Reconstruction of Manifold Distances from Noisy Observations

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 $\mu$ 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 $\mu$ 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.