Fast, Accurate Manifold Denoising by Tunneling Riemannian Optimization
This work addresses the efficiency and interpretability issues in manifold denoising for applications like diffusion models and compressed sensing, though it appears incremental by building on existing provable approaches.
The paper tackles the problem of efficiently denoising noisy data points from an unknown manifold by framing denoising as learning-to-optimize, resulting in significantly improved complexity-performance tradeoffs compared to nearest neighbor search methods.
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 $M \in \mathbb{R}^D$, 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.