Score-based diffusion models synthesize samples by reversing a stochastic process that diffuses data to noise, and are trained by minimizing a weighted combination of score matching losses. The log-likelihood of score-based diffusion models can be tractably computed through a connection to continuous normalizing flows, but log-likelihood is not directly optimized by the weighted combination of score matching losses. We show that for a specific weighting scheme, the objective upper bounds the negative log-likelihood, thus enabling approximate maximum likelihood training of score-based diffusion models. We empirically observe that maximum likelihood training consistently improves the likelihood of score-based diffusion models across multiple datasets, stochastic processes, and model architectures. Our best models achieve negative log-likelihoods of 2.83 and 3.76 bits/dim on CIFAR-10 and ImageNet 32 ˆ32 without any data augmentation, on a par with state-of-the-art autoregressive models on these tasks.

Following [2], we extend our method into multi-domain translation on AFHQ dataset, where the source domain includesCat and Wild and the target domain isDog. In this setting, similar to two-domain unpaired I2I, the EGSDE also employs an energy function pretrained on both the source andtargetdomains toguidetheinference process ofapretrained SDE.Theonlydifference is the domain-specific feature extractorEs(,) involved in the energy function is the all but the last layer of a three-class classifier rather than two-class. All experiments are repeated 5 times to eliminate randomness. The quantitative results are reported in Table 11.

Score-based generative models require guidance in order to generate plausible, on-manifold samples. The most popular guidance method, Classifier-Free Guidance (CFG), is only applicable in settings with labeled data and requires training an additional unconditional score-based model. More recently, Auto-Guidance adopts a smaller, less capable version of the original model to guide generation. While each method effectively promotes the fidelity of generated data, each requires labeled data or the training of additional models, making it challenging to guide score-based models when (labeled) training data are not available or training new models is not feasible. We make the surprising discovery that the positive curvature of log density estimates in saddle regions provides strong guidance for score-based models. Motivated by this, we develop saddle-free guidance (SFG) which maintains estimates of maximal positive curvature of the log density to guide individual score-based models. SFG has the same computational cost of classifier-free guidance, does not require additional training, and works with off-the-shelf diffusion and flow matching models. Our experiments indicate that SFG achieves state-of-the-art FID and FD-DINOv2 metrics in single-model unconditional ImageNet-512 generation. When SFG is combined with Auto-Guidance, its unconditional samples achieve general state-of-the-art in FD-DINOv2 score. Our experiments with FLUX.1-dev and Stable Diffusion v3.5 indicate that SFG boosts the diversity of output images compared to CFG while maintaining excellent prompt adherence and image fidelity.

Theories with a sign problem due to a complex action or Boltzmann weight can sometimes be numerically solved using a stochastic process in the complexified configuration space. However, the probability distribution effectively sampled by this complex Langevin process is not known a priori and notoriously hard to understand. In generative AI, diffusion models can learn distributions, or their log derivatives, from data. We explore the ability of diffusion models to learn the distributions sampled by a complex Langevin process, comparing score-based and energy-based diffusion models, and speculate about possible applications.

Generative models are a promising tool to produce cosmological simulations but face significant challenges in scalability, physical consistency, and adherence to domain symmetries, limiting their utility as alternatives to $N$-body simulations. To address these limitations, we introduce a score-based generative model with an equivariant graph neural network that simulates gravitational clustering of galaxies across cosmologies starting from an informed prior, respects periodic boundaries, and scales to full galaxy counts in simulations. A novel topology-aware noise schedule, crucial for large geometric graphs, is introduced. The proposed equivariant score-based model successfully generates full-scale cosmological point clouds of up to 600,000 halos, respects periodicity and a uniform prior, and outperforms existing diffusion models in capturing clustering statistics while offering significant computational advantages. This work advances cosmology by introducing a generative model designed to closely resemble the underlying gravitational clustering of structure formation, moving closer to physically realistic and efficient simulators for the evolution of large-scale structures in the universe.

Score-based generative models are shown to achieve remarkable empirical performances in various applications such as image generation and audio synthesis. However, a theoretical understanding of score-based diffusion models is still incomplete. Recently, Song et al. showed that the training objective of score-based

Inferring sky surface brightness distributions from noisy interferometric data in a principled statistical framework has been a key challenge in radio astronomy. In this work, we introduce Imaging for Radio Interferometry with Score-based models (IRIS). We use score-based models trained on optical images of galaxies as an expressive prior in combination with a Gaussian likelihood in the uv-space to infer images of protoplanetary disks from visibility data of the DSHARP survey conducted by ALMA. We demonstrate the advantages of this framework compared with traditional radio interferometry imaging algorithms, showing that it produces plausible posterior samples despite the use of a misspecified galaxy prior. Through coverage testing on simulations, we empirically evaluate the accuracy of this approach to generate calibrated posterior samples.

Graph Neural Networks (GNNs) have demonstrated remarkable efficacy in tackling a wide array of graph-related tasks across diverse domains. However, a significant challenge lies in their propensity to generate biased predictions, particularly with respect to sensitive node attributes such as age and gender. These biases, inherent in many machine learning models, are amplified in GNNs due to the message-passing mechanism, which allows nodes to influence each other, rendering the task of making fair predictions notably challenging. This issue is particularly pertinent in critical domains where model fairness holds paramount importance. In this paper, we propose a novel generative Fairness-Aware Subgraph Diffusion (FASD) method for unbiased GNN learning. The method initiates by strategically sampling small subgraphs from the original large input graph, and then proceeds to conduct subgraph debiasing via generative fairness-aware graph diffusion processes based on stochastic differential equations (SDEs). To effectively diffuse unfairness in the input data, we introduce additional adversary bias perturbations to the subgraphs during the forward diffusion process, and train score-based models to predict these applied perturbations, enabling them to learn the underlying dynamics of the biases present in the data. Subsequently, the trained score-based models are utilized to further debias the original subgraph samples through the reverse diffusion process. Finally, FASD induces fair node predictions on the input graph by performing standard GNN learning on the debiased subgraphs. Experimental results demonstrate the superior performance of the proposed method over state-of-the-art Fair GNN baselines across multiple benchmark datasets.