# Structure-adaptive manifold estimation

arXiv: https://arxiv.org/abs/1906.05014 · HTML: https://mlsanity.com/paper/1906.05014/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2019.

## Built on by (most cited first, 6 of 6)

- [A likelihood approach to nonparametric estimation of a singular distribution using deep generative models](https://mlsanity.com/paper/2105.04046/) (2021)
- [Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness](https://mlsanity.com/paper/2303.09863/) (2023)
- [Quadratic Matrix Factorization with Applications to Manifold Learning](https://mlsanity.com/paper/2301.12965/) (2023)
- [Spherical Rotation Dimension Reduction with Geometric Loss Functions](https://mlsanity.com/paper/2204.10975/) (2022)
- [Curvature Corrected Nonnegative Manifold Data Factorization](https://mlsanity.com/paper/2502.15124/) (2025)
- [Subspace-Constrained Quadratic Matrix Factorization: Algorithm and Applications](https://mlsanity.com/paper/2411.04717/) (2024)

## Abstract (arXiv)

We consider a problem of manifold estimation from noisy observations. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned weights become so corrupted that the averaged estimate shows very poor performance. We suggest a structure-adaptive procedure, which simultaneously reconstructs a smooth manifold and estimates projections of the point cloud onto this manifold. The proposed approach iteratively refines the weights on each step, using the structural information obtained at previous steps. After several iterations, we obtain nearly "oracle" weights, so that the final estimates are nearly efficient even in the presence of relatively large noise. In our theoretical study, we establish tight lower and upper bounds proving asymptotic optimality of the method for manifold estimation under the Hausdorff loss, provided that the noise degrades to zero fast enough.