Curvature-driven Manifold Fitting¶
Overview¶
Manifold fitting aims to reconstruct a low-dimensional manifold from high-dimensional data. This project studies the recovery of a compact \(C^3\) submanifold \(\mathcal{M} \subset \mathbb{R}^D\) with dimension \(d < D\) and positive reach \(\tau\) from noisy observations \(Y = X + \xi\), where \(X\) is uniformly distributed on \(\mathcal{M}\) and \(\xi \sim \mathcal{N}(0, \sigma^2 I_D)\) is isotropic Gaussian noise.
To project points in a tubular neighborhood of \(\mathcal{M}\) back onto the manifold, we construct a sample-based estimator using a normalized local kernel with theoretically derived bandwidth \(r = c_D \sigma\). Under suitable sample size conditions, we establish a uniform asymptotic expansion
\[ F(z) = \pi(z) + \frac{d}{2} H_{\pi(z)} \sigma^2 + O(\sigma^3), \qquad z \in \Gamma, \]
where \(\pi(z)\) is the projection of \(z\) onto \(\mathcal{M}\) and \(H_{\pi(z)}\) is the mean curvature vector at \(\pi(z)\). The resulting fitted manifold achieves second-order accuracy, and the analysis shows how curvature information can be used to improve denoising and geometric recovery under unbounded isotropic noise.
To project points in a tubular neighborhood of \(\mathcal{M}\) back onto the manifold, we construct a sample-based estimator using a normalized local kernel with theoretically derived bandwidth \(r = c_D \sigma\). Under suitable sample size conditions, we establish a uniform asymptotic expansion
\[ F(z) = \pi(z) + \frac{d}{2} H_{\pi(z)} \sigma^2 + O(\sigma^3), \qquad z \in \Gamma, \]
where \(\pi(z)\) is the projection of \(z\) onto \(\mathcal{M}\) and \(H_{\pi(z)}\) is the mean curvature vector at \(\pi(z)\). The resulting fitted manifold achieves second-order accuracy, and the analysis shows how curvature information can be used to improve denoising and geometric recovery under unbounded isotropic noise.
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