Reconstruction of Manifold Distances from Noisy Observations

TL;DR

The paper tackles the challenge of reconstructing intrinsic geodesic distances on a diameter $1$, $d$-dimensional geodesic probability space from noisy pairwise distances. It introduces two algorithms that recover true distances on a sufficiently dense subset: Algorithm 1 builds clusters by estimating $L^2$ norms of the functions $f_x(y)=\mathbb{E}d'(x,y)$ through inner products and then recovers distances with additive error $O(\varepsilon\log\varepsilon^{-1})$, achieving sample complexity $N\asymp \varepsilon^{-(2d+2)}\log(1/\varepsilon)$ and runtime $o(N^3)$ in the non-missing case; Algorithm 2 uses a regularized optimization to produce an $\varepsilon$-net with comparable recovery guarantees and a more involved runtime bound. The analysis relies on concentration of inner-product estimators, cluster containment $B(x,\delta)\subset \mathcal{C}(x)\subset B(x,4\varepsilon)$, and geometric conditions such as bounded curvature and positive injectivity radius, with extensions to missing data under robust sampling conditions. The results apply to manifolds and more general geodesic spaces, offering practical means to recover travel-time-like metrics from noisy observations, with potential applications in seismic imaging, ultrasound elastography, and related inverse problems.

Abstract

We consider the problem of reconstructing the intrinsic geometry of a manifold from noisy pairwise distance observations. Specifically, let $M$ denote a diameter 1 d-dimensional manifold and $μ$ a probability measure on $M$ that is mutually absolutely continuous with the volume measure. Suppose $X_1,\dots,X_N$ are i.i.d. samples of $μ$ and we observe noisy-distance random variables $d'(X_j, X_k)$ that are related to the true geodesic distances $d(X_j,X_k)$. With mild assumptions on the distributions and independence of the noisy distances, we develop a new framework for recovering all distances between points in a sufficiently dense subsample of $M$. Our framework improves on previous work which assumed i.i.d. additive noise with known moments. Our method is based on a new way to estimate $L_2$-norms of certain expectation-functions $f_x(y)=\mathbb{E}d'(x,y)$ and use them to build robust clusters centered at points of our sample. Using a new geometric argument, we establish that, under mild geometric assumptions--bounded curvature and positive injectivity radius--these clusters allow one to recover the true distances between points in the sample up to an additive error of $O(\varepsilon \log \varepsilon^{-1})$. We develop two distinct algorithms for producing these clusters. The first achieves a sample complexity $N \asymp \varepsilon^{-2d-2}\log(1/\varepsilon)$ and runtime $o(N^3)$. The second introduces novel geometric ideas that warrant further investigation. In the presence of missing observations, we show that a quantitative lower bound on sampling probabilities suffices to modify the cluster construction in the first algorithm and extend all recovery guarantees. Our main technical result also elucidates which properties of a manifold are necessary for the distance recovery, which suggests further extension of our techniques to a broader class of metric probability spaces.

Theorems & Definitions (48)

• Theorem 1.1
• Remark 1.2
• Definition 2.1: Orlicz norm
• Theorem 2.2: Sub-Gaussian Hoeffding Inequality
• Theorem 2.3: Hoeffding Inequality
• Theorem 2.4: Bernstein
• Proposition 2.5
• Proposition 2.6
• Definition 2.7: Manifold Recovery
• Proposition 2.8