NeurIPS 2025 Papers

We propose a principled and effective framework for one-step generative modeling. We introduce the notion of average velocity to characterize flow fields, in contrast to instantaneous velocity modeled by Flow Matching methods. A well-defined identity between average and instantaneous velocities is derived and used to guide neural network training. Our method, termed the \textit{MeanFlow} model, is self-contained and requires no pre-training, distillation, or curriculum learning. MeanFlow demonstrates strong empirical performance: it achieves an FID of 3.43 with a single function evaluation (1-NFE) on ImageNet 256$\times$256 trained from scratch, significantly outperforming previous state-of-the-art one-step diffusion/flow models. Our study substantially narrows the gap between one-step diffusion/flow models and their multi-step predecessors, and we hope it will motivate future research to revisit the foundations of these powerful models.

This paper introduces a novel iterative method for missing data imputation that sequentially reduces the mutual information between data and the corresponding missingness mask. Inspired by GAN-based approaches that train generators to decrease the predictability of missingness patterns, our method explicitly targets this reduction in mutual information. Specifically, our algorithm iteratively minimizes the KL divergence between the joint distribution of the imputed data and missingness mask, and the product of their marginals from the previous iteration. We show that the optimal imputation under this framework can be achieved by solving an ODE whose velocity field minimizes a rectified flow training objective. We further illustrate that some existing imputation techniques can be interpreted as approximate special cases of our mutual-information-reducing framework. Comprehensive experiments on synthetic and real-world datasets validate the efficacy of our proposed approach, demonstrating its superior imputation performance. Our implementation is available at \url{https://github.com/yujhml/MIRI-Imputation}.

Rectified Flows learn ODE vector fields whose trajectories are straight between source and target distributions, enabling near one-step inference. We show that this straight-path objective reveals fundamental failure modes: under deterministic training, low gradient variance drives memorization of arbitrary training pairings, even when interpolant lines between training pairs intersect. To analyze this mechanism, we study Gaussian-to-Gaussian transport and use the loss gradient variance across stochastic and deterministic regimes to characterize which vector fields optimization favors in each setting. We then show that, in a setting where all interpolating lines intersect, applying Rectified Flow yields the same specific pairings at inference as during training. More generally, we prove that a memorizing vector field exists even when training interpolants intersect, and that optimizing the straight-path objective converges to this ill-defined field. At inference, deterministic integration reproduces the exact training pairings. We validate our findings empirically on the CelebA dataset, confirming that deterministic interpolants induce memorization, while the injection of small noise restores generalization.

A reliable estimate of the full conditional distribution of a multivariate response given a set of covariates is essential in many decision-making applications. However, misspecified or miscalibrated models can lead to poor approximations of the joint distribution, resulting in unreliable predictions and suboptimal decisions. Standard recalibration methods are largely restricted to univariate settings, and while conformal prediction techniques yield multivariate regions with coverage guarantees, they do not provide an explicit form of the underlying probability distribution. We address this gap by first introducing a novel notion of latent calibration, which assesses probabilistic calibration in the latent space of conditional invertible generative models such as normalizing flows and flow matching. Second, we propose latent recalibration (LR), a post-hoc model recalibration method that learns a transformation of the latent space with finite-sample bounds on latent calibration. Unlike existing recalibration methods, LR produces a recalibrated distribution with an explicit multivariate density function while remaining computationally efficient. Extensive experiments on both tabular and image datasets show that LR consistently improves latent calibration error and the negative log-likelihood of the recalibrated models.

Counterfactual explanations play a pivotal role in explainable artificial intelligence (XAI) by offering intuitive, human-understandable alternatives that elucidate machine learning model decisions. Despite their significance, existing methods for generating counterfactuals often require constant access to the predictive model, involve computationally intensive optimization for each instance, and lack the flexibility to adapt to new user-defined constraints without retraining. In this paper, we propose DiCoFlex, a novel model-agnostic, conditional generative framework that produces multiple diverse counterfactuals in a single forward pass. Leveraging conditional normalizing flows trained solely on labeled data, DiCoFlex addresses key limitations by enabling real-time, user-driven customization of constraints such as sparsity and actionability at inference time. Extensive experiments on standard benchmark datasets show that DiCoFlex outperforms existing methods in terms of validity, diversity, proximity, and constraint adherence, making it a practical and scalable solution for counterfactual generation in sensitive decision-making domains.

Rectified flow is a generative model that learns smooth transport mappings between two distributions through an ordinary differential equation (ODE). The model learns a straight ODE by reflow steps which iteratively update the supervisory flow. It allows for a relatively simple and efficient generation of high-quality images. However, rectified flow still faces several challenges. 1) The reflow process is slow because it requires a large number of generated pairs to model the target distribution. 2) It is well known that the use of suboptimal fake samples in reflow can lead to performance degradation of the learned flow model. This issue is further exacerbated by error accumulation across reflow steps and model collapse in denoising autoencoder models caused by self-consuming training. In this work, we go one step further and empirically demonstrate that the reflow process causes the learned model to drift away from the target distribution, which in turn leads to a growing discrepancy in reconstruction error between fake and real images. We reveal the drift problem and design a new reflow step, namely the conic reflow. It supervises the model by the inversions of real data points through the previously learned model and its interpolation with random initial points. Our conic reflow leads to multiple advantages. 1) It keeps the ODE paths toward real samples, evaluated by reconstruction. 2) We use only a small number of generated samples instead of large generated samples, 600K and 4M, respectively. 3) The learned model generates images with higher quality evaluated by FID, IS, and Recall. 4) The learned flow is more straight than others, evaluated by curvature. We achieve much lower FID in both one-step and full-step generation in CIFAR-10. The conic reflow generalizes to various datasets such as LSUN Bedroom and ImageNet.

Morphing is a long-standing problem in vision and computer graphics, requir- ing a time-dependent warping for feature alignment and a blending for smooth interpolation. Recently, multilayer perceptrons (MLPs) have been explored as implicit neural representations (INRs) for modeling such deformations, due to their meshlessness and differentiability; however, extracting coherent and accurate morphings from standard MLPs typically relies on costly regularizations, which often lead to unstable training and prevent effective feature alignment. To overcome these limitations, we propose FLOWING (FLOW morphING), a framework that recasts warping as the construction of a differential vector flow, naturally ensuring continuity, invertibility, and temporal coherence by encoding structural flow prop- erties directly into the network architectures. This flow-centric approach yields principled and stable transformations, enabling accurate and structure-preserving morphing of both 2D images and 3D shapes. Extensive experiments across a range of applications—including face and image morphing, as well as Gaussian Splatting morphing—show that FLOWING achieves state-of-the-art morphing quality with faster convergence. Code and pretrained models are available in https://schardong.github.io/flowing.

We address the open question of counterfactual identification for high-dimensional multivariate outcomes from observational data. Pearl (2000) argues that counterfactuals must be identifiable (i.e., recoverable from the observed data distribution) to justify causal claims. A recent line of work on counterfactual inference shows promising results but lacks identification, undermining the causal validity of its estimates. To address this, we establish a foundation for multivariate counterfactual identification using continuous-time flows, including non-Markovian settings under standard criteria. We characterise the conditions under which flow matching yields a unique, monotone and rank-preserving counterfactual transport map with tools from dynamic optimal transport, ensuring consistent inference. Building on this, we validate the theory in controlled scenarios with counterfactual ground-truth and demonstrate improvements in axiomatic counterfactual soundness on real images.

Wasserstein gradient flow (WGF) is a common method to perform optimization over the space of probability measures. While WGF is guaranteed to converge to a first-order stationary point, for nonconvex functionals the converged solution does not necessarily satisfy the second-order optimality condition; i.e., it could converge to a saddle point. In this work, we propose a new algorithm for probability measure optimization, \emph{perturbed Wasserstein gradient flow} (PWGF), that achieves second-order optimality for general nonconvex objectives. PWGF enhances WGF by injecting noisy perturbations near saddle points via a Gaussian process-based scheme. By pushing the measure forward along a random vector field generated from a Gaussian process, PWGF helps the solution escape saddle points efficiently by perturbing the solution towards the smallest eigenvalue direction of the Wasserstein Hessian. We theoretically derive the computational complexity for PWGF to achieve a second-order stationary point. Furthermore, we prove that PWGF converges to a global optimum in polynomial time for strictly benign objectives.

A key question in sequence modeling with neural networks is how to represent and learn highly nonlinear and probabilistic state dynamics. Operator theory views such dynamics as linear maps on Hilbert spaces containing mean embedding vectors of distributions, offering an appealing but currently overlooked perspective. We propose a new approach to sequence modeling based on an operator-theoretic view of a hidden Markov model (HMM). Instead of materializing stochastic recurrence, we embed the full sequence distribution as a tensor in the product Hilbert space. A generative process is then defined as maximum mean discrepancy (MMD) gradient flow in the space of sequences. To overcome challenges with large tensors and slow sampling convergence, we introduce spectral mean flows, a novel tractable algorithm integrating two core concepts. First, we propose a new neural architecture by leveraging spectral decomposition of linear operators to derive a scalable tensor network decomposition of sequence mean embeddings. Second, we extend MMD gradient flows to time-dependent Hilbert spaces and connect them to flow matching via the continuity equation, enabling simulation-free learning and faster sampling. We demonstrate competitive results on a range of time-series modeling datasets.

Multimodal learning has gained much success in recent years. However, current multimodal fusion methods adopt the attention mechanism of Transformers to implicitly learn the underlying correlation of multimodal features. As a result, the multimodal model cannot capture the essential features of each modality, making it difficult to comprehend complex structures and correlations of multimodal inputs. This paper introduces a novel Multimodal Attention-based Normalizing Flow (MANGO) approach to developing explicit, interpretable, and tractable multimodal fusion learning. In particular, we propose a new Invertible Cross-Attention (ICA) layer to develop the Normalizing Flow-based Model for multimodal data. To efficiently capture the complex, underlying correlations in multimodal data in our proposed invertible cross-attention layer, we propose three new cross-attention mechanisms: Modality-to-Modality Cross-Attention (MMCA), Inter-Modality Cross-Attention (IMCA), and Learnable Inter-Modality Cross-Attention (LICA). Finally, we introduce a new Multimodal Attention-based Normalizing Flow to enable the scalability of our proposed method to high-dimensional multimodal data. Our experimental results on three different multimodal learning tasks, i.e., semantic segmentation, image-to-image translation, and movie genre classification, have illustrated the state-of-the-art (SoTA) performance of the proposed approach.

Quantifying uncertainty in deep regression models is important both for understanding the confidence of the model and for safe decision-making in high-risk domains. Existing approaches that yield prediction intervals overlook distributional information, neglecting the effect of multimodal or asymmetric distributions on decision-making. Similarly, full or approximated Bayesian methods, while yielding the predictive posterior density, demand major modifications to the model architecture and retraining. We introduce MCNF, a novel post hoc uncertainty quantification method that produces both prediction intervals and the full conditioned predictive distribution. MCNF operates on top of the underlying trained predictive model; thus, no predictive model retraining is needed. We provide experimental evidence that the MCNF-based uncertainty estimate is well calibrated, is competitive with state-of-the-art uncertainty quantification methods, and provides richer information for downstream decision-making tasks

Normalizing flows are deep generative models that achieve efficient likelihood estimation and sampling through invertible transformations. A key challenge is designing linear layers that enhance expressiveness while enabling efficient computation of the Jacobian determinant and inverse. In this work, we introduce a novel invertible linear layer based on the product of circulant and diagonal matrices. This decomposition provides a parameter- and computation-efficient formulation, reducing the parameter complexity from $\mathcal{O}(n^2)$ to $\mathcal{O}(mn)$ by using $m$ diagonal matrices together with $m-1$ circulant matrices, while approximating arbitrary linear transformations.Furthermore, leveraging the Fast Fourier Transform (FFT), our method reduces the time complexity of matrix inversion from $\mathcal{O}(n^{3})$ to $\mathcal{O}(mn \log n)$ and matrix log-determinant from $\mathcal{O}(n^{3})$ to $\mathcal{O}(mn)$, where $n$ is the input dimension. Building upon this, we introduce a novel normalizing flow model called Circulant-Diagonal Flow (CDFlow). Empirical results demonstrate that CDFlow excels in density estimation for natural image datasets and effectively models data with inherent periodicity. In terms of computational efficiency, our method speeds up the matrix inverse and log-determinant computations by $1.17\times$ and $4.31\times$, respectively, compared to the general dense matrix, when the number of channels is set to 96.

The study of multimodality has garnered significant interest in fields where analyzing interactions among multiple information sources can enhance predictive modeling, data fusion, and interpretability. Partial information decomposition (PID) has emerged as a useful information-theoretic framework to quantify the degree to which individual modalities independently, redundantly, or synergistically convey information about a target variable. However, existing PID methods depend on optimizing over a joint distribution constrained by estimated pairwise probability distributions, which are costly and inaccurate for continuous and high-dimensional modalities. Our first key insight is that the problem can be solved efficiently when the pairwise distributions are multivariate Gaussians, and we refer to this problem as Gaussian PID (GPID). We propose a new gradient-based algorithm that substantially enhances computational efficiency for GPID based on an alternative formulation of the underlying optimization problem. To generalize the applicability to non-Gaussian data, we learn information-preserving encoders to transform random variables of arbitrary input distributions into pairwise Gaussian random variables. Along the way, we resolved an open problem regarding the optimality of joint Gaussian solutions for GPID. Empirical validation on diverse synthetic examples demonstrates that our proposed method provides more accurate and efficient PID estimates than existing baselines. We further evaluate on a series of large-scale multimodal benchmarks to show its utility in real-world applications of quantifying PID in multimodal datasets and selecting high-performing models.

Boltzmann Generators have emerged as a promising machine learning tool for generating samples from equilibrium distributions of molecular systems using Normalizing Flows and importance weighting. Recently, Flow Matching has helped speed up Continuous Normalizing Flows (CNFs), scale them to more complex molecular systems, and minimize the length of the flow integration trajectories. We investigate the benefits of using path gradients to fine-tune CNFs initially trained by Flow Matching, in the setting where a target energy is known. Our experiments show that this hybrid approach yields up to a threefold increase in sampling efficiency for molecular systems, all while using the same model, a similar computational budget and without the need for additional sampling. Furthermore, by measuring the length of the flow trajectories during fine-tuning, we show that path gradients largely preserve the learned structure of the flow.

To accelerate diffusion model inference, numerical solvers perform poorly at extremely small steps, while distillation techniques often introduce complexity and instability. This work presents an intermediate strategy, balancing performance and cost, by learning ODE integration using loss functions derived from the derivative-integral relationship, inspired by Monte Carlo integration and Picard iteration. From a geometric perspective, the losses operate by gradually extending the tangent to the secant, thus are named as secant losses. The target of secant losses is the same as that of diffusion models, or the diffusion model itself, leading to great training stability. By fine-tuning or distillation, the secant version of EDM achieves a $10$-step FID of $2.14$ on CIFAR-10, while the secant version of SiT-XL/2 attains a $4$-step FID of $2.27$ and an $8$-step FID of $1.96$ on ImageNet-$256\times256$. Code is available at \url{https://github.com/poppuppy/secant_expectation}.

Consistency models are a class of generative models that enable few-step generation for diffusion and flow matching models. While consistency models have achieved promising results on Euclidean domains like images, their applications to Riemannian manifolds remain challenging due to the curved geometry. In this work, we propose the Riemannian Consistency Model (RCM), which, for the first time, enables few-step consistency modeling while respecting the intrinsic manifold constraint imposed by the Riemannian geometry. Leveraging the covariant derivative and exponential-map-based parameterization, we derive the closed-form solutions for both discrete- and continuous-time training objectives for RCM. We then demonstrate theoretical equivalence between the two variants of RCM: Riemannian consistency distillation (RCD) that relies on a teacher model to approximate the marginal vector field, and Riemannian consistency training (RCT) that utilizes the conditional vector field for training. We further propose a simplified training objective that eliminates the need for the complicated differential calculation. Finally, we provide a unique kinematics perspective for interpreting the RCM objective, offering new theoretical angles. Through extensive experiments, we manifest the superior generative quality of RCM in few-step generation on various non-Euclidean manifolds, including flat-tori, spheres, and the 3D rotation group SO(3), spanning a variety of crucial real-world applications such as RNA and protein generation.

Diffusion models have quickly become some of the most popular and powerful generative models for high-dimensional data. The key insight that enabled their development was the realization that access to the score---the gradient of the log-density at different noise levels---allows for sampling from data distributions by solving a reverse-time stochastic differential equation (SDE) via forward discretization, and that popular denoisers allow for unbiased estimators of this score. In this paper, we demonstrate that an alternative, backward discretization of these SDEs, using proximal maps in place of the score, leads to theoretical and practical benefits. We leverage recent results in _proximal matching_ to learn proximal operators of the log-density and, with them, develop Proximal Diffusion Models (`ProxDM`). Theoretically, we prove that $\widetilde{\mathcal O}(d/\sqrt{\varepsilon})$ steps suffice for the resulting discretization to generate an $\varepsilon$-accurate distribution w.r.t. the KL divergence. Empirically, we show that two variants of `ProxDM` achieve significantly faster convergence within just a few sampling steps compared to conventional score-matching methods.

To understand feature learning dynamics in neural networks, recent theoretical works have focused on gradient-based learning of Gaussian single-index models, where the label is a nonlinear function of a latent one-dimensional projection of the input. While the sample complexity of online SGD is determined by the *information exponent* of the link function, recent works improved this by performing multiple gradient steps on the same sample with different learning rates — yielding a *non-correlational* update rule — and instead are limited by the (potentially much smaller) *generative exponent*. However, this picture is only valid when these learning rates are sufficiently large. In this paper, we characterize the relationship between learning rate(s) and sample complexity for a broad class of gradient-based algorithms that encapsulates both correlational and non-correlational updates. We demonstrate that, in certain cases, there is a phase transition from an "information exponent regime" with small learning rate to a "generative exponent regime" with large learning rate. Our framework covers prior analyses of one-pass SGD and SGD with batch reuse, while also introducing a new layer-wise training algorithm that leverages a two-timescales approach (via different learning rates for each layer) to go beyond correlational queries without reusing samples or modifying the loss from squared error. Our theoretical study demonstrates that the choice of learning rate is as important as the design of the algorithm in achieving statistical and computational efficiency.