Deep Gaussian processes (DGPs) enable expressive hierarchical Bayesian modeling but pose substantial challenges for posterior inference, especially over inducing variables. Denoising diffusion variational inference (DDVI) addresses this by modeling the posterior as a time-reversed diffusion from a simple Gaussian prior. However, DDVI's fixed unconditional starting distribution remains far from the complex true posterior, resulting in inefficient inference trajectories and slow convergence. In this work, we propose Diffusion Bridge Variational Inference (DBVI), a principled extension of DDVI that initiates the reverse diffusion from a learnable, data-dependent initial distribution. This initialization is parameterized via an amortized neural network and progressively adapted using gradients from the ELBO objective, reducing the posterior gap and improving sample efficiency. To enable scalable amortization, we design the network to operate on the inducing inputs, which serve as structured, low-dimensional summaries of the dataset and naturally align with the inducing variables' shape. DBVI retains the mathematical elegance of DDVI, including Girsanov-based ELBOs and reverse-time SDEs,while reinterpreting the prior via a Doob-bridged diffusion process. We derive a tractable training objective under this formulation and implement DBVI for scalable inference in large-scale DGPs. Across regression, classification, and image reconstruction tasks, DBVI consistently outperforms DDVI and other variational baselines in predictive accuracy, convergence speed, and posterior quality.

We introduce a framework for designing efficient diffusion models for $d$-dimensional symmetric-space Riemannian manifolds, including the torus, sphere, special orthogonal group and unitary group. Existing manifold diffusion models often depend on heat kernels, which lack closed-form expressions and require either $d$ gradient evaluations or exponential-in-$d$ arithmetic operations per training step. We introduce a new diffusion model for symmetric manifolds with a spatially-varying covariance, allowing us to leverage a projection of Euclidean Brownian motion to bypass heat kernel computations. Our training algorithm minimizes a novel efficient objective derived via Ito's Lemma, allowing each step to run in $O(1)$ gradient evaluations and nearly-linear-in-$d$ ($O(d^{1.19})$) arithmetic operations, reducing the gap between diffusions on symmetric manifolds and Euclidean space. Manifold symmetries ensure the diffusion satisfies an "average-case" Lipschitz condition, enabling accurate and efficient sample generation. Empirically, our model outperforms prior methods in training speed and improves sample quality on synthetic datasets on the torus, special orthogonal group, and unitary group.

Diffusion models have become the de facto framework for generating new datasets. The core of these models lies in the ability to reverse a diffusion process in time. The goal of this manuscript is to explain, from a PDE perspective, how this method works and how to derive the PDE governing the reverse dynamics as well as to study its solution analytically. By linking forward and reverse dynamics, we show that the reverse process's distribution has its support contained within the original distribution. Consequently, diffusion methods, in their analytical formulation, do not inherently regularize the original distribution, and thus, there is no generalization principle. This raises a question: where does generalization arise, given that in practice it does occur? Moreover, we derive an explicit solution to the reverse process's SDE under the assumption that the starting point of the forward process is fixed. This provides a new derivation that links two popular approaches to generative diffusion models: stable diffusion (discrete dynamics) and the score-based approach (continuous dynamics). Finally, we explore the case where the original distribution consists of a finite set of data points. In this scenario, the reverse dynamics are explicit (i.e., the loss function has a clear minimizer), and solving the dynamics fails to generate new samples: the dynamics converge to the original samples. In a sense, solving the minimization problem exactly is "too good for its own good" (i.e., an overfitting regime).

Evolutionary algorithms have been successful in solving multi-objective optimization problems (MOPs). However, as a class of population-based search methodology, evolutionary algorithms require a large number of evaluations of the objective functions, preventing them from being applied to a wide range of expensive MOPs. To tackle the above challenge, this work proposes for the first time a diffusion model that can learn to perform evolutionary multi-objective search, called EmoDM. This is achieved by treating the reversed convergence process of evolutionary search as the forward diffusion and learn the noise distributions from previously solved evolutionary optimization tasks. The pre-trained EmoDM can then generate a set of non-dominated solutions for a new MOP by means of its reverse diffusion without further evolutionary search, thereby significantly reducing the required function evaluations. To enhance the scalability of EmoDM, a mutual entropy-based attention mechanism is introduced to capture the decision variables that are most important for the objectives. Experimental results demonstrate the competitiveness of EmoDM in terms of both the search performance and computational efficiency compared with state-of-the-art evolutionary algorithms in solving MOPs having up to 5000 decision variables. The pre-trained EmoDM is shown to generalize well to unseen problems, revealing its strong potential as a general and efficient MOP solver.

Since their introduction, diffusion models have quickly become the prevailing approach to generative modeling in many domains. They can be interpreted as learning the gradients of a time-varying sequence of log-probability density functions. This interpretation has motivated classifier-based and classifier-free guidance as methods for post-hoc control of diffusion models. In this work, we build upon these ideas using the score-based interpretation of diffusion models, and explore alternative ways to condition, modify, and reuse diffusion models for tasks involving compositional generation and guidance. In particular, we investigate why certain types of composition fail using current techniques and present a number of solutions. We conclude that the sampler (not the model) is responsible for this failure and propose new samplers, inspired by MCMC, which enable successful compositional generation. Further, we propose an energy-based parameterization of diffusion models which enables the use of new compositional operators and more sophisticated, Metropolis-corrected samplers. Intriguingly we find these samplers lead to notable improvements in compositional generation across a wide set of problems such as classifier-guided ImageNet modeling and compositional text-to-image generation.

The field of Generative Artificial Intelligence has witnessed remarkable progress in recent years, fueled by the advent of novel deep learning techniques. Among these advancements, diffusion-based models have emerged as a promising paradigm for generating high-quality, high-dimensional, diverse, and coherent data samples. These models leverage principles from non-equilibrium statistical mechanics to effectively reconstruct the underlying probability distribution from which a training data set was sampled. The central idea behind diffusion models is reverse diffusion. These models gradually add noise to a given data set and observe how the data vectors evolve over time.

Employing a forward diffusion chain to gradually map the data to a noise distribution, diffusion-based generative models learn how to generate the data by inferring a reverse diffusion chain. However, this approach is slow and costly because it needs many forward and reverse steps. We propose a faster and cheaper approach that adds noise not until the data become pure random noise, but until they reach a hidden noisy-data distribution that we can confidently learn. Then, we use fewer reverse steps to generate data by starting from this hidden distribution that is made similar to the noisy data. We reveal that the proposed model can be cast as an adversarial auto-encoder empowered by both the diffusion process and a learnable implicit prior. Experimental results show even with a significantly smaller number of reverse diffusion steps, the proposed truncated diffusion probabilistic models can provide consistent improvements over the non-truncated ones in terms of performance in both unconditional and text-guided image generations. Generating photo-realistic images with probabilistic models is a challenging and important task in machine learning and computer vision, with many potential applications in data augmentation, image editing, style transfer, etc. This new modeling class, which includes both score-based and diffusion-based generative models, uses noise injection to gradually corrupt the data distribution into a simple noise distribution that can be easily sampled from, and then uses a denoising network to reverse the noise injection to generate photo-realistic images. From the perspective of score matching (Hyvärinen & Dayan, 2005; Vincent, 2011) and Langevin dynamics (Neal, 2011; Welling & Teh, 2011), the denoising network is trained by matching the score function, which is the gradient of the log-density of the data, of the corrupted data distribution and that of the generator distribution at different noise levels (Song & Ermon, 2019). This training objective can also be formulated under diffusion-based generative models (Sohl-Dickstein et al., 2015; Ho et al., 2020). These two types of models have been further unified by Song et al. (2021b) under the framework of discretized stochastic differential equations.

Shared autonomy is an operational concept in which a user and an autonomous agent collaboratively control a robotic system. It provides a number of advantages over the extremes of full-teleoperation and full-autonomy in many settings. Traditional approaches to shared autonomy rely on knowledge of the environment dynamics, a discrete space of user goals that is known a priori, or knowledge of the user's policy -- assumptions that are unrealistic in many domains. Recent works relax some of these assumptions by formulating shared autonomy with model-free deep reinforcement learning (RL). In particular, they no longer need knowledge of the goal space (e.g., that the goals are discrete or constrained) or environment dynamics. However, they need knowledge of a task-specific reward function to train the policy. Unfortunately, such reward specification can be a difficult and brittle process. On top of that, the formulations inherently rely on human-in-the-loop training, and that necessitates them to prepare a policy that mimics users' behavior. In this paper, we present a new approach to shared autonomy that employs a modulation of the forward and reverse diffusion process of diffusion models. Our approach does not assume known environment dynamics or the space of user goals, and in contrast to previous work, it does not require any reward feedback, nor does it require access to the user's policy during training. Instead, our framework learns a distribution over a space of desired behaviors. It then employs a diffusion model to translate the user's actions to a sample from this distribution. Crucially, we show that it is possible to carry out this process in a manner that preserves the user's control authority. We evaluate our framework on a series of challenging continuous control tasks, and analyze its ability to effectively correct user actions while maintaining their autonomy.

How do diffusion generative models convert pure noise into meaningful images? We argue that generation involves first committing to an outline, and then to finer and finer details. The corresponding reverse diffusion process can be modeled by dynamics on a (time-dependent) high-dimensional landscape full of Gaussian-like modes, which makes the following predictions: (i) individual trajectories tend to be very low-dimensional; (ii) scene elements that vary more within training data tend to emerge earlier; and (iii) early perturbations substantially change image content more often than late perturbations. We show that the behavior of a variety of trained unconditional and conditional diffusion models like Stable Diffusion is consistent with these predictions. Finally, we use our theory to search for the latent image manifold of diffusion models, and propose a new way to generate interpretable image variations. Our viewpoint suggests generation by GANs and diffusion models have unexpected similarities.

Diffusion models have shown a great ability at bridging the performance gap between predictive and generative approaches for speech enhancement. We have shown that they may even outperform their predictive counterparts for non-additive corruption types or when they are evaluated on mismatched conditions. However, diffusion models suffer from a high computational burden, mainly as they require to run a neural network for each reverse diffusion step, whereas predictive approaches only require one pass. As diffusion models are generative approaches they may also produce vocalizing and breathing artifacts in adverse conditions. In comparison, in such difficult scenarios, predictive models typically do not produce such artifacts but tend to distort the target speech instead, thereby degrading the speech quality. In this work, we present a stochastic regeneration approach where an estimate given by a predictive model is provided as a guide for further diffusion. We show that the proposed approach uses the predictive model to remove the vocalizing and breathing artifacts while producing very high quality samples thanks to the diffusion model, even in adverse conditions. We further show that this approach enables to use lighter sampling schemes with fewer diffusion steps without sacrificing quality, thus lifting the computational burden by an order of magnitude. Source code and audio examples are available online (https://uhh.de/inf-sp-storm).