Probabilistic Invertibility of Rectified Flows Beyond Global Monotonicity

Abstract

Rectified Flow (RF) has recently emerged as an efficient alternative to diffusion-based generative modeling, with a theoretical explanation for fast sampling that hinges on invertibility properties of a rectified transport map. Existing analyses establish global invertibility of an auxiliary map $H_t(\cdot )$ under a strong uniform monotonicity condition on the endpoint Jacobian, yielding a sufficient route to straight couplings and quantitative error control. Empirical evidence, however, indicates that this monotonicity condition is frequently violated while invertibility persists for structured targets such as Gaussian mixtures.

This preprint develops a complementary “generic-in-time” theory for the Gaussian-to-Gaussian-mixture regime. The central technical observation is a transversality identity: at any collision point $H_t(x)=H_t(y)$ with $x\ne y$ and $t\in (0,1)$, the time derivative ${\partial }_t(H_t(x)-H_t(y))$ is nonzero and explicitly proportional to $-(x-y)/t$. This transversality engine implies that collision sets form a Lebesgue-null subset of $(0,1)\times B_R\times B_R$ on any compact domain $B_R$, and consequently, for Lebesgue-almost every $t\in (0,1)$, collision pairs in $B_R\times B_R$ have zero Lebesgue measure.

The argument couples (i) strip-wise flow regularity on $t\in [\epsilon ,1-\epsilon ]$ (supported by explicit mixture-score formulae) with (ii) the transversality identity and (iii) an implicit-function/Fubini slicing argument. This replaces uniform global monotonicity by an almost-sure injectivity statement aligned with the probabilistic manner in which RF is used in practice.

1. Introduction

Rectified Flow (RF) constructs a deterministic probability flow ODE that transports a simple base distribution ${\rho }_0$ (often a standard Gaussian) to a target ${\rho }_1$. In contrast to diffusion models, RF aims to learn (or approximate) velocity fields that generate near-straight trajectories between noise and data, enabling fast generation.

A recurring analytical object in RF theory is an auxiliary map $H_t(\cdot )$ that interpolates between the identity and the time-1 endpoint of the RF flow. In recent work [1], global invertibility of $H_t(\cdot )$ is obtained under a sufficient assumption imposing uniform global monotonicity through the symmetric part of the endpoint Jacobian. That route supports strong structural consequences (e.g., straight couplings under 1-RF) and quantitative error bounds. At the same time, simulations suggest that this assumption is not necessary, and an explicit open problem is raised for Gaussian mixtures.

Objective

This note develops a complementary analysis for the regime

in which the RF drift admits a closed-form score representation. The principal aim is to formalize a probabilistic notion of invertibility: for Lebesgue-almost every $t\in (0,1)$, the collision event $H_t(Z_0)=H_t(Z_0^{\prime })$ with $Z_0\ne Z_0^{\prime }$ occurs with probability zero under $(Z_0,Z_0^{\prime })\sim {\rho }_0\times {\rho }_0$.

Contributions

1. Transversality engine. A structural identity is established: at any collision point $H_t(x)=H_t(y)$ with $x\ne y$ and $t\in (0,1)$, the derivative ${\partial }_t(H_t(x)-H_t(y))$ is nonzero and equals $-(x-y)/t$. This identity is independent of monotonicity or convexity assumptions and isolates time-mixing as the mechanism that destroys persistent collisions.
2. Generic injectivity on compacts. Under standard flow regularity on strips $t\in [\epsilon ,1-\epsilon ]$, the transversality engine implies that collision sets are Lebesgue-null in $(0,1)\times B_R\times B_R$ for every radius $R$. Consequently, for Lebesgue-almost every $t$, the set of collision pairs in $B_R\times B_R$ has $2d$-dimensional Lebesgue measure zero.
3. Probabilistic invertibility under Gaussian initialization. Since ${\rho }_0$ has a smooth density, the preceding Lebesgue-null collision property yields that, for almost every $t$, the collision event under $(Z_0,Z_0^{\prime })\sim {\rho }_0\times {\rho }_0$ has probability zero on each $B_R$ and hence in the limit as $R\to \infty$.

2. Mathematical Setup

Independent coupling, interpolation, and RF drift

Let $(X_0,X_1)\sim {\rho }_0\times {\rho }_1$ with ${\rho }_0=\mathcal{N}(0,I_d)$. Define the linear interpolation

Denote by $p_t$ the density of $X_t$ and by $s_t(x):={\nabla }_x\log p_t(x)$ its score. The RF velocity field is

The RF sampling ODE is

The endpoint map is $Z_1:{\mathbb{R}}^d\to {\mathbb{R}}^d$, $Z_1(z_0):=Z_{t=1}(z_0)$.

The rectified map $H_t$

Following [1], define for each $t\in [0,1]$ the auxiliary map

Invertibility of $H_t(\cdot )$ plays a central role in straightness arguments in [1].

Gaussian-mixture targets

Let the target be a (nondegenerate) Gaussian mixture

In this regime, [1] derives a Tweedie-type identity expressing $v_t$ in terms of the score:

and provides an explicit responsibility-weighted affine form for $s_t$ (general covariance) as well as an explicit expression for ${\nabla }_x{v}_t$ on time strips. These explicit formulae imply real-analytic dependence on $x$ for each fixed $t\in (0,1)$.

Existence, uniqueness, and non-explosion

The vector field in the Tweedie identity exhibits an apparent singularity at $t=0$. The RF analysis in [1] establishes existence and uniqueness of solutions on $[0,1]$ under an Osgood non-explosion criterion and a mild moment condition. In the Gaussian mixture regime, the Osgood condition is verified explicitly for general mixtures, yielding global well-posedness of the ODE. The present work takes this well-posedness as the foundation for the endpoint map $Z_1(\cdot )$.

3. A General Transversality Theorem for Rectified Maps

Recall $H_t(z)=(1-t)z+t{Z}_1(z)$, where $Z_1$ is the time-1 endpoint map of the RF ODE. For $t\in (0,1)$ and $(x,y)\in {\mathbb{R}}^d\times {\mathbb{R}}^d$, define

A collision at time $t$ is a pair $(x,y)$ with $x\ne y$ such that $F(t,x,y)=0$.

4. Verification for Gaussian Mixture Targets

5. Relation to Monotonicity-Based Invertibility Results

The uniform monotonicity condition in [1] imposes a condition on the symmetric part of the endpoint Jacobian that yields local invertibility and properness of $H_t$, and thus global invertibility via classical global inverse function theorems. The present results provide an alternative explanation for the empirical robustness of invertibility in Gaussian-mixture settings: even when monotonicity fails, collisions are structurally transversal in time, which enforces a generic-in-time injectivity property on compacts and an almost-sure collision-free statement under Gaussian initialization.

6. Conclusion

This work develops a generic-in-time invertibility theory for the rectified map $H_t$ in the Gaussian-to-Gaussian-mixture regime. The core lemma identifies a transversality mechanism intrinsic to the convex time-mixing structure: at any collision $H_t(x)=H_t(y)$ with $x\ne y$, the derivative in $t$ is explicitly nonzero. Combined with strip-wise regularity of RF flows (supported by analytic mixture-score formulae and global well-posedness under an Osgood criterion), this mechanism yields that, for Lebesgue-almost every time $t$, collision sets on any compact domain have Lebesgue measure zero. As a result, for almost every $t$, the map $H_t$ is injective almost surely under Gaussian initialization, providing a probabilistic notion of invertibility that aligns with the operational regime of rectified flow sampling.

Several directions remain natural. Establishing full global invertibility for almost every $t$ may be approached by coercivity/properness estimates leveraging Gaussian-mixture tail structure. More broadly, generic-in-time arguments of the present type may inform sharper error bounds that depend on the geometric “straightness” of learned flows rather than uniform monotonicity.

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