rt = -1 and M 21(which is 180 rotation so all interpolants meet at (1- 1. The variance this case
one point) we have = -31 1 [1 (-3 A will be increasing for t [0, 1]. This aligns with our empirical results; however, it does not align with what is believed in the literature, that the variance would peak at = 1/2, because crossings increase variance.
2. For O the variance of the gradients will be:
Va rOT (Ś,0))
(57)
Var [(Tro((Xo) - -00 1100--11100001---0001111---0---- ()) t0))]
(58)
N Var - - (M1/22 - -
-| (1- (59)
(tCr,e (t (1- ([
(60)
4 V B (Tr((AB))
N -
(1-
N
(61)
Counter-example Let M1/2 = 21, and R = -I. Then A = I4
4 J. and B = (3t so
(1+1)*
Var VeLLOT (Ô,0) 0) 4 Varl [144)² (144)²
(3t-1)
X3" Xo]
8d (16(3--1)) which is decreasing for (1+1]
1. E [1/9, 1/3] and increasing E 1/3, 1].
Case 3 (Random):
18

---

1. For 0 the variance of the gradients will be:
4 12 c
type (θ,0)] Var[VeLMC
N 1,6 -
Xo
- -11-(())-(--) (62)
[Va((( - (arll//))iaVVaaa-----aaaa -100[(--,,,,0 ,0 - -o) C
C
Varl(-1 ,
N
(63)
because of independence of Xg, and X, Continuing:
- V (6t)) (I -1(221222 - ((M2)
(64)
(I - 1T-((14/2 [) - -
(65)
= -1 =1+ 1)) 11)
(66)
and,
Var[(I 60, ,0 - X)) =(1 -1 11-1111 T)(T + (1) 1
(67)
(1- (11-)((M1122. 1) + - -))-
(68)
Counter-example: For M 21 we have Var((1 - , = 41 which is decreasing
fort E [o, 1]. We have "ar[(I-@C, (t) (140)2 I. That is 4 12+1
which is decreasing
(1+1)2
fort E [0, 1]. We have the overall variance:
1
1 +
Var[Ve Eype (O, = 41
11 4d
(69)
1+/ (1 1)2
1+
(1 1 1)a
which is decreasing in f E [0, 1].
C Proofs ReFlow
Lemma 2 Let m and me be two distributions on R&N admitting densities, and let (Zo, Zi) be their straight line coupling. If we apply Rectified Flows again using the noise-free interpolant, recover the same coupling.
Proof. Since (Zo, Z1) is the straight coupling, for each [0, the interpolation
Z. (1 (1-t) t) Zo + Z1
oZo, measuuulll Define
b(2,t) = E[ 21 Zo 2: =]],
so that by the Doob-Dynkin lemma there is (deterministic) function satisfying
v (Z,,
E| 21 Zu
Ze] Z1 Zo almost surely.
Hence we re-solve the same linear ODE
dX,
= r (X,,t)
with Xo = Zo.
dt
the unique solution is exactly Because the noise-free Rectified Flow minimizes the mean squared drift error,
X, Zo +(1 Zo) Z..
((,, (Z1 Zn) = 0,
no further change occurs. Thus the pairing remains (Zo, Z1).
19

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Let and be two probability densities on Rd, and let T : RRD + +RD be deterministic transport map pushing To γι (so if Zo To. then Z, (Za) π1). Suppose we draw finite dataset of N i..d. samples (Z z
Proposition 2
T ( ) and for each we also sample (3.)
m time-points ( C [0. (e.g. uniformly). Let 2,(.)) = (1 {(i5)) 7)²) + {(65) 2
Define the empirical loss over this "doubly indexed" dataset by
m
200) we (2(b0)) (.-20) ||2
(ve)
N
==1
Then there exists (deterministic) vector field v attaining zero loss: Ldet (w) 0.
Proof. Our finite dataset is
D = { 2 (1-5)) 2, ((20) :i= 1...., N; 1=..,....
We may view this as drawing from the discrete joint "empirical" measure
N
P = N m
(260 20,)
1m jm1
(z ((H))) we know
For each datapoint
z
Z
E Z1 Zo
Zo Z Zi = 2(10 -(,0), Zi =z(
Let Z -UIeN.eto.1 t). Let A NueN.eto.1j(2 (() ,).
The probability of sampling ((J) E AA is O, because t lives in a two-dimensional distribution and the points of intersection are a countable union (maximum N! intersection) of ne-dimensional spaces.
E
Z1 Zo Zu = 2 Zi t= , ((,)),
Z. = Z1 Zo t= +(,) Ze -z,b01]
Hence we can define the exact conditional-exxectaaiin vector field
0(2,1) = Ep - Zo 2, = 2, 0) which on each of our training points (2, = (2( /((,,))) satisfies
u (Z {(())) = - - -30")
Substituting this v into the Monte Carlo loss gives, term by term,
|z z(dh (( (lt.5) 2 2 z ( 21 =0.
Averaging over all i,j yields T, (w) = 0.
Thus the loss can be driven exactly to zero the finite sample by choosing to interpolate the known displacements z(n) 2 at the sampled intermediate points (Z(²a)) ((..))..
Proposition 3 (Interpolating lines don't meet in high dimensions). Let To, T1 Tu(x) and yo, Y1 mo Ti (y), where mo and IT are probability distributions on R&d admitting density. Define the linear interpolants l((ta) (1 t,),, + t.y for i {0, 13.
A. For d > 2, the lines lo and I cross at = t; = E (0,1) with probability 0.
B. For d > 2 the two lines intersect forti,t E (0, 1) with probability 0.
20

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Proof. Throughout this proof, we use the following theorem: Let X be random variable with continuous probability distribution in R", and let be lower-dimensional subset of R" Since admits density, it follows that it is absolutely continuous wrt to the Lebesgue measure À then as X(A) by absolute continuity, we have that P(X E A) O.
A. Suppose the lines (t) and 1 (t) intersect some t f E (0, 1). This implies lo(t) (1())
which expands to
(1 f) + fyo = (1 1-)xz1 ty
Rearranging terms, we find
1 - (zo σ1) yo-
1/1
To compute the probability of such intersection, note that y1 must lie exactly the affine subspace defined by the above equation, which is a one-dimensional line segment in Rdd
The joint probability of the points (Io, 21, V0. v1) can be written as
1- {
(to((t intersects (t) for t (0, 1)) P( T1, 9o) P1/1
(xo 21) yo To. 21, 21,3)).
Since Ti (y) is continuous and differentiable, the probability density of y1 lying any lower- dimensional subspace (e.g., line segment) in Rd, with d > 2, is zero. Therefore,
i (zo x1) + yo To. F1 Bo) 0,
P(y
f
which implies
P((o intersects (t) at = for t E (0, 0.
Hence, the lines (t) and 4 (t) intersect with probability zero for (0, B. Suppose the lines (l)) and l (41) intersect some lo 1, [, f,forlo, E (0, 1). This implies
lofa)) (f1);
which expands to
(1 fo) + toyo = (1 in )m + lyn-
Rearranging terms, find
(1- fo)2o (1 (1) + fovo
y1
f
which for fo & E (0,1) we have that yı would belong to 2D surface. Just like before we have:
((o((fo) intersects 4 (f1) for & E (0, 1)) P(zo, 21, v0)-
P y1 (1-10)ag
(1 (1) + lovo
zo. 31, yo
0.
&
D Counter Example for straightness after one iteration
RRP, with D > 2. Let (Zo, Z1) = 1 1-ReFl ow Xo, X1) denote the coupling obtained after one ReFlow iteration. Under the assumption that the interpolant p(z, 1) (1 () (T((z) is injective in for each I, there exists vector field èn (zt, t) such that performing a second ReFlow iteration using (Zo: Z.) yields the same coupling:
Proposition 4 (Limitations of ReFlow Iterations on Noiseless Interpolants). Let mo, गi
2- R-ReFl Lowl X1) = (21, 2n)-
Moreover, this second flow in (xx, 1) generates straight- paths and achieves zero loss. Therefore, further ReFlow iterations do not alter the couplings.
21

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Figure 8: 180 rotation being realized by a continuous vector field.
Annotated Proof. Let t) be the learned vector field after CFM, with Assumption [
From
Arnold 1992], the transport map T(zo) + So v(ze, f)dt with 2g mo is injective.
Step 1: Injectivity of
Assumption: T is injective (guaranteed by properties of v, e.g., Lipschitz, linear growth).
Potential Failure: If'I is not injective, the argument fails immediately.
Step 2: Injectivity of the Interpolant p(z,t)
We claim that p(z, () (1 - 1) + (z) is also injective for each / [0, 1].
Proof (by contradiction): Suppose p(2(") = p(=0²) f) for some and 2/ (1)
+ 20
Rearranging:
T(= T(2 1- (2)) i (=0
This statement doesn't 't really contradict with T injectivity.
Failure Point: As shown in Counterexample I this is not always true. For example, if is a rotation plus translation, p(z, t) can fail to be injective even injective.
One might argue that such transport map unlikely to learned in practice; however, this not the point. Our argument is purely theoretical: injectivity of T does not imply injectivity of [22 (zo), t), and thus 1-ReFlow is insufficient even under standard regularity assumptions.
Step 3: Construction of Inverse and New Vector Field
Assuming injectivity, T(f (2., f- (2,,t)
define inverse (=1()) 20. and then in (1,,))
Dependency: This construction only works if f exists. i.e., if p(2, t) is injective.
Failure Point: If p(z,t) t) is not injective, is not well-defined, and this construction fails.
Step 4: Straight-Line Paths and Zero Loss
Because 21 2g is uniquely determined by 2,, we have f E[Va(ZZ Z,(2,]] = 0, implyyng straight paths and zero loss.
Dependency: Again, relies the injectivity of p(2, t).
22

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Counter Example 1 Let ~ (0, Ia) and Tı ~ N5, Id), and let T(ro) Risue 2o + 5. where Riso is a 180 rotation. T is injective, but the interpolant I [2g, t2)) (1 t +2) (0) is not injective (distinct can map to the same for some 1). Thus, f((1,1) 1) -nn is not well-defined, and the construction of Unew (vt, 1) fails.
Step
Assumption Needed
Where It Can Fail
1
T injective
Pathological T Nonlinear T (e.g., rotations) p(2, t) not injective
2
p(z, t) injective for
3
f=1 exists
Unique 2g for each 21
p(2, 1) not injective
Table 4: Summary of dependencies and failure points in the proof.
Final Note. This proposition only valid under the additional assumption that the interpolant p(2,0 1) is injective in for each /. The counterexample demonstrates that this is not always the case, even when injective. Therefore, care must taken before applying this argument general settings.
E
First iteration of SBM Asymmetry
To test whether our maddiet-vvariane diagnostic extends beyond Gaussians, we trained four U-Net models learn the transport map from To (0, La) to πi ~CIFAR-IO, using both noiseless and noisy interpolants (see for experimental details Appendix G.. In all settings (Figure 9. the backward field's gradient variance peaks f 0, where CIFAR- appears as input. Conversely, the forward field's variance peaks at = 1, when CIFAR-10 appears as the output.
To assess generation stability, checkpoints, sampling at regular intervals near the FID/NLL optima (TableB). We report the resulting sample variances in the table's fourth column:
repeatedly integrated 64 fixed datapoints across 100 training
Var[=f" and Var[=g"
which show consistently higher variability in the forward direction, suggesting reduced robustness during forward-time sampling.
The observed difference in sample variance across vector fields optimized near the optima may potentially impact subsequent iterations of SBM (Shi et al.) 2024y. However, precisely characterizing how this affects theoretical bounds is non-trivial and left as a promising avenue for future research.
Observation 1.
Table 5: Comparison of FID, NLL, and sample variance for CFM( = 0.05) (first iteration of SBM) and CFM (Forward and Backward). The backward vector field exhibits approximately 10 times lower variance in integrated samples compared to the forward vector field, despite both endpoints being standardized, This highlights inherent asymmetry in sampling stability between forward and backward flows.
Direction
FID4
Scaled -NL..J
Variance Std. Error
CFM( 0.05)
Backward
1,423564
0.01647 0.00024
Forward
4.250
0.36179 + 1 1.23 x 10-6
CFM
Backward
1.426562
0.01414 0.00018
Forward
4.199
0.36177 1.23 10-5
F
Synthetic Experiments Details
Experiment Setting We evaluate generative models synthetic datasets in dimensions 3 and 50. Fach dataset is constructed by sampling from a Gaussian Mixture Model (GMM) with randomly initialized means and covariances, following implementation in cinerrat_dataseee py. The
23

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Forward Gradient Variance Time
Backward Gradient Variance Time
Credant Venaatce
ent Menteree
CFM
CFM
CFM
Time (1)- From Gaussian to CIFAR-10
Time (t)- From CIFAR-10 to Gaussian
Figure 9: Comparison of gradient variance for the forward and backward passes in CFM and CFM(( 0.05) (first iteration of SBM). The schematic on the right offers intuition for how gradient variance may influence sample variance during integration. Notably, the backward-pan gradient variance peaks near the CIFAR- endpoint, yet the resulting sample variance is lower compared to the forward pass, This is possible, for instance, if the forward field increases linearly while the backward field decreases linearly. These effects are quantitatively reflected in Table T
source distribution is standard normal, and the target the GMM. We compare Conditional Flow Matching (CFM) and CFM with Stochastic Interpolation (CFM = 0.05)). both implemented as neural ODEs with time-conditioned MLP (Three-layer MLP, width 64, SELU activations) vector fields. Models are trained and evaluated on both in-sample (training) and out-of-sample (test) data. All metries are computed as described below. We have used resources from Feydy et al. [2019] to compute the distances.
Metric Descriptions
Log Probability (LogProb): Measures the average log-likelihood of generated samples under the target GMM distribution. Lower values indicate a better fit the modes.
Maximum Mean Discrepancy (MMD): A kernel-based statistical distance between two distributions, here computed using a Gaussian kernel. Lower values indicate better sample quality.
Sinkhorn Distance: An regula--eegl approximation of the Wasserstein (optimal transport) distance between empirical distributions, computed using the Sinkhorn algorithm. Lower values indicate closer distributions.
Note Log-Like Lig-Liielihood Values the positive values of the lig-likelihood (LogProb) throughout this paper, rather than the conventional negative log-likelihood (NLL). This allows for more intuitive comparison, where lower values indicate better model performance.
To improve readability and avoid confusion, report and plot
It is important note that log-likeliing likelihood (LogProb) primarily rewards models that generate samples close to the high-density regions (modes) of the target distribution, rather than accurately capturing the overall shape support of the distribution. As result, models that concentrate samples around the modes can achieve high log-likelihood scores even if they do not match the full distribution well. This should be kept in mind when interpreting LogProb values alongside other distance-based metrics.
Table 6: Comparison of CFM and CFM( 0.05) across dimensions 3 and 50.
Dimension
50
Gen
Mem
True
Data
Gen
Mem
True
Data
LogProb
4.0150
4.0156
4.1330
1.0155
51.8299
53.6502
52.5091
53.6211
1036
CFM
MMD
0.0034
1.758 x 10
0.0014
0.0032
0.0021
9.089 x 10
0.0020
0.0019
+0.003
#0000
10000
+2
=C000
Sinkhorn
0.0730
1411
1
0.0637
0.0790
1900
0.0045
14.3221
15.7400
100051
1000
2000
10152
10110
Lo110
LogProb
4.1270
4.0960
4.1330
4.0155
54.7220
53.8890
52,5094
53.6244
100091
Labis
100006
20001
1014
1811
CFM( 0.05)
MMD
0.0018
3.105
10
0.0014
0.0032
0.0020
6.09
10
0.0020
0.0019
1000044
200000
100301
1.3400
10100
Sinkhom
0.0680
3.557 x 10
0.0637
0.0790
1689
0.0304
14.3221
15.7400
24

---

How to read the table? This table summarizes several metrics for model evaluation. For readers unfamiliar with these results, here is how interpret them:
Consider the case for dimension 3. For the Gen (generated) column, we want the log-likelihood value to be closer the True value rather than the Data value. If the generated value is closer the data, it indicates memorization, rather than true generalization. For example, in CFM, the generated value much closer to the data than the true value, suggesting memorization. This discrepancy arises because negative log-likelihood (NLL) tends to favor models that sample near the training data modes, rather than those that capture the full distribution.
Similarly, in the Mem (memorization) column, a value close to the data again indicates overfitting to the training points.
For the MMD and Sinkhorn metries, these measure distances hetween newly generated trajectories (starting from training points) and the pairings from previous iterations. In both MMD and Sinkhorn, we observe that CF = 0.05) memorizes roughly ten times less than standard CFM, indicating better generalization.
25

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F1 Asymmetry in Gaussian mixtures
Experiment Settings We systematically investigated the Conditional Flow Matching (CFM) and CFM( = 0.05) approaches for learning mappings between mixtures of Gaussian distributions (GMM In each experiment, both the source and target distributions were Gaussian mixtures, with the number of components for each varied across the grid: {1, 2, 4, 8, 16, 32, 64). The neural network architecture for the vector field was a multilayer perceptron (MLP) with configurable width (w, e.g., 64 by default) and input dimension (d, e.g.. 10). Each cell in the results grid corresponds to a specific pair of source and target GMM component counts, allowing us analyze the effect of distribution complexity performed for up to 50,000 epochs using batch size of 128 and a learning rate 10 with integration performed via Neural ODEs.
learning dynamics and gradient variance. Training was
Figure 10: CFM in 10 dimensions. Each title indicates the number of Gaussian components in the source distribution multiplied by the number in the target distribution (e.g., 4x16 means 4 modes at source, 16 at target).
26

---

Lx4
2.16
Lx
1.
Figure 11: CFM combined with stochastic interpolants for mixture-to-mixture transport in 10 dimensions. Each panel is labeled as source modes x target modes.
We can notice that the variance of the gradients looks more stable in the case of the Figure II (stochastic interpolants), than in the case of Figure 10 (noisless).
G CIFAR-10 Experiment Details
Model architecture. All experiments used U-Net-based neural network (UNet Wrapper))))))) with the following configuration: imput shape (3, 32, 32), base channels 128, 2 residual blocks per level, channel multipliers [1, 2, 2). attention 16 16 resolution (4 heads, 64 head channels), and dropout rate 0.1. The model is wrapped in a Neural ODE solver (Euler method).
Training. Models were trained on the CIFAR-10 training set, using random horizontal flips and normalization to [-1, 1]. Optimization used Adam with learning rate 2 x 10 4, batch size 128, gradient clipping at 1.0, and a linear warmup over the first 5,000 steps. Each run used 400,001 steps (unless otherwise noted), with exponential moving average (EMA) of model weights (0.9999 decay). Checkpoints were saved every 20,000 steps. All experiments used data loader workers and CUDA if available.
Flow objectives. We used Conditional Flow Matching (CFM, -model cfm), Schrödinger Bridge Matching (SBM, -model sbm), and other variants.
Bidirectional setup. Both forward (Gaussian + CIFAR-10) and backward (CIFAR-10 Gaussian) models were trained independently with identical hyperparameters. For the forward model, the source is standard Gaussian noise and the target is real images; for the backward model, the roles swapped.
Hardware and runtime. CIFAR-10 10 experiments were conducted on a compute cluster equipped with NVIDIA A10 GPUs (24 GB VRAM, CUDA 12.2). Each training run was allocated single A10 GPU and typically ran for 24 hours to reach 240,000 optimization steps. These resources enabled efficient training of both forward and backward models the scale reported in the main text.
27

---

G.1
FID and Interpolant Details
For Figure 0 interpolant:
we trained four vector fields (two forward and two backward), each using different
CFM: The interpolant is deterministic,
It = (1- + tz
CFM = 0.05): The interpolant includes stochastic noise,
3z = (1 f)=o tx, + a V -111 1)
where Z I), and 0.01.
The models evaluated follows:
CFM forward FID: 4.199
CFM backward scaled NLL: 1.426
CFM 0.05) forward FID; 4.250
CFM(( 0.05) backward scaled NLL:: 1.423
G.2
Does adding powers of t help models learn better?
Grddent Vartene #1!
Figure 12:
Variance not necessarily correlated with performance. was wondering if it is worth saying that sinusoidal time embedding is better at representing function s like q(x) I+2 than polynomials. (Taylor vs Fourier)
H CelebA Experiment Details
In Table 7 we computed FID on both the training and validation sets and observed similar values, which could, in principle, indicate the trade-off between memorization and generalization. However, the CelebaOt dataset Korotin et al. [202T] is itself generated by another model, and with only 50k generated samples it likely contains overlapping images. Consequently, the training and validation splits may share duplicates, making it difficult to draw strong conclusions from the FID scores. We include these results merely as evidence that all models were trained satisfactory degree.
28

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Setting
Method
FID
CFM(( = 0)
34.36
Shuffled
CFM(a 0.05)
32.31
CFM( 0)
31,42
ReFlow
CFM(a 0.05)
42.38
Table FID scores for the 50k dataset-side setting. Lower is better.
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Reviews:
[{"review_id": "weoOph9tHa", "reviewer": "Reviewer_vzJd", "summary": "This paper proposes multiple theoretical results related to vector field learning.  They show that various ML architectures cannot approximate an optimal transport vector field without error. Next, they assert that low loss gradient variance does not necessarily correspond to an optimal learned vector field.  They provide a series of observations about the impact that stochastic interpolants have on the gradient variance, as compared to deterministic interpolants.  Finally, they present theoretical results on rectified flow models, showing that deterministic interpolants are susceptible to memorization.", "questions": "- In L137, it is not clear to me why rotation of the vector field is introduced.  There is no explanation or background provided in the text.  Can you please elaborate on why this is included?\n- What does Lemma 1 have to do with the central question that the paper is posing (i.e. what does this tell us about the optimality of vector fields)?  The lemma states that various ML architectures cannot perfectly approximate an optimal vector field — could you please clarify why this is relevant?\n- Proposition 1 — it’s unclear to me what the significance of this proposition is.  Bounds are derived for the variance of gradient.  However, the takeaways offered in the text (L162-L166) almost seem to assert that the variance of the gradient does not reliably indicate transport optimality.  Then what is the insight?\n- Section 4 — what does this section have to do with analysis of the gradient variance?", "limitations": "- Results would be more convincing from an empirical perspective if larger, more complicated datasets were used beyond CIFAR-10.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "__Strengths:__\n- The paper is written with mathematical rigor in definitions and derivations.\n- The idea of employing variance of the loss gradient as a “diagnostic” for measuring the faithfulness of learned vector fields to the true optimal transport is creative.\n- Insightful result that the introduction of noise via stochastic interpolants can ameliorate certain issues such as stagnation. \n\n__Weaknesses:__\n- The significance of the paper’s main results are not clear.  While the theoretical exposition is thorough, the paper would benefit from a more assertive stance on what these multiple results collectively signify.  After reading this paper, I do not come away feeling a greater sense of insight about the question posed in the title of this work.\n- Empirical results are evaluated only on a limited dataset (CIFAR-10).  Additional empirical validation on larger, more diverse data domains would highlight the findings more.\n- There is an overall lack of “connective tissue” in the writing.  Lemmas and propositions are offered without enough context, and there is very little flow between sections 3 and 4 (and their subsections).  It feels like an ad hoc assortment of observations rather than an intentionally constructed set of arguments supporting a central thesis.", "was_revised": true, "final_justification": "In their rebuttals, the authors have provided both additional empirical studies, as well as a detailed elaboration of what they plan to revise/add in order to strengthen the overall narrative. I believe this will make the paper much more readable, and hence more effectively reach its intended audience. My concerns are addressed, therefore I have decided to raise my initial rating by two points.", "rebuttal": "We thank the reviewer for the thoughtful feedback and for highlighting the need for clearer conceptual connections. In our revision, we will improve the paper’s structure and include new CelebA experiments (which are also detailed later in this rebuttal) that further validate our theoretical findings.\n\n---\n## Addressing Weaknesses\n>  \"significance of the paper’s main results are not clear.\" ,   \"The writing style [..] tends to assume that the reader should 'just get' why their problem is important\"\n\nOur paper sheds light on the underlying causes of common failure modes behind the state-of-the-art fixed-point generative models. Our main claim is that in noiseless transport settings, deterministic pairings and interpolants, low gradient variance does not imply $W_2$ optimality. This challenges three common assumptions:\n\n1. Gradient variance arises from interpolant intersections [3, 4]  \n2. ReFlow [5] corrects such intersections via iteration  \n3. Convergence of ReFlow is well-understood [6]\n\nWhile deterministic ReFlow can straighten paths, it lacks convergence guarantees. A now-retracted claim (with the authors highlighting flaws in the proof of this claim) [1] suggested two iterations suffice; we show that even after two steps, ReFlow may fail to yield injective couplings  (Counterexample 1). \n\nBuilding on this, we reframed Rectified Flows as an iterative process on finite datasets, as encountered in practice. This revealed that due to discretization and limited data, vector fields can skip over interpolant intersections (Figure 6), failing to correct suboptimal paths. Rather than converging to the optimal transport solution, repeated rectification can lead to memorization: the model simply reproduces the observed pairings, regardless of their quality. We formalize this in Proposition 2, showing that ReFlow can achieve zero loss and gradient variance by memorizing suboptimal deterministic couplings.\n\n>  \"Additional empirical validation\"\n\nWe now include new experiments on CelebA OT pairs from [2] that directly validate Proposition 2 in both adversarial and ReFlow-style settings.\n\n> an overall lack of “connective tissue” in the writing.\n\nWe acknowledge the reviewer’s concern, and will strengthen the paper’s narrative thread by clarifying the following schema, both textually and through an improved Figure 1:\n\n(A) Lemma 1 (Gaussian closed-form) → Prop. 1 (grad. variance bounds)  \n(B) Gaussian experiments  \n(A)+(B) ⇒ Conclusions (low grad. ≠ OT; reflow skips intersection) → (generalizes) → Prop. 2 (finite data).\n\n## Questions\n\n> Rotations in L137:\n\nThe role of rotation (L137) is to introduce a formulation for rotational vector field, which we then study the variance of in Proposition 1. \n\n> What Lemma 1 has to do with the central question that the paper is posing: \n\nThe main purpose of  Lemma 1  is to give the closed-form OT vector field formula on which Proposition 1 builds. The extra fact about exact representation is not necessary in the proof of the Proposition and could be introduced as a following remark.\n\n>  Significance of this proposition: \n\n Proposition 1  supports our empirical findings, which show that low gradient variance does not imply optimality. \n\n> Section 4 relation to the analysis:\n\nSection 4 extends our findings beyond Gaussians. Proposition 2 proves that for deterministic pairings, CFM can reach zero loss and gradient variance by memorizing, even when the vector field is not optimal.\n\nThis supports our central claim:  low gradient variance is not a reliable proxy for transport quality. We clarify this thread throughout the paper.\n\n---\n\n## New Experiments on CelebA\n\nTo empirically validate Proposition 2, we evaluate Conditional Flow Matching (CFM) on CelebA using ground-truth OT pairings from Korotin et al. [2]. Our goal is to test the central claim: when trained with deterministic interpolants, CFM is optimized to memorize its input pairings.\n\nTo test this, we compare:\n\n- **CFM (σ = 0):** trained with deterministic interpolants  \n  $x_t = (1 - t) x_0 + t x_1$\n\n- **CFM (σ = 0.05):** trained with slightly noisy interpolants  \n  $x_t = (1 - t) x_0 + t x_1 + \\sqrt{t(1 - t)} \\, \\sigma \\epsilon$\n\nThe small amount of noise ($\\sigma = 0.05$) is not introduced for performance reasons but to break the injectivity between $(x_t, t)$ and $(x_0, x_1)$—disabling the memorization pathway outlined in Proposition 2. Both models use identical U-Net architectures.\n\n---\n\n### Adversarial Pairings: CFM Optimizes for Memorization with Deterministic Interpolants\n\n**Setup:** To directly test whether CFM with deterministic interpolants ($\\sigma = 0$) is optimized to memorize suboptimal pairings, we construct an adversarial dataset:\n\n- We start from ground-truth OT pairs $(x_0, x_1)$ derived from the benchmark of Korotin et al. [2].\n- We shuffle the targets $(x_1)$, breaking the optimal structure and producing random pairings $(x_0, S(x_1))$ with no coherent transport semantics.\n\nWe compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, S(x_1))$.\n\n| Dataset     | Analysis       | Metric         | CFM (σ = 0.05) | CFM (σ = 0.0) | Advantage                         |\n| ----------- | -------------- | -------------- | -------------- | ------------- | --------------------------------- |\n| 5K Samples  | Generalization | L2 to True OT  | 34.25 ± 7.54   | 50.40 ± 16.73 | CFM (σ = 0.05) generalizes better |\n| 5K Samples  | Memorization   | L2 to Shuffled | 55.02 ± 16.51  | 28.57 ± 5.49  | CFM (σ = 0.0) memorizes more      |\n| 50K Samples | Generalization | L2 to True OT  | 30.05 ± 6.77   | 46.78 ± 14.87 | CFM (σ = 0.05) generalizes better |\n| 50K Samples | Memorization   | L2 to Shuffled | 56.48 ± 18.55  | 45.98 ± 11.85 | CFM (σ = 0.0) memorizes more      |\n\n\n**Evaluation Protocol (Lower is better):**  \nWe report L2 reconstruction error to  the true OT targets to assess generalization, and  the shuffled training targets to assess memorization.  \n\nSince the pairings are deliberately incorrect, a low L2 error to shuffled targets indicates overfitting to suboptimal data. High L2 to these targets signals resistance to memorization.\n\n**Interpretation:**  \nThe results directly support Proposition 2: CFM with deterministic interpolants ($\\sigma = 0$) minimizes loss by tightly memorizing its training targets, even when they are arbitrarily shuffled and non-optimal. In contrast, CFM with small noise ($\\sigma = 0.05$) resists this memorization trap and generalizes significantly better to the true OT targets.\n\n**Extra comments:**  \n- Memorization decreases with larger datasets, as expected, since model capacity is fixed, it becomes harder to memorize all pairings in the 50K setting.\n- However, this does not contradict Proposition 2: the optimization still favors memorization when possible. Larger datasets simply make perfect memorization less feasible, not less likely as an objective.\n\n---\n\n### Simulated 1-ReFlow: CFM Memorizes Its Own Outputs\n\n**Setup:** To evaluate whether iterative flow models suffer from self-reinforcing memorization, we simulate a 1-step ReFlow scenario—a core case of Proposition 2.\n\n- **Initial training:** We train a base CFM model ($\\sigma = 0$) on random, non-OT pairings.\n- **Integration:** Using this model, we generate new endpoints $(x_0, x^1)$ via ODE integration.\n\nWe compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, x^1)$.\n\n\n| Dataset     | Analysis       | Metric        | CFM (σ = 0.05) | CFM (σ = 0.0) | Advantage                         |\n| ----------- | -------------- | ------------- | -------------- | ------------- | --------------------------------- |\n| 5K Samples  | Generalization | L2 to True OT | 31.35 ± 7.38   | 43.35 ± 14.21 | CFM (σ = 0.05) generalizes better |\n| 5K Samples  | Memorization   | L2 to $x^1$     | 25.08 ± 8.59   | 8.63 ± 1.76   | CFM (σ = 0.0) memorizes more      |\n| 50K Samples | Generalization | L2 to True OT | 32.54 ± 8.86   | 38.16 ± 11.37 | CFM (σ = 0.05) generalizes better |\n| 50K Samples | Memorization   | L2 to $x^1$     | 16.29 ± 4.82   | 12.76 ± 4.01  | CFM (σ = 0.0) memorizes more      |\n\n\n**Evaluation Protocol (Lower is better):**  \nWe report L2 reconstruction error to the true OT targets to assess generalization, and the generated training targets $(x_0, x^1)$ to assess memorization. Since the pairings come from a previous CFM model trained on random (non-OT) data, a low L2 to $x^1$ indicates that the model is overfitting to flawed past outputs. High error rates to these targets signal that the model is resisting memorization and potentially correcting earlier mistakes.\n\n**Interpretation:**  \nThe results in the table provide further support for Proposition 2: CFM with deterministic interpolants ($\\sigma = 0$) is highly effective at memorizing the training couplings, even when those pairings are suboptimal and self-generated. In contrast, CFM ($\\sigma = 0.05$)—despite only a small amount of noise—avoids this memorization trap and learns to generalize toward the true OT map, despite never being directly trained on it.\n\n---\n\n### References\n\n[1] Saptarshi Roy, Vansh Bansal, Purnamrita Sarkar, and Alessandro Rinaldo. *2-rectifications are enough for straight flows: A theoretical insight into Wasserstein convergence.*\n\n[2] Korotin, Alexander, et al. *Do neural optimal transport solvers work? A continuous Wasserstein-2 benchmark.* NeurIPS 34 (2021): 14593–14605.\n\n[3] Fjelde, T., Mathieu, E., and Dutordoir, V. *An Introduction to Flow Matching.* MLG Blog, January 2024.\n\n[4] Gagneux, A., Martin, S., Emonet, R., Bertrand, Q., and Massias, M. *A Visual Dive into Conditional Flow Matching.* ICLR Blogposts, April 2025.\n\n[5] Liu, Xingchao, Chengyue Gong, and Qiang Liu. *Flow straight and fast: Learning to generate and transfer data with rectified flow.* arXiv:2209.03003 (2022).\n\n[6] Lee, Sangyun, Zinan Lin, and Giulia Fanti. *Improving the training of rectified flows.* NeurIPS 37 (2024): 63082–63109."}, {"review_id": "NN72Fed2Fo", "reviewer": "Reviewer_yZa2", "summary": "This paper investigates the fundamental limitations of flow-based generative models, particularly Rectified Flows and Schrödinger Bridge Matching, through a novel gradient variance analysis framework. The authors introduce a diagnostic tool that measures how learned vector fields induce transport errors by analyzing the variance of gradients during optimization. Their main theoretical contributions include proving that standard neural network architectures cannot exactly represent even simple Gaussian-to-Gaussian optimal transport, demonstrating that Rectified Flows can stagnate when reaching straight couplings and memorize deterministic pairings rather than learning optimal transport, and showing that stochastic interpolants help break these failure modes. The work challenges the common assumption that straight transport paths are always optimal and provides both theoretical insights and empirical validation on synthetic Gaussian data and CIFAR-10.", "questions": "- Can you provide results on additional real-world datasets beyond CIFAR-10? Your theoretical claims about fundamental limitations would be much stronger with broader empirical validation. Specifically, how do your findings hold for higher-resolution images, different data modalities, or more complex transport scenarios?\n- While the gradient variance diagnostic is theoretically interesting, how can practitioners actually use this tool? Can you provide concrete guidelines for when the diagnostic suggests switching to stochastic interpolants or adjusting noise levels? A practical algorithm that leverages your insights would significantly strengthen the contribution.\n- Comparison with recent advances: How do your findings relate to more recent developments in flow matching, such as consistency models or recent improvements to Rectified Flows? Have any of these approaches inadvertently addressed the memorization issues you identify?", "limitations": "Yes, the authors adequately address limitations in their conclusion, acknowledging the restricted experimental scope to CIFAR-10 and the preliminary nature of their SBM analysis.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n- I find the theoretical framework quite compelling. The gradient variance diagnostic is genuinely novel and provides insights that pure loss analysis misses - the counterintuitive finding that low variance doesn't guarantee optimal transport is particularly valuable. The mathematical rigor is impressive, especially Lemma 1's proof that MLPs cannot exactly represent the matrix inverse terms in optimal transport vector fields, and Proposition 2's formal characterization of memorization in finite datasets.\n- I think the paper makes important corrections to recent literature. Counte-rexample 1 effectively refutes claims about 1-Reflow sufficiency using rotation-based transport maps, and the theoretical explanation of why stochastic interpolants help (by breaking bijections) provides much-needed understanding of an empirically observed phenomenon.\n- The experimental design is thoughtful, particularly the synthetic Gaussian experiments that allow for analytical verification of theoretical predictions. The CIFAR-10 forward/backward asymmetry analysis provides interesting practical insights about sampling stability.\n\n# Weaknesses\n- My main concern is the limited experimental scope. While the Gaussian experiments are convincing, only CIFAR-10 is used for real-world validation. Given the paper's claims about fundamental limitations of flow models, I would expect evaluation on additional datasets to strengthen the generalizability argument. However, given the theoretical nature of this paper, I think this is more of a minor concern.\n\n- The practical impact feels somewhat incremental. While the theoretical understanding is valuable, stochastic interpolants were already used in practice (SBM), and the main practical recommendation (use noise) was already known. The gradient variance diagnostic, while theoretically interesting, doesn't clearly translate to improved training procedures.\n\n- From a clarity perspective, some technical sections are quite dense. The connection between gradient variance and actual transport quality could be made more intuitive. Figure 2's schematic helps but could be more detailed about the mechanisms involved.", "was_revised": true, "final_justification": "I maintain my original rating and leaning towards acceptance for this work.", "rebuttal": "We thank Reviewer yZa2 for the thoughtful and constructive feedback. We're glad you found the theoretical contributions and gradient variance analysis compelling. Your comments helped us clarify the connection between our results and recent developments, and motivated us to conduct additional experiments on CelebA to broaden the empirical support. \n\n---\n## Addressing Weaknesses\n\n> \"is the limited experimental scope\"\n\nWe now include new experiments on CelebA OT pairs from [1] that directly validate Proposition 2 in both adversarial and ReFlow-style settings (see the end of the rebuttal for details).\n\n> \"Practical impact feels somewhat incremental.\"\n\nWhile adding noise is not novel, our contribution lies in bridging idealized Rectified Flows (infinite data) with practical settings (finite data), where failure modes emerge:\n- no convergence to straight paths after two iterations (Counterexample 1),\n- skipping intersections at inference (Fig. 6),\n- vector fields minimizing both loss and variance reduce to memorization (Prop. 2).\nThe noise addition is not our contribution per se, but rather a principled choice aligned with the theory: Prop. 2 holds only under deterministic (noiseless) interpolants.\n\n\n>  The connection between gradient variance and actual transport quality could be made more intuitive.\n\nWe will strengthen the paper’s narrative thread by clarifying the following schema, both textually and through an improved Figure 1:\n\n(A) Lemma 1 (Gaussian closed-form) → Prop. 1 (grad. variance bounds)  \n(B) Gaussian experiments  \n(A)+(B) ⇒ Conclusions (low grad. ≠ OT; reflow skips intersection) → (generalizes) → Prop. 2 (finite data).\n\n## Questions\n\n> “Provide additional results…”\n\nWe provide additional results on CelebA (higher resolution images) OT pairs from [2] \n\n> Gradient Variance as a Tool (increase Practicality)\n\nWe agree that utilizing gradient variance as a diagnostic tool could offer a more constructive perspective on the paper. One direction (Fig. 3) is using gradient variance to tune the noise scale during training. For example, one could identify the point at which the gradient variance of a model trained with noise (e.g., SBM) becomes lower—around minima—than that of a model trained with noiseless interpolants. We view this as a separate line of inquiry. \n\nOur motivation for studying gradient variance over time stems from how the term “intersections” has been heavily emphasized in the Rectified Flow literature. In practice, however, intersections almost never occur in high-dimensional spaces with random samples (as formally proven in Proposition 3, Appendix, p. 25). Within this context, gradient variance provides a more meaningful lens for analysis than the geometry of interpolant intersections.\n\n> “Comparison, consistency models…”\n\nAlthough we did not directly evaluate consistency models in this work, prior research suggests they avoid many of the failure modes we observe in Rectified Flows. Consistency models train a one/k-step mapping, avoiding iterative dynamics that can lead to memorization, but even worse, to degradation of the target as more iterations are put in place.\n\n## New Experiments on CelebA\n\nTo empirically validate Proposition 2, we evaluate Conditional Flow Matching (CFM) on CelebA using ground-truth OT pairings from Korotin et al. [1]. Our goal is to test the central claim: when trained with deterministic interpolants, CFM is optimized to memorize its input pairings.\n\nTo test this, we compare:\n\n- **CFM (σ = 0):** trained with deterministic interpolants  \n  $x_t = (1 - t) x_0 + t x_1$\n\n- **CFM (σ = 0.05):** trained with slightly noisy interpolants  \n  $x_t = (1 - t) x_0 + t x_1 + \\sqrt{t(1 - t)} \\, \\sigma \\epsilon$\n\nThe small amount of noise ($\\sigma = 0.05$) is not introduced for performance reasons but to break the injectivity between $(x_t, t)$ and $(x_0, x_1)$—disabling the memorization pathway outlined in Proposition 2. Both models use identical U-Net architectures.\n\n---\n\n### Adversarial Pairings: CFM Optimizes for Memorization with Deterministic Interpolants\n\n**Setup:** To directly test whether CFM with deterministic interpolants ($\\sigma = 0$) is optimized to memorize suboptimal pairings, we construct an adversarial dataset:\n\n- We start from ground-truth OT pairs $(x_0, x_1)$ derived from the benchmark of Korotin et al. [2].\n- We shuffle the targets $(x_1)$, breaking the optimal structure and producing random pairings $(x_0, S(x_1))$ with no coherent transport semantics.\n\nWe compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, S(x_1))$.\n\n| Dataset     | Analysis       | Metric         | CFM (σ = 0.05) | CFM (σ = 0.0) | Advantage                         |\n| ----------- | -------------- | -------------- | -------------- | ------------- | --------------------------------- |\n| 5K Samples  | Generalization | L2 to True OT  | 34.25 ± 7.54   | 50.40 ± 16.73 | CFM (σ = 0.05) generalizes better |\n| 5K Samples  | Memorization   | L2 to Shuffled | 55.02 ± 16.51  | 28.57 ± 5.49  | CFM (σ = 0.0) memorizes more      |\n| 50K Samples | Generalization | L2 to True OT  | 30.05 ± 6.77   | 46.78 ± 14.87 | CFM (σ = 0.05) generalizes better |\n| 50K Samples | Memorization   | L2 to Shuffled | 56.48 ± 18.55  | 45.98 ± 11.85 | CFM (σ = 0.0) memorizes more      |\n\n\n**Evaluation Protocol (Lower is better):**  \nWe report L2 reconstruction error to  the true OT targets to assess generalization, and  the shuffled training targets to assess memorization.  \n\nSince the pairings are deliberately incorrect, a low L2 error to shuffled targets indicates overfitting to suboptimal data. High L2 to these targets signals resistance to memorization.\n\n**Interpretation:**  \nThe results directly support Proposition 2: CFM with deterministic interpolants ($\\sigma = 0$) minimizes loss by tightly memorizing its training targets, even when they are arbitrarily shuffled and non-optimal. In contrast, CFM with small noise ($\\sigma = 0.05$) resists this memorization trap and generalizes significantly better to the true OT targets.\n\n**Extra comments:**  \n- Memorization decreases with larger datasets, as expected, since model capacity is fixed, it becomes harder to memorize all pairings in the 50K setting.\n- However, this does not contradict Proposition 2: the optimization still favors memorization when possible. Larger datasets simply make perfect memorization less feasible, not less likely as an objective.\n\n---\n\n### Simulated 1-ReFlow: CFM Memorizes Its Own Outputs\n\n**Setup:** To evaluate whether iterative flow models suffer from self-reinforcing memorization, we simulate a 1-step ReFlow scenario—a core case of Proposition 2.\n\n- **Initial training:** We train a base CFM model ($\\sigma = 0$) on random, non-OT pairings.\n- **Integration:** Using this model, we generate new endpoints $(x_0, x^1)$ via ODE integration.\n\nWe compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, x^1)$.\n\n\n| Dataset     | Analysis       | Metric        | CFM (σ = 0.05) | CFM (σ = 0.0) | Advantage                         |\n| ----------- | -------------- | ------------- | -------------- | ------------- | --------------------------------- |\n| 5K Samples  | Generalization | L2 to True OT | 31.35 ± 7.38   | 43.35 ± 14.21 | CFM (σ = 0.05) generalizes better |\n| 5K Samples  | Memorization   | L2 to $x^1$     | 25.08 ± 8.59   | 8.63 ± 1.76   | CFM (σ = 0.0) memorizes more      |\n| 50K Samples | Generalization | L2 to True OT | 32.54 ± 8.86   | 38.16 ± 11.37 | CFM (σ = 0.05) generalizes better |\n| 50K Samples | Memorization   | L2 to $x^1$     | 16.29 ± 4.82   | 12.76 ± 4.01  | CFM (σ = 0.0) memorizes more      |\n\n\n**Evaluation Protocol (Lower is better):**  \nWe report L2 reconstruction error to the true OT targets to assess generalization, and the generated training targets $(x_0, x^1)$ to assess memorization. Since the pairings come from a previous CFM model trained on random (non-OT) data, a low L2 to $x^1$ indicates that the model is overfitting to flawed past outputs. High error rates to these targets signal that the model is resisting memorization and potentially correcting earlier mistakes.\n\n**Interpretation:**  \nThe results in the table provide further support for Proposition 2: CFM with deterministic interpolants ($\\sigma = 0$) is highly effective at memorizing the training couplings, even when those pairings are suboptimal and self-generated. In contrast, CFM ($\\sigma = 0.05$), despite only a small amount of noise, avoids this memorization trap and learns to generalize toward the true OT map, despite never being directly trained on it.\n\n---\n\n### References \n\n[1] Korotin, Alexander, et al. *Do neural optimal transport solvers work? A continuous Wasserstein-2 benchmark.* NeurIPS 34 (2021): 14593–14605."}, {"review_id": "IqhVWD2IWJ", "reviewer": "Reviewer_JSzR", "summary": "The paper investigates the common causes of model failures in class of the state of the art (SOTA) fixed point conditional generative models: Rectified Flows (ReFlows), Schrödinger Bridge Matching (SBMs). The main analysis lies in the statistical investigation of the gradient variance to decide whether the resulting vector field is optimal or not. The authors evidently demonstrate counterexamples when the low gradient variance does not coincide with the optimality of the vector field. The presented examples of SOTA models fail to learn exactly the transport between two angled Gaussians. The authors paid special attention to ReFlow model indicating its stagnation on straight couplings tending to memorize deterministic pairings. The noisy interpolants can mitigate these issues on toy examples. The empirical verification of the claimed results provided for conditional flow matching on CIFAR-10 dataset.", "questions": "1) How ReFlows would behave on datasets given the varying dataset size, dimension, noise?\n2) Can additional regularization (eg. KL on distributions) and learning rate/batch size scheduling avoid memorization effect or suboptimal vector fields?\n3) Can you provide some more (counter)examples on CFN suboptimality while achieving a stationary point on loss curve for SBM?", "limitations": "1) The work needs extended experiments with varying dataset sizes, dimensions and (likely) domain types to verify the consistence of provided claims.\n2) The work lack some thorough analysis especially towards SBMs and optimal transport (OT).\n3) The work indicates concrete limitations but hardly discuss how to fix it.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The work provides some rigorous theoretical results on counterexamples for conditional flow matching (CFN) including SBMs and ReFlows.\nThe experiments on Gaussian transport and CIFAR-10 support the claims in general.\nThe work reconsiders very common assumptions in CFNs providing counterexamples.", "was_revised": true, "final_justification": "Authors add additional experiments and theoretical justification sum up to a more solid work with a more extensive set of experiments, so I’m willing to increase my final rating to 5: Accept.", "rebuttal": "Thank you for the constructive and thoughtful review. We appreciate your interest in ReFlow’s behavior under different settings. Your feedback helped us clarify the broader implications of our theoretical findings and inspired new experimental directions.\n\n\n## Addressing Weaknesses & Limitations\n> \"The work needs extended experiments with varying dataset sizes, ...”\n> “How ReFlows would behave on datasets given ...”\n> “Some more (counter)examples on CFN suboptimality while achieving a stationary point on loss”\n\nWe now include new experiments on CelebA OT pairs from [1] that directly validate Proposition 2 in both adversarial and ReFlow-style settings, varying dataset size and noise scale. These settings allow us to probe the relationship between vector field quality and interpolation strategy, even at stationary points of the loss curve.\n\n> “The work lacks some thorough analysis, especially towards SBMs and optimal transport (OT).”\n\nTo address this, we evaluate CFM and SBM-like interpolants across four types of pairing:\n1. Random pairings  \n2. Ground-truth OT pairings  \n3. Shuffled OT pairings: $(x_0, \\mathcal{S} (\\text{OT}(x_0)))$ , where $\\mathcal S$ is a shuffling function \n4. CFM-integrated pairings\n\nWe assess how closely learned couplings align with the OT map and the pairings used during training. These experiments isolate whether models genuinely recover OT or merely memorize their training signal. We look forward to your thoughts on these results.\n\n> Work indicates concrete limitations but hardly discusses how to fix it.\n\nWe believe highlighting ReFlow’s limitations is a necessary first step toward addressing its failure modes. Existing literature largely assumes idealized conditions (e.g., infinite data, perfect discretization), while our work shows these assumptions break down in practice. Our added noise experiments—simple yet theoretically grounded—offer a promising direction for improving generalization and vector field quality. We hope this opens the door for further study.\n\n\n## Questions\n\n>  Can regularization (e.g., KL, LR scheduling) mitigate suboptimality?\n\nWe use dropout and gradient clipping, but not weight decay or KL-based regularization. While such techniques can help smooth solutions, they cannot eliminate the root issue: the learning objective permits zero-loss, zero-gradient-variance solutions that perfectly memorize mismatched pairings (see Proposition 2).\nCrucially, our findings arise even without strong regularization. The contrast between CFM (noiseless) and CFM (noisy) models—trained identically—highlights how interpolant choice fundamentally affects learning, independent of regularization strategy.\n\n---\n\n\n## New Experiments on CelebA\n\nTo empirically validate Proposition 2, we evaluate Conditional Flow Matching (CFM) on CelebA using ground-truth OT pairings from Korotin et al. [1]. Our goal is to test the central claim: when trained with deterministic interpolants, CFM is optimized to memorize its input pairings.\n\nTo test this, we compare:\n\n- **CFM (σ = 0):** trained with deterministic interpolants  \n  $x_t = (1 - t) x_0 + t x_1$\n\n- **CFM (σ = 0.05):** trained with slightly noisy interpolants  \n  $x_t = (1 - t) x_0 + t x_1 + \\sqrt{t(1 - t)} \\, \\sigma \\epsilon$\n\nThe small amount of noise ($\\sigma = 0.05$) is not introduced for performance reasons but to break the injectivity between $(x_t, t)$ and $(x_0, x_1)$—disabling the memorization pathway outlined in Proposition 2. Both models use identical U-Net architectures.\n\n---\n\n### Adversarial Pairings: CFM Optimizes for Memorization with Deterministic Interpolants\n\n**Setup:** To directly test whether CFM with deterministic interpolants ($\\sigma = 0$) is optimized to memorize suboptimal pairings, we construct an adversarial dataset:\n\n- We start from ground-truth OT pairs $(x_0, x_1)$ derived from the benchmark of Korotin et al. [2].\n- We shuffle the targets $(x_1)$, breaking the optimal structure and producing random pairings $(x_0, S(x_1))$ with no coherent transport semantics.\n\nWe compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, S(x_1))$.\n\n| Dataset     | Analysis       | Metric         | CFM (σ = 0.05) | CFM (σ = 0.0) | Advantage                         |\n| ----------- | -------------- | -------------- | -------------- | ------------- | --------------------------------- |\n| 5K Samples  | Generalization | L2 to True OT  | 34.25 ± 7.54   | 50.40 ± 16.73 | CFM (σ = 0.05) generalizes better |\n| 5K Samples  | Memorization   | L2 to Shuffled | 55.02 ± 16.51  | 28.57 ± 5.49  | CFM (σ = 0.0) memorizes more      |\n| 50K Samples | Generalization | L2 to True OT  | 30.05 ± 6.77   | 46.78 ± 14.87 | CFM (σ = 0.05) generalizes better |\n| 50K Samples | Memorization   | L2 to Shuffled | 56.48 ± 18.55  | 45.98 ± 11.85 | CFM (σ = 0.0) memorizes more      |\n\n\n**Evaluation Protocol (Lower is better):**  \nWe report L2 reconstruction error to  the true OT targets to assess generalization, and  the shuffled training targets to assess memorization.  \n\nSince the pairings are deliberately incorrect, a low L2 error to shuffled targets indicates overfitting to suboptimal data. High L2 to these targets signals resistance to memorization.\n\n**Interpretation:**  \nThe results directly support Proposition 2: CFM with deterministic interpolants ($\\sigma = 0$) minimizes loss by tightly memorizing its training targets, even when they are arbitrarily shuffled and non-optimal. In contrast, CFM with small noise ($\\sigma = 0.05$) resists this memorization trap and generalizes significantly better to the true OT targets.\n\n**Extra comments:**  \n- Memorization decreases with larger datasets, as expected, since model capacity is fixed, it becomes harder to memorize all pairings in the 50K setting.\n- However, this does not contradict Proposition 2: the optimization still favors memorization when possible. Larger datasets simply make perfect memorization less feasible, not less likely as an objective.\n\n---\n\n### Simulated 1-ReFlow: CFM Memorizes Its Own Outputs\n\n**Setup:** To evaluate whether iterative flow models suffer from self-reinforcing memorization, we simulate a 1-step ReFlow scenario—a core case of Proposition 2.\n\n- **Initial training:** We train a base CFM model ($\\sigma = 0$) on random, non-OT pairings.\n- **Integration:** Using this model, we generate new endpoints $(x_0, x^1)$ via ODE integration.\n\nWe compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, x^1)$.\n\n\n| Dataset     | Analysis       | Metric        | CFM (σ = 0.05) | CFM (σ = 0.0) | Advantage                         |\n| ----------- | -------------- | ------------- | -------------- | ------------- | --------------------------------- |\n| 5K Samples  | Generalization | L2 to True OT | 31.35 ± 7.38   | 43.35 ± 14.21 | CFM (σ = 0.05) generalizes better |\n| 5K Samples  | Memorization   | L2 to $x^1$     | 25.08 ± 8.59   | 8.63 ± 1.76   | CFM (σ = 0.0) memorizes more      |\n| 50K Samples | Generalization | L2 to True OT | 32.54 ± 8.86   | 38.16 ± 11.37 | CFM (σ = 0.05) generalizes better |\n| 50K Samples | Memorization   | L2 to $x^1$     | 16.29 ± 4.82   | 12.76 ± 4.01  | CFM (σ = 0.0) memorizes more      |\n\n\n**Evaluation Protocol (Lower is better):**  \nWe report L2 reconstruction error to the true OT targets to assess generalization, and the generated training targets $(x_0, x^1)$ to assess memorization. Since the pairings come from a previous CFM model trained on random (non-OT) data, a low L2 to $x^1$ indicates that the model is overfitting to flawed past outputs. High error rates to these targets signal that the model is resisting memorization and potentially correcting earlier mistakes.\n\n**Interpretation:**  \nThe results in the table provide further support for Proposition 2: CFM with deterministic interpolants ($\\sigma = 0$) is highly effective at memorizing the training couplings, even when those pairings are suboptimal and self-generated. In contrast, CFM ($\\sigma = 0.05$), despite only a small amount of noise, avoids this memorization trap and learns to generalize toward the true OT map, despite never being directly trained on it.\n\n---\n\n### References \n\n[1] Korotin, Alexander, et al. *Do neural optimal transport solvers work? A continuous Wasserstein-2 benchmark.* NeurIPS 34 (2021): 14593–14605."}, {"review_id": "BQlIvVU4sK", "reviewer": "Reviewer_EgrR", "summary": "This paper investigates the causes of failure modes behind Rectified Flows and Schrödinger Bridge. This paper introduces a gradient-variance diagnostic to measure the transport errors of the learned vector fields. Utilizing the Gaussian transport, the paper shows that the neural networks fail to obtain strict convergence. Moreover, the paper proves that Reflow with noiseless interpolants stagnate when arriving at straight couplings, and memorize when trained on deterministic couplings. The paper also provides a counterexample to show that 1-Reflow could fail. To address to these drawbacks, the paper claims that noise can alleviate these two effects. The paper experiments on synthetic dataset and CIFAR-10 to support this claim.", "questions": "1. The paper claims in the Gaussian setting that MLP cannot approximate without error. Is this really a critical issue? In experiments, other types of errors like discretization and sample errors also affects the final results, the approximation error could not be the main influencing factor.\n2. What is the main point of Section 3.2? If only an observation of the sample variance of the SBM is provided, it seems that this part is not strong enough to support the main findings and could be removed.\n3. The second last row of the caption of Figure 3 states that ”confidence intervals narrow with higher $\\sigma$”. This seems to be not apparent in Figure 3 since the shaded regions seems to be the same width across different $\\sigma$s?", "limitations": "The paper addressed most of the limitations. As a supplement, it remains to be examined how much drawback reflected by the Gaussian transport will appear in general flow matching methods, which is widely used in various fields.", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The perspective to study gradient variance to provide understandings on the limitations of neural networks and Rectified Flows is novel.\n2. The paper provides the counter example to show that 1-Reflow may fail in specific cases, which contributes to the study of Rectified Flows.\n\nWeaknesses:\n1. The clarity of the paper writing can be improved. The title focuses on the “gradient variance”. However, in section 4, most of the claims do not closely relate to the “gradient variance”, making the focus of the paper being obscured. In line 177-183, the key takeaways include multiple serial code (A)s and (B)s. In Proposition 1, the text and the formulas in line 157-159 are mixed together with multiple “and”s.\n2. It remains to be examined how much drawback reflected by the synthetic Gaussian transport counter examples will truly appear in general flow matching methods, which is widely used in various fields with acceptable results.\n3. The paper only presents potential understandings on the stagnation and memorization issues, but it does not provide new methods or ideas to tackle the issues while preserving the benefits of noiseless methods like Rectified Flows. Preferring stochastic flow matching is not a permanent solution, since deterministic and straight paths enjoy other advantages like faster simulation and smaller discretization errors. Moreover, the novelty of utilizing noise in generation is not that significant.\n4. It would be beneficial to experiment on more datasets to show the of the broad applicability of the proposed gradient variance study.", "was_revised": true, "final_justification": "I prefer to maintain my evaluation score. Although the authors conducted additional experiments on CelebA in the rebuttal, more discussions on the choice of $\\sigma$ could be done to increase the practicality of the proposed method. Furthermore, the current manuscript needs revision on the focus of the central contribution and the clarity. Hence, in my view, the current manuscript is not yet ready for publication.", "rebuttal": "We thank the reviewer for their thoughtful feedback. It helped us clarify the connection between Section 4 and earlier theory, refine the gradient variance discussion in Section 3.2, and reframe the Gaussian example. We also added new CelebA experiments that support our theoretical findings, especially Proposition 2.\n\n## Addressing weaknesses\n\n> “The clarity of the paper writing can be improved.”\n\nWe thank the reviewer for this valuable feedback. We will revise the manuscript to improve clarity and better emphasize the central role of gradient variance throughout. Specifically, we will improve the exposition of Proposition 1 and revise lines 177–183 to avoid overloading the narrative with serial (A)/(B) references.\n\nWe will strengthen the paper’s narrative thread and relation to gradient variance by clarifying the following schema, both textually and through an improved Figure 1:\n\n(1) Lemma 1 (Gaussian closed-form) → Prop. 1 (grad. variance bounds)  \n(2) Gaussian experiments  \n(1)+(2) ⇒ Conclusions (low grad.variance  ≠ OT; reflow skips intersection) → (generalizes) → Prop. 2 (finite data).\n\n> “...but no new method.”  \n> “Straight paths give faster simulation and lower discretization error.”\n\nWe agree that straight-line deterministic interpolants are attractive due to their simulation efficiency. While we do not propose a new method, our contribution lies in showing that this efficiency can obscure critical failure modes: in noiseless settings, models can minimize both loss and gradient variance while still learning non-optimal ($W_2$) couplings.\n\nThis challenges key assumptions in Rectified Flows:\n1. Gradient variance reflects path intersections [3,4]  \n2. ReFlow corrects such intersections via iteration [5]  \n3. Two iterations suffice for convergence [1,6]\n\nIn finite-data regimes, these assumptions break down:\n- No convergence to straight paths after two steps (Counterexample 1)  \n- Skipping intersections and vanishing variance where paths cross (Fig. 6)  \n- Loss and variance minimized via memorization of poor couplings (Prop. 2 + CelebA experiment)\n\nWhile straight paths are efficient, our results show they may never be reached in practice (Fig. 6), highlighting the need for more robust transport strategies.\n\n\n> “The novelty of utilizing noise is not that significant.”\n\nWe agree, and we do not claim that noise addition is novel. Instead, it plays a key role in our analysis by disrupting the conditions under which memorization arises (see Prop. 2, and CelebA experiments). The contrast between deterministic and noisy settings helps reveal structural limitations in the loss landscape of ReFlow.\n\n> “beneficial to experiment on more datasets”,  “to be examined how much drawback reflected by the synthetic Gaussian transport counter examples will truly appear in general flow matching methods”\n\nTo address concerns about generalization beyond synthetic settings, we provide new experiments on the CelebA benchmark. These results directly validate Proposition 2 on high-resolution, real-world data and confirm that the observed memorization dynamics persist in complex visual domains, not just toy examples.\n\n## Questions\n\n>  MLP cannot approximate without error. Is this really a critical issue?\n\nThe main purpose of Lemma 1 is to give the closed-form OT vector field and its rotated version, which Proposition 1 builds upon. The extra fact about exact representability is not needed for the proof, but shows that even Gaussians yield functions (e.g., 1/x) hard to approximate with standard networks.\n\n> Section 3.2 role?\n\nWe agree that Section 3.2 is less central to the main narrative. We will move it to the appendix and revise the surrounding text to maintain focus on the paper’s core contribution: understanding failure modes in vector field learning through gradient variance.\n\n>  Figure 3 caption\n\nThank you for catching the error in Figure 3’s caption. We meant to convey that as $\\sigma$ increases, the optimal solution shows lower variance than the vector field aligned with the pairing.\n\n## New Experiments on CelebA\n\nTo empirically validate Proposition 2, we evaluate Conditional Flow Matching (CFM) on CelebA using ground-truth OT pairings from [2]. Our goal is to test the central claim: when trained with deterministic interpolants, CFM is optimized to memorize its input pairings.\n\nTo test this, we compare:\n\n- **CFM (σ = 0):** trained with deterministic interpolants  \n  $x_t = (1 - t) x_0 + t x_1$\n\n- **CFM (σ = 0.05):** trained with slightly noisy interpolants  \n  $x_t = (1 - t) x_0 + t x_1 + \\sqrt{t(1 - t)} \\, \\sigma \\epsilon$\n\nThe small noise ($\\sigma = 0.05$) is not for performance, but to break injectivity between $(x_t, t)$ and $(x_0, x_1)$—disabling memorization (Prop. 2).\n\n\n### Adversarial Pairings: CFM Optimizes for Memorization with Deterministic Interpolants\n\n**Setup:** To directly test whether CFM with deterministic interpolants ($\\sigma = 0$) is optimized to memorize suboptimal pairings, we construct an adversarial dataset:\n\n- We start from ground-truth OT pairs $(x_0, x_1)$ derived from the benchmark of Korotin et al. [2].\n- We shuffle the targets $(x_1)$, breaking the optimal structure and producing shuffled deterministic pairings $(x_0, S(x_1))$.\n\nWe compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, S(x_1))$.\n\n| Dataset     | Analysis       | Metric         | CFM (σ = 0.05) | CFM (σ = 0.0) | Advantage                         |\n| ----------- | -------------- | -------------- | -------------- | ------------- | --------------------------------- |\n| 5K Samples  | Generalization | L2 to True OT  | 34.25 ± 7.54   | 50.40 ± 16.73 | CFM (σ = 0.05) generalizes better |\n| 5K Samples  | Memorization   | L2 to Shuffled | 55.02 ± 16.51  | 28.57 ± 5.49  | CFM (σ = 0.0) memorizes more      |\n| 50K Samples | Generalization | L2 to True OT  | 30.05 ± 6.77   | 46.78 ± 14.87 | CFM (σ = 0.05) generalizes better |\n| 50K Samples | Memorization   | L2 to Shuffled | 56.48 ± 18.55  | 45.98 ± 11.85 | CFM (σ = 0.0) memorizes more      |\n\n\n**Evaluation Protocol (Lower is better):**  \nWe report L2 reconstruction error to  the true OT targets to assess generalization, and  the shuffled training targets to assess memorization.  \n\nSince the pairings are deliberately incorrect, a low L2 error to shuffled targets indicates overfitting to suboptimal data. High L2 to these targets signals resistance to memorization.\n\n**Interpretation:**  \nThese results confirm Proposition 2: CFM with deterministic interpolants ($\\sigma = 0$) minimizes loss by memorizing training targets—even when the pairings are arbitrarily shuffled. In contrast, CFM with noise ($\\sigma = 0.05$) avoids this and generalizes better to the true OT map.\n\n**Extra comment:**  \nMemorization decreases with larger datasets because the model size is fixed, making it harder to memorize all pairings. This does not contradict Proposition 2: the objective still favors memorization when possible—larger datasets limit its feasibility, not its tendency.\n\n---\n\n### Simulated 1-ReFlow: CFM Memorizes Its Own Outputs\n\n**Setup:** To evaluate whether iterative flow models suffer from self-reinforcing memorization, we simulate a 1-step ReFlow scenario—a core case of Proposition 2.\n\n- **Initial training:** We train a base CFM model ($\\sigma = 0$) on random, non-OT pairings.\n- **Integration:** Using this model, we generate new endpoints $(x_0, x^1)$ via ODE integration.\n\nWe compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, x^1)$.\n\n\n| Dataset     | Analysis       | Metric        | CFM (σ = 0.05) | CFM (σ = 0.0) | Advantage                         |\n| ----------- | -------------- | ------------- | -------------- | ------------- | --------------------------------- |\n| 5K Samples  | Generalization | L2 to True OT | 31.35 ± 7.38   | 43.35 ± 14.21 | CFM (σ = 0.05) generalizes better |\n| 5K Samples  | Memorization   | L2 to $x^1$     | 25.08 ± 8.59   | 8.63 ± 1.76   | CFM (σ = 0.0) memorizes more      |\n| 50K Samples | Generalization | L2 to True OT | 32.54 ± 8.86   | 38.16 ± 11.37 | CFM (σ = 0.05) generalizes better |\n| 50K Samples | Memorization   | L2 to $x^1$     | 16.29 ± 4.82   | 12.76 ± 4.01  | CFM (σ = 0.0) memorizes more      |\n\n\n**Evaluation Protocol (Lower is better):**  \nWe report L2 reconstruction error to the true OT targets to assess generalization, and the generated training targets $(x_0, x^1)$ to assess memorization. Since the pairings come from a previous CFM model trained on random (non-OT) data, a low L2 to $x^1$ indicates that the model is overfitting to flawed past outputs. High error rates to these targets signal that the model is resisting memorization and potentially correcting earlier mistakes.\n\n**Interpretation:**  \nThe results in the table provide further support for Proposition 2: CFM with deterministic interpolants ($\\sigma = 0$) is highly effective at memorizing the training couplings, even when those pairings are suboptimal and self-generated. In contrast, CFM ($\\sigma = 0.05$), despite only a small amount of noise, avoids this memorization trap and learns to generalize toward the true OT map, despite never being directly trained on it.\n\n---\n\n### References\n\n[1] Saptarshi Roy, Vansh Bansal, Purnamrita Sarkar, and Alessandro Rinaldo. *2-rectifications are enough for straight flows: A theoretical insight into Wasserstein convergence.*\n\n[2] Korotin, Alexander, et al. *Do neural optimal transport solvers work? A continuous Wasserstein-2 benchmark.* NeurIPS 34 (2021): 14593–14605.\n\n[3] Fjelde, T., Mathieu, E., and Dutordoir, V. *An Introduction to Flow Matching.* MLG Blog, January 2024.\n\n[4] Gagneux, A., Martin, S., Emonet, R., Bertrand, Q., and Massias, M. *A Visual Dive into Conditional Flow Matching.* ICLR Blogposts, April 2025.\n\n[5] Liu, Xingchao, Chengyue Gong, and Qiang Liu. *Flow straight and fast: Learning to generate and transfer data with rectified flow.* \n\n[6] Lee, Sangyun, Zinan Lin, and Giulia Fanti. *Improving the training of rectified flows.* NeurIPS 37 (2024): 63082–63109."}]

Author rebuttal:

Meta-review: The paper analyzes failures of state-of-the-art fixed-point generative models—Rectified Flows and Schrödinger Bridge Matching—by introducing a gradient-variance diagnostic that quantifies transport error after integration. They also show that even low-variance (well-aligned) interpolants can still produce large transport errors. The authors establish architectural limitations, proving that standard MLP/CNN architectures cannot exactly realize Gaussian-to-Gaussian transport. It further shows that with noiseless interpolants, the fixed-point procedure either stagnates on straight pairings or memorizes deterministic non-straight pairings rather than approaching optimal transport, and proves the existence of such vector fields. The reviewers praise the strong theoretical results, clear counterexamples, and novelty of the introduced diagnostic. The authors should address concerns about the limited empirical scope by including results for additional datasets. The authors should also include the results using the FID-based benchmark in the revised manuscript, update it with the clarifications provided in the rebuttal, and incorporate the reviewers' comments to improve clarity.
Decision: Accept (spotlight)