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Fast, Accurate Manifold Denoising by Tunneling Riemannian Optimization

ICML 2025 · Learning theory and algorithms › Prediction and decision losses · approach: Metric and ordinal embedding · 0 citations (OpenAlex)

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Frames test-time manifold denoising as learning-to-optimize: online-learned Riemannian optimizers over a geometric graph built from noisy samples, with tunneling and mixed-order traversal to escape local minima.

Problem

Denoising a new noisy point lying near an unknown d-dimensional manifold quickly and accurately; existing provable methods require linear scans of the dataset (nearest neighbor/local approximation), while neural denoisers lack guarantees and efficiency.

Approach

Learn Riemannian optimizers online from noisy data over a geometric graph approximating the manifold (growing the optimizer one sample at a time), add tunnels to escape suboptimal local minimizers, and use mixed-order traversal to guarantee global optimality with near-optimal denoising error.

Key results

• Mixed-order traversal achieves near-optimal denoising error on the order of kappa sigma sqrt(d) with high probability.
• Experiments on scientific manifolds show improved complexity-performance tradeoffs compared to nearest neighbor search.

Limitations

• Relies on theoretical bounds with unspecified numerical constants c, c'.
• Test-time cost includes a tunneling/escape term that depends on a novel complexity measure.

Experimental setup

Datasets: scientific manifolds (noisy samples)

Benchmarks: scientific manifolds

Metrics: denoising error complexity-performance tradeoff

Baselines: nearest neighbor search

Builds on: Riemannian optimization moving least squares manifold denoising graph-based nearest neighbor search

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