Emiel Hoogeboom

Emiel Hoogeboom, Jonathan Heek, Tim Salimans

Currently, applying diffusion models in pixel space of high resolution images is difficult. Instead, existing approaches focus on diffusion in lower dimensional spaces (latent diffusion), or have multiple super-resolution levels of generation referred to as cascades. The downside is that these approaches add additional complexity to the diffusion framework. This paper aims to improve denoising diffusion for high resolution images while keeping the model as simple as possible. The paper is centered around the research question: How can one train a standard denoising diffusion models on high resolution images, and still obtain performance comparable to these alternate approaches? The four main findings are: 1) the noise schedule should be adjusted for high resolution images, 2) It is sufficient to scale only a particular part of the architecture, 3) dropout should be added at specific locations in the architecture, and 4) downsampling is an effective strategy to avoid high resolution feature maps. Combining these simple yet effective techniques, we achieve state-of-the-art on image generation among diffusion models without sampling modifiers on ImageNet.

Allan Jabri, Sjoerd van Steenkiste, Emiel Hoogeboom et al.

Recent progress in 3D scene understanding enables scalable learning of representations across large datasets of diverse scenes. As a consequence, generalization to unseen scenes and objects, rendering novel views from just a single or a handful of input images, and controllable scene generation that supports editing, is now possible. However, training jointly on a large number of scenes typically compromises rendering quality when compared to single-scene optimized models such as NeRFs. In this paper, we leverage recent progress in diffusion models to equip 3D scene representation learning models with the ability to render high-fidelity novel views, while retaining benefits such as object-level scene editing to a large degree. In particular, we propose DORSal, which adapts a video diffusion architecture for 3D scene generation conditioned on frozen object-centric slot-based representations of scenes. On both complex synthetic multi-object scenes and on the real-world large-scale Street View dataset, we show that DORSal enables scalable neural rendering of 3D scenes with object-level editing and improves upon existing approaches.

Emiel Hoogeboom, David Ruhe, Jonathan Heek et al.

It is currently difficult to distill discrete diffusion models. In contrast, continuous diffusion literature has many distillation approaches methods that can reduce sampling steps to a handful. Our method, Discrete Moment Matching Distillation (D-MMD), leverages ideas that have been highly successful in the continuous domain. Whereas previous discrete distillation methods collapse, D-MMD maintains high quality and diversity (given sufficient sampling steps). This is demonstrated on both text and image datasets. Moreover, the newly distilled generators can outperform their teachers.

Jonathan Heek, Emiel Hoogeboom, Thomas Mensink et al.

We present Unified Latents (UL), a framework for learning latent representations that are jointly regularized by a diffusion prior and decoded by a diffusion model. By linking the encoder's output noise to the prior's minimum noise level, we obtain a simple training objective that provides a tight upper bound on the latent bitrate. On ImageNet-512, our approach achieves competitive FID of 1.4, with high reconstruction quality (PSNR) while requiring fewer training FLOPs than models trained on Stable Diffusion latents. On Kinetics-600, we set a new state-of-the-art FVD of 1.3.

Emiel Hoogeboom, Tim Salimans

Recently, Rissanen et al., (2022) have presented a new type of diffusion process for generative modeling based on heat dissipation, or blurring, as an alternative to isotropic Gaussian diffusion. Here, we show that blurring can equivalently be defined through a Gaussian diffusion process with non-isotropic noise. In making this connection, we bridge the gap between inverse heat dissipation and denoising diffusion, and we shed light on the inductive bias that results from this modeling choice. Finally, we propose a generalized class of diffusion models that offers the best of both standard Gaussian denoising diffusion and inverse heat dissipation, which we call Blurring Diffusion Models.

Grigory Bartosh, David Ruhe, Emiel Hoogeboom et al.

Diffusion models achieve state-of-the-art generative performance but suffer from high computational costs during inference due to the repeated evaluation of a heavy neural network. In this work, we propose Dual-Rate Diffusion, a method to accelerate sampling by interleaving the execution of a heavy high-capacity context encoder and a light efficient denoising model. The context encoder is evaluated sparsely to extract high-dimensional features, which are effectively reused by the light denoising model at every step to refine the sample efficiently. This approach significantly accelerates inference without compromising sample quality. On ImageNet benchmarks, Dual-Rate Diffusion matches the performance of standard baselines while reducing computational cost by a factor of $2$-$4$. Furthermore, we demonstrate that our method is compatible with distillation techniques, such as Moment Matching Distillation, enabling further efficiency gains in few-step generation.

Gheorghe Comanici, Eric Bieber, Mike Schaekermann et al. · amazon-science, baidu

In this report, we introduce the Gemini 2.X model family: Gemini 2.5 Pro and Gemini 2.5 Flash, as well as our earlier Gemini 2.0 Flash and Flash-Lite models. Gemini 2.5 Pro is our most capable model yet, achieving SoTA performance on frontier coding and reasoning benchmarks. In addition to its incredible coding and reasoning skills, Gemini 2.5 Pro is a thinking model that excels at multimodal understanding and it is now able to process up to 3 hours of video content. Its unique combination of long context, multimodal and reasoning capabilities can be combined to unlock new agentic workflows. Gemini 2.5 Flash provides excellent reasoning abilities at a fraction of the compute and latency requirements and Gemini 2.0 Flash and Flash-Lite provide high performance at low latency and cost. Taken together, the Gemini 2.X model generation spans the full Pareto frontier of model capability vs cost, allowing users to explore the boundaries of what is possible with complex agentic problem solving.

Imagen-Team-Google, Jason Baldridge, Jakob Bauer et al.

We introduce Imagen 3, a latent diffusion model that generates high quality images from text prompts. We describe our quality and responsibility evaluations. Imagen 3 is preferred over other state-of-the-art (SOTA) models at the time of evaluation. In addition, we discuss issues around safety and representation, as well as methods we used to minimize the potential harm of our models.

David Ruhe, Jonathan Heek, Tim Salimans et al.

Diffusion models have recently been increasingly applied to temporal data such as video, fluid mechanics simulations, or climate data. These methods generally treat subsequent frames equally regarding the amount of noise in the diffusion process. This paper explores Rolling Diffusion: a new approach that uses a sliding window denoising process. It ensures that the diffusion process progressively corrupts through time by assigning more noise to frames that appear later in a sequence, reflecting greater uncertainty about the future as the generation process unfolds. Empirically, we show that when the temporal dynamics are complex, Rolling Diffusion is superior to standard diffusion. In particular, this result is demonstrated in a video prediction task using the Kinetics-600 video dataset and in a chaotic fluid dynamics forecasting experiment.

Jonathan Heek, Emiel Hoogeboom, Tim Salimans

Diffusion models are relatively easy to train but require many steps to generate samples. Consistency models are far more difficult to train, but generate samples in a single step. In this paper we propose Multistep Consistency Models: A unification between Consistency Models (Song et al., 2023) and TRACT (Berthelot et al., 2023) that can interpolate between a consistency model and a diffusion model: a trade-off between sampling speed and sampling quality. Specifically, a 1-step consistency model is a conventional consistency model whereas a $\infty$-step consistency model is a diffusion model. Multistep Consistency Models work really well in practice. By increasing the sample budget from a single step to 2-8 steps, we can train models more easily that generate higher quality samples, while retaining much of the sampling speed benefits. Notable results are 1.4 FID on Imagenet 64 in 8 step and 2.1 FID on Imagenet128 in 8 steps with consistency distillation, using simple losses without adversarial training. We also show that our method scales to a text-to-image diffusion model, generating samples that are close to the quality of the original model.

Emiel Hoogeboom, Thomas Mensink, Jonathan Heek et al.

Latent diffusion models have become the popular choice for scaling up diffusion models for high resolution image synthesis. Compared to pixel-space models that are trained end-to-end, latent models are perceived to be more efficient and to produce higher image quality at high resolution. Here we challenge these notions, and show that pixel-space models can be very competitive to latent models both in quality and efficiency, achieving 1.5 FID on ImageNet512 and new SOTA results on ImageNet128, ImageNet256 and Kinetics600. We present a simple recipe for scaling end-to-end pixel-space diffusion models to high resolutions. 1: Use the sigmoid loss-weighting (Kingma & Gao, 2023) with our prescribed hyper-parameters. 2: Use our simplified memory-efficient architecture with fewer skip-connections. 3: Scale the model to favor processing the image at a high resolution with fewer parameters, rather than using more parameters at a lower resolution. Combining these with guidance intervals, we obtain a family of pixel-space diffusion models we call Simpler Diffusion (SiD2).

Andrea Schioppa, Emiel Hoogeboom, Jonathan Heek

The rapid advancement of text-to-image Diffusion Models has led to their widespread public accessibility. However these models, trained on large internet datasets, can sometimes generate undesirable outputs. To mitigate this, approximate Machine Unlearning algorithms have been proposed to modify model weights to reduce the generation of specific types of images, characterized by samples from a ``forget distribution'', while preserving the model's ability to generate other images, characterized by samples from a ``retain distribution''. While these methods aim to minimize the influence of training data in the forget distribution without extensive additional computation, we point out that they can compromise the model's integrity by inadvertently affecting generation for images in the retain distribution. Recognizing the limitations of FID and CLIPScore in capturing these effects, we introduce a novel retention metric that directly assesses the perceptual difference between outputs generated by the original and the unlearned models. We then propose unlearning algorithms that demonstrate superior effectiveness in preserving model integrity compared to existing baselines. Given their straightforward implementation, these algorithms serve as valuable benchmarks for future advancements in approximate Machine Unlearning for Diffusion Models.

Manoj Kumar, Neil Houlsby, Emiel Hoogeboom

Generating image variations, where a model produces variations of an input image while preserving the semantic context has gained increasing attention. Current image variation techniques involve adapting a text-to-image model to reconstruct an input image conditioned on the same image. We first demonstrate that a diffusion model trained to reconstruct an input image from frozen embeddings, can reconstruct the image with minor variations. Second, inspired by how text-to-image models learn from web-scale text-image pairs, we explore a new pretraining strategy to generate image variations using a large collection of image pairs. Our diffusion model \textit{Semantica} receives a random (encoded) image from a webpage as conditional input and denoises another noisy random image from the same webpage. We carefully examine various design choices for the image encoder, given its crucial role in extracting relevant context from the input image. Once trained, \textit{Semantica} can adaptively generate new images from a dataset by simply using images from that dataset as input. Finally, we identify limitations in standard image consistency metrics for evaluating image variations and propose alternative metrics based on few-shot generation.

Tim Salimans, Thomas Mensink, Jonathan Heek et al.

We present a new method for making diffusion models faster to sample. The method distills many-step diffusion models into few-step models by matching conditional expectations of the clean data given noisy data along the sampling trajectory. Our approach extends recently proposed one-step methods to the multi-step case, and provides a new perspective by interpreting these approaches in terms of moment matching. By using up to 8 sampling steps, we obtain distilled models that outperform not only their one-step versions but also their original many-step teacher models, obtaining new state-of-the-art results on the Imagenet dataset. We also show promising results on a large text-to-image model where we achieve fast generation of high resolution images directly in image space, without needing autoencoders or upsamplers.

Emiel Hoogeboom, Eirikur Agustsson, Fabian Mentzer et al.

Despite the tremendous success of diffusion generative models in text-to-image generation, replicating this success in the domain of image compression has proven difficult. In this paper, we demonstrate that diffusion can significantly improve perceptual quality at a given bit-rate, outperforming state-of-the-art approaches PO-ELIC and HiFiC as measured by FID score. This is achieved using a simple but theoretically motivated two-stage approach combining an autoencoder targeting MSE followed by a further score-based decoder. However, as we will show, implementation details matter and the optimal design decisions can differ greatly from typical text-to-image models.

Emiel Hoogeboom, Victor Garcia Satorras, Clément Vignac et al.

This work introduces a diffusion model for molecule generation in 3D that is equivariant to Euclidean transformations. Our E(3) Equivariant Diffusion Model (EDM) learns to denoise a diffusion process with an equivariant network that jointly operates on both continuous (atom coordinates) and categorical features (atom types). In addition, we provide a probabilistic analysis which admits likelihood computation of molecules using our model. Experimentally, the proposed method significantly outperforms previous 3D molecular generative methods regarding the quality of generated samples and efficiency at training time.

Emiel Hoogeboom, Alexey A. Gritsenko, Jasmijn Bastings et al.

We introduce Autoregressive Diffusion Models (ARDMs), a model class encompassing and generalizing order-agnostic autoregressive models (Uria et al., 2014) and absorbing discrete diffusion (Austin et al., 2021), which we show are special cases of ARDMs under mild assumptions. ARDMs are simple to implement and easy to train. Unlike standard ARMs, they do not require causal masking of model representations, and can be trained using an efficient objective similar to modern probabilistic diffusion models that scales favourably to highly-dimensional data. At test time, ARDMs support parallel generation which can be adapted to fit any given generation budget. We find that ARDMs require significantly fewer steps than discrete diffusion models to attain the same performance. Finally, we apply ARDMs to lossless compression, and show that they are uniquely suited to this task. Contrary to existing approaches based on bits-back coding, ARDMs obtain compelling results not only on complete datasets, but also on compressing single data points. Moreover, this can be done using a modest number of network calls for (de)compression due to the model's adaptable parallel generation.

Alexandra Lindt, Emiel Hoogeboom

Discrete flow-based models are a recently proposed class of generative models that learn invertible transformations for discrete random variables. Since they do not require data dequantization and maximize an exact likelihood objective, they can be used in a straight-forward manner for lossless compression. In this paper, we introduce a new discrete flow-based model for categorical random variables: Discrete Denoising Flows (DDFs). In contrast with other discrete flow-based models, our model can be locally trained without introducing gradient bias. We show that DDFs outperform Discrete Flows on modeling a toy example, binary MNIST and Cityscapes segmentation maps, measured in log-likelihood.

Victor Garcia Satorras, Emiel Hoogeboom, Fabian B. Fuchs et al.

This paper introduces a generative model equivariant to Euclidean symmetries: E(n) Equivariant Normalizing Flows (E-NFs). To construct E-NFs, we take the discriminative E(n) graph neural networks and integrate them as a differential equation to obtain an invertible equivariant function: a continuous-time normalizing flow. We demonstrate that E-NFs considerably outperform baselines and existing methods from the literature on particle systems such as DW4 and LJ13, and on molecules from QM9 in terms of log-likelihood. To the best of our knowledge, this is the first flow that jointly generates molecule features and positions in 3D.

Victor Garcia Satorras, Emiel Hoogeboom, Max Welling

This paper introduces a new model to learn graph neural networks equivariant to rotations, translations, reflections and permutations called E(n)-Equivariant Graph Neural Networks (EGNNs). In contrast with existing methods, our work does not require computationally expensive higher-order representations in intermediate layers while it still achieves competitive or better performance. In addition, whereas existing methods are limited to equivariance on 3 dimensional spaces, our model is easily scaled to higher-dimensional spaces. We demonstrate the effectiveness of our method on dynamical systems modelling, representation learning in graph autoencoders and predicting molecular properties.

Emiel Hoogeboom, Didrik Nielsen, Priyank Jaini et al.

Generative flows and diffusion models have been predominantly trained on ordinal data, for example natural images. This paper introduces two extensions of flows and diffusion for categorical data such as language or image segmentation: Argmax Flows and Multinomial Diffusion. Argmax Flows are defined by a composition of a continuous distribution (such as a normalizing flow), and an argmax function. To optimize this model, we learn a probabilistic inverse for the argmax that lifts the categorical data to a continuous space. Multinomial Diffusion gradually adds categorical noise in a diffusion process, for which the generative denoising process is learned. We demonstrate that our method outperforms existing dequantization approaches on text modelling and modelling on image segmentation maps in log-likelihood.

Simon Passenheim, Emiel Hoogeboom

This paper introduces the Variational Determinant Estimator (VDE), a variational extension of the recently proposed determinant estimator discovered by arXiv:2005.06553v2. Our estimator significantly reduces the variance even for low sample sizes by combining (importance-weighted) variational inference and a family of normalizing flows which allow density estimation on hyperspheres. In the ideal case of a tight variational bound, the VDE becomes a zero variance estimator, and a single sample is sufficient for an exact (log) determinant estimate.

T. Anderson Keller, Jorn W. T. Peters, Priyank Jaini et al.

Efficient gradient computation of the Jacobian determinant term is a core problem in many machine learning settings, and especially so in the normalizing flow framework. Most proposed flow models therefore either restrict to a function class with easy evaluation of the Jacobian determinant, or an efficient estimator thereof. However, these restrictions limit the performance of such density models, frequently requiring significant depth to reach desired performance levels. In this work, we propose Self Normalizing Flows, a flexible framework for training normalizing flows by replacing expensive terms in the gradient by learned approximate inverses at each layer. This reduces the computational complexity of each layer's exact update from $\mathcal{O}(D^3)$ to $\mathcal{O}(D^2)$, allowing for the training of flow architectures which were otherwise computationally infeasible, while also providing efficient sampling. We show experimentally that such models are remarkably stable and optimize to similar data likelihood values as their exact gradient counterparts, while training more quickly and surpassing the performance of functionally constrained counterparts.

Didrik Nielsen, Priyank Jaini, Emiel Hoogeboom et al.

Normalizing flows and variational autoencoders are powerful generative models that can represent complicated density functions. However, they both impose constraints on the models: Normalizing flows use bijective transformations to model densities whereas VAEs learn stochastic transformations that are non-invertible and thus typically do not provide tractable estimates of the marginal likelihood. In this paper, we introduce SurVAE Flows: A modular framework of composable transformations that encompasses VAEs and normalizing flows. SurVAE Flows bridge the gap between normalizing flows and VAEs with surjective transformations, wherein the transformations are deterministic in one direction -- thereby allowing exact likelihood computation, and stochastic in the reverse direction -- hence providing a lower bound on the corresponding likelihood. We show that several recently proposed methods, including dequantization and augmented normalizing flows, can be expressed as SurVAE Flows. Finally, we introduce common operations such as the max value, the absolute value, sorting and stochastic permutation as composable layers in SurVAE Flows.

Emiel Hoogeboom, Victor Garcia Satorras, Jakub M. Tomczak et al.

This paper introduces a new method to build linear flows, by taking the exponential of a linear transformation. This linear transformation does not need to be invertible itself, and the exponential has the following desirable properties: it is guaranteed to be invertible, its inverse is straightforward to compute and the log Jacobian determinant is equal to the trace of the linear transformation. An important insight is that the exponential can be computed implicitly, which allows the use of convolutional layers. Using this insight, we develop new invertible transformations named convolution exponentials and graph convolution exponentials, which retain the equivariance of their underlying transformations. In addition, we generalize Sylvester Flows and propose Convolutional Sylvester Flows which are based on the generalization and the convolution exponential as basis change. Empirically, we show that the convolution exponential outperforms other linear transformations in generative flows on CIFAR10 and the graph convolution exponential improves the performance of graph normalizing flows. In addition, we show that Convolutional Sylvester Flows improve performance over residual flows as a generative flow model measured in log-likelihood.

Auke Wiggers, Emiel Hoogeboom

Autoregressive models (ARMs) currently hold state-of-the-art performance in likelihood-based modeling of image and audio data. Generally, neural network based ARMs are designed to allow fast inference, but sampling from these models is impractically slow. In this paper, we introduce the predictive sampling algorithm: a procedure that exploits the fast inference property of ARMs in order to speed up sampling, while keeping the model intact. We propose two variations of predictive sampling, namely sampling with ARM fixed-point iteration and learned forecasting modules. Their effectiveness is demonstrated in two settings: i) explicit likelihood modeling on binary MNIST, SVHN and CIFAR10, and ii) discrete latent modeling in an autoencoder trained on SVHN, CIFAR10 and Imagenet32. Empirically, we show considerable improvements over baselines in number of ARM inference calls and sampling speed.

Emiel Hoogeboom, Taco S. Cohen, Jakub M. Tomczak

Media is generally stored digitally and is therefore discrete. Many successful deep distribution models in deep learning learn a density, i.e., the distribution of a continuous random variable. Naïve optimization on discrete data leads to arbitrarily high likelihoods, and instead, it has become standard practice to add noise to datapoints. In this paper, we present a general framework for dequantization that captures existing methods as a special case. We derive two new dequantization objectives: importance-weighted (iw) dequantization and Rényi dequantization. In addition, we introduce autoregressive dequantization (ARD) for more flexible dequantization distributions. Empirically we find that iw and Rényi dequantization considerably improve performance for uniform dequantization distributions. ARD achieves a negative log-likelihood of 3.06 bits per dimension on CIFAR10, which to the best of our knowledge is state-of-the-art among distribution models that do not require autoregressive inverses for sampling.

Christina Winkler, Daniel Worrall, Emiel Hoogeboom et al.

Normalizing Flows (NFs) are able to model complicated distributions p(y) with strong inter-dimensional correlations and high multimodality by transforming a simple base density p(z) through an invertible neural network under the change of variables formula. Such behavior is desirable in multivariate structured prediction tasks, where handcrafted per-pixel loss-based methods inadequately capture strong correlations between output dimensions. We present a study of conditional normalizing flows (CNFs), a class of NFs where the base density to output space mapping is conditioned on an input x, to model conditional densities p(y|x). CNFs are efficient in sampling and inference, they can be trained with a likelihood-based objective, and CNFs, being generative flows, do not suffer from mode collapse or training instabilities. We provide an effective method to train continuous CNFs for binary problems and in particular, we apply these CNFs to super-resolution and vessel segmentation tasks demonstrating competitive performance on standard benchmark datasets in terms of likelihood and conventional metrics.

Emiel Hoogeboom, Jorn W. T. Peters, Rianne van den Berg et al.

Lossless compression methods shorten the expected representation size of data without loss of information, using a statistical model. Flow-based models are attractive in this setting because they admit exact likelihood optimization, which is equivalent to minimizing the expected number of bits per message. However, conventional flows assume continuous data, which may lead to reconstruction errors when quantized for compression. For that reason, we introduce a flow-based generative model for ordinal discrete data called Integer Discrete Flow (IDF): a bijective integer map that can learn rich transformations on high-dimensional data. As building blocks for IDFs, we introduce a flexible transformation layer called integer discrete coupling. Our experiments show that IDFs are competitive with other flow-based generative models. Furthermore, we demonstrate that IDF based compression achieves state-of-the-art lossless compression rates on CIFAR10, ImageNet32, and ImageNet64. To the best of our knowledge, this is the first lossless compression method that uses invertible neural networks.

Emiel Hoogeboom, Rianne van den Berg, Max Welling

Generative flows are attractive because they admit exact likelihood optimization and efficient image synthesis. Recently, Kingma & Dhariwal (2018) demonstrated with Glow that generative flows are capable of generating high quality images. We generalize the 1 x 1 convolutions proposed in Glow to invertible d x d convolutions, which are more flexible since they operate on both channel and spatial axes. We propose two methods to produce invertible convolutions that have receptive fields identical to standard convolutions: Emerging convolutions are obtained by chaining specific autoregressive convolutions, and periodic convolutions are decoupled in the frequency domain. Our experiments show that the flexibility of d x d convolutions significantly improves the performance of generative flow models on galaxy images, CIFAR10 and ImageNet.

Emiel Hoogeboom, Jorn W. T. Peters, Taco S. Cohen et al.

The effectiveness of Convolutional Neural Networks stems in large part from their ability to exploit the translation invariance that is inherent in many learning problems. Recently, it was shown that CNNs can exploit other invariances, such as rotation invariance, by using group convolutions instead of planar convolutions. However, for reasons of performance and ease of implementation, it has been necessary to limit the group convolution to transformations that can be applied to the filters without interpolation. Thus, for images with square pixels, only integer translations, rotations by multiples of 90 degrees, and reflections are admissible. Whereas the square tiling provides a 4-fold rotational symmetry, a hexagonal tiling of the plane has a 6-fold rotational symmetry. In this paper we show how one can efficiently implement planar convolution and group convolution over hexagonal lattices, by re-using existing highly optimized convolution routines. We find that, due to the reduced anisotropy of hexagonal filters, planar HexaConv provides better accuracy than planar convolution with square filters, given a fixed parameter budget. Furthermore, we find that the increased degree of symmetry of the hexagonal grid increases the effectiveness of group convolutions, by allowing for more parameter sharing. We show that our method significantly outperforms conventional CNNs on the AID aerial scene classification dataset, even outperforming ImageNet pre-trained models.