mlsanityData to 7 Oct 2026

Tutorial: Deriving the Standard Variational Autoencoder (VAE) Loss Function

arXiv 1907.08956 ↗First posted 21 Jul 2019cs.LG stat.MLcited by 26

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Abstract (arXiv)

In Bayesian machine learning, the posterior distribution is typically computationally intractable, hence variational inference is often required. In this approach, an evidence lower bound on the log likelihood of data is maximized during training. Variational Autoencoders (VAE) are one important example where variational inference is utilized. In this tutorial, we derive the variational lower bound loss function of the standard variational autoencoder. We do so in the instance of a gaussian latent prior and gaussian approximate posterior, under which assumptions the Kullback-Leibler term in the variational lower bound has a closed form solution. We derive essentially everything we use along the way; everything from Bayes' theorem to the Kullback-Leibler divergence.

Key facts. Cited by 26 papers in the map (data through 7 Oct 2026). First posted on arXiv 21 Jul 2019.

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Citing papers per year: 2021: 3; 2022: 5; 2023: 4; 2024: 7; 2025: 2; 2026: 5.

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  1. 2022Hybrid Predictive Coding: Inferring, Fast and Slowcited by 25
  2. 2021Factor Analysis, Probabilistic Principal Component Analysis, Variational Inference, and Variational Autoencoder: Tutorial and Surveycited by 21
  3. 2023Generalizing Across Domains in Diabetic Retinopathy via Variational Autoencoderscited by 8
  4. 2021Disentangling Generative Factors of Physical Fields Using Variational Autoencoderscited by 5
  5. 2024Paired Wasserstein Autoencoders for Conditional Samplingcited by 5
  6. 2022Variational Inference for Quantifying Inter-observer Variability in Segmentation of Anatomical Structurescited by 4
  7. 2024Deep Data Consistency: a Fast and Robust Diffusion Model-based Solver for Inverse Problemscited by 4
  8. 2022Unsupervised Clustering of Roman Potsherds via Variational Autoencoderscited by 3
  9. 2022Tackling Multimodal Device Distributions in Inverse Photonic Design using Invertible Neural Networkscited by 3
  10. 2023Data-driven Nonlinear Parametric Model Order Reduction Framework using Deep Hierarchical Variational Autoencodercited by 3
  11. 2024Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximationscited by 3
  12. 2023Uncertainty-aware transfer across tasks using hybrid model-based successor feature reinforcement learningcited by 1
14 more
  1. 2024Energy-Based Prior Latent Space Diffusion model for Reconstruction of Lumbar Vertebrae from Thick Slice MRIcited by 1
  2. 2021Maximum Entropy Auto-Encodingcited by 0
  3. 2022Variational Autoencoder Assisted Neural Network Likelihood RSRP Prediction Modelcited by 0
  4. 2023Variational Self-Supervised Contrastive Learning Using Beta Divergencecited by 0
  5. 2024Projected Belief Networks With Discriminative Alignment for Acoustic Event Classification: Rivaling State of the Art CNNscited by 0
  6. 2024CF-OPT: Counterfactual Explanations for Structured Predictioncited by 0
  7. 2024Deep evolving semi-supervised anomaly detectioncited by 0
  8. 2025Active Learning-Guided Seq2Seq Variational Autoencoder for Multi-target Inhibitor Generationcited by 0
  9. 2025Using KL-Divergence to Focus Frequency Information in Low-Light Image Enhancementcited by 0
  10. 2026Variational Contrastive Learning for Skeleton-based Action Recognitioncited by 0
  11. 2026Cryo-SWAN: the Multi-Scale Wavelet-decomposition-inspired Autoencoder Network for molecular density representation of molecular volumescited by 0
  12. 2026VAMP-Diff: VampPrior Latent Diffusion for Photoplethysmography Modelingcited by 0
  13. 2026Reconstructing and forecasting disease trajectories of patients with Alzheimer's disease using routine data in resource-constrained settingscited by 0
  14. 2026AI-Generated Image Recognition via Fusion of CNNs and Vision Transformerscited by 0