Condition Errors Refinement in Autoregressive Image Generation with Diffusion Loss
Authors: Yucheng Zhou
Hao Li
Jianbing Shen
Venue: ICLR 2026 Poster (2026)
Keywords: Language Models
Autoregressive Language Models
Autoregressive Image Generation
Primary area: foundation or frontier models, including LLMs

Abstract: Recent studies have explored autoregressive models for image generation, with promising results, and have combined diffusion models with autoregressive frameworks to optimize image generation via diffusion losses. In this study, we present a theoretical analysis of diffusion and autoregressive models with diffusion loss, highlighting the latter's advantages. We present a theoretical comparison of conditional diffusion and autoregressive diffusion with diffusion loss, demonstrating that patch denoising optimization in autoregressive models effectively mitigates condition errors and leads to a stable condition distribution. Our analysis also reveals that autoregressive condition generation refines the condition, causing the condition error influence to decay exponentially. In addition, we introduce a novel condition refinement approach based on Optimal Transport (OT) theory to address ``condition inconsistency''. We theoretically demonstrate that formulating condition refinement as a Wasserstein Gradient Flow ensures convergence toward the ideal condition distribution, effectively mitigating condition inconsistency. Experiments demonstrate the superiority of our method over diffusion and autoregressive models with diffusion loss methods.

Published as a conference paper at ICLR 2026
CONDITION ERRORS REFINEMENT IN AUTOREGRES- SIVE IMAGE GENERATION WITH DIFFUSION LOss
Yucheng Zhou", Hao Li", Jianbing Shen
SKL-IOTSC, CIS, University Macau
yucheng. cchoucconeece edu. mo, ( jianbingshen& edu
ABSTRACT
Recent studies have explored autoregressive models for image generation, with promising results, and have combined diffusion models with autoregressive frame- works to optimize image generation via diffusion losses. In this study, we present a theoretical analysis of diffusion and autoregressive models with diffusion loss, highlighting the latter's advantages. We present theoretical comparison of con- ditional diffusion and autoregressive diffusion with diffusion loss, demonstrating that patch denoising optimization in autoregressive models effectively mitigates condition errors and leads to a stable condition distribution. Our analysis also reveals that autoregressive condition generation refines the condition, causing the condition error influence to decay exponentially. In addition, we introduce novel condition refinement approach based on Optimal Transport (OT) theory to address "condition inconsistency". We theoretically demonstrate that formulating condition refinement as a Wasserstein Gradient Flow ensures convergence toward the ideal condition distribution, effectively mitigating condition inconsistency. Experiments demonstrate the superiority of our method over diffusion and autoregressive models with diffusion loss methods.
INTRODUCTION
Diffusion models have demonstrated remarkable performance in image generation and have been widely adopted across various visual generative tasks (Wei et al. 2024; Rombach al. 2022 Saharia et al. 2022) Recently, due to the impressive reasoning capabilities exhibited by large language models (LLMs), autoregressive modeling has garnered significant attention. Consequently, some studies are exploring autoregressive frameworks for image and video generation, aiming to integrate them with LLMs build more powerful multimodal models (Sun al. 2024b).
Recent advancements in autoregressive image generation have shown performance comparable to diffusion models (Sun et al.) 2024a] Tian al. 2024] Zhou et al., 2025a). However, most autore- gressive image generation methods rely on Vector Quantized Variational Autoencoders (VQ- VAES (Rombach et al., )022)) to encode visual content into discrete tokens for next-token prediction modeling. (Li et al. (024a) indicate that VQ-based image generation is sensitive to gradient approxi- mation strategies and suffers from quantization errors, and propose diffusion loss for autoregressive image generation, effectively pursuing autoregressive image generation without VQ. Nevertheless, a comparative analysis between Conditional diffusion modeling and autoregressive modeling with diffusion loss remains underexplored.
In this study, we investigate the differences between autoregressive modeling with diffusion loss and conditional diffusion modeling. Firstly, we delve into the theoretical underpinnings of patch denoising optimization in autoregressive models for condition error correction. We theoretically prove that, under standard assumptions of Markov property and Gaussian noise in diffusion modeling, the iterative patch denoising approach leads to stable condition distribution. Furthermore, our analysis
'Equal Contribution.
"Corresponding Author. This work was supported by the National Natural Science Foundation of China (No. 624B2002), the Science and Technology Development Fund of Macau SAR (FDCT) under grants 01.34/202/RRAA2 and 0102/2023RRIA,, and the Jiangyin Hi-tech Industrial Development Zone under the Taihu Innovation Scheme (FP2025-0W0--KK-IOTSSCC

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Published as a conference paper at ICLR 2026
reveals the crucial behavior of the conditional probability gradient, showing its attenuation as the condition stabilizes. Our theoretical exploration demonstrates that patch denoising in autoregressive modeling effeetively mitigates condition errois and consequently contributes to improved conditional generation quality in diffusion modeling. In addition, we theoretically demonstrate that the sequence of condition variables generated by autoregressive process effectively refines the condition, leading to a reduction in the gradient norm of the conditional probability distribution. Specifically, we demonstrate that the influence of the condition on the outcome, quantified by the gradient norm, decays exponentially towards a stationary value as the autoregressive iteration progresses.
Building upon these theoretical insights, we further analyze the issue of "condition inconsistency" in autoregressive condition generation, demonstrating how extraneous information accumulates and hinders optimal patch generation. To address this, we introduce novel condition refinement approach grounded in Optimal Transport (OT) theory. We theoretically prove that formulating condition refinement as Wasserstein Gradient Flow leads to convergence towards the ideal condition distribution, effectively mitigating condition inconsistency and ultimately enhancing the quality of patch generation within diffusion models.
In the experiments, we compare our method against other diffusion and autoregressive models with diffusion loss on ImageNet (Li et al., 2024b). Results show the superiority of our method over these competitors. We also analyze the denoising process demonstrate the effectiveness of our method in condition refinement. Our main contributions and findings are as follows:
We theoretically prove that patch denoising optimization in autoregressive models mitigates condi- tion errors and elucidates the attenuation behavior of the conditional probability gradient as the condition stabilizes.
We theoretically establish the efficacy of autoregressive condition refinement, quantifying the exponential decay of the condition's influence on the outcome as autoregressive iteration progresses to a stationary value.
We propose a condition refinement method based on Optimal Transport theory, and theoretically prove that formulating it Wasserstein Gradient Flow ensures convergence towards the ideal condition distribution.
Experiments demonstrate our method's superiority over other competitors. Extensive analysis shows the effectiveness of our method in condition refinmmnnt.
2 PRELIMINARIES
Diffusion Modeling. Diffusion models are generative frameworks that consist of forward process. The forward (diffusion) process is Markov chain that transforms data zg into Gaussian noise øT through a sequence Gaussian transitions:
T
((F1e)(20)) = II (x ((=((t=1)= = Naax; v1 1,
1=1
where 3, is predefined variance schedule with 0 < < ... < Br The reverse (denoising) process reconstructs Ty from ør via:
Po(xur) = p(zr) II (o po(x1-1l0)) (aat--ll)=NN(ppp000 - (x, t)),
(2)
where and Eg are predicted by neural network. Since the true posterior (12))1/2)))))))) is intractable,
is approximated using g( (o) during training.
The model is trained to maximize data likelihood, which is approximated via variational lower bound, This can be reformulated as mcore-matching problem, e.g.,
Ex, (x():) [[ log (ox.)((;) so(x2,
2

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Autoregressive Modeling. Autoregressive (AR) models are generative frameworks that sequentially predict each element in data sequence by conditioning on all preceding elements. These models assume that each data point z; depends only the prior points zςi {21 22, .2..... 2 The conditional and joint probabilities can be expressed as:
x = p(z1,
Tn)
II p( ==1
(4)
The generation process starts from æ1 and proceeds sequentially Tn. with each step conditioned all previously generated elements. Related work is in Appendix A
THEORETICAL ANALYSIS ON AUTOREGRESSIVE IMAGE MODELING WITH DIFFUSION LOsS
3.1 DIFFERENCE OF DIFFUSION MODELS
Diffusion models demonstrate exceptional capabilities in generating high-quality visual content. Recently, autoregressive modeling integrated with diffusion loss has shown significant potential in image generation. We will elucidate the differences between standard conditional diffusion modeling and autoregressive modeling with diffusion loss.
Conditional Difflusion Modeling. In traditional conditional diffusion models, the reverse process is conditioned on single, static condition This can be formally expressed as:
-t-1 p(=++1111 1 [32e e),
where e represents a global condition that influences every step of the denoising trajectory. When dealing with images,
extend this to individual patch 2;:
-i,--1 1 c), Vj (1....,m).
where represents the number of patches. Each patch is denoised based on the same shared condition c, irrespective of its position within the image.
Autoregressive Modeling with Diffusion Loss. Autoregressive modeling with diffusion loss allows the condition to evolve autoregressively. Instead of fixed condition c, sequence of conditions {c} depends on preceding conditions {cei} including the initial condition co. i.e.,
Ct plc. p(c,
(7)
where Cęj denotes all conditions up to time - i.e., {Co, Cl,..., G--1) ). For each patch generation Ty the reverse process is still denoising process but is guided by the dynamic condition c;, i.e.,
T ~ p(z,,,-1/Zi,..
where represents the condition for i-th patch. After generating di is passed as input to the autoregressive model along with the history of conditions (c<i+1), }, enabling the prediction of the subsequent condition C++1.
3.2 CONDITIONAL DENOISING MODEL ERROR DEFINITION
Conditional Score Matching as an Upper Bound. Score matching is central training diffusion models, and its loss is linked to the Wasserstein distance between generated and real data (Kwon et al. 2022). Conditional score matching refines this by incorporating conditioning. Understanding how conditional score matching relates to standard score matching is key to justifying its use. This section establishes that the standard score matching loss upper-bounded by conditional counterpart. This result supports the use of conditional score matching, suggesting it might lead to more controlled training process.
Theorem 1 (Conditional Score Matching Upper Bound). The standard score matching loss is
upper-bounded by the conditional score matching loss:
Ex,<p((x.) Vx. log pe - so(x,, t)][2]
< Eewpe (xxepppxx)xx |/ (xelc) so 1)
See Appendix Cioor the proof, which uses the law of total probability and Jensen's 's inequality.
3

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The conditional score matching loss serves an upper bound for the standard score matching loss. Consequently, minimizing the conditional score matching loss indirectly constrains the standard score matching loss from above.
Error in Conditional Score Matching, To analyze the error in conditional score matching, we build upon the score matching error definition from (Li Yan 2024):
T E
score = T Ex (x)s 87 (X)
(10)
To understand how conditioning affects the error structure, we need to decompose the conditional and unconditional score matching losses. Expanding these losses into their component terms allows us to identify and analyze the specific contributions of conditioning to the overall error.
Lemma (Expansion of Score Matching Loss). Expanding the square term in the score matching loss, we ger:
(ap-pi(a)) V log so(2,, }]]]
=e V|,,, log px:)) + (1so (s0( 1), Va, log pog))
(11)
Similarly, for conditional score matching:
(ex Va, (orpl())lc) so(2: 1]]
= -eepaeope [IVE,, log (oge( + [|s0(1e,) 1)1 (86 (), Vx. log logle())))) (12)
This expansion separates the loss into terms related to the score, the estimated score, and their interaction, facilitating a more gramular error analysis (Detailed Proof in Appendix(D.l)
Definition (Conditional Error Term f2). To specifically measure the impact conditioning on the true score's 's magnitude, define the conditional error term the change in the expected squared norm of the true score due to conditioning, relative to the unconditional case:
T
Σ Bes-pe(o)) (elp))
[IV (1a((()))))11)) - 1|V log (xx))))
(13)
T
This term quantifies how much the expected squared norm of the true score changes when we move from unconditional to conditional score matching. positive e would suggest that conditioning increases the magnitude of the true score, potentially indicating more complex or refined score function (Detailed Proof in Appendix (.2).
Definition 2 (Simplified Conditional Error Term Zc). For a simpler metric focused purely on the magnitude of the conditional true score, we define the simplified conditional error term ट,:
[IV=.. log (x.
66 Pe-pale))xx--iaa...
(14)
& directly measures the expected squared norm of the conditional score. Analyzing Cc. along with 6e, will help understand the behavior of the true score in conditional settings (Detailed Proof in Appendix D.3
3.3 CONDITIONAL CONTROL TERM ANALYSIS.
We first investigate the uniqueness of the conditional control term under standard diffusion assump- tions and Classifier-Free guidance. As described in Classifier-Fres guidance (Ho & Salimans 2022
Liu et al. 2023), the conditional reverse diffusion process for sampling is given by:
p(z--1lz3 1 A A/ + of (221)
Øt-1 u(xx)) of V p(c((())))) + a
(15)
4

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This shows that conditional control introduces additional term oi u, log p(c(2e) to the mean of the reverse process, compared to the unconditional diffusion sampling. To understand the impact of this conditional term, we define f((,) = V.. log Pe (z, |c,)1l2. We hypothesize that the difference between the expected value of (c:) and the expected value of the unconditional score norm isolates the contribution of this conditional control term. This is formalized in the following lemma:
Lemma 2 (Uniqueness of Conditional Control Term). Under standard diffusion assumptions and Classifier-Fie Free guidance, the difference between the expected squared norm the conditional score function and the unconditional score function isolates the contribution of the conditional control term. Let (c)) = ,V,,, log Pt (Ja/ce)]]² Then,
B[[(a)]- - R. log 2 logr(c/=:)1
(16)
where the expectation taken over #n pt(zz) and Pe(c). This lemma indicates that f f(e) through its expected difference with the unconditional score norm, precisely captures the impact of the conditional guidance term V p(ca(c)))) in the diffusion denoising process. (Proof in Appendix E
3.4 CONDITION REFINEMENT THROUGH PATCH DENOISING.
Building upon the observation that incorporating conditions can amplify errors in diffusion models, we propose an optimization strategy focused on refining the condition using patch-related corrections. Specifically, we introduce a mechanism where information from each newly generated patch is propagated to the condition of the subsequent patch through iterative update process, C++1 =cc).. This autoregressive approach aims to refine the condition during the denoising process iteratively.
To formalize our approach, we first establish the foundational assumptions under which our model operates.
Assumption 1 (Markov Property A Assumption), The reverse diffusion process adheres to the Markov property, where each state -;-1 is conditionally dependent only on the current state Zt,
Assumption (Gaussian Distribution Assumption). The conditional probability distribution pr(x(-1/2)) in the reverse diffusion process is assumed to be Gaussian, expressed as: pt (21-111;)) A2(1-1;1(21))0 o!
Assumption (Small Variance Assumption). As the number of time steps T becomes sufficiently large, the variance of of the conditional distribution pr (= is assumed to be sufficiently small, approaching zero as T'' increases. Furthermore, for simplicity, we approximate the variance each step to be equal and denote it as o².
With the above assumptions, model the patch refinement process an iterative update to the condition:
C+11 = T(c)), i
(17)
where 0:+1 represents the condition at the + 1)-th iteration, corresponding to the refinement based on the i-th patch. T is a diffusion function that governs the transition from the current condition state e; to the next state c;+1, I. encapsulating the information propagation from the generated patch to the subsequent condition. The index E N denotes the iteration step, analogous to discrete time steps. The condition update process forms a discrete-time Markov chain, as the future state c depends only the present state c;: P(c++1 [c, C1-1. co) P(c++1/c.).
From the Gaussian expansion norm in Equation equation 15 we observe that the probability dis- tribution of I, is influenced by æ; itself. Our primary goal is understand the trajectory of the conditional probability gradient as the condition e; iteratively refines through the diffusion reverse process. Therefore, we proceed to analyze how the conditional probability gradient evolves with the iterations of e, within a standard normal conditional distribution setting.
Proposition (Condition Refinement via Patch Denoising). In the diffusion denoising process, autoregressively refining the condition through patch-relates corrections using the iterative update G+1 T(cx) leads improved conditional generation quality. (DetalylD Detailedd Proof in Appendix
5

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3.5 AUTOREGRESSIVE MODELING CAN REFINE CONDITION
An autoregressive process is defined as follows:
c+1 = Σ
- + E441,
iEN
(18)
==0
Assumption 4 (Basic Assumptions for Autoregressive Process). We provide set of standard assumptions:
1. Σi=o aal] < co and suPteN |a,| < 1, i.e., the sequence (a,) is convergent.
2. {=:)2 are independent and identically distributed, following N(0 o?).
3. pa (xa(c) has continuous second-order derivatives with respect to
4. V (rrall)))) < K for some constant > O uniformly holds, i.e., is bounded.
5. (X, -I) is a separable complete metric space.
Lemma 3 (Markov Property (Meyn Tweedie 2012) (Belltt 2006), Under Assumption by defining the state vector C; = (c,, Ca-1...., Ct the sequence (c, JeN forms a strong Markov chain. Specifically:
1. The transition probability kernel P(c,41 lc,)) the augmented state space satisfies the Feller properly.
2. There exists a unique invariant probability measure π E P((P) such that
/ xxP
TP =P=m- i.e.
P(Alc)t(dc) = I(A), AA E
(19)
3. There exist constants C > and E (0, 1) such that for any initial distribution E P((P)):
|| - 1||TV Cp" π T|| Vn No
(20)
where || || denotes the total variation norm, and pm denotes the n-step transition probability kernel.
In particular, for any E No, we have:
[|C(c,) - T//TV S Cp"
(21)
where C(c,) denotes the distribution Ca. Proof can be found in Appendix
Lemma 4 (Regularity of Conditional Probability (Durrett 1996)). Under Assumption 4 there exist constants such that:
1. pr (zzlcc)) 2 for all (Ie, ci) E Xx
2. [...(( - (e.p(()))(22) s Dllc1 - cell all Iz. C1, C2 EX
Proof can be found in Appendix
Lemma (Bounded Derivative Theorem). On a fixed bounded closed interval [a, b), if the second derivative is bounded, then there exist constants M1, M2 > 0 such that:
1. sup,, (x)e)))) M1, i.e., the first derivative bounded.
2. sup,, |p((re|c)] M2. i.e., the funcitooo is bounded.
Proof can be found in Appendix
Theorem 2 (Descent of Gradient Norm in Autoregressive Process). Under Assumptions and
Lemmas 4 there exist constants M > 0 and (3 E (0, 1) such that for any 3e E X and E No:
Lz, (x s M + m
(22)
where is constant representing the stationary gradient norm (Proof in Appendix G

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4 AUTOREGRESSIVE CONDITION OPTIMIZATION
Although autoregressive methods provide contextual information, they inevitably accumulate extrane- ous noise, leading "condition inconsisteney".
Why Optimal Transport? We employ Optimal Transport (OT) to rectify this distributional drift for three theoretical reasons:
Geometric Correction: Unlike overlap- based metries (c.g.. g.. KL divergence). OT quantifies the geometric cost required to transform the noisy generated distribution back to the ideal one.
2. Least Action Principle: Formulating the refinement as a Wasserstein Gradient Flow identifies the optimal path eliminate inconsistency while preserving valid semantic information.
Convergence: The framework guarantees theoretical convergence to the stationary ideal distribu- tion, effectively acting as a mathematically grounded "denoising" step for the condition.
The full algorithm is provided in Appendix
4.1 CONDITION INCONSISTENCY IN AUTOREGRESSIVE GENERATION
The autoregressive condition generation process, as defined by Equation equation 7 sequentially con- structs conditions, aiming to capture contextual de- pendencies. However, this sequential nature can lead to conditions that are not only influenced by relevant preceding patches but also by accumulated informa- tion that is extrancous to generating the current patch. This phenomenon, which we term "condition incon- sistency", arises because the autoregressive process, while capturing dependencies, does not inherently guarantee that each generated condition c, is opti- mally focused on information strictly necessary for the corresponding patch z₁-
OT Refinement
Denoise
Refined Condition
Update
Noisy
Sinkhomn Solver
Input Condition
Latent
Sampled Prior
Lemma (Condition Information Inconsistency and Extraneous Information Accumulation), Let G;
Figure 1: The autoregressive model predicts an initial condition, which is processed by the OT Refinement module using a sampled prior derived from Algorithm The resulting refined condition then guides the Denoise MI.P for latent generation.
['((e)) be the autoregressively generated + bo(cc-1) condition for patch De, and c πI; (c.) be its pro- jection onto the minimal sufficient information sub- space I (derived from vec)). The generated e, inher- ently contains extraneous information component 1/ C. c₁. This component т is generally non-zero (i.e., [E[]|=|13] > 0), accumulating from (I-TI;) )ol(c-1) and noise components outside Iī The actual conditional distribution p(a,, (c:) deviates from the ideal p(xx)(()) ), and the conditional score Vz, log.(æz,(..) is perturbed from its optimal form under c.
Proof. Let I C Rdd denote the minimal sufficient information subspace for generating patch 2,, and कzz (.) be the orthogonal projection onto this subspace. The ideal condition of satisfies:
ci = मIę (c:) where I = (
(23)
for some basis functions } encoding relevant dependencies from preceding patches zzi. The autoregressive condition generation follows a Markov process:
C; 4 (c]- + Te(Es)
(24)
where by 2d + dd is the learned transition operator and Ho modulates the noise injection. The extraneous information component can be quantified through subspace decomposition:
7) = (1 (1-TI;)0
TT;I)GG )c
(c,, 12
(25)
K+1
7

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where {ux} forms orthonormal basis for R with the first K vectors spanning I. The -o of extrancous information ||
mell2 satisfies:
B||| E [ मτ; ( tr(Tol))
(26)
The first term represents propagated extraneous information from previous conditions, while the second term quantifies newly introduced noise. For the denoising process Dj at timestep t, the conditional score function becomes perturbed:
V 22, log ((z)(c))
V. log p(a))c)) Jn Ve, logp(a))c)
(27)
Idcal score
Perturbation term
where Jn, is the Jacobian of the perturbation. The extraneous information induces O(I deviation from the ideal denoising trajectory. The accumulated effect over N patches yields total inconsistency:
Exptal
E. (0)xx00trlls
(28)
where O denotes the Sinkhorn divergence with regularization parameter A. This completes the proof of condition information inconsistency.
4.2 OPTIMAL TRANSPORT FOR CONDITION REFINEMENT VIA WASSERSTEIN GRADIENT FLOW
Building upon the condition inconsistency analysis in Lemma 6 present a principled solution through optimal transport theory. Our approach establishes direct connections between the Wasser- stein gradient flow framework and condition refinement in autoregressive generation.
Proposition 2 (Optimal Transport as Wasserstein Gradient Flow). The condition refinement process can be formulated as a Wasserstein gradient flow that minimizes:
F(P.) = W:2 Peo
(29)
where Pe- denotes the ideal condition distribution and T-1 represents the inverse process of infor- mation accumulation in Equation equation 24 The solution admits an implementable discrete-time scheme through JKO iterations (Jordan et al. 1998):
p(k+1) aar min (( +TF((P
(30)
P
Proof. Let (RRd) denote the space of probability measures with finite second moments. We consider the energy functional:
F(P)= _ AE=r
(31)
where <(c) ||c T Tx))) encodes the inverse process regularization. The Wasserstein gradient flow O.P: = - F(P,F(P)) can be discretized via the Beddan-Kinerlehrer--OO00 (JKO) scheme:
p(++1) arg min { (P, + FPp)
(32)
The first term maintains proximity to previous iterates while the second term drives the distribution toward both the ideal condition and inverse-procee consistency.
Substituting our specific energy functional yields the update rule in Proposition 2
The optimal transport plan between ((k) and p(k+1) corresponds to the McCann interpolant:
(k++) = (N)
Th
VW2
2 lam + (V ]
(33)
Implementing this requires solving the regularized OT problem:
inf
E
(34)
-ET(PtM),P.
where i is the independent coupling and controls entropy regularization. This leads to the Sinkhorn
algorithm implementation described in Proposition
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Theorem (Convergence of Wasserstein Gradient Flow). Under the assumptions of Proposition 2
the Wasserstein gradient flow defined by the energy functional F(P) converges to the ideal condition
distribution Pc.. Specifically, for any initial distribution Pe (0) E PP2())), the sequence of distributions TR=1 generated by the JKO scheme satisfies:
{P()) 1c0
(2(P( P..) p Pcc).
(35)
where p < 1 is the contraction rate determined by the regularization parameter À and the step size Th.
Proof Sketch. The proof follows from the contractive properties of the Wasserstein gradient flow for convex energy functionals. By the JKO scheme and the regularization term AE -ll----
the sequence ( } forms Cauchy sequence in the Wasserstein space P2 (R(RR).. The contraction rate P arises from the strong convexity of the energy functional F(P) and Lipschitz continuity of the gradient flow.
Remark 1. The inverse process regularization T- directly counters the extraneous information
accumulation characterized in Equation equation By Theorem improvement in patch generation quality:
3 the refinement ensures monotonic
(*)), p(z, et)) ),
S p (pEr) p(aa)ec) p(a)]c())
(36)
where P < quantifies the contraction rate of our OT-based refinement operator.
This theorem ensures that the proposed Wasserstein gradient flow refinement process monotonically reduces the Wasserstein distance between the autoregressive condition distribution p( and the ideal condition distribution P... The contraction rate p quantifies the refinement efficiency, with smaller values of p indicating faster convergence.
Table 1: Comparison of different methods on various metrics on ImageNet 256x 256 condi- tional generation. Bascline (CDM) denotes baseline of conditional diffusion modeling.
EXPERIMENTS
5.1
EXPERIMENTAL SETTINGS
Method
FID |
18
Pre.
Rec.
Our autoregressive model is directly based GPT-XL, while the denoising module is implemented using the
LDM-4 Rombuch et al.
2013
3,60
247.7
0.87
0.48
U-VIT-I
2:29
263.9
0.81
0.62
MAR-based denoising module. For the Variational Au-
DIT-XL
DIETT
Halmmaath
Petinis
1.73
276.2
276.5
toencoder (VAF) component, we use the KL.-16 version of LDM (Rombach et al. 2022). Our experiments are conducted on the ImageNet dataset (Deng et al. 2009).
MDTWCALZZ (Loi 2t 2023
1.58
314.7
0.79
0.65
GIVT Tichankir 1 2024
3.35
0.84
0.53
MAR
1.55
303.7
0.81
0.62
De MMMMK
1.47
305.8
0.83
0.62
RAR Yu brat
1.50
306.9
with image resolutions set 256 256. For evaluation,
Baselin (CDM)
3.26
259.6
0.8
0.58
Baseline (AR)
2.02
282.6
0,80
0.39
we adopt Fréchet Inception Distance (FID) (Heusel et al.)
MAR [Li al [224g
1.55
303.7
0.81
0.62
2017). Inception Score (IS) (Salimans et al.| (016)) as well as Precision and Recall metrics (Dhariwal & Nichol, 202T). During training, the noise schedule follows a cosine shape and consists of 1000 steps. The
Oun (AR) (MAR)
1.52
317.6
0.82
0.60
1.31
324.2
learning rate set to 1 10-3, with a total of 400 epochs and a batch size of 2048. The models are trained with a 100-epoch linear learning rate warmup. We use an exponential moving average (EMA) in parameters with momentum of 0.9999.
5.2 PERFOMANCE COMPARISON
Table shows a comparison of our method against state-of-the th-ehe-rt approaches on ImageNet 256 x 256
conditional generation. Our method achieves the best FID score of 1.52, outperforming MAR (Li al. 2024a) (1.55), MDTv2-XL/2 (Gao et al. 2023
(1.58), and DiffIT (Hatamizadeh et al. 2024) (1.73).
Based on MAR. it can further reach 1.31. In IS,
Figure 2: Qualitative results 256 x 256 Ima- geNet class-conditional generation. These images are generated by Ours.
method also achieves the highest score. This demon-
strates that our model produces samples with higher

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fidelity and better alignment with the real image distribution. For Precision and Recall, it attains 0.81 and 0.63, respectively, remaining competitive with other methods. Compared to the baseline, our model exhibits significant improvements across all evaluation metrics, highlighting the effectiveness of our approach. Qualitative results are shown in Figure 2
5.3 SCALABILITY ANALYSIS
Table Comparison scalability across dif- ferent model sizes on ImageNet 256 x 256. Our method consistently achieves lower FID and higher IS compared MAR.
To investigate the scalability and robustness of our proposed method, we conducted additional evalua- tions across varying model sizes and higher image resolutions. These experiments aim to verify whether the benefits of our Condition Refinement approach persist as the model capacity increases and the gen- eration task becomes more challenging.
Size
Method
FID
IS
MAR et 2024a
2.31
281.7
208M
Ours
1.96
290.5
MAR (Li et al Ours
2024a
1.78
296.0
479M
1.59
301.5
Scalability across Model Sizes. We evaluated method against the strong baseline MAR (Li et al. 2024a) on ImageNet 256 256 using three different model scales: 208M, 479M, and 943M parameters. As presented in Table 2] our method consistently outperforms MAR across all model sizes. Notably, performance gap widens as the model size in- creases, suggesting that our autoregressive condition optimization effectively leverages larger capacities for superior generation quality.
MAR (ii al. Ours
2024a
1.55
303.7
943M
1.31
324.2
Table 3: Performance comparison on high- resolution ImageNet 512 x 512.
Method
FID +
IS 1
MAR eL 2024a Ours
1.73
279.9
1.58
302.3
figh-Resoolttin Generation. To further assess the generalization capability of our method, we extended our evaluation to a higher resolution setting on ImageNet 512 x 512 (using a model size of approximately 481M parameters). Table 3 demonstrates the superiority of our approach in high-resolution synthesis. Our method achieves an FID of 1.58 compared to 1.73 for MAR, indicating that OT-based condition refinement remains effective in mitigating inconsistencies even in higher-dimensional spaces.
5.4 CONDITION ERRORS ANALYSIS
12:
Baseline
Interdtty
SNK (50)
10
70
Figure [ggureepresente presents the denoising analysis, show- ing Signal-to-Noise Ratio (SNR) and Noise In- tensity time for both our method and base- line. The demoising process proceeds from right to left, with time steps decreasing denoising progresses. Our method consistently achieves higher SNR, with a widening gap in later stages. Similarly. the right panel shows the Noise Inten- sity. where both methods show reduction in noise denoising progresses. Consistent with the SNR analysis, our method exhibits a marginally lower Noise Intensity, especially in the earlier time steps, i.e., later stages of denoising. These results highlight the efficacy of our proposed OT-based refinement of conditional distributions and effectively mitigate the potential inconsistencies introduced by purely autoregressive methods.
Danobing Diedtiot
80
Devosits Drecticn
50
Base inc
200
Step
Figure 3: Analysis of Signal-to-Noise Ratio (SNR, Left) and Noise Intensity (Right) during denoising process of our method and the baseline. All analyses are computed in the image space after VAE decoding.
CONCLUSION
In this work, our analysis shows that patch denoising autoregressive models mitigates condition errors, stabilizing condition distribution and enhancing generation quality. Autoregressive condition generation further refines conditions, exponentially reducing error influence. To address "condition inconsistency". introduce refinement method based on Optimal Transport and prove that casting as Wasserstein Gradient Flow ensures convergence. Experimental results and analysis show the effectiveness of method.
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A RELATED WORK
A.1 DIFFUSION MODEL.
Diffusion models have emerged as a powerful generative framework, surpassing GANs (Goodfellow et al. 2020) and VAEs (Kingma & Welling, 2014) in stability and sample quality. DDPMs Yang
al. 2024b) introduced noise-based training reconstruction paradigm, later linked theoretically to Score Matching and DAEs (Vincent 2011). However, early diffusion models suffered from slow sampling due to numerous iterative steps. Improved DDPMs (Pang et al., 2024) refined noise scheduling, while DDIMs (Wei et al., 2024) accelerated generation through a non-Markovian
formulation. LDMs (Rombach et al. 2022) further optimized efficiency by applying diffusion in a dimensioall latent space. Diffusion models also exhibit theoretical advantages over GANs, notably their implicit minimization of the Wasserstein distance (Kwon et al. 2022), leading to better convergence and robustness. Enhancing conditional control remains key research focus: Classifier--ree Diffusion Guidance (Wahid et al. (025) enables flexible conditioning without external classifiers, and structure-aware adaptations (Li & Yan 2024) improve efficiency for structured data, Various applications extend their utility: Palette (Saharia et al. 2022) enhances image restoration,
GLIDE (Nchhol et al., 2022) improves text-guided synthesis, CDMs (Ho et 2022) refine images progressively, and ControlNet (Zhang et al.) 2023) integrates structural conditions for enhanced controllability. Detecting diffusion -eefusion-ggeerrteed images increasingly challenging, with studies like
Schaefer et al. (Corvi et al. 2023) highlighting the need for robust detection methods.
A.2 AUTOREGRESSIV IMAGE GENERATION
Autoregressive models, despite their effectiveness, face computational constraints due to sequential generation (Zhou al. 2024 Sun et al. 2025 Zhou et al. 2025b). Optimization efforts focus
on efficiency and scalability: LlamaGen (un et al. 2024a)
leverages large-scale training to sur- (Tian al. 2024) reduces inference latency via next-scale prediction. Spatial alignment strategies like ImageFolder (Li et al., 2024b)
pass diffusion models in quality and efficiency, while VAR
improve autoregressive modeling, and Emu3 (Wang et al., 2024) unifies token prediction across modalities. Expanding autoregression to multimodal tasks requires bridging discrete and continuous data representations. Lumina-mGPT (Liu et al.. 2024) employs a decoder-only Transformer for high-quality text-to-image synthesis, while MMAR Yaang et al. (024a) models continuous tokens to enhance understanding and generation. Traditional vector quantization autoregressive models is being reconsidered: VQ-free autoregression (Li et al. (222aa) introduces diffusion-basse per-token probabilities for efficiency, and LatentLM (Sun et al. (024h) integrates next-token diffusion for multimodal synthesis across image, speech, and text.
B LIMITATIONS
While our research provides novel theoretical insights and algorithmic advancements, it is important to acknowledge certain limitations. Specifically, our experimental evaluation has not been conducted on large-scale models due to the substantial computational resources required for such validation. Instead, our focus has been on a rigorous theoretical analysis and the development of scalable algorithms. Despite the absence of experiments on very large models, the generality and applicability of our method have been theoretically established, and our experiments on general settings support the soundness of the proposed approach. We believe future work can extend these evaluations to more resource- resource-intensive settings to further verify empirical performance at scale.
C PROOF OF THEOREM I (CONDITIONAL SCORE MATCHING UPPER BOUND)
We begin by establishing two foundational lemmas required for the proof.
Lemma 7 (Bayes' Theorem for Conditional Scores). For any measurable sets c and X4, the posterior distribution satisfies:
pe(c)x:) = (e(c))p(xxlc))
Pe Pe(c)p((x(lc)
(x()))
[P
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where the second equality explicitly shows the marginalization over d.
Lemma 8 (Jensen's s Inequality for Convex Functions). For any convex function f Rd and random variable Y with finite expectation:
(E|)) S [f())
Equality holds if and only if f is affine linear on the support of or Y is constant almost surely.
Step-by-Step Proof:
Using these lemmas, we proceed with the main proof.
Step Marginal-Conditionn Decomposition. Express the marginal distribution through condi- tioning variables:
Pr (x(x) =
(Law of total probability)
Differentiate both sides under the integral sign (valid under Dominated Convergence Theorem conditions):
[x,pe(x))= 1 Vx
Step 2: Score Function Representation.
Using Lemma
decompose the marginal score:
(x,P((x,)
Xx, log Pe(e))
Pr (x(x))
J Po(c) V (xPe((xlccdd
(x(:)
Pa Palc)pi (x,/c)
Xx. log pi (x)
Pr (xx)
p=(c(x))
(elop( [vx, (xxe/()))))]
where the critical step (line applies Lemma 7 to identify the posterior distribution.
Step 3: Jensen's Inequality Application.
Substitute into the unconditional loss:
Ex, V log - =sel? Exi |E ( log (-(e]] se |
< Ex, Belx, |/ Pe(lcc) se 2 (by Lemma 8
Here we specifically apply Lemma g with:
f(y) = lly lell (convex since = V (ogp((x:le)
|| is convex)
Step 4: Law of Total Expectation. Convert the nested expectation to a joint expectation:
Ex, Eej, Ex [.] E.rp. (xxx.xp.(11))
Thus, we obtain the final inequality:
Ex.
V logp: - 80 < Exx, V llggt(-lcc - soll
Tightness Analysis. By Lemma 8 equality holds iff Vx, log Px constant c-a.s,, which
requires (x(x/(c) = (x,) ) for all c in the support of pe. This corresponds to statistical independence X,
e.
D DETAILED DERIVATIONS FOR CONDITIONAL SCORE MATCHING ANALYSIS
D.1 PROOF OF LEMMA
Following the score matching framework from (Vincent 201T), we begin by expanding the squared norm in both unconditional and conditional score matching objectives.
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Unconditional Case:
For the unconditional score matching loss:
VV logpt(s))
so 50(21, 1)1"
(V, loge)( - s0( t))² (Va, logp((0)) - so(ar, t))
t))
= =Eg; Va, log Pe (xxx))
[S E\ll
True score norm
Learned score norm
- 2 (s0( f), Vx, P(
Aligament term
This follows directly from the identity ||a b112 =|aa||++ [|al2++11b)) 1[61|=-2a" 2aTb.
Conditional Case:
For conditional score matching, expanding the L2 norm of the training error,
have:
>/ logpe( sa 1)(²
- (onmata
Vi log
+ | t)l - (so t), V., log
| log log |1 f)]1 2 (20 1). Va, logre
( Peppe(clee)
Eenpelelsa)
|IV (2 + [I5 t)1 21 pr (s t), Vx, logp((z)lc))
D.2 DERIVATION OF ERROR DIFFERENCE & (DEFINITION
Loss Difference Analysis:
Subtracting the unconditional loss from the conditional loss (after
expansion):
Re,,, [Loss]
Regg [Loss]
Exg Ee me
|IVa, log (x)l6))11 Ez, V/, (og))())))
+ Br, ||s0|| 1|2 Er, ||s0|?
Be (80, V log (ogae())))) Ea, (so, V log px
Vanishes by tower property
The cross-terms cancel due the tower property expectation:
Ez, Eelz, (se. V logpe(z;(e))
= Ex, (se Eejzr V log px
En, (se, V Vog (x2))
where we used the identity V (og = Egju, V log po (1rlcc) from (Li & Yan 2024
Final Error Expression:
Thus, the difference reduces to:
T Σ Eg. Eeja, || [oog - |/V log (x)))l 6 T
This quantifies the excess "score energy" induced by conditioning, similar to variance decomposition probability theory.
D.3 PROPERTIES & (DEFNIITION 2
From Definition 2
we can relate ?. to c. using the law of total variance:
2 = T
Ex, [[+))
This decomposition reveals that e, contains both the intrinsic score energy from the unconditional distribution and the additional energy e, from conditioning.
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E
CONDITIONAL CONTROL TERM UNIQUENESS PROOF
Proof. We prove Lemma through three key steps:
Step 1: Bayesian Score Decomposition
Using Bayes' rule pc(xxlc) = ple (p)p((e))
we derive:
p(c)
Vx, logpe(zrlc) = Vx, logp(c)()) + Zz, logpo(z;) Vze p(c)
=0
v. log p(ck)))))))) + Vx, (o.))(r))
Step 2: Cross-Term Cancellation The squared norm decomposes as: vV, (xxlc()) = VV, log + 2V log
2(V logp(c)))) Va, logr((z)))
Taking expectation over pt (1x) and po(c): BKV 0e logp(c)zz), Ve, log Pog)((x)))] = Ex. Ee (V logp(c/<xs) Va, log (og))()))) xell
Ezz (Bl mi logp(c/<=)]] V., (x+)
=0
where the inner expectation vanishes because E. Vx, log p(c2())))) = Va, Eelp(cla))] = V,, 0.
Step 3: Variance Propagation
Combining results from Steps 1-2:
c. E V lo. E| v., (x.)(()))("
E o V logp(c/2:)]1
The scaling factor of emerges from the reverse process parameterization in (Ho & Salimans 2022 Liu et al. 2023). where the conditional mean adjustment contains an explicit of multiplier. This completes the proof that isolates the conditional control term's 's contribution.
F PROOF OF CONDITION REFNNEMENT VIA PATCH DENOISING (PROPOSITION I
To understand the long-term behavior of the condition refinement process, invoke the Markov Chain Stationary Theorem. The conditional probability density function p(æ(c) then given by: aBy invoking (Meyn & Tweedie 2012)), establish that с; iterates, its distribution converges stationary distribution. (The details of the lemma we used can be found in Appendix(K] Consequently, our analysis shifts understanding how the gradient of the conditional probability evolves when e; follows a normal distribution. To compute the gradient of the conditional probability density function p(zlc) with respect r, the joint distribution of a and follows multivariate normal distribution:
(x) )))~((
~~Ni .V
1--x(11) ((1)
Oux
Oze
(e))
Oze
Jee)
The conditional probability density function p(zlc) is:
(xr (eu + aus (c - Mee))
p(z|c)=
/ 2T
exp
0² 2(0
acc
Taking the logarithm, we obtain the log- liellikel function:
log p(xlc)
log 2m 2
(ze)
Oec
(x (Mz
Gxs (c 22
aec
2(0xz ob
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Differentiating with respect to z, while ignoring constant terms, yields:
V. logp(w/)) I (f1 + Cus (c He))
dia
Ozx
dec
The squared norm of this gradient is given by:
VV logp(p/c) x
(1z +
(c pte)
Dec o2e
Ogz
The conditional mean and the conditional variance primarily influence the gradient behavior. The conditional mean is:
Me (c- Me),
Oxe
dec
where He represents the mean c. As c iterates and reaches its stationary distribution, He converges to constant, which we denote u Mable Consequently, the conditional mean stabilizes to fixed value.
The conditional variance Tee
مem is determined by the covariance structure of the joint distribution. Since this variance does not depend on e in its stationary distribution, it remains unchanged. As reaches its stationary distribution, the deviation of from the conditional mean gradually diminishes while the variance remains constant. It leads to gradient magnitude decay, indicating attenuation of the conditional probability gradient over iterations.
G PROOF OF DESCENT OF GRADIENT NORM IN AUTOREGRESSIVE PROCESS (THEOREM 2
Proof.
Remark 2 (State Space Representation). By introducing the state vector C; (c,, c.ll.....
we can represent the original process as vector-valued first-order autoregressive process:
C--p1) :e-1:.....----1)
Ci+1 Ac + ei+1
ag
ai
ap-2 2
ap-1
Ei-1
1
0
0
0
A
Ci+1
In this representation, the conditional distribution of C depends only C,. From Assumption (I), the spectral radius of matrix A is less than 1, which ensures the stability of the process.
Let (e( fx (dd)). The log-likelihood gradient be decomposed as:
V Vz,P((Za(c.)
Vx, logg((z((cc)
(r(xx((a) lci)
(e.Pe( pr(ar((;) (e(2)(T))
(e.P((
o. (x.
(r (T)
Stationary Term
Using the triangle inequality for norms, we get:
(x. (x(()(()) p((
V ( (x(
2.. log
+
Stationary Term
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For the non-stationary term, we have:
(x( pa(F)(
Pr Pr(2el (T) P.. Pi p.(pl/
p((
(T) Vo, pe ( (x)
82
(1p (T) Vv.
V Ve, [T] + V (e (Tp pr (zz c;)]
82
[nee (V,, Va, m)]l + |V ( ( - p. (m
Using the Lipschitz property of both the conditional probability density function (x( and its gradient V.,Pt (x, with respect (with Lipschitz constant L), and the upper bound M2 for pe(x)(()) from Lemma 4
the above inequality becomes:
1
M2L |c: - call + II (t call]
82
L
M2 + px
cxll
We consider c, to be representative value from the stationary distribution æ, for simplicity we can consider Cp = E,íc]. Applying the geometric ergodicity of the Markov chain (Lemma 9 in Appendix H which gives ||c, cx S Var, (c)p', we arrive at:
T. M2 + [[..)( Σ 82
Var
For the stationary term, from Lemma 5 we have:
V (x)l))
V 2v.P((-ITT
Mi
<
= m
8
8
where constant.
Combining the estimates for both terms, we have:
M1
V log log
M2 + V..
Var, (ar,
82
MMp' +m
where M 1/2 (M₂ [VV.
Varx and = p e 1).
Thus, we obtain the desired exponential decay estimate for the gradient norm as the autoregressive process iterates. This estimate holds uniformly for all v, E X.
H GEOMETRIC ERGODICITY AND CONVERGENCE TO STATIONARY MEAN
Lemma 9 (Geometric Ergodicity and Convergence Stationary Mean), Assume that the Markov chain {cr h20 is geometrically ergodic with stationary distribution and let Ca = Ex [c]. Then, there exist constants > O and 0 < p < such that for all 2
|c; cxll S Cp's (c)
where Var, (c) = E2 |||c 12
Explanation: Lemma formalizes the geometric convergence of the Markov chain state e; towards the stationary mean Cx due to geometric ergodicity. This property ensures that the influence of the initial condition diminishes exponentially over time, allowing us to bound the distance ||c: - cx by a geometrically decaying term proportional to / /Var, (c). This justifies the transition in the eighth line of the derivation, replacing ||c cxll with term that decays geometrically with i.
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1 PROOF OF REGULARITY OF CONDITIONAL PROBABILITIES
1. Existence of lower bound: Since the conditional probability density is quadratically continuous and differentiable and is defined on tight set, there is a minimum by the extreme value theorem. Since the probability density is always positive, the minimum must greater than zero.
2. Using the boundedness of the second-order derivatives in our Assumption
for Vx 3イ O,
Consider the difference of derivatives:
Vx,Pe(Zalci) (x,pe(zz/(2))
By the median theorem in multivariable calculus, there exists point between c2 such that:
V (xellc) - (a. (x, - 02)
By taking the value of the paradigm of the above equation, we have:
V (v, Pi (z, c2)ll |I = | - 02)l
Using Multiplicative Inequality for Norms(Ia 6|| - ||a|| [[b]]and the boundedness the second-order derivatives, we obtain:
I Pe (x, (x.e(()) < (e (I 02) s K [[(1 e2)l
Thus, taking L= M, we get the final inequality:
[V Va,p((1))02)) & Dll(e 02)ll
J PROOF OF THE MARKOV PROPERTY
This paper mainly the last geometric traversal of the theorem and therefore focuses proving the geometric traversal of Markov chains. The proof is divided into three steps.
Drift Conditional Verification: Constructing Foster-Lyapunov Functions V(x) = 1 |x|2. for Vz
(X)(Xx))) = = = E' en +1 a,, |0 2 + o²
|an (1 |x| (1 11-1an 1-, 02)
(V +
wherel sup, |a,12 < 1C |as|
oc), b 1 1 0
2. Compactness:for VR > 0, the set (a: V(x) S R} is compact because it is equivalent to the closed ball |x|2 < R 1}.
3. Irregularity and continuity: Since the noise term obeys normal distribution, the transfer probability has a positive density everywhere, which guarantees strong Feller and irregularity of the chain.
According Meyn-Tweedis theory, the above condition guarantees geometric ergodicity.
K LEMMA OF MARKOV PROPERTY
Lemma 10 (Markov Chain Stationary Theorem). If a random process has transition matrix P and ergodic (i.e., any states aperiodic and irreducible), then:
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1. The limit of the n-step transition matrix exists and is given by:
T(1) mT2)
x (
((1) π
lim Pa
T(1) (2)
2. The stationary distribution (()).
.2)... satisfies the equation:
+()) = Σ R.P.s
3. a is the unique non-negative solution to the stationary equation, with T() 1.
L AUTOREGRESSIVE CONDITION OPTIMIZATION ALGORITHM
The full algorithm integrates three key components: (1) autoregressive condition generation, (2) diffusion-bass denoising, and (3) optimal transport refinement. The pseudocode below specifies the detailed computational workflow.
Algorithm 1 Autoregressive Condition Optimization (ACO) with Denoising Integration
Require: Initial condition cy Φ0(c;-1, <<ss;; Diffusion model (Dr) with noise levels {B:} Target latent distribution Ps+, OT parameters \, e, Kinki Learning rate schedule {n% gradient clipping threshold 5
Ensure: Optimized condition et, generated latent z!r)
1: Initialize: c()) + co. (0) 10,
2: for k 0 to K do
3: Denoising trajectory:
4: for = T to do
5: (k, + (2(2(1,0), (()) (DDIM }pdate}
6: end for
7: Inverse process alignment:
8:
((c("))= (kc(k) (2(1 +allV,T-111
9: Optimal transport computation:
Sample reference latents {zj} ~ P..
11:
Compute pairwise cost matrix: Cmn
(k.0) 2m
+사 ||c(0)
T
Latent matching
Condition consistency
Entropy-regularized OT: 13: Initialize (0) t 1, (0) +-
to Kinng, do for (1) Kallll D B..0 17
15:
16:
( + K.(ul(T)), J00y
pik)
end for
18:
(R) & diag(u(Kum) K. diag(u!Ksm)
Gradient computation & update:
20:
VeLor + (()) o Dc(k)
ac
86 VoLrey + acte)
22: Viotal <- Clip(V Crip(V.Lor + VeLreg:7 T)
23:
c(++)) +- (K)) 1i6 Viotal
end for
25: Return cg +- (K), zi t- (K.0)
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L.1 IMPLEMENTATION DETAILS
Target Distribution Estimation: Maintain an EMA of generated latents:
P!u = (1
1 + ((-(( -
6(2b (0)
with = 0.1 and buffer size B = 2048.
Adaptive Entropy Regularization: Schedule e during Sinkhorn iterations:
k
Emax (Emax Eminn)
K
Stochastic Optimization: Use Adam optimizer with:
Tie = 7/o min(1, 1, v Kwaarm/
where kwar 100 controls learning rate warmup.
L.2 CONVERGENCE ANALYSIS
The algorithm maintains the convergence guarantee in Theorem 3
through:
1. Monotonic Improvement: For Lyapunov function
V4- = (2(P2 (.. + Aeg (((c))
we have Ve+1 S Vk T/s VV O
2. Error Propagation Bound: Approximation error from Sinkhorn iterations satisfies
(*) >"I1e Cp
with c+0 where 8 is the minimum cost matrix entry.
3. Stability Condition: Gradient clipping ensures Lipschitz continuity:
| ||2 S r(1 LT
where Ly-1 is the Lipschitz constant of the inverse process.
M BOUNDED THEOREM
Function (r(zalc) in the fixed interval [a, b] has second order derivatives V2 (2epe and 22, (xxpcc)) is bounded, so we have K > O,
V2, (e,eel)) S K, Vxy E [a,
THE FIRST DERIVATIVE BOUNDED
We use the Mean Value Theorem to show that first-order derivatives are bounded. According to the mean value theorem, if 24, x (a, b). there exists a point & (xz, af), then:
Ve.Po(z//c) V...((n.))) = V2.,o(((c)(a) - xx).
(xrlol s C, so we have:
- -(( 1,, xx)1 s C1 -
This shows that V (xxlc)) is Lipschitz continuous and the Lipschitz constant C.So (I.lc) is bounded. Next, we give specifie boundedness estimates Taking a, we have:
V (v.P(( - V..Pr(alc) < CC -
Since Jx:, - b a, we get:
(x.p( S Vz,p((alc) C(b-ol
Therefore, there exists constant M1 (ale)( + C(b - a) such that for all De [a,
[z,Po(ze(o) Mi.
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THE ORIGINAL FUNCTION IS BOUNDED
According to the Fundamental Theorem of Calculus, we have:
pe(at(c) - pr(alc) = S. Va.Po(ylc)dy.
Since (aree( S M1 for all xa E (a, b). we can estimate the above integral:
(pe(cssc)) - pp((ale) = 1 V.,pe(ylc) dy S (o. dy S Mi - al.
Since [rg al s a, we have:
(e(xxcc) pr(alc)] S M1 a).
Therefore, the original function (= is bounded and:
(xx( c)l - [p(cc] Mı a)-
Thus, there exists the constant M2 = [p((ac)) + Mi (b - a) such that for all æt E [a, bj: (x. v M2-
Thus, in a fixed interval, bounded second-order derivative is bounded by bounded first-order derivative, and the original function is bounded by the proof.
N THE USE OF LARGE LANGUAGE MODELS
In this paper, ChatGPT was employed to assist in polishing the writing. The model was used as language aid improve clarity, grammar, and readability of the text, while ensuring that the academic content and arguments remain entirely the work of the author.
O TABLE OF NOTATIONS
To facilitate easier reading of the theoretical sections and provide quick reference for the mathemat- ical symbols used throughout the paper, we summarize the key notations in Table
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Table 4: Summary of Notations
Symbol
Description
Diffusion Autoregressive Basies
To, TT
The original data (image) and the Gaussian noise at time T.
#17
The sequence of latent variables in the forward diffusion process.
g( (x+-1)
The forward diffusion transition kernel.
po ( (2x))
The reverse (denoising) process approximated by the network.
so (x(2,)) t)
The score function predicted by the neural network.
A global, static condition (in standard conditional diffusion).
a
The autoregressively generated condition for the i-th patch.
The i-th image patch.
vei
The set of patches preceding i.e., { , -4-1
to, To
The transition operator and noise modulation in the AR condition process.
Error Analysis & Theory
cc
Conditional Error Term (Eq. 13). Measures the change in expected squared norm due to conditioning.
&
Simplified Conditional Error Term (Eq. 14). Directly measures the ex- pected squared norm of the conditional score.
T(c))
The transition function representing patch-based condition refinement.
of V., logp(c)
The conditional guidance term in the reverse process mean.
M, B, m
Constants and decay rate describing the descent of the gradient norm (Theo- rem 2).
Condition Refinement (Optimal Transport)
c
The ideal condition for patch 2;.
I
The minimal sufficient information subspace for patch V;.
wr;
Orthogonal projection onto the subspace IÇ"
m
Extraneous Information Component (Eq. 25). The deviation from the ideal condition (7): C; -c)).
Wa(:.))
The 2- 2-ssssernteinn distance between probability distributions.
F(P)
The free energy functional minimized by the Wasserstein Gradient Flow.
T-1
The inverse process regularization operator.
((k)
The probability distribution of the condition at refinement step k.
P
The contraction rate of the Wasserstein Gradient Flow (Theorem 3).
25


Reviews:
[{"review_id": "noLRUz8liw", "reviewer": "Reviewer_65aN", "summary": "The paper conducts a thorough theoretical analysis of autoregressive models with diffusion loss, contrasting them with standard conditional diffusion models. The central thesis is that the patch-by-patch denoising optimization in autoregressive frameworks serves as an effective mechanism for refining the guiding condition, leading to a more stable condition distribution and mitigating errors. This work provides a solid theoretical foundation for understanding and improving conditional autoregressive generation by framing condition refinement as a distribution-level optimization problem.", "questions": "1. Could you provide a more intuitive, high-level explanation at the beginning of Section 4 to bridge the gap between \"condition inconsistency\" and the OT framework?\n2. What is the computational overhead of the proposed OT refinement step? How does it affect the overall image generation latency compared to the baseline MAR model?\n3. The concept of \"extraneous information\" is central to your motivation. Is it possible to visualize this phenomenon? For instance, by projecting the condition vectors (c_i) before and after refinement into a 2D space, or by showing how the generated patch changes with and without refinement at an intermediate step. This would make the problem much more tangible.\n4. The OT optimization introduces several hyperparameters (e.g., λ, ηk). How sensitive is the model's performance to the choice of these parameters? Is there a robust range for their values?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper's primary strength lies in its rigorous theoretical contributions. The formalization of the condition refinement process as a Wasserstein Gradient Flow is both elegant and novel, providing a principled guarantee of convergence that is often missing in heuristic-based approaches. The detailed lemmas and theorems build a convincing mathematical argument.\n2. The paper is logically well-organized. It seamlessly transitions from a comparative analysis of diffusion models, to the definition of conditional error, to the proposal of the OT-based solution, and finally to empirical validation. The argument is self-contained and easy to follow for readers with the requisite background.", "weaknesses": "1. Limited Experimental Scale: As the authors acknowledge in Appendix B, the experiments are confined to the 256x256 resolution on ImageNet. While this is a standard benchmark, state-of-the-art generative modeling research is increasingly focused on higher resolutions and larger models. The absence of such experiments may leave questions about the method's scalability and generalizability.\n2. Readability and Accessibility: The theoretical sections are dense and assume significant familiarity with advanced mathematical concepts like Optimal Transport and Wasserstein Gradient Flows. While precise, this may limit the paper's accessibility. A more intuitive explanation or a high-level overview of why OT is the right tool for this problem could broaden the paper's impact.\n3. Lack of Ablation Studies: The paper presents the final model's performance but would benefit from ablation studies that isolate the impact of the core contribution—the OT-based refinement. For instance, an experiment comparing the full model against a version without the WGF optimization would more clearly quantify the gains from this specific module.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oCpYoEqqm0", "reviewer": "Reviewer_Tnpe", "summary": "This paper focuses on condition error issues in autoregressive image generation with diffusion loss. It first conducts a theoretical analysis of diffusion models and autoregressive models with diffusion loss, demonstrating that patch denoising optimization in autoregressive models can effectively mitigate condition errors and form a stable condition distribution, while autoregressive condition generation refines conditions to make condition error influence decay exponentially. It further proposes a OT-Based refinement approach, theoretically proving that formulating this refinement as a Wasserstein Gradient Flow ensures convergence toward the ideal condition distribution. Experiments on ImageNet show the superiority of the proposed method over existing diffusion and autoregressive models with diffusion loss methods.", "questions": "1. See the weaknesses section for detail.\n\n2. The proposed theory and method are developed only in the context of autoregressive models with diffusion loss. It remains unclear whether the approach has broader applicability beyond this specific model class.\n\n3. In Table 1, the distinction between the two baselines is unclear, and it is not stated which baseline “Ours” is built upon. The difference between Ours and Ours (MAR) is also ambiguous. Without proper citations or explanations, the table is confusing to readers.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The theoretical framework for autoregressive image modeling with diffusion loss is both sound and novel. The theory is rigorous and clearly connects diffusion loss to autoregressive conditional modeling. The mathematical exposition is clear and technically solid.\n\n2. The proposed ideas of autoregressive patch-wise denoising and OT-based condition refinement are conceptually well-motivated.\n\n3. The paper provides a rigorous theoretical analysis demonstrating that the patch-wise denoising optimization in autoregressive models effectively mitigates condition errors.\n\n4. It further establishes a mathematically consistent framework linking energy optimization to Optimal Transport (OT) regularization, offering a clear and unified theoretical explanation of the condition error phenomenon.", "weaknesses": "1. The empirical validation does not fully match the strength of the theory. The main theory predicts (i) conditional score norm decays exponentially as AR iterations progress and (ii) OT refinement decreases condition inconsistency (Sinkhorn divergence) monotonically. The paper lacks direct empirical plots that verify these claims.\n\n2. Lack of important experiments at higher resolutions, such as the ImageNet 512 × 512 experiment.\n\n3. The comparison against stronger and more recent baselines (after 2025) is missing, weakening the empirical significance of the claims.\n\n4. A figure of the framework is needed to outline the designed methods, including autoregressive patch denoising and OT-based methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hIMEmOS04g", "reviewer": "Reviewer_KiVa", "summary": "This paper addresses an important and quite novel problem in autoregressive (AR) image generation. The authors point out that the conditional errors accumulated from previously generated patches may cause inconsistency, and they theoretically analyze how the patch denoising process in AR models can alleviate such conditional errors. Moreover, they prove that the influence of conditional error decays exponentially with iterations. The proposed conditional optimization method based on Optimal Transport (OT) and formulated as Wasserstein Gradient Flow (WGF) is elegant, and the paper shows it can converge toward the ideal conditional distribution.", "questions": "Mainly about the points above, especially clarification of baselines and computational complexity.\n\nOverall, I think this paper has strong theoretical contribution and interesting methodology. With a bit more experiment and clarification, it could be a quite nice work.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "（1）The theoretical part is quite impressive. Especially Theorem 2 gives a deep understanding about how the conditional influence (the gradient norm) exponentially decays, which brings new insights into the stability of AR generative models. The combination of OT and WGF for refining conditional distribution looks creative and convincing.\n\n（2）The experimental results are strong. On ImageNet 256x256, the FID score reaches 1.31, which is very competitive compared with existing works.", "weaknesses": "（1） Algorithm 1 seems to describe a nested loop structure. I am a bit worry that the computation cost could be large, maybe even K times T slower than the standard AR model. Some clarification or runtime comparison could be helpful.\n\n(2) It is a bit unclear what “Baseline (CDM)” (FID 3.26) and “Baseline” (FID 2.02) exactly mean. Does “Baseline” refer to the AR model without OT refinement? Since the paper’s best FID (1.31) is quite good, some ablation study would help to show how much improvement really comes from the proposed OT method, rather than from the backbone MAR model itself.\n\n（3）The experiments are mainly done on ImageNet. It would be nice if the authors could test at least one more dataset or show more model comparison to make the results more solid.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fugoOE7SDi", "reviewer": "Reviewer_i7t2", "summary": "The paper theoretically and empirically investigates how autoregressive image generators incorporating diffusion loss can mitigate conditional inconsistency during generation. It provides a rigorous analysis comparing conditional diffusion models and AR diffusion with diffusion loss, proving that autoregressive patch denoising refines condition distributions and that the influence of condition errors decays exponentially during iteration. To address residual condition inconsistency, the authors introduce a condition refinement method based on Optimal Transport formulated as a Wasserstein Gradient Flow, proving convergence toward the ideal condition distribution. Experiments on ImageNet show superior FID and IS scores over existing diffusion and AR baselines, supporting their theoretical findings.", "questions": "Same as weaknesses section.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper presents a solid theoretical framework that connects autoregressive modeling, diffusion loss, and conditional refinement through explicit mathematical proofs and lemmas.\n\n2. It introduces a novel condition refinement approach using Optimal Transport and Wasserstein Gradient Flow, offering a principled solution to condition inconsistency.\n\n3. The quantitative performance on ImageNet is competitive or superior to major diffusion and AR baselines, confirming practical benefits of the proposed method.", "weaknesses": "1. The experimental scope is limited: evaluations are conducted only on ImageNet 256×256 and with moderate-scale models, without testing scalability to larger or multimodal setups.\n\n2. Despite extensive theory, the method’s implementation details (e.g., computational cost of OT refinement, convergence sensitivity to hyper-parameters) are underexplained.\n\n3. Some notation and theoretical transitions are difficult to follow and may reduce accessibility for non-mathematical readers.\n\n4. The impact of the OT regularization term on actual generation diversity and efficiency is not sufficiently analyzed, only FID/IS metrics are shown.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)