Author Description

Normalizing Flows, Diffusion Normalizing Flows And Variational Autoencoders Are Powerful Generative Models. This Element Provides A Unified Framework To Handle These Approaches Via Markov Chains. The Authors Consider Stochastic Normalizing Flows As A Pair Of Markov Chains Fulfilling Some Properties, And Show How Many State-of-the-art Models For Data Generation Fit Into This Framework. Indeed Numerical Simulations Show That Including Stochastic Layers Improves The Expressivity Of The Network And Allows For Generating Multimodal Distributions From Unimodal Ones. The Markov Chains Point Of View Enables The Coupling Of Both Deterministic Layers As Invertible Neural Networks And Stochastic Layers As Metropolis-hasting Layers, Langevin Layers, Variational Autoencoders And Diffusion Normalizing Flows In A Mathematically Sound Way. The Authors' Framework Establishes A Useful Mathematical Tool To Combine The Various Approaches. Cover -- Title Page -- Copyright Page -- Generalized Normalizing Flows Via Markov Chains -- Contents -- 1 Introduction -- Related Work -- 2 Preliminaries -- Basics Of Probability -- Markov Kernels -- Kullback-leibler Divergence -- Wasserstein Distance -- 3 Normalizing Flows -- 4 Stochastic Normalizing Flows -- 4.1 Training Snfs -- 5 Stochastic Layers -- 5.1 Langevin Layer -- 5.2 Metropolis-hastings Layer -- 5.3 Metropolis-adjusted Langevin Layer -- 5.4 Vae Layer -- Autoencoders -- Variational Autoenconders Via Markov Kernels -- Vaes As One-layer Snfs -- 5.5 Diffusion-normalizing Flow Layer -- 5.6 Training Of Stochastic Layers -- 6 Conditional Generative Modeling -- 6.1 Conditional Normalizing Flows -- 6.2 Conditional Snfs -- 6.3 Conditional Vaes -- 7 Numerical Results -- 7.1 Posterior Approximation For Gaussian Mixtures -- 7.2 Example From Scatterometry -- 7.3 Image Generation Via 2d Energy Modeling -- 8 Conclusions And Open Questions -- Appendix I Invertible Neural Networks -- I.1 Invertible Residual Networks -- I.2 Inns -- I.3 Continuous Normalizing