# Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problems

arXiv: https://arxiv.org/abs/2507.07291 · HTML: https://mlsanity.com/paper/2507.07291/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 66 references that are in the map.
- First posted on arXiv 9 Jul 2025.

## Builds on (most cited first, 29 of 29)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Variational Deep Embedding: An Unsupervised and Generative Approach to Clustering](https://mlsanity.com/paper/1611.05148/) (2016)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [LassoNet: A Neural Network with Feature Sparsity](https://mlsanity.com/paper/1907.12207/) (2019)
- [Identifiability of deep generative models without auxiliary information](https://mlsanity.com/paper/2206.10044/) (2022)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [Variational Autoencoder with Learned Latent Structure](https://mlsanity.com/paper/2006.10597/) (2020)
- [Blind Image Deconvolution using Deep Generative Priors](https://mlsanity.com/paper/1802.04073/) (2018)
- [LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood](https://mlsanity.com/paper/2206.14882/) (2022)
- [Losing dimensions: Geometric memorization in generative diffusion](https://mlsanity.com/paper/2410.08727/) (2024)
- [Multi-Facet Clustering Variational Autoencoders](https://mlsanity.com/paper/2106.05241/) (2021)
- [Disconnected Manifold Learning for Generative Adversarial Networks](https://mlsanity.com/paper/1806.00880/) (2018)
- [PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularization](https://mlsanity.com/paper/2205.12021/) (2022)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [Geometry-Aware Maximum Likelihood Estimation of Intrinsic Dimension](https://mlsanity.com/paper/1904.06151/) (2019)
- [Canonical normalizing flows for manifold learning](https://mlsanity.com/paper/2310.12743/) (2023)
- [Manifold Learning Benefits GANs](https://mlsanity.com/paper/2112.12618/) (2021)
- [Wasserstein Patch Prior for Image Superresolution](https://mlsanity.com/paper/2109.12880/) (2021)
- [Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems](https://mlsanity.com/paper/2305.01090/) (2023)
- [Intrinsic dimension estimation for locally undersampled data](https://mlsanity.com/paper/1906.07670/) (2019)
- [Manipulating Identical Filter Redundancy for Efficient Pruning on Deep and Complicated CNN](https://mlsanity.com/paper/2107.14444/) (2021)
- [Robust estimation of the intrinsic dimension of data sets with quantum cognition machine learning](https://mlsanity.com/paper/2409.12805/) (2024)
- [InfoCatVAE: Representation Learning with Categorical Variational Autoencoders](https://mlsanity.com/paper/1806.08240/) (2018)
- [Non-linear, Sparse Dimensionality Reduction via Path Lasso Penalized Autoencoders](https://mlsanity.com/paper/2102.10873/) (2021)

## Built on by (most cited first, 1 of 1)

- [A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifolds](https://mlsanity.com/paper/2608.23798/) (2026)

## Abstract (arXiv)

High-dimensional datasets often exhibit low-dimensional geometric structures, as suggested by the manifold hypothesis, which implies that data lie on a smooth manifold embedded in a higher-dimensional ambient space. While this insight underpins many advances in machine learning and inverse problems, fully leveraging it requires to deal with three key tasks: estimating the intrinsic dimension (ID) of the manifold, constructing appropriate local coordinates, and learning mappings between ambient and manifold spaces. In this work, we propose a framework that addresses all these challenges using a Mixture of Variational Autoencoders (VAEs) and tools from Riemannian geometry. We specifically focus on estimating the ID of datasets by analyzing the numerical rank of the VAE decoder pullback metric. The estimated ID guides the construction of an atlas of local charts using a mixture of invertible VAEs, enabling accurate manifold parameterization and efficient inference. We how this approach enhances solutions to ill-posed inverse problems, particularly in biomedical imaging, by enforcing that reconstructions lie on the learned manifold. Lastly, we explore the impact of network pruning on manifold geometry and reconstruction quality, showing that the intrinsic dimension serves as an effective proxy for monitoring model capacity.