Denoising Score-Matching for Uncertainty Quantification in Inverse Problems
CEA（法国原子能和替代能源委员会）；Inria（法国国家信息与自动化研究所）；AIM；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎-萨克雷大学）；Université Paris Diderot（巴黎狄德罗大学）；Sorbonne Paris Cité（索邦巴黎西岱大学）
Zaccharie Ramzi, Benjamin Remy, Francois Lanusse, Jean-Luc Starck, Philippe Ciuciu
Deep neural networks have proven extremely efficient at solving a wide rangeof inverse problems, but most often the uncertainty on the solution they provideis hard to quantify. In this work, we propose a generic Bayesian framework forsolving inverse problems, in which we limit the use of deep neural networks tolearning a prior distribution on the signals to recover. We adopt recent denoisingscore matching techniques to learn this prior from data, and subsequently use it aspart of an annealed Hamiltonian Monte-Carlo scheme to sample the full posteriorof image inverse problems. We apply this framework to Magnetic ResonanceImage (MRI) reconstruction and illustrate how this approach not only yields highquality reconstructions but can also be used to assess the uncertainty on particularfeatures of a reconstructed image.
Epistemic Uncertainty Quantification for Pre-trained VLMs via Riemannian Flow Matching
Department of Information Technology, Uppsala University, Uppsala, Sweden（瑞典乌普萨拉大学信息科技系）；Science for Life Laboratory, Uppsala University, Uppsala, Sweden（瑞典乌普萨拉大学生命科学实验室）
We present high quality image synthesis results using diffusion probabilistic models, a class of latent variable models inspired by considerations from nonequilibrium thermodynamics. Our best results are obtained by training on a weighted variational bound designed according to a novel connection between diffusion probabilistic models and denoising score matching with Langevin dynamics, and our models naturally admit a progressive lossy decompression scheme that can be interpreted as a generalization of autoregressive decoding. On the unconditional CIFAR10 dataset, we obtain an Inception score…
We propose a novel unpaired image-to-image translation method that uses denoising diffusion probabilistic models without requiring adversarial training. Our method, UNpaired Image Translation with Denoising Diffusion Probabilistic Models (UNIT-DDPM), trains a generative model to infer the joint distribution of images over both domains as a Markov chain by minimising a denoising score matching objective conditioned on the other domain. In particular, we update both domain translation models simultaneously, and we generate target domain images by a denoising Markov Chain Monte Carlo approach tha…
Harvard Data Science Initiative（哈佛数据科学倡议）；Harvard University（哈佛大学）
Uncertainty quantification has been a core of the statistical machine learning, but its computational bottleneck has been a serious challenge for both Bayesians and frequentists. We propose a model-based framework in quantifying uncertainty, called predictive-matching Generative Parameter Sampler (GPS). This procedure considers an Uncertainty Quantification (UQ) distribution on the targeted parameter, which matches the corresponding predictive distribution to the observed data. This framework adopts a hierarchical modeling perspective such that each observation is modeled by an individual para…
Indiana University Bloomington（印第安纳大学伯明顿分校）；Uppsala University（乌普萨拉大学）；Linköping University（林雪平大学）
Diffusion MRI (dMRI) is the only non-invasive technique sensitive to tissue micro-architecture, which can, in turn, be used to reconstruct tissue microstructure and white matter pathways. The accuracy of such tasks is hampered by the low signal-to-noise ratio in dMRI. Today, the noise is characterized mainly by visual inspection of residual maps and estimated standard deviation. However, it is hard to estimate the impact of noise on downstream tasks based only on such qualitative assessments. To address this issue, we introduce a novel metric, Noise Uncertainty Quantification (NUQ), for quanti…
Redwood Center for Theoretical Neuroscience（雷德伍德理论神经科学中心）；University of California Berkeley（加州大学伯克利分校）；Berkeley AI Research（伯克利人工智能研究院）；Helen Wills Neuroscience Institute（海伦·威尔斯神经科学研究所）
Energy-Based Models (EBMs) assign unnormalized log-probability to data samples. This functionality has a variety of applications, such as sample synthesis, data denoising, sample restoration, outlier detection, Bayesian reasoning, and many more. But training of EBMs using standard maximum likelihood is extremely slow because it requires sampling from the model distribution. Score matching potentially alleviates this problem. In particular, denoising score matching \citep{vincent2011connection} has been successfully used to train EBMs. Using noisy data samples with one fixed noise level, these …
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
Tweedie distributions are a special case of exponential dispersion models, which are often used in classical statistics as distributions for generalized linear models. Here, we reveal that Tweedie distributions also play key roles in modern deep learning era, leading to a distribution independent self-supervised image denoising formula without clean reference images. Specifically, by combining with the recent Noise2Score self-supervised image denoising approach and the saddle point approximation of Tweedie distribution, we can provide a general closed-form denoising formula that can be used fo…
Score-based model research in the last few years has produced state of the art generative models by employing Gaussian denoising score-matching (DSM). However, the Gaussian noise assumption has several high-dimensional limitations, motivating a more concrete route toward even higher dimension PDF estimation in future. We outline this limitation, before extending the theory to a broader family of noising distributions -- namely, the generalised normal distribution. To theoretically ground this, we relax a key assumption in (denoising) score matching theory, demonstrating that distributions whic…
Tsinghua University（清华大学）；Peng Cheng Laboratory（鹏城实验室）；Renmin University of China（中国人民大学）
Score-based generative models have excellent performance in terms of generation quality and likelihood. They model the data distribution by matching a parameterized score network with first-order data score functions. The score network can be used to define an ODE ("score-based diffusion ODE") for exact likelihood evaluation. However, the relationship between the likelihood of the ODE and the score matching objective is unclear. In this work, we prove that matching the first-order score is not sufficient to maximize the likelihood of the ODE, by showing a gap between the maximum likelihood and…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
The density estimation is one of the core problems in statistics. Despite this, existing techniques like maximum likelihood estimation are computationally inefficient due to the intractability of the normalizing constant. For this reason an interest to score matching has increased being independent on the normalizing constant. However, such estimator is consistent only for distributions with the full space support. One of the approaches to make it consistent is to add noise to the input data which is called Denoising Score Matching. In this work we derive analytical expression for the Denoisin…
University of Montreal（蒙特利尔大学）；Microsoft Research Lab（微软研究院实验室）
Denoising Score Matching with Annealed Langevin Sampling (DSM-ALS) has recently found success in generative modeling. The approach works by first training a neural network to estimate the score of a distribution, and then using Langevin dynamics to sample from the data distribution assumed by the score network. Despite the convincing visual quality of samples, this method appears to perform worse than Generative Adversarial Networks (GANs) under the Fréchet Inception Distance, a standard metric for generative models. We show that this apparent gap vanishes when denoising the final Langevin sam…
The first order derivative of a data density can be estimated efficiently by denoising score matching, and has become an important component in many applications, such as image generation and audio synthesis. Higher order derivatives provide additional local information about the data distribution and enable new applications. Although they can be estimated via automatic differentiation of a learned density model, this can amplify estimation errors and is expensive in high dimensional settings. To overcome these limitations, we propose a method to directly estimate high order derivatives (score…
National Tsing Hua University（国立清华大学）；Mediatek Inc.（联发科股份有限公司）
Many existing conditional score-based data generation methods utilize Bayes' theorem to decompose the gradients of a log posterior density into a mixture of scores. These methods facilitate the training procedure of conditional score models, as a mixture of scores can be separately estimated using a score model and a classifier. However, our analysis indicates that the training objectives for the classifier in these methods may lead to a serious score mismatch issue, which corresponds to the situation that the estimated scores deviate from the true ones. Such an issue causes the samples to be …
Iterated Denoising Energy Matching for Sampling from Boltzmann Densities
Mila – Québec AI Institute（米拉-魁北克人工智能研究所）；McGill University（麦吉尔大学）；Dreamfold；Université de Montréal（蒙特利尔大学）；University of Oxford（牛津大学）；Ciela Institute（Ciela研究所）；Center for Computational Astrophysics, Flatiron Institute（弗拉铁隆研究所计算天体物理中心）；Jagiellonian University（雅盖隆大学）；CIFAR（加拿大高级研究所）
The multivariate linear regression model with shuffled data and additive Gaussian noise arises in various correspondence estimation and matching problems. Focusing on the denoising aspect of this problem, we provide a characterization the minimax error rate that is sharp up to logarithmic factors. We also analyze the performance of two versions of a computationally efficient estimator, and establish their consistency for a large range of input parameters. Finally, we provide an exact algorithm for the noiseless problem and demonstrate its performance on an image point-cloud matching task. Our …
Nonlinear denoising score matching for enhanced learning of structured distributions
Department of Mathematics（数学系）；Texas State University（德克萨斯州立大学）；Department of Mathematics and Statistics（数学与统计学系）；University of Massachusetts Amherst（马萨诸塞大学阿姆赫斯特分校）；Division of Applied Mathematics（应用数学系）；Brown University（布朗大学）；School of Mathematics（数学学院）；Georgia Institute of Technology（佐治亚理工学院）
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…
Application of Bounded Total Variation Denoising in Urban Traffic Analysis
1 School of Mathematical Sciences, University of Chinese Academy of Sciences；LSEC, Institute of Computational Mathematics；Scientific/Engineering Computing, Academy of Mathematics；Systems Science, Beijing 100190, China 2 NCMIS \& LSEC, Institute of Computational Mathematics；School of Mathematical Sciences, University of Chinese Academy of Sciences
