An Introduction to Deep Generative Modeling (2103.05180v2)

Abstract: Deep generative models (DGM) are neural networks with many hidden layers trained to approximate complicated, high-dimensional probability distributions using a large number of samples. When trained successfully, we can use the DGMs to estimate the likelihood of each observation and to create new samples from the underlying distribution. Developing DGMs has become one of the most hotly researched fields in artificial intelligence in recent years. The literature on DGMs has become vast and is growing rapidly. Some advances have even reached the public sphere, for example, the recent successes in generating realistic-looking images, voices, or movies; so-called deep fakes. Despite these successes, several mathematical and practical issues limit the broader use of DGMs: given a specific dataset, it remains challenging to design and train a DGM and even more challenging to find out why a particular model is or is not effective. To help advance the theoretical understanding of DGMs, we introduce DGMs and provide a concise mathematical framework for modeling the three most popular approaches: normalizing flows (NF), variational autoencoders (VAE), and generative adversarial networks (GAN). We illustrate the advantages and disadvantages of these basic approaches using numerical experiments. Our goal is to enable and motivate the reader to contribute to this proliferating research area. Our presentation also emphasizes relations between generative modeling and optimal transport.

An Expert Overview of "An Introduction to Deep Generative Modeling"

The paper "An Introduction to Deep Generative Modeling" by Lars Ruthotto and Eldad Haber provides a comprehensive introduction to the field of deep generative models (DGMs). DGMs, as outlined in the work, utilize neural networks to approximate complex, high-dimensional probability distributions. The authors focus on three prevalent approaches: normalizing flows (NF), variational autoencoders (VAE), and generative adversarial networks (GAN), providing insights into their theoretical underpinnings and practical applications.

Key Concepts and Mathematical Framework

The authors establish a mathematical framework for DGMs, aiming to deepen the reader's understanding of how these models work. The goal is to learn a representation of an intractable distribution using a finite sample set, which is a classic problem in probability and statistics, complicated by high-dimensional spaces inherent in modern datasets.

Challenges in Training Deep Generative Models

The paper elucidates several core challenges in DGM training:

1. Ill-Posedness of Training: Identifying a unique probability distribution from finite samples is fundamentally ill-posed. Success heavily depends on hyperparameters like network design, training objectives, and regularization strategies.
2. Similarity Metrics: For effective training, it is essential to quantify the similarity between generated samples and the original dataset. This often involves inverting the generator or conducting distribution comparisons, both of which present distinct challenges.
3. Latent Space Dimension: Determining the dimensionality of the latent space is crucial, as under- or overestimations can significantly affect the generator's performance.

Methodologies

1. Normalizing Flows (NF):

• NFs are constructed by concatenating invertible transformations, allowing direct computation of likelihoods via change of variables, thus enabling efficient maximum likelihood training.
• Ruthotto and Haber discuss finite and continuous NFs, contrasting methods like non-linear independent components estimation (NICE) and real NVP flow with continuous dynamical system representations. The latter, when coupled with optimal transport regularization, provides robust solutions in certain scenarios.

2. Variational Autoencoders (VAE):

• VAEs address the challenge of non-invertibility by approximating the posterior distribution of latent variables. Incorporating a second 'encoder' network, VAEs provide a tractable way to optimize likelihood lower bounds (ELBO).
• The paper highlights the tension between reconstruction accuracy and regularization, impacting the VAE's ability to generalize beyond training distributions.

3. Generative Adversarial Networks (GAN):

• GANs are unique in leveraging adversarial concepts, training a discriminator in tandem with the generator to refine sample quality. The authors delve into discriminators based on binary classification and transport cost metrics (Wasserstein GANs).
• The intrinsic challenge here is the minimax optimization and the potential for mode collapse, demanding careful hyperparameter tuning and monitoring.

Implications and Future Directions

The discussion highlights practical implications, notably the potential of DGMs in scientific domains where data representations might be inherently complex. The interplay between theoretical advances, such as improved metrics for distribution comparison and computational innovations, promises continued evolution in this field. Future research could enhance domain-specific customization of DGMs, integrating existing domain knowledge, thus expanding their applicability beyond traditional generative tasks.

In summary, Ruthotto and Haber's paper serves as a foundational text for researchers seeking to engage with or contribute to the rapidly evolving arena of deep generative modeling, offering both a rigorous mathematical exposition and a critical examination of contemporary challenges and methodologies.