# Learning Manifold Dimensions with Conditional Variational Autoencoders

arXiv: https://arxiv.org/abs/2302.11756 · HTML: https://mlsanity.com/paper/2302.11756/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 34 references that are in the map.
- First posted on arXiv 23 Feb 2023.

## Builds on (most cited first, 15 of 15)

- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Toward Multimodal Image-to-Image Translation](https://mlsanity.com/paper/1711.11586/) (2017)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Diagnosing and Enhancing VAE Models](https://mlsanity.com/paper/1903.05789/) (2019)
- [A Hierarchical Latent Variable Encoder-Decoder Model for Generating Dialogues](https://mlsanity.com/paper/1605.06069/) (2016)
- [Generating Diverse Structure for Image Inpainting With Hierarchical VQ-VAE](https://mlsanity.com/paper/2103.10022/) (2021)
- [Amortized Causal Discovery: Learning to Infer Causal Graphs from Time-Series Data](https://mlsanity.com/paper/2006.10833/) (2020)
- [Taming VAEs](https://mlsanity.com/paper/1810.00597/) (2018)
- [PixelVAE: A Latent Variable Model for Natural Images](https://mlsanity.com/paper/1611.05013/) (2016)
- [Resampled Priors for Variational Autoencoders](https://mlsanity.com/paper/1810.11428/) (2018)
- [Leveraging the Exact Likelihood of Deep Latent Variable Models](https://mlsanity.com/paper/1802.04826/) (2018)

## Built on by (most cited first, 20 of 20)

- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [A Geometric Framework for Understanding Memorization in Generative Models](https://mlsanity.com/paper/2411.00113/) (2024)
- [A Review of Bayesian Uncertainty Quantification in Deep Probabilistic Image Segmentation](https://mlsanity.com/paper/2411.16370/) (2024)
- [Hitchhiker's guide on the relation of Energy-Based Models with other generative models, sampling and statistical physics: a comprehensive review](https://mlsanity.com/paper/2406.13661/) (2024)
- [Learning multi-modal generative models with permutation-invariant encoders and tighter variational objectives](https://mlsanity.com/paper/2309.00380/) (2023)
- [Reducing Class-wise Confusion for Incremental Learning with Disentangled Manifolds](https://mlsanity.com/paper/2503.17677/) (2025)
- [Deep Generative Models: Complexity, Dimensionality, and Approximation](https://mlsanity.com/paper/2504.00820/) (2025)
- [From NeurODEs to AutoencODEs: a mean-field control framework for width-varying Neural Networks](https://mlsanity.com/paper/2307.02279/) (2023)
- [Distributional Autoencoders Know the Score](https://mlsanity.com/paper/2502.11583/) (2025)
- [Sparse Autoencoders, Again?](https://mlsanity.com/paper/2506.04859/) (2025)
- [On Convolutions, Intrinsic Dimension, and Diffusion Models](https://mlsanity.com/paper/2506.20705/) (2025)
- [Investigating and Improving Latent Density Segmentation Models for Aleatoric Uncertainty Quantification in Medical Imaging](https://mlsanity.com/paper/2307.16694/) (2023)
- [A Wiener Process Perspective on Local Intrinsic Dimension Estimation Methods](https://mlsanity.com/paper/2406.17125/) (2024)
- [From Points to Spheres: A Geometric Reinterpretation of Variational Autoencoders](https://mlsanity.com/paper/2507.17255/) (2025)
- [Knowledge Abstraction for Knowledge-based Semantic Communication: A Generative Causality Invariant Approach](https://mlsanity.com/paper/2507.17784/) (2025)
- [HomID : Benchmarking Intrinsic Dimension Estimators on Homogenous Manifolds with Anisotropic Embeddings](https://mlsanity.com/paper/2510.01335/) (2025)
- [Aligning the Unseen in Attributed Graphs: Interplay between Graph Geometry and Node Attributes Manifold](https://mlsanity.com/paper/2601.22806/) (2026)
- [Commutator-Induced Uncertainty in VAEs](https://mlsanity.com/paper/2605.23449/) (2026)
- [FiGuRO: Intrinsic Dimension Estimation for Multi-Modal Data](https://mlsanity.com/paper/2608.10857/) (2026)

## Abstract (arXiv)

Although the variational autoencoder (VAE) and its conditional extension (CVAE) are capable of state-of-the-art results across multiple domains, their precise behavior is still not fully understood, particularly in the context of data (like images) that lie on or near a low-dimensional manifold. For example, while prior work has suggested that the globally optimal VAE solution can learn the correct manifold dimension, a necessary (but not sufficient) condition for producing samples from the true data distribution, this has never been rigorously proven. Moreover, it remains unclear how such considerations would change when various types of conditioning variables are introduced, or when the data support is extended to a union of manifolds (e.g., as is likely the case for MNIST digits and related). In this work, we address these points by first proving that VAE global minima are indeed capable of recovering the correct manifold dimension. We then extend this result to more general CVAEs, demonstrating practical scenarios whereby the conditioning variables allow the model to adaptively learn manifolds of varying dimension across samples. Our analyses, which have practical implications for various CVAE design choices, are also supported by numerical results on both synthetic and real-world datasets.