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Autoregressive Quantile Networks for Generative Modeling

Georg Ostrovski, Will Dabney, Rémi Munos

TL;DR

The paper introduces autoregressive implicit quantile networks (AIQN), a quantile-regression-based approach to generative modeling that avoids KL divergence and aligns with perceptual metrics. By extending IQN to an autoregressive framework (PixelIQN) and conditioning on quantile inputs, the method achieves higher perceptual quality without sacrificing diversity, as demonstrated on CIFAR-10 and ImageNet 32x32—outperforming PixelCNN and comparable models on standard metrics like Inception Score and FID. The work links quantile regression with optimal transport through quantile divergence and shows potential extensions to latent-variable spaces (AIQN-VAE), offering a simple, robust alternative to likelihood- and GAN-based methods. Overall, AIQN provides a flexible, architecture-agnostic path to better perceptual fidelity in generative modeling.

Abstract

We introduce autoregressive implicit quantile networks (AIQN), a fundamentally different approach to generative modeling than those commonly used, that implicitly captures the distribution using quantile regression. AIQN is able to achieve superior perceptual quality and improvements in evaluation metrics, without incurring a loss of sample diversity. The method can be applied to many existing models and architectures. In this work we extend the PixelCNN model with AIQN and demonstrate results on CIFAR-10 and ImageNet using Inception score, FID, non-cherry-picked samples, and inpainting results. We consistently observe that AIQN yields a highly stable algorithm that improves perceptual quality while maintaining a highly diverse distribution.

Autoregressive Quantile Networks for Generative Modeling

TL;DR

The paper introduces autoregressive implicit quantile networks (AIQN), a quantile-regression-based approach to generative modeling that avoids KL divergence and aligns with perceptual metrics. By extending IQN to an autoregressive framework (PixelIQN) and conditioning on quantile inputs, the method achieves higher perceptual quality without sacrificing diversity, as demonstrated on CIFAR-10 and ImageNet 32x32—outperforming PixelCNN and comparable models on standard metrics like Inception Score and FID. The work links quantile regression with optimal transport through quantile divergence and shows potential extensions to latent-variable spaces (AIQN-VAE), offering a simple, robust alternative to likelihood- and GAN-based methods. Overall, AIQN provides a flexible, architecture-agnostic path to better perceptual fidelity in generative modeling.

Abstract

We introduce autoregressive implicit quantile networks (AIQN), a fundamentally different approach to generative modeling than those commonly used, that implicitly captures the distribution using quantile regression. AIQN is able to achieve superior perceptual quality and improvements in evaluation metrics, without incurring a loss of sample diversity. The method can be applied to many existing models and architectures. In this work we extend the PixelCNN model with AIQN and demonstrate results on CIFAR-10 and ImageNet using Inception score, FID, non-cherry-picked samples, and inpainting results. We consistently observe that AIQN yields a highly stable algorithm that improves perceptual quality while maintaining a highly diverse distribution.

Paper Structure

This paper contains 11 sections, 1 theorem, 23 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Distance Metrics and Loss Functions
  4. Quantile Regression
  5. Autoregressive Implicit Quantiles
  6. Quantile Regression and the Wasserstein
  7. Quantile Density Function
  8. PixelIQN
  9. CIFAR-10
  10. ImageNet 32x32
  11. Discussion and Conclusions

Key Result

Proposition 1

For any distributions $P$ and $Q$, define the quantile divergence Then the expected quantile loss of a quantile function $\bar{Q}$ implicitly defining the distribution $Q$ satisfies where $h(P)$ does not depend on $Q$.

Figures (13)

  • Figure 1: Categorization of generative models by discretization vs. reparameterization and loss functions by likelihood-based vs. quantile-regression-based.
  • Figure 2: Illustration of Gated PixelCNN layer block for PixelIQN. Dashed line shows boundary of standard Gated PixelCNN, with $v$ the vertical and $h$ the horizontal stack. Conditioning on $\tau$ is identical to the location-dependent conditioning in Gated PixelCNN.
  • Figure 3: CIFAR-10: Real example images (left), samples generated by PixelCNN (center), and samples generated by PixelIQN (right).
  • Figure 4: Evaluations by Inception score (higher is better) and FID (lower is better) on CIFAR-10 and ImageNet 32x32. Dotted lines correspond to models trained with class-label conditioning.
  • Figure 5: Inception score and FID for CIFAR-10 and ImageNet. WGAN and DC-GAN results taken from wganradford2015unsupervised. PixelIQN(l) is the small 15-layer version of the model. Models marked $*$ refer to class-conditional training.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof