Diffusion models have emerged as a powerful tool for image generation and denoising. Typically, generative models learn a trajectory between the starting noise distribution and the target data distribution. Recently Liu et al. (2023b) designed a novel alternative generative model Rectified Flow(RF), which aims to learn straight flow trajectories from noise to data using a sequence of convex optimization problems with close ties to optimal transport. If the trajectory is curved, one must use many Euler discretization steps or novel strategies, such as exponential integrators, to achieve a satisfactory generation quality. In contrast, RF has been shown to theoretically straighten the trajectory through successive rectifications, reducing the number of function evaluations (NFEs) while sampling. It has also been shown empirically that RF may improve the straightness in two rectifications if one can solve the underlying optimization problem within a sufficiently small error. In this paper, we make two key theoretical contributions: 1) we provide the first theoretical analysis of theWasserstein distance between the sampling distribution of RF and the target distribution. Our error rate is characterized by the number of discretization steps and a new formulation of straightness stronger than that in the original work. 2) under a mild regularity assumption, we show that for a rectified flow from a Gaussian to any general target distribution with finite first moment (e.g. mixture of Gaussians), two rectifications are sufficient to achieve a straight flow, which is in line with the previous empirical findings. Additionally, we also present empirical results on both simulated and real datasets to validate our theoretical findings.
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Deep Learning · Graph Neural Networks
Equivariance encodes known symmetries into neural networks, often enhancing generalization. However, equivariant networks cannot *break* symmetries: the output of an equivariant network must, by definition, have at least the same self-symmetries as its input. This poses an important problem, both (1) for prediction tasks on domains where self-symmetries are common, and (2) for generative models, which must break symmetries in order to reconstruct from highly symmetric latent spaces. This fundamental limitation can in fact be addressed by considering *equivariant conditional distributions*, instead of equivariant functions. We therefore present novel theoretical results that establish necessary and sufficient conditions for representing such distributions. Concretely, this representation provides a practical framework for breaking symmetries in any equivariant network via randomized canonicalization. Our method, SymPE (Symmetry-breaking Positional Encodings), admits a simple interpretation in terms of positional encodings. This approach expands the representational power of equivariant networks while retaining the inductive bias of symmetry, which we justify through generalization bounds. Experimental results demonstrate that SymPE significantly improves performance of group-equivariant and graph neural networks across diffusion models for graphs, graph autoencoders, and lattice spin system modeling.
Diffusion Generative Modeling for Spatially Resolved Gene Expression Inference from Histology Images
Applications · Genetics, Cell Biology, Health, etc
Spatial Transcriptomics (ST) allows a high-resolution measurement of RNA sequence abundance by systematically connecting cell morphology depicted in Hematoxylin and eosin (H\&E) stained histology images to spatially resolved gene expressions. ST is a time-consuming, expensive yet powerful experimental technique that provides new opportunities to understand cancer mechanisms at a fine-grained molecular level, which is critical for uncovering new approaches for disease diagnosis and treatments. Here, we present $\textbf{Stem}$ ($\underline{\textbf{S}}$pa$\underline{\textbf{T}}$ially resolved gene $\underline{\textbf{E}}$xpression inference with diffusion $\underline{\textbf{M}}$odel), a novel computational tool that leverages a conditional diffusion generative model to enable in silico gene expression inference from H&E stained images. Through better capturing the inherent stochasticity and heterogeneity in ST data, $\textbf{Stem}$ achieves state-of-the-art performance on spatial gene expression prediction and generates biologically meaningful gene profiles for new H&E stained images at test time. We evaluate the proposed algorithm on datasets with various tissue sources and sequencing platforms, where it demonstrates clear improvement over existing approaches. $\textbf{Stem}$ generates high-fidelity gene expression predictions that share similar gene variation levels as ground truth data, suggesting that our method preserves the underlying biological heterogeneity. Our proposed pipeline opens up the possibility of analyzing existing, easily accessible H&E stained histology images from a genomics point of view without physically performing gene expression profiling and empowers potential biological discovery from H&E stained histology images. Code is available at: https://github.com/SichenZhu/Stem.
Deep Learning · Generative Models and Autoencoders
The generative modeling of data on manifolds is an important task, for which diffusion models in flat spaces typically need nontrivial adaptations. This article demonstrates how a technique called `trivialization' can transfer the effectiveness of diffusion models in Euclidean spaces to Lie groups. In particular, an auxiliary momentum variable was algorithmically introduced to help transport the position variable between data distribution and a fixed, easy-to-sample distribution. Normally, this would incur further difficulty for manifold data because momentum lives in a space that changes with the position. However, our trivialization technique creates a new momentum variable that stays in a simple fixed vector space. This design, together with a manifold preserving integrator, simplifies implementation and avoids inaccuracies created by approximations such as projections to tangent space and manifold, which were typically used in prior work, hence facilitating generation with high-fidelity and efficiency. The resulting method achieves state-of-the-art performance on protein and RNA torsion angle generation and sophisticated torus datasets. We also, arguably for the first time, tackle the generation of data on high-dimensional Special Orthogonal and Unitary groups, the latter essential for quantum problems. Code is available at https://github.com/yuchen-zhu-zyc/TDM.
Miscellaneous Aspects of Machine Learning · Unsupervised and Semi-supervised Learning
We present SlotAdapt, an object-centric learning method that combines slot attention with pretrained diffusion models by introducing adapters for slot-based conditioning. Our method preserves the generative power of pretrained diffusion models, while avoiding their text-centric conditioning bias. We also incorporate an additional guidance loss into our architecture to align cross-attention from adapter layers with slot attention. This enhances the alignment of our model with the objects in the input image without using external supervision. Experimental results show that our method outperforms state-of-the-art techniques in object discovery and image generation tasks across multiple datasets, including those with real images. Furthermore, we demonstrate through experiments that our method performs remarkably well on complex real-world images for compositional generation, in contrast to other slot-based generative methods in the literature. The project page can be found at https://kaanakan.github.io/SlotAdapt/
Social Aspects · Fairness, Equity, Justice and Safety
Recent advances in diffusion models have significantly enhanced their ability to generate high-quality images and videos, but they have also increased the risk of producing unsafe content. Existing unlearning/editing-based methods for safe generation remove harmful concepts from models but face several challenges: (1) They cannot instantly remove harmful or undesirable concepts (e.g., artist styles) without additional training. (2) Their safe generation capabilities depend on collected training data. (3) They alter model weights, risking degradation in quality for content unrelated to the targeted toxic concepts. To address these challenges, we propose SAFREE, a novel, training-free approach for safe text-to-image and video generation, that does not alter the model's weights. Specifically, we detect a subspace corresponding to a set of toxic concepts in the text embedding space and steer prompt token embeddings away from this subspace, thereby filtering out harmful content while preserving intended semantics. To balance the trade-off between filtering toxicity and preserving safe concepts, SAFREE incorporates a novel self-validating filtering mechanism that dynamically adjusts the denoising steps when applying the filtered embeddings. Additionally, we incorporate adaptive re-attention mechanisms within the diffusion latent space to selectively diminish the influence of features related to toxic concepts at the pixel level. By integrating filtering across both textual embedding and visual latent spaces, SAFREE ensures coherent safety checking, preserving the fidelity, quality, and safety of the generated outputs. Empirically, SAFREE achieves state-of-the-art performance in suppressing unsafe content in T2I generation (reducing it by 22% across 5 datasets) compared to other training-free methods and effectively filters targeted concepts, e.g., specific artist styles, while maintaining high-quality output. It also shows competitive results against training-based methods. We further extend SAFREE to various T2I backbones and T2V tasks, showcasing its flexibility and generalization. As generative AI rapidly evolves, SAFREE provides a robust and adaptable safeguard for ensuring safe visual generation.
Theory · Probabilistic Methods
Diffusion probabilistic models have achieved mainstream success in many generative modeling tasks, from image generation to inverse problem solving. A distinct feature of these models is that they correspond to deep hierarchical latent variable models optimizing a variational evidence lower bound (ELBO) on the data likelihood. Drawing on a basic connection between likelihood modeling and compression, we explore the potential of diffusion models for progressive coding, resulting in a sequence of bits that can be incrementally transmitted and decoded with progressively improving reconstruction quality. Unlike prior work based on Gaussian diffusion or conditional diffusion models, we propose a new form of diffusion model with uniform noise in the forward process, whose negative ELBO corresponds to the end-to-end compression cost using universal quantization. We obtain promising first results on image compression, achieving competitive rate-distortion-realism results on a wide range of bit-rates with a single model, bringing neural codecs a step closer to practical deployment. Our code can be found at https://github.com/mandt-lab/uqdm.
Theory · Probabilistic Methods
Diffusion probabilistic models have achieved mainstream success in many generative modeling tasks, from image generation to inverse problem solving. A distinct feature of these models is that they correspond to deep hierarchical latent variable models optimizing a variational evidence lower bound (ELBO) on the data likelihood. Drawing on a basic connection between likelihood modeling and compression, we explore the potential of diffusion models for progressive coding, resulting in a sequence of bits that can be incrementally transmitted and decoded with progressively improving reconstruction quality. Unlike prior work based on Gaussian diffusion or conditional diffusion models, we propose a new form of diffusion model with uniform noise in the forward process, whose negative ELBO corresponds to the end-to-end compression cost using universal quantization. We obtain promising first results on image compression, achieving competitive rate-distortion-realism results on a wide range of bit-rates with a single model, bringing neural codecs a step closer to practical deployment. Our code can be found at https://github.com/mandt-lab/uqdm.
Deep Learning · Theory
We consider a class of conditional forward-backward diffusion models for conditional generative modeling, that is, generating new data given a covariate (or control variable). To formally study the theoretical properties of these conditional generative models, we adopt a statistical framework of distribution regression to characterize the large sample properties of the conditional distribution estimators induced by these conditional forward-backward diffusion models. Here, the conditional distribution of data is assumed to smoothly change over the covariate. In particular, our derived convergence rate is minimax-optimal under the total variation metric within the regimes covered by the existing literature. Additionally, we extend our theory by allowing both the data and the covariate variable to potentially admit a low-dimensional manifold structure. In this scenario, we demonstrate that the conditional forward-backward diffusion model can adapt to both manifold structures, meaning that the derived estimation error bound (under the Wasserstein metric) depends only on the intrinsic dimensionalities of the data and the covariate.
Applications · Genetics, Cell Biology, Health, etc
Functional magnetic resonance imaging (fMRI) is essential for mapping brain activity but faces challenges like lengthy acquisition time and sensitivity to patient movement, limiting its clinical and machine learning applications. While generative models such as diffusion models can synthesize fMRI signals to alleviate these issues, they often underperform due to neglecting the brain's complex structural and dynamic properties.To address these limitations, we propose the Physiological Dynamics-Driven Hierarchical Diffusion Model, a novel framework integrating two key brain physiological properties into the diffusion process: brain hierarchical regional interactions and multifractal dynamics. To model complex interactions among brain regions, we construct hypergraphs based on the prior knowledge of brain functional parcellation reflected by resting-state functional connectivity (rsFC). This enables the aggregation of fMRI signals across multiple scales and generates hierarchical signals. Additionally, by incorporating the prediction of two key dynamics properties of fMRI—the multifractal spectrum and generalized Hurst exponent—our framework effectively guides the diffusion process, ensuring the preservation of the scale-invariant characteristics inherent in real fMRI data.Our framework employs progressive diffusion generation, with signals representing broader brain region information conditioning those that capture localized details, and unifies multiple inputs during denoising for balanced integration.Experiments demonstrate that our model generates physiologically realistic fMRI signals, potentially reducing acquisition time and enhancing data quality, benefiting clinical diagnostics and machine learning in neuroscience.
The diffusion sampling process faces a persistent challenge stemming from its incoherence, attributable to varying noise directions across different timesteps.Our Representative Guidance (RepG) offers a new perspective to address this issue by reformulating the sampling process with a coherent direction toward a representative target.From this perspective, classic classifier guidance reveals its drawback in lacking meaningful representative information, as the features it relies on are optimized for discrimination and tend to highlight only a narrow set of class-specific cues. This focus often sacrifices diversity and increases the risk of adversarial generation.In contrast, we leverage self-supervised representations as the coherent target and treat sampling as a downstream task—one that focuses on refining image details and correcting generation errors, rather than settling for oversimplified outputs.Our Representative Guidance achieves superior performance and demonstrates the potential of pre-trained self-supervised models in guiding diffusion sampling. Our findings show that RepG not only significantly improves vanilla diffusion sampling, but also surpasses state-of-the-art benchmarks when combined with classifier-free guidance.
Applications · Computer Vision
Face recognition (FR) stands as one of the most crucial applications in computer vision. The accuracy of FR models has significantly improved in recent years due to the availability of large-scale human face datasets. However, directly using these datasets can inevitably lead to privacy and legal problems. Generating synthetic data to train FR models is a feasible solution to circumvent these issues. While existing synthetic-based face recognition methods have made significant progress in generating identity-preserving images, they are severely plagued by context overfitting, resulting in a lack of intra-class diversity of generated images and poor face recognition performance. In this paper, we propose a framework to $\textbf{U}$nleash model $\textbf{I}$nherent capabilities to enhance intra-class diversity for synthetic face recognition, shorted as $\textbf{UIFace}$. Our framework first train a diffusion model that can perform denoising conditioned on either identity contexts or a learnable empty context. The former generates identity-preserving images but lacks variations, while the latter exploits the model's intrinsic ability to synthesize intra-class-diversified images but with random identities. Then we adopt a novel two-stage denoising strategy to fully leverage the strengths of both type of contexts, resulting in images that are diverse as well as identity-preserving. Moreover, an attention injection module is introduced to further augment the intra-class variations by utilizing attention maps from the empty context to guide the denoising process in ID-conditioned generation. Experiments show that our method significantly surpasses previous approaches with even less training data and half the size of synthetic dataset. More surprisingly, the proposed $\textbf{UIFace}$ even achieves comparable performance of FR models trained on real datasets when we increase the number of synthetic identities.
Applications · Chemistry and Drug Discovery
Molecular structure elucidation is a fundamental step in understanding chemical phenomena, with applications in identifying molecules in natural products, lab syntheses, forensic samples, and the interstellar medium.We consider the task of predicting a molecule's all-atom 3D structure given only its molecular formula and moments of inertia, motivated by the ability of rotational spectroscopy to measure these moments.While existing generative models can conditionally sample 3D structures with approximately correct moments, this soft conditioning fails to leverage the many digits of precision afforded by experimental rotational spectroscopy.To address this, we first show that the space of $n$-atom point clouds with a fixed set of moments of inertia is embedded in the Stiefel manifold $\mathrm{St}(n, 4)$.We then propose Stiefel Flow Matching as a generative model for elucidating 3D structure under exact moment constraints.Additionally, we learn simpler and shorter flows by finding approximate solutions for equivariant optimal transport on the Stiefel manifold.Empirically, enforcing exact moment constraints allows Stiefel Flow Matching to achieve higher success rates and faster sampling than Euclidean diffusion models, even on high-dimensional manifolds corresponding to large molecules in the GEOM dataset.
Reinforcement Learning · Batch/Offline
Offline model-based reinforcement learning (RL) offers a principled approach to using a learned dynamics model as a simulator to optimize a control policy. Despite the near-optimal performance of existing approaches on benchmarks with high-quality datasets, most struggle on datasets with low state-action space coverage or suboptimal demonstrations.We develop a novel offline model-based RL approach that particularly shines in low-quality data regimes while maintaining competitive performance on high-quality datasets.Neural Stochastic Differential Equations for Uncertainty-aware, Offline RL (NUNO) learns a dynamics model as neural stochastic differential equations (SDE), where its drift term can leverage prior physics knowledge as inductive bias.In parallel, its diffusion term provides distance-aware estimates of model uncertainty by matching the dynamics' underlying stochasticity near the training data regime while providing high but bounded estimates beyond it.To address the so-called model exploitation problem in offline model-based RL, NUNO builds on existing studies by penalizing and adaptively truncating neural SDE's rollouts according to uncertainty estimates.Our empirical results in D4RL and NeoRL MuJoCo benchmarks evidence that NUNO outperforms state-of-the-art methods in low-quality datasets by up to 93% while matching or surpassing their performance by up to 55% in some high-quality counterparts.
Applications · Physics
The rapid development of AI highlights the pressing need for sustainable energy, a critical global challenge for decades. Nuclear fusion, generally seen as a promising solution, has been the focus of intensive research for nearly a century, with investments reaching hundreds of billions of dollars. Recent advancements in Inertial Confinement Fusion (ICF) have drawn significant attention to fusion research, in which Laser-Plasma Interaction (LPI) is critical for ensuring fusion stability and efficiency. However, the complexity of LPI makes analytical approaches impractical, leaving researchers dependent on extremely computationally intensive Particle-in-Cell (PIC) simulations to generate data, posing a significant bottleneck to the advancement of fusion research. In response, this work introduces Diff-PIC, a novel framework that leverages conditional diffusion models as a computationally efficient alternative to PIC simulations for generating high-fidelity scientific LPI data. In this work, physical patterns captured by PIC simulations are distilled into diffusion models associated with two tailored enhancements: (1) To effectively capture the complex relationships between physical parameters and their corresponding outcomes, the parameters are encoded in a physically informed manner. (2) To further enhance efficiency while maintaining physical validity, the rectified flow technique is employed to transform our model into a one-step conditional diffusion model. Experimental results show that Diff-PIC achieves a $\sim$16,200$\times$ speedup compared to traditional PIC on a 100 picosecond simulation, while delivering superior accuracy compared to other data generation approaches.
Deep Learning · Robustness
Recently Diffusion-based Purification (DiffPure) has been recognized as an effective defense method against adversarial examples. However, we find DiffPure which directly employs the original pre-trained diffusion models for adversarial purification, to be suboptimal. This is due to an inherent trade-off between noise purification performance and data recovery quality. Additionally, the reliability of existing evaluations for DiffPure is questionable, as they rely on weak adaptive attacks. In this work, we propose a novel Adversarial Diffusion Bridge Model, termed ADBM. ADBM directly constructs a reverse bridge from the diffused adversarial data back to its original clean examples, enhancing the purification capabilities of the original diffusion models. Through theoretical analysis and experimental validation across various scenarios, ADBM has proven to be a superior and robust defense mechanism, offering significant promise for practical applications. Code is available at https://github.com/LixiaoTHU/ADBM.
Deep Learning · Generative Models and Autoencoders
We propose SymDiff, a method for constructing equivariant diffusion models using the framework of stochastic symmetrisation. SymDiff resembles a learned data augmentation that is deployed at sampling time, and is lightweight, computationally efficient, and easy to implement on top of arbitrary off-the-shelf models. In contrast to previous work, SymDiff typically does not require any neural network components that are intrinsically equivariant, avoiding the need for complex parameterisations or the use of higher-order geometric features. Instead, our method can leverage highly scalable modern architectures as drop-in replacements for these more constrained alternatives. We show that this additional flexibility yields significant empirical benefit for E(3)-equivariant molecular generation. To the best of our knowledge, this is the first application of symmetrisation to generative modelling, suggesting its potential in this domain more generally.
Theory · Everything Else
High-dimensional data must be highly structured to be learnable. Although the compositional and hierarchical nature of data is often put forward to explain learnability, quantitative measurements establishing these properties are scarce. Likewise, accessing the latent variables underlying such a data structure remains a challenge. In this work, we show that forward-backward experiments in diffusion-based models, where data is noised and then denoised to generate new samples, are a promising tool to probe the latent structure of data. We predict in simple hierarchical models that, in this process, changes in data occur by correlated chunks, with a length scale that diverges at a noise level where a phase transition is known to take place. Remarkably, we confirm this prediction in both text and image datasets using state-of-the-art diffusion models. Our results show how latent variable changes manifest in the data and establish how to measure these effects in real data using diffusion models.
Theory · Learning Theory
Score-based diffusion models have achieved remarkable empirical performance in the field of machine learning and artificial intelligence for their ability to generate high-quality new data instances from complex distributions. Improving our understanding of diffusion models, including mainly convergence analysis for such models, has attracted a lot of interests. Despite a lot of theoretical attempts, there still exists significant gap between theory and practice. Towards to close this gap, we establish an iteration complexity at the order of $d^{1/3}\varepsilon^{-2/3}$, which is better than $d^{5/12}\varepsilon^{-1}$, the best known complexity achieved before our work. This convergence analysis is based on a randomized midpoint method, which is first proposed for log-concave sampling (Shen & Lee, 2019), and then extended to diffusion models by Gupta et al. (2024). Our theory accommodates $\varepsilon$-accurate score estimates, and does not require log-concavity on the target distribution. Moreover, the algorithm can also be parallelized to run in only $O(\log^2(d/\varepsilon))$ parallel rounds in a similar way to prior works.
Applications · Chemistry and Drug Discovery
We present the Illustrated AlphaFold, a visual walkthrough of the architecture and information flow of AlphaFold 3. We explain every model component and training detail, with particular focus on the advances since AlphaFold 2 – including the unified tokenization scheme that extends to DNA, RNA, and small molecules, as well as the novel diffusion-based structural module. Finally, we include some musings on the ML lessons learned from studying AlphaFold 3.