# Manifold Denoising by Nonlinear Robust Principal Component Analysis

arXiv: https://arxiv.org/abs/1911.03831 · HTML: https://mlsanity.com/paper/1911.03831/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Nov 2019.

## Built on by (most cited first, 3 of 3)

- [Manifold embedding data-driven mechanics](https://mlsanity.com/paper/2112.09842/) (2021)
- [Robust Autoencoders for Collective Corruption Removal](https://mlsanity.com/paper/2303.02828/) (2023)
- [The Central Spanning Tree Problem](https://mlsanity.com/paper/2404.06447/) (2024)

## Abstract (arXiv)

This paper extends robust principal component analysis (RPCA) to nonlinear manifolds. Suppose that the observed data matrix is the sum of a sparse component and a component drawn from some low dimensional manifold. Is it possible to separate them by using similar ideas as RPCA? Is there any benefit in treating the manifold as a whole as opposed to treating each local region independently? We answer these two questions affirmatively by proposing and analyzing an optimization framework that separates the sparse component from the manifold under noisy data. Theoretical error bounds are provided when the tangent spaces of the manifold satisfy certain incoherence conditions. We also provide a near optimal choice of the tuning parameters for the proposed optimization formulation with the help of a new curvature estimation method. The efficacy of our method is demonstrated on both synthetic and real datasets.