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1. Analytical Probability Distributions and EM-Learning for Deep Generative Networks

   Balestriero, Randall; Paris, Sebastien; Baraniuk, Richard G. (January 2020, Advances in neural information processing systems)

   Deep Generative Networks (DGNs) with probabilistic modeling of their output and latent space are currently trained via Variational Autoencoders (VAEs). In the absence of a known analytical form for the posterior and likelihood expectation, VAEs resort to approximations, including (Amortized) Variational Inference (AVI) and Monte-Carlo (MC) sampling. We exploit the Continuous Piecewise Affine (CPA) property of modern DGNs to derive their posterior and marginal distributions as well as the latter's first moments. These findings enable us to derive an analytical Expectation-Maximization (EM) algorithm that enables gradient-free DGN learning. We demonstrate empirically that EM training of DGNs produces greater likelihood than VAE training. Our findings will guide the design of new VAE AVI that better approximate the true posterior and open avenues to apply standard statistical tools for model comparison, anomaly detection, and missing data imputation. 

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2. Analytical Probability Distributions and EM-Learning for Deep Generative Networks

   Balestriero, Randall; Paris, Sebastien; Baraniuk, Richard G. (December 2020, Advances in neural information processing systems)

   null (Ed.)

   Deep Generative Networks (DGNs) with probabilistic modeling of their output and latent space are currently trained via Variational Autoencoders (VAEs). In the absence of a known analytical form for the posterior and likelihood expectation, VAEs resort to approximations, including (Amortized) Variational Inference (AVI) and Monte-Carlo (MC) sampling. We exploit the Continuous Piecewise Affine (CPA) property of modern DGNs to derive their posterior and marginal distributions as well as the latter's first moments. These findings enable us to derive an analytical Expectation-Maximization (EM) algorithm that enables gradient-free DGN learning. We demonstrate empirically that EM training of DGNs produces greater likelihood than VAE training. Our findings will guide the design of new VAE AVI that better approximate the true posterior and open avenues to apply standard statistical tools for model comparison, anomaly detection, and missing data imputation. 

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3. Implications of data topology for deep generative models

   https://doi.org/10.3389/fcomp.2024.1260604  

   Jin, Yinzhu; McDaniel, Rory; Tatro, N Joseph; Catanzaro, Michael J; Smith, Abraham D; Bendich, Paul; Dwyer, Matthew B; Fletcher, P Thomas (August 2024, Frontiers in Computer Science)

   Many deep generative models, such as variational autoencoders (VAEs) and generative adversarial networks (GANs), learn an immersion mapping from a standard normal distribution in a low-dimensional latent space into a higher-dimensional data space. As such, these mappings are only capable of producing simple data topologies, i.e., those equivalent to an immersion of Euclidean space. In this work, we demonstrate the limitations of such latent space generative models when trained on data distributions with non-trivial topologies. We do this by training these models on synthetic image datasets with known topologies (spheres, torii, etc.). We then show how this results in failures of both data generation as well as data interpolation. Next, we compare this behavior to two classes of deep generative models that in principle allow for more complex data topologies. First, we look at chart autoencoders (CAEs), which construct a smooth data manifold from multiple latent space chart mappings. Second, we explore score-based models, e.g., denoising diffusion probabilistic models, which estimate gradients of the data distribution without resorting to an explicit mapping to a latent space. Our results show that these models do demonstrate improved ability over latent space models in modeling data distributions with complex topologies, however, challenges still remain. 

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4. Generative text modeling through short run inference

   Pang, B.; Nijkamp, E.; Han, T.; Wu, Y. N. (January 2021, Conference of the European Chapter of the Association for Computational Linguistics)

   Latent variable models for text, when trained successfully, accurately model the data distribution and capture global semantic and syntactic features of sentences. The prominent approach to train such models is variational autoencoders (VAE). It is nevertheless challenging to train and often results in a trivial local optimum where the latent variable is ignored and its posterior collapses into the prior, an issue known as posterior collapse. Various techniques have been proposed to mitigate this issue. Most of them focus on improving the inference model to yield latent codes of higher quality. The present work proposes a short run dynamics for inference. It is initialized from the prior distribution of the latent variable and then runs a small number (e.g., 20) of Langevin dynamics steps guided by its posterior distribution. The major advantage of our method is that it does not require a separate inference model or assume simple geometry of the posterior distribution, thus rendering an automatic, natural and flexible inference engine. We show that the models trained with short run dynamics more accurately model the data, compared to strong language model and VAE baselines, and exhibit no sign of posterior collapse. Analyses of the latent space show that interpolation in the latent space is able to generate coherent sentences with smooth transition and demonstrate improved classification over strong baselines with latent features from unsupervised pretraining. These results together expose a well-structured latent space of our generative model. 

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5. Learning joint latent space EBM prior model for multi-layer generator

   https://doi.org/10.1109/CVPR52729.2023.00351  

   Cui, J; Wu, Y N; Han, T (June 2023, Conference on Computer Vision and Pattern Recognition (CVPR 2023))

   This paper studies the fundamental problem of learning multi-layer generator models. The multi-layer generator model builds multiple layers of latent variables as a prior model on top of the generator, which benefits learning complex data distribution and hierarchical representations. However, such a prior model usually focuses on modeling inter-layer relations between latent variables by assuming non-informative (conditional) Gaussian distributions, which can be limited in model expressivity. To tackle this issue and learn more expressive prior models, we propose an energy-based model (EBM) on the joint latent space over all layers of latent variables with the multi-layer generator as its backbone. Such joint latent space EBM prior model captures the intra-layer contextual relations at each layer through layer-wise energy terms, and latent variables across different layers are jointly corrected. We develop a joint training scheme via maximum likelihood estimation (MLE), which involves Markov Chain Monte Carlo (MCMC) sampling for both prior and posterior distributions of the latent variables from different layers. To ensure efficient inference and learning, we further propose a variational training scheme where an inference model is used to amortize the costly posterior MCMC sampling. Our experiments demonstrate that the learned model can be expressive in generating high-quality images and capturing hierarchical features for better outlier detection. 

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