NeurIPS 2026 论文列表

Conference on Neural Information Processing Systems

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6,437篇论文

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    6,09894.7%

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MvFFN: Multi-view Floor-Plan Feed-Forward Network for Unposed Wide-Baseline Panorama Layout Reconstruction

Yuguang Li, Yichuan (Ethan) Deng, Zixuan Liu, Ivaylo Boyadzhiev, Xiuyuan Wu, Xinyu Zou, Linda Shapiro, Alex Colburn

展开摘要

Reconstructing accurate camera poses and floor-plan layouts from sparse, unposed, wide-baseline RGB panoramas remains a challenging open problem. Direct multi-room floor-plan prediction is especially difficult in this setting: pose, geometry, semantics, and global layout structure are tightly coupled, yet a complete floor-plan is hard to learn as a single global output. Our key insight is to keep the model's predictions strictly local. Multi-view Ordered Wall Instance Segmentation (MOIS) is designed around this concept. Each panorama column predicts only the wall it observes plus the next two adjacent walls in clockwise order. The global wall partition, room layouts, and shared coordinates emerge from deterministic aggregation of overlapping local predictions. Building on MOIS, we propose Multi-view floor-plan Feed-Forward Network (MvFFN), which jointly predicts coarse camera poses, dense global geometry, and per-wall semantic outputs. A downstream Layout Chaining (LC) post processing pipeline then assembles these predictions into multi-room floor-plans with shared global coordinates, consistent wall identities, and connected room structure. The dense MOIS predictions recover cross-view wall identity and local within-room connectivity more accurately than composed modular baselines, and the assembled layouts substantially outperform prior methods on wall and junction level accuracy. We also introduce the ZInD-CrossView dataset, which augments ZInD with globally unique wall instances and cross-view segmentation labels for this task.

Poster
Mutual Predictability Decomposition: Learning Interpretable Cross-Set Structure via Bi-Directional Prediction

Zixi Qin, Wanqing Li, Yuanhao Zhuo, Guoxin Su

展开摘要

Understanding relationships between variable sets is fundamental in machine learning, particularly in medical data analysis. Existing methods typically provide only coarse shared/private partitioning of variables, limiting their ability to reveal finer-grained predictive structures across views. We propose Mutual Predictability Decomposition (MPD), a framework that decomposes the variables in each view into three subsets: mutually predictable variables that can be inferred across views, auxiliary variables that are themselves not cross-view predictable but are necessary for predicting the mutually predictable subset, and unpredictable variables that do not contribute to cross-view predictability. MPD is formulated as a coupled bi-directional prediction problem with sparse variable assignment, and optimized through a differentiable relaxation implemented using gated neural networks. Experiments on synthetic and real-world medical datasets demonstrate that MPD recovers interpretable cross-view structures, improves cross-view prediction, and provides more informative decompositions than existing relationship analysis methods. Code will be available on GitHub upon acceptance.

Poster
Mutable Transcripts: Mitigating Context Pollution through Editable Conversation State

Dan Barry, Andrew Hines

展开摘要

Contemporary large language model (LLM) chat systems treat conversation history as an immutable sequence of turns that defines the model’s working context. However, user intent in real interactions is not static: it evolves through correction, refinement, and shifting constraints. This mismatch between dynamic intent and static transcripts can result in context pollution, where outdated or irrelevant information persists and continues to influence subsequent responses. We introduce mutable transcripts, a new interaction paradigm that enables users to revise prior turns through natural language edit requests, allowing the conversation history itself to be updated rather than appended. This reframes the transcript from a passive record into an editable representation of conversational state. We present a working prototype that integrates transcript-level revision into a standard chat interface and evaluate its feasibility through a controlled user study (n=17) and an illustrative transcript analysis of representative interaction scenarios. Participants significantly preferred mutable transcripts over standard chat across measures of clarity, confidence, and ease of use, with reduced intent to restart conversations. Transcript analysis of representative user study conversations shows that mutable transcripts can reduce conversation length and eliminate obsolete retained context. These findings provide initial evidence that user-driven revision of conversational history can improve interaction quality and help maintain a more current representation of user intent.

Poster
Muon Does Not Converge on Convex Lipschitz Functions

Tetiana Parshakova, Ahmed Khaled, Michael Crawshaw, Guillaume Garrigos, Robert Gower

展开摘要

Muon and its variants have shown strong empirical performance in a variety of deep learning tasks. Existing convergence analyses of Muon rely on smoothness assumptions, though arguably the most successful function class for developing deep learning methods (such as AdaGrad, Shampoo, Schedule-Free and more) has been the class of convex and Lipschitz functions. In this paper we question whether the classical convex Lipschitz model is a useful one for understanding Muon. Our answer is no. We show that Muon does not converge on the class of convex and Lipschitz functions, regardless of the choice of learning rate schedule. We also show that error feedback restores convergence of Muon and all the non-Euclidean subgradient methods with momentum. However, this theoretical fix using error feedback degrades the performance of Muon in two representative settings for image classification (CIFAR-10) and language modeling (nanoGPT on FineWeb-Edu 10B). Our conclusion is that convex Lipschitz theory, despite having a prominent role in the design of practical methods for deep learning, is not the most suited one for Muon. This suggests that Muon's success must come from structure absent from this model, most plausibly related to smoothness.

Poster
Multivariate Time Series Forecasting needs Cross Variable Loss

Kuiye Ding, Yifan Hu, Hanchen Wang, Hao Xue

展开摘要

Multivariate time series forecasting presents unique challenges because future variables often co-evolve under shared system dynamics. While existing studies mainly focus on cross-variable dependencies in historical observations, dependencies among future values are much less explored. Specifically, modern forecasting models largely follow the Direct Forecasting (DF) paradigm, generating multi-step forecasts with point-wise objectives that do not explicitly constrain cross-variable structure. In this work, we show that the DF objective is mismatched in the presence of cross-variable and lagged dependencies, revealing an objective gap. To address this issue, we propose **C**ross-**V**ariable **Loss** (CvLoss), a plug-in structural regularizer that constrains forecast residuals on a cross-variable graph. CvLoss penalizes inconsistent edge-wise residual differences over forecast patches, encouraging consistency across both synchronous and asynchronous interactions. Our experiments show that CvLoss consistently improves competitive forecasting models, outperforms representative learning objectives, and is compatible with a variety of forecasting backbones. Code is available at: https://anonymous.4open.science/r/CvLoss.

Poster
Multistage Defer Trees for Hybrid Interpretability: If at First You Can't Succeed, Tree Again

Zakk Heile, Hayden McTavish, Margo Seltzer, Cynthia Rudin

展开摘要

Recent work has shown that well-optimized individual decision trees can match complex black box models in some settings, primarily in noisy domains. For the remaining settings, however, complex ensembled compositions of trees often achieve higher accuracy at the cost of interpretability, leaving practitioners with difficult modeling decisions along an accuracy-interpretability tradeoff. Ideally, we would like to classify as much of the data as possible with one or a small number of trees, achieving interpretability for most samples while maintaining state-of-the-art accuracy. We introduce Multistage Defer Trees: a sequence of sparse decision trees that each make predictions for most samples, while deferring a small proportion to the next tree in the sequence or, ultimately, to a black box. We demonstrate that we can train this model class to match the performance of complex tree-based ensembles while routing most samples through only one or a small number of sparse decision trees. We discuss a range of techniques for training these models while maintaining simplicity. Our method expands the accuracy--interpretability frontier in settings where single-tree methods remain insufficient, demonstrating that even when complex models are necessary, they need not be fully opaque.

Poster
Multiscale Microenvironment Vector Space Projection for Uncovering Diverse Pathological Biomarker

Zixiu Ding, Wenxuan Hao, Liya Ding, Yuan Dai, Jiazhen Yang, Shuwen Han, Lingxiang Jia, Zunlei Feng

展开摘要

Due to the challenge of uncovering biologically meaningful spatial tissue patterns from standard H\&E-stained whole-slide images (WSIs) without relying on multiplexed staining or fixed-scale modeling, a unified framework is proposed for spatial biomarker discovery that integrates cell classification, adaptive multi-scale microenvironment construction, and colocalization pattern analysis. Cell-level predictions assign discrete, semantically meaningful types that serve as the basis for microenvironment modeling, where physical adjacency initializes local ecological features and a combination of spatial competition-based seed selection with homogeneity-constrained recursive region growing allows microenvironments to expand from locally pure regions in a data-driven manner, capturing heterogeneity across scales without predefined spatial ranges. Microenvironment-level cell composition features are then used to construct a colocalization feature space, explicitly modeling spatial relationships among cell types, and differential biomarkers are identified via a distributional comparison framework that systematically detects group-specific colocalization patterns. Experiments demonstrate that this framework reliably discovers discriminative and biologically interpretable spatial biomarkers from H\&E images alone, exhibiting robustness in tissues with complex structures and heterogeneous scales, and offering a flexible, extensible paradigm for spatial phenotypic biomarker discovery in standard histopathology.

Poster
Multiple Descent of Generalization Curve for Optimally Regularized Ridge Regression

Maxim Bochkov, Fedor Noskov

展开摘要

It is known that optimally regularized ridge regression can exhibit non-monotone dependence of the generalization risk on the sample size. We show that this behavior can be arbitrarily intricate in finite dimensions: for every $d \ge 2$, there exists a random design linear model in $\mathbb{R}^d$ whose generalization risk, viewed as a function of the sample size, exhibits at least $d - 1$ local maxima.

Poster
Multimodal LLMs Outperform Pathology Foundation Models in Cross-Domain Histological Similarity

Yishu Zhang, Yun Li, Daiwei Zhang

展开摘要

State-of-the-art pathology foundation models, trained on millions of histology tiles, can fail to preserve tissue similarity when comparisons cross slide or institution boundaries. We show that general-purpose multimodal LLMs, without being trained as pathology foundation models, consistently outperform these specialized models in cross-domain histological similarity judgments. Using a relative similarity framework that we release as the MOSAIC (Model Similarity Assessment across Institutions and Cohorts) benchmark, we evaluate 17 models across 6 datasets and find that pathology encoders often rank same-institution, different-disease tiles as more similar than same-disease, different-institution tiles, a clinically dangerous failure mode invisible to standard within-domain evaluations. LLMs appear less susceptible to this failure, likely because they perform semantic visual comparison of morphology and tissue architecture rather than relying on shortcut features tied to acquisition context. Scaling training data does not resolve the problem for pathology encoders, implicating the learning objective rather than data coverage. Our results expose a fundamental robustness gap in current pathology foundation models and establish multimodal LLMs as a viable alternative for cross-institutional retrieval, dataset harmonization, and multi-site quality control. Code and data will be released upon acceptance.

Poster
Multimodal Foundation Agents Should Use Brain Data as Privileged Supervision

Richard Csaky

展开摘要

This position paper argues that multimodal foundation models should use brain recordings as privileged internal-state supervision during training. External behavior alone leaves human-like cognition, robustness, and reasoning weakly constrained. Neural recordings can serve as a training-time modality that shapes representations, latent-state dynamics, private computation policies such as reasoning tokens, and robustness before distillation into agents that operate from ordinary sensory, language, and action streams. We develop this position through Foundational Human Modeling (FHM), a program centered on scaling generative brain models, aligning heterogeneous neural modalities through shared latent geometry, inserting brain tokens into unified multimodal token streams, and testing whether the privileged channel improves distilled agents after removal. The argument is grounded in privileged-information theory and prior work showing that neural or human-derived targets can improve robustness, semantic transfer, data efficiency, and behavioral similarity after auxiliary signals are removed.

Poster
Multimodal Context-Aware Human Motion Generation with Language, Vision, and Object

Junyu Shi, Yong Sun, Zhiyuan Zhang, Lijiang Liu, Yuxin He, Zhengjie Zhang, Qiang Nie

展开摘要

Motion generation has made substantial progress in synthesizing human motion from language, yet text alone remains ambiguous for specifying fine-grained spatial, temporal, and interaction details. Visual observations offer a powerful source of complementary context: even sparse images can reveal intermediate body states, object affordances, and interaction relations that are difficult to specify precisely in language. Whatmore, contextual constraints like visual, object, and partial-motion conditions can ground motion genetration with higher physical fidelity and controllability. We therefore study a unified multimodal context-aware framework for motion generation under diverse conditions. The key challenge is that heterogeneous conditions impose constraints at different spatial-temporal granularities and exhibit token- and phase-dependent relevance. Moreover, incorporating multiple modalities without a robust motion prior may entangle modality-specific semantics and cause negative transfer. In these regards, we combine global multimodal modulation with fine-grained motion token-level context selection, enabling the model to adaptively exploit relevant signals during generation. We further adopt progressive training that first learns a strong language-conditioned motion prior, then extends to multimodal context with different modality combinations. To support multimodal training, we introduce Mo900H, a large-scale benchmark integrating 21 motion datasets with over 900 hours of human motion. Our method reduces FID by 21.1% and 48.3% compared with SOTA methods on the HumanML3D and Mo900H datasets, while also improving motion captioning and enabling diverse multimodal-conditioned generation.

Poster
Multilingual Safety Alignment via Self-Distillation

Ruiyang Qin, Qingzhuo Wang, Dongrui Liu, Qiang Li, Zhihua Wei, Wen Shen

展开摘要

Large language models (LLMs) exhibit severe multilingual safety misalignment: they possess strong safeguards in high-resource languages but remain highly vulnerable to jailbreak attacks in low-resource languages. Current safety alignment methods generally rely on high-quality response data for each target language, which is expensive and difficult to generate. In this paper, we propose a cross-lingual safeguard transfer framework named Multilingual Self-Distillation (**MSD**). This framework transfers an LLM's inherent safety capabilities from high-resource (*e.g.*, English) to low-resource (*e.g.*, Javanese) languages, overcoming the need for response data in any language. Our framework is flexible and can be integrated with different self-distillation strategies. Specifically, we implement two concrete methods---on-policy MSD and off-policy MSD---both of which enable effective cross-lingual safety transfer using only multilingual queries. Furthermore, we propose Dual-Perspective Safety Weighting (DPSW), a divergence measure to optimize the distillation objective. By jointly considering the perspectives of both the teacher and the student, DPSW adaptively increases the penalty weights on safety-critical tokens while reducing the weights on non-critical tokens. Extensive experiments on representative LLMs across diverse multilingual jailbreak and utility benchmarks demonstrate that our method consistently achieves superior multilingual safety performance. Notably, it generalizes effectively to more challenging datasets and unseen languages while preserving the model's general capabilities.

Poster
Multilevel and Sequential Monte Carlo for Training-Free Diffusion Guidance

Aidan Gleich, Scott C Schmidler

展开摘要

We address the problem of accurate, training-free guidance for conditional generation in trained diffusion models. Existing methods typically rely on point-estimates to approximate the posterior score, often resulting in biased approximations that fail to capture multimodality inherent to the reverse process of diffusion models. We propose a sequential Monte Carlo (SMC) framework that integrates over the full denoising distribution to construct an unbiased Monte Carlo estimator of $p_\theta(y|x_t)$, avoiding the bias of existing training-free guidance methods. To ensure computational tractability, we incorporate variance-reduction schemes based on MultiLevel Monte Carlo (MLMC). We evaluate on three image generation tasks. On CIFAR-10 and ImageNet classifier guidance, our approach improves accuracy by 18 and 38 points respectively over the strongest training-free baseline. On CelebA-HQ joint-attribute guidance, three existing methods (DPS, LGD, TFG) used within our framework all outperform the closest SMC-based comparator (TDS) by 25-37 points, and improve over the strongest baseline by 2-14 points.

Poster
Multilateral Resistance-Guided Graph Message Passing for Trade Flow Prediction

Quansi Li, Haijing Yu

展开摘要

Trade flows are a central variable for studying international trade networks. Traditional economic models, grounded in economic theory and mathematical derivation, provide structured representations of bilateral trade flows. However, these representations often rely on fixed functional forms, have limited ability to incorporate high-dimensional features, do not directly capture trade inertia, and scale poorly to large and dynamic trade networks. In recent years, graph neural networks have been increasingly used to address some of these limitations because they are naturally compatible with networked trade data and offer strong scalability. However, the feature aggregation process in existing graph neural models remains weakly interpretable from an economic perspective. To address this limitation, we propose MRTGNN, a graph neural network model built on a theory-guided MRTGNN layer derived from the fixed-point equations of multilateral resistance in structural gravity. Theoretically, we prove that an $L$-layer MRTGNN can approximate $L$ iterations of a multilateral-resistance fixed-point operator on a compact domain, with an explicit bound on the iteration error. Empirically, we extensively evaluate MRTGNN on panel data covering 176 countries from 2002 to 2019 against various representative baselines. The full model achieves the best RMSE, $R^2$, and Spearman rank correlation across the reported baselines. Under counterfactual tariff shocks involving China, it produces a Network Spillover Ratio one order of magnitude higher than a generic graph-attention baseline. These results show that theory-guided message passing improves trade flow prediction while enabling more interpretable counterfactual reasoning in global trade networks.

Poster
Multiform Attack for Transferable Cross-Modal Person Re-Identification

Yunpeng Gong, Can Yang, Qingyuan Zeng, Dejun Xu, Zhiming Luo, Zhenzhong Wang, Min Jiang

展开摘要

Cross-modal person re-identification (ReID) adversarial attacks face a fundamental generalization dilemma: existing methods suffer from source-pair overfitting, where the learned perturbations become entangled with the feature covariance and alignment patterns of the source modality pair, thereby limiting their transferability under heterogeneous domain shifts. We argue that the core issue is the difficulty of reducing source-pair-specific bias while maintaining attack effectiveness. To address this, we propose Multiform Attack (MA). The framework first learns a universal attack direction via Mahalanobis-guided gradient optimization to capture the intrinsic covariance structure of the feature manifold, though this perturbation may still retain source-pair-specific bias. To overcome this bias and achieve cross-distribution generalization, the second stage leverages multiple heterogeneous source distributions to optimize for generalization, performing sparse, structure-sensitive residual correction in the discrete pixel space via attention-shift-filtered multi-objective evolutionary search. This two-stage design models the attack as a base direction plus a structured residual, which helps reduce overfitting to the source modality pair and improve transferability to unseen modalities. Extensive experiments show that MA achieves superior transferability across unseen modalities, datasets, and model architectures.

Poster
Multidimensional Observer Model and Perceptual Dimensions of Human Image Quality Assessment

sheng zhao, Weikai Lin, Yuhao Zhu

展开摘要

Judging image quality is not only ecologically relevant to everyday humans tasks, but also underpins many machine vision tasks such as image generation. This paper proposes a framework to understand the inherent perceptual space underlying image quality judgment in humans. We propose a multi-dimensional observer model that represents images as distributions in a latent perceptual space and that models human judgment as comparing noisy samples. Being constrained by neural representations in the primate ventral stream and fit to large-scale behavioral data, the model enables analysis of perceptual structure while matching the predictive power of existing metrics. Using this model, we find that the perceptual spaces needed to account for image quality judgment in humans are extremely low-dimensional compared to the image space even when considering its sparsity. The exact structure of the space (e.g., dimensionalities, information encoded) varies between low-level and high-level quality judgments, suggesting that, despite a shared retinal encoding in the beginning, humans selectively construct task-dependent perceptual spaces in visual decision making.

Poster
Multiclass Classification with Rare Useful Features: Fundamental Limits and the Optimality of Diversity Pursuit Higher Criticism

Gabriel Moryoussef, Alon Kipnis

展开摘要

We consider classification in high-dimensional feature spaces where only a vanishing fraction of features are useful and the number of classes may be large. While this rare-useful-feature regime is well understood in the binary setting, its multiclass version raises new questions: a feature can be useful for class discrimination without separating any specific pair of classes. We focus on a rare feature diversity model, in which most features are useless, but a small fraction promote discrimination among classes in a way that can be detected by a per-feature diversity test, such as one-way ANOVA or the Kruskal--Wallis test. For this setting, we propose Diversity Pursuit Higher Criticism (DP-HC), a feature-screening procedure that applies a higher-criticism threshold to the collection of p-values obtained from per-feature diversity tests. We justify DP-HC under a multiclass rare/weak Gaussian feature model with class-mean contrasts of fixed magnitude in random directions, extending the binary model of Donoho and Jin (2008). When the number of classes is fixed, our model recovers the binary phase transition. When the number of classes grows logarithmically with the number of features, however, we derive a new and strictly harder phase transition, driven by a second-order variance term that the fixed-class analysis does not capture. We show that whenever classification beyond chance is possible, DP-HC with ANOVA, followed by a simple classifier based on similarity to class means, achieves asymptotically perfect accuracy. Experiments on real multiclass classification benchmarks demonstrate substantial improvements, and numerical simulations confirm that the derived phase-transition curve accurately predicts the finite-sample behavior of DP-HC in the growing-class regime.

Poster
MultiTalk: Scaling Full-Duplex Speech Models to Long, Multi-Party, Bilingual Conversation

Ke Wang, Houxing Ren, Zimu Lu, Yunqiao Yang, Zhuofan Zong, Mingjie Zhan, Hongsheng Li

展开摘要

End-to-end full-duplex speech models have brought open-source machine conversation close to human fluency, yet existing systems fall short of real-world deployment in two entangled respects: long-context conversational robustness and multi-party interaction capability. Realistic settings, including meetings, group lessons, family dinners, and social-robot reception, are inherently long-horizon and multi-party at the same time, requiring a single model to perceive, attribute, contextualize, and respond across multiple speakers over extended durations. Progress along these axes is bottlenecked by both data and evaluation. On the data side, open multi-party conversational speech corpora total only a few hundred hours and are not designed for codec-frame-level full-duplex modeling. On the evaluation side, existing long-audio benchmarks focus on passive listening, while existing speech-to-speech benchmarks remain dyadic and short. In this work, we extend the Moshi paradigm along long-horizon and multi-party axes simultaneously, in both English and Chinese, with three contributions. First, we release an open data engine and a 57.6 k-hour corpus for long, multi-party, bilingual full-duplex dialogue. The engine produces parallel-stream audio with controllable length, participant count, conversational dynamics (turn-taking, overlap, backchannels, interruption, addressee shifts, long-range co-reference), and language (English, Chinese, and intra-sentential code-switching), exceeding all prior open multi-party conversational speech corpora by more than an order of magnitude. Second, we introduce MultiTalkBench, the first benchmark to jointly evaluate long-form full-duplex dialogue with an average duration of 32.6 minutes, multi-party, and bilingual full-duplex dialogue, with explicit probes for long-range entity tracking, topic coherence, and addressee selection. Third, we train a bilingual Moshi-style full-duplex model on the released corpus that sustains coherent multi-party English-Chinese conversation over extended durations, substantially outperforming open-source baselines including Moshi, MiniCPM-o-4.5, and Qwen3-Omni-30B-A3B-Instruct.

Poster
MultiSTEVE-1s: A Model Zoo and Interpretability Suite for Instruction-Following Vision Agents

Karolis Jucys, George Adamopoulos, Özgür Şimşek

展开摘要

A striking case of goal misgeneralization was previously observed in OpenAI's Minecraft agent VPT: it killed villagers standing under some leaves, mistaking them for trees. Although this agent was released publicly, enabling white-box interpretability research, few open-weight model organisms of misalignment exist outside the LLM space. In this work, we release MultiSTEVE-1s -- a model zoo of 140 fine-tuned versions of VPT, over 1,000 training checkpoints, and an interpretability suite for analysing them. Specifically, we use the STEVE-1 training procedure to add instruction-following capabilities to VPT with fixed hyperparameters and controlled variations in training randomness. We demonstrate the utility of MultiSTEVE-1s by showcasing the research it enables. First, for some training runs, the only difference is a least-significant bit flip in a single initialised weight. Others differ in the full randomness for weight initialisation and data. Yet, the single bit-flip setting produces agents that act nearly as differently from each other as the full randomness ones. Second, we use our interpretability suite to show that several known VPT attention heads retain their roles after STEVE-1 fine-tuning, while attention strength to the same behaviourally meaningful frame can vary substantially across agents and checkpoints. Finally, even though the agents are similarly capable in in-distribution tasks, the out-of-distribution behaviour of villager killing can differ substantially between them -- in one setting, an agent kills villagers less than 5\% of the time, while another agent kills them nearly 50\% of the time. Our results show the value of studying multiple similarly trained agents, rather than acting like a behavioural biology lab with only one rat.

Poster
Multi-step Consistency Models: A Complete Error Theory and Optimal Step Selection

Yanshu Wang, Rui Liu, Tong Yang

展开摘要

Consistency models achieve fast image synthesis by chaining a small number of neural evaluations along a diffusion trajectory, yet the distribution error of $N$-step sampling for $N \geq 3$ has lacked a rigorous characterisation. We establish a tight Wasserstein-$p$ error bound for arbitrary $N$: the total error equals a Lipschitz-weighted sum of per-step local consistency errors, $\sum_{k=1}^{N} L_x^{N-k} \delta_k$, making the propagation mechanism explicit. From this bound we derive a closed-form optimal step count $N^{\star}$ whose structure bifurcates on whether the spatial Lipschitz constant $L_x$ exceeds one: for $L_x > 1$, exponential error growth yields $N^{\star} \approx (\ln L_x)^{-1} \log(\beta\gamma(L_x-1)/(c \ln L_x))$; for $L_x < 1$, geometric-series saturation gives $N^{\star} \approx (\beta\gamma(1-L_x)/(c \, |\ln L_x|))^{1/(\gamma+1)}$. The two-regime structure accounts for ECT's best FID of $2.73$ at $N=2$: with $L_x \approx 0.85 < 1$ (Case 2), the formula gives $N^{\star} \approx 1.7$. Experiments on CIFAR-10 reproduce the predicted U-shaped FID curve with $N^{\star} = 8$ on a capacity-constrained model in which error accumulation is easier to see.

Poster