Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator

Abstract

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m-dimensional submanifold ${\mathcal {M}}$ in $\mathbb {R}^d$ as the sample size n increases and the neighborhood size h tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of $O\Big (\big (\frac{\log n}{n}\big )^\frac{1}{2m}\Big )$ to the eigenvalues and eigenfunctions of the weighted Laplace–Beltrami operator of ${\mathcal {M}}$. No information on the submanifold ${\mathcal {M}}$ is needed in the construction of the graph or the “out-of-sample extension” of the eigenvectors. Of independent interest is a generalization of the rate of convergence of empirical measures on submanifolds in $\mathbb {R}^d$ in infinity transportation distance.

Appendices

Proofs of Propositions in Sect. 1.6

Proof

(of Proposition 1) The first claim follows immediately from (1.33). To deduce the second part, let $q_1, q_2 \in B_{\mathcal {M}}(p,\frac{r}{2})$. Consider a smooth curve $\tilde{\gamma }:[0,1] \rightarrow {\mathcal {M}}$ connecting $q_1$ and $q_2$, i.e., $\tilde{\gamma }(0)=q_1$ and $\tilde{\gamma }(i)=q_2$. We observe that if $\tilde{\gamma }$ is not contained in $B_{\mathcal {M}}(p,r)$, then

$$\begin{aligned} d(q_1,q_2) \le d(q_1,p) + d(q_2,p) < r \le {{\,\mathrm{Length}\,}}(\tilde{\gamma }). \end{aligned}$$

In fact, to deduce that $r \le {{\,\mathrm{Length}\,}}(\tilde{\gamma })$ let $s \in (0,1)$ be such that $\tilde{\gamma }(s) \not \in B_{\mathcal {M}}(p,r)$. It is straightforward to see that the length of the restriction of $\tilde{\gamma }$ to the interval [0, s] is larger than the distance between $\tilde{\gamma }(s)$ and $\partial B_{\mathcal {M}}(p,\frac{r}{2})$, which in turn is larger than $\frac{r}{2}$. Similarly, the length of the restriction of $\tilde{\gamma }$ to the interval [s, 1] is larger than $\frac{r}{2}$. Hence, $r \le {{\,\mathrm{Length}\,}}(\tilde{\gamma })$ as desired.

Now, let $\tilde{\gamma }$ be a smooth curve realizing the distance between $q_1$ and $q_2$ (which after appropriate normalization has to be a geodesic). From the previous observation, we see that $\tilde{\gamma }$ is contained in $B_{\mathcal {M}}(p, r)$. Consider $\gamma :=\exp _p^{-1} \circ \tilde{\gamma }$, where we note that $\exp ^{-1}_p$ is well defined along $\tilde{\gamma }$ given that $r \le i_0$. From the first part of the proposition, we deduce that

$$\begin{aligned} \frac{1}{2} d(\exp _p^{-1}(q_1), \exp _p^{-1}(q_2)) \le \frac{1}{2} {{\,\mathrm{Length}\,}}(\gamma ) \le {{\,\mathrm{Length}\,}}(\tilde{\gamma }) = d(q_1,q_2). \end{aligned}$$

Finally, for an arbitrary smooth curve $\gamma :[0,1] \rightarrow B(r) \subseteq T_p{\mathcal {M}}$ with $\gamma (0)=\exp ^{-1}_p(q_1)$ and $\gamma (i)= \exp ^{-1}_p(q_2)$ we have

$$\begin{aligned} d(q_1,q_2) \le {{\,\mathrm{Length}\,}}(\exp _p \circ \gamma ) \le 2 {{\,\mathrm{Length}\,}}(\gamma ). \end{aligned}$$

Taking the infimum on the right-hand side over all such curves $\gamma$, we deduce that $d(q_1,q_2) \le 2 d(\exp ^{-1}_p(q_1),\exp ^{-1}_p(q_2))$. This completes the proof. $\square$

Proof

(Proof of Proposition 2) The inequality $\left|x-y \right| \le d(x,y)$ is trivial. To show the other inequality, we note that since $\left|x-y \right| \le \frac{R}{2}$, it follows from [19, Prop 6.3] that

$$\begin{aligned} d(x,y) \le R- R\sqrt{1- \frac{2\left|x-y \right|}{R}}. \end{aligned}$$

Using the fact that for every $t \in [0,1]$, $\sqrt{1-t}\ge 1- \frac{1}{2} t - \frac{1}{2} t^2$

$$\begin{aligned} d(x,y)\le & {} R- R\left( 1- \frac{\left|x-y \right|}{R} - \frac{2}{R^2} \left|x-y \right|^2 \right) \nonumber \\= & {} \left|x-y \right|+ \frac{2}{R}\left|x-y \right|^2 \le 2 |x-y|. \end{aligned}$$

(A.1)

To improve the error estimate, let $L=d(x,y)$ and let $\gamma :[0,L] \rightarrow {\mathcal {M}}$ be an arc-length-parameterized length-minimizing geodesic between x and y. Heuristically, $\gamma$ is a “straight” line in ${\mathcal {M}}$, and thus, its curvature in $\mathbb {R}^d$ is bounded by the maximal principal curvature of ${\mathcal {M}}$ in $\mathbb {R}^d$, which is bounded by $\frac{1}{R}$. More precisely, we claim that

$$\begin{aligned} \left|\ddot{\gamma }(t) \right| \le \frac{1}{R} \qquad \text {for all } t \in [0, L]. \end{aligned}$$

(A.2)

This statement follows from [19, Prop 6.1] (and is used in the proof of Proposition 6.3 of [19]). Using translation, we can assume that $x=0$. Furthermore, note that that ${\dot{\gamma }}(t) \cdot \ddot{\gamma }(t)=0$ for all t. Thus,

$$\begin{aligned} \begin{aligned} \left|x-y \right| = \left|\gamma (L) \right|&\ge \gamma (L) \cdot {\dot{\gamma }}(L) = \int _0^L {\dot{\gamma }}(s) \cdot {\dot{\gamma }}(L) \hbox {d}s \\&= \int _0^L \left( {\dot{\gamma }}(L) - \int _s^L \ddot{\gamma }(r) \hbox {d}r \right) \cdot {\dot{\gamma }}(L) \,\hbox {d}s \\&= L - \int _0^L \int _s^L \int _r^L \ddot{\gamma }(r) \cdot \ddot{\gamma }(z) \hbox {d}z \hbox {d}r \hbox {d}s \ge L - \frac{L^3}{R^2} \end{aligned} \end{aligned}$$

(A.3)

Combining with (A.1) implies $L \le |x-y| + \frac{8}{R^2} |x-y|^3$. $\square$

Kernel-Density Estimates via Transportation

Here, we use the estimates on infinity transportation distance established in Sect. 2 to show the kernel density estimates we need. While the estimates we prove are not optimal, they do not affect the rate of convergence of eigenvalues and eigenfunctions in our main theorems. We chose to present the proof as follows as it highlights how the optimal transportation estimates can be used to provide general kernel-density estimates in a simple and direct way.

Lemma 18

Consider $\eta : \mathbb {R}\rightarrow \mathbb {R}$, nonincreasing, supported on [0, 1], and normalized: $\int _{\mathbb {R}^m} \eta (\left|x \right|) \hbox {d}x =1$. Consider $h>0$ satisfying Assumption 3. Then, (1.12) holds. That is, there exists a universal constant $C>0$ such that

$$\begin{aligned} \max _{i=1, \dots , n} |m_i - p({\mathbf {x}}_i) |\le CL_p h + C \alpha \eta (0) m \omega _m \frac{\varepsilon }{h} + C \alpha m \left( K + \frac{1}{R^2} \right) h^2 , \end{aligned}$$

(B.1)

where $\varepsilon$ is the $\infty$-OT distance between $\mu _n$ and $\mu$ (see Sect. 2).

The weights $\mathbf {m}$ are defined by

$$\begin{aligned} m_i = \frac{1}{n h^m } \sum _{j=1}^n \eta \left( \frac{|x_i - x_j |}{h} \right) , \quad i =1 , \dots ,n , \end{aligned}$$

p is the density of $\mu$ with respect to ${\mathcal {M}}$’s volume form. We remark that we do not require $\eta$ to be Lipschitz on [0, 1].

Proof

First, notice that for every i, j with $|x_i - x_j |\le h$ we have $|x_i - x_j |\le \frac{R}{2}$, and hence, Proposition 2 implies that

$$\begin{aligned} d(x_i, x_j) \le |x_i - x_j |+ \frac{8}{R^2} |x_i - x_j |^3 \le \left( 1+ \frac{8 h^2}{R^2} \right) |x_i- x_j |. \end{aligned}$$

Therefore, for every i, j and every $y \in U_j$,

$$\begin{aligned} \eta \left( \frac{|x_i - x_j |}{ h} \right) \le \eta \left( \frac{d (x_i , x_j) }{ {{\hat{h}}}} \right) \le \eta \left( \frac{(d( x_i, y) - \varepsilon )_+ }{ {\hat{h}}} \right) , \end{aligned}$$

where we recall that $\varepsilon$ is the $\infty$-OT distance between $\mu _n$ and $\mu$ and where ${\hat{h}}:= h+ \frac{27 h^3}{R^2}$. From this, it follows that

$$\begin{aligned} \begin{aligned} m_i = \frac{1}{n h^m} \sum _{j=1}^{n} \eta \left( \frac{|{\mathbf {x}}_i - {\mathbf {x}}_j |}{h} \right)&\le \frac{1}{ h^m} \int _{{\mathcal {M}}} \eta \left( \frac{ (d( x_i, y) -\varepsilon )_+}{ {{\hat{h}}}} \right) p(y) \hbox {dVol}(y)\\&\le (p(x_i) + 10L_p h) \frac{1}{ h^m} \\&\quad \int _{{\mathcal {M}}} \eta \left( \frac{(d (x_i , y ) -\varepsilon )_+}{{{\hat{h}}}} \right) \hbox {dVol}(y), \end{aligned} \end{aligned}$$

(B.2)

where the last inequality follows using the Lipschitz continuity of p, the fact that $\varepsilon < h$ and the fact that $h < \frac{R}{2}$ (so that in particular ${\hat{h}} + \varepsilon < 10 h$). Now,

$$\begin{aligned} \begin{aligned} \frac{1}{ h^m} \int _{{\mathcal {M}}} \eta \left( \frac{ ( d(x_i , y) -\varepsilon )_+}{{{\hat{h}}}} \right) \hbox {dVol}(y)&= \frac{1}{ h^m} \int _{B( {\hat{h}} + \varepsilon )} \eta \left( \frac{(|z |-\varepsilon )_+}{{\hat{h}}} \right) J_{x_i}(z) \hbox {d}z\\&\le \frac{1}{ h^m} \int _{B( {\hat{h}} + \varepsilon )} \eta \left( \frac{(|z |-\varepsilon )_+}{{{\hat{h}}}} \right) J_{x_i}(z) \hbox {d} z\\&\le (1+ C mK h^2) \frac{1}{ h^m}\\&\quad \int _{B( {\hat{h}} + \varepsilon )} \eta \left( \frac{(|z |-\varepsilon )_+}{ {{\hat{h}}}} \right) \hbox {d} z, \end{aligned} \end{aligned}$$

(B.3)

where C is a universal constant. The last integral above can be estimated as follows

$$\begin{aligned} \begin{aligned} \frac{1}{h^m}\int _{\mathbb {R}^m} \eta \left( \frac{(|z |- \varepsilon )_+}{ {{\hat{h}}}} \right) \hbox {d}z&= \eta (0) \omega _m \frac{\varepsilon ^m}{h^m} + \frac{1}{h^m}\int _{B( {\hat{h}}+\varepsilon ) \setminus B( \varepsilon )} \eta \left( \frac{|z |-\varepsilon }{ {{\hat{h}}}} \right) \hbox {d}z\\&= \eta (0) \omega _m \frac{\varepsilon ^m}{h^m} + \frac{{\hat{h}}^m}{h^m}\int _{0}^{1} m \omega _m \left( r+ \frac{\varepsilon }{{{\hat{h}}}} \right) ^{m-1} \eta \left( r \right) \hbox {d}r\\&\le \eta (0) \omega _m \frac{\varepsilon ^m}{h^m} + \left( 1 + \frac{16 mh^2 }{R^2}\right) \\&\quad \int _{0}^{1} m \omega _m \left( r+ \frac{\varepsilon }{ h} \right) ^{m-1} \eta \left( r \right) \hbox {d}r \end{aligned} \end{aligned}$$

(B.4)

Using the binomial theorem, we obtain

$$\begin{aligned} \begin{aligned} m \omega _m\! \int _{0}^1 \! \left( r+ \frac{\varepsilon }{h}\right) ^{m-1} \!\eta (r)\hbox {d}r&\le m \omega _m \int _{0}^1 \! r^{m-1} \eta (r) \hbox {d}r + m \omega _m \eta (0) \\&\quad \sum _{k=1}^{m-1} \left( {\begin{array}{c}m-1\\ k\end{array}}\right) \left( \frac{\varepsilon }{h} \right) ^k \!\frac{1}{m-k}\\&= 1 + \omega _m \eta (0) \sum _{k=1}^{m-1} \left( {\begin{array}{c}m\\ k\end{array}}\right) \left( \frac{\varepsilon }{h} \right) ^k\\&= 1+ \omega _m \eta (0) \left( \left( 1+ \frac{\varepsilon }{h} \right) ^{m} - 1 - \frac{\varepsilon ^m}{h^m} \right) \\&\le 1+ 2 m\eta (0)\omega _m\frac{\varepsilon }{h} - \eta (0) \omega _m\frac{\varepsilon ^m}{h^m} \end{aligned} \end{aligned}$$

where in the first equality we have used the fact that $\eta$ was assumed to be normalized, and in the last inequality, we have used

$$\begin{aligned} (1+s)^m \le 1 + 2ms \quad \text { whenever } 0 \le s < \frac{1}{m}. \end{aligned}$$

Combining (B.2), (B.3) and (B.4), we conclude that

$$\begin{aligned} m_i - p(x_i) \le p(x_i) + CL_p h + C \alpha \eta (0) m \omega _m \frac{\varepsilon }{h} +C\alpha m \left( K + \frac{1}{R^2} \right) h^2, \end{aligned}$$

for a universal constant $C>0$.

In a similar fashion, we can find an upper bound for $p(x_i) -m_i$. Indeed, observe that for every i, j and $y\in U_i$ we have

$$\begin{aligned} \eta \left( \frac{|x_i - x_j |}{h} \right) \ge \eta \left( \frac{d(x_i, x_j) }{h} \right) \ge \eta \left( \frac{d(x_i, y) + \varepsilon }{h} \right) \end{aligned}$$

and so

$$\begin{aligned} \begin{aligned} m_i&\ge \frac{1}{h^m} \int _{{\mathcal {M}}} \eta \left( \frac{d(x_i, y) + \varepsilon }{h} \right) p(y) \hbox {dVol}(y)\\&\ge \frac{1}{h^m} \int _{{\mathcal {M}}} \eta \left( \frac{d(x_i, y) + \varepsilon }{h} \right) (p(x_i)- L_p d(x_i, y)) \hbox {dVol}(y)\\&\ge (p(x_i)- L_p h ) \frac{1}{h^m} \int _{{\mathcal {M}}} \eta \left( \frac{d(x_i, y) + \varepsilon }{h} \right) \hbox {dVol}(y). \end{aligned} \end{aligned}$$

(B.5)

The above integral can be estimated from below by

$$\begin{aligned} \begin{aligned} \frac{1}{h^m}\int _{{\mathcal {M}}} \eta \left( \frac{d(x_i, y) + \varepsilon }{h} \right) \hbox {dVol}(y)&= \frac{1}{h^m} \int _{B( h- \varepsilon )} \eta \left( \frac{|z |+ \varepsilon }{h} \right) J_{x_i}(z) \hbox {d} z\\&\ge (1- Cm K h^2) \frac{1}{h^m} \int _{B( h- \varepsilon )} \eta \left( \frac{|z |+ \varepsilon }{h} \right) \hbox {d} z\\&= (1- Cm K h^2) \int _{\varepsilon /h}^1 m \omega _m \eta (r)(r - \frac{\varepsilon }{h})^{m-1}\hbox {d}r \end{aligned} \end{aligned}$$

(B.6)

where the second equality follows using polar coordinates and a change in variables; the last inequality follows from the fact that $\eta$ is assumed to be normalized. In turn,

$$\begin{aligned} \begin{aligned} \int _{\varepsilon /h}^1 m \omega _m \eta (r)\left( r - \frac{\varepsilon }{h}\right) ^{m-1}\hbox {d}r&\ge \int _{\varepsilon /h}^1 m \omega _m \eta (r)r^{m-1} \hbox {d}r\\&\quad - m \omega _m \frac{\varepsilon }{h}\int _{\varepsilon /h}^1 (m-1) \eta (r)r^{m-2} \hbox {d}r\\&\ge 1- 2\eta (0)m \omega _m \frac{\varepsilon }{h}, \end{aligned} \end{aligned}$$

where we have used the fact that $\eta$ was assumed to be normalized. Combining the above inequalities, we deduce that

$$\begin{aligned} p(x_i) - m_i \le L_p h + C \alpha m \omega _m K h^2 + C \alpha m \omega _m \eta (0) \frac{\varepsilon }{h}. \end{aligned}$$

$\square$