Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel

TL;DR

This work advances inverse problems on closed Riemannian manifolds by addressing partial data scenarios for both distance measurements and heat-kernel observations. It develops a two-pronged approach: (i) reconstructing a manifold from partial distance data using local coordinate frames built from distance measurements, with explicit bi-Lipschitz stability under bounded geometry; (ii) recovering the manifold from noisy heat-kernel data with separated sources and observations under non-negative Ricci curvature, yielding a stability estimate and uniqueness in the noiseless limit. The results connect manifold learning with rigorous geometric analysis, providing explicit error bounds and demonstrating that the manifold structure is recoverable up to diffeomorphism even with substantial data incompleteness. A novel feature is the separation between source and observation sets in the heat-kernel problem, and the framework offers implications for conditional stability under invariant geometric constraints and for practical manifold learning tasks.

Abstract

We consider how a closed Riemannian manifold $M$ and its metric tensor $g$ can be approximately reconstructed from local distance measurements. Moreover, we consider an inverse problem of determining $(M,g)$ from limited knowledge on the heat kernel. In the part 1 of the paper, we considered the approximate construction of a smooth manifold in the case when one is given the noisy distances $\tilde d(x,y)=d(x,y)+\varepsilon_{x,y}$ for all points $x,y\in X$, where $X$ is a $δ$-dense subset of $M$ and $|\varepsilon_{x,y}|<δ$. In this part 2 of the paper, we consider a similar problem with partial data, that is, the approximate construction of the manifold $(M,g)$ when we are given $\tilde d(x,y)$ for $x\in X$ and $y \in U\cap X$, where $U$ is an open subset of $M$. In addition, we consider the inverse problem of determining the manifold $(M,g)$ with non-negative Ricci curvature from noisy observations of the heat kernel $G(y,z,t)$. We show that a manifold approximating $(M,g)$ can be determined in a stable way, when for some unknown source points $z_j$ in $X\setminus U$, we are given the values of the heat kernel $G(y,z_k,t)$ for $y\in X\cap U$ and $t\in (0,1)$ with a multiplicative noise. We also give a uniqueness result for the inverse problem in the case when the data does not contain noise and consider applications in manifold learning. A novel feature of the inverse problem for the heat kernel is that the set $M\setminus U$ containing the sources and the observation set $U$ are disjoint.

Figures (7)

• Figure 1: On a closed manifold $(M,g)$, we consider the distances $d(x_i,y_j)$ from the blue points $y_j\in Y$ in the open subset $U\subset M$ to the red points $x_i$ filling the manifold $M$. These distances with measurement errors $\varepsilon_{i,j}$ define the noisy distance vectors $\vec{R}_{i}=[\widehat{R}_{i,j}]$, where $\widehat{R}_{i,j}=d(x_i,y_j)+\varepsilon_{i,j}$ and $|\varepsilon_{i,j}|<\varepsilon_1$. The inverse problem is to construct an approximation of the manifold $(M,g)$ from these data.
• Figure 2: The Riemannian normal coordinates on the manifold $M$ when $\hbox{dim}\,(M)=2$. These coordinates are centered at the point $x_{{i_0}}$ and are associated to the basis $\{v_1,v_2\}$. We compute approximately the coordinates of the points $x_\ell=\exp_{x_{i_0}}(\xi_{\ell})$, that is, $X(x_\ell)=(X_1(x_\ell),X_2(x_\ell))\in {\Bbb {R}}^2$ and the metric tensor $g_{jk}$ at the point $x_{{i_0}}$.
• Figure 3: Left: Setting of Lemma \ref{['lem: First variation lemma A']}. The green vertical line corresponds to the boundary of the set $U$ when the lemma is applied. Right: Setting of Lemma \ref{['lem: First variation lemma delta']}.
• Figure 4: An auxiliary figure for Proposition \ref{['lem: inner product lemma']}.
• Figure 5: Left: Distance-minimizing paths $\gamma_i$ and a shortcut $\mu_{\tau}$ in Lemma \ref{['observing same geodesic']}; the angles of $\mu$ and $\gamma_i$ at $y_1$ are close to $\pi$. Right: Setting of Lemma \ref{['lemma-angle-separation']}.

Theorems & Definitions (24)

• Lemma 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 2.1
• Corollary 2.2
• Definition 2.3
• Theorem 2.4
• Corollary 2.5
• proof
• Lemma 3.1