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Appendix to Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures
A Missing definitions and proofs 16
A.1 Finitely presentable modules 16
A.2 Discriminative power of descriptors 17
A.3 References for Proposition 1 of the main article 17
A.4 Definition of the Hilbert decomposition signed measure 17
A.5 The Hilbert decomposition signed measure as a signed barcode 18
A.6 Proof of Theorem 1 of the main article 19
A.7 Proof of Lemma 1 of the main article 19
A.8 Lipschitz-stability of vectorizations 20
B Review of numerical descriptors of persistence modules 22
C Further details about experiments 23
C.1 Hyperparameter choices 23
C.2 One-parameter experiments 23
C.3 Further graph experiments 23
C.4 Pointcloud classification filtering Rips by density 23
C.5 The enrichment factor 25
D Runtime experiments 25
D.1 Runtime of computation of Hilbert decomposition signed measure 25
D.2 Runtime of whole pipeline 26
E Stability experiments 26
A Missing definitions and proofs
For an introduction to multiparameter persistence, see, e.g., [10]. We mostly follow the conventions from [12, 55].
A.1 Finitely presentable modules
Let  $x \in \mathbb{R}^n$ . We denote by  $P_x: \mathbb{R}^n \longrightarrow \mathrm{vec}$  the persistence module such that  $P_x(y) = \mathbb{k}$  if  $y \geq x$  and  $P_x(y) = 0$  otherwise, with all structure morphisms that are not forced to be zero being the identity  $\mathbb{k} \longrightarrow \mathbb{k}$ .
Definition 8. A persistence module is finitely generated projective if it is a finite direct sum of modules of the form  $P_x$ . A persistence module is finitely presentable if it is isomorphic to the cokernel of a morphism between two finitely generated projective modules.
The intuition behind the notion of finitely presentable persistence module is the following. By definition, any such module is isomorphic to the cokernel of a morphism  $\bigoplus_{j\in J}P_{y_j}\to \bigoplus_{i\in I}P_{x_i}$ . The elements  $1\in \mathbb{k} = P_{x_i}(x_i)$  are often referred to as the generators of the persistence module—and in TDA they correspond to topological features being born—while the elements  $1\in \mathbb{k} = P_{y_j}(y_j)$  are referred to as the relations—and in TDA they correspond to topological features dying or merging.
It is worth mentioning that most construction related to TDA produce finitely presentable modules:
Lemma 2. If  $(S, f: S \longrightarrow \mathbb{R}^n)$  is a finite filtered simplicial complex, then  $H_i(f)$  is finitely presentable for all  $i \in \mathbb{N}$ .
Proof. Let  $S_{k}$  denote the set of  $k$ -dimensional simplices of  $S$ . By definition of homology, the persistence module  $H_{i}(f)$  is the cokernel of a morphism  $\bigoplus_{\tau \in S_{i+1}} P_{f(\tau)} \to K$ , where  $K$  is the kernel of a morphism  $\bigoplus_{\tau \in S_{i}} P_{f(\tau)} \to \bigoplus_{\tau \in S_{i-1}} P_{f(\tau)}$ . Since  $K$  is the kernel of a morphism between finitely presentable modules, it is itself finitely presentable (see, e.g., [9, Lemma 3.14]), it follows that  $H_{i}(f)$  is finitely presentable, since it is the cokernel of a morphism between finitely presentable modules (see, e.g., [51, Exercise 4.8]).
Example 4. Let  $a < b \in \mathbb{R}$ . Let  $\mathbb{k}_{[a,b)}: \mathbb{R} \longrightarrow \operatorname{vec}$  be the one-parameter persistence module such that  $\mathbb{k}_{[a,b)}(x) = \mathbb{k}$  if  $x \in [a,b)$  and  $\mathbb{k}_{[a,b)}(x) = 0$  if  $x \notin [a,b)$ ; and, if  $a \leq x \leq y < b$ , then  $\mathbb{k}_{[a,b)}(x \leq y)$  is the identity linear map  $\mathbb{k} \longrightarrow \mathbb{k}$ , and is the zero linear map otherwise. The module  $\mathbb{k}_{[a,b)}$  is often referred to as the interval module with support  $[a,b)$ . This interval module is finitely presentable, since it is isomorphic to the cokernel of any non-zero map  $P_b \longrightarrow P_a$ .
An example of a persistence module that is not finitely presentable is any interval module supported over an open interval, such as  $\mathbb{k}_{(a,b)}$ .
A.2 Discriminative power of descriptors
It is well-known [19, Theorem 12] that the one-parameter barcode is equivalent to the rank invariant, which is a descriptor that readily generalizes to the multiparameter case, as follows.
Definition 9 ([19]). The rank invariant of a persistence module  $M: \mathbb{R}^n \longrightarrow \mathrm{vec}$  is the function which maps each pair of comparable elements  $x \leq y \in \mathbb{R}^n$  to the rank of the linear map  $M(x) \longrightarrow M(y)$ .
Proposition 2. The Hilbert function is a strictly weaker descriptor than the rank invariant. In particular, the descriptors of [20, 76, 79] are more discriminative than the Hilbert decomposition signed measure.
Proof. The persistence modules  $M = \mathbb{k}_{[0,\infty)}$  and  $N = \mathbb{k}_{[0,1)} \oplus \mathbb{k}_{[1,\infty)}$  have the same Hilbert function  $\mathbb{R} \longrightarrow \mathbb{Z}$ , since in both cases the Hilbert function is just the indicator function of the set  $[0,\infty)$ . However, the rank of the linear map  $M(0 \leq 2): M(0) \longrightarrow M(2)$  is one, while the rank of the linear map  $N(0 \leq 2): N(0) \longrightarrow N(2)$  is zero, proving the first claim. Since the invariants of [20, 76, 79] determine the rank invariant of (at least some) one-parameter slices, it follows that they are strictly more discriminative than the Hilbert function.
A.3 References for Proposition 1 of the main article
Both statements are well known. For the first statement, see, e.g., [57, Proposition 2.1], and, for the second one, see, e.g., [57, Remark 2.30].
A.4 Definition of the Hilbert decomposition signed measure
For intuition about the concepts in this section, and their connection to [55], see Appendix A.5. Here, to be self-contained, we unfold the definitions of [55].
Lemma 3 (cf. [55, Proposition 5.2]). Let  $M: \mathbb{R}^n \to \mathrm{vec}$  be finitely presentable. There exists a pair of finitely generated projective modules  $(P, Q)$  such that, as functions  $\mathbb{R}^n \to \mathbb{Z}$ , we have
$$
\dim (M) = \dim (P) - \dim (Q).
$$
Note that the above decomposition of the Hilbert function is not unique; in fact, there are infinitely many of these decompositions, given by different pairs  $(P, Q)$ . Nevertheless, as we will show below, they all yield the same, unique, Hilbert decomposition signed measure. For this we will use the following well known fact from geometric measure theory.
Lemma 4. To prove that  $\mu = \nu \in \mathcal{M}(\mathbb{R}^n)$ , it is enough to show that, for every  $x\in \mathbb{R}^n$ , we have
$$
\mu \big (\{y \in \mathbb {R} ^ {n}: y \leq x \} \big) = \nu \big (\{y \in \mathbb {R} ^ {n}: y \leq x \} \big),
$$
Proof. Since sets of the form  $\{y\in \mathbb{R}^n:y\leq x\}$  generate the Borel sigma-algebra, and the measures in  $\mathcal{M}(\mathbb{R}^n)$  are necessarily finite, the result follows from a standard application of the  $\pi -\lambda$  theorem [37, Theorem A.1.4]. For instance, one can follow the proof of [37, Theorem A.1.5], by noting that positivity of the measures is not required for the proof to work.
The Hilbert decomposition signed measure is built as a sum of signed Dirac measures, one for each summand of  $P$  and  $Q$  in the decomposition of Lemma 3. The following lemma justifies this insight.
Lemma 5. Let  $x \in \mathbb{R}^n$ , let  $P_x: \mathbb{R}^n \longrightarrow \mathrm{vec}$  be the corresponding finitely generated projective module, and let  $\delta_x$  be the corresponding Dirac measure. Then  $\dim(P_x)(y) = \delta_x(\{w \in \mathbb{R}^n : w \leq y\})$ .
Proof. Note that, by definition of  $P_x$ , we have  $\dim(P_x)(y) = 1$  if  $x \leq y$  and  $\dim(P_x)(y) = 0$  otherwise. Similarly,  $\delta_x(\{w : w \leq y\}) = 1$  if  $x \leq y$  and  $\delta_x(\{w : w \leq y\}) = 0$  otherwise. The result follows.
We now have the required ingredients to define the Hilbert decomposition signed measure formally. While the statement itself only claims existence and uniqueness, the proof actually builds the measure explicitly as a finite sum of signed Dirac measures (see Eq. (1)), as explained above.
Proposition 3. If  $M: \mathbb{R}^n \to \mathrm{vec}$  is a unique point measure  $\mu_M \in \mathcal{M}(\mathbb{R}^n)$  with
$$
\dim (M (x)) = \mu_ {M} \left(\left\{y \in \mathbb {R} ^ {n}: y \leq x \right\}\right), f o r a l l x \in \mathbb {R} ^ {n}.
$$
Proof. Let  $(P, Q)$  be as in Lemma 3. Let  $\{x_i\}_{i \in I}$  be such that  $P \cong \bigoplus_{i \in I} P_{x_i}$  and let  $\{y_j\}_{j \in J}$  be such that  $Q \cong \bigoplus_{j \in J} P_{y_j}$ . To show existence, define the measure
$$
\mu_ {M} = \sum_ {i \in I} \delta_ {x _ {i}} - \sum_ {j \in J} \delta_ {y _ {j}}. \tag {1}
$$
$$
\begin{array}{l} \dim (M (z)) = \dim (P (z)) - \dim (Q (z)) \\ = | \left\{i \in I: x _ {i} \leq z \right\} | - | \left\{j \in J: y _ {j} \leq z \right\} | \\ = \mu_ {M} (\{w \in \mathbb {R} ^ {n}: w \leq z \}), \\ \end{array}
$$
where in the second equality we used Lemma 5. Uniqueness follows from Lemma 4.
A.5 The Hilbert decomposition signed measure as a signed barcode
We clarify the connection between the Hilbert decomposition signed measure and the concept of signed barcode of [55].
Let  $M: \mathbb{R}^n \longrightarrow \mathrm{vec}$  be finitely presentable. A pair of finitely generated projective modules  $(P, Q)$  as in Lemma 3 is what is called a Hilbert decomposition of  $\dim(M)$  in [55]. Since the modules  $P$  and  $Q$  are, by assumption, finitely generated projective, there must exist multisets  $\{x_i\}_{i \in I}$  and  $\{y_j\}_{j \in J}$  of elements of  $\mathbb{R}^n$  such that  $P \cong \bigoplus_{i \in I} P_{x_i}$  and  $Q \cong \bigoplus_{j \in J} P_{y_j}$ . A pair of multisets of elements of  $\mathbb{R}^n$  is what is called a signed barcode in [55]. The intuition is that each element  $x$  of  $\mathbb{R}^n$  determines its upset  $\{y \in \mathbb{R}^n : x \leq y\}$ , which coincides with the support of the module  $P_x$  (by Lemma 5). Thus, the supports of the summands of  $P$  (resp.  $Q$ ) are interpreted as the positive (resp. negative) bars of the signed barcode  $(\{x_i\}_{i \in I}, \{y_j\}_{j \in J})$ . See Figs. 2 and 3 for illustrations.
Figure 2: Left to right: The filtered simplicial complex of Fig. 1 of the main article; its  $0th$  dimensional homology persistence module  $H_0(f)$ ; the Hilbert function of  $H_0(f)$ ; and a decomposition of the Hilbert function of  $H_0(f)$  as a linear combination of Hilbert functions of finitely generated projective persistence modules:  $\dim(H_0(f)) = \dim(P_{(0,1)} \oplus P_{(1,0)} \oplus P_{(2,2)}) - \dim(P_{(2,1)} \oplus P_{(1,2)})$ .
Figure 3: A Hilbert decomposition (in the sense of [55]) of the module of Fig. 2, and the corresponding Hilbert decomposition signed measure. In the Hilbert decomposition, the supports of the persistence modules in yellow are interpreted as positive bars, while the supports of the persistence modules in blue are interpreted as negative bars. Since bars corresponding to finitely generated projective modules are of the form  $\{y \in \mathbb{R}^n : y \geq x\}$  for some  $x \in \mathbb{R}^n$ , they are uniquely characterized by their corresponding  $x \in \mathbb{R}^n$ ; this is why bars in [55] are just taken to be points of  $\mathbb{R}^n$ .
A.6 Proof of Theorem 1 of the main article
Claim (1.) follows by combining the stability of sublevel set homology in the presentation distance [9, Theorem 1.9  $(i)]$  for  $p = 1$  with the stability of bigraded Betti numbers with respect to the presentation distance [55, Theorem 1.5], and using the fact that the signed 1-Wasserstein distance between the signed barcodes of [55] is equal to the distance induced by the Kantorovich-Rubinstein norm between the signed measures associated to the signed barcodes, which is proven in [55, Proposition 6.11].
We now prove claim (2.), using definitions from [55]. Consider the following two signed barcodes (in the sense of [55]):
$$
(\mathcal{B}_{+},\mathcal{B}_{-}):= \left( \begin{array}{c}\{f(\tau)\}_{\substack{\tau \in S\text{s.t.}\\ \dim (\tau)\\ \text{is even}}},  \{f(\tau)\}_{\substack{\tau \in S\text{s.t.}\\ \dim (\tau)\\ \text{is odd}}}\right),  (\mathcal{C}_{+},\mathcal{C}_{-}):= \left( \begin{array}{c}\{g(\tau)\}_{\substack{\tau \in S\text{s.t.}\\ \dim (\tau)\\ \text{is even}}},  \{g(\tau)\}_{\substack{\tau \in S\text{s.t.}\\ \dim (\tau)\\ \text{is odd}}}\end{array} \right)
$$
It is clear that the signed measures associated to these signed barcodes, in the sense of [55, Section 6.4], are equal to  $\mu_{\chi(f)}$  and  $\mu_{\chi(g)}$ , respectively. By [55, Proposition 6.11] and [55, Definition 6.1], it is then enough to prove that there exists a bijection  $F: \mathcal{B}_{+} \cup \mathcal{C}_{-} \longrightarrow \mathcal{C}_{+} \cup \mathcal{B}_{-}$  such that
$$
\sum_ {i \in \mathcal {B} _ {+} \cup \mathcal {C} _ {-}} \| i - F (i) \| _ {1} \leq \sum_ {\tau \in S} \| f (\tau) - g (\tau) \| _ {1}.
$$
To construct such a bijection, we simply map  $f(\tau) \in \mathcal{B}_{+}$  to  $g(\tau) \in \mathcal{C}_{+}$  when  $\dim(\tau)$  is even, and  $g(\tau) \in \mathcal{C}_{-}$  to  $f(\tau) \in \mathcal{B}_{-}$  when  $\dim(\tau)$  is odd.
We remark that the content of claim (2.) is essentially the same as that of [42, Lemma 12], although they use a slightly different terminology.
A.7 Proof of Lemma 1 of the main article
It is well-known [72, Theorem 12.4.1] that, if  $S$  is a finite simplicial complex, then
$$
\sum_ {i \in \mathbb {N}} (- 1) ^ {i} \dim (H _ {i} (S; \Bbbk)) = \sum_ {\tau \in S} (- 1) ^ {\dim (\tau)}.
$$
It follows from this and Proposition 3 that, if  $(S,f)$  is a filtered simplicial complex, with  $f:S\longrightarrow \mathbb{R}^n$  and  $x\in \mathbb{R}^n$ , then
$$
\mu_{\chi (f)}\left(\{y\in \mathbb{R}^{n}:y\leq x\}\right) = \sum_{\substack{\tau \in S\\ f(\tau)\leq x}}(-1)^{\dim (\tau)}.
$$
Now note that the right-hand side is also equal to the measure of the set  $\{y\in \mathbb{R}^n:y\leq x\}$  with respect to the signed measure  $\sum_{\tau \in S}(-1)^{\dim (\tau)}\delta_{f(\tau)}$ . The result then follows from Lemma 4.
A.8 Lipschitz-stability of vectorizations
Proof of Theorem 2 of the main article. Let  $\lambda = \mu - \nu$  and let  $\psi$  be a coupling between  $\lambda^{+}$  and  $\lambda^{-}$ . Then, for every  $x \in \mathbb{R}^n$ , we have
$$
(K * \mu - K * \nu) (x) = (K * \lambda^ {+} - K * \lambda^ {-}) (x) = \int_ {\mathbb {R} ^ {n}} K (x - y) d \mu (y) - \int_ {\mathbb {R} ^ {n}} K (x - z) d \nu (z),
$$
which is equal to  $\int_{\mathbb{R}^n\times \mathbb{R}^n}\bigl (K(x - y) - K(x - z)\bigr)d\psi (y,z)$ , since  $K(x - y)$  does not depend on  $z$  and  $K(x - z)$  does not depend on  $y$ . Thus,
$$
\begin{array}{l} \| K * \mu - K * \nu \| _ {2} ^ {2} = \left(\int_ {\mathbb {R} ^ {n}} \left(\int_ {\mathbb {R} ^ {n} \times \mathbb {R} ^ {n}} (K (x - y) - K (x - z)) d \psi (y, z)\right) ^ {2} d x\right) ^ {1 / 2} \\ \leq \int_ {\mathbb {R} ^ {n} \times \mathbb {R} ^ {n}} \left(\int_ {\mathbb {R} ^ {n}} (K (x - y) - K (x - z)) ^ {2} d x\right) ^ {1 / 2} d \psi (y, z) \\ = \int_ {\mathbb {R} ^ {n} \times \mathbb {R} ^ {n}} \| K _ {y} - K _ {z} \| _ {2} d \psi (y, z) \\ \leq \int_ {\mathbb {R} ^ {n} \times \mathbb {R} ^ {n}} c \| y - z \| _ {2} d \psi (y, z), \\ \end{array}
$$
where in the first inequality we used Minkowski's integral inequality [44, Theorem 202], and in the second inequality we used the hypothesis about  $K$ . Since the coupling  $\psi$  is arbitrary, we have  $\| K * \mu - K * \nu \|_2 \leq c \| \lambda \|_2^{\mathsf{KR}} = c \| \mu - \nu \|_2^{\mathsf{KR}}$ , as required.
Proposition 4. Let  $K: \mathbb{R}^n \to \mathbb{R}$  be a Gaussian kernel with zero mean and covariance  $\Sigma$ . For all  $y, z \in \mathbb{R}^n$ , we have  $\| K_y - K_z\|_2 \leq c \| y - z\|_2$ , with
$$
c = \frac {\left\| \Sigma^ {- 1} \right\| _ {2} ^ {1 / 2}}{\sqrt {2} \pi^ {n / 4} \det (\Sigma) ^ {1 / 4}},
$$
where  $\| \Sigma^{-1}\| _2$  denotes the operator norm of  $\Sigma^{-1}$  associated to the Euclidean norm.
$$
K (x) = \frac {\exp \left(- \frac {1}{2} \| x \| _ {\Sigma^ {- 1}} ^ {2}\right)}{\sqrt {(2 \pi) ^ {n} \det (\Sigma)}},
$$
where  $\| -\|_{\Sigma^{-1}}$  denotes the norm associated to the quadratic form  $\Sigma^{-1}$ .
We start by noting that  $\| K_y - K_z\| _2^2 = 2\| K\| _2^2 -2\langle K_y,K_z\rangle$ . Now, a standard computation shows that
$$
\| K \| _ {2} ^ {2} = \int_ {\mathbb {R} ^ {n}} \frac {\exp (- \| x \| _ {\Sigma - 1} ^ {2})}{(2 \pi) ^ {n} \det (\Sigma)} d x = \frac {1}{\pi^ {n / 2} \det (\Sigma) ^ {1 / 2}}.
$$
The term  $\langle K_y, K_z \rangle$  can be computed similarly, using the formula for the product of Gaussian densities [56, Section 8.1.8]:
$$
\begin{array}{l} \langle K _ {y}, K _ {z} \rangle = \int_ {\mathbb {R} ^ {n}} K _ {y} (x) K _ {z} (x) d x \\ = \int_ {\mathbb {R} ^ {n}} \frac {\exp \left(- \| x - \frac {y + z}{2} \| _ {\Sigma^ {- 1}} ^ {2} - \| \frac {y - z}{2} \| _ {\Sigma^ {- 1}} ^ {2}\right)}{(2 \pi) ^ {n} \det (\Sigma)} d x \\ = \exp \left(- \left\| \frac {y - z}{2} \right\| _ {\Sigma^ {- 1}} ^ {2}\right) \int_ {\mathbb {R} ^ {n}} \frac {\exp \left(- \left\| x - \frac {y + z}{2} \right\| _ {\Sigma^ {- 1}} ^ {2}\right)}{(2 \pi) ^ {n} \det (\Sigma)} d x \\ = \frac {\exp \left(- \left\| \frac {y - z}{2} \right\| _ {\Sigma^ {- 1}} ^ {2}\right)}{\pi^ {n / 2} \det (\Sigma) ^ {1 / 2}}. \\ \end{array}
$$
$$
\| K _ {y} - K _ {z} \| _ {2} ^ {2} = 2 \| K \| _ {2} ^ {2} - 2 \langle K _ {y}, K _ {z} \rangle = 2 \frac {1 - \exp \left(- \left\| \frac {y - z}{2} \right\| _ {\Sigma^ {- 1}} ^ {2}\right)}{\pi^ {n / 2} \det (\Sigma) ^ {1 / 2}} \leq 2 \frac {\left\| \frac {y - z}{2} \right\| _ {\Sigma^ {- 1}} ^ {2}}{\pi^ {n / 2} \det (\Sigma) ^ {1 / 2}},
$$
where, for the inequality, we use the fact that the function  $z \mapsto 1 - \exp(-z)$  is 1-Lipschitz when restricted to  $\mathbb{R}_{\geq 0}$ . The result now follows from the fact that  $\|y - z\|_{\Sigma^{-1}}^2 \leq \| \Sigma^{-1}\|_2 \|y - z\|_2^2$ , by a standard property of the operator norm.
Proposition 5. Let  $K: \mathbb{R}^n \to \mathbb{R}$  be a Lipschitz and of compact support. For all  $y, z \in \mathbb{R}^n$ , we have  $\| K_y - K_z\|_2 \leq c \| y - z\|_2$ , with  $c = a \cdot (2|\operatorname{supp}(K)|)^{1/2}$ , where  $|\operatorname{supp}(K)|$  denotes the Lebesgue measure of the support of  $K$ .
Proof. Note that  $|K_y(x) - K_z(x)| = |K(x - y) - K(x - z)| \leq a\| (x - y) - (x - z)\|_2 = a\| y - z\|_2$ , by assumption. Now
$$
\begin{array}{l} \| K _ {y} - K _ {z} \| _ {2} ^ {2} = \int_ {\mathbb {R} ^ {n}} \left(K _ {y} (x) - K _ {z} (x)\right) ^ {2} d x \\ = \int_ {\operatorname {s u p p} \left(K _ {y}\right) \cup \operatorname {s u p p} \left(K _ {z}\right)} \left(K _ {y} (x) - K _ {z} (x)\right) ^ {2} d x \\ \leq \int_ {\operatorname {s u p p} \left(K _ {y}\right) \cup \operatorname {s u p p} \left(K _ {z}\right)} a ^ {2} \| y - z \| _ {2} ^ {2} d x \\ = a ^ {2} \| y - z \| _ {2} ^ {2} \int_ {\operatorname {s u p p} \left(K _ {y}\right) \cup \operatorname {s u p p} \left(K _ {z}\right)} d x \\ \leq a ^ {2} \cdot 2 | \operatorname {s u p p} (K) | \cdot \| y - z \| _ {2} ^ {2} \\ \end{array}
$$
Lemma 6. Let  $\alpha$  be a measure on the  $(n - 1)$ -dimensional sphere  $S^{n - 1}$ . The function  $SW^{\alpha}$  is conditionally negative semi-definite on the set  $\mathcal{M}_0(\mathbb{R}^n)$ .