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This paper studies a training method to jointly estimate an energy-based model and a flow-based model, in which the two models are iteratively updated based on a shared adversarial value function. This joint training method has the following traits. (1) The update of the energy-based model is based on noise contrastive estimation, with the flow model serving as a strong noise distribution. (2) The update of the flow model approximately minimizes the Jensen-Shannon divergence between the flow model and the data distribution. (3) Unlike generative adversarial networks (GAN) which estimates an implicit probability distribution defined by a generator model, our method estimates two explicit probabilistic distributions on the data. Using the proposed method we demonstrate a significant improvement on the synthesis quality of the flow model, and show the effectiveness of unsupervised feature learning by the learned energy-based model. Furthermore, the proposed training method can be easily adapted to semi-supervised learning. We achieve competitive results to the state-of-the-art semi-supervised learning methods.
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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 models learn fast and yield good results in data denoising citep{saremi2019neural}. However, demonstrations of such models in high quality sample synthesis of high dimensional data were lacking. Recently, citet{song2019generative} have shown that a generative model trained by denoising score matching accomplishes excellent sample synthesis, when trained with data samples corrupted with multiple levels of noise. Here we provide analysis and empirical evidence showing that training with multiple noise levels is necessary when the data dimension is high. Leveraging this insight, we propose a novel EBM trained with multi-scale denoising score matching. Our model exhibits data generation performance comparable to state-of-the-art techniques such as GANs, and sets a new baseline for EBMs. The proposed model also provides density information and performs well in an image inpainting task.
A new low-dimensional parameterization based on principal component analysis (PCA) and convolutional neural networks (CNN) is developed to represent complex geological models. The CNN-PCA method is inspired by recent developments in computer vision using deep learning. CNN-PCA can be viewed as a generalization of an existing optimization-based PCA (O-PCA) method. Both CNN-PCA and O-PCA entail post-processing a PCA model to better honor complex geological features. In CNN-PCA, rather than use a histogram-based regularization as in O-PCA, a new regularization involving a set of metrics for multipoint statistics is introduced. The metrics are based on summary statistics of the nonlinear filter responses of geological models to a pre-trained deep CNN. In addition, in the CNN-PCA formulation presented here, a convolutional neural network is trained as an explicit transform function that can post-process PCA models quickly. CNN-PCA is shown to provide both unconditional and conditional realizations that honor the geological features present in reference SGeMS geostatistical realizations for a binary channelized system. Flow statistics obtained through simulation of random CNN-PCA models closely match results for random SGeMS models for a demanding case in which O-PCA models lead to significant discrepancies. Results for history matching are also presented. In this assessment CNN-PCA is applied with derivative-free optimization, and a subspace randomized maximum likelihood method is used to provide multiple posterior models. Data assimilation and significant uncertainty reduction are achieved for existing wells, and physically reasonable predictions are also obtained for new wells. Finally, the CNN-PCA method is extended to a more complex non-stationary bimodal deltaic fan system, and is shown to provide high-quality realizations for this challenging example.
This paper proposes a multi-grid method for learning energy-based generative ConvNet models of images. For each grid, we learn an energy-based probabilistic model where the energy function is defined by a bottom-up convolutional neural network (ConvNet or CNN). Learning such a model requires generating synthesized examples from the model. Within each iteration of our learning algorithm, for each observed training image, we generate synthesized images at multiple grids by initializing the finite-step MCMC sampling from a minimal 1 x 1 version of the training image. The synthesized image at each subsequent grid is obtained by a finite-step MCMC initialized from the synthesized image generated at the previous coarser grid. After obtaining the synthesized examples, the parameters of the models at multiple grids are updated separately and simultaneously based on the differences between synthesized and observed examples. We show that this multi-grid method can learn realistic energy-based generative ConvNet models, and it outperforms the original contrastive divergence (CD) and persistent CD.
Contrastive divergence is a popular method of training energy-based models, but is known to have difficulties with training stability. We propose an adaptation to improve contrastive divergence training by scrutinizing a gradient term that is difficult to calculate and is often left out for convenience. We show that this gradient term is numerically significant and in practice is important to avoid training instabilities, while being tractable to estimate. We further highlight how data augmentation and multi-scale processing can be used to improve model robustness and generation quality. Finally, we empirically evaluate stability of model architectures and show improved performance on a host of benchmarks and use cases,such as image generation, OOD detection, and compositional generation.
Selecting input variables or design points for statistical models has been of great interest in adaptive design and active learning. Motivated by two scientific examples, this paper presents a strategy of selecting the design points for a regression model when the underlying regression function is discontinuous. The first example we undertook was for the purpose of accelerating imaging speed in a high resolution material imaging; the second was use of sequential design for the purpose of mapping a chemical phase diagram. In both examples, the underlying regression functions have discontinuities, so many of the existing design optimization approaches cannot be applied because they mostly assume a continuous regression function. Although some existing adaptive design strategies developed from treed regression models can handle the discontinuities, the Bayesian approaches come with computationally expensive Markov Chain Monte Carlo techniques for posterior inferences and subsequent design point selections, which is not appropriate for the first motivating example that requires computation at least faster than the original imaging speed. In addition, the treed models are based on the domain partitioning that are inefficient when the discontinuities occurs over complex sub-domain boundaries. We propose a simple and effective adaptive design strategy for a regression analysis with discontinuities: some statistical properties with a fixed design will be presented first, and then these properties will be used to propose a new criterion of selecting the design points for the regression analysis. Sequential design with the new criterion will be presented with comprehensive simulated examples, and its application to the two motivating examples will be presented.
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