• I is the interval $[0, +\infty)$ or $[0, T]$ for $T \ge 0$.

• $E = \mathbb{R}^n$ is the Euclidean space, $M(E)$ the vector space of $n \times n$ matrices, $\mathscr{G}_d(E)$ the Grassmannian of $d$-planes in $E$.

• A is a subset of E, med(A) denotes its medial axis, reach(A) its reach. For every $x \in E$, dist(x, A) is the distance from x to A.

• For $x, y \in E$, $x \perp y$ denotes the orthogonality of $x$ and $y$

• If $x, y \in E$, $x \otimes y = x'y \in M(E)$ is the outer product, and $x^{\otimes 2} = x \otimes x$.

• $\|\cdot\|$ is the Euclidean norm on $E$ and $\langle\cdot, \cdot\rangle$ the corresponding inner product, $\|\cdot\|_F$ the Frobenius norm on $M(E)$, $\|\cdot\|_\gamma$ the $\gamma$-norm on $E \times M(E)$ (defined in Subsect. 4.1).

• $W_p$ is the $p$-Wasserstein distance between measures on $E$, $W_{p,\gamma}$ is the $(p, \gamma)$-Wasserstein distance between measures on $E \times M(E)$ (defined in Subsect. 4.1).

• $\mathcal{H}^d$ is the $d$-dimensional Hausdorff measure on $E$ or on a subspace $T \subset E$ (not renormalized).

• If $\mu$ is a measure of positive finite mass, $|\mu|$ denotes its mass, $\bar{\mu} = \frac{1}{|\mu|}\mu$ is the associated probability measure, $\tilde{\mu}$ denotes the associated lifted measure (introduced in Subsect. 4.1).

• $1_A$ is the indicator function of a measurable set $A$.

• If $T$ is a subspace of $E$, $p_T$ denotes the orthogonal projection matrix on $T$.

• $\mathcal{B}(x,r)$ and $\overline{\mathcal{B}}(x,r)$ denote the open and closed balls of $E$, $\partial \mathcal{B}(x,r)$ the sphere. $V_d$ and $S_{d-1}$ denote $\mathcal{H}^d(\mathcal{B}(0,1))$ and $\mathcal{H}^{d-1}(\partial \mathcal{B}(0,1))$ (note that $S_{d-1} = dV_d$).

• $\mathcal{M}_0$ is a Riemannian manifold, and $\mathcal{B}_{\mathcal{M}_0}(x,r)$ and $\overline{\mathcal{B}}_{\mathcal{M}_0}(x,r)$ denote the open and closed geodesics balls. For $x_0, y_0 \in \mathcal{M}_0$, $d_{\mathcal{M}_0}(x_0, y_0)$ denotes the geodesic distance.

• If $T$ is a subspace of $E$, $\mathcal{B}_T(x,r)$ and $\overline{\mathcal{B}}_T(x,r)$ denote the open and closed balls of $T$ for the Euclidean distance.

• If $f$ is a map with values in $\mathbb{R}$ and $t \in \mathbb{R}$, $f^t$ denotes the sublevel set $f^t = f^{-1}((-\infty, t])$.

## B Table of constants

In the following table, each constant is preceded by the result where it appeared first. If a constant is defined from the others, it is indicated here. The indices are arbitrary and only reflect the order of apparition of each result.
<table>
  <thead>
    <tr>
      <th>Index</th>
      <th>Result</th>
      <th>Constant</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>1.</td>
      <td>Corollary 2.9</td>
      <td>a, c<sub>1</sub> = 8diam(supp(μ)) + 5, c'<sub>1</sub> = 2a<sup>-1</sup></td>
    </tr>
    <tr>
      <td>2.</td>
      <td>Hypothesis 2</td>
      <td>ρ</td>
    </tr>
    <tr>
      <td>3.</td>
      <td>Hypothesis 3</td>
      <td>L<sub>0</sub>, f<sub>min</sub>, f<sub>max</sub></td>
    </tr>
    <tr>
      <td>4.</td>
      <td>Hypothesis 4</td>
      <td>c<sub>4</sub>, r<sub>4</sub></td>
    </tr>
    <tr>
      <td>5.</td>
      <td>Lemma 3.4</td>
      <td>c<sub>5</sub>: t ↦ 1/4 (1 - √1 - 2t)</td>
    </tr>
    <tr>
      <td>6.</td>
      <td>Lemma 3.9</td>
      <td>J<sub>min</sub> = (23/24)<sup>d</sup>, J<sub>max</sub> = (5/4)<sup>d</sup></td>
    </tr>
    <tr>
      <td>7.</td>
      <td>Lemma 3.11</td>
      <td>c<sub>7</sub> = 4L<sub>0</sub>J<sub>max</sub> + d/2ρf<sub>max</sub></td>
    </tr>
    <tr>
      <td>8.</td>
      <td>Proposition 3.13</td>
      <td>c<sub>8</sub> = c<sub>7</sub> + f<sub>max</sub>J<sub>max</sub>d2<sup>d</sup>ρ</td>
    </tr>
    <tr>
      <td>9.</td>
      <td>Proposition 3.13, Hypothesis 5</td>
      <td>c<sub>9</sub> = f<sub>min</sub>J<sub>min</sub>V<sub>d</sub></td>
    </tr>
    <tr>
      <td>10.</td>
      <td>Proposition 3.13, Hypothesis 6</td>
      <td>c<sub>10</sub> = d2<sup>d</sup>f<sub>max</sub>J<sub>max</sub>V<sub>d</sub></td>
    </tr>
    <tr>
      <td>11.</td>
      <td>Proposition 3.14, Hypothesis 7</td>
      <td>c<sub>11</sub> = f<sub>max</sub>J<sub>max</sub>/f<sub>min</sub>J<sub>min</sub> (ρ/√4 - √13)<sup>d</sup> d2<sup>d</sup>√3</td>
    </tr>
    <tr>
      <td>12.</td>
      <td>Subsect. 3.3</td>
      <td>Δ, Δ<sub>0</sub>, Θ</td>
    </tr>
    <tr>
      <td>13.</td>
      <td>Proposition 3.19</td>
      <td>r<sub>13</sub> = min( sin(Θ)/8ρ, sin(Θ)<sup>2</sup>/4, Δ<sub>0</sub> sin(Θ)/4, Δ)</td>
    </tr>
    <tr>
      <td>14.</td>
      <td>Proposition 4.1</td>
      <td>c<sub>13</sub> = (2/sin(Θ))<sup>α</sup> V<sub>α</sub>f<sub>max</sub>H<sub>M<sub>0</sub></sub><sup>d'</sup>(N<sub>0</sub>)</td>
    </tr>
    <tr>
      <td>15.</td>
      <td>Lemma 4.5</td>
      <td>c<sub>14</sub> = 6ρ + 1/(f<sub>min</sub>J<sub>min</sub>) (4c<sub>7</sub> + f<sub>max</sub>2<sup>d</sup>dρ + c<sub>8</sub>),</td>
    </tr>
    <tr>
      <td>16.</td>
      <td>Lemma 4.6</td>
      <td>c<sub>15</sub> = 2(1 + 4(5/3d)<sup>d-1</sup>) c<sub>10</sub>/c<sub>9</sub></td>
    </tr>
    <tr>
      <td>17.</td>
      <td>Lemma 4.7</td>
      <td>c<sub>16</sub> = (2 + 2(5/3d)<sup>d-1/2</sup>) c<sub>11</sub>/c<sub>9</sub></td>
    </tr>
    <tr>
      <td>18.</td>
      <td>Lemma 4.8</td>
      <td>c<sub>17</sub> = 2<sup>d-1</sup>/c<sub>9</sub> + 2(12/3d)<sup>d-1</sup>(c<sub>10</sub>+1)/c<sub>9</sub> + 2<sup>d+3</sup>(3/3d)<sup>d-1</sup>(c<sub>10</sub>+1)/c<sub>9</sub></td>
    </tr>
    <tr>
      <td>19.</td>
      <td>Proposition 4.11</td>
      <td>c<sub>18</sub> = 2<sup>d-2</sup>/c<sub>9</sub> + 4(3/3d)<sup>d-1/2</sup>(c<sub>11</sub>+4<sup>d-1/2</sup>)/c<sub>9</sub> + 2·4<sup>d</sup>(2c<sub>11</sub>(3/3d)<sup>d-1/2</sup>+1)/c<sub>9</sub></td>
    </tr>
    <tr>
      <td>20.</td>
      <td>Lemma 4.13</td>
      <td>c<sub>19</sub> = 4(1 + c<sub>20</sub>), c'<sub>19</sub> = 4c<sub>18</sub></td>
    </tr>
    <tr>
      <td>21.</td>
      <td>Theorem 4.14</td>
      <td>c<sub>20</sub> = 3 + c<sub>15</sub> + c<sub>16</sub> + c<sub>17</sub></td>
    </tr>
    <tr>
      <td>22.</td>
      <td>Corollary 4.17</td>
      <td>c<sub>21</sub> = 2 + 1/2c'<sub>19</sub> = 2(1 + c<sub>18</sub>)</td>
    </tr>
    <tr>
      <td>23.</td>
      <td>Subsect. 5.2</td>
      <td>c<sub>22</sub> = c<sub>21</sub>(c<sub>4</sub>)<sup>1/2</sup> + c<sub>19</sub> + c<sub>14</sub></td>
    </tr>
    <tr>
      <td>24.</td>
      <td>Corollary 5.5</td>
      <td>ρ<sub>γ</sub>, f̃<sub>min,γ</sub>, χ<sub>23,γ</sub> = f̃<sub>min,γ</sub>J<sub>min</sub>V<sub>d</sub></td>
    </tr>
    <tr>
      <td>24.</td>
      <td>Corollary 5.5</td>
      <td>χ<sub>1,γ</sub> = 8diam(M) + 8γ + 5, χ'<sub>1,γ</sub> = 2(χ<sub>23,γ</sub>)<sup>-1/2</sup></td>
    </tr>
  </tbody>
</table>

C Supplementary material for Sect. 2

*Proof of Lemma 2.4.* The proof is based on the following observations. We can use the triangle inequality, then the Pythagorean Theorem with ⟨v, y - x⟩ = 0 and Lemma 2.3 Point 1 to get

$$
\begin{align*}
\|\gamma(t) - x\| &\le \|(y + tv) - x\| + \|\gamma(t) - (y + tv)\| \\
&\le \sqrt{\|tv\|^2 + \|y - x\|^2} + \frac{\rho}{2}t^2 \\
&= \sqrt{t^2 + l^2} + \frac{\rho}{2}t^2.
\end{align*}
$$
For any $r \le \frac{1}{\rho}$, consider the equation

$$ \sqrt{t^2 + l^2} + \frac{\rho}{2} t^2 = r. \qquad (48) $$

By squaring this equality, we get $(\frac{\rho}{2})^2 t^4 - (1 + \rho r)t^2 + (r^2 - t^2) = 0$. By considering the polynomial $T \mapsto (\frac{\rho}{2})^2 T^2 - (1 + \rho r)T + (r^2 - t^2)$, whose discriminant is $1 + 2\rho r + (\rho t)^2 > 0$, we see that the solutions of Equation (48) are

$$ T_1 = \frac{\sqrt{2}}{\rho} \sqrt{1 + \rho r - \sqrt{1 + 2\rho r + \rho^2 l^2}} \quad \text{and} \quad T'_1 = \frac{\sqrt{2}}{\rho} \sqrt{1 + \rho r + \sqrt{1 + 2\rho r + \rho^2 l^2}}. $$

Following the same ideas, one obtains

$$ \|\gamma(t) - x\| \ge \sqrt{t^2 + l^2} - \frac{\rho}{2} t^2. $$

Moreover, the equation

$$ \sqrt{t^2 + l^2} - \frac{\rho}{2} t^2 = r \qquad (49) $$

admits the following roots:

$$ T_2 = \frac{\sqrt{2}}{\rho} \sqrt{1 - \rho r - \sqrt{1 - 2\rho r + \rho^2 l^2}} \quad \text{and} \quad T'_2 = \frac{\sqrt{2}}{\rho} \sqrt{1 - \rho r + \sqrt{1 - 2\rho r + \rho^2 l^2}}. $$

We now prove the five points successively.

**Point 1.** Observe that $\dot{\phi}(t) = 2\langle\dot{\gamma}(t), \gamma(t) - x\rangle$, and that

$$ \ddot{\phi}(t) = 2\langle\dot{\gamma}(t), \dot{\gamma}(t)\rangle + 2\langle\dot{\gamma}(t), \gamma(t) - x\rangle. $$

By Cauchy-Schwarz inequality, $\langle\dot{\gamma}(t), \gamma(t) - x\rangle \ge -\|\dot{\gamma}(t)\|\|\gamma(t) - x\|$. Note that $\langle\dot{\gamma}(t), \dot{\gamma}(t)\rangle = 1$ since $\gamma$ is parametrized by arc-length, and that $\|\dot{\gamma}(t)\| \le \rho$ by Equation (1). Hence we get

$$ \ddot{\phi}(t) \ge 2(1 - \rho \|\gamma(t) - x\|). \qquad (50) $$

Consider Equation (48) with $r = \frac{1}{\rho}$. We see that $\|\gamma(t) - x\| \le \frac{1}{\rho}$ when $t$ is lower than

$$ T_1 = \frac{\sqrt{2}}{\rho} \sqrt{2 - \sqrt{3 + \rho^2 l^2}}. $$

In this case, $\ddot{\phi}(t) \ge 0$ according to Equation (50). Since $\dot{\phi}(0) = 0$, we deduce that $\phi$ is increasing on $[0, T_1]$.

**Point 2.** As we have seen with Equation (49), we have $\|\gamma(t) - x\| > r$ when $t \in (T_2, T'_2)$. In order to give an upper bound on $T_2$, we use the inequality $\sqrt{b} - \sqrt{a} = \frac{1}{\sqrt{a+b}}(b-a) \le \frac{1}{\sqrt{b}}(b-a)$, where $a < b$, to get

$$ 1 - \rho r - \sqrt{1 - 2\rho r + \rho^2 l^2} \le \frac{1}{1 - \rho r} \rho^2 (r^2 - l^2) $$

and we conclude that $T_2 \le \frac{\sqrt{2}}{\sqrt{1-\rho r}} \sqrt{r^2 - l^2}$. Since $r \le \frac{1}{2\rho}$, we obtain $T_2 \le 2\sqrt{r^2 - l^2}$.

**Point 3.** When $l=0$, algebraic manipulations show that $T_2 = \frac{1}{\rho}(1 - \sqrt{1 - 2\rho r})$ and $T'_2 = \frac{1}{\rho}(1 + \sqrt{1 - 2\rho r})$.

**Point 4.** Consider the map $\phi: t \mapsto \|\gamma(t) - x\|^2$. By definition of $b$, for all $t \in (0, b)$, we have $\|\gamma(t) - x\| \le r$. Hence Equation (50) gives $\ddot{\phi}(t) \ge 2(1 - \rho r)$. It follows that $\dot{\phi}(t) \ge 2(1 - \rho r)t$, and that

$$ \begin{aligned} \phi(b) - \phi(a) &= \int_a^b \dot{\phi}(t) dt \ge \int_a^b 2(1 - \rho r) t dt \\ &= (1 - \rho r)(b^2 - a^2). \end{aligned} $$
Note that $r^2 = \phi(b)$. Besides, $s^2 = \phi(a)$ or $s^2 < \phi(a)$, depending on whether $s \ge l$ or $s < l$. In both cases, we have $r^2 - s^2 \ge \phi(b) - \phi(a)$, and we deduce that

$$r^2 - s^2 \ge (1 - \rho r)(b^2 - a^2).$$

Writing $r^2 - s^2 = (r+s)(r-s)$ and $b^2 - a^2 = (b+a)(b-a)$ leads to

$$b-a \le \frac{r+s}{b+a} \frac{1}{1-\rho r} (r-s). \qquad (51)$$

Now, let us give a lower bound on $b$. According to Equation (48), $b$ is lower bounded by $T_1 = \frac{\sqrt{2}}{\rho}\sqrt{1+\rho r - \sqrt{1+2\rho r + \rho^2 l^2}}$. Using the inequality $\sqrt{b} - \sqrt{a} = \frac{1}{\sqrt{b+\sqrt{a}}}(b-a) \ge \frac{1}{2\sqrt{b}}(b-a)$, where $a < b$, we get

$$1 + \rho r - \sqrt{1 + 2\rho r + \rho^2 l^2} \ge \frac{1}{2(1 + \rho r)} \rho^2 (r^2 - l^2),$$

and we conclude that $b \ge (1 + \rho r)^{-\frac{1}{2}}\sqrt{r^2 - s^2}$. Injecting $b+a \ge b \ge (1 + \rho r)^{-\frac{1}{2}}\sqrt{r^2 - s^2}$ in Equation (51) yields

$$b(v) - a(v) \le \frac{(1 + \rho r)^{\frac{1}{2}}}{1 - \rho r} \sqrt{r^2 - s^2}.$$

Under the hypothesis $r \le \frac{1}{2\rho}$, we get $b-a \le \sqrt{6\sqrt{r^2-s^2}}$.

**Point 5.** When $l=0$, we have $b(v)+a(v) \ge r+s$. Hence Equation (51) yields $b(v)-a(v) \le \frac{1}{1-\rho r}(r-s)$. Using $r \le \frac{1}{2\rho}$, we obtain $b(v)-a(v) \le 2(r-s)$. $\square$

of Corollary 2.9. We shall first study an intermediate quantity. Let $\mu$ be a probability measure on $E = \mathbb{R}^n$, $m \in (0,1)$, and $d_{\mu,m}$ the corresponding DTM. Consider the quantity $c(\mu,m)$ is defined as

$$c(\mu, m) = \sup_{x \in \text{supp}(\mu)} d_{\mu,m}(x).$$

Suppose that $\mu$ satisfies the following for $r < (\frac{m}{a})^{\frac{1}{d}}: \forall x \in \text{supp}(\mu), \mu(\mathcal{B}(x,r)) \ge ar^d$. Let us show that $c(\mu,m) \le Cm^{\frac{1}{d}}$ with $C = a^{-\frac{1}{d}}$. By definition,

$$\delta_{\mu,t}(x) = \inf \{r \ge 0 \mid \mu(\overline{\mathcal{B}}(x,r)) > t\} \quad \text{and} \quad d_{\mu,m}^2(x) = \frac{1}{m} \int_0^m \delta_{\mu,t}^2(x) dt.$$

Using the assumption $\mu(\mathcal{B}(x,r)) \ge ar^d$ for all $x \in \text{supp}(\mu)$, we get $\delta_{\mu,t}(x) \le (\frac{t}{a})^{\frac{1}{d}}$, and a simple computation yields

$$d_{\mu,m}^2(x) \le \frac{d}{d+2} \left(\frac{t}{a}\right)^{\frac{2}{d}} \le \left(\frac{t}{a}\right)^{\frac{2}{d}},$$

which yields the result.

We can now prove the corollary. Let $\pi$ be an optimal transport plan for $w = W_2(\mu, v)$. Denote $\alpha = w^{\frac{1}{2}}$ and $D = \text{diam}(\text{supp}(\mu))$. Define $\pi'$ to be $\pi$ restricted to the set $\{x, y \in E \mid \|x-y\| < \alpha\}$. We denote its marginals $\mu'$ and $v'$. By Markov inequality, $1 - |\pi'| \le \frac{w^2}{\alpha^2} = w$, where we recall that $|\pi'|$ denotes the total mass of $\pi'$. Consider the probability measures $\bar{\mu}' = \frac{1}{|\pi'|}\mu'$ and $\bar{v}' = \frac{1}{|v'|}v'$. Let us show that we have

$$W_2(\mu, \bar{\mu}') = 2D\alpha, \quad W_2(\bar{\mu}', \bar{v}') \le \alpha \quad \text{and} \quad W_2(v, \bar{v}') \le 2(1+D)\alpha. \qquad (52)$$

The first inequality is an application of Lemma 4.4:

$$W_2(\mu, \bar{\mu}') \le 2(1 - |\mu'|)^{\frac{1}{2}} D = 2(1 - |\pi'|)^{\frac{1}{2}} D \le 2w^{\frac{1}{2}} D.$$
To obtain the second inequality, we write

$$
\begin{aligned}
W_2^2(\overline{\mu'}, \overline{v'}) &= \int \|x-y\|^2 d\overline{\pi'}(x,y) = \int \|x-y\| \frac{d\pi'(x,y)}{|\pi'|} \\
&\le \frac{1}{|\pi'|} \int \|x-y\| d\pi(x,y).
\end{aligned}
$$

Hence Jensen inequality leads to $W_2(\overline{\mu'}, \overline{v'}) \le \frac{w}{|\pi'|^2}$. Since $1 - |\pi'| \le w$, we have $\frac{w}{|\pi'|^2} \le \frac{w}{1-w}$, and the assumption $w \le \frac{1}{4}$ yields $\frac{w}{1-w} \le \alpha$. This proves the second point. Finally, we obtain the third inequality by applying the triangle inequality:

$$
W_2(\nu, \bar{\nu}') \le W_2(\nu, \mu) + W_2(\mu, \bar{\mu}') + W_2(\bar{\mu}', \nu').
$$

Next, let us deduce that

$$
\begin{array}{l}
c(\overline{\mu'}, m) \le c(\mu) + m^{-\frac{1}{2}} 2D\alpha \\
\text{and} \\
c(\overline{v'}, m) \le c(\mu, m) + \left(m^{-\frac{1}{2}} + m^{-\frac{1}{2}} 2D + 1\right) \alpha.
\end{array}
\qquad (53)
$$

The first inequality follows from the stability of the DTM (see Equation (3)):

$$
c(\overline{\mu'}, m) = \sup_{x \in \text{supp}(\overline{\mu'})} d_{\overline{\mu'}}(x) \le \sup_{x \in \text{supp}(\overline{\mu'})} d_{\mu}(x) + m^{-\frac{1}{2}} W_2(\overline{\mu'}, \mu),
$$

and we conclude with $W_2(\overline{\mu}, \overline{\mu'}) = 2D\alpha$. In order to prove the second inequality, we also use Equation (3):

$$
c(\overline{v'}, m) = \sup_{x \in \text{supp}(\overline{v'})} d_{\overline{v'}}(x) \le \sup_{x \in \text{supp}(\overline{v'})} d_{\overline{\mu'}}(x) + m^{-\frac{1}{2}} W_2(\overline{\mu'}, \overline{v'}).
$$

Since $\pi'$ has support included in $\{x, y \in E \mid \|x-y\| < \alpha\}$, we use the fact that the DTM is 1-Lipschitz to obtain

$$
\sup_{x \in \text{supp}(\overline{v'})} d_{\overline{\mu'}}(x) \le \sup_{x \in \text{supp}(\overline{\mu'})} d_{\overline{\mu'}}(x) + \alpha = c(\mu', m) + \alpha
$$

and we deduce

$$
\begin{aligned}
c(\overline{v'}, m) &\le c(\mu', m) + \alpha + m^{-\frac{1}{2}} W_2(\overline{\mu'}, \overline{v'}) \\
&\le c(\mu, m) + (m^{-\frac{1}{2}} + m^{-\frac{1}{2}} 2D + 1)\alpha.
\end{aligned}
$$

We can now conclude with Theorem 2.8. In our context, it reads

$$
\begin{aligned}
d_i(W[\mu], W[v]) &\le m^{-\frac{1}{2}} W_2(\mu, \mu') + m^{-\frac{1}{2}} W_2(\overline{\mu'}, \overline{v'}) + m^{-1} W_2(v, \overline{v'}) + c(\overline{\mu'}, m) + c(\overline{v'}, m) \\
&\le (m^{-\frac{1}{2}}(4D+1) + 4(D+1))\alpha + 2c(\mu, m),
\end{aligned}
$$

where we used Equations (52) and (53) on the last line. Since $m \le 1$, we can simplify this expression into

$$
d_i(W[\mu], W[v]) \le m^{-\frac{1}{2}}(8D+5)\alpha + 2c(\mu, m).
$$

We conclude the proof by using the inequality $c(\mu, m) \le a^{-\frac{1}{d}} m^{\frac{1}{d}}$ shown at the beginning of the proof. $\square$
References

[1] Allen Hatcher. *Algebraic Topology*. Cambridge University Press, 2002.

[2] Partha Niyogi, Stephen Smale, and Shmuel Weinberger. Finding the homology of submanifolds with high confidence from random samples. *Discrete & Computational Geometry*, 39(1-3):419–441, 2008.

[3] Frédéric Chazal and André Lieutier. Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees. *Computational Geometry*, 40(2):156–170, 2008.

[4] Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, and Larry Wasserman. Homotopy reconstruction via the cech complex and the vietoris-rips complex. *arXiv preprint arXiv:1903.06955*, 2019.

[5] Frédéric Chazal, David Cohen-Steiner, and André Lieutier. A sampling theory for compact sets in Euclidean space. *Discrete & Computational Geometry*, 41(3):461–479, 2009.

[6] Dominique Attali, André Lieutier, and David Salinas. Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes. *Computational Geometry*, 46(4):448–465, 2013.

[7] Sara Kalisnik and Davorin Lesnik. Finding the homology of manifolds using ellipsoids. *arXiv preprint arXiv:2006.09194*, 2020.

[8] Herbert Edelsbrunner. The union of balls and its dual shape. In *Proceedings of the ninth annual symposium on Computational geometry*, pages 218–231, 1993.

[9] Herbert Edelsbrunner and Ernst P Mücke. Three-dimensional alpha shapes. *ACM Transactions on Graphics (TOG)*, 13(1):43–72, 1994.

[10] Vin de Silva and Gunnar E Carlsson. Topological estimation using witness complexes. In *PBG*, pages 157–166, 2004.

[11] Dominique Attali, Herbert Edelsbrunner, and Yuriy Mileyko. Weak witnesses for delaunay triangulations of submanifolds. In *Proceedings of the 2007 ACM symposium on Solid and physical modeling*, pages 143–150, 2007.

[12] Jean-Daniel Boissonnat and Arijit Ghosh. Manifold reconstruction using tangential Delaunay complexes. *Discrete & Computational Geometry*, 51(1):221–267, 2014.

[13] Frédéric Chazal and André Lieutier. Stability and computation of topological invariants of solids in $\mathbb{R}^n$. *Discrete & Computational Geometry*, 37(4):601–617, 2007.

[14] Frédéric Chazal and Steve Yann Oudot. Towards persistence-based reconstruction in Euclidean spaces. In *Proceedings of the twenty-fourth annual symposium on Computational geometry*, SCG '08, pages 232–241, New York, NY, USA, 2008. ACM.

[15] Brittany Terese Fasy, Rafal Komendarczyk, Sushovan Majhi, and Carola Wenk. On the reconstruction of geodesic subspaces of $\mathbb{R}^n$. *arXiv preprint arXiv:1810.10144*, 2018.

[16] Herbert Edelsbrunner and John Harer. *Computational topology: an introduction*. American Mathematical Soc., 2010.

[17] Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec. *Geometric and topological inference*, volume 57. Cambridge University Press, 2018.

[18] Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. *The Structure and Stability of Persistence Modules*. SpringerBriefs in Mathematics, 2016.
[19] Peter Bubenik, Gunnar Carlsson, Peter T Kim, and Zhi-Ming Luo. Statistical topology via morse theory persistence and nonparametric estimation. *Algebraic methods in statistics and probability II*, 516:75–92, 2010.

[20] Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, and Aarti Singh. Confidence sets for persistence diagrams. *The Annals of Statistics*, 42(6):2301–2339, 2014.

[21] Katharine Turner, Yuriy Mileyko, Sayan Mukherjee, and John Harer. Fréchet means for distributions of persistence diagrams. *Discrete & Computational Geometry*, 52(1):44–70, 2014.

[22] F. Chazal, D. Cohen-Steiner, and Q. Mérigot. Geometric inference for probability measures. *Journal on Found. of Comp. Mathematics*, 11(6):733–751, 2011.

[23] J. Phillips, B. Wang, and Y Zheng. Geometric inference on kernel density estimates. In *Proc. 31st Annu. Sympos. Comput. Geom (SoCG 2015)*, pages 857–871, 2015.

[24] Leonidas Guibas, Dmitriy Morozov, and Quentin Mérigot. Witnessed k-distance. *Discrete & Computational Geometry*, 49(1):22–45, 2013.

[25] Mickaël Buchet, Frédéric Chazal, Steve Y Oudot, and Donald R Sheehy. Efficient and robust persistent homology for measures. *Computational Geometry*, 58:70–96, 2016.

[26] Hirokazu Anai, Frédéric Chazal, Marc Glisse, Yuichi Ike, Hiroya Inakoshi, Raphaël Tinarrage, and Yuhei Umeda. DTM-based filtrations. In *Topological Data Analysis*, pages 33–66. Springer, 2020.

[27] Yong Wang, Yuan Jiang, Yi Wu, and Zhi-Hua Zhou. Spectral clustering on multiple manifolds. *IEEE Transactions on Neural Networks*, 22(7):1149–1161, 2011.

[28] Dian Gong, Xuemei Zhao, and Gérard Medioni. Robust multiple manifolds structure learning. *arXiv preprint arXiv:1206.4624*, 2012.

[29] Ery Arias-Castro, Gilad Lerman, and Teng Zhang. Spectral clustering based on local PCA. *The Journal of Machine Learning Research*, 18(1):253–309, 2017.

[30] Siu-Wing Cheng and Man-Kwun Chiu. Tangent estimation from point samples. *Discrete & Computational Geometry*, 56(3):505–557, 2016.

[31] Eddie Aamari, Jisu Kim, Frédéric Chazal, Bertrand Michel, Alessandro Rinaldo, and Larry Wasserman. Estimating the Reach of a Manifold. *Electronic journal of statistics*, 2019.

[32] Eddie Aamari and Clément Levrard. Nonasymptotic rates for manifold, tangent space and curvature estimation. *The Annals of Statistics*, 47(1):177–204, 2019.

[33] Amit Singer and H-T Wu. Vector diffusion maps and the connection laplacian. *Communications on pure and applied mathematics*, 65(8):1067–1144, 2012.

[34] Jisu Kim, Alessandro Rinaldo, and Larry Wasserman. Minimax rates for estimating the dimension of a manifold. *arXiv preprint arXiv:1605.01011*, 2016.

[35] Vladimir I Koltchinskii. Empirical geometry of multivariate data: a deconvolution approach. *Annals of statistics*, pages 591–629, 2000.

[36] Anna V Little, Jason Lee, Yoon-Mo Jung, and Mauro Maggioni. Estimation of intrinsic dimensionality of samples from noisy low-dimensional manifolds in high dimensions with multiscale SVD. In *2009 IEEE/SP 15th Workshop on Statistical Signal Processing*, pages 85–88. IEEE, 2009.

[37] Philippos Mordohai and Gérard Medioni. Dimensionality estimation, manifold learning and function approximation using tensor voting. *Journal of Machine Learning Research*, 11(1), 2010.
[38] Diego H Díaz Martínez, Facundo Mémoli, and Washington Mio. The shape of data and probability measures. *Applied and Computational Harmonic Analysis*, 2018.

[39] Facundo Memoli, Zane Smith, and Zhengchao Wan. The Wasserstein transform. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, *Proceedings of the 36th International Conference on Machine Learning*, volume 97 of *Proceedings of Machine Learning Research*, pages 4496–4504, Long Beach, California, USA, 09–15 Jun 2019. PMLR.

[40] Blanche Buet, Gian Paolo Leonardi, and Simon Masnou. A varifold approach to surface approximation. *Archive for Rational Mechanics and Analysis*, 226(2):639–694, 2017.

[41] Blanche Buet, Gian Paolo Leonardi, and Simon Masnou. Weak and approximate curvatures of a measure: a varifold perspective. *arXiv preprint arXiv:1904.05930*, 2019.

[42] Nicolas Charon and Alain Trouvé. The varifold representation of nonoriented shapes for diffeomorphic registration. *SIAM Journal on Imaging Sciences*, 6(4):2547–2580, 2013.

[43] Herbert Federer. Curvature measures. *Transactions of the American Mathematical Society*, 93(3):418–491, 1959.

[44] M.P. do Carmo. *Riemannian Geometry*. Mathematics (Boston, Mass.). Birkhäuser, 1992.

[45] Frank Morgan. *Geometric measure theory: a beginner's guide*. Academic press, 2016.

[46] Alexander Lytchak. Almost convex subsets. *Geometriae Dedicata*, 115(1):201–218, 2005.

[47] Jean-Daniel Boissonnat, André Lieutier, and Mathijs Wintraecken. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. *Journal of Applied and Computational Topology*, 3(1-2):29–58, 2019.

[48] Stephanie B Alexander and Richard L Bishop. Gauss equation and injectivity radii for subspaces in spaces of curvature bounded above. *Geometriae Dedicata*, 117(1):65–84, 2006.

[49] Eddie Aamari. *Vitesses de convergence en inférence géométrique*. PhD thesis, Université Paris-Saclay, 2018.

[50] Magnus Botnan and William Crawley-Boevey. Decomposition of persistence modules. *Proceedings of the American Mathematical Society*, 148(11):4581–4596, 2020.

[51] Ralph J Herbert. *Multiple points of immersed manifolds*, volume 250. American Mathematical Soc., 1981.

[52] Alfred Gray. *Tubes*, volume 221. Birkhäuser, 2012.

[53] Raphaël Tinarrage. Computing persistent Stiefel-Whitney classes of line bundles. *Journal of Applied and Computational Topology*, pages 1–61, 2021.