Reconstruction of a Riemannian manifold from noisy intrinsic distances

TL;DR

This paper demonstrates that a Riemannian manifold can be approximately reconstructed from noisy, incomplete intrinsic distance data of randomly sampled points, with high probability as sample size increases.

Contribution

It introduces a method to reconstruct manifolds from noisy, incomplete geodesic distances, extending the geometric Whitney problem to stochastic measurement errors.

Findings

Reconstruction is feasible with large probability given enough samples.

The method handles random noise in distance measurements.

It addresses missing data scenarios, including lack of information on distant points.

Abstract

We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian manifold $(M,g)$ is considered as an abstract metric space with intrinsic distances, not as an embedded submanifold of an ambient Euclidean space. Let $\{X_1,X_2,\dots,X_N\}$ bea set of $N$ sample points sampled randomly from an unknown Riemannian $M$ manifold. We assume that we are given the numbers $D_{jk}=d_M(X_j,X_k)+\eta_{jk}$, where $j,k\in \{1,2,\dots,N\}$. Here, $d_M(X_j,X_k)$ are geodesic distances, $\eta_{jk}$ are independent, identically distributed random variables such that $\mathbb E e^{|\eta_{jk}|}$ is finite. We show that when $N$ is large enough, it is possible to construct an approximation of the Riemannian manifold $(M,g)$ with a large probability. This problem is a generalization of the geometric Whitney problem with random measurement errors. We consider also the case when the information on noisy distance $D_{jk}$ of points $X_j$ and $X_k$ is missing with some probability. In particular, we consider the case when we have no information on points that are far away.

Equations

$$i)\Vert {Sec}_M{\Vert }_{L^{\infty }(M)}\le {\Lambda }^2,ii)diam(M)\le D,iii)inj(M)\ge i_0,$$

$$0<{\rho }_{min}\le \frac{d\mu }{d{V}_g}\le {\rho }_{max},where{\rho }_{min},{\rho }_{max}\in {\mathbb{R}}_{+}.$$

$$\mathbb{E}{\eta }_{jk}=0,\mathbb{E}({\eta }_{jk}^2)={\sigma }^2,\mathbb{E}{e}^{|{\eta }_{jk}|}=\beta .$$

$$D_{jk}=d_M(X_j,X_k)+{\eta }_{jk}.$$

$$\displaystyle\overline{D}_{jk}=D_{jk}^{\hbox{\tiny(partial\ data)}}=\left\{\begin{array}[]{cl}D_{jk}&\hbox{if $Y_{jk}=1$,}\\ \hbox{`missing'}&\hbox{if $Y_{jk}=0$,}\end{array}\right.$$

$$\mathbb{P}(Y_{jk}=1|X_j,X_k)=\Phi (X_j,X_k).$$

$${\Phi }^1(0)={\varphi }_0,\Vert {\Phi }^1{\Vert }_{C^1(\mathbb{R})}\le H,c_1{\Phi }^1(d_M(x,y))\le \Phi (x,y)\le c_2{\Phi }^1(d_M(x,y)),x,y\in M,$$

$$N=\bigg{\lfloor}C_{0}\delta^{{-3n}}\bigg{(}\log^{2}(\frac{1}{\theta})+\log^{8}(\frac{1}{{{\delta}}})\bigg{)}\bigg{\rfloor}$$

$$\frac{1}{L}\le \frac{d_M(F(x),F(y))}{d_{M^{\ast }}(x,y)}\le L,for\; allx,y\in M^{\ast },$$

$$inj(M^{\ast })\ge \min \{(C_1\Lambda {)}^{-1},(1-C_1\delta )inj(M)\}.$$

$$I^{(0)}=\{1,2,\dots ,N_0\},I^{(1)}=\{N_0+1,\dots ,N_0+N_1\},I^{(2)}=\{N_0+N_1+1,\dots ,N_0+N_1+N_2\}.$$

$$k_{\Phi }(y,z)={\int }_M|d_M(y,x)-d_M(z,x){|}^2\Phi (y,x)\Phi (x,z)d\mu (x),$$

$$A_{\Phi }(y,z)$$

$$Q(y_1,y_2)=\frac{V_{\Phi }(y_1,y_2)}{W_{\Phi }(y_1,y_2)},$$

$$V_{\Phi }(y_1,y_2)={\int }_M{\beta }_1(\frac{A_{\Phi }(y_1,z)}{b}){\psi }_{\rho }(k_{\Phi }(y_1,z))\Phi (z,y_2)d_M(z,y_2)d\mu (z),$$

$$W_{\Phi }(y_1,y_2)={\int }_M{\beta }_1(\frac{A_{\Phi }(y_1,z)}{b}){\psi }_{\rho }(k_{\Phi }(y_1,z))\Phi (z,y_2)d\mu (z),$$

$$D_{\Phi }(y_1,\rho )=\{z\in M:k_{\Phi }(y,z)<{\rho }^2,A_{\Phi }(y_1,z)\ge b\}.$$

$$\displaystyle d^{app}(y_{1},y_{2})=\left\{\begin{array}[]{cl}{Q}(y_{1},y_{2}),&\hbox{if $W_{\Phi}(y_{1},y_{2})>u$},\\ D,&\hbox{otherwise.}\\ \end{array}\right.$$

$$\underset{y_j\in {\mathbb{R}}^m}{\min }(\sum _{i,j=1}^N(\Vert y_i-y_j{\Vert }_{{\mathbb{R}}^m}-d_{ij}{)}^2),$$

$$\Vert F(x)-F(y){\Vert }_{{\mathbb{R}}^m}\approx d_M(x,y),x,y\in M,$$

$$\displaystyle\hbox{vol}_{g}(M)\leq V_{0}=v(n,-\Lambda^{2},D)\leq{\omega_{n}}\bigg{(}\frac{\sinh(\Lambda D)}{\Lambda D}\bigg{)}^{n-1}D^{n}$$

$$\frac{{vol}_g(B(x,\rho ))}{{vol}_g(M)}\ge \frac{v(n,-{\Lambda }^2,\rho )}{v(n,-{\Lambda }^2,D)}\ge {\hat{c}}_3{\rho }^n,\mu (B(x,\rho ))\ge c_3{\rho }^n,{\hat{c}}_3=\frac{{\omega }_n}{V_0},c_3=\frac{{\rho }_{min}}{{\rho }_{max}}{\hat{c}}_3,$$

$$r_0=\min (\frac{1}{2H}{\varphi }_0,i_0,\frac{\pi }{2\Lambda }),r_1=\frac{1}{2}{r}_0,{\varphi }_1=\frac{c_1}{2}{\varphi }_0.$$

$$\Phi (x,y)\ge c_1{\Phi }^1(r_0)\ge {\varphi }_1,ford(x,y)\le r_0.$$

$$\displaystyle N_{0}=\lfloor C_{2}\delta_{1}^{-n}({\log(\frac{1}{\theta})}+\log(\frac{1}{\delta_{1}}))\rfloor,\quad N\geq N_{0}+\lfloor C_{2}\varepsilon_{1}^{-2n}\bigg{(}\log^{2}(\frac{1}{\theta})+\log^{2}(\frac{1}{\delta_{1}})+\log^{8}(\frac{1}{\varepsilon_{1}})\bigg{)}\rfloor.$$

$$|d^{(a)}(X_j,X_{j^{\prime }})-d_M(X_j,X_{j^{\prime }})|\le {\epsilon }_1,if{d}_M(X_j,X_{j^{\prime }})<r_1,$$

$$d^{(a)}(X_j,X_{j^{\prime }})\ge r_1-{\epsilon }_1,if{d}_M(X_j,X_{j^{\prime }})\ge r_1.$$

$$N_0\ge C_3{\delta }_1^{-n}(\log ({\delta }_1^{-1})+\log ({\theta }^{-1})),$$

$$m\le m_0=1/(c_3({\delta }_1/6{)}^n)=6^n{c}_3^{-1}{\delta }_1^{-n}=C_5{\delta }_1^{-n}.$$

$$\mu (W_k)\ge c_3({\delta }_1/6{)}^{-n}\ge 1/(C_{6}m)$$

$$\mathbb{P}\left[E_i^r\right]\le {\left(1-\frac{1}{C_{6}m}\right)}^r\le {\left(1-\frac{1}{m_1}\right)}^r\le e^{-r/m_1}.$$

$$\mathbb{P}\left[T>b{m}_1\log m_1\right]=\mathbb{P}\left[\bigcup _{i=1}^m{E}_i^r\right]\le \frac{m_1}{C_6}\cdot \mathbb{P}[E_1^r]\le \frac{1}{C_6}{m}_1^{-b+1}.$$

Full text

Keywords: Inverse problems, Manifold learning, Geometric Whitney problem.

1. Introduction

Let $M$ be a manifold and $g$ an intrinsic Riemannian metric on it. Assume that one is given distances, $d_{M}(X_{j},X_{k})$, with random measurement errors, between points in a randomly sampled set $\{X_{1},X_{2},\dots,X_{N}\}$ of points of $M$. In this paper we ask, how one can construct a Riemannian manifold $(M^{*},g^{*})$ from these data so that with a large probability, the distance (in Lipschitz-sense) of the constructed manifold $(M^{*},g^{*})$ to the original Riemannian manifold $(M,g)$ can be estimated. The need of constructing the non-Euclidean, intrinsic metric is encountered in many applications, e.g., in medical and seismic imaging, discussed in Section 1.4.

In the traditional manifold learning, for instance by using the ISOMAP algorithm introduced in the seminal paper [53], one often aims to map points $X_{j}$ to points $Y_{j}=F(X_{j})$ in an Euclidean space ${\mathbb{R}}^{m}$, where $m\geq n$ is as small as possible, so that the Euclidean distances $\|Y_{j}-Y_{k}\|_{{\mathbb{R}}^{m}}$ are close to the intrinsic distances $d_{M}(X_{j},X_{k})$ and find a submanifold $\widetilde{M}\subset{\mathbb{R}}^{m}$ that is close to the points $Y_{j}$. However, even in the ideal case when one is given an infinite set of points $X_{j}$ that form a dense subset of a smooth manifold $M$ and one has no measurement errors, finding a map $F:M\to{\mathbb{R}}^{m}$ for which the embedded manifold $F(M)=\widetilde{M}\subset{\mathbb{R}}^{m}$ is isometric to $(M,g)$ is numerically a very difficult task as it means finding a map which existence is proved by the Nash embedding theorem, see [39, 40] and [55] on numerical techniques based on Nash embedding theorem. One can overcome this difficulty by formulating the problem in a coordinate invariant way: Given the geodesic distances of points sampled from a Riemannian manifold $(M,g)$, construct a manifold $M^{*}$ with an intrinsic metric tensor $g^{*}$ so that Lipschitz distance of $(M^{*},g^{*})$ to the original manifold $(M,g)$ is small. This problem was studied in [42, 29] using diffusion maps [9, 10]. In this paper we consider this problem when distances are given with random errors and use metric geometry to construct $(M^{*},g^{*})$ so that the distance of $(M^{*},g^{*})$ and $(M,g)$ can be estimated with a large probability. We emphasise that we consider $M^{*}$ as an abstract manifold, that is not isometricly embedded to an Euclidean space, but where the metric is given by a metric tensor $g^{*}$ that is constructed from the above data.

1.1. The main result

Let $n\geq 2$ be an integer, $\Lambda>0,$ $D>0$, and $i_{0}>0$. Let $(M,g)$ be a compact Riemannian manifold of dimension $n$ such that

[TABLE]

where $\hbox{Sec}_{M}$ is the sectional curvature of $(M,g)$, $\,\hbox{diam}\,(M)$ is the diameter of $(M,g)$ and $\hbox{inj}\,(M)$ is the injectivity radius of $(M,g)$, that is, the minimal radius of Riemannian normal coordinates. Let $d_{M}(x,y)$ denote the intrinsic (or geodesic) distance of the points $x,y\in M$ determined by the metric tensor $g$ corresponding to the line element $ds^{2}=g_{jk}(x)dx^{j}dx^{k}$. Here and below, we use Einstein’s summation convention and sum over indexes appearing as super and sub-indexes.

Let $(\Omega,\Sigma,\mathbb{P})$ be a complete probability space, ${\mathcal{B}}$ be the $\sigma$-algebra of Borel sets on $M$, and $\mu:{\mathcal{B}}\to[0,1]$ be a probability measure on $M$. Let $dV_{g}$ be Riemannian volume on $(M,g)$. Assume that the Radon-Nikodym derivative of $\mu$ satisfies

[TABLE]

Definition 1.1**.**

Let $X_{j}$, $j=1,2,\dots,N$ be independent, identically distributed (i.i.d.) random variables having distribution $\mu$. Let $\sigma>0,$ $\beta>1$, and $\eta_{jk}$ be random variables satisfying

[TABLE]

We assume that all random variables $\eta_{jk}$ and $X_{j}$ are independent. Let

[TABLE]

be the geodesic distances of points $X_{j}$ and $X_{k}$ measured with errors $\eta_{jk}$.

Note that the above assumptions are satisfied when $\eta_{jk}\sim N(0,\sigma^{2})$ are i.i.d. Gaussian random variables and $\beta\leq 2e^{\sigma^{2}}$. We are mostly interested in a case when $\sigma$ is fixed and $N$ is large.

Definition 1.2**.**

The partial data is given by

[TABLE]

where $Y_{jk}$ are random variables taking values in $\{0,1\}$ and $j,k\in\{1,2,\dots,N\}$. We assume that $Y_{jk}$ are independent of random variables $X_{j^{\prime}}$ for all $j^{\prime}\in\{1,2,\dots,N\}\setminus\{j,k\}$ and of $\eta_{j^{\prime\prime}k^{\prime\prime}}$ for all $j^{\prime\prime}$ and $k^{\prime\prime}$. Above, the value ‘missing’, can be replaced by a large real value, e.g. by $2D$.

Assume that the conditional probability of the event $\{Y_{jk}=1\}$, when $X_{j}$ and $X_{k}$ are known, is

[TABLE]

More precisely, when ${\mathcal{B}}_{jk}\subset\Sigma$ is the $\sigma$-algebra generated by the random variables $X_{j}$ and $X_{k}$, above in formula (1.6) we use notation $\mathbb{P}(Y_{jk}=1\,|\,X_{j},X_{k})=\mathbb{P}(Y_{jk}=1\,|\,\mathcal{B}_{jk})$. Here, $\Phi:M\times M\to[0,1]$ is a measurable function such that there is a function $\Phi^{1}:[0,\infty)\to[0,1]$ so that $s\mapsto\Phi^{1}(s)$ is non-increasing and

[TABLE]

where $0<c_{1}<1<c_{2}$ and $\phi_{0}\in{\mathbb{R}}_{+}$.

Below, for $t\in{\mathbb{R}}$ we denote by $\lfloor t\rfloor$ the largest integer $m$ such that $m\leq t$.

We will show that probabilistic considerations involving the above data, combined with the deterministic results in [24] (where we considered small deterministic errors) and Appendix A, yield that one can construct a smooth manifold $(M^{*},g^{*})$ that approximates the original manifold $(M,g)$. The proofs of Theorem 1 below and the results in [24] give a procedure, which the output is a submanifold $M^{*}\subset{\mathbb{R}}^{d}$ (where $d$ depends only on $n,D,\Lambda,$ and $i_{0}$) and a metric tensor $g^{*}$ on $M^{*}$.

Theorem 1**.**

Let $n\geq 2$, $D,\Lambda,i_{0},\rho_{min},\rho_{max},\sigma,\beta,c_{1},c_{2},H,\phi_{0}>0$ be given. Then there are $\delta_{0}>0$, and $C_{0}>0$, depending on $n,D,\Lambda,i_{0},\rho_{min},\rho_{max},\sigma,\beta,c_{1},c_{2},H,\phi_{0}$, and there is ${C_{1}}>0$, depending on $n$, such that the following holds for $\theta\in(0,\frac{1}{2})$.

Let $M$ be a compact $n$-dimensional manifold satisfying (1.1), $0<{{\delta}}<\delta_{0}$, and

[TABLE]

and $\overline{D}_{jk}$, $j,k=1,2,\dots,N$ be as in Definitions 1.1 and 1.2. Suppose that one is given samples of the random variables $\overline{D}_{jk}$ for $j,k=1,2,\dots,N$. Then withß a probability larger than $1-\theta$ one can construct a compact, smooth $n$-dimensional Riemannian manifold $(M^{*},g^{*})$ that approximates the manifold $(M,g)$ in the following way:

1. (1)

There is a diffeomorphism $F:M^{*}\to M$ satisfying

[TABLE]

where $L=1+C_{1}{{\delta}}$, that is, the Lipschitz distance of the metric spaces $(M^{*},g^{*})$ and $(M,g)$ satisfies $d_{Lip}((M^{*},g^{*}),(M,g))\leq\log L$. 2. (2)

The sectional curvature $\hbox{Sec}_{M^{*}}$ of ${M^{*}}$ satisfies $|\hbox{Sec}_{M^{*}}|\leq C_{1}\Lambda^{2}$. 3. (3)

The injectivity radius $\hbox{inj}({M^{*}})$ of ${M^{*}}$ satisfies

[TABLE]

We note that the knowledge of the authors, the results are new also in the case when there is no missing data, that is, $\Phi(x,y)=1$ for all $x,y\in M$.

Remark 1.3*.*

Theorem 1 concerns the regime where the noise level $\sigma$ is a fixed constant, the number of points $N$ is large, and we are interested in the situation where we want the probability $\theta$ of a wrong final reconstruction to be very small. This is reflected by the fact that the probability $\theta$ of obtaining a wrong reconstruction appears only in the logarithmic term $\log(\theta^{-1})$.

1.2. Idea of the proof and three nets of points on the manifold

Let us assume that $N=N_{0}+N_{1}+N_{2}$, where $N_{0},N_{1},N_{2}\in{\mathbb{Z}}_{+}$. We are interested in the case when $N_{2}>N_{1}>N_{0}$. We consider the random set $S_{0}=\{X_{1},\dots,X_{N_{0}}\}$ as a coarse net on $M$ and compute approximate distances between the points in the net $S_{0}$ by using the auxiliary nets $S_{1}=\{X_{N_{0}+1},\dots,X_{N_{0}+N_{1}}\}$ and $S_{2}=\{X_{N_{0}+N_{1}+1},\dots,X_{N_{0}+N_{1}+N_{2}}\}$. These random sets correspond to the index sets

[TABLE]

Recall that $X_{j}$ are independent random variables, taking values on $M$, with the distribution $\mu$.

Below, we say that a set $Y\subset M$ is $\delta-$dense in $M$ if for all $p\in M$ there is $y\in Y$ such such that $d_{M}(p,y)<\delta$. A $\delta-$dense subset $S$ of $M$ is often called a $\delta-$net.

Let us give an overview of the ideas on the proof of Theorem 1.

First, we use the “densest net” $S_{2}$ to compute approximately the numbers

[TABLE]

for $y$ and $z$ in the “medium dense net” $S_{1}$, see Prop. 3.5. This corresponds to taking average of function $|d_{M}(y,x)-d_{M}(z,x)|^{2}$ over all those sample points $x\in S_{2}$ for which the distances $d_{M}(y,x)$ and $d_{M}(z,x)$ are not missing, see (3.12).

Note that when the product $\Phi(y,x)\Phi(x,z)$ is small, there are only a small amount of sample points $x\in S_{2}$ for which the value of the function $|d_{M}(y,x)-d_{M}(z,x)|^{2}$ can be computed, and then the estimator for the function $k_{\Phi}(y,z)$ is not reliable. Thus the reliability of the estimator for the function $k_{\Phi}(y,z)$ is measured by

[TABLE]

Indeed, when $A_{\Phi}(y,z)$ is larger than some threshold value $b>0$, the obtained estimator for the function $k_{\Phi}(y,z)$ is reliable with a large probability. When we compute an estimator for the function $k_{\Phi}(y,z)$ using a sampling imitating the integral in (1.9), we can compute also an estimator for $A_{\Phi}(y,z)$, see (3.13).

Figure 1. *Left: The set $D_{\Phi}(y_{1},\rho)$ satisfies $B_{M}(y,c_{5}^{-1}\rho)\subset D_{\Phi}(y_{1},\rho)\subset B_{M}(y,\rho)$ and thus $D_{\Phi}(y_{1},\rho)$ can be considered as an approximate $\rho$-neighbourhood of the the point $y$. Right: The approximate distance $d^{app}(y_{1},y_{2})$ in the formula (1.15) is the average of distances from $y_{2}$ to the points in the neighbourhood $D_{\Phi}(y_{1},\rho)$. Later, we approximate $d^{app}(y_{1},y_{2})$ by taking the average of distances of $y_{2}$ to the points in $S_{1}\cap D_{\Phi}(y_{1},\rho)$, where $S_{1}$ is the medium dense net of sample points. *

Second, we are going to use the set $S_{1}$ to compute the approximate distances $d^{app}(y_{1},y_{2})$ of the points $y_{1}$ and $y_{2}$ in the “coarse net” $S_{0}$ using reliable distances $k_{\Phi}(y,z)^{1/2}$. We do this by computing estimators for the functions (see Definition 4.2)

[TABLE]

where $\beta_{1}\in C^{\infty}({\mathbb{R}})$ is a cut-off function such that $\beta_{1}(t)=0$ for $t<1$ and $\beta_{1}(t)=1$ for $t>2$, and $\psi_{\rho}\in C^{\infty}({\mathbb{R}})$ is a cut-off function such that $\psi_{\rho}(s)=1$ for $s<\rho^{2}$ and $\psi_{\rho}(s)=0$ for $s>2\rho^{2}$. Here, $\beta_{1}({A_{\Phi}(y_{1},z)}/b)\psi_{\rho}(k_{\Phi}(y,z))$ is the smoothened version of the indicator function of the set

[TABLE]

The set is $D_{\Phi}(y_{1},\rho)$ is a Lipschitz approximation the union of the ball $B_{M}(y_{1},\rho)$.

Then, roughly speaking, we compute an estimator for the function $V_{\Phi}(y_{1},y_{2})$ computing averages of function $\beta_{1}({A_{\Phi}(y_{1},z)}/b)d_{M}(z,y_{2})$ over all sample points $z$ the medium net $S_{1}$ that are in the set $D_{\Phi}(y_{1},\rho)$ and for which data on the distance $d_{M}(z,y_{2})$ is not missing. At the same time, we compute an estimator for the function $W_{\Phi}(y_{1},y_{2})$ by computing averages of function $\beta_{1}({A_{\Phi}(y_{1},z)}/b)\psi_{\rho}(k_{\Phi}(y,z))$ over the same sample points. The idea is that when $W_{\Phi}(y_{1},y_{2})$ is larger than some threshold $u>0$, the estimators computed from random data for the functions $V_{\Phi}(y_{1},y_{2})$, $W_{\Phi}(y_{1},y_{2})$, and $Q(y_{1},y_{2})$ are reliable with a large probability. Then, we define for $y_{1},y_{2}\in S_{0}$

[TABLE]

Then there is $r_{1}$ such that $d^{app}(y_{1},y_{2})$ approximate the true distance $d_{M}(y_{1},y_{2})$ with a small error $\varepsilon_{1}$ when $d_{M}(y_{1},y_{2})<r_{1}$, and moreover, if $d_{M}(y_{1},y_{2})\geq r_{1}$, then $d^{app}(y_{1},y_{2})>r_{1}-\varepsilon_{1}$. In other words, with a large probability, we can construct the distances $d_{M}(y_{1},y_{2})$ with small errors for all points $y_{1}$ and $y_{2}$ in $S_{0}$ that close to each other.

After the above constructions, we will use Proposition 4.10 in Appendix A, concerning a reconstruction of a Riemannian manifold when we are given distances with small (deterministic) errors. This result is an improved version of the earlier results given in [24, Corollary 1.10].

1.3. Earlier results for submanifolds of ${\mathbb{R}}^{n}$ and graphs of functions.

In dimensionality reduction and in the traditional manifold learning, the aim is to transform data, consisting of points in a $d$-dimensional space that are near an $n$-dimensional submanifold $M$, where $d>>n$ into a set of points in a low dimensional space ${\mathbb{R}}^{m}$ close to an $n$-dimensional submanifold, where $d>m\geq n$. During transformation all of them try to preserve some geometric properties, such as appropriately measured distances between points of the original data set, see [7, 8, 53]. Perhaps the most basic of such methods is ‘Principal Component Analysis’ (PCA), [44, 50] where one projects the data points onto the span of the $n$ eigenvectors corresponding to the $n$ largest eigenvalues of the ($d\times d$) covariance matrix of the data points.

In the case of ‘Multi Dimensional Scaling’ (MDS) [11], the pairwise distances between points are attempted to be preserved. One minimizes a certain ‘stress function’ which captures the total error in pairwise distances between the data points and between their lower dimensional counterparts. For instance, given points $(x_{j})_{j=1}^{N}$, $x_{j}\in{\mathbb{R}}^{d}$, one tries to find $(y_{j})_{j=1}^{N}$, $x_{j}\in{\mathbb{R}}^{m}$, that is an (approximate) minimizer of a stress function

[TABLE]

where $d_{ij}=\|x_{i}-x_{j}\|_{{\mathbb{R}}^{d}}$ are the Euclidean distances of points $x_{i}$ and $x_{j}$.

‘ISOMAP’ [53] attempts to improve on MDS by trying to capture geodesic distances between points while projecting. For each data point $x_{i}$ in the data set $\mathcal{X}=(x_{j})_{j=1}^{N}$, $x_{j}\in{\mathbb{R}}^{d}$, a ‘neighbourhood graph’ is constructed using the $K$-neighbours of $x_{i}$, that is, the $K$ nearest points of $\mathcal{X}$ to $x_{i}$, the edges carrying the length between points. Now the shortest distance between points is computed in the resulting global graph containing all the neighbourhood graphs using a standard graph theoretic algorithm such as Dijkstra’s. Let $D^{G}=[d^{G}_{ij}]$ be the $N\times N$ matrix of graph distances. Then MDS is used to find $(y_{j})_{j=1}^{N}$, $x_{j}\in{\mathbb{R}}^{m}$ that (approximately) solve the minimization problem (1.16) with distances $d_{ij}$ replaced by $d^{G}_{ij}$. If the data set $\mathcal{X}=(x_{j})_{j=1}^{N}$ consists of $\delta$-dense set points of a submanifold $M\subset{\mathbb{R}}^{d}$ with small $\delta$, then ISOMAP tries to find an approximation for isometric embedding, that is, a map $F:M\to{\mathbb{R}}^{m}$ for which

[TABLE]

where $d_{M}(x,y)$ is the intrinsic distance of the points $x$ and $y$ of the isometrically embedded manifold $M\subset{\mathbb{R}}^{d}$. Observe that one can not have equality $\|F(x)-F(y)\|_{{\mathbb{R}}^{m}}=d_{M}(x,y)$ in (1.17) unless all unit speed geodesics of $M$ are mapped to Euclidean lines (parametrized by the Euclidean length) in the map $F$, implying that $M$ has to be a flat manifold. Thus, even if the data points are contained in a submanifold $M$ and the number $N$ of data points grows and if $d_{ij}$ are equal to the intrinsic distances $d_{M}(x_{i},x_{j})$, the ISOMAP algorithm does not produce a manifold which intrinsic distances are the same as those of the original manifold $M$ when the manifold $M$ has non-zero curvature. The convexity and flatness conditions that guarantee that the ISOMAP algorithm reconstructs the original manifold are studied in [3, 16, 18] for ISOMAP and in [56] for the continuum ISOMAP.

In the seminal papers [9, 10] on ‘Diffusion Maps’, a complete graph is built on the data points sampled from a manifold $(M,g)$ and each edge is assigned a weight $a(x,y)$ that is a Gaussian function of the distance of the points $x$ and $y$. The normalized version of the kernel $a(x,y)$ defines a diffusion operator on $M$ and this operator has eigenfunctions $\phi_{j}(x)$. These functions can be used to construct a non-isometric embedding $x\mapsto(\phi_{j}(x))_{j=1}^{m}$ of the manifold $M$ into ${\mathbb{R}}^{m}$. This construction is continued in [42] by computing an approximation the metric tensor $g$ by using finite differences to find the Laplacian of the products of the local coordinate functions. When there are no errors in the data, this construction is shown in [29] to converge to a correct limit as the number of points sampled from the manifold $M$ tends to infinity. Other topological embedding methods for manifolds, based on heat kernels and eigenfunctions, have been developed in [2, 17, 30]. Moreover, locally linear construction methods are studied in [4, 36, 37, 48, 57].

The construction of a surface approximating a set of points in ${\mathbb{R}}^{m}$ is closely related to the classical Whitney’s problem. This problem is the construction of a function $F(x)\in C^{m}(\overline{S}),$ where $S\subset{\mathbb{R}}^{n}$ is open, which is equal to a given function $f(x)$ on $K$, where $K\subset S$. This problem has been studied in different norms in [20, 21, 22, 25] and the interpolation results on Whitney problem have been applied for manifold of a submanifold of ${\mathbb{R}}^{m}$ in [23, 26].

1.4. Applications of new results with intrinsic distances

A. Submanifolds of Banach spaces. In several imaging problems the images are considered as elements of Banach spaces that have no inner product structure. For example, in many applications the 2-dimensional images (corresponding photographs) or 3-dimensional objects, are modeled by functions $u:D\to{\mathbb{R}}$, $D\subset{\mathbb{R}}^{m}$, $m=2,3$ that are in the space of functions of bounded variation, $BV(D)$ or in Besov spaces $B^{s}_{11}(D)$, [49, 34]. For example, in medical imaging, such functions are used to model piecewise constant or piecewise smooth functions that correspond to the structure of the human body with internal organs with sharp jumps of density at the boundaries of the organs.

B. Physical models with non-Euclidean metric. Many inverse problems can be formulated as geometric problems where the goal is to determine the underlying manifold structure. For example, by probing a medium with waves one can measure the travel times between points. This defines a non-Euclidean metric called the travel time metric $g$. Recovering the wave speed function inside the medium is equivalent to the determination of a Riemannian manifold from external or boundary measurements. The relation of the boundary measurements to the distances between the points in an $\delta$-net in the interior of the manifold is considered e.g. in [1, 15, 32, 33]. Determining the wave speed of elastic waves inside a body is a central problem in seismic imaging of the Earth, [52, 54]. An example, in medical imaging, where the physical structures are represented using abstract Riemannian manifolds is ultrasound imaging where the acoustic properties of the inside of a body is imaged. The typical ultrasound images correspond to image of the body represented in the non-Euclidean travel time coordinates, of more precisely, in the Riemannian normal coordinates of the travel time metric [46]. In these coordinates, the image rays, that is, the geodesics from the location of the source device are straight lines. Theorem 1 can be applied when we can measure travel times of waves between points. For instance, for the earthquakes the travel times of the surface waves can be directly measured and one can use these data to deduce the properties of the Earth close to the surface. Another example is the Magnetic Resonance Imaging (MRI) based Elastography in medical imaging, where the elastic properties of the body of a patient are determined by observing the propagation of elastic waves sent into the body [28].

Models involving local and missing data are encountered in sensor technology, e.g. in the radio frequency identification (RFID) or in smart dust sensors, where a large number of low quality sensor send signals, either to receivers or to each others. On the earlier results on missing data in manifold learning, see e.g. [6, 19].

1.5. Notations

For a Riemannian manifold $(M,g)$ satisfying (1.1), let $\hbox{vol}_{g}$ be the Riemannian volume on $(M,g)$, and $p\in M$, the exponential map $\exp_{p}:T_{p}M\to M$ defines a smooth surjective map $\exp_{p}:\{\xi\in T_{p}M:\ \|\xi\|_{g}<D+1\}\to M$. By Bishop-Gromov inequality [45, Chapter 9, Lem. 1.6], the function $r\mapsto\hbox{vol}_{g}(B(x,r))/v(n,-\Lambda^{2},r)$ is non-increasing and bounded by 1, where $\hbox{vol}_{g}(B(x,r))$ is the volume of the ball $B(x,r)\subset M$ and $v(n,-\Lambda^{2},R)$ is the volume of the ball of radius $r$ in the hyperbolic space of dimension $n$ having curvature $-\Lambda^{2}$. Hence

[TABLE]

where $\omega_{n}$ is the volume of the unit ball in ${\mathbb{R}}^{n}$, see [45, Ch. 6, Cor. 2.4]. Moreover,

[TABLE]

where we use the fact that $v(n,-\Lambda^{2},\rho)\geq\omega_{n}\rho^{n}$. Let

[TABLE]

Then Definition 1.2 implies

[TABLE]

Acknowledgements. The authors express their gratitude to the Fields Institute, where parts of this work have been done. Ch.F.  was partly supported by AFOSR, grant DMS-1265524, and NSF, grant FA9550-12-1-0425. S.I. was partly supported by RFBR, grants 14-01-00062 and 17-01-00128-A, M.L. was supported by AF, grants 284715, 312110. and H.N. was partly supported by NSF grant DMS-1620102 and a Ramanujan Fellowship.

2. Reformulation of the main result with several parameters

Our aim is to prove the following result that yields of Theorem 1 when it is combined with the results in Appendix A on manifold reconstruction with small deterministic errors.

Theorem 2**.**

Let $n\geq 2$ and $D,\Lambda,i_{0},\rho_{min},\rho_{max},c_{0},c_{1},H,\sigma,\beta>0$ be given. Then there are $C_{2}>1$, $\widehat{\varepsilon}_{1}<1$, and $\widehat{\delta}_{1}<1$ depending on $n,D,\Lambda,i_{0},\rho_{min},\rho_{max},c_{0},c_{1},H,\sigma,\beta$, such that the following holds for $\varepsilon_{1}\leq\widehat{\varepsilon}_{1},\delta_{1}\leq\widehat{\delta}_{1},$ and $\theta\in(0,1/2)$: Let

[TABLE]

Also, let $X_{j}$ and $\overline{D}_{jk}$, $j,k=1,2,\dots,N$ be as in Definitions 1.1 and 1.2. Suppose that we are given samples of the random variables $\overline{D}_{jk}$ for $j,k=1,2,\dots,N$. Let $r_{1}$ be given in (1.20). Then with a probability larger than $1-\theta$ the set $\{X_{j}:\ j=1,2,\dots,N_{0}\}$ is a $\delta_{1}$-net in $M$ and one can determine the approximate distances $d^{(a)}(X_{j},X_{j^{\prime}})$ so that the following holds:

For all $j,j^{\prime}\in\{1,2,\dots,N_{0}\}$,

[TABLE]

*In the case when the probability $\Phi(x,y)$, for that the information on the distance of $x$ and $y$ is not missing, is bounded from below by a positive constant, that is, $\Phi(x,y)\geq\phi_{2},$ for all $(x,y)\in M\times M$, the inequality (2.2) holds for all $X_{j},X_{j^{\prime}}$, $j,j^{\prime}\in\{1,2,\dots,N_{0}\}$. *

We note when $\Phi(x,y)$ is bounded from below by a positive constant, $\Phi(x,y)\geq c_{1}\phi_{0},$ we can choose the function $\Phi^{1}$ to be equal to the constant $\phi_{0}$. Without loss of generality, we can assume that in this case $\phi_{2}=c_{1}\phi_{0}$.

2.1. Probability that the sample points form a dense net

First we estimate the probability that the set $S_{0}=\{X_{1},\dots,X_{N_{0}}\}$ is a $\delta_{1}$-net, using standard methods based on the collectors problem.

Lemma 2.1**.**

There is $C_{3}\geq 10$ such if $\theta\in(0,\frac{1}{2})$, $\delta_{1}\in(0,\frac{D}{2})$, and

[TABLE]

then the probability that the set $\{X_{j}:\ j=1,2,\dots,N_{0}\}$ is a $\delta_{1}$-net in $M$ is larger than $1-\frac{1}{2}\theta$.

Proof. Recall that by (1.18), we have $\hbox{vol}_{g}(M)\leq V_{0}.$ Also, by (1.19), the $\mu-$volume of a metric ball $B_{M}(x,\delta_{1}/6)\subset M$ is bounded from below by $c_{3}(\delta_{1}/6)^{n}$.

Let $\{z_{1},\dots,z_{m}\}$ be a maximal $(\delta_{1}/3)$-separated subset in $M$. Then

[TABLE]

Let $V_{k}=\{y\in M:\ \hbox{dist}(y,z_{k})<\hbox{dist}(y,z_{j})\hbox{ for all }j\not=k\}$ be the open Voronoi sets corresponding to points $z_{k}$ and let $W_{k}$, $k=1,2,\dots,m$, be such disjoint sets that $V_{k}\subset W_{k}\subset\overline{V}_{k}$ and that the union of the sets $W_{k}$ is $M$. Note that then the balls $B_{M}(z_{j},\delta_{1}/6)$ are disjoint and thus there is $C_{6}=C_{6}(n,\Lambda,D,i_{0},\rho_{max},\rho_{min})$ so that

[TABLE]

for all $k=1,2,\dots,m$. Observe that $\,\hbox{diam}\,(W_{k})<\delta_{1}$, and that if for all $k=1,2,\dots,m$ there is $X_{j}$, $j\leq N_{0}$ such that $X_{j}\in W_{k}$, then the set $\{X_{j}:\ j=1,2,\dots,N_{0}\}$ is a $\delta_{1}$-net in $M$.

We can use the classical collectors problem to estimate the probability of the event $A_{m,N_{0}}$ that all sets $W_{k}$ contain at least one point $X_{j},$ $j=1,2,\dots,N_{0}$. The tail estimates are used to give a solution for this problem, and for the convenience of the reader we give the details of this below (see also [12, 13, 41] for related results).

Let us choose an infinite sequence of i.i.d. random variables $X_{1},X_{2},\dots$ having distribution $\mu$ on $M$ and let $T$ be the smallest number such that all sets $W_{k}$, $k=1,2,\dots,m$ contain at least one point $X_{j}$, $j=1,2,\dots,T$. Let ${E}_{i}^{r}$ denote the event that the $i$-th set $W_{i}$ does not contain any of the first $r$ points $X_{1},\dots,X_{r}$. Let $m_{1}=C_{6}m_{0}\geq C_{6}m$ and $b>1$. Then

[TABLE]

For $r=\lfloor bm_{1}\log m_{1}\rfloor+1$, we have $\mathbb{P}\left[{E}_{i}^{r}\right]\leq e^{(-bm_{1}\log m_{1})/m_{1}}=m_{1}^{-b}.$ Then,

[TABLE]

Hence, when $b=1+\frac{\log(2(C_{6}\theta)^{-1})}{\log m_{1}}$, we have

[TABLE]

Observe that $m_{1}=C_{6}C_{5}\delta_{1}^{-n}$ and

[TABLE]

Therefore, when (2.4) is valid with a suitable $C_{3}$, we have $N_{0}>bm_{1}\log m_{1}$. This implies $\mathbb{P}(A_{m,N_{0}})=\mathbb{P}(T\leq N_{0})\geq 1-\frac{\theta}{2}.$ $\square$

3. The modified $L^{2}$-norm of the differences of the distance functions

Let $y,z\in M$ be (deterministic) points on $M$. Denote

[TABLE]

Definition 3.1**.**

For $x\in M$, let $r_{x}\in C(M)$ we define the distance function

[TABLE]

For $y,z\in M$, let

[TABLE]

The map $R:M\to L^{\infty}(M)$, given by $R(x)=r_{x}$, defines an isometric embedding $R:M\to R(M)\subset L^{\infty}(M)$. Below we will consider this map in different functions spaces. The function $A(y,z)$ measures the relative density of the points $x\in M$ for which the both distances $d_{M}(x,y)$ and $d_{M}(x,z)$ are non-missing in the data that is given to us. In the next lemmas we analyze these functions. Recall that $r_{1}=r_{0}/2$.

Lemma 3.2**.**

If $d_{M}(y,z)\leq r_{0}$ then

[TABLE]

Proof. Let $x,y\in M$ be such that $\ell=d_{M}(y,z)\leq r_{1}$. Also, let $[yz]=\gamma_{y,\xi}([0,\ell])$, $\xi\in S_{y}M$ be a length minimizing geodesic from $y$ to $z$. Let $q=\gamma_{y,\xi}(\ell/2)$. Using properties of $\Phi$ given in (1.21) (see also (1.6)-(1.7)), we see that for all $x\in B_{M}(q,r_{1})$ we have $x\in B_{M}(y,r_{0})$ and $x\in B_{M}(z,r_{0})$, and so $\Phi_{y}(x)\Phi_{z}(x)\geq{}{}\phi_{1}^{2}$. Observe that

[TABLE]

Hence,

[TABLE]

$\square$

Let $X$ have distribution $\mu$. Let $Y_{X,y}$ be a random variable, taking values in $\{0,1\}$, that is 1 with probability $\Phi(X,y)$ and $Y_{X,z}$ be a random variable, taking values in $\{0,1\}$, that is 1 with probability $\Phi(X,z)$. Also, let $\eta,\eta^{\prime}$ have the zero mean and variance $\sigma^{2}$, be such that all $X$, $\eta$, $\eta^{\prime}$ are independent random variables. We assume that under the condition that $X$ is given, random variables $Y_{X,y}$, $Y_{X,z}$, $\eta$, and $\eta^{\prime}$ are independent.

Lemma 3.3**.**

Let $y,z\in M.$ We have

[TABLE]

Proof. We denote $R_{y}(X)=d_{M}(y,X)+\eta$ and $R_{z}(X)=d_{M}(z,X)+\eta^{\prime}.$ Then

[TABLE]

$\square$

3.1. Deterministic estimates for the rough distance function

In this subsection, we consider the rough distance function $k_{\Phi}(y,z)$.

In the study of metric spaces, Kuratowski observed that the map $R(x)=r_{x}$ defines an isometric embedding $R:M\to R(M)\subset C(M)$ of the manifold $M$ into the vector space $C(M)$. When there are no missing data, that is, $\Phi=1$, the following proposition show that the map $\overline{R}:M\to L^{2}(M)$, given by $\overline{R}(x)=r_{x}$ defines a bi-Lipschitz embedding $\overline{R}:M\to\overline{R}(M)\subset L^{2}(M)$. Note that here $L^{2}(M)$ is the space $L^{2}(M,d\mu)$, where $\mu$ is a probability measure on $M$.

Proposition 3.4**.**

There is a constant $c_{5}\in(0,1)$ such that

[TABLE]

Due to this, we call the map $\overline{R}:M\to L^{2}(M)$ the $L^{2}$-Kuratowski embedding. The proof of the Proposition 3.4 is the special case of the Proposition 3.5 when $\Phi=1$, claims (i)-(ii), given below.

Proposition 3.5**.**

(i) We have for all $y,z\in M$ the inequality

[TABLE]

(ii) Let $\widehat{c}_{4}=\frac{1}{4}\min(c_{2}Hr_{1},c_{4}).$ There is $c_{5}\leq 1$ such that if $A(y,z)\geq\widehat{c}_{4}$ then

[TABLE]

(iii) For all $y,z\in M$ satisfying $d_{M}(y,z)\leq r_{1}$, where $r_{1}$ is defined in (1.20), we have $A(y,z)\geq c_{4}\geq\widehat{c}_{4},$ and so the inequality (3.6) is valid.

By this proposition, if $A(y,z)\geq\widehat{c}_{4}$ then $k_{\Phi}(y,z)^{1/2}$ approximates $d_{M}(y,z)$.

Proof. (i) We have by triangular inequality

[TABLE]

As $A(y,z)\leq 1$, this proves the inequality (3.5).

(ii) To prove the inequality in (3.6), we use the following (well known) corollary of Toponogov’s theorem. Similar kind of formulas are used in Section 4.5 of [5]. However, we present the results in the form needed later and give the proof for the convenience of the reader.

Lemma 3.6**.**

*Let $M$ be a Riemannian manifold with sectional curvature bounded below by $-\Lambda^{2}$. Let $x,y,z\in M$ and $\beta=\angle xyz$ be the angle of the length minimizing curves $[xy]$ and $[yz]$ at $y$. Assume that $d_{M}(y,z)\leq\frac{1}{2}d_{M}(x,y)$ and $d_{M}(x,y)\leq\frac{2}{3}\min(i_{0},\pi/(2\Lambda))$. Then *

[TABLE]

Proof. Let $\gamma_{y,\xi}([0,\ell])$ be a distance minimizing geodesic from $y$ to $z$, where $|\xi|=1$ and $\ell=d_{M}(y,z)$. Consider functions

[TABLE]

Let $\ell_{0}=\min(i_{0},\pi/(2\Lambda))$. Observe that then $d_{M}(\gamma_{y,\xi}(s),x)\leq\ell_{0}$ for all $s\in[0,\ell]$.

The gradient of $F(p)$ at $p\in B(0,\ell_{0})$ is equal to the normal vector $\nu$ of the sphere $\Sigma=\partial B(x,r)$, where $r=d_{M}(p,x)$, at the point $p$ and the Hessian of $F$ at $p$ and the shape operator $S(p)$ of the sphere $\Sigma$ have the relation Hess$(F)(\xi,\eta)=g(S(p)\xi,\eta)$, $\xi,\eta\in T_{p}M$, where $g:T_{p}M\times T_{p}M\to{\mathbb{R}}$ is the quadratic form determined by the metric tensor $g$, see [45]. By the standard comparision estimates [45, Ch. 6, Thm. 2.1], in the space $T_{p}\Sigma$ we have

[TABLE]

As $\frac{d}{dt}(\tan(t))=1/\cos^{2}(t)$, the mean value theorem implies that for $0<s<\pi/(2\Lambda)$ we have $\tan(\Lambda s)\geq\Lambda s,$ so that $\|S(p)\|\leq 1/F(p).$

Since $\partial_{s}f(s)=g(\nabla F(\gamma_{y,\xi}(s)),\dot{\gamma}_{y,\xi}(s))=g(\nu(\gamma_{y,\xi}(s)),\dot{\gamma}_{y,\xi}(s)),$ where $\nu(x)=\nabla F(x)$ is the normal of the sphere $\partial B(y,s)$ at the point $x=\gamma_{y,\xi}(s)$. Moreover, $f(0)=d_{M}(x,y)$ and $\partial_{s}f(0)=g(-\dot{\gamma}_{y,\xi}(s),\dot{\gamma}_{y,\xi}(s))=-\cos\beta.$ Also, since

[TABLE]

where $\gamma=\gamma_{y,\xi}$, we have

[TABLE]

Hence, using Taylor’s series we see that

[TABLE]

This proves the claim. $\square$

Next we continue the proof of inequality (3.6) We consider the claim in two cases:

Case 1. Assume that $d_{M}(y,z)\geq r_{1}/16$. We show that there $c_{5}^{\prime}>0$ such that

[TABLE]

Proof in Case 1. Assume that $0<d_{M}(y,z)<r_{1}/16$. Then, if $x\in M$ is such that $r_{1}/2<d_{M}(x,z)<r_{1}$, we have $d_{M}(x,y)\geq r_{1}/4$ and by (3.7),

[TABLE]

where $\beta$ is the angle $\angle xyz$. When $\beta<\pi/4$, this yields

[TABLE]

Thus, let

[TABLE]

Then

[TABLE]

where by [45, Cor. 2.4 in Chapter 6.2], see also (1.19), (1.20), there is $c_{3}^{\prime}=c_{3}^{\prime}(n,\Lambda)>0$ such that

[TABLE]

Thus there exists $c_{5}^{\prime}$ such that (3.9) is valid.

Case 2. Assume that $d_{M}(y,z)\geq r_{1}/16$. Then we show that there $c_{5}^{\prime\prime}$ such that

[TABLE]

Proof in Case 2. The assumption $d_{M}(y,z)\geq r_{1}/16$ and definition of $\widehat{c}_{4}$ imply that $4\widehat{c}_{4}c_{2}^{-1}H^{-1}\leq r_{1}$ so that $d_{M}(y,z)\geq\frac{1}{4}\widehat{c}_{4}c_{2}^{-1}H^{-1}$. Denote $d_{M}(y,z)=\ell$. Then $\ell\geq 2a$, where $a=\frac{1}{8}\widehat{c}_{4}c_{2}^{-1}H^{-1}.$

Assume that $A(y,z)\geq\widehat{c}_{4}$. Since $d_{M}(y,x)+d_{M}(x,z)\geq d_{M}(y,z)=\ell$, we see that for all $x\in M$ we have either $d_{M}(x,y)\geq\ell/2$ or $d_{M}(x,z)\geq\ell/2$. Let us assume that the latter is true. As $s\mapsto\Phi^{1}(s)$ is non-increasing and $\Phi$ takes values in $[0,1]$, we have $\Phi_{y}(x)\Phi_{z}(x)\leq\Phi_{z}(x)\leq c_{2}\Phi^{1}(d_{M}(x,z))\leq c_{2}\Phi^{1}(\ell/2).$ This yields that

[TABLE]

so that $\Phi^{1}(\ell/2)\geq\widehat{c}_{4}/c_{2}$.

Let $[yz]=\gamma_{y,\xi}([0,\ell])$, $\xi\in S_{y}M$ be a length minimizing geodesic from $y$ to $z$. Let $q=\gamma_{y,\xi}(\ell/2)$, $p=\gamma_{y,\xi}(\ell/2-a)$, and $r=a/2$. When $d_{M}(x,p)<r$, we have

[TABLE]

so that $d_{M}(z,x)-d_{M}(y,x)\geq a.$ Recall that $\Phi(x,y)\geq c_{1}\Phi^{1}(d_{M}(x,y))$, $\Phi^{1}:[0,\infty)\to[0,1]$ and $\|\Phi^{1}\|_{C^{1}({\mathbb{R}})}\leq H$. As $\Phi^{1}_{y}(q)=\Phi^{1}_{z}(q)=\Phi^{1}(\ell/2)\geq\widehat{c}_{4}/c_{2}$ and $B_{M}(p,r)\subset B_{M}(q,\frac{3}{2}a)$, we have for all $x\in B_{M}(p,r)$

[TABLE]

Then, as $\ell/D\leq 1$,

[TABLE]

where $c_{5}^{\prime\prime}=(16^{-n-3}{\rho_{min}}c_{1}^{2}c_{2}^{-n+4}\widehat{c}_{3}H^{-(n+2)}(\widehat{c}_{4})^{n+4}D^{-2})^{1/2}$. This yields (ii) with $c_{5}=\min(c_{5}^{\prime},c_{5}^{\prime\prime})$.

(iii) As $c_{1}\leq 1$, Lemma 3.2 implies that if $d_{M}(y,z)\leq r_{1}$, then $A(y,z)\geq c_{4}\geq\widehat{c}_{4}.$ $\square$

3.2. Probabilistic estimates

for the rough distance function

In this subsection, we consider the rough distance function $k_{\Phi}(X_{j},X_{k})^{1/2}$.

Next we determine approximately $k_{\Phi}(X_{j},X_{k})=\|(r_{X_{j}}-r_{X_{k}})\Phi^{1/2}_{X_{j}}\Phi^{1/2}_{X_{k}}\|_{L^{2}(M,d\mu)}^{2}$ for $(j,k)\in{\mathcal{I}}^{(0)}\times{\mathcal{I}}^{(1)}$ using averaging over the data on the “densest net” $S_{2}=\{X_{j}:\ j\in{\mathcal{I}}^{(2)}\}$.

For $(j,k)\in I^{(0)}\times I^{(1)}$, we consider the random variables

[TABLE]

We also consider the random variables

[TABLE]

Roughly speaking, below $T_{jk}$ measures how well $K_{jk}^{L}$ approximates $k_{\Phi}(X_{j},X_{k})$ that further approximates $d_{M}(X_{j},X_{k})^{2}$.

3.2.1. Probabilistic notations

To introduce some notations, let us assume for simplicity that the complete probability space $(\Omega,\Sigma,\mathbb{P})$ can be represented as a product $\Omega=\Omega_{1}\times\Omega_{2}\times\Omega_{3}$ such that $\mathbb{P}=\mathbb{P}_{1}\times\mathbb{P}_{2}\times\mathbb{P}_{3}$ and $\omega=(\omega_{1},\omega_{2},\omega_{3})\in\Omega_{1}\times\Omega_{2}\times\Omega_{3}$. We assume that $\eta=\eta(\omega_{1})$, $\eta=\eta^{\prime}(\omega_{2})$ are random variables with variance $\sigma$, and $X=X(\omega_{3})$. We assume that $X$ is a random variables having distribution $\mu$. Assume that $X$, $\eta$ and $\eta^{\prime}$ are independent.

For an integrable function $F(\omega_{1},\omega_{2},\omega_{3})=f(\eta(\omega_{1}),\eta^{\prime}(\omega_{2}),X(\omega_{3}))$ we denote

[TABLE]

As $X$ is a random variable, we have that also $\mathbb{E}_{\eta,\eta^{\prime}}f(\eta,\eta^{\prime},X)$ is a random variable. The expectation $\mathbb{E}_{\eta,\eta^{\prime}}f(\eta,\eta^{\prime},X)$ over variables ${\eta,\eta^{\prime}}$ is function of $X$, and thus it can be considered as the expectation of $f(\eta,\eta^{\prime},X)$ under the condition that $X$ is known.

Below, we consider conditional expectations using $\sigma$-algebras. Let ${\mathcal{B}}_{X}\subset\Sigma$ be $\sigma$-algebra generated by the random variable $X:\Omega\to{\mathbb{R}}$, that is, the $\sigma$-algebra generated by sets $X^{-1}(S)\subset\Omega$, where $S\subset{\mathbb{R}}$ is an open set, see [31, Ch. 5]. We recall that $\mathbb{E}(F|\mathcal{B}_{X})(\omega)$ is the $\mathcal{B}_{X}$-measurable random variable that satisfies

[TABLE]

for all sets $S\in\mathcal{B}_{X}$. In formula (3.14), $\mathbb{E}_{\eta,\eta^{\prime}}F$ is in fact equal to the conditional expectation $\mathbb{E}(F|{\mathcal{B}_{X}})=\mathbb{E}(F|{\mathcal{B}_{X}})(\omega)$ of the random variable $F$ with respect to the $\sigma$-algebra $\mathcal{B}_{X}\subset\Sigma$, that is, we have $(\mathbb{E}_{\eta,\eta^{\prime}}F)(\omega_{3})=\mathbb{E}(F|{\mathcal{B}_{X}})(\omega)$ with $\omega=(\omega_{1},\omega_{2},\omega_{3}).$ As $X$ is a random variable, $\mathbb{E}(Z\,|{\mathcal{B}}_{X})$ is a random variable, too. Recall also the notation $\mathbb{P}(A\,|{\mathcal{B}}_{X})=\mathbb{E}(1_{A}\,|{\mathcal{B}}_{X})$ for an event $A\in\Sigma$, where $1_{A}(\omega)$ is the indicator function of the set $A\subset\Omega$. Below we use several times the fact that

[TABLE]

Below, we will consider the $\sigma$-algebra ${\mathcal{B}}_{j}\subset\Sigma$ generated by the random variable $X_{j}:\Omega\to{\mathbb{R}}$. We also consider the $\sigma$-algebra ${\mathcal{B}}_{jk}$ generated by the random variables $X_{j}$ and $X_{k}$.

By Lemma 3.3, the conditional expectation of $K_{jk}$, under the condition that $X_{j}$ and $X_{k}$ are known, satisfies \mathbb{E}\bigg{(}K_{jk}\,\bigg{|}{\mathcal{B}}_{jk}\bigg{)}=k_{\Phi}(X_{j},X_{k}) where $k_{\Phi}(X_{j},X_{k})=\|(r_{X_{j}}-r_{X_{k}})\Phi^{1/2}_{X_{j}}\Phi^{1/2}_{X_{k}}\|_{L^{2}(M,d\mu)}^{2}$ is a random variable.

3.2.2. Probabilistic estimates for rough distances $K^{L}_{jk}$ and reliability values $T_{jk}$

Below use the following form of Hoeffding’s inequality.

Lemma 3.7** (Hoeffding’s inequality [27]).**

Let $Z_{1},\dots,Z_{N}$ be $N$ i.i.d. copies of the random variable $Z$ satisfying $0\leq Z\leq L$, where $L>0$. Then, for $\varepsilon>0$, we have

[TABLE]

Below, we will show that $K^{L}_{jk}$, defined in (3.12), can be considered to be an approximation of $k_{\Phi}(X_{j},X_{k})$ which further approximates $d_{M}(X_{j},X_{k})^{2}$ when $A(X_{j},X_{k})$ is larger than a suitable threshold value. Let

[TABLE]

For $j\in{\mathcal{I}}^{(0)}$, we consider the events $\mathcal{E}_{j}^{(1)}\subset\Omega$ and $\mathcal{E}^{(1)}\subset\Omega$, defined by

[TABLE]

Below, we use a smooth cut-off functions

[TABLE]

where $\psi_{1}\in C_{0}^{\infty}({\mathbb{R}})$ satisfies $\hbox{supp }(\psi_{1})\subset(-2,2)$ and $\psi_{1}(t)=1$ for $-1\leq t\leq 1$ and $0\leq\psi_{1}(t)\leq 1$ for all $t\in{\mathbb{R}},$ $\psi_{1}(-t)=\psi_{1}(t)$, $\|\psi_{1}\|_{C^{1}({\mathbb{R}})}\leq 2$, and the function $\psi_{1}|_{{\mathbb{R}}_{+}}$ is non-increasing.

Note that if $\beta_{1}(T_{jk}(bN_{2})^{-1})>0$ then $\frac{T_{jk}}{bN_{2}}\geq 1.$ Also, if $\mathcal{E}^{(1)}_{j}$ happens and $A_{jk}\geq c_{4}$ then

[TABLE]

Moreover, if $\frac{1}{b}\,\frac{T_{jk}}{N_{2}}\geq 1,$ then (3.18) implies

[TABLE]

For $y,z\in M$, let $Y(z,y)$ be a random variable that is 1 with probability $\Phi(y,z)$ and 0 with probability $1-\Phi(y,z)$. Assume that for $y,z\in M$ the random variables $Y(y,z)$ are independent. Let

[TABLE]

As $\mathbb{E}\frac{1}{N_{2}}T(y,z)=A(y,z)$, Hoeffding’s inequality implies

[TABLE]

As $T_{jk}$ and $T(X_{j},X_{k})$ have the same distributions and $A_{jk}=A(X_{j},X_{k})$ for $j\in{\mathcal{I}}^{(0)}$ and $k\in{\mathcal{I}}^{(1)}$, inequality (3.21) implies for the conditional probability, under the condition that $X_{j}$ and $X_{k}$ are known, that

[TABLE]

Thus we have by (3.15) $\mathbb{P}(\mathcal{E}^{(1)}_{j})\geq 1-2N_{1}\exp(-2N_{2}\varepsilon_{3}^{2}).$ Hence,

[TABLE]

We recall that by Lemma 3.2 and Proposition 3.5,

[TABLE]

Lemma 3.8**.**

Let $L>2\max(D^{2},\sigma)$, $\varepsilon_{2}>0$, and $\varepsilon(L):=\beta e^{-(L^{1/2}-D)/2}(D^{2}+6\beta^{2})$ and consider the events $\mathcal{E}_{j}^{(2)}\subset\Omega$, ${j\in I^{(0)}}$, and $\mathcal{E}^{(2)}\subset\Omega$,

[TABLE]

*Then *

[TABLE]

Proof. Denote $\eta_{jk\ell}=\eta_{j\ell}-\eta_{k\ell}$ and $D_{jk\ell}=d_{M}(X_{j},X_{\ell})-d_{M}(X_{k},X_{\ell})$. Then

[TABLE]

Let ${r}=L^{1/2}$. Observe that $|D_{jk\ell}|\leq D$, so that if $|D_{jk\ell}+\eta_{jk\ell}|\geq{r}$ then $|\eta_{jk\ell}|>{r}-D$. By (3.26), $Y=e^{|\eta_{jk\ell}|}$ satisfies

[TABLE]

Thus, using the fact that $\eta_{jk\ell}$ and $D_{jk\ell}$ are independent, we see using Schwartz inequality that

[TABLE]

By Lemma 3.3, the above shows that

[TABLE]

satisfies

[TABLE]

By arguing as in (3.21)-(3.22), we see that Hoeffding’s inequality implies

[TABLE]

Using this, (3.15) and (3.27), and summing over $j\in I^{(0)}$, we obtain (3.25). $\square$

4. Determination of the approximate distances in the coarse net

Next we assume that

[TABLE]

Next we define ${Q}_{j,j^{\prime}}$ that will turn out to the approximate distances $d_{M}(X_{j},X_{j^{\prime}})$ for points $X_{j}$ and $X_{j^{\prime}}$, where $j,j^{\prime}\in I^{(0)}$, that are sufficiently close to each other.

Definition 4.1**.**

Let $\rho\in(0,1)$ satisfy (4.1). For $j,j^{\prime}\in I^{(0)}$, let

[TABLE]

In the case when ${W}_{j,j^{\prime}}$ is zero, we define ${Q}_{j,j^{\prime}}$ to be $D$. We define for $j,j^{\prime}\in I^{(0)}$

[TABLE]

Roughly speaking, above the function $\beta_{1}(T_{jk}(bN_{2})^{-1})$ measures the reliability of the terms $\psi_{\rho}(K_{jk}^{L})$ in formulas (4.3) and (4.4), and ${W}_{j,j^{\prime}}$ measures reliability of ${Q}_{j,j^{\prime}}$ in formula (4.7). The numbers $d^{app}(X_{j},X_{j^{\prime}})$ will be the final approximation for the distances $d_{M}(X_{j},X_{j^{\prime}})$ for all pairs $(X_{j},X_{j^{\prime}})$ of points that are close to each other. Observe that ${Q}_{j,j^{\prime}}$ and ${W}_{j,j^{\prime}}$ can be computed from the given data.

For technical purposes, we define deterministic (indexed with (d)) and random (indexed with (r)) functions

[TABLE]

The motivation behind defining functions $W^{(d),-}$ and $W^{(r),-}$ is that we can use Hoeffding’s inequality to estimate how close $W^{(d),-}(X_{j},X_{j^{\prime}})$ and $W^{(r),-}(X_{j},X_{j^{\prime}})$ are when $X_{j}$ and $X_{j^{\prime}}$ are known. Also, we show that we have $W_{j,j^{\prime}}\geq W^{(r),-}(X_{j},X_{j^{\prime}})$ with a large probability.

Lemma 4.2**.**

If $d_{M}(x,z)<r_{1}$, then

[TABLE]

Moreover, when we have $\Phi(x,y)\geq c_{1}\phi_{0}$ for all $x,y\in M$, the inequality (4.10) holds for all $x,y\in M$.

Proof. Let

[TABLE]

and recall that by (4.1), $\rho\leq r_{1}$.

When $d_{M}(x,y)<\rho/4$, by Lemma 3.5 (iii), we have that $A(x,y)\geq c_{4}$. Also, in the case when $\Phi(x,y)\geq c_{1}\phi_{0}\geq\phi_{1}$ for all $(x,y)\in M\times M$, we have $A(x,y)\geq c_{4}$. Thus in both cases, $\beta_{1}(A(y,x)b^{-1})=1$. Moreover, by 3.5 (i), we have $k_{\Phi}(y,x)\leq(d_{M}(y,x))^{2}<({\rho/2})^{2},$ so that $\psi_{{\rho/2}}(k_{\Phi}(y,x))=1$.

Also, by (1.21), when $d_{M}(x,z)<r_{1}=r_{0}/2$, we have $\Phi({z,x})\geq\phi_{1}$.

Thus, when $d_{M}(x,z)<r_{1}$ or $\Phi(x,y)\geq c_{1}\phi_{0}\geq\phi_{1}$, and we have $d_{M}(x,y)<\rho/4$, it holds that $w_{-}(x,y,z)\geq\phi_{1}.$ Hence we see that

[TABLE]

$\square$

Let us write for $j,j^{\prime}\in I^{(0)}$

[TABLE]

and consider the terms ${Q}^{1}_{j,j^{\prime}}$ and ${Q}^{2}_{j,j^{\prime}}$ separately.

First we will show that ${Q}_{j,j^{\prime}}^{2}$ is small with a large probability when $W_{j,j^{\prime}}\geq u_{2}$. To that end, let $h_{0}>0$ and ${\mathcal{E}}^{(3)}_{j,j^{\prime}}\subset\Omega$, $(j,j^{\prime})\in I^{(0)}\textfractionsolidus\times I^{(0)}$, and ${\mathcal{E}}^{(3)}\subset\Omega$ be the events

[TABLE]

Lemma 4.3**.**

For any $h_{0}\in(0,1)$ we have

[TABLE]

Proof. Let us next recall some basic facts: Let $a=(a_{k})_{k=1}^{N_{1}}$ satisfy $0\leq a_{k}\leq 1$ and

[TABLE]

where $\eta_{k}$ are i.i.d. variables, $\mathbb{E}\eta_{k}=0$ and $\mathbb{E}e^{|\eta_{k}|}\leq\beta$. Since $0\leq a_{k}\leq 1$, we have $a_{k}^{2}\leq a_{k}$, so that $\sum_{k=1}^{N_{1}}a_{k}^{2}\leq\sum_{k=1}^{N_{1}}a_{k},$ and

[TABLE]

Then using Jensen’s inequality for the random variable $R=e^{\eta_{k}}$ with concave function $h:[0,\infty)\to{\mathbb{R}}$, $h(s)=s^{t}$ with $t\in[0,1]$ (for which the standard Jensen’s inequality reverses), we obtain $\mathbb{E}(R^{t})\leq(\mathbb{E}R)^{t}$. This yields

[TABLE]

for $t\in[0,1]$. Hence, as $\mathbb{E}\eta_{k}=0$, the moment generating function of $\eta_{k}$, $M(t)=\mathbb{E}(\exp(t\eta_{k}))$ satisfies, by considerations involving Taylor series and the mean value theorem, $|M_{k}(t)-1|\leq c^{\prime}t^{2},\quad t\in[0,1],$ where $c^{\prime}\leq e^{2\beta}$. Then, by using the independency of random variables $\eta_{k}$, we see that the moment generating function of $V_{N_{1}}$ satisfies for $s\in[0,{{Z}}_{N_{1}}]$

[TABLE]

where we use the fact that $\sum_{k=1}^{N_{1}}(a_{k}/Z_{N_{1}})^{2}=1/S_{N_{1}}$. Similarly, by considering $-\eta_{k}$ instead of $\eta_{k}$, we see that $\mathbb{E}\exp(-sV_{N_{1}})\leq\exp(e^{2\beta}\frac{{s}^{2}}{S_{N_{1}}})\leq\exp(e^{2\beta}\frac{{s}^{2}}{Z_{N_{1}}})$.

Then we see that for $0<\lambda\leq 1$ and $s\in[0,{{Z}}_{N_{1}}]$,

[TABLE]

When $s=\lambda Z_{N_{1}}e^{-2\beta}/2$, the above implies

[TABLE]

Next we consider a fixed $j,j^{\prime}\in I^{(0)}$ and let

[TABLE]

for $k\in I^{(1)}$, $\lambda=h_{0}$, and let ${{Z}}_{N_{1}}=N_{1}{W}_{j,j^{\prime}}$ be defined analogously to (4.14). Then we see that ${Q}_{j,j^{\prime}}^{2}=V_{N_{1}}$, where $V_{N_{1}}$ is be defined analogously to (4.14) with $\eta_{k}=\eta_{k,j^{\prime}}$. Note that $\eta_{k,j^{\prime}}$ are independent of variables $T_{jk}$, $K^{L}_{jk}$, and $Y_{j^{\prime},k}$. Also, $V_{N_{1}}$ is a function of these variables. Let $\mathcal{B}^{*}$ be the $\sigma-$algebra generated by all random variables $T_{jk},$ $K^{L}_{jk}$, and $Y_{j^{\prime},k}$, $j,j^{\prime}\in I^{(0)}$, $k\in I^{(2)}$. As $Z_{N_{1}}$ is measurable with respect to the $\sigma-$algebra $\mathcal{B}^{*}$, by applying (3.15) and (4.16), we see

[TABLE]

By doing this analysis for all $j,j^{\prime}\in I^{(0)}$ and summing up the results, we obtain the claim. $\square$

Next we analyse ${Q}_{j,j^{\prime}}^{1}$. We assume that $L$ is so large and $\varepsilon_{2},\varepsilon_{3}$ are so small that

[TABLE]

We denote, see (4.8),

[TABLE]

Let us consider the events ${\mathcal{E}}^{(4)}_{j,j^{\prime}}\subset\Omega$ and ${\mathcal{E}}^{(4)}\subset\Omega$,

[TABLE]

Observe that

[TABLE]

We see using Hoeffding’s inequality that for $y,z\in M$ that

[TABLE]

Thus we see using (4.18) for all $j,j^{\prime}\in I^{(0)}$ and (3.15), and summing the results,

[TABLE]

Below, to shorten notations, let us denote $\mathcal{E}^{(5)}=\bigcap_{k\in\{1,2,3,4\}}\mathcal{E}^{(k)}.$

Lemma 4.4**.**

Assume the event ${\mathcal{E}}^{(5)}$ happens. Then, if (4.17) is valid, we have for all $(j,j^{\prime})\in I^{(0)}\times I^{(0)}$ the implication $({W^{(d),-}_{j,j^{\prime}}\geq u_{0})\implies(W_{j,j^{\prime}}\geq u_{2}}).$

Proof. As $\mathcal{E}^{(5)}$ happens, also the event $\mathcal{E}^{(2)}$ happens. If $0\leq s\leq\frac{1}{20}\rho^{2}$, then $\psi_{\rho}(s-\frac{1}{10^{2}}\rho^{2})=1$ and $\psi_{{\rho/2}}(s)=1$. On the other hand, if $s>\frac{1}{20}\rho^{2}$, we have $4(s-\frac{1}{10^{2}}\rho^{2})>s$. Thus, as the function $\psi_{1}|_{{\mathbb{R}}_{+}}$ is non-increasing,

[TABLE]

for all $s\geq 0$. Hence, we have

[TABLE]

Thus, as ${\mathcal{E}}^{(2)}$ happens,

[TABLE]

When $\mathcal{E}^{(5)}$ happens, also $\mathcal{E}^{(4)}$ happens, and we have the implication $(W^{(d),-}_{j,j^{\prime}}\geq u_{0})\implies(W^{(r),-}_{j,j^{\prime}}\geq u_{1}).$

Recall that $\|\beta_{1}\|_{C^{1}({\mathbb{R}})}\leq 2$. Thus, when $\mathcal{E}^{(1)}$ happens,

[TABLE]

Then

[TABLE]

This and (4.19) imply

[TABLE]

Recall that by the assumption of the claim, $\varepsilon_{3}<\frac{b}{4}u_{1}$. Computing the average of the inequalities (4.21) times $Y_{j^{\prime},k}$ over $k\in I^{(1)}$, we obtain

[TABLE]

Thus when ${\mathcal{E}}^{(5)}$ and thus ${\mathcal{E}}^{(4)}$ happen, we have

[TABLE]

$\square$

Lemma 4.5**.**

When the event ${\mathcal{E}}^{(5)}$ happens and (4.17) is valid, it holds for all $j,j^{\prime}\in I^{(0)}$ that if $W(X_{j},X_{j^{\prime}})\geq u_{2}$, then

[TABLE]

Proof. Below, in this proof assume that the event $\mathcal{E}^{(5)}$ happens. We will first show using Lemma 3.8 that if $W(X_{j},X_{j^{\prime}})\geq u_{2}$, then

[TABLE]

Consider next the indexes $(j,k)\in I^{(0)}\times I^{(1)}$ for which $\psi_{\rho}(K^{L}_{jk})>0$ and ${\beta_{1}(T_{jk}(bN_{2})^{-1})})>0$. Then $\psi_{\rho}(K^{L}_{jk})>0$ implies $K^{L}_{jk}<2\rho^{2},$ and hence, as the event $\mathcal{E}^{(2)}$ happens,

[TABLE]

Moreover, if ${\beta_{1}(T_{jk}(bN_{2})^{-1})})>0$ then $T_{jk}N_{2}^{-1}>b$. As the event $\mathcal{E}^{(5)}$ happens,we have

[TABLE]

see Proposition 3.5 and (3.16). Then by Proposition 3.5,

[TABLE]

that implies with (4.24) that

[TABLE]

We denote $v_{j,j^{\prime},k}=\beta_{1}(T_{jk}(bN_{2})^{-1})\psi_{\rho}(K_{jk}^{L})Y_{j^{\prime}k}$ and observe that if $v_{j,j^{\prime},k}>0$ then the distance $d_{M}(X_{j},X_{k})$ satisfies (4.26).

Assume next that $W_{j,j^{\prime}}\geq u_{2}$. Then, we have ${\sum_{k\in I^{(1)}}v_{j,j^{\prime},k}}=W(X_{j},X_{j^{\prime}})\geq u_{2}>0$ and

[TABLE]

This means that we can consider $Q^{1}(X_{j},X_{j^{\prime}})$ as a weighted average of distances $d_{M}(X_{k},X_{j^{\prime}})$. Hence, when the event ${\mathcal{E}}^{(5)}$ happens, we see that for all $(j,j^{\prime})\in I^{(0)}\times I^{(0)}$ such that $W_{j,j^{\prime}}\geq u_{2}$ we have

[TABLE]

where we have used (4.26) in the last inequality. By (4.17), here $\varepsilon_{2}+\varepsilon(L)<\frac{1}{100}\rho^{2}$ and the inequality (4.23) follows.

As we have assumed that the event $\mathcal{E}^{(5)}$ happens, also the event $\mathcal{E}^{(3)}$ happens, see (4.12). Thus if ${W}_{j,j^{\prime}}\geq u_{2}$, then ${Q}^{2}_{j,j^{\prime}}<h_{0}$. Combining the above, we see that

[TABLE]

$\square$

The above proposition means that if $W_{j,j^{\prime}}\geq u_{2}$, then the number ${Q}_{j,j^{\prime}}$ approximates $d_{M}(X_{j},X_{j^{\prime}})$ with a large probability when all parameters are suitably chosen.

Next we consider the proof of Theorem 2. We use below

[TABLE]

Note that $\rho\leq 1$. Then $\varepsilon(L)\leq\rho^{2}/200$, implying that $\varepsilon_{2}+\varepsilon(L)\leq\rho^{2}/100.$ Below we assume that $\varepsilon_{1}<\widehat{\varepsilon}_{1}$, where $\widehat{\varepsilon}_{1}=\min(1,8c_{5}({\phi_{1}{c_{3}}})^{-1/n}).$ Then we have $\varepsilon_{3}<\frac{1}{4}c_{4}$, see (3.16).

Below, see Lemma 2.1, let $N_{0}$ be

[TABLE]

Next we consider the probability of $\mathcal{E}^{(5)}$. We see that $\mathbb{P}(\mathcal{E}^{(5)})\geq 1-p^{(5)}$, where $p^{(5)}=p^{(1)}+p^{(2)}+p^{(3)}+p^{(4)}$. Next we consider there probabilities one by one.

Lemma 4.6**.**

There is $C_{7}>1$ such that we have $p^{(3)}<\theta/8$, when

[TABLE]

Proof. Below, we use that $t\leq-\log(1-t)$ for $0<t<1$. We see that $p^{(3)}<\theta/8$ if

[TABLE]

Next we use that $t\log t\leq t^{2}$, so that for $t>e$ we have $\log(t\log t)\leq 2\log t$. Also, recall that $N_{0}$ is given in (4.28). Then we see that

[TABLE]

where $C_{9}$ is suitable. Thus (4.30) is valid when $N_{1}\geq P_{1}$. This yields that claim. $\square$

Lemma 4.7**.**

There is $C_{10}>2C_{7}$ such that we have $p^{(4)}<\theta/8$ when

[TABLE]

Note that when $N_{1}$ is chosen as in (4.31), also the inequality (4.29) is valid.

Proof. Next we estimate $p^{(4)}=2N_{0}^{2}\exp(-2N_{1}(u_{1}/2)^{2}).$ We see that $p^{(4)}<\theta/8$ if

[TABLE]

Also, we see that

[TABLE]

where $C_{10}$ is suitably chosen. Thus (4.32) is valid when $N_{1}\geq P_{2}$. $\square$

Lemma 4.8**.**

There is $C_{12}>0$ such that $p^{(2)}<\theta/8$ when

[TABLE]

Proof. Let $N_{0}$ and $N_{1}$ are given in (4.28) and (4.31). We see that $p^{(2)}\leq\theta/8$ if

[TABLE]

We have

[TABLE]

where $C_{12}$ and $C_{13}$ are suitable. Thus (4.34) is valid when $N_{2}\geq P_{3}$. This yields the claim. $\square$

Lemma 4.9**.**

There is $C_{14}>0$ such that we have $p^{(1)}<\theta/8$ when

[TABLE]

Proof. Using (4.27), we see that the inequality

[TABLE]

is valid when $N_{2}\geq R_{4}={2^{4n+7}}{\phi_{1}^{-2}{c_{3}^{-2}}c_{4}^{-2}\rho^{-2n}}\,\log(\frac{16N_{0}N_{1}}{\theta}).$ We see that

[TABLE]

where $C_{14}$ is suitable. Thus (4.36) is valid when $N_{2}\geq P_{4}$. This yields the claim. $\square$

Next we prove Theorems 1 and 2.

Proof (of Theorem 2). We observe that when $\mathcal{E}^{(5)}$ happens, by Lemma 4.2 and Lemma 4.4, for all $X_{j}$ and $X_{j^{\prime}}$ such that $d_{M}(X_{j},X_{j^{\prime}})<r_{1}$ we have $W_{j,j^{\prime}}\geq u_{2}$.

Let $N_{0}$ and $N_{1}$ be given in (4.28) and (4.31). The conditions (4.35) and (4.33) are valid when we choose a suitable $C_{15}>1$ and

[TABLE]

Then $p^{(5)}=p^{(1)}+p^{(2)}+p^{(3)}+p^{(4)}\leq\frac{1}{2}\theta.$ This and Lemma 2.1 prove that with probability $1-\theta$ we have for all $(j,j^{\prime})\in I^{(0)}\times I^{(0)}$ that inequality (2.2) holds when $d_{M}(X_{j},X_{j^{\prime}})<r_{1}$, and the inequality (2.3) holds when $d_{M}(X_{j},X_{j^{\prime}})\geq r_{1}$.

In the case when $\Phi(x,y)\geq c_{1}\phi_{0},$ for $(x,y)\in M\times M$, we see that when the events ${\mathcal{E}}^{(5)}$ happens, Lemmas 4.2 and 4.4 yield that we will have with probability $1-\theta$ that ${W}_{j,j^{\prime}}>{u_{2}}$ for all $(j,j^{\prime})$. This implies that the inequality (2.2) holds for all pairs $(X_{j},X_{j^{\prime}})$ with $j,j^{\prime}\in\{1,2,\dots,N_{0}\}$. $\square$

Proof (of Thm. 1) Let $\varepsilon_{1}={{\delta}}^{3/2}$ and $\delta_{1}={\Lambda^{2/3}\delta^{1/2}}/20$. For $N_{0},N_{1}$ given in (4.28) and (4.31),

[TABLE]

with suitable $C_{16}$ and $C_{17}$. Moreover, $N_{2}$ satisfies (4.37) when we choose a suitable $C_{18}$ and

[TABLE]

Let $\widehat{\delta}=\varepsilon_{1}=\delta^{3/2}$ and $\widehat{r}=(\widehat{\delta}/\Lambda^{2})^{1/3}={\Lambda^{2/3}\delta^{1/2}}$. Then by Theorem 2, with probability $1-\theta$ the set $\mathcal{X}=\{X_{j}:\ j=1,2,\dots,N_{0}\}$ is a $\delta_{1}$-dense subset of $M$ and the approximate distances $\widetilde{d}(X_{j},X_{j^{\prime}})=d^{app}(X_{j},X_{j^{\prime}})$, $j,j^{\prime}\in\{1,2,\dots N_{0}\}$, see (4.7), satisfy the conditions given in Proposition 4.10 in Appendix A. Thus with probability $1-\theta$ can apply Proposition 4.10 (see also [24, Corollary 1.10]) with $\widehat{\delta}=\delta^{3/2}$ and $\widehat{r}=(\widehat{\delta}/\Lambda^{2})^{1/3}$ to construct a Riemannian manifold $(M^{*},g^{*})$ that approximates the original manifold $(M,g)$ so that the claims (1)-(3) in Theorem 1 are satisfied. $\square$

Appendix A: Reconstruction of a manifold with a small deterministic errors

Here, we give results on the reconstruction of a Riemannian manifold when one is given distances with small deterministic errors. The following result is an improvement of Corollary 1.10 in [24].

Proposition 4.10**.**

There are $C_{n}^{\prime}>0$, depending on $n$, and $c_{1}^{\prime}(n,K)>0$, depending on $n,K$, such that the following holds: Let $0<\widehat{\delta}<c_{1}^{\prime}(n,K)$, $\widehat{r}=(\widehat{\delta}/K)^{1/3}$ and $M$ be a compact $n$-dimensional manifold with $|\operatorname{Sec}(M)|\leq K$ and $\operatorname{inj}(M)>2\widehat{r}$. Let $\mathcal{X}=\{x_{j}\}_{j=1}^{N}$ be an $\widehat{r}/20$-dense subset of $M$. Moreover, let $\widetilde{d}\colon\mathcal{X}\times\mathcal{X}\to{\mathbb{R}}_{+}\cup\{0\}$ be an approximate local distance function that satisfies

[TABLE]

Then, given the values $\widetilde{d}(x_{j},x_{k})$, $j,k=1,2,\dots,N$, one can construct a compact $n$-dimensional Riemannian manifold $(M^{*},g^{*})$ such that:

1. (1)

There is a diffeomorphism $F:M^{*}\to M$ satisfying

[TABLE] 2. (2)

$|\operatorname{Sec}(M^{*})|\leq C_{n}^{\prime}K$. 3. (3)

The injectivity radius $\operatorname{inj}(M^{*})$ of $M^{*}$ satisfies

[TABLE]

Proof.

A result similar to the claim is proven in [24, Corollary 1.10] under the assumption that the set $\mathcal{X}$ is a $\widehat{\delta}$-dense subset of $M$, instead of $\widehat{r}/20$-dense as it is assumed in the claim. Moreover, by [24, Corollary 1.10], it is enough to construct numbers $\widetilde{D}_{j,k}$, $j,k=1,2,\dots,\widetilde{N}$, such that the following is true: There a $\widehat{\delta}$-net $\mathcal{Y}=\{y_{j}:\ j=1,2,\dots,\widetilde{N}\}\subset M$ such that the conditions (4.40) and (4.41) are valid for the function $\widetilde{d}^{\prime}\colon\mathcal{Y}\times\mathcal{Y}\to{\mathbb{R}}_{+}\cup\{0\}$ defined by $\widetilde{d}^{\prime}(y_{j},y_{k})=\widetilde{D}_{j,k}$.

Next we construct the required $\widehat{\delta}$-net $\mathcal{Y}\subset M$ and an approximate distance function $\widetilde{d}^{\prime}$ on $\mathcal{Y}\times\mathcal{Y}$. We assume that $c_{1}^{\prime}(n,K)$ is chosen so small that $\widehat{\delta}/\widehat{r}=K\widehat{r}^{2}<\frac{1}{150}$.

For $p\in M$ we denote by $E_{p}$ the restriction of the Riemannian exponential map $\exp_{p}$ to the $\widehat{r}$-ball in $T_{p}M$ centered at the origin. This restriction is a diffeomorphism onto the $\widehat{r}$-ball centered at $p$ in $M$. It distorts distances by at most $\frac{1}{2}\widehat{\delta}$, namely for all $u,v\in T_{p}M$ such that $|u|,|v|<\widehat{r}$ we have

[TABLE]

This inequality holds as long as $\operatorname{inj}(M)>2\widehat{r}$ and $K\widehat{r}^{2}<\pi/2$, see [24, Section 4] for a proof.

For every $p\in\mathcal{X}$, define $X_{p}=\{x\in\mathcal{X}:\widetilde{d}(p,x)<\widehat{r}/6-\widehat{\delta}\}$. By (4.40) and (4.41), $X_{p}$ is contained in the $\widehat{r}/6$-neighborhood of $p$. Define $\widetilde{X}_{p}=E_{p}^{-1}(X_{p})$, let $\widetilde{V}_{p}$ be the convex hull of $\widetilde{X}_{p}$ in $T_{p}M$, and $V_{p}=E_{p}(\widetilde{V}_{p})$. Since $\mathcal{X}$ is $\widehat{r}/20$-dense in $M$, (4.40) and (4.42) imply that for every $u\in T_{p}M$ such that $|u|<\widehat{r}/6-\widehat{r}/20-2\widehat{\delta}$ there exists $v\in\widetilde{X}_{p}$ such that $|u-v|<\widehat{r}/20+\widehat{\delta}/2$. This implies that $\widetilde{V}_{p}$ contains the ball of radius $\widehat{r}/6-2\widehat{r}/20-2\widehat{\delta}-\widehat{\delta}/2>\widehat{r}/20$ centered at the origin. (Here we use the assumption that $\widehat{\delta}/\widehat{r}<\frac{1}{150}$). Hence $V_{p}$ contains the $\widehat{r}/20$-ball centered at $p$ and therefore $\bigcup_{p\in\mathcal{X}}V_{p}=M$.

We represent points of $\widetilde{V}_{p}$ as linear combinations of points of $\widetilde{X}_{p}$ as follows. Let $X_{p}^{n}$ be the set of all $n$-tuples of points of $X_{p}$ and $\Delta^{n}$ the standard coordinate simplex in ${\mathbb{R}}^{n}$:

[TABLE]

For $\alpha=(a_{1},\dots,a_{n})\in X_{p}^{n}$ and $\tau=(t_{1},\dots,t_{n})\in\Delta^{n}$, let

[TABLE]

This defines a map $S_{p}\colon X_{p}^{n}\times\Delta^{n}\to T_{p}M$. For a fixed $\alpha=(a_{1},\dots,a_{n})\in X_{p}^{n}$, the range of $S_{p}(\alpha,\cdot)$ is a (possibly degenerate) affine simplex in $T_{p}M$ with vertices $0,E_{p}^{-1}(a_{1}),\dots,E_{p}^{-1}(a_{n})$. Since $0\in\widetilde{X}_{p}$, the union of all such affine simplices is precisely the convex hull of $\widetilde{X}_{p}$. Thus $S_{p}(X_{p}^{n}\times\Delta^{n})=\widetilde{V}_{p}$.

Fix an $\varepsilon^{\prime}$-dense finite set $\Sigma\subset\Delta^{n}$, where $\varepsilon^{\prime}=\widehat{\delta}/(3\widehat{r}\sqrt{n})$, and define $Y_{p}=E_{p}(S_{p}(X_{p}^{n}\times\Sigma))\subset M.$ Since $\widetilde{V}_{p}$ is contained in the $\widehat{r}/6$-ball, $S_{p}(\alpha,\cdot)$ is Lipschitz with Lipschitz constant $\widehat{r}\sqrt{n}/6$. Therefore $S_{p}(X_{p}^{n}\times\Sigma)$ is $\widehat{\delta}/2$-dense in $\widetilde{V}_{p}$. Hence, by (4.42), $Y_{p}$ is $\widehat{\delta}$-dense in $V_{p}$.

Now define $\mathcal{Y}\subset M$ by $\mathcal{Y}=\bigcup_{p\in\mathcal{X}}Y_{p}$. Since the sets $V_{p}$ cover $M$ and $Y_{p}$ is $\widehat{\delta}$-dense in $V_{p}$ for each $p$, $\mathcal{Y}$ is a $\widehat{\delta}$-net in $M$. The points of $\mathcal{Y}$ are indexed by triples $(p,\alpha,\tau)$ where $p\in\mathcal{X}$, $\alpha\in X_{p}^{n}$, $\tau\in\Sigma$. This index set can be enumerated algorithmically using the known data.

Our first goal is to compute approximate squared distances $Q(x,y)$ between sufficiently close pairs of points $x,y\in\mathcal{Y}$. Fix $p,q\in\mathcal{X}$ such that $\widetilde{d}(p,q)<2\widehat{r}/3-2\widehat{\delta}$ (the case $p=q$ is not excluded). By (4.40) and (4.41) we have $d_{M}(p,q)<2\widehat{r}/3-\widehat{\delta}$. Hence, by the triangle inequality, $d_{M}(x,y)<\widehat{r}-\widehat{\delta}$ for all $x\in V_{p}$ and $y\in V_{q}$, In particular $V_{q}$ is contained in the range of $E_{p}$. By (4.42),

[TABLE]

We compute the values $Q(x,y)$ for all $x\in X_{p}\cup Y_{p}$ and $y\in X_{q}\cup Y_{q}$ in several steps. First consider $x\in X_{p}$ and $y\in X_{q}$. In this case we simply define $Q(x,y)=\widetilde{d}(x,y)^{2}$. Then, by (4.40),

[TABLE]

Hence, by (4.43),

[TABLE]

Now consider $x\in Y_{p}$ and $y\in X_{q}$. By the definition of $Y_{p}$ we have $x=E_{p}(S_{p}(\alpha,\tau))$ for some $\alpha=(a_{1},\dots,a_{n})\in X_{p}^{n}$ and $\tau=(t_{1},\dots,t_{n})\in\Sigma$. We define $Q(x,y)$ using the values of $Q$ that we have from the previous step. Introduce the following notation: $a_{0}=p$, $t_{0}=1-\sum_{i=1}^{n}t_{i}$, $v_{i}=E_{p}^{-1}(a_{i})$ for $i=0,\dots,n$ (in particular $v_{0}=0$), $v=E_{p}^{-1}(x)=\sum t_{i}v_{i}$, and $w=E_{p}^{-1}(y)$. In this notation, $v-w=\sum_{i=0}^{n}t_{i}(v_{i}-w)$, hence

[TABLE]

where $\langle\,\cdotp,\,\cdotp\rangle$ is the scalar product in $T_{p}M$. With this identity in mind, we define

[TABLE]

Since $y\in X_{q}$ and $a_{i},a_{j}\in X_{p}$, the values $Q(a_{i},y)$, $Q(a_{k},y)$ and $Q(a_{i},a_{j})$ are defined in the previous step (in the case of $Q(a_{i},a_{j})$, this is the previous step with $q=p$). Since $\sum_{i=0}^{n}t_{i}=1$, applying (4.45) to the terms in the right-hand side of (4.46) yields that

[TABLE]

Therefore, by (4.43),

[TABLE]

We now have the values $Q(x,y)$ satisfying (4.48) for all $x\in Y_{p}$ and $y\in X_{q}$. Exchanging the roles of $p$ and $q$ we similarly find $Q(x,y)$ for all $x\in X_{p}$ and $y\in Y_{q}$.

Finally, consider $x\in Y_{p}$ and $y\in Y_{q}$. Again, let $x=E_{p}(S_{p}(\alpha,\tau))$ where $\alpha=(a_{1},\dots,a_{n})\in X_{p}^{n}$ and $\tau=(t_{1},\dots,t_{n})\in\Sigma$, and define $a_{0}=p$ and $t_{0}=1-\sum_{i=1}^{n}t_{i}$. From the previous steps we already have values $Q(a_{i},y)$ and $Q(a_{i},a_{j})$. Therefore we can define $Q(x,y)$ by the same formula (4.46). Then, starting from (4.48) instead of (4.44), we obtain the same estimates as above but with different constants: (4.45) with $7\widehat{\delta}\widehat{r}$ in the right-hand side, (4.47) with $11\widehat{\delta}\widehat{r}$ in the right-hand side, and finally (4.48) with $12\widehat{\delta}\widehat{r}$ in the right-hand side:

[TABLE]

Now one might take the square root of $Q(x,y)$ as an approximate distance between $x$ and $y$; however this approximation is not good enough. For a better one, we use an algorithm described in [24, §2.4] to construct a map $F\colon Y_{p}\cup Y_{q}\to{\mathbb{R}}^{n}$ that preserves distances up to an error $O(\widehat{\delta})$. Let us outline how the algorithm works in the present set-up.

First define an approximate scalar product $P(x,y)$ for all pairs $x,y\in Y_{p}\cup Y_{q}$ by

[TABLE]

By (4.43), (4.49) and the Euclidean identity $\langle u,v\rangle=\tfrac{1}{2}(|u|^{2}+|v|^{2}-|u-v|^{2})$ for $u,v\in T_{p}M$, this approximates the scalar product of $E_{p}^{-1}(x)$ and $E_{p}^{-1}(y)$ in $T_{p}M$:

[TABLE]

Then, since $E_{p}^{-1}(X_{p})$ is a $\widehat{\delta}/2$-net in $E_{p}^{-1}(V_{p})$ and the latter contains the $\widehat{r}/6$-ball centered at the origin, we can find points $a_{1},\dots,a_{n}\in Y_{p}$ such that the vectors $v_{i}:=E_{p}^{-1}(a_{i})$, $i=1,\dots,n$, approximate an orthonormal basis of $T_{p}M$ rescaled by the factor $\widehat{r}/6$:

[TABLE]

where $\delta_{ij}$ is the Kronecker delta and $C_{1}=C_{1}(n)>0$ is a suitable constant. (A straightforward modification of the algorithm from [24, §2.4] can be used to find such points efficiently).

The inequalities (4.51) imply that the linear map $L\colon T_{p}M\to{\mathbb{R}}^{n}$ defined by

[TABLE]

is $(C_{2}\widehat{\delta}/\widehat{r})$-close in the operator norm to a linear isometry from $T_{p}M$ to ${\mathbb{R}}^{n}$ for some constant $C_{2}=C_{2}(n)>1$, see [24, Lemma 2.6]. Hence $L$ distorts distances within the $\widehat{r}$-ball by at most $2C_{2}\widehat{\delta}$. We approximate $L\circ E_{p}^{-1}$ by a map $F\colon Y_{p}\cup Y_{q}\to{\mathbb{R}}^{n}$ defined by

[TABLE]

and compute $\widetilde{d}^{\prime}(x,y)=|F(x)-F(y)|$ for all $x,y\in Y_{p}\cap Y_{q}$. By (4.50) we have $|F(x)-L(E_{p}^{-1}(x))|<120\sqrt{n}\,\widehat{\delta}$ for all $x,y\in Y_{p}\cap Y_{q}$. Hence, by (4.42) and the above mentioned property of $L$,

[TABLE]

where $C_{4}=2C_{2}+120\sqrt{n}+1$.

The domain of the function $\widetilde{d}^{\prime}$ defined by the above procedure includes all pairs $x,y\in\mathcal{Y}$ such that $d_{M}(x,y)<\widehat{r}/4$. Indeed, if $d_{M}(x,y)<\widehat{r}/4$ and $p,q\in X$ are such that $x\in Y_{p}$, $y\in Y_{q}$, then by the triangle inequality we have $d_{M}(p,q)<\widehat{r}/4+2\widehat{r}/6<2\widehat{r}/3-3\widehat{\delta}$, and hence, by (4.40), $\widetilde{d}(p,q)<2\widehat{r}/3-2\widehat{\delta}$. Thus for any such pair $x,y$ the value $\widetilde{d}^{\prime}(x,y)$ is defined and satisfies (4.52).

To finish the construction, set $\widetilde{d}^{\prime}(x,y)=\widehat{r}$ for all remaining pairs $x,y\in\mathcal{Y}$. Now the function $\widetilde{d}^{\prime}$ is defined on $\mathcal{Y}\times\mathcal{Y}$ and it satisfies the assumptions [24, Corollary 1.10] for $\widehat{r}^{\prime}=\widehat{r}/4$ in place of $\widehat{r}$, $\widehat{\delta}^{\prime}=C_{4}\widehat{\delta}$ in place of $\widehat{\delta}$, and $K^{\prime}=\widehat{\delta}^{\prime}/(\widehat{r}^{\prime})^{3}=2^{6}C_{4}K$ in place of $K$. Applying [24, Corollary 1.10] with these modified parameters finishes the proof of Proposition 4.10. ∎

The constructions in Prop. 4.10 and [24, Corollary 1.10] are algorithmic, for the details, see [24].