Estimation of Riemannian Quantities from Noisy Data via Density Derivatives

Abstract

We study the recovery of geometric structure from data generated by convolving the uniform measure on a smooth compact submanifold $M\subset\mathbb{R}^D$ with ambient Gaussian noise. Our main result is that several fundamental Riemannian quantities of $M$, including tangent spaces, the intrinsic dimension, and the second fundamental form, are identifiable from derivatives of the noisy density. We first derive uniform small-noise expansions of the data density and its derivatives in a tubular neighborhood of $M$. These expansions show that, at the population level, tangent spaces can be recovered from the density Hessian with $O(σ^2)$ error, while the intrinsic dimension can be estimated consistently. We further construct estimators for the second fundamental form from density derivatives, obtaining $O(d(y,M)+σ)$ and $O(d(y,M)+σ^2)$ errors for hypersurfaces and submanifolds with arbitrary codimension. At the sample level, we estimate the density and its derivatives by kernel methods in the ambient space and plug them into the population constructions, yielding uniform nonparametric rates in the ambient dimension. Finally, we show that these density-based constructions admit a geometric interpretation through density-induced ambient metrics, linking the geometry of $M$ to ambient geodesic structure.

Figures (8)

• Figure 1: Illustration of Riemannian geometry estimation from noisy data. The underlying submanifold $M$ (black) is observed through noisy samples (gray). For a point $y$ near $M$, let $\pi(y)$ denote the projection of $y$ onto $M$. For a tangent vector $u$, the true tangent space and second fundamental form at $\pi(y)$ are $T_{\pi(y)}M$ and $\Pi_{\pi(y)}(u,u)$, respectively, while the estimated tangent space and second fundamental form at $y$ are denoted by $\widetilde{T}_yM$ and $\widetilde{\Pi}_y(u,u)$.
• Figure 2: Tangent space estimation on a noisy torus. Left: noisy samples $\mathcal{Y}$ (gray), together with the true tangent planes (orange) and the estimated tangent planes (blue) at representative points on the torus for $N=1000$ and $\sigma=0.05$. Middle and right: box plots of the tangent space error $\mathrm{err}_{\mathrm{tan}}$ over the evaluation set $\mathcal{W}$ as functions of the noise level $\sigma$ for fixed $N = 10^4$ and of the sample size $N$ for fixed $\sigma = 0.05$, respectively.
• Figure 3: Comparison of tangent space estimators on the torus. Left: box plots of tangent space errors for $N = 1000$ and $\sigma = 0.05$, comparing LPCA, the diffusion-geometry estimator JI24, and the proposed Hessian-based estimator. Middle and right: errors of LPCA and of the proposed estimator as functions of the noise level $\sigma$ for fixed $N = 3000$ and of the sample size $N$ for fixed $\sigma = 0.05$, respectively.
• Figure 4: Mean curvature estimation on the torus. Left: norm of the true mean curvature vectors (orange) and of the estimated mean curvature vectors (blue) on the torus for $N=1000$ and $\sigma=0.05$; darker colors indicate larger norm. Middle and right: box plots of the curvature error $\mathrm{err}_{\mathrm{curv}}$ over the evaluation set $\mathcal{W}$ as functions of the noise level $\sigma$ for fixed $N = 10^4$ and of the sample size $N$ for fixed $\sigma = 0.05$, respectively.
• Figure 5: Comparison of mean curvature estimators on the torus. Left: box plots of curvature errors for $N = 1000$ and $\sigma = 0.05$, comparing the Weingarten map estimator of CY19, the diffusion-geometry estimator JI24, and the proposed hypersurface estimator. Middle and right: curvature errors of JI24 and of the proposed estimator as functions of the noise level $\sigma$ for fixed $N = 3000$ and of the sample size $N$ for fixed $\sigma = 0.05$, respectively.

Theorems & Definitions (43)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• Corollary 3.2
• Lemma 3.3
• Theorem 3.4
• Theorem 3.5