Fast, Accurate Manifold Denoising by Tunneling Riemannian Optimization

• URL: http://arxiv.org/abs/2502.16819v1
• Date: Mon, 24 Feb 2025 04:02:16 GMT
• Title: Fast, Accurate Manifold Denoising by Tunneling Riemannian Optimization
• Authors: Shiyu Wang, Mariam Avagyan, Yihan Shen, Arnaud Lamy, Tingran Wang, Szabolcs Márka, Zsuzsa Márka, John Wright,
• Abstract summary: We consider the problem of efficiently denoising a noisy new data point sampled from an unknown $d$-dimensional manifold $M in mathbbRD$, using only noisy samples.<n>This work proposes a framework for test-time efficient manifold denoising, by framing the concept of "learning-to-denoise" as "learning-to-optimize"
• Score: 4.597774455074727
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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