Beyond Straightness: Non-Crossing Flow Matching via Quantile AlignTree Coupling

# Beyond Straightness: Non-Crossing Flow Matching via Quantile AlignTree Coupling
Source: [https://arxiv.org/html/2609.03443](https://arxiv.org/html/2609.03443)
Junyi LinAffiliation:Institute of Statistics and Big Data, Renmin University of China, Beijing, ChinaEmail:[junyilin@ruc\.edu\.cn](mailto:[email protected])Mengyu LiAffiliation:Department of Statistics and Data Science, Tsinghua University, Beijing, ChinaEmail:[2025104242@ruc\.edu\.cn](mailto:[email protected])Jingxuan HuAffiliation:Institute of Statistics and Big Data, Renmin University of China, Beijing, ChinaEmail:[chengmeng@ruc\.edu\.cn](mailto:[email protected])Kejun He††thanks:Corresponding authorAffiliation:Institute of Statistics and Big Data, Renmin University of China, Beijing, ChinaEmail:[kejunhe@ruc\.edu\.cn](mailto:[email protected])Cheng Meng††thanks:Corresponding authorAffiliation:Institute of Statistics and Big Data, Renmin University of China, Beijing, ChinaEmail:[mengyuli@tsinghua\.edu\.cn](mailto:)

###### Abstract

The performance of Flow Matching largely depends on the quality of the coupling between the source and target distributions\. However, independent coupling often leads to path crossings and local velocity ambiguity, while OT\-based couplings typically incur high construction costs\. To address this challenge, we proposeQuantile AlignTree Flow Matching \(QAT\-FM\), an efficient structured coupling strategy that constructs a hierarchical coupling between a Gaussian prior and the target data distribution via a quantile\-aligned tree structure\. QAT\-FM constructs the coupling in𝒪⁡\(N​d​log⁡N\)\\mathcal\{O\}\(Nd\\log N\)time and supports per\-pair source sampling with𝒪⁡\(d\)\\mathcal\{O\}\(d\)complexity, enabling scalable training for large\-scale high\-dimensional generative tasks\. Theoretically, we prove that the QAT coupling satisfies marginal consistency, induces non\-crossing linear interpolation paths, and consistently improves path separation at intermediate times compared with independent coupling, thereby alleviating local velocity ambiguity\. QAT\-FM further extends naturally to conditional generation, enabling structured conditional coupling while preserving global Gaussian alignment\. Experiments across diverse benchmark datasets demonstrate that QAT\-FM achieves competitive generative performance while substantially reducing coupling construction cost\.

## 1Introduction

In recent years, continuous\-time flow\-based generative models have become increasingly prominent in generative modeling\([Chen et al\., 2018](https://arxiv.org/html/2609.03443#bib.bib2);[Grathwohl et al\., 2019](https://arxiv.org/html/2609.03443#bib.bib11);[Lipman et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib1);[Liu et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib3)\)\. A representative class is continuous normalizing flows \(CNFs\), which define continuous transformations as the solution flows of ordinary differential equations \(ODEs\) whose time\-dependent velocity fields are parameterized by neural networks\. During sampling, new data are generated by numerically integrating the learned ODE dynamics\. To improve training efficiency, Flow Matching\([Lipman et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib1)\)directly supervises the velocity field with targets derived from pre\-specified conditional probability paths, avoiding the repeated numerical ODE integration required by conventional CNF training, enabling scalable training of generative models\. Flow Matching has achieved competitive performance across diverse domains, including image generation\([Ma et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib12);[Esser et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib13)\), speech synthesis\([Le et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib14);[Liu et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib15)\), and text generation\([Hu et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib16);[Hu et al\., 2026](https://arxiv.org/html/2609.03443#bib.bib17)\)\.

Although Flow Matching improves the scalability of training continuous\-time generative models, its performance remains sensitive to the conditional probability paths and source–target couplings constructed during training\([Lipman et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib1);[Tong et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib6);[Pooladian et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib5)\)\. In common linear\-path formulations, Flow Matching trains the velocity field by regressing target velocities along linear interpolation paths between noise and data samples; different coupling choices therefore directly alter the structure of the supervision signal\. When training paths cross, or pass too closely through the same intermediate region with substantially different target velocities,*velocity ambiguity*arises locally; under mean\-squared\-error training, these inconsistent velocity directions are averaged, increasing the difficulty of fitting the velocity field\([Guo and Schwing, 2025](https://arxiv.org/html/2609.03443#bib.bib18)\)\. By contrast, training paths with the*non\-crossing*property or good*path separation*reduce this local ambiguity, yielding a more deterministic mapping from intermediate states to target velocities and thereby improving training stability\. In conditional generation, coupling strategies must also preserve the alignment between samples and conditions, since couplings that ignore the conditioning information may produce geometrically favorable but conditionally inconsistent supervision signals\([Chemseddine et al\., 2025](https://arxiv.org/html/2609.03443#bib.bib19);[Cheng and Schwing, 2025](https://arxiv.org/html/2609.03443#bib.bib20)\)\.

Several existing approaches seek to alleviate path crossing and velocity ambiguity by improving coupling quality\. The reflow procedure in Rectified Flow\([Liu et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib3)\)reconstructs pairings between noise and data using a learned model, yielding more deterministic and straighter generation trajectories; however, this requires additional sampling and re\-training steps, imposing substantial computational overhead\. OT\-based methods\([Villani and others, 2009](https://arxiv.org/html/2609.03443#bib.bib4)\)construct geometrically optimal pairings by minimizing transport cost; under a quadratic cost, the induced displacement interpolation exhibits a favorable non\-crossing structure that alleviates velocity ambiguity\. Nonetheless, exact empirical OT is computationally expensive and difficult to scale to large datasets; in conditional generation, maintaining conditioning consistency typically requires condition\-aware constraints or condition\-wise couplings, further increasing algorithmic complexity\. Mini\-batch OT\([Pooladian et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib5);[Tong et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib6)\)substantially reduces this cost by solving OT within each mini\-batch, but the resulting coupling is primarily local and cannot strictly preserve the non\-crossing structure at the full\-dataset level\. Most of the above methods construct couplings only over finite Gaussian samples; SD\-FM\([Mousavi\-Hosseini et al\., 2026](https://arxiv.org/html/2609.03443#bib.bib7)\)further improves the alignment of continuous Gaussian information via gradient\-based optimization, but its iterative optimization incurs substantial sampling or optimization cost and does not yield an explicit low\-cost global coupling construction\.

Table 1:Comparison of Flow Matching coupling strategies\.- 1Notation\.NN: dataset size;BB: mini\-batch size;dd: data dimension;KK: SD optimization steps;EE: reflow rounds;Θ\\Theta: one network forward pass\. Complexities are reported up to constant factors\.
- 2✓✓indicates that the property is also preserved in conditional generation\.

Table[1](https://arxiv.org/html/2609.03443#S1.T1)compares representative Flow Matching coupling strategies along five axes: coupling construction cost \(Build Cost\), sampling cost \(Sample Cost\), whether the coupling induces non\-crossing paths at the full\-dataset level \(Non\-Cross\), whether it improves path separation \(Path Sep\.\), and whether the source marginal is aligned with a continuous Gaussian prior \(Gaussian Align\.\)\. As shown, prior methods generally trade off computational efficiency, path geometry, and Gaussian\-prior alignment\. To address these limitations, we proposeQuantile AlignTree Flow Matching \(QAT\-FM\), a low\-cost structured coupling strategy that achieves non\-crossing paths, improved path separation, and continuous Gaussian\-prior alignment, with several of these properties naturally extending to conditional generation\.

Contributions\.Our main contributions are as follows:

- 1\)We proposeQuantile AlignTree \(QAT\), an efficient semi\-discrete transport construction based on binary tree\-structured quantile alignment\. QAT induces an exact coupling between an empirical data distribution and a continuous Gaussian prior\. With construction complexity𝒪⁡\(N​d​log⁡N\)\\mathcal\{O\}\(Nd\\log N\)and per\-pair sampling complexity𝒪⁡\(d\)\\mathcal\{O\}\(d\), QAT scales to large\-scale, high\-dimensional generative modeling tasks\.
- 2\)The QAT coupling supports multiple tree\-splitting strategies and can naturally incorporate conditioning information into both tree construction and coupling\.
- 3\)We prove that QAT induces favorable path structure: the resulting linear interpolation paths satisfy the*non\-crossing*property, and path separation at intermediate times is uniformly greater than that of standard independent\-coupling flow matching, alleviating local velocity ambiguity\.
- 4\)We conduct systematic experiments on low\-dimensional synthetic data, unconditional generation, discrete class\-conditional generation, continuous text\-conditional generation, and large\-scale image synthesis\. Results show that QAT\-FM achieves competitive generation quality with substantially lower coupling construction complexity than OT\-based methods\.

## 2Preliminaries

#### Notation\.

Let𝒫⁡\(ℝd\)\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\)denote the set of Borel probability measures onℝd\\mathbb\{R\}^\{d\}\. Forα∈𝒫⁡\(ℝd\)\\alpha\\in\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\)andβ∈𝒫⁡\(ℝd′\)\\beta\\in\\mathcal\{P\}\(\\mathbb\{R\}^\{d^\{\\prime\}\}\), a joint distributionπ∈𝒫⁡\(ℝd×ℝd′\)\\pi\\in\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d^\{\\prime\}\}\)is called a*coupling*ofα\\alphaandβ\\betaif its marginals areα\\alphaandβ\\beta; we writeΠ⁡\(α,β\)\\Pi\(\\alpha,\\beta\)for the collection of all such couplings\. For a vector𝒙∈ℝd\\bm\{x\}\\in\\mathbb\{R\}^\{d\}, let𝒙\(j\)\\bm\{x\}^\{\(j\)\}denote itsjj\-th coordinate\. Forρ∈𝒫⁡\(ℝd\)\\rho\\in\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\), letρ\(j\)\\rho^\{\(j\)\}denote itsjj\-th marginal distribution,Fρ\(j\)F\_\{\\rho\}^\{\(j\)\}the cumulative distribution function \(CDF\) ofρ\(j\)\\rho^\{\(j\)\}, and\(Fρ\(j\)\)−1\\bigl\(F\_\{\\rho\}^\{\(j\)\}\\bigr\)^\{\-1\}its generalized inverse\. For a measurable mapT:ℝd→ℝkT:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{k\}, we writeT\#​ρT\_\{\\\#\}\\rhofor the pushforward ofρ\\rhounderTT\.

#### Flow\-based Generative Modeling\.

Given a time\-dependent velocity fieldv:\[0,1\]×ℝd→ℝdv:\[0,1\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}, writevt​\(x\):=v​\(t,x\)v\_\{t\}\(x\):=v\(t,x\)\. The*flow map*ϕt:ℝd→ℝd\\phi\_\{t\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}induced byvtv\_\{t\}is defined as the solution to the ODE

dd​t​ϕt​\(x\)=vt​\(ϕt​\(x\)\),ϕ0​\(x\)=x\.\\frac\{d\}\{dt\}\\phi\_\{t\}\(x\)=v\_\{t\}\(\\phi\_\{t\}\(x\)\),\\qquad\\phi\_\{0\}\(x\)=x\.Settingρt=\(ϕt\)\#​γ\\rho\_\{t\}=\(\\phi\_\{t\}\)\_\{\\\#\}\\gamma, whereγ\\gammadenotes the source distribution, the pair\(ρt,vt\)\(\\rho\_\{t\},v\_\{t\}\)satisfies the continuity equation

∂tρt\+∇⋅\(ρt​vt\)=0,\\partial\_\{t\}\\rho\_\{t\}\+\\nabla\\cdot\(\\rho\_\{t\}v\_\{t\}\)=0,\(1\)withρ0=γ\\rho\_\{0\}=\\gamma\. When\(ϕ1\)\#​γ=μ\(\\phi\_\{1\}\)\_\{\\\#\}\\gamma=\\mu, the flow transports the sourceγ\\gammato the target distributionμ\\mu\. The goal of flow\-based generative modeling is therefore to learn a parameterized velocity fieldvθv\_\{\\theta\}whose induced flow map satisfies\(ϕ1\)\#​γ=μ\(\\phi\_\{1\}\)\_\{\\\#\}\\gamma=\\mu\.

Flow Matching\([Lipman et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib1)\)provides a tractable training procedure for this velocity field\. It specifies a probability path\(ρt\)t∈\[0,1\]\(\\rho\_\{t\}\)\_\{t\\in\[0,1\]\}connectingγ\\gammatoμ\\mutogether with a compatible target velocity fieldutu\_\{t\}\. The parameterized velocity field is then learned via the regression objective

ℒFM​\(θ\)=𝔼t∼Unif⁡\[0,1\]xt∼ρt​‖vθ​\(t,xt\)−ut​\(xt\)‖2\.\\mathcal\{L\}\_\{\\mathrm\{FM\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\mathrm\{Unif\}\[0,1\]\\\\ x\_\{t\}\\sim\\rho\_\{t\}\\end\{subarray\}\}\\left\\\|v\_\{\\theta\}\(t,x\_\{t\}\)\-u\_\{t\}\(x\_\{t\}\)\\right\\\|^\{2\}\.\(2\)This formulation avoids backpropagating through numerical ODE solvers during training\.

#### Conditional Flow Matching\.

In practice, the marginal velocity fieldutu\_\{t\}is often intractable to construct directly\. Conditional Flow Matching111Here, “conditional” refers to conditioning on paired endpoints\. To avoid ambiguity with conditional generation, we refer to this formulation simply as Flow Matching \(FM\) throughout the remainder of the paper unless otherwise specified\.\(CFM\)\([Lipman et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib1);[Tong et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib6)\)addresses this by conditioning on paired endpoints: givenπ∈Π⁡\(γ,μ\)\\pi\\in\\Pi\(\\gamma,\\mu\), draw\(x0,x1\)∼π\(x\_\{0\},x\_\{1\}\)\\sim\\pi, specify a conditional pathpt\(⋅∣x0,x1\)p\_\{t\}\(\\cdot\\mid x\_\{0\},x\_\{1\}\)and its compatible velocityut\(⋅∣x0,x1\)u\_\{t\}\(\\cdot\\mid x\_\{0\},x\_\{1\}\), and trainvθ​\(t,xt\)v\_\{\\theta\}\(t,x\_\{t\}\)to matchut​\(xt∣x0,x1\)u\_\{t\}\(x\_\{t\}\\mid x\_\{0\},x\_\{1\}\), wherext∼pt\(⋅∣x0,x1\)x\_\{t\}\\sim p\_\{t\}\(\\cdot\\mid x\_\{0\},x\_\{1\}\)\. A canonical and widely used choice is the linear interpolation path

xt=\(1−t\)​x0\+t​x1,ut​\(xt∣x0,x1\)=x1−x0,x\_\{t\}=\(1\-t\)x\_\{0\}\+tx\_\{1\},\\qquad u\_\{t\}\(x\_\{t\}\\mid x\_\{0\},x\_\{1\}\)=x\_\{1\}\-x\_\{0\},under which the training objective reduces to

ℒCFM​\(θ\)=𝔼t∼Unif⁡\[0,1\]\(x0,x1\)∼π​‖vθ​\(t,\(1−t\)​x0\+t​x1\)−\(x1−x0\)‖2\.\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\mathrm\{Unif\}\[0,1\]\\\\ \(x\_\{0\},x\_\{1\}\)\\sim\\pi\\end\{subarray\}\}\\left\\\|v\_\{\\theta\}\\\!\\left\(t,\(1\-t\)x\_\{0\}\+tx\_\{1\}\\right\)\-\(x\_\{1\}\-x\_\{0\}\)\\right\\\|^\{2\}\.\(3\)This reveals that the endpoint pairing is governed by the couplingπ∈Π⁡\(γ,μ\)\\pi\\in\\Pi\(\\gamma,\\mu\)\. The simplest choice is the independent couplingπ=γ⊗μ\\pi=\\gamma\\otimes\\mu\. A more principled alternative is the*OT coupling*

πOT∈arg⁡min⁡∫ℝd×ℝdπ∈Π⁡\(γ,μ\)⁡‖x0−x1‖2​𝑑π​\(x0,x1\),\\pi\_\{\\mathrm\{OT\}\}\\in\\arg\\min\_\{\\pi\\in\\Pi\(\\gamma,\\mu\)\}\\int\_\{\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\}\\\|x\_\{0\}\-x\_\{1\}\\\|^\{2\}\\,d\\pi\(x\_\{0\},x\_\{1\}\),\(4\)which under the linear path tends to produce shorter, geometrically natural paired trajectories\([Pooladian et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib5);[Tong et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib6)\)\. However, computing the exact global OT coupling is expensive at large scale; in practice, mini\-batch OT is used as an approximation, reducing computational cost at the expense of coupling exactness and cross\-batch consistency\. For the special caseγ=𝒩⁡\(0,Id\)\\gamma=\\mathcal\{N\}\(0,I\_\{d\}\),[Mousavi\-Hosseini et al\. \(2026\)](https://arxiv.org/html/2609.03443#bib.bib7)proposes solving a semi\-discrete OT problem in \([4](https://arxiv.org/html/2609.03443#S2.E4)\) to better align the continuous Gaussian source with the empirical target distribution\.

#### Data Partitioning by Trees\.

The preceding discussion shows that the choice of couplingπ\\pidirectly governs the geometry of particle trajectories in FM, while exact global OT coupling is prohibitively expensive at large scale\([Zhang et al\., 2025](https://arxiv.org/html/2609.03443#bib.bib21);[Mousavi\-Hosseini et al\., 2026](https://arxiv.org/html/2609.03443#bib.bib7)\)\. This motivates binary tree\-based space partitioning, which provides a simple way to organize data hierarchically and will serve as the foundation of our coupling construction\. Given a data space𝒳⊆ℝd\\mathcal\{X\}\\subseteq\\mathbb\{R\}^\{d\}and a dataset𝒟=\{xi\}i=1N⊆𝒳\\mathcal\{D\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}\\subseteq\\mathcal\{X\}, letμ^𝒟=N−1​∑i=1Nδxi\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}=N^\{\-1\}\\sum\_\{i=1\}^\{N\}\\delta\_\{x\_\{i\}\}denote its empirical distribution\. A*binary partition tree*TTconsists of an internal node setℐ\\mathcal\{I\}and a leaf node setℒ\\mathcal\{L\}\. Each nodeη\\etais associated with a regionRη⊆𝒳R\_\{\\eta\}\\subseteq\\mathcal\{X\}and data subset𝒟η=𝒟∩Rη\\mathcal\{D\}\_\{\\eta\}=\\mathcal\{D\}\\cap R\_\{\\eta\}, where the rootrrsatisfiesRr=𝒳R\_\{r\}=\\mathcal\{X\}\.

For each internal nodeη∈ℐ\\eta\\in\\mathcal\{I\}, letμ^η=\|𝒟η\|−1​∑x∈𝒟ηδx\\widehat\{\\mu\}\_\{\\eta\}=\|\\mathcal\{D\}\_\{\\eta\}\|^\{\-1\}\\sum\_\{x\\in\\mathcal\{D\}\_\{\\eta\}\}\\delta\_\{x\}denote the conditional empirical measure on𝒟η\\mathcal\{D\}\_\{\\eta\}\. A*split strategy*selects a split coordinateκη∈\[d\]\\kappa\_\{\\eta\}\\in\[d\]and thresholdsη∈ℝs\_\{\\eta\}\\in\\mathbb\{R\}based onμ^η\\widehat\{\\mu\}\_\{\\eta\}, partitioningRηR\_\{\\eta\}into

Rη−=\{x∈Rη:x\(κη\)≤sη\},Rη\+=\{x∈Rη:x\(κη\)\>sη\},R\_\{\\eta\}^\{\-\}=\\\{x\\in R\_\{\\eta\}:x^\{\(\\kappa\_\{\\eta\}\)\}\\leq s\_\{\\eta\}\\\},\\qquad R\_\{\\eta\}^\{\+\}=\\\{x\\in R\_\{\\eta\}:x^\{\(\\kappa\_\{\\eta\}\)\}\>s\_\{\\eta\}\\\},and correspondingly partitioning𝒟η\\mathcal\{D\}\_\{\\eta\}into𝒟η−=𝒟η∩Rη−\\mathcal\{D\}\_\{\\eta\}^\{\-\}=\\mathcal\{D\}\_\{\\eta\}\\cap R\_\{\\eta\}^\{\-\}and𝒟η\+=𝒟η∩Rη\+\\mathcal\{D\}\_\{\\eta\}^\{\+\}=\\mathcal\{D\}\_\{\\eta\}\\cap R\_\{\\eta\}^\{\+\}\. After recursive splitting, the leaf regions form a partition of𝒳\\mathcal\{X\}, denotedℛ⁡\(T\):=\{Rℓ:ℓ∈ℒ\}\\mathcal\{R\}\(T\):=\\\{R\_\{\\ell\}:\\ell\\in\\mathcal\{L\}\\\}\.

## 3Quantile AlignTree Flow Matching

### 3\.1Quantile AlignTree Coupling

The preceding analysis shows that in the linear FM objective, the couplingπ\\pigoverns how source samples are paired with target samples, directly shaping the geometry of flow paths\. The key idea of QAT is to build a binary partition tree over the empirical data distribution and transfer its topology to the source distribution by matching the split mass ratios at each internal node\. Figure[1](https://arxiv.org/html/2609.03443#S3.F1)summarizes the resulting coupling and sampling procedure\.

Letμ^𝒟\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}denote the empirical measure of dataset𝒟=\{xi\}i=1N⊆ℝd\\mathcal\{D\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}\\subseteq\\mathbb\{R\}^\{d\}, and letγ∈𝒫⁡\(ℝd\)\\gamma\\in\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\)be the source distribution, e\.g\.,γ=𝒩⁡\(0,Id\)\\gamma=\\mathcal\{N\}\(0,I\_\{d\}\)\. Following the binary partition tree construction of Section[2](https://arxiv.org/html/2609.03443#S2), we build a data treeT⁡\(μ^𝒟\)T\(\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\)onμ^𝒟\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\. For each internal nodeη∈ℐ\\eta\\in\\mathcal\{I\}, let\(κη,sη\)\(\\kappa\_\{\\eta\},s\_\{\\eta\}\)denote its split coordinate and threshold, and letμ^η\\widehat\{\\mu\}\_\{\\eta\}be the conditional empirical measure at that node\. Based on the data tree, we define the Quantile AlignTree as follows\.

###### Definition 1\(Quantile AlignTree \(QAT\)\)

Given the data treeT⁡\(μ^𝒟\)T\(\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\)and source distributionγ\\gamma, the*Quantile AlignTree*ofγ\\gammawith respect to the data tree, denotedT~γ​\(T\)\\widetilde\{T\}\_\{\\gamma\}\(T\), is constructed recursively with the same topology asT⁡\(μ^𝒟\)T\(\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\)\. At each internal nodeη\\eta, the split coordinateκη\\kappa\_\{\\eta\}is inherited from the data tree, but the data\-side thresholdsηs\_\{\\eta\}is replaced by the source\-side aligned threshold

s~η=\(Fγη\(κη\)\)−1​\(Fμ^η\(κη\)​\(sη\)\),\\widetilde\{s\}\_\{\\eta\}=\\left\(F\_\{\\gamma\_\{\\eta\}\}^\{\(\\kappa\_\{\\eta\}\)\}\\right\)^\{\-1\}\\\!\\left\(F\_\{\\widehat\{\\mu\}\_\{\\eta\}\}^\{\(\\kappa\_\{\\eta\}\)\}\(s\_\{\\eta\}\)\\right\),\(5\)whereγη\\gamma\_\{\\eta\}denotes the conditional distribution ofγ\\gammaon the current source\-side regionR~η\\widetilde\{R\}\_\{\\eta\}\. The source\-side region is then partitioned as

R~η−=\{z∈R~η:z\(κη\)≤s~η\},R~η\+=\{z∈R~η:z\(κη\)\>s~η\},\\widetilde\{R\}\_\{\\eta\}^\{\-\}=\\\{z\\in\\widetilde\{R\}\_\{\\eta\}:z^\{\(\\kappa\_\{\\eta\}\)\}\\leq\\widetilde\{s\}\_\{\\eta\}\\\},\\qquad\\widetilde\{R\}\_\{\\eta\}^\{\+\}=\\\{z\\in\\widetilde\{R\}\_\{\\eta\}:z^\{\(\\kappa\_\{\\eta\}\)\}\>\\widetilde\{s\}\_\{\\eta\}\\\},and the left and right child nodes are constructed recursively\.

The quantile\-based aligned threshold \([5](https://arxiv.org/html/2609.03443#S3.E5)\) guarantees that the source side and the data side carry equal mass ratios at every split\. Specifically, when the source\-side conditional marginal is continuous,

γη​\(R~η−\)=μ^η​\(Rη−\),γη​\(R~η\+\)=μ^η​\(Rη\+\)\.\\gamma\_\{\\eta\}\(\\widetilde\{R\}\_\{\\eta\}^\{\-\}\)=\\widehat\{\\mu\}\_\{\\eta\}\(R\_\{\\eta\}^\{\-\}\),\\qquad\\gamma\_\{\\eta\}\(\\widetilde\{R\}\_\{\\eta\}^\{\+\}\)=\\widehat\{\\mu\}\_\{\\eta\}\(R\_\{\\eta\}^\{\+\}\)\.Consequently, the data tree and the aligned source tree carry equal mass at corresponding leaf nodes, naturally inducing a coupling fromγ\\gammatoμ^𝒟\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\.

![Refer to caption](https://arxiv.org/html/2609.03443v1/pipeline.png)Figure 1:Overview of the QAT coupling and sampling procedure\. \(*Left*\) QAT builds a binary partition tree over the data using feature\-based or condition\-aware splits, and transfers the same tree topology to the Gaussian source by quantile alignment, yielding the structured couplingπQAT\\pi\_\{\\mathrm\{QAT\}\}\. \(*Right*\) each data leaf is paired with an aligned Gaussian source region\. Since these source regions are axis\-aligned hyperrectangles, sampling from the corresponding QAT source conditional reduces to parallel per\-coordinate truncated normal sampling\.###### Definition 2\(QAT Coupling\)

Letℒ\\mathcal\{L\}be the nonempty leaf node set of data treeT⁡\(μ^𝒟\)T\(\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\)\. For each leafℓ∈ℒ\\ell\\in\\mathcal\{L\}, let𝒟ℓ=𝒟∩Rℓ\\mathcal\{D\}\_\{\\ell\}=\\mathcal\{D\}\\cap R\_\{\\ell\}andαℓ:=1N​\|𝒟ℓ\|\\alpha\_\{\\ell\}:=\\tfrac\{1\}\{N\}\|\\mathcal\{D\}\_\{\\ell\}\|\. Letμ^ℓ=\|𝒟ℓ\|−1​∑xi∈𝒟ℓδxi\\widehat\{\\mu\}\_\{\\ell\}=\|\\mathcal\{D\}\_\{\\ell\}\|^\{\-1\}\\sum\_\{x\_\{i\}\\in\\mathcal\{D\}\_\{\\ell\}\}\\delta\_\{x\_\{i\}\}denote the conditional empirical measure on𝒟ℓ\\mathcal\{D\}\_\{\\ell\}, and letγℓ\\gamma\_\{\\ell\}denote the conditional source measure associated with leafℓ\\ellin the Quantile AlignTree\. Define the*QAT coupling*as

πQAT:=∑ℓ∈ℒαℓ​γℓ⊗μ^ℓ\.\\pi\_\{\\mathrm\{QAT\}\}:=\\sum\_\{\\ell\\in\\mathcal\{L\}\}\\alpha\_\{\\ell\}\\,\\gamma\_\{\\ell\}\\otimes\\widehat\{\\mu\}\_\{\\ell\}\.\(6\)ThenπQAT∈Π⁡\(γ,μ^𝒟\)\\pi\_\{\\mathrm\{QAT\}\}\\in\\Pi\(\\gamma,\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\)\.

Intuitively, the QAT coupling matches each data leaf𝒟ℓ\\mathcal\{D\}\_\{\\ell\}with an aligned source region\. To draw a training pair\(x0,x1\)∼πQAT\(x\_\{0\},x\_\{1\}\)\\sim\\pi\_\{\\mathrm\{QAT\}\}, one samples a leafℓ\\ellwith probabilityαℓ\\alpha\_\{\\ell\}, drawsx0∼γℓx\_\{0\}\\sim\\gamma\_\{\\ell\}from the corresponding conditional source distribution, and drawsx1∼μ^ℓx\_\{1\}\\sim\\widehat\{\\mu\}\_\{\\ell\}from the conditional empirical distribution on that leaf\.

### 3\.2QAT Flow Matching

Substituting the QAT coupling into the linear FM objective \([3](https://arxiv.org/html/2609.03443#S2.E3)\) yields the QAT\-FM training objective\. Letμ^𝒟\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}be the empirical data distribution,γ\\gammathe source distribution, andπQAT∈Π⁡\(γ,μ^𝒟\)\\pi\_\{\\mathrm\{QAT\}\}\\in\\Pi\(\\gamma,\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\)the QAT coupling of Definition[2](https://arxiv.org/html/2609.03443#Thmdefinition2)\. QAT\-FM learns the velocity field via

ℒQAT​\-​FM​\(θ\):=𝔼t∼Unif⁡\[0,1\]\(x0,x1\)∼πQAT​‖vθ​\(t,\(1−t\)​x0\+t​x1\)−\(x1−x0\)‖2\.\\mathcal\{L\}\_\{\\mathrm\{QAT\\text\{\-\}FM\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\mathrm\{Unif\}\[0,1\]\\\\ \(x\_\{0\},x\_\{1\}\)\\sim\\pi\_\{\\mathrm\{QAT\}\}\\end\{subarray\}\}\\left\\\|v\_\{\\theta\}\\bigl\(t,\(1\-t\)x\_\{0\}\+tx\_\{1\}\\bigr\)\-\(x\_\{1\}\-x\_\{0\}\)\\right\\\|^\{2\}\.\(7\)Compared to standard FM, QAT\-FM differs only in the sample pairing:πQAT\\pi\_\{\\mathrm\{QAT\}\}replaces the independent coupling, while the linear interpolation path and velocity regression target remain unchanged\. This structured pairing yields the following geometric properties\.

###### Theorem 1\(Non\-Crossing Property\)

LetπQAT∈Π⁡\(γ,μ^𝒟\)\\pi\_\{\\mathrm\{QAT\}\}\\in\\Pi\(\\gamma,\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\)be the QAT coupling\. Consider two paired samples\(x0,x1\),\(x0′,x1′\)∈supp⁡\(πQAT\)\(x\_\{0\},x\_\{1\}\),\(x\_\{0\}^\{\\prime\},x\_\{1\}^\{\\prime\}\)\\in\\mathrm\{supp\}\(\\pi\_\{\\mathrm\{QAT\}\}\)\. Ifx1x\_\{1\}andx1′x\_\{1\}^\{\\prime\}belong to two distinct leaf nodesℓ≠ℓ′\\ell\\neq\\ell^\{\\prime\}of the data tree, then the corresponding linear interpolation paths

xt=\(1−t\)​x0\+t​x1,xt′=\(1−t\)​x0′\+t​x1′x\_\{t\}=\(1\-t\)x\_\{0\}\+tx\_\{1\},\\qquad x\_\{t\}^\{\\prime\}=\(1\-t\)x\_\{0\}^\{\\prime\}\+tx\_\{1\}^\{\\prime\}do not intersect, i\.e\.,xt≠xt′x\_\{t\}\\neq x\_\{t\}^\{\\prime\}for allt∈\[0,1\]t\\in\[0,1\], almost surely\.

Theorem[1](https://arxiv.org/html/2609.03443#Thmtheorem1)establishes that paths associated with distinct target leaves are strictly non\-crossing\. This property alleviates the velocity ambiguity that can arise from overlapping training paths: since the FM objective fitsvθv\_\{\\theta\}pointwise at eacht∈\[0,1\]t\\in\[0,1\], two training paths satisfyingxt=xt′x\_\{t\}=x\_\{t\}^\{\\prime\}butx1−x0≠x1′−x0′x\_\{1\}\-x\_\{0\}\\neq x\_\{1\}^\{\\prime\}\-x\_\{0\}^\{\\prime\}impose conflicting supervision signals at the same spatial location, increasing the difficulty of fittingvθv\_\{\\theta\}\. Paths that remain well\-separated at intermediate times are therefore desirable\. Beyond preventing cross\-leaf path crossings, the QAT coupling also consistently improves the expected inter\-path distance, as the following theorem quantifies\.

###### Theorem 2\(Improved Path Separation\)

Letγ=𝒩⁡\(0,Id\)\\gamma=\\mathcal\{N\}\(0,I\_\{d\}\), letπQAT∈Π⁡\(γ,μ^𝒟\)\\pi\_\{\\mathrm\{QAT\}\}\\in\\Pi\(\\gamma,\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\)be the QAT coupling, and letπind:=γ⊗μ^𝒟\\pi\_\{\\mathrm\{ind\}\}:=\\gamma\\otimes\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}be the independent coupling\. For any couplingπ\\pi, let𝔼π⊗2\\mathbb\{E\}\_\{\\pi^\{\\otimes 2\}\}denote expectation over two independent pairs\(x0,x1\),\(x0′,x1′\)∼π\(x\_\{0\},x\_\{1\}\),\(x\_\{0\}^\{\\prime\},x\_\{1\}^\{\\prime\}\)\\sim\\pi, and definext=\(1−t\)​x0\+t​x1x\_\{t\}=\(1\-t\)x\_\{0\}\+tx\_\{1\}andxt′=\(1−t\)​x0′\+t​x1′x\_\{t\}^\{\\prime\}=\(1\-t\)x\_\{0\}^\{\\prime\}\+tx\_\{1\}^\{\\prime\}\. Then for everyt∈\(0,1\)t\\in\(0,1\),

𝔼πQAT⊗2​\[‖xt−xt′‖22\]−𝔼πind⊗2​\[‖xt−xt′‖22\]=4​t​\(1−t\)​ΔQAT,\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{QAT\}\}^\{\\otimes 2\}\}\\\!\\left\[\\\|x\_\{t\}\-x\_\{t\}^\{\\prime\}\\\|\_\{2\}^\{2\}\\right\]\-\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{ind\}\}^\{\\otimes 2\}\}\\\!\\left\[\\\|x\_\{t\}\-x\_\{t\}^\{\\prime\}\\\|\_\{2\}^\{2\}\\right\]=4t\(1\-t\)\\,\\Delta\_\{\\mathrm\{QAT\}\},whereΔQAT\\Delta\_\{\\mathrm\{QAT\}\}is the node\-level path\-separation gain induced by the QAT tree:

ΔQAT:=∑η∈ℐαη​qη​\(1−qη\)​\(𝔼γη\+​\[X\(κη\)\]−𝔼γη−​\[X\(κη\)\]\)​\(𝔼μ^η\+​\[X\(κη\)\]−𝔼μ^η−​\[X\(κη\)\]\)\.\\Delta\_\{\\mathrm\{QAT\}\}:=\\sum\_\{\\eta\\in\\mathcal\{I\}\}\\alpha\_\{\\eta\}q\_\{\\eta\}\(1\-q\_\{\\eta\}\)\\Bigl\(\\mathbb\{E\}\_\{\\gamma\_\{\\eta^\{\+\}\}\}\\\!\\bigl\[X^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\-\\mathbb\{E\}\_\{\\gamma\_\{\\eta^\{\-\}\}\}\\\!\\bigl\[X^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\\Bigr\)\\Bigl\(\\mathbb\{E\}\_\{\\widehat\{\\mu\}\_\{\\eta^\{\+\}\}\}\\\!\\bigl\[X^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\-\\mathbb\{E\}\_\{\\widehat\{\\mu\}\_\{\\eta^\{\-\}\}\}\\\!\\bigl\[X^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\\Bigr\)\.\(8\)Hereαη:=μ^𝒟​\(Rη\)\\alpha\_\{\\eta\}:=\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\(R\_\{\\eta\}\)is the data mass at nodeη\\eta,qη:=Fμ^η\(κη\)​\(sη\)q\_\{\\eta\}:=F\_\{\\widehat\{\\mu\}\_\{\\eta\}\}^\{\(\\kappa\_\{\\eta\}\)\}\(s\_\{\\eta\}\)is the conditional left\-child mass fraction at split coordinateκη\\kappa\_\{\\eta\}, andγη±\\gamma\_\{\\eta^\{\\pm\}\},μ^η±\\widehat\{\\mu\}\_\{\\eta^\{\\pm\}\}denote the conditional source and empirical distributions on the left and right child regions ofη\\eta\. Under non\-degenerate splits,ΔQAT\>0\\Delta\_\{\\mathrm\{QAT\}\}\>0, so the QAT coupling strictly dominates the independent coupling in expected inter\-path separation\.

### 3\.3Conditional QAT Strategy

In conditional generation tasks, data samples carry conditioning variables\. Let\{\(xi,ci\)\}i=1N⊆ℝd×ℝm\\\{\(x\_\{i\},c\_\{i\}\)\\\}\_\{i=1\}^\{N\}\\subseteq\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{m\}be the joint dataset,μ^𝒟C:=N−1​∑i=1Nδ\(xi,ci\)\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{C\}\}:=N^\{\-1\}\\sum\_\{i=1\}^\{N\}\\delta\_\{\(x\_\{i\},c\_\{i\}\)\}the joint empirical distribution, andμ^C:=N−1​∑i=1Nδci\\widehat\{\\mu\}\_\{C\}:=N^\{\-1\}\\sum\_\{i=1\}^\{N\}\\delta\_\{c\_\{i\}\}the marginal over condition space\. A standard OT coupling based only on data\-space distances can ignore the alignment between samples and their conditioning variables\. Alternatively, solving OT directly between the joint sourceγ⊗μ^C\\gamma\\otimes\\widehat\{\\mu\}\_\{C\}and the joint empirical distributionμ^𝒟C\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{C\}\}increases the optimization dimension and computational cost\([Cheng and Schwing, 2025](https://arxiv.org/html/2609.03443#bib.bib20);[Mousavi\-Hosseini et al\., 2026](https://arxiv.org/html/2609.03443#bib.bib7)\)\. We therefore propose the following Conditional QAT strategy\.

###### Definition 3\(Conditional QAT\)

Let\{ci\}i=1N⊆ℝm\\\{c\_\{i\}\\\}\_\{i=1\}^\{N\}\\subseteq\\mathbb\{R\}^\{m\}be conditioning variables\. Conditional QAT constructs a data tree on the joint samples\{\(xi,ci\)\}i=1N⊆ℝd\+m\\\{\(x\_\{i\},c\_\{i\}\)\\\}\_\{i=1\}^\{N\}\\subseteq\\mathbb\{R\}^\{d\+m\}\. For each internal nodeη\\eta, letκη∈\[d\+m\]\\kappa\_\{\\eta\}\\in\[d\+m\]denote its split coordinate\. When building the source\-side AlignTree, ifκη≤d\\kappa\_\{\\eta\}\\leq d\(a data coordinate\), the source side is partitioned using the standard QAT aligned threshold in \([5](https://arxiv.org/html/2609.03443#S3.E5)\); ifκη\>d\\kappa\_\{\\eta\}\>d\(a condition coordinate\), no split is applied to the source and both child nodes inherit the parent source region, the split is used only to refine the target\-side grouping:R~η−=R~η\+=R~η\\widetilde\{R\}\_\{\\eta^\{\-\}\}=\\widetilde\{R\}\_\{\\eta^\{\+\}\}=\\widetilde\{R\}\_\{\\eta\}\.

The Conditional QAT coupling takes the same form as Definition[2](https://arxiv.org/html/2609.03443#Thmdefinition2), with the target empirical distribution replaced by the joint distributionμ^𝒟C\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{C\}\}:

πCQAT=∑ℓ∈ℒCαℓ​γℓ⊗μ^𝒟C,ℓ∈Π⁡\(γ,μ^𝒟C\),\\pi\_\{\\mathrm\{CQAT\}\}=\\sum\_\{\\ell\\in\\mathcal\{L\}\_\{C\}\}\\alpha\_\{\\ell\}\\,\\gamma\_\{\\ell\}\\otimes\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{C\},\\ell\}\\in\\Pi\(\\gamma,\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{C\}\}\),whereℒC\\mathcal\{L\}\_\{C\}is the set of nonempty leaves of the joint data tree, andμ^𝒟C,ℓ\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{C\},\\ell\}is the conditional joint empirical distribution at leafℓ\\ell\. A sample fromπCQAT\\pi\_\{\\mathrm\{CQAT\}\}can be written as\(x0,\(x1,c\)\)\(x\_\{0\},\(x\_\{1\},c\)\), where the linear FM path is applied only in the data space betweenx0x\_\{0\}andx1x\_\{1\}, whileccis provided as conditioning input to the velocity model\. By encoding conditional structure through leaf\-level matching in the joint tree, this construction avoids explicitly solving the product\-space OT problem betweenγ⊗μ^C\\gamma\\otimes\\widehat\{\\mu\}\_\{C\}andμ^𝒟C\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{C\}\}, while retaining the efficient sampling form of unconditional QAT\.

### 3\.4Efficient QAT Construction and Sampling

Whenγ=𝒩⁡\(0,Id\)\\gamma=\\mathcal\{N\}\(0,I\_\{d\}\), each source\-side leaf regionR~ℓ\\widetilde\{R\}\_\{\\ell\}is fully characterized by its coordinate\-wise bounds\(aℓ,bℓ\)\(a\_\{\\ell\},b\_\{\\ell\}\)\(Remark[1](https://arxiv.org/html/2609.03443#Thmremark1)\)\. Regardless of whether the data tree is built from unconditional or conditional samples, each leafℓ\\ellstores only the data index set𝒮ℓ\\mathcal\{S\}\_\{\\ell\}and the source\-side bounds\(aℓ,bℓ\)\(a\_\{\\ell\},b\_\{\\ell\}\), which fully specify the leaf\-wise product couplingγℓ⊗μ^ℓ\\gamma\_\{\\ell\}\\otimes\\widehat\{\\mu\}\_\{\\ell\}\. Training pairs are drawn by sampling a leafℓ\\ellproportionally to its mass, then independently drawing a source point from the corresponding truncated Gaussian region and a target point from𝒮ℓ\\mathcal\{S\}\_\{\\ell\}\. The full construction, sampling algorithm, and complexity analysis are given in Appendix[B](https://arxiv.org/html/2609.03443#A2)\.

#### Split Strategy\.

QAT is compatible with a variety of split strategies\. Throughout this work we adopt a maximum\-variance rule: for each nodeη\\eta, the split coordinate is the dimension of highest empirical variance and the split threshold is the empirical mean along that coordinate,

κη=arg⁡maxj​Varμ^η⁡\(Y\(j\)\),sη=𝔼μ^η​\[Y\(κη\)\],\\kappa\_\{\\eta\}=\\arg\\max\_\{j\}\\operatorname\{Var\}\_\{\\widehat\{\\mu\}\_\{\\eta\}\}\\\!\\big\(Y^\{\(j\)\}\\big\),\\qquad s\_\{\\eta\}=\\mathbb\{E\}\_\{\\widehat\{\\mu\}\_\{\\eta\}\}\\\!\\big\[Y^\{\(\\kappa\_\{\\eta\}\)\}\\big\],whereY=XY=Xin the unconditional case andY=\(X,C\)Y=\(X,C\)in the conditional case\. For discrete empirical data, we recurse until each leaf contains at most one sample; in the conditional setting, the terminal splitting phase prioritizes data coordinates over condition coordinates, maximizing the number of source\-side splits and thereby increasing the path\-separation gain of Theorem[2](https://arxiv.org/html/2609.03443#Thmtheorem2)\.

#### High\-Dimensional Data Handling\.

For high\-dimensional data, axis\-aligned partitioning in the raw data space may fail to capture the principal directions of variation\. We therefore first rotate the data via an orthogonal transformP∈ℝd×dP\\in\\mathbb\{R\}^\{d\\times d\}, settingzi=P​xiz\_\{i\}=Px\_\{i\}and constructing the QAT coupling on the top\-kksubspacezi\(1:k\)z\_\{i\}^\{\(1:k\)\}\. By the rotational invariance of the standard Gaussian,P\#​𝒩​\(0,Id\)=𝒩⁡\(0,Id\)P\_\{\\\#\}\\mathcal\{N\}\(0,I\_\{d\}\)=\\mathcal\{N\}\(0,I\_\{d\}\)for any orthogonalPP\. Training then proceeds by sampling\(ϵ\(1:k\),zi\(1:k\)\)∼πQAT\(\\epsilon^\{\(1:k\)\},z\_\{i\}^\{\(1:k\)\}\)\\sim\\pi\_\{\\mathrm\{QAT\}\}, independently drawingϵ\(−k\)∼𝒩⁡\(0,Id−k\)\\epsilon^\{\(\-k\)\}\\sim\\mathcal\{N\}\(0,I\_\{d\-k\}\), and setting

x0=P⊤\(ϵ\(1:k\),ϵ\(−k\)\)\.x\_\{0\}=P^\{\\top\}\\\!\\big\(\\epsilon^\{\(1:k\)\},\\epsilon^\{\(\-k\)\}\\big\)\.The marginal distribution ofx0x\_\{0\}remains𝒩⁡\(0,Id\)\\mathcal\{N\}\(0,I\_\{d\}\), while the QAT tree operates entirely within the low\-dimensional subspace\. In practice,PPis composed of a Patch\-Hadamard transform and a PCA rotation at the mean\-pooling scale, concentrating the principal data variation into a small number of coordinates\. Details are given in Appendix[B\.3](https://arxiv.org/html/2609.03443#A2.SS3)\.

## 4Experiments

We evaluate QAT\-FM through a series of experiments: we begin with path visualizations on two\-dimensional synthetic data, then proceed to high\-dimensional image generation benchmarks covering unconditional generation \(CIFAR\-10\), class\-conditional generation \(ImageNet\-32\), text\-conditional generation \(CelebA\-64\), and large\-scale latent\-space generation \(ImageNet\-256\)\. Additional implementation details are provided in Appendix[C](https://arxiv.org/html/2609.03443#A3); extended experimental results appear in Appendix[D](https://arxiv.org/html/2609.03443#A4)\.

### 4\.12D Synthetic Experiments

![Refer to caption](https://arxiv.org/html/2609.03443v1/2d_results.png)Figure 2:2D checkerboard results\.\(*Left*\) sample evolution induced by different coupling methods fromt=0t=0tot=1t=1\. \(*Right*\) generated samples with varying numbers of function evaluations \(NFE\)\.Following[Lipman et al\. \(2023\)](https://arxiv.org/html/2609.03443#bib.bib1);[Pooladian et al\. \(2023\)](https://arxiv.org/html/2609.03443#bib.bib5), we compare different coupling methods on two\-dimensional checkerboard data\. Four methods are evaluated: I\-FM, OT\-FM\([Tong et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib6)\), SD\-FM\([Mousavi\-Hosseini et al\., 2026](https://arxiv.org/html/2609.03443#bib.bib7)\), and the proposed QAT\-FM\. All methods share the same network architecture and training configuration\.

Figure[2](https://arxiv.org/html/2609.03443#S4.F2)\(left\) shows sample evolution fromt=0t=0tot=1t=1\. OT\-FM and SD\-FM capture the global structure of the target distribution earlier than I\-FM, while QAT\-FM yields better local separation and more organized intermediate transport\. Figure[2](https://arxiv.org/html/2609.03443#S4.F2)\(right\) shows generated samples with varying NFE\. QAT\-FM recovers a clear checkerboard structure at small NFE, indicating that its structured coupling improves few\-step sample quality\.

### 4\.2Unconditional Generation on CIFAR\-10

We evaluate QAT\-FM on unconditional generation using CIFAR\-10\([Krizhevsky et al\., 2009](https://arxiv.org/html/2609.03443#bib.bib9)\), which contains50,00050\{,\}000RGB training images of size32×3232\\times 32across1010categories\. The velocity\-field network follows the architecture of[Tong et al\. \(2024\)](https://arxiv.org/html/2609.03443#bib.bib6); training details are provided in Appendix[C](https://arxiv.org/html/2609.03443#A3)\. We generate50,00050\{,\}000samples using several numerical solvers and compute FID\([Heusel et al\., 2017](https://arxiv.org/html/2609.03443#bib.bib8)\)\. Results are averaged over55random seeds and reported with standard deviations\.

Table 2:FID results on unconditional CIFAR\-10 generation at100100k training steps\.
Figure 3:FID \(Dopri5\) vs\. steps\.
Table[2](https://arxiv.org/html/2609.03443#S4.T2)reports CIFAR\-10 results at100100k training steps\. Under few\-step Euler sampling, OT\-FM performs best, while QAT\-FM substantially improves over I\-FM and SD\-FM, indicating that structured coupling can improve generation quality under coarse numerical integration\. Under the adaptive Dopri5 solver, QAT\-FM achieves the lowest FID, showing competitive final generation quality in the unconditional setting\. Figure[3](https://arxiv.org/html/2609.03443#S4.F3)further shows that QAT\-FM maintains strong performance during late\-stage training\.

#### Ablation Studies\.

Our main experiments use patch sizep=4p=4, PCA dimensionk=32k=32, and mean\-based tree splitting\. To assess the sensitivity of these choices, we conduct three ablation studies: \(1\) varying the PCA dimensionk∈\{32,64,128\}k\\in\\\{32,64,128\\\}; \(2\) varying the patch sizep∈\{1,2,4\}p\\in\\\{1,2,4\\\}; and \(3\) comparing three split strategies—mean split, median split, and random split with ratio∼𝒰⁡\(0\.25,0\.75\)\\sim\\mathcal\{U\}\(0\.25,0\.75\)\. Results are summarized in Table[3](https://arxiv.org/html/2609.03443#S4.T3)\.

Table 3:Ablation studies on PCA dimension, patch size, and tree splitting strategy\.
As shown in Table[3](https://arxiv.org/html/2609.03443#S4.T3), QAT\-FM is insensitive to the choice of PCA dimension, patch size, and tree splitting strategy; FID and NFE vary only slightly across all configurations\. These results suggest that the performance gains of QAT\-FM do not rely on carefully tuned tree construction hyperparameters, but stem primarily from the global tree\-structured quantile alignment mechanism\.

### 4\.3Conditional Generation Experiments

#### Class\-conditional generation\.

We evaluate QAT\-FM on class\-conditional generation using ImageNet\-1k\([Deng et al\., 2009](https://arxiv.org/html/2609.03443#bib.bib10)\), which contains1,0001\{,\}000categories and over11M training images\. All images are resized to32×3232\\times 32, with random horizontal flipping used for augmentation\. We build the Conditional QAT coupling using both original and horizontally flipped training images\. At evaluation, we generate5050K samples and compute FID against the training set\. Results are averaged over55random seeds and reported with standard deviations\.

Table 4:FID results on class\-conditional ImageNet\-32×3232\\times 32generation\.Table[4](https://arxiv.org/html/2609.03443#S4.T4)reports FID on class\-conditional ImageNet\-32×3232\\times 32generation\. Both C2OT and QAT\-FM improve few\-step generation over I\-FM, demonstrating the effectiveness of condition\-aware coupling\. While C2OT performs best under small Euler step counts, its advantage reverses as the solver becomes more accurate\. By contrast, QAT\-FM remains competitive in few\-step sampling and achieves the best performance under larger\-step Euler solvers and Dopri5\. This indicates that Conditional QAT provides more stable supervision for high\-quality conditional FM training\.

#### Text\-conditional generation\.

We further evaluate QAT\-FM on text\-conditional CelebA\-64×6464\\times 64generation\. CelebA\([Liu et al\., 2015](https://arxiv.org/html/2609.03443#bib.bib23)\)contains over200200K celebrity face images with attribute annotations, and CelebA\-Dialog\([Jiang et al\., 2021](https://arxiv.org/html/2609.03443#bib.bib24)\)provides fine\-grained natural\-language descriptions for these images\. We use the text captions as conditioning inputs\. Following C2OT\([Cheng and Schwing, 2025](https://arxiv.org/html/2609.03443#bib.bib20)\), each caption is encoded by a CLIP\([Radford et al\., 2021](https://arxiv.org/html/2609.03443#bib.bib25)\)\-like DFN text encoder\([Fang et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib26)\), and the resulting embedding is used as the condition variable\. We compare QAT\-FM with I\-FM, OT\-FM, and C2OT under the same text\-conditioning setting\. At evaluation, we generate5050K samples and report FID for image fidelity and SigLIP\-2\([Tschannen et al\., 2025](https://arxiv.org/html/2609.03443#bib.bib27)\)CLIP score for text\-image alignment\.

\(a\)Quantitative results\.

![Refer to caption](https://arxiv.org/html/2609.03443v1/celeba_vis.png)

\(b\)Qualitative samples\.

Figure 4:Text\-conditional generation results on CelebA\-64×6464\\times 64\. \(a\) Quantitative comparison in terms of FID and CLIP score under different solvers\. \(b\) Qualitative samples generated by Euler\-2 and Euler\-100 using the caption:“The full face of this female is beamed with happiness\. This person looks very young and has no bangs\. There is not any glasses on her face\.”Figure[4](https://arxiv.org/html/2609.03443#S4.F4)shows that QAT\-FM consistently outperforms I\-FM, OT\-FM, and C2OT in FID across different solvers, while also requiring the lowest average NFE\. Meanwhile, QAT\-FM attains the same CLIP scores as I\-FM and C2OT, and higher CLIP scores than OT\-FM, indicating that the improvement in image fidelity does not come at the cost of weaker text\-image alignment\. The qualitative samples further show that QAT\-FM produces more coherent faces under Euler\-2 and sharper details under Euler\-100, consistent with the quantitative results\.

![Refer to caption](https://arxiv.org/html/2609.03443v1/lightingdit.png)Figure 5:Qualitative comparison between I\-FM and QAT\-FM on ImageNet\-256×256256\\times 256class\-conditional generation using Euler\-250 sampling\. Each column uses the same input condition\.Table 5:Class\-conditional generation results on ImageNet\-256×256256\\times 256latent generation\.
#### Large\-scale latent generation\.

To assess scalability, we further evaluate QAT\-FM on class\-conditional ImageNet\-256×256256\\times 256latent generation\([Rombach et al\., 2022](https://arxiv.org/html/2609.03443#bib.bib28);[Ma et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib12)\)\. Following[Yao et al\. \(2025\)](https://arxiv.org/html/2609.03443#bib.bib29), images are encoded into VA\-VAE latents, and the flow model is trained in the latent space\. We construct the Conditional QAT directly over the VAE latents, while following the network architecture, training hyperparameters, and evaluation protocol of[Yao et al\. \(2025\)](https://arxiv.org/html/2609.03443#bib.bib29)\. Due to computational constraints, we train for6464epochs using the official recommended setting\. The implementation and evaluation details are provided in Appendix[C](https://arxiv.org/html/2609.03443#A3)\.

Table[5](https://arxiv.org/html/2609.03443#S4.T5)shows that QAT\-FM consistently improves FID and Inception Score over I\-FM across all solver step counts, while maintaining competitive sFID, precision, and recall\. The improvements hold in both few\-step and high\-accuracy sampling regimes, indicating that the QAT coupling benefits latent\-space flow training beyond a particular solver choice\. Figure[5](https://arxiv.org/html/2609.03443#S4.F5)provides qualitative examples under Euler\-250 sampling\. These results suggest that QAT\-FM can be integrated into modern latent\-space flow backbones with minimal architectural changes while improving generation quality\.

## 5Conclusion

We study the prior\-data coupling problem in Flow Matching and propose QAT\-FM, a simple yet effective coupling strategy based on tree\-structured quantile alignment\. QAT\-FM avoids the high cost of OT\-based coupling while preserving favorable path geometry, including non\-crossing paths and improved path separation\. It also extends naturally to conditional generation through condition\-aware tree construction\. Extensive experiments verify the effectiveness and scalability of our method across synthetic data, image generation, and large\-scale latent\-space generation\. Overall, QAT\-FM provides an efficient and structured coupling mechanism for improving Flow Matching training\.

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## Appendix AProofs

### A\.1Proof of Theorem[1](https://arxiv.org/html/2609.03443#Thmtheorem1)

Proof\.By the binary tree structure, for any two distinct data leavesℓ≠ℓ′\\ell\\neq\\ell^\{\\prime\}, there exists a unique first internal nodeη\\etaat which the root\-to\-ℓ\\elland root\-to\-ℓ′\\ell^\{\\prime\}paths diverge\. Without loss of generality, assumeℓ\\elllies in the left subtree ofη\\etaandℓ′\\ell^\{\\prime\}lies in the right subtree\.

By the construction of the data tree, nodeη\\etais split along coordinateκη\\kappa\_\{\\eta\}with thresholdsηs\_\{\\eta\}, so its left and right child regions are

Rη−=\{x∈Rη:x\(κη\)≤sη\},Rη\+=\{x∈Rη:x\(κη\)\>sη\}\.R\_\{\\eta^\{\-\}\}=\\\{x\\in R\_\{\\eta\}:x^\{\(\\kappa\_\{\\eta\}\)\}\\leq s\_\{\\eta\}\\\},\\qquad R\_\{\\eta^\{\+\}\}=\\\{x\\in R\_\{\\eta\}:x^\{\(\\kappa\_\{\\eta\}\)\}\>s\_\{\\eta\}\\\}\.Sincex1∈Rη−x\_\{1\}\\in R\_\{\\eta^\{\-\}\}andx1′∈Rη\+x\_\{1\}^\{\\prime\}\\in R\_\{\\eta^\{\+\}\},

x1\(κη\)≤sη<\(x1′\)\(κη\)\.x\_\{1\}^\{\(\\kappa\_\{\\eta\}\)\}\\leq s\_\{\\eta\}<\(x\_\{1\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\.\(9\)
On the source side, QAT reuses the same split coordinateκη\\kappa\_\{\\eta\}and replaces the data\-side thresholdsηs\_\{\\eta\}by the aligned thresholds~η\\widetilde\{s\}\_\{\\eta\}\. The corresponding source\-side child regions are

R~η−=\{z∈R~η:z\(κη\)≤s~η\},R~η\+=\{z∈R~η:z\(κη\)\>s~η\}\.\\widetilde\{R\}\_\{\\eta^\{\-\}\}=\\\{z\\in\\widetilde\{R\}\_\{\\eta\}:z^\{\(\\kappa\_\{\\eta\}\)\}\\leq\\widetilde\{s\}\_\{\\eta\}\\\},\\qquad\\widetilde\{R\}\_\{\\eta^\{\+\}\}=\\\{z\\in\\widetilde\{R\}\_\{\\eta\}:z^\{\(\\kappa\_\{\\eta\}\)\}\>\\widetilde\{s\}\_\{\\eta\}\\\}\.Since QAT matches each data leaf with the corresponding source leaf region, the source samples paired with leavesℓ\\ellandℓ′\\ell^\{\\prime\}satisfyx0∈R~η−x\_\{0\}\\in\\widetilde\{R\}\_\{\\eta^\{\-\}\}andx0′∈R~η\+x\_\{0\}^\{\\prime\}\\in\\widetilde\{R\}\_\{\\eta^\{\+\}\}\. Hence

x0\(κη\)≤s~η<\(x0′\)\(κη\)\.x\_\{0\}^\{\(\\kappa\_\{\\eta\}\)\}\\leq\\widetilde\{s\}\_\{\\eta\}<\(x\_\{0\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\.\(10\)
For anyt∈\[0,1\]t\\in\[0,1\], the two linear interpolation paths are

xt=\(1−t\)​x0\+t​x1,xt′=\(1−t\)​x0′\+t​x1′\.x\_\{t\}=\(1\-t\)x\_\{0\}\+tx\_\{1\},\\qquad x\_\{t\}^\{\\prime\}=\(1\-t\)x\_\{0\}^\{\\prime\}\+tx\_\{1\}^\{\\prime\}\.Examining theκη\\kappa\_\{\\eta\}\-th coordinate,

\(xt′\)\(κη\)−xt\(κη\)=\(1−t\)​\{\(x0′\)\(κη\)−x0\(κη\)\}\+t⁡\{\(x1′\)\(κη\)−x1\(κη\)\}\.\(x\_\{t\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\-x\_\{t\}^\{\(\\kappa\_\{\\eta\}\)\}=\(1\-t\)\\bigl\\\{\(x\_\{0\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\-x\_\{0\}^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\\\}\+t\\bigl\\\{\(x\_\{1\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\-x\_\{1\}^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\\\}\.By \([9](https://arxiv.org/html/2609.03443#A1.E9)\) and \([10](https://arxiv.org/html/2609.03443#A1.E10)\), both differences on the right\-hand side are strictly positive\. Therefore,

\(xt′\)\(κη\)−xt\(κη\)\>0,∀t∈\[0,1\],\(x\_\{t\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\-x\_\{t\}^\{\(\\kappa\_\{\\eta\}\)\}\>0,\\qquad\\forall\\,t\\in\[0,1\],soxt′≠xtx\_\{t\}^\{\\prime\}\\neq x\_\{t\}for allt∈\[0,1\]t\\in\[0,1\]\. This proves that linear interpolation paths from two distinct QAT leaves are strictly non\-coincident almost surely\.□\\square

### A\.2Proof of Theorem[2](https://arxiv.org/html/2609.03443#Thmtheorem2)

Proof\.For any couplingπ∈Π⁡\(γ,μ^𝒟\)\\pi\\in\\Pi\(\\gamma,\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\), let\(X0,X1\)\(X\_\{0\},X\_\{1\}\)and\(X0′,X1′\)\(X\_\{0\}^\{\\prime\},X\_\{1\}^\{\\prime\}\)be two independent pairs drawn fromπ\\pi\. By the linear interpolationXt−Xt′=\(1−t\)​\(X0−X0′\)\+t⁡\(X1−X1′\)X\_\{t\}\-X\_\{t\}^\{\\prime\}=\(1\-t\)\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)\+t\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\),

𝔼π⊗2​‖Xt−Xt′‖22=\\displaystyle\\mathbb\{E\}\_\{\\pi^\{\\otimes 2\}\}\\\|X\_\{t\}\-X\_\{t\}^\{\\prime\}\\\|\_\{2\}^\{2\}=\(1−t\)2​𝔼​‖X0−X0′‖22\+t2​𝔼​‖X1−X1′‖22\\displaystyle\(1\-t\)^\{2\}\\mathbb\{E\}\\\|X\_\{0\}\-X\_\{0\}^\{\\prime\}\\\|\_\{2\}^\{2\}\+t^\{2\}\\mathbb\{E\}\\\|X\_\{1\}\-X\_\{1\}^\{\\prime\}\\\|\_\{2\}^\{2\}\(11\)\+2​t​\(1−t\)​𝔼π⊗2​\[\(X0−X0′\)⊤​\(X1−X1′\)\]\.\\displaystyle\+2t\(1\-t\)\\mathbb\{E\}\_\{\\pi^\{\\otimes 2\}\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\right\]\.The first two terms depend only on the marginal distributionsγ\\gammaandμ^𝒟\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}, and are therefore identical underπQAT\\pi\_\{\\mathrm\{QAT\}\}andπind\\pi\_\{\\mathrm\{ind\}\}\. The difference between the two couplings arises entirely from the cross term\. Under the independent couplingπind=γ⊗μ^𝒟\\pi\_\{\\mathrm\{ind\}\}=\\gamma\\otimes\\widehat\{\\mu\}\_\{\\mathcal\{D\}\},X0−X0′X\_\{0\}\-X\_\{0\}^\{\\prime\}andX1−X1′X\_\{1\}\-X\_\{1\}^\{\\prime\}are mutually independent with zero mean, giving

𝔼πind⊗2​\[\(X0−X0′\)⊤​\(X1−X1′\)\]=0\.\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{ind\}\}^\{\\otimes 2\}\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\right\]=0\.\(12\)
Recursive analysis of the QAT cross term\.For each nodeη\\eta, letπη\\pi\_\{\\eta\}denote the normalized QAT coupling conditioned on nodeη\\eta’s region, and let𝔼η\\mathbb\{E\}\_\{\\eta\}denote expectation when both pairs are drawn independently fromπη\\pi\_\{\\eta\}\. Define

Γη:=𝔼η​\[\(X0−X0′\)⊤​\(X1−X1′\)\]\.\\Gamma\_\{\\eta\}:=\\mathbb\{E\}\_\{\\eta\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\right\]\.Ifη\\etais a leaf, the QAT coupling within that leaf isγη⊗μ^η\\gamma\_\{\\eta\}\\otimes\\widehat\{\\mu\}\_\{\\eta\}, soX0−X0′X\_\{0\}\-X\_\{0\}^\{\\prime\}andX1−X1′X\_\{1\}\-X\_\{1\}^\{\\prime\}are independent with zero mean, givingΓη=0\\Gamma\_\{\\eta\}=0\.

For any internal nodeη∈ℐ\\eta\\in\\mathcal\{I\}with left and right childrenη−\\eta^\{\-\}andη\+\\eta^\{\+\}, let

Zη:=𝟏​\{X0∈R~η−,X1∈Rη−\},Zη′:=𝟏​\{X0′∈R~η−,X1′∈Rη−\}Z\_\{\\eta\}:=\\mathbf\{1\}\\\{X\_\{0\}\\in\\widetilde\{R\}\_\{\\eta^\{\-\}\},\\,X\_\{1\}\\in R\_\{\\eta^\{\-\}\}\\\},\\qquad Z\_\{\\eta\}^\{\\prime\}:=\\mathbf\{1\}\\\{X\_\{0\}^\{\\prime\}\\in\\widetilde\{R\}\_\{\\eta^\{\-\}\},\\,X\_\{1\}^\{\\prime\}\\in R\_\{\\eta^\{\-\}\}\\\}indicate whether the first and second pairs fall into the left child\. Writepη−:=μ^η​\(Rη−\)p\_\{\\eta\}^\{\-\}:=\\widehat\{\\mu\}\_\{\\eta\}\(R\_\{\\eta^\{\-\}\}\)andpη\+:=μ^η​\(Rη\+\)p\_\{\\eta\}^\{\+\}:=\\widehat\{\\mu\}\_\{\\eta\}\(R\_\{\\eta^\{\+\}\}\)\. By the law of total expectation,

Γη=\\displaystyle\\Gamma\_\{\\eta\}=𝔼\[𝔼η\[\(X0−X0′\)⊤\(X1−X1′\)\|Zη,Zη′\]\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\mathbb\{E\}\_\{\\eta\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\,\\middle\|\\,Z\_\{\\eta\},Z\_\{\\eta\}^\{\\prime\}\\right\]\\right\]\(13\)=\\displaystyle=\(pη−\)2​Γη−\+\(pη\+\)2​Γη\+\\displaystyle\(p\_\{\\eta\}^\{\-\}\)^\{2\}\\Gamma\_\{\\eta^\{\-\}\}\+\(p\_\{\\eta\}^\{\+\}\)^\{2\}\\Gamma\_\{\\eta^\{\+\}\}\+pη−pη\+𝔼η\[\(X0−X0′\)⊤\(X1−X1′\)\|Zη=1,Zη′=0\]\\displaystyle\+p\_\{\\eta\}^\{\-\}p\_\{\\eta\}^\{\+\}\\,\\mathbb\{E\}\_\{\\eta\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\,\\middle\|\\,Z\_\{\\eta\}=1,\\,Z\_\{\\eta\}^\{\\prime\}=0\\right\]\+pη\+pη−𝔼η\[\(X0−X0′\)⊤\(X1−X1′\)\|Zη=0,Zη′=1\]\.\\displaystyle\+p\_\{\\eta\}^\{\+\}p\_\{\\eta\}^\{\-\}\\,\\mathbb\{E\}\_\{\\eta\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\,\\middle\|\\,Z\_\{\\eta\}=0,\\,Z\_\{\\eta\}^\{\\prime\}=1\\right\]\.
When both pairs fall into the same child, they contribute\(pη−\)2​Γη−\(p\_\{\\eta\}^\{\-\}\)^\{2\}\\Gamma\_\{\\eta^\{\-\}\}and\(pη\+\)2​Γη\+\(p\_\{\\eta\}^\{\+\}\)^\{2\}\\Gamma\_\{\\eta^\{\+\}\}respectively\. For the cross\-subtree terms, writem0,η±:=𝔼γη±​\[X\]m\_\{0,\\eta\}^\{\\pm\}:=\\mathbb\{E\}\_\{\\gamma\_\{\\eta^\{\\pm\}\}\}\[X\]andm1,η±:=𝔼μ^η±​\[X\]m\_\{1,\\eta\}^\{\\pm\}:=\\mathbb\{E\}\_\{\\widehat\{\\mu\}\_\{\\eta^\{\\pm\}\}\}\[X\]\. Taking the caseZη=0Z\_\{\\eta\}=0,Zη′=1Z\_\{\\eta\}^\{\\prime\}=1\(first pair in right child, second in left\), the cross\-subtree contribution is

pη\+pη−𝔼η\[\(X0−X0′\)⊤\(X1−X1′\)\|Zη=0,Zη′=1\]\\displaystyle p\_\{\\eta\}^\{\+\}p\_\{\\eta\}^\{\-\}\\,\\mathbb\{E\}\_\{\\eta\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\,\\middle\|\\,Z\_\{\\eta\}=0,\\,Z\_\{\\eta\}^\{\\prime\}=1\\right\]=\\displaystyle=pη\+​pη−​\{𝔼η\+​\[X0⊤​X1\]−\(m0,η\+\)⊤​m1,η−−\(m0,η−\)⊤​m1,η\+\+𝔼η−​\[X0⊤​X1\]\}\.\\displaystyle p\_\{\\eta\}^\{\+\}p\_\{\\eta\}^\{\-\}\\bigg\\\{\\mathbb\{E\}\_\{\\eta^\{\+\}\}\[X\_\{0\}^\{\\top\}X\_\{1\}\]\-\(m\_\{0,\\eta\}^\{\+\}\)^\{\\top\}m\_\{1,\\eta\}^\{\-\}\-\(m\_\{0,\\eta\}^\{\-\}\)^\{\\top\}m\_\{1,\\eta\}^\{\+\}\+\\mathbb\{E\}\_\{\\eta^\{\-\}\}\[X\_\{0\}^\{\\top\}X\_\{1\}\]\\bigg\\\}\.SinceΓη±=2​\{𝔼η±​\[X0⊤​X1\]−\(m0,η±\)⊤​m1,η±\}\\Gamma\_\{\\eta^\{\\pm\}\}=2\\bigl\\\{\\mathbb\{E\}\_\{\\eta^\{\\pm\}\}\[X\_\{0\}^\{\\top\}X\_\{1\}\]\-\(m\_\{0,\\eta\}^\{\\pm\}\)^\{\\top\}m\_\{1,\\eta\}^\{\\pm\}\\bigr\\\}, the above simplifies to

pη\+​pη−​\{12​Γη\+\+12​Γη−\+\(m0,η\+−m0,η−\)⊤​\(m1,η\+−m1,η−\)\}\.p\_\{\\eta\}^\{\+\}p\_\{\\eta\}^\{\-\}\\bigg\\\{\\frac\{1\}\{2\}\\Gamma\_\{\\eta^\{\+\}\}\+\\frac\{1\}\{2\}\\Gamma\_\{\\eta^\{\-\}\}\+\\left\(m\_\{0,\\eta\}^\{\+\}\-m\_\{0,\\eta\}^\{\-\}\\right\)^\{\\top\}\\left\(m\_\{1,\\eta\}^\{\+\}\-m\_\{1,\\eta\}^\{\-\}\\right\)\\bigg\\\}\.The caseZη=1Z\_\{\\eta\}=1,Zη′=0Z\_\{\\eta\}^\{\\prime\}=0contributes the same amount\. Substituting into \([13](https://arxiv.org/html/2609.03443#A1.E13)\),

Γη=pη−​Γη−\+pη\+​Γη\+\+2​pη−​pη\+​\(m0,η\+−m0,η−\)⊤​\(m1,η\+−m1,η−\)\.\\Gamma\_\{\\eta\}=\\,p\_\{\\eta\}^\{\-\}\\Gamma\_\{\\eta^\{\-\}\}\+p\_\{\\eta\}^\{\+\}\\Gamma\_\{\\eta^\{\+\}\}\+2p\_\{\\eta\}^\{\-\}p\_\{\\eta\}^\{\+\}\\left\(m\_\{0,\\eta\}^\{\+\}\-m\_\{0,\\eta\}^\{\-\}\\right\)^\{\\top\}\\left\(m\_\{1,\\eta\}^\{\+\}\-m\_\{1,\\eta\}^\{\-\}\\right\)\.\(14\)
Reduction to split\-coordinate mean differences\.Since the source distribution is𝒩⁡\(0,Id\)\\mathcal\{N\}\(0,I\_\{d\}\)and all QAT splits are axis\-aligned, the source regionsR~η−\\widetilde\{R\}\_\{\\eta^\{\-\}\}andR~η\+\\widetilde\{R\}\_\{\\eta^\{\+\}\}differ only in theκη\\kappa\_\{\\eta\}\-th coordinate interval\. By Gaussian coordinate independence, for anyj≠κηj\\neq\\kappa\_\{\\eta\},

\(projj\)\#​γη\+=\(projj\)\#​γη−,\(\\mathrm\{proj\}\_\{j\}\)\_\{\\\#\}\\gamma\_\{\\eta^\{\+\}\}=\(\\mathrm\{proj\}\_\{j\}\)\_\{\\\#\}\\gamma\_\{\\eta^\{\-\}\},so\(m0,η\+\)\(j\)=\(m0,η−\)\(j\)\(m\_\{0,\\eta\}^\{\+\}\)^\{\(j\)\}=\(m\_\{0,\\eta\}^\{\-\}\)^\{\(j\)\}for allj≠κηj\\neq\\kappa\_\{\\eta\}\. Writing

Δη\\displaystyle\\Delta\_\{\\eta\}:=\(m0,η\+−m0,η−\)⊤​\(m1,η\+−m1,η−\)\\displaystyle:=\\left\(m\_\{0,\\eta\}^\{\+\}\-m\_\{0,\\eta\}^\{\-\}\\right\)^\{\\top\}\\left\(m\_\{1,\\eta\}^\{\+\}\-m\_\{1,\\eta\}^\{\-\}\\right\)=\[\(m0,η\+\)\(κη\)−\(m0,η−\)\(κη\)\]​\[\(m1,η\+\)\(κη\)−\(m1,η−\)\(κη\)\],\\displaystyle=\\,\\left\[\(m\_\{0,\\eta\}^\{\+\}\)^\{\(\\kappa\_\{\\eta\}\)\}\-\(m\_\{0,\\eta\}^\{\-\}\)^\{\(\\kappa\_\{\\eta\}\)\}\\right\]\\left\[\(m\_\{1,\\eta\}^\{\+\}\)^\{\(\\kappa\_\{\\eta\}\)\}\-\(m\_\{1,\\eta\}^\{\-\}\)^\{\(\\kappa\_\{\\eta\}\)\}\\right\],equation \([14](https://arxiv.org/html/2609.03443#A1.E14)\) becomes

Γη=pη−​Γη−\+pη\+​Γη\+\+2​pη−​pη\+​Δη\.\\Gamma\_\{\\eta\}=p\_\{\\eta\}^\{\-\}\\Gamma\_\{\\eta^\{\-\}\}\+p\_\{\\eta\}^\{\+\}\\Gamma\_\{\\eta^\{\+\}\}\+2p\_\{\\eta\}^\{\-\}p\_\{\\eta\}^\{\+\}\\Delta\_\{\\eta\}\.\(15\)
Telescoping from the root\.Letαη:=μ^𝒟​\(Rη\)\\alpha\_\{\\eta\}:=\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\(R\_\{\\eta\}\),αη−:=μ^𝒟​\(Rη−\)\\alpha\_\{\\eta^\{\-\}\}:=\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\(R\_\{\\eta^\{\-\}\}\),αη\+:=μ^𝒟​\(Rη\+\)\\alpha\_\{\\eta^\{\+\}\}:=\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}\(R\_\{\\eta^\{\+\}\}\)\. Sincepη−=αη−/αηp\_\{\\eta\}^\{\-\}=\\alpha\_\{\\eta^\{\-\}\}/\\alpha\_\{\\eta\}andpη\+=αη\+/αηp\_\{\\eta\}^\{\+\}=\\alpha\_\{\\eta^\{\+\}\}/\\alpha\_\{\\eta\}, multiplying \([15](https://arxiv.org/html/2609.03443#A1.E15)\) byαη\\alpha\_\{\\eta\}gives

αη​Γη=αη−​Γη−\+αη\+​Γη\+\+2​αη−​αη\+αη​Δη\.\\alpha\_\{\\eta\}\\Gamma\_\{\\eta\}=\\alpha\_\{\\eta^\{\-\}\}\\Gamma\_\{\\eta^\{\-\}\}\+\\alpha\_\{\\eta^\{\+\}\}\\Gamma\_\{\\eta^\{\+\}\}\+2\\frac\{\\alpha\_\{\\eta^\{\-\}\}\\alpha\_\{\\eta^\{\+\}\}\}\{\\alpha\_\{\\eta\}\}\\Delta\_\{\\eta\}\.\(16\)Telescoping from the rootrr\(whereαr=1\\alpha\_\{r\}=1\) and usingΓℓ=0\\Gamma\_\{\\ell\}=0at all leaves,

𝔼πQAT⊗2​\[\(X0−X0′\)⊤​\(X1−X1′\)\]=Γr=2​∑η∈ℐαη−​αη\+αη​Δη\.\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{QAT\}\}^\{\\otimes 2\}\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\right\]=\\Gamma\_\{r\}=2\\sum\_\{\\eta\\in\\mathcal\{I\}\}\\frac\{\\alpha\_\{\\eta^\{\-\}\}\\alpha\_\{\\eta^\{\+\}\}\}\{\\alpha\_\{\\eta\}\}\\Delta\_\{\\eta\}\.\(17\)Settingqη:=αη−/αηq\_\{\\eta\}:=\\alpha\_\{\\eta^\{\-\}\}/\\alpha\_\{\\eta\}, so thatαη−​αη\+αη=αη​qη​\(1−qη\)\\frac\{\\alpha\_\{\\eta^\{\-\}\}\\alpha\_\{\\eta^\{\+\}\}\}\{\\alpha\_\{\\eta\}\}=\\alpha\_\{\\eta\}q\_\{\\eta\}\(1\-q\_\{\\eta\}\),

𝔼πQAT⊗2​\[\(X0−X0′\)⊤​\(X1−X1′\)\]=2​ΔQAT\.\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{QAT\}\}^\{\\otimes 2\}\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\right\]=2\\Delta\_\{\\mathrm\{QAT\}\}\.\(18\)
Combining \([11](https://arxiv.org/html/2609.03443#A1.E11)\), \([12](https://arxiv.org/html/2609.03443#A1.E12)\), and \([18](https://arxiv.org/html/2609.03443#A1.E18)\),

𝔼πQAT⊗2​\[‖Xt−Xt′‖22\]−𝔼πind⊗2​\[‖Xt−Xt′‖22\]\\displaystyle\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{QAT\}\}^\{\\otimes 2\}\}\\\!\\left\[\\\|X\_\{t\}\-X\_\{t\}^\{\\prime\}\\\|\_\{2\}^\{2\}\\right\]\-\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{ind\}\}^\{\\otimes 2\}\}\\\!\\left\[\\\|X\_\{t\}\-X\_\{t\}^\{\\prime\}\\\|\_\{2\}^\{2\}\\right\]\(19\)=\\displaystyle=2​t​\(1−t\)​𝔼πQAT⊗2​\[\(X0−X0′\)⊤​\(X1−X1′\)\]\\displaystyle 2t\(1\-t\)\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{QAT\}\}^\{\\otimes 2\}\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\right\]=\\displaystyle=4​t​\(1−t\)​ΔQAT\.\\displaystyle 4t\(1\-t\)\\Delta\_\{\\mathrm\{QAT\}\}\.Under non\-degenerate splits, each effective split node satisfiesΔη≥0\\Delta\_\{\\eta\}\\geq 0; if at least one node hasΔη\>0\\Delta\_\{\\eta\}\>0, thenΔQAT\>0\\Delta\_\{\\mathrm\{QAT\}\}\>0\. This completes the proof\.□\\square

### A\.3Properties of CQAT

Let𝒟c=\{\(xi,ci\)\}i=1N⊆ℝd×ℝm\\mathcal\{D\}\_\{c\}=\\\{\(x\_\{i\},c\_\{i\}\)\\\}\_\{i=1\}^\{N\}\\subseteq\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{m\}be the conditional dataset with joint empirical distributionμ^𝒟c\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{c\}\}\. Denote the Conditional QAT coupling byπCQAT∈Π⁡\(γ,μ^𝒟c\)\\pi\_\{\\mathrm\{CQAT\}\}\\in\\Pi\(\\gamma,\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{c\}\}\), whereγ=𝒩⁡\(0,Id\)\\gamma=\\mathcal\{N\}\(0,I\_\{d\}\)\. Throughout,X1X\_\{1\}refers to the data\-coordinate component of a joint sample\(X1,C\)\(X\_\{1\},C\)\.

###### Proposition 1\(Non\-crossing property of CQAT\)

Letℓ≠ℓ′\\ell\\neq\\ell^\{\\prime\}be two leaf nodes of the Conditional QAT tree, and letη\\etabe the first internal node at which the root\-to\-ℓ\\elland root\-to\-ℓ′\\ell^\{\\prime\}paths diverge\. Ifη\\etais a data\-coordinate split node, i\.e\.,κη≤d\\kappa\_\{\\eta\}\\leq d, then the linear interpolation paths drawn from the two leaf couplings do not intersect\.

Proof\.Without loss of generality, assumeℓ\\elllies in the left subtreeη−\\eta^\{\-\}andℓ′\\ell^\{\\prime\}in the right subtreeη\+\\eta^\{\+\}\. Sinceκη≤d\\kappa\_\{\\eta\}\\leq d, the CQAT tree performs a data\-coordinate split atη\\eta\. By the tree construction, for any\(x1,c1\)∈Rℓ\(x\_\{1\},c\_\{1\}\)\\in R\_\{\\ell\}and\(x1′,c1′\)∈Rℓ′\(x\_\{1\}^\{\\prime\},c\_\{1\}^\{\\prime\}\)\\in R\_\{\\ell^\{\\prime\}\},

x1\(κη\)≤sη<\(x1′\)\(κη\)\.x\_\{1\}^\{\(\\kappa\_\{\\eta\}\)\}\\leq s\_\{\\eta\}<\(x\_\{1\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\.Moreover, sinceη\\etais a data\-coordinate split node, the source\-side region is simultaneously partitioned: for anyx0∈R~ℓx\_\{0\}\\in\\widetilde\{R\}\_\{\\ell\}andx0′∈R~ℓ′x\_\{0\}^\{\\prime\}\\in\\widetilde\{R\}\_\{\\ell^\{\\prime\}\},

x0\(κη\)≤s~η<\(x0′\)\(κη\)\.x\_\{0\}^\{\(\\kappa\_\{\\eta\}\)\}\\leq\\widetilde\{s\}\_\{\\eta\}<\(x\_\{0\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\.Hence, for allt∈\[0,1\]t\\in\[0,1\],

\(xt′\)\(κη\)−xt\(κη\)=\(1−t\)​\[\(x0′\)\(κη\)−x0\(κη\)\]\+t⁡\[\(x1′\)\(κη\)−x1\(κη\)\]\>0,\(x\_\{t\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\-x\_\{t\}^\{\(\\kappa\_\{\\eta\}\)\}=\(1\-t\)\\bigl\[\(x\_\{0\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\-x\_\{0\}^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\+t\\bigl\[\(x\_\{1\}^\{\\prime\}\)^\{\(\\kappa\_\{\\eta\}\)\}\-x\_\{1\}^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\>0,which impliesxt′≠xtx\_\{t\}^\{\\prime\}\\neq x\_\{t\}for allt∈\[0,1\]t\\in\[0,1\]\.□\\square

###### Proposition 2\(Path separation property of CQAT\)

Letxt=\(1−t\)​x0\+t​x1x\_\{t\}=\(1\-t\)x\_\{0\}\+tx\_\{1\}andxt′=\(1−t\)​x0′\+t​x1′x\_\{t\}^\{\\prime\}=\(1\-t\)x\_\{0\}^\{\\prime\}\+tx\_\{1\}^\{\\prime\}, wherex1,x1′x\_\{1\},x\_\{1\}^\{\\prime\}denote the data\-coordinate components of joint samples\. For anyt∈\(0,1\)t\\in\(0,1\),

𝔼πCQAT⊗2​\[‖xt−xt′‖22\]−𝔼πind⊗2​\[‖xt−xt′‖22\]=4​t​\(1−t\)​ΔCQAT,\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{CQAT\}\}^\{\\otimes 2\}\}\\\!\\left\[\\\|x\_\{t\}\-x\_\{t\}^\{\\prime\}\\\|\_\{2\}^\{2\}\\right\]\-\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{ind\}\}^\{\\otimes 2\}\}\\\!\\left\[\\\|x\_\{t\}\-x\_\{t\}^\{\\prime\}\\\|\_\{2\}^\{2\}\\right\]=4t\(1\-t\)\\Delta\_\{\\mathrm\{CQAT\}\},whereΔCQAT\\Delta\_\{\\mathrm\{CQAT\}\}is the node\-level separation induced by the Conditional QAT tree structure:

ΔCQAT\\displaystyle\\Delta\_\{\\mathrm\{CQAT\}\}:=∑η∈ℐc:κη≤dαηqη\(1−qη\)\(𝔼γη\+\[X\(κη\)\]−𝔼γη−\[X\(κη\)\]\)\\displaystyle:=\\sum\_\{\\eta\\in\\mathcal\{I\}\_\{c\}:\\,\\kappa\_\{\\eta\}\\leq d\}\\alpha\_\{\\eta\}q\_\{\\eta\}\(1\-q\_\{\\eta\}\)\\Bigl\(\\mathbb\{E\}\_\{\\gamma\_\{\\eta^\{\+\}\}\}\\\!\\bigl\[X^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\-\\mathbb\{E\}\_\{\\gamma\_\{\\eta^\{\-\}\}\}\\\!\\bigl\[X^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\\Bigr\)\(20\)×\(𝔼μ^𝒟c,η\+​\[X\(κη\)\]−𝔼μ^𝒟c,η−​\[X\(κη\)\]\)\.\\displaystyle\\times\\Bigl\(\\mathbb\{E\}\_\{\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{c\},\\eta^\{\+\}\}\}\\\!\\bigl\[X^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\-\\mathbb\{E\}\_\{\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{c\},\\eta^\{\-\}\}\}\\\!\\bigl\[X^\{\(\\kappa\_\{\\eta\}\)\}\\bigr\]\\Bigr\)\.Hereℐc\\mathcal\{I\}\_\{c\}is the set of internal nodes of the Conditional QAT tree;αη:=μ^𝒟c​\(Rη\)\\alpha\_\{\\eta\}:=\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{c\}\}\(R\_\{\\eta\}\)is the joint data mass at nodeη\\eta; andqη:=μ^𝒟c,η​\(Rη−\)q\_\{\\eta\}:=\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{c\},\\eta\}\(R\_\{\\eta^\{\-\}\}\)is the conditional mass of the left child\.

Proof\.The telescoping recursion \([17](https://arxiv.org/html/2609.03443#A1.E17)\) in the proof of Theorem[2](https://arxiv.org/html/2609.03443#Thmtheorem2)relies only on the left\-right child decomposition at each internal node and the product structure of the leaf couplings\. Replacing the QAT tree with the Conditional QAT tree andμ^𝒟\\widehat\{\\mu\}\_\{\\mathcal\{D\}\}with the joint empirical distributionμ^𝒟c\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{c\}\}, the same argument gives

𝔼πCQAT⊗2​\[\(X0−X0′\)⊤​\(X1−X1′\)\]=2​∑η∈ℐcαη−​αη\+αη​Δη,\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{CQAT\}\}^\{\\otimes 2\}\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\right\]=2\\sum\_\{\\eta\\in\\mathcal\{I\}\_\{c\}\}\\frac\{\\alpha\_\{\\eta^\{\-\}\}\\alpha\_\{\\eta^\{\+\}\}\}\{\\alpha\_\{\\eta\}\}\\Delta\_\{\\eta\},whereαη:=μ^𝒟c​\(Rη\)\\alpha\_\{\\eta\}:=\\widehat\{\\mu\}\_\{\\mathcal\{D\}\_\{c\}\}\(R\_\{\\eta\}\)andΔη\\Delta\_\{\\eta\}is the local cross\-term contribution at nodeη\\eta\.

It remains to show that condition\-coordinate split nodes contribute zero\. Ifκη\>d\\kappa\_\{\\eta\}\>d, the CQAT tree does not partition the source\-side region atη\\eta, so both child nodes inherit the same source region:γη−=γη\+\\gamma\_\{\\eta^\{\-\}\}=\\gamma\_\{\\eta^\{\+\}\}\. Consequently,

𝔼γη\+​\[X\]=𝔼γη−​\[X\],\\mathbb\{E\}\_\{\\gamma\_\{\\eta^\{\+\}\}\}\[X\]=\\mathbb\{E\}\_\{\\gamma\_\{\\eta^\{\-\}\}\}\[X\],and the source\-side mean difference vanishes, givingΔη=0\\Delta\_\{\\eta\}=0\. Therefore, only data\-coordinate split nodes \(κη≤d\\kappa\_\{\\eta\}\\leq d\) contribute, and the sum reduces to

𝔼πCQAT⊗2​\[\(X0−X0′\)⊤​\(X1−X1′\)\]=2​ΔCQAT\.\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{CQAT\}\}^\{\\otimes 2\}\}\\\!\\left\[\(X\_\{0\}\-X\_\{0\}^\{\\prime\}\)^\{\\top\}\(X\_\{1\}\-X\_\{1\}^\{\\prime\}\)\\right\]=2\\Delta\_\{\\mathrm\{CQAT\}\}\.Combining with the linear interpolation expansion \([11](https://arxiv.org/html/2609.03443#A1.E11)\) and the fact that the cross\-term vanishes under the independent couplingπind\\pi\_\{\\mathrm\{ind\}\}, we obtain

𝔼πCQAT⊗2​\[‖Xt−Xt′‖22\]−𝔼πind⊗2​\[‖Xt−Xt′‖22\]=4​t​\(1−t\)​ΔCQAT\.\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{CQAT\}\}^\{\\otimes 2\}\}\\\!\\left\[\\\|X\_\{t\}\-X\_\{t\}^\{\\prime\}\\\|\_\{2\}^\{2\}\\right\]\-\\mathbb\{E\}\_\{\\pi\_\{\\mathrm\{ind\}\}^\{\\otimes 2\}\}\\\!\\left\[\\\|X\_\{t\}\-X\_\{t\}^\{\\prime\}\\\|\_\{2\}^\{2\}\\right\]=4t\(1\-t\)\\Delta\_\{\\mathrm\{CQAT\}\}\.Under non\-degenerate splits, if at least one data\-coordinate split node contributes positively, thenΔCQAT\>0\\Delta\_\{\\mathrm\{CQAT\}\}\>0\. This completes the proof\.□\\square

## Appendix BAlgorithms

### B\.1QAT Coupling Construction and Sampling

This section presents pseudocode for QAT coupling construction and training\-pair sampling\. QAT usesγ=𝒩⁡\(0,Id\)\\gamma=\\mathcal\{N\}\(0,I\_\{d\}\)as the source distribution and maintains source\-side coordinate bounds\(aℓ,bℓ\)∈\(ℝ∪\{±∞\}\)d×\(ℝ∪\{±∞\}\)d\(a\_\{\\ell\},b\_\{\\ell\}\)\\in\(\\mathbb\{R\}\\cup\\\{\\pm\\infty\\\}\)^\{d\}\\times\(\\mathbb\{R\}\\cup\\\{\\pm\\infty\\\}\)^\{d\}at each leaf node\. Throughout,Φ\\PhiandΦ−1\\Phi^\{\-1\}denote the standard\-normal CDF and its inverse, respectively\.

Algorithm 1QAT Coupling Construction1:Dataset

𝒟=\{xi\}i=1N⊆ℝd\\mathcal\{D\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}\\subseteq\\mathbb\{R\}^\{d\}
2:

𝒮r←\{1,…,N\}\\mathcal\{S\}\_\{r\}\\leftarrow\\\{1,\\ldots,N\\\},

ar←\(−∞,…,−∞\)a\_\{r\}\\leftarrow\(\-\\infty,\\ldots,\-\\infty\),

br←\(\+∞,…,\+∞\)b\_\{r\}\\leftarrow\(\+\\infty,\\ldots,\+\\infty\)
3:

𝒬←\{r\}\\mathcal\{Q\}\\leftarrow\\\{r\\\},

ℒ←∅\\mathcal\{L\}\\leftarrow\\varnothing
4:while

𝒬≠∅\\mathcal\{Q\}\\neq\\varnothingdo

5:Pop one node

η\\etafrom

𝒬\\mathcal\{Q\}
6:if

\|𝒮η\|≤1\|\\mathcal\{S\}\_\{\\eta\}\|\\leq 1then

7:Add

\(𝒮η,aη,bη\)\(\\mathcal\{S\}\_\{\\eta\},\\,a\_\{\\eta\},\\,b\_\{\\eta\}\)to

ℒ\\mathcal\{L\}
8:else

9:

κη←argmaxj∈\[d\]Var\{xi:i∈𝒮η\}\(x\(j\)\)\\kappa\_\{\\eta\}\\leftarrow\\arg\\max\_\{j\\in\[d\]\}\\operatorname\{Var\}\_\{\\\{x\_\{i\}:i\\in\\mathcal\{S\}\_\{\\eta\}\\\}\}\\\!\(x^\{\(j\)\}\),

sη←1\|𝒮η\|​∑i∈𝒮ηxi\(κη\)s\_\{\\eta\}\\leftarrow\\frac\{1\}\{\|\\mathcal\{S\}\_\{\\eta\}\|\}\\sum\_\{i\\in\\mathcal\{S\}\_\{\\eta\}\}x\_\{i\}^\{\(\\kappa\_\{\\eta\}\)\}
10:

𝒮η−←\{i∈𝒮η:xi\(κη\)≤sη\}\\mathcal\{S\}\_\{\\eta^\{\-\}\}\\leftarrow\\\{i\\in\\mathcal\{S\}\_\{\\eta\}:x\_\{i\}^\{\(\\kappa\_\{\\eta\}\)\}\\leq s\_\{\\eta\}\\\},

𝒮η\+←𝒮η∖𝒮η−\\mathcal\{S\}\_\{\\eta^\{\+\}\}\\leftarrow\\mathcal\{S\}\_\{\\eta\}\\setminus\\mathcal\{S\}\_\{\\eta^\{\-\}\},

qη←\|𝒮η−\|/\|𝒮η\|q\_\{\\eta\}\\leftarrow\|\\mathcal\{S\}\_\{\\eta^\{\-\}\}\|/\|\\mathcal\{S\}\_\{\\eta\}\|
11:

s~η←Φ−1​\(Φ⁡\(aη\(κη\)\)\+qη⋅\(Φ⁡\(bη\(κη\)\)−Φ⁡\(aη\(κη\)\)\)\)\\widetilde\{s\}\_\{\\eta\}\\leftarrow\\Phi^\{\-1\}\\\!\\bigl\(\\Phi\(a\_\{\\eta\}^\{\(\\kappa\_\{\\eta\}\)\}\)\+q\_\{\\eta\}\\cdot\\bigl\(\\Phi\(b\_\{\\eta\}^\{\(\\kappa\_\{\\eta\}\)\}\)\-\\Phi\(a\_\{\\eta\}^\{\(\\kappa\_\{\\eta\}\)\}\)\\bigr\)\\bigr\)
12:

\(aη−,bη−\)←\(aη,bη\)\(a\_\{\\eta^\{\-\}\},b\_\{\\eta^\{\-\}\}\)\\leftarrow\(a\_\{\\eta\},b\_\{\\eta\}\);

\(aη\+,bη\+\)←\(aη,bη\)\(a\_\{\\eta^\{\+\}\},b\_\{\\eta^\{\+\}\}\)\\leftarrow\(a\_\{\\eta\},b\_\{\\eta\}\)
13:

bη−\(κη\)←s~ηb\_\{\\eta^\{\-\}\}^\{\(\\kappa\_\{\\eta\}\)\}\\leftarrow\\widetilde\{s\}\_\{\\eta\};

aη\+\(κη\)←s~ηa\_\{\\eta^\{\+\}\}^\{\(\\kappa\_\{\\eta\}\)\}\\leftarrow\\widetilde\{s\}\_\{\\eta\}
14:Push

η−\\eta^\{\-\}and

η\+\\eta^\{\+\}into

𝒬\\mathcal\{Q\}
15:endif

16:endwhile

17:return

ℒ=\{\(𝒮ℓ,aℓ,bℓ\):ℓ∈ℒ\}\\mathcal\{L\}=\\\{\(\\mathcal\{S\}\_\{\\ell\},\\,a\_\{\\ell\},\\,b\_\{\\ell\}\):\\ell\\in\\mathcal\{L\}\\\}

Because the construction terminates when\|𝒮ℓ\|≤1\|\\mathcal\{S\}\_\{\\ell\}\|\\leq 1, each non\-empty leaf holds exactly one training sample\. Every leaf therefore carries mass1/N1/N, and sampling a leaf is equivalent to drawing a data index uniformly at random\.

Algorithm 2QAT Coupling Training\-Pair Sampling1:Leaf list

ℒ=\{\(𝒮ℓ,aℓ,bℓ\):ℓ∈ℒ\}\\mathcal\{L\}=\\\{\(\\mathcal\{S\}\_\{\\ell\},\\,a\_\{\\ell\},\\,b\_\{\\ell\}\):\\ell\\in\\mathcal\{L\}\\\}
2:Sample

ℓ∼Unif⁡\(ℒ\)\\ell\\sim\\operatorname\{Unif\}\(\\mathcal\{L\}\); let

𝒮ℓ=\{i\}\\mathcal\{S\}\_\{\\ell\}=\\\{i\\\}and set

x←xix\\leftarrow x\_\{i\}
3:for

j=1,…,dj=1,\\ldots,ddo

4:Sample

uj∼Unif⁡\(Φ⁡\(aℓ\(j\)\),Φ⁡\(bℓ\(j\)\)\)u\_\{j\}\\sim\\operatorname\{Unif\}\\bigl\(\\Phi\(a\_\{\\ell\}^\{\(j\)\}\),\\,\\Phi\(b\_\{\\ell\}^\{\(j\)\}\)\\bigr\); set

ϵ\(j\)←Φ−1​\(uj\)\\epsilon^\{\(j\)\}\\leftarrow\\Phi^\{\-1\}\(u\_\{j\}\)
5:endfor

6:return

\(ϵ,x\)\(\\epsilon,\\,x\)

In flow matching training, the sampled pair\(ϵ,x\)\(\\epsilon,x\)serves as\(x0,x1\)\(x\_\{0\},x\_\{1\}\)\.

#### Complexity analysis\.

Assuming the tree is approximately balanced, it has𝒪⁡\(log⁡N\)\\mathcal\{O\}\(\\log N\)levels\. At each level, computing coordinate means and variances over allNNsamples costs𝒪⁡\(N​d\)\\mathcal\{O\}\(Nd\), giving a total construction cost of𝒪⁡\(N​d​log⁡N\)\\mathcal\{O\}\(Nd\\log N\)\. Sampling a single training pair requires generatingddindependent truncated\-Gaussian variates via the inverse\-CDF method, each inO⁡\(1\)O\(1\), for a per\-sample cost of𝒪⁡\(d\)\\mathcal\{O\}\(d\)\. Epoch\-level sampling therefore costs𝒪⁡\(N​d\)\\mathcal\{O\}\(Nd\), matching standard Gaussian sampling in asymptotic complexity\.

#### Vectorized implementation\.

In practice, the source\-side bounds for each sample’s corresponding leaf are precomputed and stacked into two matrices𝐀,𝐁∈\(ℝ∪\{±∞\}\)N×d\\mathbf\{A\},\\mathbf\{B\}\\in\(\\mathbb\{R\}\\cup\\\{\\pm\\infty\\\}\)^\{N\\times d\}indexed by data order\. Given a batch of indicesℐ=\{ib\}b=1B\\mathcal\{I\}=\\\{i\_\{b\}\\\}\_\{b=1\}^\{B\}, one retrieves𝐀ℐ,:\\mathbf\{A\}\_\{\\mathcal\{I\},:\}and𝐁ℐ,:\\mathbf\{B\}\_\{\\mathcal\{I\},:\}directly and drawsUb​j∼Unif⁡\(Φ⁡\(𝐀ib,j\),Φ⁡\(𝐁ib,j\)\)U\_\{bj\}\\sim\\operatorname\{Unif\}\\bigl\(\\Phi\(\\mathbf\{A\}\_\{i\_\{b\},j\}\),\\Phi\(\\mathbf\{B\}\_\{i\_\{b\},j\}\)\\bigr\)in parallel, then setsϵb​j=Φ−1​\(Ub​j\)\\epsilon\_\{bj\}=\\Phi^\{\-1\}\(U\_\{bj\}\)\. The entire batch sampling step is therefore fully vectorizable\.

### B\.2CQAT Coupling Construction and Sampling

In CQAT, the condition variable is embedded as a Euclidean vector and concatenated with the data feature to formyi=\(xi,ci\)∈ℝd\+my\_\{i\}=\(x\_\{i\},c\_\{i\}\)\\in\\mathbb\{R\}^\{d\+m\}; the algorithm then builds a tree directly on the joint sample set𝒟c=\{yi\}i=1N\\mathcal\{D\}\_\{c\}=\\\{y\_\{i\}\\\}\_\{i=1\}^\{N\}\. To prevent late\-stage splits driven purely by condition coordinates — which partition the data index set without updating the source\-side bounds — we restrict the candidate split coordinates to\[d\]\[d\]whenever the node sample count falls below a thresholdndatan\_\{\\mathrm\{data\}\}\. Algorithm[3](https://arxiv.org/html/2609.03443#alg3)gives the full construction\.

Algorithm 3CQAT Coupling Construction1:Joint dataset

𝒟c=\{yi\}i=1N⊆ℝd\+m\\mathcal\{D\}\_\{c\}=\\\{y\_\{i\}\\\}\_\{i=1\}^\{N\}\\subseteq\\mathbb\{R\}^\{d\+m\}, data\-split threshold

ndatan\_\{\\mathrm\{data\}\}
2:

𝒮r←\{1,…,N\}\\mathcal\{S\}\_\{r\}\\leftarrow\\\{1,\\ldots,N\\\},

ar←\(−∞,…,−∞\)a\_\{r\}\\leftarrow\(\-\\infty,\\ldots,\-\\infty\),

br←\(\+∞,…,\+∞\)b\_\{r\}\\leftarrow\(\+\\infty,\\ldots,\+\\infty\)
3:

𝒬←\{r\}\\mathcal\{Q\}\\leftarrow\\\{r\\\},

ℒ←∅\\mathcal\{L\}\\leftarrow\\varnothing
4:while

𝒬≠∅\\mathcal\{Q\}\\neq\\varnothingdo

5:Pop one node

η\\etafrom

𝒬\\mathcal\{Q\}
6:if

\|𝒮η\|≤1\|\\mathcal\{S\}\_\{\\eta\}\|\\leq 1then

7:Add

\(𝒮η,aη,bη\)\(\\mathcal\{S\}\_\{\\eta\},\\,a\_\{\\eta\},\\,b\_\{\\eta\}\)to

ℒ\\mathcal\{L\}
8:else

9:

𝒥η←\[d\]\\mathcal\{J\}\_\{\\eta\}\\leftarrow\[d\]if

\|𝒮η\|≤ndata\|\\mathcal\{S\}\_\{\\eta\}\|\\leq n\_\{\\mathrm\{data\}\}, else

𝒥η←\[d\+m\]\\mathcal\{J\}\_\{\\eta\}\\leftarrow\[d\+m\]
10:

κη←argmaxj∈𝒥ηVar\{yi:i∈𝒮η\}\(y\(j\)\)\\kappa\_\{\\eta\}\\leftarrow\\arg\\max\_\{j\\in\\mathcal\{J\}\_\{\\eta\}\}\\operatorname\{Var\}\_\{\\\{y\_\{i\}:i\\in\\mathcal\{S\}\_\{\\eta\}\\\}\}\\\!\(y^\{\(j\)\}\),

sη←1\|𝒮η\|​∑i∈𝒮ηyi\(κη\)s\_\{\\eta\}\\leftarrow\\frac\{1\}\{\|\\mathcal\{S\}\_\{\\eta\}\|\}\\sum\_\{i\\in\\mathcal\{S\}\_\{\\eta\}\}y\_\{i\}^\{\(\\kappa\_\{\\eta\}\)\}
11:

𝒮η−←\{i∈𝒮η:yi\(κη\)≤sη\}\\mathcal\{S\}\_\{\\eta^\{\-\}\}\\leftarrow\\\{i\\in\\mathcal\{S\}\_\{\\eta\}:y\_\{i\}^\{\(\\kappa\_\{\\eta\}\)\}\\leq s\_\{\\eta\}\\\},

𝒮η\+←𝒮η∖𝒮η−\\mathcal\{S\}\_\{\\eta^\{\+\}\}\\leftarrow\\mathcal\{S\}\_\{\\eta\}\\setminus\\mathcal\{S\}\_\{\\eta^\{\-\}\}
12:

\(aη−,bη−\)←\(aη,bη\)\(a\_\{\\eta^\{\-\}\},b\_\{\\eta^\{\-\}\}\)\\leftarrow\(a\_\{\\eta\},b\_\{\\eta\}\);

\(aη\+,bη\+\)←\(aη,bη\)\(a\_\{\\eta^\{\+\}\},b\_\{\\eta^\{\+\}\}\)\\leftarrow\(a\_\{\\eta\},b\_\{\\eta\}\)
13:if

κη≤d\\kappa\_\{\\eta\}\\leq dthen

14:

qη←\|𝒮η−\|/\|𝒮η\|q\_\{\\eta\}\\leftarrow\|\\mathcal\{S\}\_\{\\eta^\{\-\}\}\|/\|\\mathcal\{S\}\_\{\\eta\}\|
15:

s~η←Φ−1​\(Φ⁡\(aη\(κη\)\)\+qη⋅\(Φ⁡\(bη\(κη\)\)−Φ⁡\(aη\(κη\)\)\)\)\\widetilde\{s\}\_\{\\eta\}\\leftarrow\\Phi^\{\-1\}\\\!\\bigl\(\\Phi\(a\_\{\\eta\}^\{\(\\kappa\_\{\\eta\}\)\}\)\+q\_\{\\eta\}\\cdot\\bigl\(\\Phi\(b\_\{\\eta\}^\{\(\\kappa\_\{\\eta\}\)\}\)\-\\Phi\(a\_\{\\eta\}^\{\(\\kappa\_\{\\eta\}\)\}\)\\bigr\)\\bigr\)
16:

bη−\(κη\)←s~ηb\_\{\\eta^\{\-\}\}^\{\(\\kappa\_\{\\eta\}\)\}\\leftarrow\\widetilde\{s\}\_\{\\eta\};

aη\+\(κη\)←s~ηa\_\{\\eta^\{\+\}\}^\{\(\\kappa\_\{\\eta\}\)\}\\leftarrow\\widetilde\{s\}\_\{\\eta\}
17:endif

18:Push

η−\\eta^\{\-\}and

η\+\\eta^\{\+\}into

𝒬\\mathcal\{Q\}
19:endif

20:endwhile

21:return

ℒ=\{\(𝒮ℓ,aℓ,bℓ\):ℓ∈ℒ\}\\mathcal\{L\}=\\\{\(\\mathcal\{S\}\_\{\\ell\},\\,a\_\{\\ell\},\\,b\_\{\\ell\}\):\\ell\\in\\mathcal\{L\}\\\}

#### Scale balancing\.

Since CQAT builds on data and condition coordinates jointly, scale imbalance betweenxix\_\{i\}andcic\_\{i\}may bias split selection toward high\-variance condition dimensions\. We therefore build the tree on\(X,w​C\)\(X,\\,wC\)\. For discrete class labels, we setw=∞w=\\infty, which is formally equivalent to first partitioning all samples by class and then constructing an independent QAT within each class subtree — matching the approach of C2OT\([Cheng and Schwing, 2025](https://arxiv.org/html/2609.03443#bib.bib20)\)\. For continuous conditions, we adopt the variance\-normalized weight

w=\(∑jVar⁡\(X\(j\)\)∑kVar⁡\(C\(k\)\)\)1/2,w=\\left\(\\frac\{\\sum\_\{j\}\\operatorname\{Var\}\(X^\{\(j\)\}\)\}\{\\sum\_\{k\}\\operatorname\{Var\}\(C^\{\(k\)\}\)\}\\right\)^\{1/2\},\(21\)so that the rescaled condition variable has aggregate variance comparable to that of the data features\.

#### Complexity analysis\.

Building the tree overd\+md\+mcoordinates gives a construction cost of𝒪⁡\(N⁡\(d\+m\)​log⁡N\)\\mathcal\{O\}\(N\(d\+m\)\\log N\)under a balanced\-tree assumption\. Condition\-coordinate splits \(κη\>d\\kappa\_\{\\eta\}\>d\) partition the data index set without touching the source\-side bounds; data\-coordinate splits \(κη≤d\\kappa\_\{\\eta\}\\leq d\) perform quantile alignment exactly as in QAT\. At training time, Algorithm[2](https://arxiv.org/html/2609.03443#alg2)applies directly: one returns the data samplexix\_\{i\}together with its paired condition variablecic\_\{i\}\. Since source sampling remains confined todd\-dimensional Gaussian space, the per\-sample cost is𝒪⁡\(d\)\\mathcal\{O\}\(d\), unchanged from QAT\.

### B\.3High\-Dimensional Image Data Processing

For high\-dimensional image data, we do not build the QAT directly in raw pixel space\. Instead, we first apply an orthogonal transform to concentrate the principal variation into the leading coordinates\. Crucially, only a pure rotation is applied — no mean subtraction — so that if the source is𝒩⁡\(0,Id\)\\mathcal\{N\}\(0,I\_\{d\}\), it remains standard Gaussian under both the forward and inverse transforms\.

#### Patch\-Hadamard transform\.

Let the input image have spatial resolutionH×WH\\times WwithCCchannels, flattened toxi∈ℝdx\_\{i\}\\in\\mathbb\{R\}^\{d\}\(d=H​W​Cd=HWC\)\. For patch sizepp, the image is divided intod′=\(H/p\)​\(W/p\)​Cd^\{\\prime\}=\(H/p\)\(W/p\)Cnon\-overlapping patches, each of spatial dimensionr=p2r=p^\{2\}\. A normalized Hadamard matrixHpatch∈ℝr×rH\_\{\\mathrm\{patch\}\}\\in\\mathbb\{R\}^\{r\\times r\}\([Horadam, 2012](https://arxiv.org/html/2609.03443#bib.bib22)\)is applied independently to each patch\. Because the normalized Hadamard matrix is orthogonal with constant directionr−1/2𝟏rr^\{\-1/2\}\\mathbf\{1\}\_\{r\}, extracting the coefficient along this direction is equivalent to patch\-level mean pooling\. We denote the resulting mean\-pooled representation asx¯i∈ℝd′\\bar\{x\}\_\{i\}\\in\\mathbb\{R\}^\{d^\{\\prime\}\}\.

LetHp∈ℝd×dH\_\{p\}\\in\\mathbb\{R\}^\{d\\times d\}denote the global patch\-wise Hadamard transform induced by patch sizepp\. Since each per\-patch transform is orthogonal,HpH\_\{p\}is itself orthogonal\. We order the output coordinates so that the firstd′d^\{\\prime\}entries ofHp​xiH\_\{p\}x\_\{i\}correspond to the constant\-direction coefficients across all patches\. In practice, if only the low\-dimensional tree representation is needed,x¯i\\bar\{x\}\_\{i\}can be obtained directly by patch\-level mean pooling, with PCA directions estimated on thisd′d^\{\\prime\}\-dimensional representation\.

#### PCA rotation and low\-dimensional QAT\.

Patch\-Hadamard pooling avoids the cost of PCA in the fulldd\-dimensional pixel space\. We estimate the PCA rotation matrix𝐏∈ℝd′×d′\\mathbf\{P\}\\in\\mathbb\{R\}^\{d^\{\\prime\}\\times d^\{\\prime\}\}from\{x¯i\}i=1N\\\{\\bar\{x\}\_\{i\}\\\}\_\{i=1\}^\{N\}without mean shift\. Settingzi=𝐏​x¯i∈ℝd′z\_\{i\}=\\mathbf\{P\}\\bar\{x\}\_\{i\}\\in\\mathbb\{R\}^\{d^\{\\prime\}\}, we use only the leadingkkcoordinateszi\(1:k\)z\_\{i\}^\{\(1:k\)\}to construct the QAT coupling\. Algorithm[4](https://arxiv.org/html/2609.03443#alg4)summarizes the full construction pipeline\.

Algorithm 4High\-Dimensional QAT Construction1:Image dataset

𝒟=\{xi\}i=1N\\mathcal\{D\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}, patch size

pp, reduced dimension

kk
2:

d′←\(H/p\)​\(W/p\)​Cd^\{\\prime\}\\leftarrow\(H/p\)\(W/p\)C
3:

x¯i←\(Hpxi\)\(1:d′\)\\bar\{x\}\_\{i\}\\leftarrow\(H\_\{p\}x\_\{i\}\)^\{\(1:d^\{\\prime\}\)\}, for

i=1,…,Ni=1,\\ldots,N
4:Estimate PCA rotation

𝐏∈ℝd′×d′\\mathbf\{P\}\\in\\mathbb\{R\}^\{d^\{\\prime\}\\times d^\{\\prime\}\}from

\{x¯i\}i=1N\\\{\\bar\{x\}\_\{i\}\\\}\_\{i=1\}^\{N\}without mean shift

5:

z~i←\(𝐏x¯i\)\(1:k\)\\widetilde\{z\}\_\{i\}\\leftarrow\(\\mathbf\{P\}\\bar\{x\}\_\{i\}\)^\{\(1:k\)\}, for

i=1,…,Ni=1,\\ldots,N
6:Run Algorithm[1](https://arxiv.org/html/2609.03443#alg1)on

\{z~i\}i=1N\\\{\\widetilde\{z\}\_\{i\}\\\}\_\{i=1\}^\{N\}
7:return

ℒ\\mathcal\{L\},

𝐏\\mathbf\{P\},

HpH\_\{p\}

#### Inverse\-transform sampling\.

The low\-dimensional QAT constructs a coupling over the leadingkkPCA\-pooled coordinates, whose source marginal is𝒩⁡\(0,Ik\)\\mathcal\{N\}\(0,I\_\{k\}\)\. To recover a source sample in the originaldd\-dimensional pixel space, we fill the remaining coordinates with independent standard Gaussian noise: first thed′−kd^\{\\prime\}\-kresidual PCA\-pooled coordinates, then the high\-frequency Hadamard coefficients for each patch \(all per\-patch dimensions except the constant direction\)\. Since𝐏\\mathbf\{P\}andHpH\_\{p\}are both orthogonal and all unfilled coordinates are drawn as independent standard Gaussians, the resulting source sample has marginal exactly𝒩⁡\(0,Id\)\\mathcal\{N\}\(0,I\_\{d\}\)\.

Algorithm 5High\-Dimensional QAT Sampling1:

𝒟=\{xi\}i=1N\\mathcal\{D\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}, leaf list

ℒ=\{\(𝒮ℓ,aℓ,bℓ\):ℓ∈ℒ\}\\mathcal\{L\}=\\\{\(\\mathcal\{S\}\_\{\\ell\},\\,a\_\{\\ell\},\\,b\_\{\\ell\}\):\\ell\\in\\mathcal\{L\}\\\},

𝐏\\mathbf\{P\},

HpH\_\{p\},

d′d^\{\\prime\},

kk
2:Sample one leaf

ℓ∼Unif⁡\(ℒ\)\\ell\\sim\\operatorname\{Unif\}\(\\mathcal\{L\}\); let

𝒮ℓ=\{i\}\\mathcal\{S\}\_\{\\ell\}=\\\{i\\\}and set

x←xix\\leftarrow x\_\{i\}
3:for

j=1,…,kj=1,\\ldots,kdo

4:Sample

uj∼Unif⁡\(Φ⁡\(aℓ\(j\)\),Φ⁡\(bℓ\(j\)\)\)u\_\{j\}\\sim\\operatorname\{Unif\}\\bigl\(\\Phi\(a\_\{\\ell\}^\{\(j\)\}\),\\,\\Phi\(b\_\{\\ell\}^\{\(j\)\}\)\\bigr\); set

ϵ\(j\)←Φ−1​\(uj\)\\epsilon^\{\(j\)\}\\leftarrow\\Phi^\{\-1\}\(u\_\{j\}\)
5:endfor

6:Sample

ϵ\(k\+1:d′\)∼𝒩\(0,Id′−k\)\\epsilon^\{\(k\+1:d^\{\\prime\}\)\}\\sim\\mathcal\{N\}\(0,I\_\{d^\{\\prime\}\-k\}\)
7:

ϵpool←\(ϵ\(1:k\),ϵ\(k\+1:d′\)\)\\epsilon\_\{\\mathrm\{pool\}\}\\leftarrow\\bigl\(\\epsilon^\{\(1:k\)\},\\,\\epsilon^\{\(k\+1:d^\{\\prime\}\)\}\\bigr\)
8:

ϵ¯←𝐏⊤​ϵpool\\bar\{\\epsilon\}\\leftarrow\\mathbf\{P\}^\{\\top\}\\epsilon\_\{\\mathrm\{pool\}\}
9:Sample

ϵhigh∼𝒩⁡\(0,Id−d′\)\\epsilon\_\{\\mathrm\{high\}\}\\sim\\mathcal\{N\}\(0,I\_\{d\-d^\{\\prime\}\}\)
10:

ϵH←\(ϵ¯,ϵhigh\)\\epsilon\_\{H\}\\leftarrow\\bigl\(\\bar\{\\epsilon\},\\,\\epsilon\_\{\\mathrm\{high\}\}\\bigr\)
11:

x0←Hp⊤​ϵHx\_\{0\}\\leftarrow H\_\{p\}^\{\\top\}\\epsilon\_\{H\}
12:return

\(x0,x\)\(x\_\{0\},\\,x\)

#### Complexity analysis\.

Letd=H​W​Cd=HWC,r=p2r=p^\{2\}, andd′=\(H/p\)​\(W/p\)​Cd^\{\\prime\}=\(H/p\)\(W/p\)C\. Computing patch\-level mean pooling costs𝒪⁡\(N​d\)\\mathcal\{O\}\(Nd\); the full patch\-wise Hadamard transform can be applied via the Fast Walsh\-Hadamard Transform\([Horadam, 2012](https://arxiv.org/html/2609.03443#bib.bib22)\)in𝒪⁡\(N​d​log⁡r\)\\mathcal\{O\}\(Nd\\log r\)\. PCA is estimated on thed′d^\{\\prime\}\-dimensional pooled representation, and the low\-dimensional QAT is built inkk\-dimensional space, giving tree construction cost𝒪⁡\(N​k​log⁡N\)\\mathcal\{O\}\(Nk\\log N\)\. At sampling time, the inverse PCA rotation costs𝒪⁡\(d′2\)\\mathcal\{O\}\(d^\{\\prime 2\}\)per sample and the inverse patch\-wise Hadamard transform costs𝒪⁡\(d​log⁡r\)\\mathcal\{O\}\(d\\log r\), giving a total per\-sample cost of𝒪⁡\(d\+d′2\+d​log⁡r\)\\mathcal\{O\}\(d\+d^\{\\prime 2\}\+d\\log r\); batch sampling is fully vectorizable\.

## Appendix CImplementation Details

This section details the per\-experiment training configurations\. Unless stated otherwise, all methods share the same network architecture, training configuration, and sampling protocol, differing only in the coupling strategy used to form training pairs: I\-FM uses independent coupling, OT\-FM uses mini\-batch OT coupling, and QAT\-FM uses the QAT or Conditional QAT coupling proposed in this work\.

### C\.1Two\-Dimensional Experiment

#### Data\.

Following[Lipman et al\. \(2023\)](https://arxiv.org/html/2609.03443#bib.bib1);[Pooladian et al\. \(2023\)](https://arxiv.org/html/2609.03443#bib.bib5), we consider transport from a two\-dimensional standard Gaussian source distribution to a checkerboard target distribution\. Training samples are drawn independently from the source and target; the flow paths learned by I\-FM, OT\-FM, SD\-FM, and QAT\-FM are then compared under the same network architecture and optimization configuration\.

#### Network and training\.

Following[Lipman et al\. \(2024\)](https://arxiv.org/html/2609.03443#bib.bib30), the velocity field is parameterized by a five\-layer MLP with hidden width512512and Swish activations\([Ramachandran et al\., 2017](https://arxiv.org/html/2609.03443#bib.bib31)\)\. All methods are trained for2020K iterations with batch size256256and Adam\([Kingma and Ba, 2015](https://arxiv.org/html/2609.03443#bib.bib34)\)at learning rate10−310^\{\-3\}\. For QAT\-FM,5050K samples are drawn from the checkerboard target to build the QAT tree, and training pairs are sampled from the induced coupling\. For OT\-FM, exact EMD is solved within each mini\-batch using POT\([Flamary et al\., 2021](https://arxiv.org/html/2609.03443#bib.bib32)\)\. For SD\-FM, a discrete reference measure is built from5050K target samples and the corresponding regularized problem is solved with entropy parameter0\.050\.05; we optimize with Adam, as the SGD\-based strategy of[Genevay et al\. \(2016\)](https://arxiv.org/html/2609.03443#bib.bib33);[Mousavi\-Hosseini et al\. \(2026\)](https://arxiv.org/html/2609.03443#bib.bib7)proved unstable and slow to converge in this setting\.

Table 6:Hyperparameter settings for image\-space experiments\.

### C\.2CIFAR\-10

We conduct unconditional image generation on CIFAR\-10\([Krizhevsky et al\., 2009](https://arxiv.org/html/2609.03443#bib.bib9)\), using the U\-Net architecture of[Tong et al\. \(2024\)](https://arxiv.org/html/2609.03443#bib.bib6)\. Main network and training hyperparameters are listed in Table[6](https://arxiv.org/html/2609.03443#A3.T6)\. We train with Adam, a linear warmup to2×10−42\\times 10^\{\-4\}over55k steps \(constant thereafter\), gradient norm clipping at1\.01\.0, and an EMA decay of0\.99990\.9999\.

For QAT\-FM, the QAT coupling is built from the50,00050\{,\}000CIFAR\-10 training images together with their horizontally flipped augmentations, givingn=100,000n=100\{,\}000tree samples\. Images are first processed by patch\-level mean pooling with patch sizep=4p=4; PCA rotation is then estimated on the pooled representation, and the leadingk=32k=32coordinates are retained for tree construction\. The tree splits on the maximum\-variance coordinate at the empirical mean threshold and recurses until each leaf contains at most one sample\. At evaluation,5050K images are generated, and FID is computed usingclean\-fid\([Parmar et al\., 2022](https://arxiv.org/html/2609.03443#bib.bib36)\), following the protocol of[Tong et al\. \(2024\)](https://arxiv.org/html/2609.03443#bib.bib6)\.

### C\.3ImageNet\-32

We evaluate class\-conditional generation on ImageNet\-1k\([Deng et al\., 2009](https://arxiv.org/html/2609.03443#bib.bib10)\)at32×3232\\times 32resolution\. All images are resized to32×3232\\times 32with random horizontal flipping, and class labels are provided as condition inputs to the velocity\-field network\. We follow the class\-conditional U\-Net architecture of[Tong et al\. \(2024\)](https://arxiv.org/html/2609.03443#bib.bib6); main hyperparameters are listed in Table[6](https://arxiv.org/html/2609.03443#A3.T6)\. Optimizer, gradient clipping, and EMA settings match the CIFAR\-10 setup\. The learning rate warms up linearly to1×10−41\\times 10^\{\-4\}over2020k steps and then decays linearly to3×10−53\\times 10^\{\-5\}\.

For QAT\-FM, the Conditional QAT coupling is built from ImageNet\-1k training images and their horizontal\-flip augmentations, yielding over22M tree samples\. Patch\-level mean pooling \(p=4p=4\) and PCA rotation are applied as in the CIFAR\-10 setup, retaining the leadingk=32k=32coordinates\. Since the condition is a discrete class label, training samples are first partitioned by class and an independent QAT coupling is constructed within each class subset\. For C2OT\([Cheng and Schwing, 2025](https://arxiv.org/html/2609.03443#bib.bib20)\), we follow its oversampling strategy with ratio5050, yielding a local OT batch size of128×50=6400128\\times 50=6400per GPU\. Following the C2OT setting for class\-conditional generation, we use its conditional weighting scheme and set the condition weight to10810^\{8\}\.

At evaluation,5050K image samples are generated to estimate Inception feature statistics of the generated distribution; reference statistics are computed from all∼1\.28\{\\sim\}1\.28M ImageNet\-1k training images, following[Dhariwal and Nichol \(2021\)](https://arxiv.org/html/2609.03443#bib.bib35);[Ma et al\. \(2024\)](https://arxiv.org/html/2609.03443#bib.bib12)\.

### C\.4CelebA\-64

#### Data\.

We evaluate text\-conditional generation on CelebA\([Liu et al\., 2015](https://arxiv.org/html/2609.03443#bib.bib23)\)at64×6464\\times 64resolution\. Since CelebA does not provide text annotations by default, we use the natural\-language descriptions from CelebA\-Dialog\([Jiang et al\., 2021](https://arxiv.org/html/2609.03443#bib.bib24)\)as text conditions\.

#### Network and Training\.

We use the U\-Net architecture with main hyperparameters in Table[6](https://arxiv.org/html/2609.03443#A3.T6)\. Following[Cheng and Schwing \(2025\)](https://arxiv.org/html/2609.03443#bib.bib20), scale\-shift normalization is applied in the U\-Net residual blocks, and text embeddings are projected to the conditioning dimension via a linear layer before being fed to the network\. Optimizer, gradient clipping, and EMA settings match the CIFAR\-10 experiment; the learning rate warms up linearly to1×10−41\\times 10^\{\-4\}over1010k steps and remains constant thereafter\.

For QAT\-FM, a Conditional QAT coupling is constructed in the joint space of low\-dimensional image representations and text embeddings\. The image\-side representation is obtained via patch\-level mean pooling \(p=4p=4\) and PCA rotation, retaining the leadingk=32k=32coordinates\. Text conditions are encoded by aDFN5B\-CLIP\-ViT\-H\-14text encoder\([Fang et al\., 2024](https://arxiv.org/html/2609.03443#bib.bib26)\); the resulting embeddings are projected to3232dimensions via PCA and used as condition\-side features in the joint tree\. To prevent late\-stage splits driven purely by condition coordinates, candidate split dimensions are restricted to data coordinates when a node’s sample count falls belowndata=128n\_\{\\mathrm\{data\}\}=128\. For C2OT\([Cheng and Schwing, 2025](https://arxiv.org/html/2609.03443#bib.bib20)\), we follow its oversampling strategy with ratio5050, a local OT batch size of64×50=320064\\times 50=3200per GPU, which substantially increases the coupling cost; see Appendix[D\.1](https://arxiv.org/html/2609.03443#A4.SS1)\. For a fair conditional comparison, we use the Euclidean distance between text embeddings as the conditional distance and apply the weighting scheme in \([21](https://arxiv.org/html/2609.03443#A2.E21)\)\.

#### Evaluation\.

At evaluation,5050K images are generated and assessed along two axes: image fidelity via FID against training\-set reference statistics\([Ning et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib38);[Wang et al\., 2023](https://arxiv.org/html/2609.03443#bib.bib37)\), and text\-image alignment via CLIP score computed with thesiglip2\-base\-patch16\-224checkpoint of SigLIP\-2\([Tschannen et al\., 2025](https://arxiv.org/html/2609.03443#bib.bib27)\)\.

### C\.5ImageNet\-256

We conduct class\-conditional latent\-space generation on ImageNet\-256×256256\\times 256\([Deng et al\., 2009](https://arxiv.org/html/2609.03443#bib.bib10)\)\. Images are encoded by VA\-VAE\([Yao et al\., 2025](https://arxiv.org/html/2609.03443#bib.bib29)\)into latent representations, and a flow\-based LightningDiT\([Yao et al\., 2025](https://arxiv.org/html/2609.03443#bib.bib29)\)is trained in the resulting latent space with class labels as condition inputs\. All network architecture, optimizer, learning\-rate schedule, EMA, precision, and sampling configurations follow the official implementation of[Yao et al\. \(2025\)](https://arxiv.org/html/2609.03443#bib.bib29); due to computational constraints, we adopt the recommended lightweight training setting of6464epochs\.

For QAT\-FM, the Conditional QAT coupling is built directly over VA\-VAE latent representations\. PCA projection is applied, retaining the leadingk=256k=256coordinates for tree construction; patch\-level mean pooling is omitted since VA\-VAE already maps images to a compact latent space\. The tree splits on the maximum\-variance coordinate at the empirical mean and recurses until each leaf contains at most one sample\. Classifier\-free guidance is applied during inference with a scale of1010\.

At evaluation, we follow the protocol of[Yao et al\. \(2025\)](https://arxiv.org/html/2609.03443#bib.bib29)and compute FID against the ImageNet\-256×256256\\times 256reference statistics from guided\-diffusion\([Dhariwal and Nichol, 2021](https://arxiv.org/html/2609.03443#bib.bib35)\)\. We additionally report IS, sFID, Precision, and Recall\.

## Appendix DExtended Experimental Results

This section provides supplementary results comprising a runtime comparison of coupling strategies and additional visualization of generated samples\. Appendix[D\.1](https://arxiv.org/html/2609.03443#A4.SS1)benchmarks the wall\-clock time of each coupling approach at both construction and sampling phases; the subsequent four subsections present additional generated samples from CIFAR\-10, ImageNet\-32×3232\\times 32, CelebA\-64×6464\\times 64, and ImageNet\-256×256256\\times 256\.

### D\.1Runtime Comparison

We compare the wall\-clock time of different coupling strategies at both construction and sampling phases\. All experiments fix the feature dimension atd=32d=32and vary the dataset sizenn\. To reflect the repeated pairing demand of FM training, we report cumulative times over100100training epochs\. For OT\-based methods, source points are resampled from a standard Gaussian and the coupling is recomputed from scratch at every epoch; by contrast, the QAT and SD\-FM couplings are constructed once offline and reused throughout training\. Full OT is solved with theEMDsolver from POT\([Flamary et al\., 2021](https://arxiv.org/html/2609.03443#bib.bib32)\); mini\-batch OT \(OTBatch\) usesot\.dist\_batchto compute batch pairwise distances, fully exploiting GPU parallelism\. Entropy regularization withε=0\.05\\varepsilon=0\.05is applied for both SD\-FM and batch OT\. SD\-FM optimizes a semi\-discrete coupling with SGD\([Mousavi\-Hosseini et al\., 2026](https://arxiv.org/html/2609.03443#bib.bib7)\), running until convergence or100100optimization epochs\.

Figure 6:Runtime comparison of different coupling strategies\. Left: total coupling construction time over100100epochs\. Right: total sampling time over100100epochs\.Figure[6](https://arxiv.org/html/2609.03443#A4.F6)reports cumulative construction and sampling times over100100epochs\. Full OT becomes rapidly infeasible asnngrows; batch OT reduces the per\-epoch cost but must recompute the coupling every epoch, keeping its total time high\. SD\-FM requires only a single construction pass, but the semi\-discrete optimization itself is expensive\. By contrast, QAT constructs the tree and records the Gaussian bounds at each leaf in a single offline pass, achieving substantially lower construction time at all scales\. At sampling time, QAT draws source points by querying leaf bounds and sampling from truncated Gaussians, placing its per\-sample cost close to that of independent \(random\) sampling\. OT\-based methods must sample from the coupling’s joint distribution, and SD\-FM requires a nearest\-neighbor lookup over all training points at each draw—both become prohibitive at largenn\. Overall, QAT achieves structured coupling at a sampling efficiency comparable to independent coupling\.

### D\.2CIFAR\-10

![Refer to caption](https://arxiv.org/html/2609.03443v1/cifar10_fvis.png)Figure 7:Additional generated samples for unconditional CIFAR\-10 generation\. Rows correspond to I\-FM, OT\-FM, SD\-FM, and QAT\-FM; columns compare three sampling configurations:Euler\-2,Euler\-100, anddopri5\. All images are randomly generated without cherry\-picking\.
### D\.3ImageNet\-32

![Refer to caption](https://arxiv.org/html/2609.03443v1/img32_fvis.png)Figure 8:Additional generated samples for class\-conditional ImageNet\-32×3232\\times 32generation\. The four row blocks correspond to I\-FM, OT\-FM, C2OT, and QAT\-FM, respectively\. The three column blocks correspond to macaw, sports car, and pizza class conditions\. Within each class block, the eight columns show samples generated withEuler\-1,Euler\-2,Euler\-5,Euler\-10,Euler\-20,Euler\-50,Euler\-100, andDopri5, respectively\. All images are randomly generated without cherry\-picking\.
### D\.4CelebA\-64

![Refer to caption](https://arxiv.org/html/2609.03443v1/celeba_fvis.png)Figure 9:Additional generated samples for text\-conditional CelebA\-64×6464\\times 64generation\. Each horizontal block corresponds to one CelebA\-Dialog text condition; the three columns show results underEuler\-2,Euler\-100, andDopri5; within each block, the four rows correspond to I\-FM, OT\-FM, C2OT and QAT\-FM\. All images are randomly generated without cherry\-picking\.
### D\.5ImageNet\-256

![Refer to caption](https://arxiv.org/html/2609.03443v1/img256_fvis.png)Figure 10:Additional generated samples for class\-conditional latent\-space generation on ImageNet\-256×256256\\times 256\. The top and bottom groups show samples from I\-FM and QAT\-FM, respectively, generated under the same random seed and class conditions\. All images are randomly generated without cherry\-picking\.