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Likelihood-Free Variational Autoencoders

Chen Xu, Qiang Wang, Lijun Sun

TL;DR

This work proposes EnVAE, a novel likelihood-free generative framework that has a deterministic decoder and employs the energy score--a proper scoring rule--to build the reconstruction loss, which enables likelihood-free inference without requiring explicit parametric density functions.

Abstract

Variational Autoencoders (VAEs) typically rely on a probabilistic decoder with a predefined likelihood, most commonly an isotropic Gaussian, to model the data conditional on latent variables. While convenient for optimization, this choice often leads to likelihood misspecification, resulting in blurry reconstructions and poor data fidelity, especially for high-dimensional data such as images. In this work, we propose EnVAE, a novel likelihood-free generative framework that has a deterministic decoder and employs the energy score--a proper scoring rule--to build the reconstruction loss. This enables likelihood-free inference without requiring explicit parametric density functions. To address the computational inefficiency of the energy score, we introduce a fast variant, FEnVAE, based on the local smoothness of the decoder and the sharpness of the posterior distribution of latent variables. This yields an efficient single-sample training objective that integrates seamlessly into existing VAE pipelines with minimal overhead. Empirical results on standard benchmarks demonstrate that EnVAE achieves superior reconstruction and generation quality compared to likelihood-based baselines. Our framework offers a general, scalable, and statistically principled alternative for flexible and nonparametric distribution learning in generative modeling.

Likelihood-Free Variational Autoencoders

TL;DR

This work proposes EnVAE, a novel likelihood-free generative framework that has a deterministic decoder and employs the energy score--a proper scoring rule--to build the reconstruction loss, which enables likelihood-free inference without requiring explicit parametric density functions.

Abstract

Variational Autoencoders (VAEs) typically rely on a probabilistic decoder with a predefined likelihood, most commonly an isotropic Gaussian, to model the data conditional on latent variables. While convenient for optimization, this choice often leads to likelihood misspecification, resulting in blurry reconstructions and poor data fidelity, especially for high-dimensional data such as images. In this work, we propose EnVAE, a novel likelihood-free generative framework that has a deterministic decoder and employs the energy score--a proper scoring rule--to build the reconstruction loss. This enables likelihood-free inference without requiring explicit parametric density functions. To address the computational inefficiency of the energy score, we introduce a fast variant, FEnVAE, based on the local smoothness of the decoder and the sharpness of the posterior distribution of latent variables. This yields an efficient single-sample training objective that integrates seamlessly into existing VAE pipelines with minimal overhead. Empirical results on standard benchmarks demonstrate that EnVAE achieves superior reconstruction and generation quality compared to likelihood-based baselines. Our framework offers a general, scalable, and statistically principled alternative for flexible and nonparametric distribution learning in generative modeling.

Paper Structure

This paper contains 32 sections, 18 equations, 26 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. Variational Autoencoder with Likelihood-free Inference
  5. A Fast Single-sampling Variant Mediated by the Local Linearity Approximation
  6. Implementation of $\textit{EnVAE}$ and $\textit{FEnVAE}$
  7. Experiment
  8. Image Generation and Reconstruction Analysis
  9. Reconstruction and Residual Error Distribution Analysis
  10. Efficiency and Applicability Analysis of FEnVAE
  11. Comparison of Generation Performance and Computational Efficiency
  12. Validity of Local Linear Approximation in FEnVAE.
  13. Conclusion
  14. Appendix
  15. Additional Related Work
  16. ...and 17 more sections

Figures (26)

  • Figure 1: Implementations of EnVAE and FEnVAE. (a). The architecture of EnVAE. Multiple samples of the latent variables are drawn and decoded. (b). The architecture of FEnVAE. A single sample of the latent variable is drawn and decoded into three outputs.
  • Figure 2: Reconstruction image samples.
  • Figure 3: Generation image samples.
  • Figure 4: Uncertainty visualization and residual distribution of reconstructed images. (a). Uncertainty visualization (RGB). (b). Pixel-wise normalized residual distribution of reconstructed images.
  • Figure 5: Comparison of performance and efficiency between EnVAE and FEnVAE
  • ...and 21 more figures