Though denoising diffusion probabilistic models (DDPMs) have achieved remarkable generation results, the low sampling efficiency of DDPMs still limits further applications. Since DDPMs can be formulated as diffusion ordinary differential equations (ODEs), various fast sampling methods can be derived from solving diffusion ODEs. However, we notice that previous sampling methods with fixed analytical form are not robust with the error in the noise estimated from pretrained diffusion models. In this work, we construct an error-robust Adams solver (ERA-Solver), which utilizes the implicit Adams numerical method that consists of a predictor and a corrector. Different from the traditional predictor based on explicit Adams methods, we leverage a Lagrange interpolation function as the predictor, which is further enhanced with an error-robust strategy to adaptively select the Lagrange bases with lower error in the estimated noise. Experiments on Cifar10, LSUN-Church, and LSUN-Bedroom datasets demonstrate that our proposed ERA-Solver achieves 5.14, 9.42, and 9.69 Fenchel Inception Distance (FID) for image generation, with only 10 network evaluations.
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Performing super-resolution of a depth image using the guidance from an RGB image is a problem that concerns several fields, such as robotics, medical imaging, and remote sensing. While deep learning methods have achieved good results in this problem, recent work highlighted the value of combining modern methods with more formal frameworks. In this work, we propose a novel approach which combines guided anisotropic diffusion with a deep convolutional network and advances the state of the art for guided depth super-resolution. The edge transferring/enhancing properties of the diffusion are boosted by the contextual reasoning capabilities of modern networks, and a strict adjustment step guarantees perfect adherence to the source image. We achieve unprecedented results in three commonly used benchmarks for guided depth super-resolution. The performance gain compared to other methods is the largest at larger scales, such as x32 scaling. Code (//github.com/prs-eth/Diffusion-Super-Resolution) for the proposed method is available to promote reproducibility of our results.
Diffusion-based generative models have recently emerged as powerful solutions for high-quality synthesis in multiple domains. Leveraging the bidirectional Markov chains, diffusion probabilistic models generate samples by inferring the reversed Markov chain based on the learned distribution mapping at the forward diffusion process. In this work, we propose Modiff, a conditional paradigm that benefits from the denoising diffusion probabilistic model (DDPM) to tackle the problem of realistic and diverse action-conditioned 3D skeleton-based motion generation. We are a pioneering attempt that uses DDPM to synthesize a variable number of motion sequences conditioned on a categorical action. We evaluate our approach on the large-scale NTU RGB+D dataset and show improvements over state-of-the-art motion generation methods.
Generative image modeling enables a wide range of applications but raises ethical concerns about responsible deployment. This paper introduces an active strategy combining image watermarking and Latent Diffusion Models. The goal is for all generated images to conceal an invisible watermark allowing for future detection and/or identification. The method quickly fine-tunes the latent decoder of the image generator, conditioned on a binary signature. A pre-trained watermark extractor recovers the hidden signature from any generated image and a statistical test then determines whether it comes from the generative model. We evaluate the invisibility and robustness of the watermarks on a variety of generation tasks, showing that Stable Signature works even after the images are modified. For instance, it detects the origin of an image generated from a text prompt, then cropped to keep $10\%$ of the content, with $90$+$\%$ accuracy at a false positive rate below 10$^{-6}$.
Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.
In this paper, a sampling-based trajectory planning algorithm for a laboratory-scale 3D gantry crane in an environment with static obstacles and subject to bounds on the velocity and acceleration of the gantry crane system is presented. The focus is on developing a fast motion planning algorithm for differentially flat systems, where intermediate results can be stored and reused for further tasks, such as replanning. The proposed approach is based on the informed optimal rapidly exploring random tree algorithm (informed RRT*), which is utilized to build trajectory trees that are reused for replanning when the start and/or target states change. In contrast to state-of-the-art approaches, the proposed motion planning algorithm incorporates a linear quadratic minimum time (LQTM) local planner. Thus, dynamic properties such as time optimality and the smoothness of the trajectory are directly considered in the proposed algorithm. Moreover, by integrating the branch-and-bound method to perform the pruning process on the trajectory tree, the proposed algorithm can eliminate points in the tree that do not contribute to finding better solutions. This helps to curb memory consumption and reduce the computational complexity during motion (re)planning. Simulation results for a validated mathematical model of a 3D gantry crane show the feasibility of the proposed approach.
Large neural networks can improve the accuracy and generalization on tasks across many domains. However, this trend cannot continue indefinitely due to limited hardware memory. As a result, researchers have devised a number of memory optimization methods (MOMs) to alleviate the memory bottleneck, such as gradient checkpointing, quantization, and swapping. In this work, we study memory optimization methods and show that, although these strategies indeed lower peak memory usage, they can actually decrease training throughput by up to 9.3x. To provide practical guidelines for practitioners, we propose a simple but effective performance model PAPAYA to quantitatively explain the memory and training time trade-off. PAPAYA can be used to determine when to apply the various memory optimization methods in training different models. We outline the circumstances in which memory optimization techniques are more advantageous based on derived implications from PAPAYA. We assess the accuracy of PAPAYA and the derived implications on a variety of machine models, showing that it achieves over 0.97 R score on predicting the peak memory/throughput, and accurately predicts the effectiveness of MOMs across five evaluated models on vision and NLP tasks.
MRI synthesis promises to mitigate the challenge of missing MRI modality in clinical practice. Diffusion model has emerged as an effective technique for image synthesis by modelling complex and variable data distributions. However, most diffusion-based MRI synthesis models are using a single modality. As they operate in the original image domain, they are memory-intensive and less feasible for multi-modal synthesis. Moreover, they often fail to preserve the anatomical structure in MRI. Further, balancing the multiple conditions from multi-modal MRI inputs is crucial for multi-modal synthesis. Here, we propose the first diffusion-based multi-modality MRI synthesis model, namely Conditioned Latent Diffusion Model (CoLa-Diff). To reduce memory consumption, we design CoLa-Diff to operate in the latent space. We propose a novel network architecture, e.g., similar cooperative filtering, to solve the possible compression and noise in latent space. To better maintain the anatomical structure, brain region masks are introduced as the priors of density distributions to guide diffusion process. We further present auto-weight adaptation to employ multi-modal information effectively. Our experiments demonstrate that CoLa-Diff outperforms other state-of-the-art MRI synthesis methods, promising to serve as an effective tool for multi-modal MRI synthesis.
In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly $1$ for the approximate solution. The proof is based on new H\"older continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain $H^1$ stability, higher moment $H^1$ stability, $L^2$ stability, and higher moment $L^2$ stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
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