Creating noise from data is easy; creating data from noise is generative modeling. We present a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by slowly removing the noise. Crucially, the reverse-time SDE depends only on the time-dependent gradient field (\aka, score) of the perturbed data distribution. By leveraging advances in score-based generative modeling, we can accurately estimate these scores with neural networks, and use numerical SDE solvers to generate samples. We show that this framework encapsulates previous approaches in score-based generative modeling and diffusion probabilistic modeling, allowing for new sampling procedures and new modeling capabilities. In particular, we introduce a predictor-corrector framework to correct errors in the evolution of the discretized reverse-time SDE. We also derive an equivalent neural ODE that samples from the same distribution as the SDE, but additionally enables exact likelihood computation, and improved sampling efficiency. In addition, we provide a new way to solve inverse problems with score-based models, as demonstrated with experiments on class-conditional generation, image inpainting, and colorization. Combined with multiple architectural improvements, we achieve record-breaking performance for unconditional image generation on CIFAR-10 with an Inception score of 9.89 and FID of 2.20, a competitive likelihood of 2.99 bits/dim, and demonstrate high fidelity generation of 1024 x 1024 images for the first time from a score-based generative model.
Czech Technical University in Prague（布拉格捷克理工大学）；Cisco Systems, Inc.（思科系统公司）
Reconstruction error is a prevalent score used to identify anomalous samples when data are modeled by generative models, such as (variational) auto-encoders or generative adversarial networks. This score relies on the assumption that normal samples are located on a manifold and all anomalous samples are located outside. Since the manifold can be learned only where the training data lie, there are no guarantees how the reconstruction error behaves elsewhere and the score, therefore, seems to be ill-defined. This work defines an anomaly score that is theoretically compatible with generative mode…
Multivariate time series prediction has attracted a lot of attention because of its wide applications such as intelligence transportation, AIOps. Generative models have achieved impressive results in time series modeling because they can model data distribution and take noise into consideration. However, many existing works can not be widely used because of the constraints of functional form of generative models or the sensitivity to hyperparameters. In this paper, we propose ScoreGrad, a multivariate probabilistic time series forecasting framework based on continuous energy-based generative m…
Score calibration enables automatic speaker recognizers to make cost-effective accept / reject decisions. Traditional calibration requires supervised data, which is an expensive resource. We propose a 2-component GMM for unsupervised calibration and demonstrate good performance relative to a supervised baseline on NIST SRE'10 and SRE'12. A Bayesian analysis demonstrates that the uncertainty associated with the unsupervised calibration parameter estimates is surprisingly small.
Learning generative models for graph-structured data is challenging because graphs are discrete, combinatorial, and the underlying data distribution is invariant to the ordering of nodes. However, most of the existing generative models for graphs are not invariant to the chosen ordering, which might lead to an undesirable bias in the learned distribution. To address this difficulty, we propose a permutation invariant approach to modeling graphs, using the recent framework of score-based generative modeling. In particular, we design a permutation equivariant, multi-channel graph neural network …
Score-based generative models (SGMs) have recently demonstrated impressive results in terms of both sample quality and distribution coverage. However, they are usually applied directly in data space and often require thousands of network evaluations for sampling. Here, we propose the Latent Score-based Generative Model (LSGM), a novel approach that trains SGMs in a latent space, relying on the variational autoencoder framework. Moving from data to latent space allows us to train more expressive generative models, apply SGMs to non-continuous data, and learn smoother SGMs in a smaller space, re…
Deep generative models (DGMs) of images are now sufficiently mature that they produce nearly photorealistic samples and obtain scores similar to the data distribution on heuristics such as Frechet Inception Distance (FID). These results, especially on large-scale datasets such as ImageNet, suggest that DGMs are learning the data distribution in a perceptually meaningful space and can be used in downstream tasks. To test this latter hypothesis, we use class-conditional generative models from a number of model classes---variational autoencoders, autoregressive models, and generative adversarial …
ENS（巴黎高等师范学院）；CNRS（法国国家科学研究中心）；PSL University（巴黎文理研究大学）；University of Oxford（牛津大学）
Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by dist…
KU Leuven（鲁汶大学）；Univ. Lille（里尔大学）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）
Commonly, machine learning models minimize an empirical expectation. As a result, the trained models typically perform well for the majority of the data but the performance may deteriorate in less dense regions of the dataset. This issue also arises in generative modeling. A generative model may overlook underrepresented modes that are less frequent in the empirical data distribution. This problem is known as complete mode coverage. We propose a sampling procedure based on ridge leverage scores which significantly improves mode coverage when compared to standard methods and can easily be combi…
NVIDIA（英伟达）；University of Waterloo（滑铁卢大学）；Vector Institute（矢量研究所）
Score-based generative models (SGMs) have demonstrated remarkable synthesis quality. SGMs rely on a diffusion process that gradually perturbs the data towards a tractable distribution, while the generative model learns to denoise. The complexity of this denoising task is, apart from the data distribution itself, uniquely determined by the diffusion process. We argue that current SGMs employ overly simplistic diffusions, leading to unnecessarily complex denoising processes, which limit generative modeling performance. Based on connections to statistical mechanics, we propose a novel critically-…
Score-based generative models can produce high quality image samples comparable to GANs, without requiring adversarial optimization. However, existing training procedures are limited to images of low resolution (typically below 32x32), and can be unstable under some settings. We provide a new theoretical analysis of learning and sampling from score models in high dimensional spaces, explaining existing failure modes and motivating new solutions that generalize across datasets. To enhance stability, we also propose to maintain an exponential moving average of model weights. With these improveme…
On Forgetting and Stability of Score-based Generative models
Sorbonne Université and Université Paris Cité, CNRS, LPSM（索邦大学和巴黎大学，CNRS，LPSM）；Center for Statistics and Images, Mines Paris, PSL University（统计与图像中心，巴黎 Mines，PSL 大学）；CREST, ENSAE Paris, Institut Polytechnique de Paris（CREST，ENSAE 巴黎，巴黎理工学院）
Compared to the existing function-based models in deep generative modeling, the recently proposed diffusion models have achieved outstanding performance with a stochastic-process-based approach. But a long sampling time is required for this approach due to many timesteps for discretization. Schrödinger bridge (SB)-based models attempt to tackle this problem by training bidirectional stochastic processes between distributions. However, they still have a slow sampling speed compared to generative models such as generative adversarial networks. And due to the training of the bidirectional stochas…
UC Berkeley（加州大学伯克利分校）；Prescient Design, Genentech, Roche（罗氏基因泰克Prescient Design部门）；NNAISENSE（NNAISENSE公司）；Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
We consider the problem of generative modeling based on smoothing an unknown density of interest in $\mathbb{R}^d$ using factorial kernels with $M$ independent Gaussian channels with equal noise levels introduced by Saremi and Srivastava (2022). First, we fully characterize the time complexity of learning the resulting smoothed density in $\mathbb{R}^{Md}$, called M-density, by deriving a universal form for its parametrization in which the score function is by construction permutation equivariant. Next, we study the time complexity of sampling an M-density by analyzing its condition number for…
ENS（巴黎高等师范学院）；CNRS（法国国家科学研究中心）；PSL University（巴黎文理研究大学）；Collège de France（法兰西公学院）；Flatiron Institute（弗莱提荣研究所）
Score-based generative models (SGMs) synthesize new data samples from Gaussian white noise by running a time-reversed Stochastic Differential Equation (SDE) whose drift coefficient depends on some probabilistic score. The discretization of such SDEs typically requires a large number of time steps and hence a high computational cost. This is because of ill-conditioning properties of the score that we analyze mathematically. We show that SGMs can be considerably accelerated, by factorizing the data distribution into a product of conditional probabilities of wavelet coefficients across scales. Th…
Several machine learning applications involve the optimization of higher-order derivatives (e.g., gradients of gradients) during training, which can be expensive in respect to memory and computation even with automatic differentiation. As a typical example in generative modeling, score matching (SM) involves the optimization of the trace of a Hessian. To improve computing efficiency, we rewrite the SM objective and its variants in terms of directional derivatives, and present a generic strategy to efficiently approximate any-order directional derivative with finite difference (FD). Our approxi…
We propose a theoretical framework for thinking about score normalization, which confirms that normalization is not needed under (admittedly fragile) ideal conditions. If, however, these conditions are not met, e.g. under data-set shift between training and runtime, our theory reveals dependencies between scores that could be exploited by strategies such as score normalization. Indeed, it has been demonstrated over and over experimentally, that various ad-hoc score normalization recipes do work. We present a first attempt at using probability theory to design a generative score-space normaliza…
Accurate and automated detection of anomalous samples in a natural image dataset can be accomplished with a probabilistic model for end-to-end modeling of images. Such images have heterogeneous complexity, however, and a probabilistic model overlooks simply shaped objects with small anomalies. This is because the probabilistic model assigns undesirably lower likelihoods to complexly shaped objects that are nevertheless consistent with set standards. To overcome this difficulty, we propose an unregularized score for deep generative models (DGMs), which are generative models leveraging deep neur…
In this work, we investigated the application of score-based gradient learning in discriminative and generative classification settings. Score function can be used to characterize data distribution as an alternative to density. It can be efficiently learned via score matching, and used to flexibly generate credible samples to enhance discriminative classification quality, to recover density and to build generative classifiers. We analysed the decision theories involving score-based representations, and performed experiments on simulated and real-world datasets, demonstrating its effectiveness …
Deep generative models are powerful tools that have produced impressive results in recent years. These advances have been for the most part empirically driven, making it essential that we use high quality evaluation metrics. In this paper, we provide new insights into the Inception Score, a recently proposed and widely used evaluation metric for generative models, and demonstrate that it fails to provide useful guidance when comparing models. We discuss both suboptimalities of the metric itself and issues with its application. Finally, we call for researchers to be more systematic and careful …
Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds
National University of Singapore（国立新加坡大学）；The University of Tokyo（东京大学）；RIKEN Center for Advanced Intelligence Project（RIKEN高级智能项目中心）；Agency for Science, Technology and Research (A ⋆ \star STAR)（科技研究局）；Nanyang Technological University（南洋理工大学）
Score-of-Mixture Training: Training One-Step Generative Models Made Simple via Score Estimation of Mixture Distributions
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts, USA（电气工程与计算机科学系，麻省理工学院（MIT），剑桥，马萨诸塞州，美国）
提出Score-of-Mixture Training (SMT)框架，通过最小化$α$-skew Jensen-Shannon散度并估计混合分布得分来训练单步生成模型，支持从零开始训练与预训练扩散模型蒸馏，实验证明其在CIFAR-10和ImageNet 64x64上表现优异。
ProGen: Revisiting Probabilistic Spatial-Temporal Time Series Forecasting from a Continuous Generative Perspective Using Stochastic Differential Equations
Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
Discrete-time diffusion-based generative models and score matching methods have shown promising results in modeling high-dimensional image data. Recently, Song et al. (2021) show that diffusion processes that transform data into noise can be reversed via learning the score function, i.e. the gradient of the log-density of the perturbed data. They propose to plug the learned score function into an inverse formula to define a generative diffusion process. Despite the empirical success, a theoretical underpinning of this procedure is still lacking. In this work, we approach the (continuous-time) …
University of Tübingen（蒂宾根大学）；Stanford University（斯坦福大学）；Norwegian University of Science and Technology（挪威科技大学）
The tremendous success of generative models in recent years raises the question whether they can also be used to perform classification. Generative models have been used as adversarially robust classifiers on simple datasets such as MNIST, but this robustness has not been observed on more complex datasets like CIFAR-10. Additionally, on natural image datasets, previous results have suggested a trade-off between the likelihood of the data and classification accuracy. In this work, we investigate score-based generative models as classifiers for natural images. We show that these models not only …
Wasserstein proximal operators describe score-based generative models and resolve memorization
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of South Carolina, Columbia（南卡罗来纳大学哥伦比亚分校）
Julia Computing, Inc（Julia计算公司）；Deep Forest Sciences Inc.（深林科学公司）；Carnegie Mellon University（卡内基梅隆大学）
Recent advances in generative models have made exploring design spaces easier for de novo molecule generation. However, popular generative models like GANs and normalizing flows face challenges such as training instabilities due to adversarial training and architectural constraints, respectively. Score-based generative models sidestep these challenges by modelling the gradient of the log probability density using a score function approximation, as opposed to modelling the density function directly, and sampling from it using annealed Langevin Dynamics. We believe that score-based generative mo…
Flow-based generative models leverage invertible generator functions to fit a distribution to the training data using maximum likelihood. Despite their use in several application domains, robustness of these models to adversarial attacks has hardly been explored. In this paper, we study adversarial robustness of flow-based generative models both theoretically (for some simple models) and empirically (for more complex ones). First, we consider a linear flow-based generative model and compute optimal sample-specific and universal adversarial perturbations that maximally decrease the likelihood s…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Institute of Numerical Mathematics RAS（俄罗斯科学院数值数学研究所）
One of the biggest challenges in the research of generative adversarial networks (GANs) is assessing the quality of generated samples and detecting various levels of mode collapse. In this work, we construct a novel measure of performance of a GAN by comparing geometrical properties of the underlying data manifold and the generated one, which provides both qualitative and quantitative means for evaluation. Our algorithm can be applied to datasets of an arbitrary nature and is not limited to visual data. We test the obtained metric on various real-life models and datasets and demonstrate that o…
How to model distribution of sequential data, including but not limited to speech and human motions, is an important ongoing research problem. It has been demonstrated that model capacity can be significantly enhanced by introducing stochastic latent variables in the hidden states of recurrent neural networks. Simultaneously, WaveNet, equipped with dilated convolutions, achieves astonishing empirical performance in natural speech generation task. In this paper, we combine the ideas from both stochastic latent variables and dilated convolutions, and propose a new architecture to model sequentia…
Deep probabilistic generative models enable modeling the likelihoods of very high dimensional data. An important application of generative modeling should be the ability to detect out-of-distribution (OOD) samples by setting a threshold on the likelihood. However, some recent studies show that probabilistic generative models can, in some cases, assign higher likelihoods on certain types of OOD samples, making the OOD detection rules based on likelihood threshold problematic. To address this issue, several OOD detection methods have been proposed for deep generative models. In this paper, we ma…
We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small $L^2$ error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following i…
National Energy Research Scientific Computing Center（国家能源研究科学计算中心）；Berkeley Lab（伯克利实验室）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Berkeley Institute for Data Science（伯克利数据科学研究所）；University of California, Berkeley（加州大学伯克利分校）
Score-based generative models are a new class of generative algorithms that have been shown to produce realistic images even in high dimensional spaces, currently surpassing other state-of-the-art models for different benchmark categories and applications. In this work we introduce CaloScore, a score-based generative model for collider physics applied to calorimeter shower generation. Three different diffusion models are investigated using the Fast Calorimeter Simulation Challenge 2022 dataset. CaloScore is the first application of a score-based generative model in collider physics and is able…
MIT（麻省理工学院）；IBM Research（IBM研究院）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
We study convergence of a generative modeling method that first estimates the score function of the distribution using Denoising Auto-Encoders (DAE) or Denoising Score Matching (DSM) and then employs Langevin diffusion for sampling. We show that both DAE and DSM provide estimates of the score of the Gaussian smoothed population density, allowing us to apply the machinery of Empirical Processes. We overcome the challenge of relying only on $L^2$ bounds on the score estimation error and provide finite-sample bounds in the Wasserstein distance between the law of the population distribution and th…
