Diffusion models achieve state-of-the-art generation quality across many applications, but their ability to capture rare or extreme events in heavy-tailed distributions remains unclear. In this work, we show that traditional diffusion and flow-matching models with standard Gaussian priors fail to capture heavy-tailed behavior. We address this by repurposing the diffusion framework for heavy-tail estimation using multivariate Student-t distributions. We develop a tailored perturbation kernel and derive the denoising posterior based on the conditional Student-t distribution for the backward process. Inspired by $\gamma$-divergence for heavy-tailed distributions, we derive a training objective for heavy-tailed denoisers. The resulting framework introduces controllable tail generation using only a single scalar hyperparameter, making it easily tunable for diverse real-world distributions. As specific instantiations of our framework, we introduce t-EDM and t-Flow, extensions of existing diffusion and flow models that employ a Student-t prior. Remarkably, our approach is readily compatible with standard Gaussian diffusion models and requires only minimal code changes. Empirically, we show that our t-EDM and t-Flow outperform standard diffusion models in heavy-tail estimation on high-resolution weather datasets in which generating rare and extreme events is crucial.

Score Distillation Sampling (SDS) has been pivotal for leveraging pre-trained diffusion models in downstream tasks such as inverse problems, but it faces two major challenges: $(i)$ mode collapse and $(ii)$ latent space inversion, which become more pronounced in high-dimensional data. To address mode collapse, we introduce a novel variational framework for posterior sampling. Utilizing the Wasserstein gradient flow interpretation of SDS, we propose a multimodal variational approximation with a \emph{repulsion} mechanism that promotes diversity among particles by penalizing pairwise kernel-based similarity. This repulsion acts as a simple regularizer, encouraging a more diverse set of solutions. To mitigate latent space ambiguity, we extend this framework with an \emph{augmented} variational distribution that disentangles the latent and data. This repulsive augmented formulation balances computational efficiency, quality, and diversity. Extensive experiments on linear and nonlinear inverse tasks with high-resolution images ($512 \times 512$) using pre-trained Stable Diffusion models demonstrate the effectiveness of our approach.

Conditional diffusion and flow models are effective for super-resolving small-scale details in natural images. However, in physical sciences such as weather, three major challenges arise: (i) spatially misaligned input-output distributions (PDEs at different resolutions lead to divergent trajectories), (ii) misaligned and distinct input-output channels (channel synthesis), (iii) several channels with diverse stochasticity scales (multiscale). To address these, we propose to first encode inputs into a latent base distribution that is closer to the target, then apply Flow Matching to generate small-scale physics. The encoder captures deterministic components, while Flow Matching adds stochastic details. To handle uncertainty in the deterministic part, we inject noise via an adaptive noise scaling mechanism, dynamically adjusted by maximum-likelihood estimates of the encoder’s predictions. Experiments on real-world weather data (including super-resolution from 25 km to 2 km scales in Taiwan) and in synthetic Kolmogorov flow datasets show that our proposed Adaptive Flow Matching (AFM) framework outperforms existing methods and produces better-calibrated ensembles.

Diffusion models are a powerful tool for probabilistic forecasting, yet most applications in high-dimensional complex systems predict future states individually. This approach struggles to model complex temporal dependencies and fails to explicitly account for the progressive growth of uncertainty inherent to the systems. While rolling diffusion frameworks, which apply increasing noise to forecasts at longer lead times, have been proposed to address this, their integration with state-of-the-art, high-fidelity diffusion techniques remains a significant challenge. We tackle this problem by introducing Elucidated Rolling Diffusion Models (ERDM), the first framework to successfully unify a rolling forecast structure with the principled, performant design of Elucidated Diffusion Models (EDM). To do this, we adapt the core EDM components--its noise schedule, network preconditioning, and Heun sampler--to the rolling forecast setting. The success of this integration is driven by three key contributions: $(i)$ a novel loss weighting scheme that focuses model capacity on the mid-range forecast horizons where determinism gives way to stochasticity; $(ii)$ an efficient initialization strategy using a pre-trained EDM for the initial window; and $(iii)$ a bespoke hybrid sequence architecture for robust spatiotemporal feature extraction under progressive denoising. On 2D Navier–Stokes simulations and ERA5 global weather forecasting at $1.5^\circ$ resolution, ERDM consistently outperforms key diffusion-based baselines, including conditional autoregressive EDM. ERDM offers a flexible and powerful general framework for tackling diffusion-based dynamics forecasting problems where modeling uncertainty propagation is paramount.

The iterative and stochastic nature of diffusion models enables *test-time scaling*, whereby spending additional compute during denoising generates higher-fidelity samples. Increasing the number of denoising steps is the primary scaling axis, but this yields quickly diminishing returns. Instead optimizing the *noise trajectory*—the sequence of injected noise vectors—is promising, as the specific noise realizations critically affect sample quality; but this is challenging due to a high-dimensional search space, complex noise-outcome interactions, and costly trajectory evaluations. We address this by first casting diffusion as a Markov Decision Process (MDP) with a terminal reward, showing tree-search methods such as Monte Carlo tree search (MCTS) to be meaningful but impractical. To balance performance and efficiency, we then resort to a relaxation of MDP, where we view denoising as a sequence of independent *contextual bandits*. This allows us to introduce an $\epsilon$-greedy search algorithm that *globally explores* at extreme timesteps and *locally exploits* during the intermediate steps where de-mixing occurs. Experiments on EDM and Stable Diffusion reveal state-of-the-art scores for class-conditioned/text-to-image generation, exceeding baselines by up to $164$% and matching/exceeding MCTS performance. To our knowledge, this is the first practical method for test-time noise *trajectory* optimization of *arbitrary (non-differentiable)* rewards.

Accelerated MRI protocols routinely involve a predefined sampling pattern that undersamples the k-space. Finding an optimal pattern can enhance the reconstruction quality, however this optimization is a challenging task. To address this challenge, we introduce a novel deep learning framework, AutoSamp, based on variational information maximization that enables joint optimization of sampling pattern and reconstruction of MRI scans. We represent the encoder as a non-uniform Fast Fourier Transform that allows continuous optimization of k-space sample locations on a non-Cartesian plane, and the decoder as a deep reconstruction network. Experiments on public 3D acquired MRI datasets show improved reconstruction quality of the proposed AutoSamp method over the prevailing variable density and variable density Poisson disc sampling for both compressed sensing and deep learning reconstructions. We demonstrate that our data-driven sampling optimization method achieves 4.4dB, 2.0dB, 0.75dB, 0.7dB PSNR improvements over reconstruction with Poisson Disc masks for acceleration factors of R =5, 10, 15, 25, respectively. Prospectively accelerated acquisitions with 3D FSE sequences using our optimized sampling patterns exhibit improved image quality and sharpness. Furthermore, we analyze the characteristics of the learned sampling patterns with respect to changes in acceleration factor, measurement noise, underlying anatomy, and coil sensitivities. We show that all these factors contribute to the optimization result by affecting the sampling density, k-space coverage and point spread functions of the learned sampling patterns.

Diffusion models have emerged as a key pillar of foundation models in visual domains. One of their critical applications is to universally solve different downstream inverse tasks via a single diffusion prior without re-training for each task. Most inverse tasks can be formulated as inferring a posterior distribution over data (e.g., a full image) given a measurement (e.g., a masked image). This is however challenging in diffusion models since the nonlinear and iterative nature of the diffusion process renders the posterior intractable. To cope with this challenge, we propose a variational approach that by design seeks to approximate the true posterior distribution. We show that our approach naturally leads to regularization by denoising diffusion process (RED-diff) where denoisers at different timesteps concurrently impose different structural constraints over the image. To gauge the contribution of denoisers from different timesteps, we propose a weighting mechanism based on signal-to-noise-ratio (SNR). Our approach provides a new variational perspective for solving inverse problems with diffusion models, allowing us to formulate sampling as stochastic optimization, where one can simply apply off-the-shelf solvers with lightweight iterates. Our experiments for various linear and nonlinear image restoration tasks demonstrate the strengths of our method compared with state-of-the-art sampling-based diffusion models. The code is available online \footnote{\url{https://github.com/NVlabs/RED-diff}}.

It has been demonstrated that the training problem for a variety of (non) linear two-layer neural networks (such as two-layer perceptrons, convolutional networks, and self-attention) can be posed as equivalent convex optimization problems, with an induced regularizer which encourages low rank. However, this regularizer becomes prohibitively expensive to compute at moderate scales, impeding training convex neural networks. To this end, we propose applying the Burer-Monteiro factorization to convex neural networks, which for the first time enables a Burer-Monteiro perspective on neural networks with non-linearities. This factorization leads to an equivalent yet computationally tractable non-convex alternative with no spurious local minima. We develop a novel relative optimality bound of stationary points of the Burer-Monteiro factorization, providing verifiable conditions under which any stationary point is a global optimum. Further, for the first time, we show that linear self-attention with sufficiently many heads has no spurious local minima. Our experiments validate the novel relative optimality bound and the utility of the Burer-Monteiro factorization for scaling convex neural networks.

Existing text-to-image models struggle to follow complex text prompts, raising the need for extra grounding inputs for better controllability. In this work, we propose to decompose a scene into visual primitives - denoted as dense blob representations - that contain fine-grained details of the scene while being modular, human-interpretable, and easy-to-construct. Based on blob representations, we develop a blob-grounded text-to-image diffusion model, termed BlobGEN, for compositional generation. Particularly, we introduce a new masked cross-attention module to disentangle the fusion between blob representations and visual features. To leverage the compositionality of large language models (LLMs), we introduce a new in-context learning approach to generate blob representations from text prompts. Our extensive experiments show that BlobGEN achieves superior zero-shot generation quality and better layout-guided controllability on MS-COCO. When augmented by LLMs, our method exhibits superior numerical and spatial correctness on compositional image generation benchmarks.

Using image models naively for solving inverse video problems often suffers from flickering, texture-sticking, and temporal inconsistency in generated videos. To tackle these problems, in this paper, we view frames as continuous functions in the 2D space, and videos as a sequence of continuous warping transformations between different frames. This perspective allows us to train function space diffusion models only on **images** and utilize them to solve temporally correlated inverse problems. The function space diffusion models need to be equivariant with respect to the underlying spatial transformations. To ensure temporal consistency, we introduce a simple post-hoc test-time guidance towards (self)-equivariant solutions. Our method allows us to deploy state-of-the-art latent diffusion models such as Stable Diffusion XL to solve video inverse problems. We demonstrate the effectiveness of our method for video inpainting and $8\times$ video super-resolution, outperforming existing techniques based on noise transformations. We provide generated video results in the following URL: https://giannisdaras.github.io/warped_diffusion.github.io/.

We consider guiding denoising diffusion models with general differentiable loss functions in a plug-and-play fashion, enabling controllable generation without additional training. This paradigm, termed Loss-Guided Diffusion (LGD), can easily be integrated into all diffusion models and leverage various efficient samplers. Despite the benefits, the resulting guidance term is, unfortunately, an intractable integral and needs to be approximated. Existing methods compute the guidance term based on a point estimate. However, we show that such approaches have significant errors over the scale of the approximations. To address this issue, we propose a Monte Carlo method that uses multiple samples from a suitable distribution to reduce bias. Our method is effective in various synthetic and real-world settings, including image super-resolution, text or label-conditional image generation, and controllable motion synthesis. Notably, we show how our method can be applied to control a pretrained motion diffusion model to follow certain paths and avoid obstacles that are proven challenging to prior methods.

Vision transformers using self-attention or its proposed alternatives have demonstrated promising results in many image related tasks. However, the underpinning inductive bias of attention is not well understood. To address this issue, this paper analyzes attention through the lens of convex duality. For the non-linear dot-product self-attention, and alternative mechanisms such as MLP-mixer and Fourier Neural Operator (FNO), we derive equivalent finite-dimensional convex problems that are interpretable and solvable to global optimality. The convex programs lead to block nuclear-norm regularization that promotes low rank in the latent feature and token dimensions. In particular, we show how self-attention networks implicitly clusters the tokens, based on their latent similarity. We conduct experiments for transferring a pre-trained transformer backbone for CIFAR-100 classification by fine-tuning a variety of convex attention heads. The results indicate the merits of the bias induced by attention compared with the existing MLP or linear heads.

Reliable MRI is crucial for accurate interpretation in therapeutic and diagnostic tasks. However, undersampling during MRI acquisition as well as the overparameterized and non-transparent nature of deep learning (DL) leaves substantial uncertainty about the accuracy of DL reconstruction. With this in mind, this study aims to quantify the uncertainty in image recovery with DL models. To this end, we first leverage variational autoencoders (VAEs) to develop a probabilistic reconstruction scheme that maps out (low-quality) short scans with aliasing artifacts to the diagnostic-quality ones. The VAE encodes the acquisition uncertainty in a latent code and naturally offers a posterior of the image from which one can generate pixel variance maps using Monte-Carlo sampling. Accurately predicting risk requires knowledge of the bias as well, for which we leverage Stein's Unbiased Risk Estimator (SURE) as a proxy for mean-squared-error (MSE). A range of empirical experiments is performed for Knee MRI reconstruction under different training losses (adversarial and pixel-wise) and unrolled recurrent network architectures. Our key observations indicate that: 1) adversarial losses introduce more uncertainty; and 2) recurrent unrolled nets reduce the prediction uncertainty and risk.

Lack of ground-truth MR images impedes the common supervised training of neural networks for image reconstruction. To cope with this challenge, this article leverages unpaired adversarial training for reconstruction networks, where the inputs are undersampled k-space and naively reconstructed images from one dataset, and the labels are high-quality images from another dataset. The reconstruction networks consist of a generator which suppresses the input image artifacts, and a discriminator using a pool of (unpaired) labels to adjust the reconstruction quality. The generator is an unrolled neural network - a cascade of convolutional and data consistency layers. The discriminator is also a multilayer CNN that plays the role of a critic scoring the quality of reconstructed images based on the Wasserstein distance. Our experiments with knee MRI datasets demonstrate that the proposed unpaired training enables diagnostic-quality reconstruction when high-quality image labels are not available for the input types of interest, or when the amount of labels is small. In addition, our adversarial training scheme can achieve better image quality (as rated by expert radiologists) compared with the paired training schemes with pixel-wise loss.

Synthesizing larger texture images from a smaller exemplar is an important task in graphics and vision. The conventional CNNs, recently adopted for synthesis, require to train and test on the same set of images and fail to generalize to unseen images. This is mainly because those CNNs fully rely on convolutional and upsampling layers that operate locally and not suitable for a task as global as texture synthesis. In this work, inspired by the repetitive nature of texture patterns, we find that texture synthesis can be viewed as (local) \textit{upsampling} in the Fast Fourier Transform (FFT) domain. However, FFT of natural images exhibits high dynamic range and lacks local correlations. Therefore, to train CNNs we design a framework to perform FFT upsampling in feature space using deformable convolutions. Such design allows our framework to generalize to unseen images, and synthesize textures in a single pass. Extensive evaluations confirm that our method achieves state-of-the-art performance both quantitatively and qualitatively.

Undersampled magnetic resonance image (MRI) reconstruction is typically an ill-posed linear inverse task. The time and resource intensive computations require tradeoffs between accuracy and speed. In addition, state-of-the-art compressed sensing (CS) analytics are not cognizant of the image diagnostic quality. To address these challenges, we propose a novel CS framework that uses generative adversarial networks (GAN) to model the (low-dimensional) manifold of high-quality MR images. Leveraging a mixture of least-squares (LS) GANs and pixel-wise ℓ1/ℓ2cost, a deep residual network with skip connections is trained as the generator that learns to remove the aliasing artifacts by projecting onto the image manifold. The LSGAN learns the texture details, while the ℓ1/ℓ2cost suppresses high-frequency noise. A discriminator network, which is a multilayer convolutional neural network (CNN), plays the role of a perceptual cost that is then jointly trained based on high-quality MR images to score the quality of retrieved images. In the operational phase, an initial aliased estimate (e.g., simply obtained by zero-filling) is propagated into the trained generator to output the desired reconstruction. This demands a very low computational overhead. Extensive evaluations are performed on a large contrast-enhanced MR dataset of pediatric patients. Images rated by expert radiologists corroborate that GANCS retrieves higher quality images with improved fine texture details compared with conventional Wavelet-based and dictionary-learning-based CS schemes as well as with deep-learning-based schemes using pixel-wise training. In addition, it offers reconstruction times of under a few milliseconds, which are two orders of magnitude faster than the current state-of-the-art CS-MRI schemes.

Recovering high-resolution images from limited sensory data typically leads to a serious ill-posed inverse problem, demanding inversion algorithms that effectively capture the prior information. Learning a good inverse mapping from training data faces severe challenges, including: (i) scarcity of training data; (ii) need for plausible reconstructions that are physically feasible; (iii) need for fast reconstruction, especially in real-time applications. We develop a successful system solving all these challenges, using as basic architecture the repetitive application of alternating proximal and data fidelity constraints. We learn a proximal map that works well with real images based on residual networks with recurrent blocks. Extensive experiments are carried out under different settings: (a) reconstructing abdominal MRI of pediatric patients from highly undersampled k-space data and (b) super-resolving natural face images. Our key findings include: 1. a recurrent ResNet with a single residual block (10-fold repetition) yields an effective proximal which accurately reveals MR image details. 2. Our architecture significantly outperforms conventional non-recurrent deep ResNets by 2dB SNR; it is also trained much more rapidly. 3. It outperforms state-of-the-art compressed-sensing Wavelet-based methods by 4dB SNR, with 100x speedups in reconstruction time.

Mapping origin-destination (OD) network traffic is pivotal for network management and proactive security tasks. However, lack of sufficient flow-level measurements as well as potential anomalies pose major challenges towards this goal. Leveraging the spatiotemporal correlation of nominal traffic, and the sparse nature of anomalies, this paper brings forth a novel framework to map out nominal and anomalous traffic, which treats jointly important network monitoring tasks including traffic estimation, anomaly detection, and traffic interpolation. To this end, a convex program is first formulated with nuclear and l1-norm regularization to effect sparsity and low rank for the nominal and anomalous traffic with only the link counts and a small subset of OD-flow counts. Analysis and simulations confirm that the proposed estimator can exactly recover sufficiently low-dimensional nominal traffic and sporadic anomalies so long as the routing paths are sufficiently “spread-out” across the network, and an adequate amount of flow counts are randomly sampled. The results offer valuable insights about data acquisition strategies and network scenaria giving rise to accurate traffic estimation. For practical networks where the aforementioned conditions are possibly violated, the inherent spatiotemporal traffic patterns are taken into account by adopting a Bayesian approach along with a bilinear characterization of the nuclear and l1 norms. The resultant nonconvex program involves quadratic regularizers with correlation matrices, learned systematically from (cyclo)stationary historical data. Alternating-minimization based algorithms with provable convergence are also developed to procure the estimates. Insightful tests with synthetic and real Internet data corroborate the effectiveness of the novel schemes.

Accurate estimation of origin-to-destination (OD) traffic flows provides valuable input for network management tasks. However, lack of flow-level observations as well as intentional and unintentional anomalies pose major challenges toward achieving this goal. Leveraging the low intrinsic-dimensionality of OD flows and the sparse nature of anomalies, this paper proposes a convex program with nuclear-norm and ℓ1-norm regularization terms to estimate the nominal and anomalous traffic components, using a small subset of (possibly anomalous) flow counts in addition to link counts. Analysis and simulations confirm that the said estimator can exactly recover sufficiently low-dimensional nominal traffic and sparse enough anomalies when the routing matrix is column-incoherent, and an adequate amount of flow counts are randomly sampled. The results offer valuable insights about the measurement types and network scenaria giving rise to accurate traffic estimation. Tests with real Internet data corroborate the effectiveness of the novel estimator.

Extracting latent low-dimensional structure from high-dimensional data is of paramount importance in timely inference tasks encountered with `Big Data' analytics. However, increasingly noisy, heterogeneous, and incomplete datasets as well as the need for real-time processing pose major challenges towards achieving this goal. In this context, the fresh look advocated here permeates benefits from rank minimization to track low-dimensional subspaces from incomplete data. Leveraging the low-dimensionality of the subspace sought, a novel estimator is proposed based on an exponentially-weighted least-squares criterion regularized with the nuclear norm. After recasting the non-separable nuclear norm into a form amenable to online optimization, a real-time algorithm is developed and its convergence established under simplifying technical assumptions. The novel subspace tracker can asymptotically offer the well-documented performance guarantees of the batch nuclear-norm regularized estimator. Simulated tests with real Internet data confirm the efficacy of the proposed algorithm in tracking the traffic subspace, and its superior performance relative to state-of-the-art alternatives.

Given the noiseless superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the low-rank and sparse components becomes possible. This fundamental identifiability issue arises with traffic anomaly detection in backbone networks, and subsumes compressed sensing as well as the timely low-rank plus sparse matrix recovery tasks encountered in matrix decomposition problems. Leveraging the ability of l1and nuclear norms to recover sparse and low-rank matrices, a convex program is formulated to estimate the unknowns. Analysis and simulations confirm that the said convex program can recover the unknowns for sufficiently low-rank and sparse enough components, along with a compression matrix possessing an isometry property when restricted to operate on sparse vectors. When the low-rank, sparse, and compression matrices are drawn from certain random ensembles, it is established that exact recovery is possible with high probability. First-order algorithms are developed to solve the nonsmooth convex optimization problem with provable iteration complexity guarantees. Insightful tests with synthetic and real network data corroborate the effectiveness of the novel approach in unveiling traffic anomalies across flows and time, and its ability to outperform existing alternatives.