# Fast, Accurate Manifold Denoising by Tunneling Riemannian Optimization

arXiv: https://arxiv.org/abs/2502.16819 · HTML: https://mlsanity.com/paper/2502.16819/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 24 Feb 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 8 of 8)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs](https://mlsanity.com/paper/1603.09320/) (2016)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Manifold Approximation by Moving Least-Squares Projection (MMLS)](https://mlsanity.com/paper/1606.07104/) (2016)
- [Dense residual Transformer for image denoising](https://mlsanity.com/paper/2205.06944/) (2022)
- [Approximating the Riemannian Metric from Point Clouds via Manifold Moving Least Squares](https://mlsanity.com/paper/2007.09885/) (2020)
- [TpopT: Efficient Trainable Template Optimization on Low-Dimensional Manifolds](https://mlsanity.com/paper/2310.10039/) (2023)

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

- [Local Averaging Accurately Distills Manifold Structure From Noisy Data](https://mlsanity.com/paper/2506.18761/) (2025)

## Abstract (arXiv)

Learned denoisers play a fundamental role in various signal generation (e.g., diffusion models) and reconstruction (e.g., compressed sensing) architectures, whose success derives from their ability to leverage low-dimensional structure in data. Existing denoising methods, however, either rely on local approximations that require a linear scan of the entire dataset or treat denoising as generic function approximation problems, often sacrificing efficiency and interpretability. We consider the problem of efficiently denoising a new noisy data point sampled from an unknown d-dimensional manifold M ∈ ℝD, using only noisy samples. This work proposes a framework for test-time efficient manifold denoising, by framing the concept of "learning-to-denoise" as "learning-to-optimize". We have two technical innovations: (i) online learning methods which learn to optimize over the manifold of clean signals using only noisy data, effectively "growing" an optimizer one sample at a time. (ii) mixed-order methods which guarantee that the learned optimizers achieve global optimality, ensuring both efficiency and near-optimal denoising performance. We corroborate these claims with theoretical analyses of both the complexity and denoising performance of mixed-order traversal. Our experiments on scientific manifolds demonstrate significantly improved complexity-performance tradeoffs compared to nearest neighbor search, which underpins existing provable denoising approaches based on exhaustive search.