SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In this section, we provide theoretical and experimental evidence that the multiparameter persistence image (MPI) [10], which is another decomposition-based representation from the literature, suffers from lack of stability as it does not enjoy guarantees such as the ones of S-CDRs (see Theorem 1). There are two main sources of instability in the MPI. The first one is due to the discretization induced by the lines: since it is obtained by placing Gaussian functions on top of slices of the intervals in the decompositions, if the Gaussian bandwidth  $\sigma$  (see previous work, third item in Section 3.1) is too small w.r.t. the distance between consecutive lines in  $L$ , discretization effects can arise, as illustrated in Figure 6, in which the intervals do not appear as continuous shapes.
The second problem comes from the weight function: it is easy to build decompositions that are close in the interleaving distance, yet whose interval volumes are very different. See [5, Figure 15], and Figure 7, in which we show a family of modules  $\mathbb{M}^{\epsilon}$  (left) made of a single interval built from connecting two squares with a bridge of diameter  $\epsilon > 0$ . We also show the distances between  $\mathbb{M}^{\epsilon}$  and the limit module  $\mathbb{M}$  made of two squares with no bridge, through their MPI and S-CDRs (right). Even though the interleaving distance between  $\mathbb{M}^{\epsilon}$  and  $\mathbb{M}$  goes to zero as  $\epsilon$  goes to zero, the distances between MPI representations converge to a positive constant, whereas the distances between S-CDRs goes effectively to zero.
We also show that this lack of stability can even prevent convergence. In Figure 8, we repeated the setup of Section 4.1 on a synthetic data set sampled from a noisy annulus in the Euclidean plane. As
Figure 6: Examples of numerical artifacts occurring when the number of lines is too small w.r.t. the Gaussian bandwidth, for two different families of lines.
Figure 7: Example of unstability of MPI.
one can clearly see, convergence does not happen for MPI as the Euclidean distance to the limit MPI representation associated to the maximal number of subsample points suddenly drops to zero after an erratic behavior, while S-CDRs exhibit a gradual decrease in agreement with Theorem 1.
Figure 8: Example of lack of convergence for MPI.
C Simplicial homology
In this section, we recall the basics of simplicial homology with coefficients in  $\mathbb{Z} / 2\mathbb{Z}$ , which we use for practical computations. A more detailed presentation can be found in [30, Chapter 1]. The basic bricks of simplicial (persistent) homology are simplicial complexes, which are combinatorial models of topological spaces that can be stored and processed numerically.
Definition 3. Given a set of points  $X_{n} \coloneqq \{x_{1},\ldots ,x_{n}\}$  sampled in a topological space  $X$ , an abstract simplicial complex built from  $X_{n}$  is a family  $S(X_{n})$  of subsets of  $X_{n}$  such that:
- if  $\tau \in S(X_n)$  and  $\sigma \subseteq \tau$ , then  $\sigma \in S(X_n)$ , and
- if  $\sigma, \tau \in S(X_n)$ , then either  $\sigma \cap \tau \in S(X_n)$  or  $\sigma \cap \tau = \emptyset$ .
Each element  $\sigma \in S(X_n)$  is called a simplex of  $S(X_n)$ , and the dimension of a simplex is defined as  $\dim(\sigma) := \operatorname{card}(\sigma) - 1$ . Simplices of dimension 0 are called vertices.
An important linear operator on simplices is the so-called boundary operator. Roughly speaking, it turns a simplex into the chain of its faces, where a chain is a formal sum of simplices. The set of chains has a group structure, denoted by  $Z_{*}(S(X_{n}))$ .
Definition 4. Given a simplex  $\sigma \coloneqq [x_{i_1},\ldots ,x_{i_p}]$ , the boundary operator  $\partial$  is defined as
$$
\partial (\sigma) := \sum_ {j = 1} ^ {p} \left[ x _ {i _ {1}}, \dots , x _ {i _ {j - 1}}, x _ {i _ {j + 1}}, \dots , x _ {i _ {p}} \right].
$$
In other words, it is the chain constructed from  $\sigma$  by removing one vertex at a time. This operator  $\partial$  can then be extended straightforwardly to chains by linearity.
Given a simplicial complex, a topological feature is defined as a cycle, i.e., a chain such that each simplex in its boundary appears an even number of times. In order to formalize this property, we remove a simplex in a chain every time it appears twice, and we let a cycle be a chain  $c$  s.t.  $\partial (c) = 0$ .
Now, one can easily check that  $\partial \circ \partial (c) = 0$  for any chain  $c$ , i.e., the boundary of a cycle is always a cycle. Hence, one wants to exclude cycles that are boundaries, since they correspond somehow to trivial cycles. Again, such boundaries form a group that is denoted by  $B_{*}(S(X_{n}))$ .
Definition 5. The homology group in dimension  $k$  is the quotient group
$$
H _ {k} (S (X _ {n})) := \frac {Z _ {k} (S (X _ {n})}{B _ {k} (S (X _ {n}))}.
$$
In other words, it is the group (one can actually show it is a vector space) of cycles made of simplices of dimension  $k$  that are not boundaries.
See Figure 9 for an illustration of these definitions. Finally, given a filtered simplicial complex (with a filtration defined as in Section 2), computing its associated persistence barcode using the simplicial homology functor can be done with several softwares, such as, e.g., Gudhi [36].
Figure 9: Example of simplicial complex  $S$  built from eight points, and made of eight vertices (simplices of dimension 0), fourteen edges (simplices of dimension 1) and six triangles (simplices of dimension 2). The purple path is a cycle; indeed  $\partial ([v_1, v_7] + [v_7, v_4] + [v_4, v_6] + [v_6, v_1]) = [v_1] + [v_7] + [v_7] + [v_4] + [v_4] + [v_6] + [v_6] + [v_1] = 0$  since every vertex appears twice. Similarly, the blue path is a cycle as well. However, both paths represent the same topological feature, in the sense that they belong to the same equivalence class of  $H_{1}(S)$ , since their sum is exactly the boundary of the 2-chain comprised of the six triangles of the complex, i.e., they differ only by a trivial cycle. Hence, the dimension of  $H_{1}(S)$  is 1.
D Modules, interleaving and bottleneck distances
In this section, we provide a more formalized version of multiparameter persistence modules and their associated distances. Strictly speaking, multiparameter persistence modules are nothing but a parametrized family of vector spaces obtained from applying the homology functor to a multifiltration, as explained in Section 2.
Definition 6. A multiparameter persistence module  $\mathbb{M}$  is a family of vector spaces  $\{M(\alpha):\alpha \in \mathbb{R}^n\}$ , together with linear transformations, also called transition maps,  $\varphi_{\alpha}^{\beta}:\mathbb{M}(\alpha)\to \mathbb{M}(\beta)$  for any  $\alpha \leq \beta$  (where  $\leq$  denotes the partial order of  $\mathbb{R}^n$ ), that satisfy  $\varphi_{\alpha}^{\gamma} = \varphi_{\beta}^{\gamma}\circ \varphi_{\alpha}^{\beta}$  for any  $\alpha \leq \beta \leq \gamma$ .
Of particular interest are interval modules, since they are easier to work with.
Definition 7. An interval module  $\mathbb{M}$  is a multiparameter persistence module such that:
- its dimension is at most 1:  $\dim (\mathbb{M}(\alpha))\leq 1$  for any  $\alpha \in \mathbb{R}^n$ , and
- its support  $\operatorname{supp}(\mathbb{M}) \coloneqq \{\alpha \in \mathbb{R}^n : \dim(\mathbb{M}(\alpha)) = 1\}$  is an interval of  $\mathbb{R}^n$
where an interval of  $\mathbb{R}^n$  is a subset of  $I\subseteq \mathbb{R}^n$  that satisfy:
- (convexity) if  $p, q \in I$  and  $p \leq r \leq q$  then  $r \in I$ , and
- (connectivity) if  $p, q \in I$ , then there exists a finite sequence  $r_1, r_2, \ldots, r_m \in I$ , for some  $m \in \mathbb{N}$ , such that  $p \sim r_1 \sim r_2 \sim \dots \sim r_m \sim q$ , where  $\sim$  can be either  $\leq$  or  $\geq$ .
In the main body of this article, we study representations for candidate decompositions of modules, i.e., direct sums<sup>4</sup> of interval modules that approximate the original modules.
Multiparameter persistence modules can be compared with the interleaving distance [27].
Definition 8 (Interleaving distance). Given  $\varepsilon >0$ , two multiparameter persistence modules  $\mathbb{M}$  and  $\mathbb{M}'$  are  $\varepsilon$ -interleaved if there exist two morphisms  $f\colon \mathbb{M}\to \mathbb{M}_{\varepsilon}'$  and  $g\colon \mathbb{M}'\to \mathbb{M}_{\varepsilon}$  such that  $g_{.+ \varepsilon} \circ f_{.} = \varphi_{.}^{+2\varepsilon}$  and  $f_{.+ \varepsilon} \circ g_{.} = \psi_{.}^{+2\varepsilon}$ , where  $\mathbb{M}_{\varepsilon}$  is the shifted module  $\{\mathbb{M}(x + \varepsilon)\}_{x\in \mathbb{R}^n}$ ,  $\varepsilon = (\varepsilon, \ldots, \varepsilon) \in \mathbb{R}^n$ , and  $\varphi$  and  $\psi$  are the transition maps of  $\mathbb{M}$  and  $\mathbb{M}'$  respectively. The interleaving distance between two multiparameter persistence modules  $\mathbb{M}$  and  $\mathbb{M}'$  is then defined as  $d_{\mathrm{I}}(\mathbb{M},\mathbb{M}') := \inf \left\{\varepsilon \geq 0:\mathbb{M}\text{and}\mathbb{M}'\text{are}\varepsilon\text{-interleaved}\right\}$ .
The main property of this distance is that it is stable for multi-filtrations that are obtained from the sublevel sets of functions. More precisely, given two continuous functions  $f, g: S \to \mathbb{R}^n$  defined on a simplicial complex  $S$ , let  $\mathbb{M}(f), \mathbb{M}(g)$  denote the multiparameter persistence modules obtained from the corresponding multifiltrations  $\{S_x^f := \{\sigma \in S: f(\sigma) \leq x\}\}_{x \in \mathbb{R}^n}$  and  $\{S_x^g := \{\sigma \in S: g(\sigma) \leq x\}\}_{x \in \mathbb{R}^n}$ . Then, one has [27, Theorem 5.3]:
$$
d _ {\mathrm {I}} (\mathbb {M} (f), \mathbb {M} (g)) \leq \| f - g \| _ {\infty}. \tag {9}
$$
Another usual distance is the bottleneck distance [5, Section 2.3]. Intuitively, it relies on decompositions of the modules into direct sums of indecomposable summands<sup>5</sup> (which are not necessarily intervals), and is defined as the largest interleaving distance between summands that are matched under some matching.
Definition 9 (Bottleneck distance). Given two multisets  $A$  and  $B$ ,  $\mu \colon A \nrightarrow B$  is called a matching if there exist  $A' \subseteq A$  and  $B' \subseteq B$  such that  $\mu \colon A' \to B'$  is a bijection. The subset  $A' := \operatorname{coim}(\mu)$  (resp.  $B' := \operatorname{im}(\mu)$ ) is called the coimage (resp. image) of  $\mu$ .
Let  $\mathbb{M} \cong \bigoplus_{i \in \mathcal{I}} M_i$  and  $\mathbb{M}' \cong \bigoplus_{j \in \mathcal{J}} M_j'$  be two multiparameter persistence modules. Given  $\varepsilon \geq 0$ , the modules  $\mathbb{M}$  and  $\mathbb{M}'$  are  $\varepsilon$ -matched if there exists a matching  $\mu: \mathcal{I} \nrightarrow \mathcal{J}$  such that  $M_i$  and  $M_{\mu(i)}'$  are  $\varepsilon$ -interleaved for all  $i \in \operatorname{coim}(\mu)$ , and  $M_i$  (resp.  $M_j'$ ) is  $\varepsilon$ -interleaved with the null module  $\mathbf{0}$  for all  $i \in \mathcal{I} \backslash \operatorname{coim}(\mu)$  (resp.  $j \in \mathcal{J} \backslash \operatorname{im}(\mu)$ ).
The bottleneck distance between two multiparameter persistence modules  $\mathbb{M}$  and  $\mathbb{M}'$  is then defined as  $d_{\mathrm{B}}(\mathbb{M},\mathbb{M}')\coloneqq \inf \left\{\varepsilon \geq 0:\mathbb{M}\text{and}\mathbb{M}'\text{are}\varepsilon \text{-matched}\right\}$ .
<sup>4</sup>Since multiparameter persistence modules are essentially families of vector spaces connected by transition maps, they admit direct sum decompositions, pretty much like usual vector spaces do.
5Recall that a module  $\mathbb{M}$  is indecomposable if  $\mathbb{M}\cong A\oplus B\Rightarrow \mathbb{M}\simeq A$  or  $\mathbb{M}\simeq B$
Since a matching between the decompositions of two multiparameter persistence modules induces an interleaving between the modules themselves, it follows that  $d_{\mathrm{I}} \leq d_{\mathrm{B}}$ . Note also that  $d_{\mathrm{B}}$  can actually be arbitrarily larger than  $d_{\mathrm{I}}$ , as showcased in [5, Section 9].
E The fibered barcode and its properties
Definition 10. Let  $n \in \mathbb{N}^*$  and  $F = \{F^{(1)}, \ldots, F^{(n)}\}$  be a multifiltration on a topological space  $X$ . Let  $\mathbf{e}, b \in \mathbb{R}^n$ , and  $\ell_{\mathbf{e},b} : \mathbb{R} \to \mathbb{R}^n$  be the line in  $\mathbb{R}^n$  defined with  $\ell_{\mathbf{e},b}(t) = t \cdot \mathbf{e} + b$ , that is,  $\ell_{\mathbf{e},b}$  is the line of direction  $\mathbf{e}$  passing through  $b$ . Let  $F_{\mathbf{e},b} : \mathbb{R} \to \mathcal{P}(X)$  defined with  $F_{\mathbf{e},b}(t) = \bigcap_{i=1}^{n} F^{(i)}([\ell_{\mathbf{e},b}(t)]_i)$ , where  $[\cdot]_i$  denotes the  $i$ -th coordinate. Then, each  $F_{\mathbf{e},b}$  is a single-parameter filtration and has a corresponding persistence barcode  $B_{\mathbf{e},b}$ . The set  $\mathcal{B}(F) = \{B_{\mathbf{e},b} : \mathbf{e}, b \in \mathbb{R}^n\}$  is called the fibered barcode of  $F$ .
The two following lemmas from [25] describe two useful properties of the fibered barcode.
Lemma 1 (Lemma 1 in [25]). Let  $\mathbf{e}, b \in \mathbb{R}^n$  and  $\ell_{\mathbf{e},b}$  be the corresponding line. Let  $\hat{e} = \min_i[\mathbf{e}]_i$ . Let  $F, F'$  be two multi-filtrations,  $\mathbb{M}, \mathbb{M}'$  be the corresponding persistence modules and  $B_{\mathbf{e},b} \in \mathcal{B}(F)$  and  $B_{\mathbf{e},b}' \in \mathcal{B}(F')$  be the corresponding barcodes in the fibered barcodes of  $F$  and  $F'$ . Then, the following stability property holds:
$$
d _ {\mathrm {B}} \left(B _ {\mathbf {e}, b}, B _ {\mathbf {e}, b} ^ {\prime}\right) \leq \frac {d _ {\mathrm {I}} \left(\mathbb {M} , \mathbb {M} ^ {\prime}\right)}{\hat {e}}. \tag {10}
$$
Lemma 2 (Lemma 2 in [25]). Let  $\mathbf{e},\mathbf{e}^{\prime},b,b^{\prime}\in \mathbb{R}^{n}$  and  $\ell_{\mathbf{e},b},\ell_{\mathbf{e}^{\prime},b^{\prime}}$  be the corresponding lines. Let  $\hat{e} = \min_{i}[\mathbf{e}]_{i}$  and  $\hat{e}^{\prime} = \min_{i}[\mathbf{e}^{\prime}]_{i}$ . Let  $F$  be a multi-filtration,  $\mathbb{M}$  be the corresponding persistence module and  $B_{\mathbf{e},b},B_{\mathbf{e}^{\prime},b^{\prime}}\in \mathcal{B}(F)$  be the corresponding barcodes in the fibered barcode of  $F$ . Assume  $\mathbb{M}$  is decomposable  $\mathbb{M} = \oplus_{i = 1}^{m}M_{i}$ , and let  $K > 0$  such that  $M_{i}\subseteq B_{\infty}(0,K):= \{x\in \mathbb{R}^{n}:||x||_{\infty}\leq K\}$  for all  $i\in [[1,m]]$ . Then, the following stability property holds:
$$
d _ {\mathrm {B}} \left(B _ {\mathbf {e}, b}, B _ {\mathbf {e} ^ {\prime}, b ^ {\prime}}\right) \leq \frac {\left(K + \max  \left\{\| b \| _ {\infty} , \| b ^ {\prime} \| _ {\infty} \right\}\right) \cdot \| \mathbf {e} - \mathbf {e} ^ {\prime} \| _ {\infty} + \| b - b ^ {\prime} \| _ {\infty}}{\hat {e} \cdot \hat {e} ^ {\prime}}. \tag {11}
$$
F Proof of Theorem 1
Our proof is based on several lemmas. In the first one, we focus on the S-CDR weight function  $w$  as defined in Definition 2.
Lemma 3. Let  $M$  and  $M'$  be two interval modules with compact support. Then, one has
$$
d _ {1} (M, 0) = \frac {1}{2} \sup  _ {b, d \in \operatorname {s u p p} (M)} \min  _ {j} \left(d _ {j} - b _ {j}\right) _ {+} = w (M). \tag {12}
$$
Furthermore, one has the equality
$$
\left| w (M) - w \left(M ^ {\prime}\right) \right| \leq d _ {1} \left(M, M ^ {\prime}\right). \tag {13}
$$
Proof. We first show Equation (12) with two inequalities.
First inequality:  $\leq$  Let  $M$  be an interval module. If  $d_{\mathrm{I}}(M,0) = 0$ , then the inequality is trivial, so we now assume that  $d_{\mathrm{I}}(M,0) > 0$ . Let  $\delta > 0$  such that  $\delta < d_{\mathrm{I}}(M,0)$ . By definition of  $d_{\mathrm{I}}$ , the identity morphism  $M \to M_{2\delta}$  cannot be factorized by 0. This implies the existence of some  $b \in \mathbb{R}^n$  such that  $\operatorname{rank}(M(b) \to M(b + 2\delta)) > 0$ ; in particular,  $b, b + 2\delta \in \operatorname{supp}(M)$ . Making  $\delta$  converge to  $d_{\mathrm{I}}(M,0)$  yields the desired inequality.
Second inequality:  $\geq$  Let  $(K_{n})_{n\in \mathbb{N}}$  be a compact interval exhaustion of  $\operatorname {supp}(M)$ , and  $b_{n},d_{n}\in K_{n}$  be two points that achieve the maximum in
$$
\frac{1}{2}\sup_{b,d\in K_{n}}\min_{j}(d_{j} - b_{j})_{+}.
$$
Now, by functoriality of persistence modules, we can assume without loss of generality that  $b_{n}$  and  $d_{n}$  are on the same diagonal line (indeed, if they are not, it is possible to transform  $d_{n}$  into
$\tilde{d}_n$  such that  $b_n$  and  $\tilde{d}_n$  are on the same diagonal line and also achieve the supremum). Thus,  $\mathrm{rank}(M(b_n) \to M(d_n)) > 0$ , and  $d_{\mathrm{I}}(M,0) \geq \frac{1}{2} \| d_n - b_n \|_{\infty}$ . Taking the limit over  $n \in \mathbb{N}$  leads to the desired inequality.
Inequality (13) follows directly from the triangle inequality applied on  $d_{\mathrm{I}}$ .
In the following lemma, we rewrite volumes of interval module supports using interleaving distances.
Lemma 4. Let  $M$  be an interval module, and  $R \subseteq \mathbb{R}^n$  be a compact rectangle, with  $n \geq 2$ . Then, one has:
$$
\operatorname {v o l} \left(\operatorname {s u p p} (M) \cap R\right) = 2 \int_ {\{y \in \mathbb {R} ^ {n}: y _ {n} = 0 \}} d _ {\mathrm {I}} \left(M \big | _ {l _ {y} \cap R}, 0\right) \mathrm {d} \lambda^ {n - 1} (y)
$$
where  $l_y$  is the diagonal line crossing  $y$ , and  $\lambda^{n-1}$  denotes the Lebesgue measure in  $\mathbb{R}^{n-1}$ .
Proof. Using the change of variables  $y_{i} = x_{i} - x_{n}$  and  $t = x_{n}$  (which has a trivial Jacobian) yields the following inequalities:
$$
\begin{array}{l} \operatorname {v o l} (\operatorname {s u p p} (M) \cap R) = \int_ {\operatorname {s u p p} (M) \cap R} \mathrm {d} \lambda^ {n} (x) \\ = \int_ {\{y \in \mathbb {R} ^ {n}: y _ {n} = 0 \}} \int_ {t \in \mathbb {R}} \mathbb {1} _ {\operatorname {s u p p} (M) \cap R} (y + t) d t d \lambda^ {n - 1} (y) \\ = \int_ {\left\{y \in \mathbb {R} ^ {n}: y _ {n} = 0 \right\}} \operatorname {d i a m} _ {\| \cdot \| _ {\infty}} \left(\operatorname {s u p p} (M) \cap l _ {y} \cap R\right) d \lambda^ {n - 1} (y) \\ \end{array}
$$
where  $l_{y}$  is the diagonal line passing through  $y$ . Now, since  $M$  is an interval module, one has  $\mathrm{diam}_{\| \cdot \|_{\infty}}(\operatorname{supp}(M) \cap l_{y} \cap R) = 2d_{\mathrm{I}}(M|_{l_{y} \cap R}, 0)$ , which concludes the proof.
In the following proposition, we provide stability bounds for single interval modules.
Proposition 1. If  $M$  and  $M'$  are two interval modules, then for any  $\delta > 0$  and S-CDR parameter  $\phi_{\delta}$  in Definition 2, one has:
1.  $0 \leq \phi_{\delta}(M)(x) \leq \frac{w(M)}{\delta} \land 1$ , for any  $x \in \mathbb{R}^n$ ,
2.  $\| \phi_{\delta}(M) - \phi_{\delta}(M^{\prime})\|_{\infty}\leq 2(d_{\mathrm{I}}(M,M^{\prime})\wedge \delta) / \delta .$
Proof. Claim 1. is a simple consequence of Equation (12).
Claim 2. for S-CDR parameter (a) is a simple consequence of the triangle inequality.
Let us prove Claim 2. for (b). Let  $x\in \mathbb{R}^n$  and  $\delta >0$  .One has:
$$
\begin{array}{l} \left| \phi_ {\delta} (M) (x) - \phi_ {\delta} \left(M ^ {\prime}\right) (x) \right| \leq \frac {2}{(2 \delta) ^ {n}} \int_ {\{y: y _ {n} = 0 \}} \left| d _ {\mathrm {I}} \left(M \big | _ {l _ {y} \cap R _ {x, \delta}}, 0\right) - d _ {\mathrm {I}} \left(M ^ {\prime} \big | _ {l _ {y} \cap R _ {x, \delta}}, 0\right) \right| \mathrm {d} \lambda^ {n - 1} (y) \\ \leq \frac {2}{(2 \delta) ^ {n}} \int_ {\{y: y _ {n} = 0 \}} d _ {\mathrm {I}} \left(M \big | _ {l _ {y} \cap R _ {x, \delta}}, M ^ {\prime} \big | _ {l _ {y} \cap R _ {x, \delta}}\right) \mathrm {d} \lambda^ {n - 1} (y) \\ \leq 2 \left(d _ {\mathrm {I}} \left(M, M ^ {\prime}\right) \wedge \delta\right) / \delta , \\ \end{array}
$$
where the first inequality comes from Lemma 4, the second inequality is an application of the triangle inequality, and the third inequality comes from Lemma 1.
Finally, let us prove Claim 2. for (c). Let  $x \in \mathbb{R}^n$  and  $\delta > 0$ . Let  $b \leq d \in \operatorname{supp}(M) \cap R_{x,\delta}$ . Let also  $\gamma > 0$ . Then, using Lemma 4, one has:
$$
\begin{array}{l} \frac {1}{(2 \delta) ^ {n}} \operatorname {v o l} (\operatorname {s u p p} (M) \cap R _ {b, d}) = \frac {2}{(2 \delta) ^ {n}} \int_ {\{y \in \mathbb {R} ^ {n}: y _ {n} = 0 \}} d _ {\mathrm {I}} \left(M \mid_ {R _ {b, d} \cap l _ {y}}, 0\right) \mathrm {d} \lambda^ {n - 1} (y) \\ \leq \frac {2}{\delta} \gamma + \frac {2}{(2 \delta) ^ {n}} \int_ {\{y \in \mathbb {R} ^ {n}: y _ {n} = 0 \}} d _ {\mathrm {I}} (M \big | _ {R _ {b + \gamma , d - \gamma} \cap l _ {y}}, 0) \mathrm {d} \lambda^ {n - 1} (y), \\ \end{array}
$$
using the convention  $R_{a,b} = \varnothing$  if  $a \not\leq b$ . Now, set  $\gamma := d_{\mathrm{I}}(M|_{R_{x,\delta}}, M'|_{R_{x,\delta}})$ . If  $b + \gamma$  or  $d - \gamma \notin \operatorname{supp}(M')$  then  $d_{\mathrm{I}}(M|_{R_{x,\delta}}, M'|_{R_{x,\delta}}) = \gamma > d_{\mathrm{I}}(M, M')$  which is impossible. Thus,
$$
\begin{array}{l} \frac {1}{(2 \delta) ^ {n}} \operatorname {v o l} \left(R _ {b, d}\right) \leq 2 d _ {\mathbf {I}} \left(M \mid_ {R _ {x, \delta}}, M ^ {\prime} \mid_ {R _ {x, \delta}}\right) / \delta \\ + \sup  _ {a, c \in R _ {x, \delta} \cap \operatorname {s u p p} (M ^ {\prime})} \frac {2}{(2 \delta) ^ {n}} \int_ {\{y \in \mathbb {R} ^ {n}: y _ {n} = 0 \}} d _ {\mathrm {I}} \left(M \mid_ {R _ {a, c} \cap l _ {y}}, 0\right) \mathrm {d} \lambda^ {n - 1} (y) \\ = 2 d _ {\mathrm {I}} \left(M \mid_ {R _ {x, \delta}}, M ^ {\prime} \mid_ {R _ {x, \delta}}\right) / \delta + \phi_ {\delta} \left(M ^ {\prime}\right) (x) \\ \end{array}
$$
Finally, taking the supremum on  $b \leq d \in \operatorname{supp}(M) \cap R_{x,\delta}$  yields
$$
\phi_ {\delta} (M) (x) - \phi_ {\delta} (M ^ {\prime}) (x) \leq 2 d _ {\mathrm {I}} \left(M \big | _ {R _ {x, \delta}}, M ^ {\prime} \big | _ {R _ {x, \delta}}\right) / \delta \leq 2 \left(d _ {\mathrm {I}} (M, M ^ {\prime}) \wedge \delta\right) / \delta .
$$
The desired inequality follows by symmetry on  $M$  and  $M^{\prime}$
Equipped with these results, we can finally prove Theorem 1.
Proof. Theorem 1.
Let  $\mathbb{M} = \oplus_{i=1}^{m} M_i$  and  $\mathbb{M}' = \oplus_{j=1}^{m'} M_j'$  be two modules that are decomposable into interval modules and  $x \in \mathbb{R}^n$ .
Inequality 5. To simplify notations, we define the following:  $w_{i} \coloneqq w(M_{i})$ ,  $\phi_{i,x} \coloneqq \phi_{\delta}(M_i)(x)$  and  $w_{j}^{\prime} \coloneqq w(M_{j}^{\prime})$ ,  $\phi_{j,x}^{\prime} \coloneqq \phi_{\delta}(x,M_{j}^{\prime})$ . Let us also assume without loss of generality that the indices are consistent with a matching achieving the bottleneck distance. In other words, the bottleneck distance is achieved for a matching that matches  $M_{i}$  with  $M_{i}^{\prime}$  for every  $i$  (up to adding 0 modules in the decompositions of  $\mathbb{M}$  and  $\mathbb{M}^{\prime}$  so that  $m = m^{\prime}$ ). Finally, assume without loss of generality that  $\sum_{i}w_{i}^{\prime}\geq \sum_{i}w_{i}$ . Then, one has:
$$
\begin{array}{l} | V _ {1, \delta} (\mathbb {M}) (x) - V _ {1, \delta} (\mathbb {M} ^ {\prime}) (x) | = \left| \frac {1}{\sum_ {i} w _ {i}} \sum_ {i} w _ {i} \phi_ {i, x} - \frac {1}{\sum_ {i} w _ {i} ^ {\prime}} \sum_ {i} w _ {i} ^ {\prime} \phi_ {i, x} ^ {\prime} \right| \\ \leq \frac {1}{\sum_ {i} w _ {i} ^ {\prime}} \left| \sum_ {i} w _ {i} \phi_ {i, x} - \sum_ {i} w _ {i} ^ {\prime} \phi_ {i, x} ^ {\prime} \right| + \left| \frac {1}{\sum_ {i} w _ {i}} - \frac {1}{\sum_ {i} w _ {i} ^ {\prime}} \right| \left| \sum_ {i} w _ {i} \phi_ {i, x} \right|. \\ \end{array}
$$
Now, for any index  $i$ , since  $d_{\mathrm{I}}(M_i,M_i') \leq d_{\mathrm{B}}(\mathbb{M},\mathbb{M}')$  and  $|w_{i} - w_{i}^{\prime}| \leq d_{\mathrm{I}}(M_i,M_i^{\prime}) \leq d_{\mathrm{B}}(\mathbb{M},\mathbb{M}^{\prime})$  by Lemma 3, Proposition 1 ensures that: