PROJECTED LATENT MARKOV CHAIN MONTE CARLO: CONDITIONAL SAMPLING OF NORMALIZING FLOWS

Abstract

We introduce Projected Latent Markov Chain Monte Carlo (PL-MCMC), a technique for sampling from the exact conditional distributions learned by normalizing flows. As a conditional sampling method, PL-MCMC enables Monte Carlo Expectation Maximization (MC-EM) training of normalizing flows from incomplete data. Through experimental tests applying normalizing flows to missing data tasks for a variety of data sets, we demonstrate the efficacy of PL-MCMC for conditional sampling from normalizing flows.

1. INTRODUCTION

Conditional sampling from modeled joint probability distributions offers a statistical framework for approaching tasks involving missing and incomplete data. Deep generative models have demonstrated an exceptional capability for approximating the distributions governing complex data. Brief analysis illustrates a fundamental guarantee for generative models: the inaccuracy (i.e. divergence from ground truth) of a generative model's approximated joint distribution upper bounds the expected inaccuracies of the conditional distributions known by the model, as shown in Appendix A. Although this guarantee holds for all generative models, specialized variants are typically used to approach tasks involving the conditional distributions among modeled variables, due to the difficulty in accessing the conditional distributions known by unspecialized generative models. Quite often, otherwise well trained generative models possess a capability for conditional inference that is regrettably locked away from our access. Normalizing flow architectures like RealNVP (Dinh et al., 2014) and GLOW (Kingma & Dhariwal, 2018) have demonstrated accurate and expressive generative performance, showing great promise for application to missing data tasks. Additionally, by enabling the calculation of exact likelihoods, normalizing flows offer convenient mathematical properties for approaching exact conditional sampling. We are therefore motivated to develop techniques for sampling from the exact conditional distributions known by normalizing flows. In this paper, we propose Projected Latent Markov Chain Monte Carlo (PL-MCMC), a conditional sampling technique that takes advantage of the convenient mathematical structure of normalizing flows by defining a Markov Chain within a flow's latent space and accepting proposed transitions based on the likelihood of the resulting imputation. In principle, PL-MCMC enables exact conditional sampling without requiring specialized architecture, training history, or external inference machinery. Our Contributions: We prove that a Metropolis-Hastings implementation of our proposed PL-MCMC technique is asymptotically guaranteed to sample from the exact conditional distributions known by any normalizing flow satisfying very mild positivity and smoothness requirements. We then describe how to use PL-MCMC to perform Monte Carlo Expectation Maximization (MC-EM) training of normalizing flows from incomplete training data. To illustrate and demonstrate aspects of the technique, we perform a series of experiments utilizing PL-MCMC to complete CIFAR-10 images, CelebA images, and MNIST digits affected by missing data. Finally, we perform a series of experiments training non-specialized normalizing flows to model MNIST digits and continuous UCI datasets from incomplete training data to verify the performance of the proposed method. Through these experimental results, we find that PL-MCMC holds great practical promise for tasks requiring conditional sampling from normalizing flows.

