Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domains
Authors: Yunrui Guan
Krishna Balasubramanian
Shiqian Ma
Venue: ICLR 2026 Poster (2026)
Keywords: Flow Matching
Constrained Generative Modeling
Wasserstein Convergence Rates
Primary area: learning theory

Abstract: We study generative modeling on convex domains using flow matching and mirror maps, and identify two fundamental challenges. First, standard log-barrier mirror maps induce heavy-tailed dual distributions, leading to ill-posed dynamics. Second, coupling with Gaussian priors performs poorly when matching heavy-tailed targets. To address these issues, we propose Mirror Flow Matching based on a \emph{regularized mirror map} that controls dual tail behavior and guarantees finite moments, together with coupling to a Student-$t$ prior that aligns with heavy-tailed targets and stabilizes training. We provide theoretical guarantees, including spatial Lipschitzness and temporal regularity of the velocity field, Wasserstein convergence rates for flow matching with Student-$t$ priors and primal-space guarantees for constrained generation, under $\varepsilon$-accurate learned velocity fields. Empirically, our method outperforms baselines in synthetic convex-domain simulations and achieves competitive sample quality on real-world constrained generative tasks.

Published as a conference paper at ICLR 2026
MIRROR FLOW MATCHING WITH HEAVY-TAILED PRI- ORS FOR GENERATIVE MODELING ON CONVEX Do- MAINS
Yunrui Guan
Krishnakumar Balasubramanian Department of Statisties
Shiqian Ma
Department of CMOR Rice University Houston, TX 77005 yge3erice.ed edu
Department of CMOR Rice University Houston, TX 77005
University of California, Davis Davis, CA 95616
kba la @ala@ucceo suudaviis edu
sqma @ rice....oo ce edu
ABSTRACT
We study generative modeling on convex domains using flow matching and mirror maps, and identify two fundamental challenges. First, standard log-barrier mirror maps induce heavy-tailed dual distributions, leading to ill-posed dynamics. Second, coupling with Gaussian priors performs poorly when matching heavy-tailed targets. To address these issues, we propose Mirror Flow Matching based on regularized mirror map that controls dual tail behavior and guarantees finite moments, together with coupling to Student-t prior that aligns with heavy-tailed targets and stabilizes training. We provide theoretical guarantees, including spatial Lipschitzness and temporal regularity of the velocity field, Wasserstein convergence rates for flow matching with Student-l. priors and primal-space guarantees for constrained genera- tion, under e-accurate learned velocity fields. Empirically, our method outperforms baselines in synthetic convex-domes simulations and achieves competitive sample quality on real-world constrained generative tasks.
INTRODUCTION
Flow matching (Lipman et al. 2023 Liu et al. 2023c Albergo et al. 2023; Albergo & Vanden-
Eijnden 2023 Tong et al. 2024| Chen & Lipman (024) has emerged powerful framework
for generative modeling, unifying scsre-based diffusion and optimal transport approaches under a single perspective. The central idea in flow matching is to construct continuous-timg deterministic flow that transports a simple prior distribution (e.g., Gaussian) to a complex target distribution, by learning its velocity field. Formally, given random variables Xo To and X₃ Tı, both supported on Rdd, we seek dime-dependent vector field v Rd x [0, 1] + Id such that the solution of the ODE dX,, v(x,, f) dt, with Xo no, satisfies X1 πi A simple construction based on straight-line interpolation X, (1 - ()Xu + LX2, which yields the conditional velocity field o" v"(x, () E[X, Xo X; xr. This vector field e" minimizes the regression loss min, []],((X, t) making the optimal velocity field for the interpolation path. Since computing exactly intractable, modern flow-mathhing methods approximate with a neural network and simulate the ODE numerically. This pathwise formulation leads to scalable training objectives, principled continuous-time generative processes, and improved sample quality.
_ x, 12]
Constrained Flow Matching. In many applications, the target is supported on constrained do- mains like polytope, simplex, positive semidefinite matrices, rather than the full Fuclidean space. Examples include molecular generation, where atoms and bonds must satisfy physical stability constraints (Fishman et al. 2023b), preference alignment (Kim et al.] (024)). policy optimization and physical constraints for robotics (Zhang et al. 2025 [Utaarsh et al.) (025)) and watermarked
content generation (Liu et al.] 2023a Standard now-baseed methods fail in this setting: projecting unconstrained samples back onto the domain distorts the distribution.
Related works. Several strategies address the challenge in constrained flow matching, including reflection beleettion-based methods (Lou & Ermon 2023 Fishman et al.) 2023a Xie et al. 2024] Chhiittoper al. 2024) that keep trajectories inside the domain using boundary normals; mirror-map diffusion

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models (Liu et al. 2023a Feng al. 2025) that transform constrained problems into unconstrained ones using mirror-mapy pauge--map approaches (Li al. 2025) that enforce fcasibility via reflections; and -estatce-penalty methods (Huan et al. 2025; Khalafi et al., 2024) that penalize distance to the constraint set, at notable computational cost. Despite this progress, no framework yet ensures constraint satisfaction while providing convergence rates for flow matching.
In this work, we focus on the development of mirror flow matching, where the velocity field is adapted to the geometry of the constraint set. Formally, let K. {δ)(x) < 0, ó Rd R, i = 1,... m where Φ; are smooth convex functions, be a closed convex set, and suppose the target distribution Th is supported on K. Our approach is based on constructing a mirror map V : K R where V/ K + IR is a strictly convex, differentiable potential. The mirror map transports points from the constrained primal space K. to an unconstrained dual space. In this dual space, one can perform standard (unconstrained) flow matching, i.e., define Z. = VV((X,, and evolve it via dZ, = D (Z, 1) dt with Zo = (Xo)). where nD is velocity field learned by minimizing the unconstrained flow matching objective. The primal trajectory is then recovered by mapping back using the inverse mirror map X; = ( This mirror-descent-basee formulation ensures that the entire trajectory (et)) remains in K. while leveraging the flexibility of unconstrained flow matching in the dual space. Thus, mirror flow matching combines geometry-awan sampling with scalable learning, broadening the applicability of flow models to structured domains that naturally arise in the aforementioned application areas.
1.1 CHALLENGES AND SOLUTIONS
Methodological Challenges.
Extending flow matching constrained domains via mirror maps introduces key challenges. First, the transformed target distribution in the dual space may have heavy tails, causing standard mirror maps (e.g., log-barrier) to violate moment conditions required for well-posed flow ODEs (Figure red dots). We address this with a regularized mirror map that controls heavy tails and ensures finite p-th moments for all p 2 1 (Figure [ blue dots), stabilizing training. Second, Gaussian priors often mismatch the heavy-tailed dual distributions; we instead adopt a Student-t prior, improving alignment, sample quality, and stability. Together, these modifications overcome limitations of standard log- lor-berrier and Gaussian priors, yielding high-fidelity constrained generative modeling. A visual illustration is provided in Appendix Section A
Theoretical Challenges. In addition to the methodological issues above, theoretical analysis of mirror flow matching poses challenges. Rigorous error bounds for the sampling stage require the velocity field vzz, t) to be Lipschitz in z (Benton et al. 2024 Bansal et al. 2024 Zhou & Liu 2025 Gao et al. 2024). ). while ODE discretization error further requires Lipschitz continuity in both x and t (Bansal al. 2024 Zhou & Liu 2025). However, the dual velocity field 80 t) is generally not
Lipschitz over E 0. Partial progress includes spatial Lipschitzness on f E [0. T C [0, under
bounded , (Benton et al. 2024 Zhou & Liu 2025) or Gaussian-like like ạ (Gao et al. 2024 In general,
unbounded IT can induce polynomial growth in V 2 L)]l as ||=||| grows and singularities near I 1, motivating early stopping. Recent work Fendero-Encinar al. 2025) leverages Log-Sobolev inequalities to establish spatial Lipschitzness, though t-Lipschitzness is not addressed. We overcome this challenge by using t-distribution as priors. While such priors have been explored empirically (for example, (Pandey et al. 2025) Appendix B)), our motivation comes from addressing the above theoretical challenge.
Contributions. In this work, we introduce flow matching with Student-t prior (see Section B) and provide new theoretical guarantees establishing both spatial Lipschitzness and temporal regularity (see Proposition D. This result enables us to obtain explicit error bounds under substantially more general target distributions (see Theorem 3 in the dual Euclidean space under the assumption that the learned velocity fields approximates the true dynamics up to e-accuracy. Finally in Theorem 4 we further prove primal-spent guarantees for constrained dynamics.
INGREDIENTS FOR DESIGNING MIRROR FLOW MATCHING
2.1 INGREDIENT 1: THE MIRROR MAP
Before introducing our proposed mirror map, we first explain why the classical log- log-barrier is not suitable in our setting. The main issue arises from our first identified challenge: ensuring the existence
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of moments. As the following general result shows, if the log-barrier transformation induces heavy tails, then even low-order moments (e.g., the second moment) may fail to exist.
Lemma 2.1. Let be a random variable in Re with law P. Then, (i) if P(
R) > Ci for
some constant C > 0, then ||[[Y||?] does not exist, and (ūi) if P( 2 R) S C/R with B > then [||||||]] is finite.
In addition to controlling tails, we would also like the geometry induced by the mirror map to have a desirable metric property: the metric in the dual space should be stronger than that in the primal space. Formally, we require
Poosed
|| ull S Lo (+) (V)ll Y
(1)
for some constant Lv > 0. To see why this is important, we first recall some definitions of p- Wasserstein distance in primal space and dual space. Let V. μ be two probability measures on K. Then we have:
W)(,, inf Bylll =l!"),
7EI(U,))
Figure 1: Dual space distribu- tion comparison between the log barrier and our mirror map (k 0.5). The primal distri- bution is truncated Gaussian mixture within a polytope (see Appendix AA). The log barrier yields heavy-tailed distribu- tion, while our mirror map pro- duces a much lighter tail.
W Wn,(( inf (IV(() (.(I)
TET(Ms)
where 7 E pL) means y coupling of v. The first one is just the Wasserstein distance for K. under Euclidean distance, and the second one is actually the Wasserstein distance in the dual space. To see this, let v', v', denote the distribution of v, μ in dual space, i.e. v' = (V 4// and μí = (V Then we have W2.0 (p, v) W2 v"). We remark that W2 was used to ana- lyze the convergence of mirror Langevin algorithm (e.g., see Li et al. (2022).
In general, an upper bound for W2, (v, 1i) doesn't 't directly imply an error bound for (2(u, p)) in the primal space. But under inequality ① Wasserstein distances in the primal space can be controlled by those in the dual space:
W2(u, W2(,; =
inf Eylll= - v (EI(()H)
inf E, 7EI(USH)
V (x) VV W2, W 1)?.
Inequality [1 is equivalent to V4 being Log Lipsschit.. Since A and A are inverses of each other, this condition is in turn equivalent to Vi being strongly convex. However, classical mirror maps generally only strictly convex, not strongly convex. As result, Ly be arbitrarily large in certain domains; for instance, even for simple 2D polytopes with three facets (d = 2, m = 3), the constant L may blow up (see Example 5 the Appendix).
These observations suggest that we need to design new mirror map that balances tail behavior and convexity. In particular, the desired mirror map should satisfy the following goals:
1. Transform the constrained distribution into unconstrained distribution on Rd
2. Ensure that key moments (e.g., the second moment) of the transformed distribution exist.
3. Be strongly convex, so that convergence guarantees in the dual Euclidean metric can be transferred to guarantees in the primal Euclidean metric.
Motivated by the mirror-map framework of Vural et al. (2022), we propose in Proposition 2.2 modified log- barrierre that achieves these properties.
Proposition 2.2. Let K. = (x)(x) < 0, Vi E [m]]. where Φ; are smooth convex functions with bounded gradient. Let v(()) = - - Em (-(x(())) + =1al- Then we have Wz(U, n) S W2, (v, p). Denote Ks = {x E KC =o(()) 2 8). Let X be a random variable on KC whose law is denoted as P. Assume there exists positive constants Cx 3,50 s.t. for all 0 < 5 < öu it
holds that P(KC(K&) < Cx Then there exists some constant C' s.t. in the dual space R2 for all R 2 / 18% (here C" is some constant that depends on K.), (X)] 2 R) I C/RBle. By choosing K < B/p, we can guarantee B||l (X))PP exists.
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Specific examples (including Lg ball and polytopes) are discussed in Appendix Section B We verify that the boundary- measure condition P(K (ss) I Cx8 is natural in typical cases.
Example (Uniform distribution on the cube). Let = [-1, 1] and let P the uniform distribution E K. d(x, ak) 2 83. Then the boundary layer has -2-26) = 1 (1-8)4, 8) Using the first-order expansion (1 (1-8)² R nd
on K. Define the d-interior as K = fx
probability mass ((KA))))
1 1-dō, we obtain P(C \ KCs) R d5. Hence the condition P(A \ KCs) Cx holds with / 1 and Cx= = d. This shows the assumption mild and satisfied by standard convex bodies such as the cube
under uniform measure.
2.2 INGREDIENT 2: THE PRIOR DISTRIBUTION
For flow matching, let the target distribution be denoted by X1 ~ with density and let the initial distribution (prior) be Xo ~ To. The evolution between mo and アı is described by time-dependent vector field, where v(x, t) denotes the true vector field. Considering straight-line interpolation, by definition, the velocity field a point (x, is the conditional expectation of the instantaneous displacement along this interpolation: vxr = E[X,
Xo X; = z). To make this expression explicit (Karras al. 2022 Wan et al. 2025), ), note that the interpolation relation X, (1 X) + L can be inverted to obtain Xo I-1 Xx - -X)). Substituting this into the displacement X1 Xy yields X₁ Xu X + 1-t 1- Taking conditional expectation given X₁ 2,, we obtain the foosed-form expression for the true velocity field:
v(x,
Xi X,
E[Xi X2 = =+].
Thus, the vector field v(z, t) consists of two interpretable terms: a deterministic contraction term z|| that directs the flow
-I- that pulls z toward the origin, and prediction term E[, X, = toward the target distribution m.
A crucial modeling choice in flow matching is the prior distribution. The choice of the prior distribution affects this conditonal expectation ][X1 X, = æ] significantly. While Gaussian priors are the standard choice in unconstrained generative modeling, they are poorly suited when the target distribution exhibits heavy tails. The following example illustrates this pathology. Denote standard Student distribution (2) Cr.a(1 + |
Example 2. Consider the ame-iimeniional target density X, p(x) x (1 + 2 1 Suppose we use Gaussian prior Xo ~ ~N 1). Then the conditional distribution of X; given an interpolated point X, = x, is given by
p(X,|X, =)) x g(z1) = exp (- (tz) (-(tz) - x)²
2(1 1)2
This conditional distribution develops two modes: one near æ 0 and another near #1 ~ x/t. Although the + 0 limit will not cause singularity (Wan et al. 2025). we emphasize that for large values of ||=]|| the vector field would scales as cxp(x²) for some small values tf implying that the true velocity field u(z, t) can blow up upper-xxponentilll in x. Furthermore, discussed in Wan jet al.] (2025); Zhou & Liu (2025), singularities exist as / - 1. By contrast, if we replace the Gaussian prior with heavy-tailed Student-/ prior (e.g., with 1), the conditional density becomes
p(X,|X,- = x g(z1) (1
1))
2
for which the dominant mode remains near even as x being large, over E ( this case, the conditional expectation does not explode with x, and the resulting velocity field remains controlled. See Appendix Section
C for a visualization.
This example highlights& a key principle: when the target distribution is heavier-tailed than the prior, the conditional distribution is likely to have mode that is dominant near for some values of Then the induced velocity field can diverge at large ||æ]|, producing ill ill-posed dynamics and complicating error analysis. In particular, such blow-ups directly cause the Lipschitz constant of v(x, 1) to diverge
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Vaual and Thoeetory frimal Space
Tagestry Dusl
-2
(a) Primal space trajectory
(b) Dual space trajectory
Figure 2: Visualization of interpolations in primal and dual spaces Straight line interpolation in the dual space (Figure (b)) corresponds to curved "geodesic" interpolation in primal space Figure (a)).
as ||~||| + oG, necessitating additional assumptions the tail of data distribution (e.g., bounded support) (Benton et al., 2024 Bansal et al. 2024 Gao et al. 2024; Zhou & Liu 2025). Choosing a Student-/ prior prevents these blow-ups by making the data distribution to dominate the tail behavior of the conditional distribution, suppressing the mode near z/t. In this way. the mode near zero will be dominant, ensuring controlled velocity fields, finite-moment guarantees of the interpolation conditional distribution, and stability in both theoretical analysis and practical training.
3 MIRROR FLOW MATCHING
Recall from Section 2 that we discussed choices of mirror maps for closed convex sets of the form K. = {x E Rd db (xr < O, Vi [ml]] In mirror flow matching, both the prior mo and the target Ii are required to be supported on I.. The objective is to learn a continuous-time flow X₁ defined by the ODE ai X, = v (X,,t)
vX(,, t) with Xo ~ To( that transports Ty to T over the interval t E [0, 1].
Mirror flow matching achieves this transport by interpolating in a transformed (mirror) space. Given a mirror map Vv, we map x E IC into the dual space via 2 = V (z). As shown in Li et al. (022)) the dual Euclidean space (R Ia) is isometric to the primal space equipped with the squared Hessian metric (K,, (V2 2)) We denote these metrics as gf = (D2 and g Ia. The procedure is then V (x)). (2.) Perform flow matching in the dual space using straight-line interpolation Z, = (1 - 2) + 1Z,. (3.) After generating samples 2 in the dual space, map them back to primal space using the inverse mirror map V (2). In particular,
follows: (1.) Map primal data to dual space:
interpolation in primal space is defined as X, V u (Zi), which can be interpreted as the geodesic interpolation between Xo and X, under the squared Hessian metric. See Figure for an illustrative trajectory visualization in both the primal and dual spaces.
Relation between dual and primal velocity fields. Consider a dual-space flow Z, defined by vector field v By direct differentiation, the corresponding primal velocity field is
P (X..t) = d x, V (z+)
dt 7.) = V2 (Z:) P (Ze,t).
(2)
dt
The flow matching objective in the dual space is min E, Z. [|le (lo, (Z,,t) d 1
2, (1-1-10 t)Zg + tZ
(3)
whose solution is known to be the conditional expectation v (2 El Z, 2, = Liu et al. 2023b). The following proposition establishes the equivalence between primal and dual formulation.
Proposition 3.1. Learning vector field in the dual Euclidean spuce (Rdd, Ia) is equivalent to learning a vector field in the primal spuce (K,, (V2)))) Specifically,
min 110 (X.,
d 3 x and min E
d [100 (Ze,t)
de 2
are equivalent, with the correspondence e t) (²)())) (x,)))(((x))) Moreover, the primal flow matching objective is solved by o (xx, E X, X, = x]]
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Algorithm I Mirror Flow matching with Student t distribution
1: Map data distribution from K. to I using VV, obtain samples for Z1.
2: Learn
vector field (2, with prior Tu(
td
mingo 11.zo~=f.2100t
(Z) Zo)ll²] where Ze tZ + (1 Zb
3: Choose step size h for Euler discretization s.t. à is integer. Choose T E (0, 1) as early stopping time, satisfying f. E.
4: Perform Euler discretization sample from of with constant step size h, up time T:
5: Generate 5 ~ D
6: for k = 0 to T - do
h
7: EA(K+1) Akk + hō hk)
8: end for
9: Denote the obtained sample by äт ä7.
10: Map samples Br back to K. using V to obtain Zy.
This result shows that training in the dual space with straight-line interpolation is equivalent to training the primal space with geodesic interpolation under the squared Hessian metric. From an algorithmic standpoint, this equivalence is highly convenient: we can train the dual-space vector field v which is simpler due to its Euclidean geometry, and recover the primal vector field v" by the transformation in equation 2 Thus, the difficult geometry of IC is automatically handled by the mirror map, while optimization is carried out an unconstrained Euclidean space.
The algorithmic procedure for mirror flow matching is summarized in Algorithm This pipeline leverages the simplicity of Euclidean training in dual space, while ensuring that the generated samples respect the original convex constraints in primal space. Here, h denotes the step size (in sampling stage) and < denotes the terminal time if early stopping is adopted.
4 THEORETICAL RESULTS
In this section, provide theoretical analysis of error bounds for flow matching. A key component of our analysis is the accuracy of the neural network used to approximate the target velocity field. We adopt the following assumption, which is standard in the literature on flow-based generative modeling (see, e.g., [enton et al. (2024); Bansal et al. (2024); Li et al. (2025) as well as in the study of diffusion models (see, e.g., Chen al. (2023): Li et al. (2024)). Theoretical justification for this assumption can be found in Wang et al. (022)); Zhou & Liu (2025). where the authors establish
that such an e-level approximation error can be achieved by a neural network under suitable training conditions.
Assumption 1. (Neural Network Estimation Error) Let v(æ, t) denote the true velocity field and b(x,t) t) neural network approximation. We assume that the approximation error bounded in mean square, i.e., [[]lu(x,t) 6(z, t)
Intuitively, Assumption I states that the learned velocity field is close to the true velocity field in an average sense across both space and time. The parameter therefore quantifies the quality of the neural network approximation: smaller e implies more accurate approximation, which directly translates into higher fidelity of the generated samples.
4.1 GUARANTEES FOR EUCLIDEAN FLOW MATCHING WITH T-DISTRIBUTION PRIORS
In this subsection, provide eπror analysis for flow matching in Euclidean space when the prior distribution is chosen to be Student-t distribution (henceforth referred to as f-Flow). Our analysis applies to the general framework of flow matching with straight-line interpolation, and is not restricted to the mirror flow matching setup. To maintain notation consistency, we denote random variables as Z Rdd with density .Tp. We begin by introducing the assumptions required.
Assumption 2 (Finite Moments). Let Zo denote the prior (chosen as Student-t) random variable and Z, denote the target random variable, both supported on Rd., We assume that they have finite second moments, i.e. Zl|ZZ| [|2] oo, E. E [|2] co, which is necessary for well-defiedneess.

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Assumption 3 (Polynomial Tail Bound). Let (x) denote the probability density function of the data distribution supported on TR. It is assumed to satisfy: (1) For ||>||| 2 1, we have xf (P(x)) and (2) For ||a]|| < 1, we have "I (x) S Cu-
S T
C
The above assumption allows the target distribution to be heavy-tailed. covering wide range of realistic distributions. We next establish Lipschitz guarantees for the true vector field, showing that under Assumption 3 the velocity field induced by t-Flow is both spatially Lipschitz, and admits a controlled temporal derivative, which is crucial for bounding the discretization error in Theorem 3
Proposition 4.1. Let uD be the minimizer of the I-Flow objective (Equation 3). Under Assumption
with a 2 2d y + 2, there exist constants B1, B3 such that, for all t [0, T):
1. (Spatial Lipschitznss)) The vector field v o (2,t) t) is LL-isschitz in 2, with L1 == d+y
(I-T)= By.
2. (Temporal Regularity) The time derivative of the velocity field is bounded as
1) $ (1
B1 - T)² |>|| (1-
1
1
+ d 3/D
V 2(1 - 2(1-T)2 (B2 + +3B))
1
-
The proof is deferred to Appendix G.1 To the best of our knowledge, the only prior work that controlled the temporal Lipschitzness of the vector field in order to bound discretization error is Zhou & Liu 2025). However, their analysis required the data distribution have bounded support,
whereas our result only assumes polynomial tail bound. For spatial Lipschitzness, existing results
either imposed stronger conditions on the data distribution (Zhou & Liu 2025 Benton al.) 2024]
Gao et al. 2024) or studied different problem settings (Cordero-Ennciar et al. 2025). We can now quantify the discretization error of t-Flow.
Theorem 3 (Discretization Error of t-Flow). Consider t-Flow in Euclidean space. Let xp denote the data distribution supported on Rd, and fip be the law of generated sample 2T obtained by Fuler discretization with constant step size h, up to time T (see line of Algorithm [. Under Assumption 3 with 2 2d + v/ + 2, Assumption [ and Assumption 7 there exists a constant D3, depending polynomially on T-T, d, v, and on B1, B2, E|| Z1 1 E Zu 112), such that
6L1
W2
VDD +2 + (1
T)
2 (E 12]
+ 1IIl
L
The proof is provided in Appendix G.2] The error bound consists of two terms. The first term captures the discretization error (from Euler steps of size h) and the neural network approximation error (measured by e). Both vanish as h - 0 and e - 0.The second term corresponds to early stopping error, which decreases to zero as T - 1. Thus, by taking T close to 1 and ensuring accurate vector field approximation with sufficiently small step size, we can guarantee high-quality samples.
We now compare our result with recent related works. [ansall et al. (024) did not analyze the Lipschitz properties of the velocity field, but instead imposed them assumptions, /Zhou & Liu (2025) established both spatial and temporal Lipschitzness and further analyzed neural network approximation, but required the data distribution to have bounded support. We note that the exponen- tial dependence on the spatial Lipschitz constant L2 arises due to non-convexity, and also appears in existing analyses (Bansal af.) 2024) Zhou & Liu| 2025). It is plausible that this exponential dependency could be improved to polynomial dependence by following the probabilistic coupling strategy Chen et al. (2023), though the resulting algorithm not purely deterministic.
4.2 PRIMAL SPACE GUARANTEE FOR MIRROR FLOW MATCHING
We next obtain the following primal space guarantee. First note that that the primal space (K., gP) and the dual space (Rd,, _) are isometric. Hence, we have the following result.
Lemma 4.2. If the vector field DD Ln Lipschitz in the dual space (R& ,)), it L2 Lipschitz in the primal space (K, gf) (under the squared Hessian metric).
To relate Assumption 3 with the distribution in primal space, we impose the following condition.
Assumption 4. (Primal Space Probability Density Function). Denote T (a) as the probability density function for oh in the primal space, under Euclidean metric. Assume that T Eua (x) smooth and that there exists a small constant dn such that sup Pouc S Opar 8 Vo S d0-
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Theorem 4. Let कf be the law of output samples generated by Algorithm 1 (i.e., the law of Fr in Line 10 Under Assumption 1 anddd 4 with R S 24+v+2 and we further require z. there exists
constant L1, D3 and M 2 21 [12] B such that
W
+1 T)M.
1
The proof is provided in Appendix
G.3 and essentially follows by Proposition 2.2 and Theorem
EXPERIMENTS
We demonstrate the effectiveness of our approach by performing numerical simulation (see section 5.1) and real world data experiments AFHQv2 dataset (see section (.2). The numerical simulation is performed on a personal laptop using a CPU. The real world data experiments were performed on single A100 GPU.
5.1 NUMERICAL SIMULATION
We build on the experimental setup of Li et al. (2025) and conduct numerical simulations on two representative constrained generative modeling tasks. The first task is a 10-dimensional polytope problem, defined as {z E R a < bj = .. 30}, with constraints loaded from pre-specified data file from et al. (2025). The target distribution is a uniform mixture of Gaussians, where the means are partly sampled at random and partly human-designed to stress-test the model -3) E 2 and covariances are fixed to 0.4/ The second task is
(e.g., (-3, -3, 6-dimensional L2 ball problem, defined as {x E R : || 1|2 < 25}, with target distributions generated in the same manner as in the polytope case. For both tasks, we used a simple MLP network with 4 layers, and hidden layer size being 128. We used ELU activation function. We perform 10, 000 training iterations.
We implemented method with x 0.3 and used t-Flow prior with = 10. As shown in Table T and Table [ our approach consistently outperforms both Gauge Flow Matching (Li et al.) 2025) and Reflected Flow Matching (RFM) (Xie et al., 2024). Across both tasks, method achieves
lower KL divergence and smaller Maximum Mean Discrepancy (MMD) values, while simultaneously guaranteeing sample feasibility. For the L2 ball case, Gauge Flow Matching omitted since coincides with Reflected Flow Matching.
We also evaluated the performance of our approach under different choices of μ, k for a 10 dimen- sional polytope task. The results are presented in Figure 3 In addition, we visualized the dual space distribution justify that our mirror map doesn't induce heavy tail, whereas the log-barries does; See Appendix D Empirically, we observed that t-Flow outperforms G-Flow. Also, larger values of would induce a tail that is heavier than smaller values of k. The result indicates that a large v would require smaller which is consistent with our theoretical findings.
(a) MMD vs Kappa
Figure 3: We test the performance of t-Flow and G-Flow for different values of k. For t-Flow, we compare the performance among different choices of (degree of freedom). We RFM as baseline comparison. Networks are trained with 40, 000 iterations.
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Table Performance comparison with 10-dimensional polytope constraints. Results based on
average of 10 runs, each run with an average of 10, 000 samples. MMD values are scaled by 10
Method
MMD +.
KL. Divergence +
Feasibility
Mirror t-Flow
0.995 + 0.021
1.424 + 0.037
100%
Mirror G-Flow
1.006 + 0.016
1.447 = 0.046
100%
Gauge Vanilla (Li et al.
2025
+.001
5.023 0.073
95.257 + 0.150%
Gauge Reflect (Li et al. 2025
1.830 0.011
5.057 0.075
100%
RFM (Xie et 2024
1.217
2.034 0.052
100%
MDM [Luu Ju et al.
2023a)
1.258 + 0.013
1.708 0.054
100%
Table 2: Performance comparison on 6-dimensional L2 ball constraints. Results are based on an average of 10 runs, each run with an average of 10, 000 samples. MMD values are scaled by 10-2
Method
MMD +
KL Divergence +
Feasibility
Mirror t-Flow
5.329 + 0.101
0.162 0.011
100%
Mirror G-Flow
6.244 + 0.286
0.176 + 0.015
100%
RFM (Xie et al.
2024)
5.935 + 0.222
0.285 + 0.012
100%
MDM Liu et al.
2023a
36.156 + 0.102
8.017 = 0.046
100%
We also implemented MDM (Liu et al. 2023a for both tasks. We remark that the original MDM paper, the author only provided closed-form formula of their mirror map (and its inverse) under specific assumptions of the polytope, which can't be applied for an arbitrary polytope. For an implementation of log-barrier, the inverse mirror map is difficult implemented MDM with regularized log-barrier. For L2 ball case, we implemented MDM with the closed form mirror map provided in Liu al. (2023a). We observed that the neural network failed to learn useful information, which is likely due to the heavy tailed nature of the inducede dual space distribution.
solve, and therefore
These experiments highlight the advantages of our method. By jointly choosing mirror maps and priors based on careful analysis, our approach achieves superior performance on numerical benchmarks while preserving feasibility by construction. The ability to obtain tighter divergence metries under strict feasibility underscores its promise for high-dimensional constrained generative modeling, demonstrating robustness across geometries (polytope vs. Lg ball) and scalability to practical domains where constraints are central.
5.2 REAL-DATA APPLICATION: WATERMARKED IMAGE GENERATION
Following Liu et al. (023a), we evaluate our method on the task of 64 x 64 watermarked image generation using the AFHQv2 dataset. We begin by generating parameters (a,, (i). which serve as user-specific private keys. These parameters define polytope K. = {x < a, < 6.j. where an image can be vectorized and checked for feasibility: an image lying inside K verifiably generated by the model. During training, we first watermark the AFHQv2 images by projecting them (with added noise) onto the polytope, thereby producing a watermarked dataset. We then use these watermarked images as training data and compare the performance of Mirror Diffusion Models (MDM) (Liu et al. 2023ab with our proposed Mirror (-Flow approach.
A crucial component is the initialization used for the models. We first train both methods with random neural network initialization under limited training budget (24 hours). We set the mirror map parameter
0.1 for our method, with random initialization. We first report the CMMD metric (Jayasumana et al. 2024). CMMD combines CLIP embedding with Maximum Mean Discrepancy metric and is considered more reliable than FID for evaluating generative models. With 10, 000 gen- erated images, our approach achieves a CMMD score of 0.177, which competitive with the MDM baseline (Liu et al. 2023a), calculated be 0.152. Nevertheless, shown in Figure 4(a)

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Published as a conference paper at ICLR 2026
Table 3: Performance comparison watermarked image generation the AFHQ2 dataset. Both implementations are initialized at EDM (Karras et al. 2022) checkpoint, For MDM, we use the code from Liu et al. (2023a). For flow matching, we apply the training framework from Lee al. (2024).
Method
FID (50k).
CMMD
training time
Mirror Flow (x = 0.05)
4.27
0.023
hours
Mirrod Diffusion Model (Liu et al. 2023a)
7.29
0.170
13 hours
(a) With random initialization
(b) With EDM checkpoint initialization
Figure 4: Samples of generated watermarked images from the AFHQv2 64 x 64 dataset. Constraint satisfaction were checked with built-in functions of Liu et al. 2023a).
already produces visually high-quality samples within a limited training budget, demonstrating strong potential for further improvements with better initializations.
Towards that, in Table 3 we next report results when the models are initialized at EDM (Krrra cct al. 2022) checkpoint for AFHQv2 dataset; the corresponding sample images are displayed in in Figure 4bb) We note that in this case, method achieves superior CMMD and FID scores, requiring a smaller amount of training time. Finally, we remark that we initialize at the checkpoint for a flow matching model from Lee al. (2024), the FID (50k) achieve 3.14 after 1.5 hours of training. This value is similar to 3.05 reported in Liu et al.] scheduled number of iterations could result in an estimated training time up several hundred hours in our experimental setup.
(2023a), while fully executing their
CONCLUSION
We introduced I-Flow, a flow-matching framework with Student-/ priors, and established rigorous guarantees both spatial Lipschitzness and temporal regularity of the underlying velocity field. Our analysis yielded the first error bounds for flow matching under polynomial tail assumptions, thereby extending prior results beyond bounded-support assumptions. We further demonstrated that t-Flow provides robust sample quality in practice, particularly in scenarios where Gaussian priors fail to capture heavy-tailed structures. Beyond technical guarantees, our results emphasize that successful generative modeling complex domains requires careful co-design of mirror maps and priors, rather than defaulting to standard choices. This perspective opens up several promising avenues. One direction is exploring adaptive choices of degrees of freedom in the (-prior could yield even more flexibility, enabling flows that automatically adapt to local tail behavior of the data. Another is extending t-Flow to constrained domains with non-convex geometry, potentially leveraging landing techniques. On the theory front, improving the exponential dependence on Lipschitz constants, for example via probabilistic couplings randomized flow strategies is interesting. Finally integrating t-Flow with hybrid diffusion-floww architectures and energy-based models offers yet another exciting path, combining the complementary strengths of these paradigms.
ACKNOWLEDGMENNT
Research of Krishnakumar Balasubramanian supported in part by National Science Foundation (NSF) grant DMS DMS-2413426. Research of Shiqian Ma was supported in part by NSF grants CCF. 2311275 and
and ONR grant N00014-24-1-2700
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REPRODUCIBILITY STATEMENT
Proofs for the theoretical results are presented in the Appendix. Codes to reproduce the experimental results are provided in the supplementary material. LLM usage: LLM was used only polish the writing.
REFERENCES
Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions. arXiv preprint arXiv:2303.08797 2023.
Michael Samuel Albergo and Eric Vanden-Eijnden. Building normalizing flows with stochastic interpolants. In The Eleventh International Conference on Learning Representations, 2023. URL
https://openren
net www.oooodddeggeeeeaa..
Vansh Bansal, Saptarshi Roy, Purnamrita Sarkar, and Alessandro Rinaldo. On the wasserstein convergence and straightness of rectified flow. arXiv preprint 1.Xiv:2410.1194499 2024.
Joe Benton, George Deligiannidis, and Arnaud Doucet. Error bounds for flow matching meth- ods. Transactions on Machine Learning Research, 2024. ISSN 2835-8856. URL https
Ricky T. Q. Chen and Yaron Lipman. Flow matching on general geometries. In The Twelfth International Conference on Learning Representations, 2024. URL htttp::
net/ (tt.:/ooonnndddohhll11l1
Sitan Chen, Sinho Chewi, Holden Lee, Yuanzhi Li, Jianfeng Lu, and Adil Salim. The probability flow ode is provably fast. Advances in Neural Information Processing Systems, 36:68552-685 2023.
Jacob Christopher, Stephen Baek, and Nando Fioretto. Constrained synthesis with projected diffusion models. Advances in Neural Information Processing Systems, 37:89307-89333,, 2024.
Paula Cordero-Enciiar, O Deniz Akyildiz, and Andrew B Duncan. Non-asymptotic analysis of diffusion annealed langevin monte carlo for generative modelling, arXiv preprint arXiv:2502.09:066 2025.
Berthy Feng, Ricardo Baptista, and Katherine Bouman. Neural approximate mirror maps for con- strained diffusion models. In The Thirteenth International Conference Learning Representations, 2025. htttptts///
ew net/forumt t
Nic Fishman, Leo Klarner, Valentin De Bortoli, Emile Mathieu, and Michael John Hutchinson. Diffusion models for constrained domains. Transactions on Machine Learning Research, 2023a. ISSN 2835-8856. URL Certification.
net.e/ewwweeeeeeeeeeeeeeeeeeeeee Eorum?id XuWIFO4VGO o
Expert
Nic Fishman, Leo Klarner, Emile Mathicu, Michael Hutchinson, and Valentin De Bortoli. Metropolis sampling for constrained diffusion models. Advances in Neural Information Processing Systems, (1662266663332 2023b.
Yuan Gao, Jian Huang, and Yuling Jiao. Gaussian interpolation flows. Journal of Machine Learning Research, 25(253):1 2024.
Zhengyan Huan, Jacob Boerma, Li-Ping Liu, and Shuchin Aeron. Efficient constraint-aware flow matching via randomized exploration. arXiv preprint arXiv:2508. 13316, 2025.
Tuomas Hytönen, Jan Van Neerven, Mark Veraar, and Lutz Weis. Analysis in Banach spaces, volume 1. Springer, 2016.
Sadeep Jayasumana, Srikumar Ramalingam, Andreas Veit, Daniel Glasner, Ayan Chakrabarti, and Sanjiv Kumar. Rethinking fid: Towards a better evaluation metric for image generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9307-9315, 2024.
11

---

Published as a conference paper at ICLR 2026
Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. Advances in neural information processing systems, 35:26565-26577, 2022.
Shervin Khalafi, Dongsheng Ding, and Alejandro Ribeiro. Constrained diffusion models via dual training. Advances in Neural Information Processing Systems, 37:26543-2255 -6676,, 2024.
Minu Kim, Yongsik Lee, Sehyeok Kang, Jihwan Oh, Song Chong, and Se-Young Yun. Preference alignment with flow matching. Advances in Neural Information Processing Systems, 37:35140- 35164, 2024.
Sangyun Lee, Zinan Lin, and Giulia Fanti. Improving the training of rectified flows. Advances in neural information processing systems, 17:63082-63100 2024.
Gen Li, Yuting Wei, Yuxin Chen, and Yuejie Chi. Towards non-asymptotic convergence for diffusion- based generative models. In The Twelfth International Conference on Learning Representations, 2024.
Ruilin Li, Molei Tao, Santosh S Vempala, and Andre Wibisono. The mirror langevin algorithm converges with vanishing bias. In International Conference on Algorithmic Learning Theory, pp. 718-742. PMLR, 2022.
Xinpeng Li, Enming Liang, and Minghua Chen. Gauge flow matching for efficient constrained generative modeling over general convex set. ICLR 2025 Workshop on Deep Generative Model in Machine Learning: Theory, Principle and Efficacy, 2025.
Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference Learming Repre- sentations, 2023. URL
www.:nnee//ooiiioooomoo PgVMdDCJTV
Guan-Horng Liu, Tianrong Chen, Evangelos Theodorou, and Molei Tao. Mirror diffusion models for constrained and watermarked generation. Advances in Neural Information Processing Systems, 36: 42898 42917, 2023a.
Xingchao Liu, Chengyue Gong, and qiang liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. In The Eleventh International Conference on Learning Representations, 2023b. URL http:/ttss//pppeeeiiiiiie net forum? wow.mmiidxx//TTTteeee
Xingchao Liu, Chengyue Gong, and qiang liu. Flow straight and fast: Learning generate and transfer data with rectified flow, In The Eleventh International Conference on Learning Representations, 2023c. URL httpttttts:::peeeeeeeeeee
net/ hwtp:/ee//orr///. www.:::dddo///itlmaa..
Aaron Lou and Stefano Ermon. Reflected diffusion models. International Conference on Machine Learning, pp.
PMLR, 2023.
Kushagra Pandey, Jaideep Pathak, Yilun Xu, Stephan Mandt, Michael Pritchard, Arash Vahdat, and Morteza Mardani. Heavy-tailed diffusion models. In The Thirteenth International Confer- ence on Learning Representations, 2025. URL atttps://openeevieee net/ forum?id=
toz10EN4qp
Alexander Tong, Kilian Fatras, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector- Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. Transactions on Machine Learning Research, 2024. ISSN 2835- 8856. URL http:httt:/:oeeeeeeee
net wwwp://eeeooooooteeeteemmamaaea (ttprrmmmmmccCDSmcccccc
Expert Certification.
Utkarsh Utkarsh, Pengfei Cai, Alan Edelman, Rafael Rackauckas, thysics-constraing flow matching: Sampling generative models with hard constraints. arXiv preprini 2501v:2500604471 2025.
and Christopher Vincent
Nuri Mert Vural, Lu Yu, Krishna Balasubramanian, Stanislav Volgushev, and Murat A Erdogdu. Mirror descent strikes again: Optimal stochastic convex optimization under infinite noise variance. In Conference Learning Theory, pp. 65-102. PMLR, 2022.
12

---

Published as a conference paper at ICLR 2026
Zhengchao Wan, Qingsong Wang, Gal Mishne, and Yusu Wang. Elucidating flow matching ODE dynamics via data geometry and denoisers. In Forty-second International Conference on Machine Learning, 2025. httpst::: httpps::/ppenreeeee....... www.review.eeeeeeeeeep net www.://oooondddddnneedeaeeeeoee
Yuqing Wang, Ye He, and Molei Tao. Evaluating the design space of diffusion-based generative models. Advances Neural Information Processing Systems, 37:1 1307-993999, 2024.
Tianyu Xie, Yu Zhu, Longlin Yu, Tong Yang, Ziheng Cheng, Shiyue Zhang, Xiangyu Zhang, and Cheng Zhang, Reflected flow matching. In Forty-first International Conference on Machine Learning, 2024. URL https: battps://openrevtatt
net Seerumiid-Sf5KAAA
Qinglun Zhang, Zhen Liu, Haoqiang Fan, Guanghui Liu, Bing Zeng, and Shuaicheng Liu. Flowpolicy: Enabling fast and robust 3d flow-based policy via consistency flow matching for robot manipulation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pp. 14754-14762, 2025.
Zhengyu Zhou and Weiwei Liu. An error analysis of flow matching for deep generative mod- cling. In Forty-second International Conference Machine Learning, 2025. URL https: 1ttp://www...../rommmo11-2222222
A VISUAL ILLUSTRATION OF METHODOLOGICAA CHALLENGES
We illustrate the benefits of our approach in Figure Ax < b) with
The constraint set is polytope K (xr IR?
1
5
1133
A
T =(10 30
90 5).
The target и is a mixture of three Gaussians, truncated to KC: 1/1100 0J"", diag (8, 2)) with weight 0.6, ([-115,-10)) "0]",dho diag (1, 1)) with weight 0.2, and ( 3)² diag (0.5, 0.25)) with weight 0.2. We compare G-flow (Gaussian prior) and t-flow (Student-f prior) under both the log-barrier mir- ror map and our proposed regularized map (Figures 5(c). alongside samples from the true target (Figure 5(a)) Vector fields were parameterized by neural networks and simulated via Euler discretization (h 0.1) with early stopping. As shown in Figure 5 our approach achieves robust mode recovery and faithful constrained sampling, consistently outperforming Gaussian-bass flow methods.
B EXAMPLES VERIFYING PROPOSITION 2.2
Proposition 2.2 can be specialized to several classical examples of convex sets.
1. L2 ball. Consider the closed Euclidean ball K. = {o E Rd: ||ج||| < R}. Define the mirror potential (z) - (R² ||x]|2) I- 1|2 In this case the barrier function Φ(x) = ||>|| R².
which is clearly smooth and convex. Moreover, its gradient is bounded on K, satisfying the required assumptions.
2. Polytope. Let K = fx E Rdd of < b, vi E [m]] be polytope defined by linear inequalities. xÿ. Here the barrier functions are (x) = of a b, Each db is affine (hence smooth and convex), with Hessian (2)) O. and its gradient is bounded uniformly. Thus the conditions are again satisfied.
Define the potential (xx) = -n T (f''"=")''= + شز
C VISUAL ILLUSTRATION CORRESPONDING SECTION 2.2
We illustrate the blow-up phenomenon discussed in Section 2.22 In Figure (a)
illustrate the t + 0 limit, blows-up for small t, and t + limit respectively, for the G-llow. hh corresponding Figure 7(d) 7 7(f) for the t-flow is more benign.
7( 7
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Drouad Tram Relereee
Drawoted Sorvares Ge en log Havee
Genecited Sareples Eiloo log Harner
Them
633
632
EX
900
6001
(a) Ground truth
(b) G-flow Log Barrier
(c) t-flow Log Barrier
Genertted Samplen laye curmire m0g
Greecrnd Sarelle Hike eur
S30
600
100
(d) G-flow proposed mirror map
(e) T-flow proposed mirror map
Figure 5: Figure 5(a) shows the ground-truth reference distribution. Figures 5(b) and ((c) illustrate that the bog-barrier method performs poorly (both with G or t-flow), while Figure 5(d) demonstrates that G-flow (with our mirror map) fails to capture the mode centered near (-10, Oj. In contrast, Figure ((e) shows that t-flow with our mirror map covers the target distribution better. All results are
obtained with discretization step size h = 0.1. See also Figure 6 for a zoomed-in illustration near the boundary.
Groud taflerene
Ganelzed Saripie
Seneraed Sanga
1.75
0003
0000
0.000
0002
0002
0/001
6000
13000 13-01- 12..0-2.-2223333 1225-13.80
-31001575-11 50-325- 2..3-2255 22--2-2----22525-20
(a) Ground truth
(b) G-flow proposed mirror map
(c) t-flow proposed mirror map
Figure 6: We generate a total of 10,000 samples, but for visualization we only display those lying in the boundary region distribution. Figures 6(b) and 6(c) demonstrate that, near the boundary, t-flow provides a closer approximation the ground truth than G-flow.
14, 12), y E [0, 2]. Figure 6(a) shows the ground-truth reference
D ADDITIONAL EXPERIMENTAL RESULTS
We visualize the dual space distribution induced by different mirror maps. To plot the figure, generate 10, 000 true samples inside a 10 dimensional polytope, and map them to a dual space using different mirror maps. Then select two-dimensional subspace y2 ys of the dual space R 10 and visualize the samples in this subspace. See Figure g We observe that the log- barrier indeed produces heavy tails, whereas mirror map with 0.2 doesn't.
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01x) + cxp
Re 7. 0.02, 5
X1-2
[10 1 oxp
215-17
B, 0.05
20g
A1-17
2.0
3.0
1.2
1.5
2.0
1.0
0.8
1.0
0.6
3.0
0.4
0.2
0.0
100
100
200
(a) G Prior:
(b) Prior: small
(c) G Prior: large large
010
F.c 0.02,
011
t 0.05 5
b. m 0.9, 10000
ple.1
0.035
0.030
0.030
0.0
0.025
0.020
0.020
0.015
0.035
04
0.010
0.010
0.005
0.000
0.000
0
100
700
100
600
0
500
100
150
700
2500
5000
7500
10000
12500
Xg
(d) (Prior:
(e) Prior: small
Prior: large large
Figure 7: Illustration for Example 2
(i) Figures ( and 7 demonstrate that in the limit / + 0, the
distribution remains well-behaved and does not blow up. (ii) Figure ( shows that for sufficiently
large values of (here choose moderately large x for readability), there exists small value of t such that the flow with a Gaussian prior diverges. (iii) Figure ( illustrates that such divergence does occur when using Student-f prior. (iv) Finally, Figures [[c) and ((f) that i approaches 1, the Gaussian-prior flow becomes unstable, whereas the Student-t prior remains stable.
2D projection space samples
2D projection space samples dims
200
Log Barrerr
Log Barr ar
150
x 0.7
0.2
K=0.3
0=
100
50
-50
-10
-100
-150
-30
-200
-40
-50
(a) Dual Space Distribution
(b) Dual Space Distribution (Zoomed)
Figure 8: We compare the difference hetween our mirror map (with different k), and the log-barrier.
E PROOFS FOR SECTION
Proof. [Proof of Lemma 2.1
Recall that for any one dimensional random variable X, we have
10
P(X 2 t)dt
/"
E E
BII.
L 1 xx
=
-B1/ /// dt E
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1. First claim.
Assume P(I 2 R) 2 C Hence know (where = 11/p, so that di pppp-dss) (ds)
P 2 2tdt = P(] d/ = / P([ 2 s)ps ds
B|
Ds -ds {0 Cps
The integral does not converge.
2. Second claim.
Assume P(I R) I
C R
P 2 l)dl= // P(IY 2 Dpy / P(IY 2 8)pbP ds
B|J
SBDD" ds 10 Cps 1ps--1-Ads.
Ju
The integral converges p p -1, i.e. -1,
p.
Example 5. Let K. C I be triangle defined by the following inequalities:
100x, 0.01æ² S 1,
100z1
0.01x² I 1,
-1 S 0.
Recall that for each constraint af'x S by we can define ()) = log (h; -af af.f. x:).
Then the log-barrier
(x)) = E, tis (x)). Take derivative, we obtain Vi((x)
E. b. ai.
Vubki, Σ a; 100k, - 0.01
100
1
100
bi 0.
0.01
100k; + 0.01
-0.01
1100 (1-100k; -0001,)) +0.k,
2 200k
0.02
Consider two points (k1,,1)) k2), (k1, k2)) _2 in the dual space:
(VVb()),))) (ki, -k2) 0.01 100k, 0.01k, 100k, +0.01k,
0.041
Hence
V (ki(k - Vib(ki, -
hk1,,22) (k1, k22 I
0.01( (T- 0.01kk) (1 1000, +0.01le)
1.041,
2 10
2k²
(1 - 100k; - 0.01k2) 100k, + 0.01
When (k1, O, we have - kz)1l
+2 10
k2) -2)
The above example shows that, there are cases when the polytope is "ill-shaped", and leading to very large Lg.
Proof. [Proof of Proposition 2.2
We have
m
VV(()) Σ
V(V(2)= = [ooo
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Note that (2)((x) = 0 due to convexity of o;. So we have (x) č I. It follows that W2(2, 1) I (2,
Furthermore, (x:) > i- (-),(a))
+ we know
[V & |||||| + Σ |I (x)) (V - ||]|| +
Σ 8"
Vo:(
Since assumed (z) are of bounded gradient, we know [
C for some C².
Denote
C'
2 sup |V R 8
xEks
Hence we know, Ro, is such that for E Rd |||||| & Ro 2 (VE(K)). It follows that
C
P(IV Ro,N)= S P(CAK()) S Ox8
R
8/x
where note that
Cx
ROTE ( 80/K
F PROOFS FOR SECTION 3
We first provide some definitions related to conditional expectation in an abstract vector space. We follow the notation in Hytönen al. (2016). Let (S, sf) be measurable space, and X Banach space. (S X) denote the linear space of all p-measurable functions from S to X, with [g [1] < co. When is sub-o-algebra of o/ LP X) represent the Lp space w.r.t. (S, F 3. Definition F.1. (Hytönen et al. 2016 Theorem 2.6.23 and Proposition 2.6.31)
μ is o-finite on the sub-algebra F, then every f L): admits a unique conditional expectation with respect to F. It satisfies
[ Elfl fdy, VF E $
Furthermore, let g E FF; and that f E (S; X2) be a-integrable over F. Let Xi Y be bounded bi-linear map. Then B(g, f) E (S; Y) o-integrable over F, and we have
BiB(g, NI B(g. E
Proof. [Proof of Proposition
3.1
In primal space, the corresponding interpolation would be 46 (Z,) v (2,) d d z
X,
di
dt
Recall that the two minimization problems are:
min E
10 (Xe, t)
Xe X
and minE [io( - di
Z, 1230
dt
respectively.
Recall that 26 evaluated at x is the inverse of V evaluated at z = Velz), i.c. Viv(I)) = 2V2 (Vv(+)))-- Hence we obtain (2((x)) (2) Z. _ Z. Condition on X, = v, we have
d x1113 x = (PP(((Xnt)))
d X1, u (X,,t) t)
d
1|u (X,,)))) dt
x. dt"
dt
d dt
=(V))()))² (X(,t) - 8200(22
Ze, v (X V220
Zı
=g (2((E)) dt
d Mi (ob (x,
d 7 (V
d
dt
dt
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Hence get |10 ( (z, t) d X, 113. (2)e (z,t) 2 equivalently (lu?(2, d
d Zell Zelly = 1 (z,t कi x11 10
So we get (2, (a (x,1), o (3))
The equivalence follows from the change of variable formula.
Now we show the last claim. Now consider 9 to be the sigma algebra corresponding to X, 2. Note that each tangent space T.M is Hilbert space, with Riemannian metric g. Then for any Y (that is measurable in G) we have
E[ll d
E[ d
|B[1 X₁ Mlbce)l
X,
X,|X, x] + B dt
X, = a] -11g(r)))
dt
dt
E[ll d
E[ d dt
X;
X1|X; x =)11((>) + E|| X, X = x]
Y (1)))))
dt
+ 2B[ d
l8l d di
X,
XolXx x]. E X, X, x| Y
Since = E oe X x| - Y is measurable in G, we have
d EK dt
d
X₁
LE|
di
X x}. = EE(( Xr fJ B[(s] X,|X; = x).
= |B X -1 X,|X, = x]], f) = O.
where the last equality is by tower property (Hytönen et al. Hence get
2016 Proposition 2.6.33).
Elll d X, y(lyae)]
= B[l d
E d
X.
X,|X, ) + B[[
X, X, X, x] -
dt
dt
2 Blll d
E d
X.
X,|X;
=0]1 =)]1 G.
dt
dt
It follows that among all Y being measurable in G, the choice Y = E X,|X, =x] minimize the problem. Hence v vx, () E X,|X,
G PROOFS FOR SECTION 4
We start with several intermediate results. Define Pt.: (1+ 11-11l " - p(=)) p))=))
Through-
Sae +1+ (P)) p(=)d=
out this section, to make the notation compatible with (21), we use p denote the probability density function the data distribution, supported on Euclidean space.
Proposition G.1. Under Assumption with a 2 2d + v + 2, there exists a constant that doesn' depend on t, for all t [).
B
E
[1 -1-T)p+
In other words, we have that, for all E (0, 1), there exists B1, B2 independent of x, so that
sup
Ep... ||| [|] < B1, Vx,
(E[O,T]
sup E Ep... ||| [| < Bz, Vx. rE[O.T]
Proof. [Proof of Proposition G.1
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To derive the desired upper bound, we aim to upper bound (2)) We first derive a lower bound on the normalizing constant:
1 1 " p(z1)
121
(2
* p(z2)d21 2
udd p(z)
ddz
J2d ( ? + 1 7 - 21 11?
+ 21 1|
||22115Ro
(171)
C²
(2-1)2
2(
2 (1
2
(2 1 |al| + Ro)
Bo
(1-1)2 +
1
+ Ro)²
1
(1-t) t)²
44
2 (1 t)² + (11==] || + tRo)
We will split I into different regions, and derive upper bounds of Pt,. (31) for each of them.
1. Region 1 1| =1ll S ||=|| + Ro. We remark that it suffices to consider |-| 2 2Ro so that
1 ||2||| $ 11 - 21
ll
S
|all| 24
+ Ro. which implies ||
||
2
|||||| Ra 2 Ru-
2t
Otherwise, if ||=|| 2Ro.
||
1 S 1 21 <
||=||| + Ro. i.e., we have that
2t
21
|||||| S 7Ro so that SBq (R= 4il +Ro)
|| 2 (21 dzz
49
which is of constant order.
Now since
[=| 21
2 2g, we alternatively have
(14 + -11
I -- 1:1
1 1-t 113) ued
p(21)dz
(1 -12
rdd
1
(1
-))²
b44
2
2
2
2
(1 - -)²
(11=)] + &tRo)²
2
t)²
x
Using
(1 + //
σ tz
(12)
4 p(=1)
1 1-t t
p(21) 5
1
C u4a
utd
(1 + 1
x t.1
1-t
1 (1 + w/
1-t
12)
2
|
PTd
(L-1)
Cuia
(22)
|
(1-1
+
21 (?
|
1-t
||
we obtain
(1 + 113) p(21)
C rt
(1 t)2
(21)
2=4
Jga(1 (1 - p(z)dz
|| vld
(1 f)2 + 3 3 1
2( -1)²
1)2 + |arl1 C udd
(1 - 1)2 2
19

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Published as a conference paper at ICLR 2026
When 1 10-20 21 S ||a|| + Ro.
||21|| Pt.1 Pe. (2))d2
B (R= 1111 -Ro)
12/ (1 - (2=112(111 +) (1- B1ae1 C ) d21
$2
BB 1Ill +Ra)
(1 t1-
|| || u4a
||:||
|| (1 (1 - +)2 3 |
C "2d
<2Vol(BB
(R= (RB(B(R-
+ Ro))
sup
2t
Bg (R= 4:1 +
(1-
| vー
- ||,=11
C Coia
sup
(1 T)2 |21| u+d
B (R= 41
)
$2CB|1ar|1 1
9 (1 T) ||||| -
C vii
2
ta
3C (4t)
<20B||0011 1 [d
(1 - 7)2(20))
=||z|||
20 20 ايي
(1 - Tuutd
where observe that 2
- 2 =
and
v4d
u4d
[2]
||||||
||=|||
20
uid
= =||z||2 (102110-00022000 -||x||/dtuu2-a
(a- 2-o---2--
2n
2n 2d
To control the second moment so that it doesn't 't explode with || we need a 2d u 2
2. Region 2 _ - 21|l= 2 # |||+||| + Ro and ||=||| 2 1
For this case, we can have a sharper upper bound on (2))
(1 + x t2 122)
+ p(21)
1
2
ped
(() <
(1 + 1
122)
(1 + 2
1- 121
2n red
T-t
1-t
(1-1) (T712
Cu4z
vtd
(1 111 -- tai 112) ||
111 21 22) ||||||
PTJ
(م
(1- f)2
C Cota
(=:2 + 3 20 ||||||
+ R)
||
((1 - t)² + 111 + )2)
21
Hence
(1+ 111 -
1
(2))
1-6
Sa (1+ + 1 1: 12 12) - p(z)dz
(1 - t)²
v44 पकनत
(1 11)
2( (1-1)² + | |0||| + R2 |||||| Pid
2
1)2 + 1 [[[=]] + 12))
(1- (1= + (11=l + +2
Cvia
= 22( (1 - t)2 + 11 11 + tRo) )2 ||||||
utd
We see that for | - 21 2 3 | + Ro. Pt. (21) has a polynomial tail bound that doesn't 't
depend on x, f. Thus 5 11 4 || + and 11 || (zz can be bounded by some constant that doesn 't depend on 21, d:
f l =x 112 de [|++==]+++ Ro and 1|=1
|| I1 (r SC²/ /
11211121 1 2 dar.
20

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The convergence of the integral is equivalent to the convergence of _ pd-at! When a - 2d + 2, converges.
3. Region 3 || zıll ž E || |ل||||||||| + Ro and | s 1.
We simply have J1+ - de || Ro and ||ll|:i || pr, (z1) S 1.
With the above Proposition, we can prove Lemma G.2 and G.3 which are the key ingredients in proving the Lipschitzness of v.
Lemma G.2. Under Assumption
we have
+ d 2/V
v+ d 2/D
V V.E = =I]1
E B S
1-T B1, V2 [0.T].
Proof. [Proof of Lemma G.2
E 12 = x| = V JR 21 + 1 1----111 (13) 12) p(z)) )zz
Sge 1 (12) p(z)dz
(1 1
12)
red
2 p(21)
JR (1 + 11 1-t [12) 4 p(z)dz dž1
Rd
T p(z1) Saa (1 + 31 1-) p(z)dz de1
x 121 113)
= Rd
21
(1((1
+
|||
1'
J2n(1 (1 + + --1 44 p(z)dz)
(1 b1l - 113) - p(2))
21
Vx
(1
+ -
-
tz 112)
and p(z)dz)
2 d21-
Rad
Rd
t
(zz + + (12) - p(=)dd
Observe that
x
t21 112) utd
1
v+d
1 x 121 (12) 2 -1 (V. x
1 T 121 11?)
(1+
2
1
1-t t
1 t
/ + d
(1+
-1
1
112) un -1
(2 x t2 1
2
1 1-t
1
1-t t
-
1
x -t
t21 (12)
Pad 1/ + d
t21
1
(I
t
1 1+
t21
1
1-t
1-t
Hence
V.EIZIZ = x] | (1 1 x x- - 12ı 122 - d
1
r t2
1-t
2
t
1-
1-1
p(21) Seul (1 + - 112) _ p(z)dd
dzi
(a (1 p
(1-0)=113) =14 p(2)d2
1 x 1z 112)
V + d
r - lz 1 1- 1 p(2)ddz)
+
(1 +
112
RB
Rd
1 t
(1 + +1l
122) rid p(z1)
T-t
z dz1.
[gc
311 [22) 11)- p(2)dz
21

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(1+ 14 1- ||² - p(21)
Recall that we use the notation (2))
[1+
T-r
--)------------- - p())dz
1 T
f
v+d d
1
x 121 -
V E Z =x] =
JRd
21
v
1
11
x -- 12 I-1
1 1-t
1-t/ t\
(1)=
+ /
u d
1
1 12
21
1- 1. 1 1 _
gd
Rd
1 + 7 ||
In general, we have E H[ = E[(X- BX B| Let X = 21 and
Y = 2ナd
T To bound V IZ = zz), we consider any unit vector ui
1
T V
E
71 7
= 2/10- = Blu²( (x- E[X]- ( - -[Y]) < - - B [XI]|| IY - [Y]]]]
S [|[ + [EXIIIIXX + ||B[У]I]] S B X Y || + XX
We have
+ d
1 121
V +d d
o 121
| v(1 t)² 1+ 1|| 13
v (1 1)
|| - 12) 113
At (1 t)2 = b| t2 1 ||||| reach maximum,
sup || ++d d V0 v 2(1 - t)
Therefore
d 2
V 2 =I]|= S v 1 1-t t E (eute)))lll
Lemma G.3. Under Assumption 3 with a 2 2d + 2, we have
V' ++d 3/v
1 + d
3/2
El = =11
[ 21 |2] + 3 3B =
(B2 + 3BB) V2 [0,
Ot
V 2(1 t)2
2(1 3(1-T) T722
Proof. [Proof of Lemma G.3
La (1 + 1 - (|2] p(21)
= x]
Se (1 + -)) p())) p(z)dz
(1+ [|2) " p(21)
21
Sg (1 + - [2]) ب p(z)dz
d=
Rd
p(z1) Spu 9- (12) " p(2)ds
21
1
- t2
1-t
122
2 de1
Ot
1/
(Jac + Hll - 112) - p(2)dz)
-1a
21
(1 -1
1
σ - tz 112)
udd p(z)dz)
(1 + bll سーاي (13) " p(21)
dz1-
Rd
Ø1
Rd
1-1 1
(Jzu (1 + | (I2) " p(z)dz)
Observe that
0
(1
+
1
x 1:2
(2) 12)- -/1
0t
1
1 - t
v+ d
x -- 1.2 [ -بب
1 x 12 112)
(1
-11
( 11
2
1-1
1 - I.
= V ++d d
+ -1 x tzı
[32]
(2
1
x 12ı r I - 21
2
1-1
1-1
(1 - 1)2
tz1 (12)
a d
1
1
= (1+ + -1
1)3 13 (1a1l
- o (0 + 1) + ||| 113)
1
v 1 +
1
2 (1
22

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Published as a conference paper at ICLR 2026
Hence
ZZ =a]
=/re 21 (1+ 1
-| - 12
1 122)
Wdd v +d d
- 12 (1 t)3 (1|x||2 - xf1 + t) + +1112111))
V
-
p(21) Sgal (1 + -
pl )2 dz1
(Jaa +
112
- -Jee 21
(1+ -+ 11
121
+ d
1
| (1 t)a (11=11 -2Fx(1 + t) + (122)12)((())))))
1
-
+ +
(1 + - 112)- p(z1) )
2 dz1
(Jeal1 + 211 (13) 1 p(2)dz
(1+ وير " p(=))
Define (t.)(21)
1 (2) - p(z)d=
Z/ = x]]
Ot
= =/re 21
v+dd
11 1 1 2-1811 12 (1 - t)3 (110|12 - x(1 t) + =1122113) (),((2))d2
Aud
(1 J v
u+d d
1
(1
12 (1 - t)3 (1|=|| - x x(1+ +
1 +
1 -
Define x= 21, Y = utd
(1-1)a (||==| - 2f'a(1 +t) + 1 Note that
1
1+ -
12
||||| = || 4+ +d d
(x )22)² )z (x - 21)
V
+
(1
1)3
1-1
V + || L2
||00 #1ll
d 1
| 1z ||| 21|l
d
< v
||||||
(1 - 1)3
1+
2 1
v t (1 - t)2 + -|| tz
where observe that if ||
1
|al| we have ||0 110--11 211l S 2||2 - t22ll- Then | I
vtd If y² 1-t"
2
|| 2 1 |||%||
||M|| < 1/ v VD 1
+ d Vv 1
d3
V 2(1 t) 1-1 |1
1-111=
21l1 S
1 2 (1 -1-17 (1= + |alll) & ||a|||
V
2 (1 - f)²²
Recall: [[[X]- - B[X] = E[(x - EX - E[Y])] Therefore
② B Z, x||| < EE (X- E(X])(Y-E(Y)) & B|||X E[X]|| I| - E|ylll|
0t
< I [[[X]))(IX + | [ID] = |X||| |Y]| + || || II]
< 1 t)² (B1| 1| 3E ||,1|111)) 111²).
The following Lemma will be used when analyzing the discretization error.
Lemma G.4. Under Assumption 3 with 2 2d + v + 2 and Assumption depends polynomially in 1-T d, v and B1, B2, [8[] Z, [|] [1| Zo |11 S.1.
2 there exists D3 that
[[llo(Z,,)) rl Zz 13]
< h22 Dz-
23

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Published as a conference paper at ICLR 2026
Proof. [Proof of Lemma G.4]
By chain rule,
d
0
8 0t 4,
di (21,,
(Z,,
"Z,,
Ot
dr
and therefore (note that 7 = v v(,211) 1))
θ
0 Ze, t)ll - || 11 (Z
t)ll +
u (Zz,t)) 10((z, t)ll-
dr
Recall
1
1 ++ d
3/8
ol0, 1)l
(1 - 7)2 ||a||| + (1 - 2)
B1 +
v 2(1 - T)2 (B2 +3B/) 1i [0,
0t
1
+ v+d d 2/v
| (z t)ll
B1, Vt E [0,
1
v (1 TT2
and
+ 1
|u((z t)ll
E = =|لl S 1-T T (1æ=) + 1 = =|]l)
(1|=]) + Bi|l
Z = =x])
1-7 T [||=|| + B1).
T
Hence we have
1 (Z), -
(1 (1-T)² |Z + (1 (1- T)2
B1 +
1
V v d
2(1 T)2 (B2+3B)
3/D
1-T T
+ (1- v v+d d 2
1
1
B1
(1
B1) Vt E [0,
(1 T)²
1-7
follows that there exists D1, D2 (that depends polynomially in ,-
d, v, B1, B2) s.t.
_
v t)| D.1 D2, e [0.7].
dt
Recall that Law (Z)) = (17) + (1 1)). Hence
B|| ||| = B |2 + (1 - t) %ol 2| = 2 |1I + (1 1) + 26(1 - t)E 21]
$ 2E 7 |1 + Z
which
implies
there
exists
Da
(that
depends
polynomially
in
1-7, d, v, B1 B2, E||| Z [1 E[| Zo | 1 s.t.
d E[|
v oZ t)13 < D3.
dt
By Jensen's inequality,
Elll [( Z2, u( Z,, t =E[| '' ( d v(Zz. s) ds|l1 S (t (t-
1' d
2 v(Z..
ds]
ds
ds
I 1''l|| d
v (Z, 1 < h2
dt
24

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G.1 PROOF OF PROPOSITION 4.1
Proof. [Proof of Proposition
4.1 Using Lemma G.2
1
[Vz0(2, t)]l
||
V.EIZi1Z = =x]]
1
+ V v 2/
B1 Vt E [0.1].
T
(1 T)2
Notice that
0z, 1) =
1 +
1-1 BIZiIZ = =z))
01
01
(1 - t)2 2 +
B|Zz| Ze = =/ +
1 - E Ze 2].
(1 -1-t) t)2
Using Lemma G.3
we have
1
1 1 v+d 3/D
|| Ot
02,
(2,,t)l t)ll s
|2|
+
B1
(B + B2+) vt E [0,
(1 - T)2
(1 (1- 2)
1 1 T 2(1 T)²
G.2 PROOF OF THEOREM 3
Proof. [Proof of Theorem
Define
dz, v Ze. t)dt, Zo no.
dY, 6 4, to dlt, Yo Zo.
By direct computation,
7t d V.) = 2(2 - -Y,,((z,1)) - G(F,,,
dt 14 - = 2(/4 2 Y,, dt dt
=2(7, Y v,, 7z, 1) /4, 1:)) + 2,7 V,, 1,, ((,,,, 44) v(Fr,,4)) ())
+ 2(4 Ye. v(V t,, 1,) +(V t,5 4)).
Using Young's inequality, bound the rest of the terms as follows.
1. We bound the first term. By Lemma G.4
[ - Y. (( f)
vl f)H1
1
<L - y.1 + Blllu(Zr (Ze, t) - v(Zr,, =
L
1
E - v. +
112 D2
L
2. We bound the second term.
(E - Y, Di Zt, ti) vY +.))
< EL v.1 +
Ell (Ze, t.) -YYY,, (Y,,,t,)) 11
L
L - -.1 +
L Ze. -r.177.
L
Here we used Proposition 4.1
25

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Published as a conference paper at ICLR 2026
3. We bound the third term. Recall that assumed (1]] 6(2; t)l3 1 E?. Then
2(Z - Ye, o(Y e t2) 6(Y,, t,))
< LL, - Y.ll²] + Blllu(Y (Y,,, 44) - 8(Y 411
T11
SL,BilZ, - -Y,1 +
_
1
TA
Together,
di E11 - Y.11 - 31, ++11 + L
(h² + LE - -.,1171 + =-)
Define
1= + 11 6
Then
E1 2111 - 1
3
flit 23l1ll44100 (K) dt
Sc Z, - -Yr, 1r,1131 +
L b1:
23L2h
<031 Zt, 12,1891 +
K
L2
=63 Zz, - 101
1
(h² D3 + =²) + (c3Lrh 1)) - -Y,113
L
3Lih -1
5(208Lih Z) Y 1 +
(h22 (h²Ds =²)).
L
For Ai41 S (203tth - 1)A, +
cdihh B with Ao = 0, we have
1.
1-1 aith 1)² 203
1
1 - (203k1h 1)"h carch 1
(2e - 1)" -
Ag
B
B
B.
L3
1 (2 - 1) Li
2L3
In general, for x E [0, 1] we have ex < 11+ 2x. Hence 20 1 S 3lag 6L,h S 12L
And we get (2e 1)" s (1 + h)" s (1 + 126x )m S 12111
n
Hence
12L
BillZr - Yr
(h + t2)
L
This implies
cL2
W2(
+D #
S
v
11
Consequently,
Wa P ( <
V1 + + +
T)
I
G.3 PROOF OF THEOREM 4
Proof. [Proof of Lemma
Note that we have
||u" (z1, P v ("(xz, (lg
P viy (z)) P(2, (gp()))
-100((2,t)) "3 t)
= -1100(E11))) (' 0 (g)())))
=Ilo"(za,t) - (222)t)))
26

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Published as a conference paper at ICLR 2026
where P denotes parallel transport from to y. This proves the result.
Lemma G.5. Under Assumption 4 For 2240-2 can guaranlee Assumption 3 holds with 2d + v 2.
Proof. Using the change of variable formula, together with the fact that the determinant of matrix equals the product of all its eigenvalues, we know
da d (x) det (2
2 d
P (2))
where n "uuc " Hess denotes the probability density function of the target distribution in primal space, under the Euclidean metric and squared Hessian metric, respectively. Furthermore, the isometric mapping from primal space to dual space guarantees that
17 Euc (x) 2 x (z).
Notice that
C' 8N
sup [VV S
Since assumed SuPrexV,s (x) S Opp 87, have
C'
2,, 22 2 8x (2) Bxc s 07
Using 87
//K, we conclude that there exists some C > s.t.
(c) S |=|p/0 Vz 1.
To guarantee 7/K 2 +2 need K S 2d4042'
Proof. [Proof of Theorem 4
Using Lemma G.5 we know Assumption 3 holld with a 2d + ++2
The result follows from applying Proposition 2.2 and Theorem
27


Reviews:
[{"review_id": "XtaMgsEmpP", "reviewer": "Reviewer_H5Xj", "summary": "The paper “Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domains”proposes a framework for generative modeling under convex constraints using flow matching enhanced by mirror maps and Student-t priors. The authors identify that traditional log-barrier mirror maps lead to heavy-tailed dual distributions and ill-posed dynamics, while Gaussian priors fail to align with such heavy-tailed targets. To address these issues, they introduce a regularized mirror map ensuring finite moments and strong convexity, coupled with Student-t priors that stabilize training and maintain well-behaved velocity fields. Theoretical contributions include proofs of spatial Lipschitzness, temporal regularity, and Wasserstein convergence bounds for flow matching with heavy-tailed priors, extended to the constrained (primal) domain. Empirically, the method achieves superior or competitive performance compared to reflection- and gauge-based baselines on constrained generative tasks—such as sampling within polytopes, L2  balls, and watermarked image generation—while guaranteeing feasibility and improving robustness on complex convex domains.", "questions": "- How should the number of degrees of freedom of the Student t prior be set?\n\n- What if the distribution has mass at the boundary of the constraint set? The mirror transformation would send such points to infinity, so how can we model them?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- The paper is rigorous and very well written. \n\n- It makes relevant methodological contributions: particularly on how to perform flow matching with a mirror map framework, on how to choose the mirror map, and on how to set the initial distribution in order to handle target distributions with heavy tails, which is important as the distributions in the dual space can be heavy-tailed.\n\n- The paper contains experiments in which the target distributions are over polytopes, and watermarked image generation experiments, although the images are low-quality.", "weaknesses": "- The applications of the work are not very compelling and may not justify the complexity of the framework. \n\n- In the experimental section, the authors compare their performance to Mirror Diffusion Models, which is a previous work on using mirror maps for diffusion models. However, there is not comparison to Mirror Diffusion Models in the paper. Since Flow Matching and Diffusion Models are very related, it would be useful to clarify the differences between Mirror Flow Matching and Mirror Diffusion Models.\n\n- More generally, the paper is lacking a related work section that places it in the context of previous works that leverage mirror maps for diffusion/flows.\n\n- The paper can be considered incremental in that it is not the first diffusion/flows paper to rely on mirror maps, but it does clarify some design choices to make them work better.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "j3S3nfUiDB", "reviewer": "Reviewer_9fkz", "summary": "This work aims to improve the mirror flow matching algorithm — which is used to generate samples from a distribution supported on a convex domain — by proposing a new mirror map and using the Student-t distribution as the prior distribution.\n\nThey show on artificial data that the log-barrier transformation, which is commonly used as a mirror map, can induce heavy tails when pushing the target distribution into the dual space. To replace it, they introduce a new mirror map that ensures the existence of the moment of order p of the pushforward, by this new map, of probability measures satisfying the “boundary-measure condition.” This mirror map also has the property of being strongly convex, which allows transferring convergence guarantees from the dual space to the primal space.\n\nThey also illustrate, with a well-chosen 1D example, that for target distributions with heavy tails, the Gaussian prior is unsuited for the flow matching framework, and they propose using a Student-t prior instead. Still with the Student-t prior, they obtain Lipschitzness guarantees both in time and space for the true velocity field in the general flow matching framework when the target has a polynomial tail bound. This allows them to derive an upper bound on the Wasserstein distance between the true data distribution and the intermediate distributions generated by the learned velocity field of the general flow matching algorithm. Using the strong convexity property of the mirror map, they transfer this discretization bound from the dual space to the primal space.\n\nThey tested this new mirror flow matching approach on simulated data and on the FHQv2 dataset by generating watermarked images.", "questions": "- Proposition 2.2 assumes that the newly defined mirror map does not induce heavy tails and guarantees the existence of moments of the pushforward probabilities that initially (before being pushed forward by the mirror map) satisfy the boundary-measure condition. It would be interesting to see, in practice, for examples where this assumption is violated, whether the proposed mirror map induces lighter tails than the log-barrier map. You already verified this for the experiment described in Appendix A — do you observe the same result (i.e., that your proposed mirror map induces lighter tails than the log-barrier map) for the target distributions described in the experiments from Sections 5.1 and 5.2?\n\n- Line 202: you claim, “This example highlights a key principle: when the target distribution is heavier-tailed than the prior, the conditional distribution is likely to have a mode that is dominant near $x_t$\n for some values of $t$” Do you know of other examples or references that could help generalize this principle beyond the pathological case presented in Example 2?\n\n- Can you justify your choice of baselines for the experiments detailed in Sections 5.1 and 5.2?\n\n- How did you set the hyperparameters $\\kappa$ and $\\nu$ in both experiments? What happens for other values of these hyperparameters? How sensitive is the algorithm to them? \n\n- What is the architecture of the neural network used to parameterize the vector field trained in the dual space?\n\n- Can you provide more information about the results reported in Table 1 (e.g., the number of samples used to compute the MMD and KL divergence metrics)? How is Feasibility defined?\n\n- Finally, can you comment further on the fact that your results are not as good as those of the MDM baseline for the watermarked image generation task? Do you think this might be due to suboptimal choices of the hyperparameters $\\kappa$ and $\\nu$?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well written and a pleasure to read. The main new ideas concerning mirror flow matching — the mirror map and the new Student-t prior — are well motivated, and the arguments given to introduce each of them are fully valid.\n\nIn addition, the theoretical results on the Lipschitzness guarantees, both in time and in space, of the velocity field when the prior is a Student-t distribution (Proposition 4.1) are new and relevant for the general flow matching framework, as they require lighter constraints on the target distribution than prior works. The same holds for the discretization upper bound in Theorem 3, which is also relevant for the general flow matching framework.", "weaknesses": "While the theoretical sections are very rigorous, the experimental section is not and should be rewritten. The choice of baselines is not justified, and the baselines differ between Experiments 5.1 and 5.2. The choices of the hyperparameters $\\kappa$ and $\\nu$ are also not justified, and their values change between the two experiments. The architecture of the neural network used to estimate flow matching in the dual space is not mentioned. The comparison with other baselines is insufficiently detailed (e.g., the number of samples used to compute the MMD and KL divergence metrics in Table 1 is not specified). Finally, the notion of Feasibility (Table 1) is defined in the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fLM5CshJMC", "reviewer": "Reviewer_hXik", "summary": "This work proposes mirror flow matching, which addresses previous limitations of mirror diffusion models (Liu et al., 2023a) and presents a comprehensive theoretical framework for this approach. The key contributions include: (1) a modified mirror map designed to guarantee Lipschitz conditions, and (2) the use of a Student's t-distribution heavy-tailed prior for training the flow matching model in the transformed dual space. The empirical study demonstrates improvements compared to baseline methods, such as reflect flow matching and gauge flow matching.", "questions": "1. In the original mirror diffusion model work, the authors consider specific linear constraints with orthonormality conditions, which admit a closed-form inverse. However, such requirements are not explicitly stated in this work. Can the authors specify the forward and inverse mirror maps used in this work and clarify what conditions are necessary for closed-form inverses?\n2. What is the role of the heavy-tailed prior distribution in Proposition 4.1? How does it contribute to the theoretical guarantees or empirical performance?\n3. Why does RFM not achieve a 100% feasibility rate in Table 1? \n4. In Proposition 3.1, what is the prior distribution for the primal and dual spaces, respectively? \n\n\nOverall, this work is theoretically solid. I will adjust my rating upon clarification of the applicability and details of the mirror map stated above.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The theoretical analysis is comprehensive and solid, including spatial and temporal regularity results and final distribution error bounds under discretization. These theoretical contributions may be of independent interest beyond this work.", "weaknesses": "1. The mirror map is limited to simple sets with closed-form expressions. Despite the comprehensive theoretical analysis of this framework, its practical applicability remains limited due to these restrictions.\n2. The additional benefits in the empirical study of the proposed approach compared to Gaussian priors are minor. Moreover, the experiments lack a comparison to the traditional mirror diffusion models (in Tables 1 and 2), which serve as the main baseline work discussed throughout the paper.\n3. The expression of the inverse mirror map is unclear. The authors should provide a table with explicit forms of both the forward and inverse mirror maps for different convex sets to improve clarity and reproducibility.\n4. It is unclear how the Lipschitz conditions in Equation (1) are satisfied given Proposition 2.2, which only provides a bound on the expectation of the Lipschitz constant, while Equation (1) requires pointwise Lipschitz conditions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)