SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof. Let  $a_1, \ldots, a_k \in \mathbb{R}$  such that  $\sum_{i} a_i = 0$  and let  $\mu_1, \ldots, \mu_k \in \mathcal{M}_0(\mathbb{R}^n)$ . Let  $\theta \in S^{n-1}$ , let  $\nu_i = \pi_*^\theta \mu_i$ , and let  $\lambda_i = \nu_i^+ + \sum_{\ell \neq i} \nu_\ell^-$ . Note that  $\lambda_i$  is a positive measure on  $\mathbb{R}$  for all  $i$ , and that  $\lambda_i(\mathbb{R}) = \nu_i^+(\mathbb{R}) + \sum_{\ell \neq i} \nu_\ell^-(\mathbb{R}) = \nu_i^+(\mathbb{R}) + \sum_{\ell \neq i} \nu_\ell^+(\mathbb{R}) = \sum_\ell \nu_\ell^+(\mathbb{R}) = m$  is independent of  $i$ . Then
$$
\begin{array}{l} \left\| \nu_ {i} - \nu_ {j} \right\| ^ {\mathsf {K R}} = \left\| \nu_ {i} ^ {+} + \nu_ {j} ^ {-} + \sum_ {i \neq \ell \neq j} \nu_ {\ell} ^ {-} - \left(\nu_ {j} ^ {+} + \nu_ {i} ^ {-} + \sum_ {i \neq \ell \neq j} \nu_ {\ell} ^ {-}\right) \right\| ^ {\mathsf {K R}} \\ = \left\| \lambda_ {i} - \lambda_ {j} \right\| ^ {\mathsf {K R}}. \\ \end{array}
$$
It follows that  $\sum_{i,j}a_ia_j\| \nu_i - \nu_j\| ^{\mathsf{KR}} = \sum_{i,j}a_ia_j\| \lambda_i - \lambda_j\| ^{\mathsf{KR}}\leq 0$  since the Wasserstein distance on positive measures of a fixed total mass  $m$  on the real line is conditionally negative semi-definite [22, Proposition 2.1 (i)], as it is isometric to an  $L^1$ -distance [57, Remark 2.30]. The result then follows by integrating over  $\theta \in S^{n - 1}$ .
Proof of Theorem 3. We start with the stability of the sliced Wasserstein distance. By linearity of integration, it is enough to prove that  $\left\| \pi_{*}^{\theta}\mu -\pi_{*}^{\theta}\nu \right\|^{K\mathsf{R}}\leq \| \mu -\nu \| _2^{K\mathsf{R}}$  for every  $\theta \in S^{n - 1}$ , which follows directly from the fact that orthogonal projection onto  $L(\theta)$  is a 1-Lipschitz map when using Euclidean distances.
To prove the existence of a Hilbert space  $\mathcal{H}$  and a map  $\Phi_{SW}^{\alpha}:\mathcal{M}_0(\mathbb{R}^n)\to \mathcal{H}$  such that, for all  $\mu ,\nu \in \mathcal{M}_0(\mathbb{R}^n)$ , we have  $k_{SW}^{\alpha}(\mu ,\nu) = \langle \Phi_{SW}^{\alpha}(\mu),\Phi_{SW}^{\alpha}(\nu)\rangle_{\mathcal{H}}$ , we apply [7, Theorem 3.2.2, p.74]; this requires us to prove that  $SW^{\alpha}$  is a conditionally negative semi-definite distance, which we do in Lemma 6. To conclude, we must show that  $\| \Phi_{SW}^{\alpha}(\mu) - \Phi_{SW}^{\alpha}(\nu)\|_{\mathcal{H}}\leq 2\cdot SW^{\alpha}(\mu ,\nu)$ . This follows from the fact that  $\| \Phi_{SW}^{\alpha}(\mu) - \Phi_{SW}^{\alpha}(\nu)\|_{\mathcal{H}} = 2 - 2\cdot k_{SW}^{\alpha}(\mu ,\nu)$  and that the function  $z\mapsto 1 - \exp (-z)$  is 1-Lipschitz when restricted to  $\mathbb{R}_{\geq 0}$ .
B Review of numerical descriptors of persistence modules
Hilbert function. In the one-parameter case, the Hilbert function is known as the Betti curve; see, e.g., [74], and [73] for an extension beyond the one-parameter case. The Hilbert function is itself a numerical descriptor, so it can be used as a feature to train vector-based machine learning models; it has been used in the multiparameter case in [34], where it performs favorably when compared to application-specific state-of-the-art methods for drug discovery tasks.
Euler characteristic. The (pointwise) Euler characteristic of a (multi)filtered simplicial complex is also readily a numerical invariant. It has been used to train machine leaning models in the multiparameter case in [6, 42].
Barcode-based. Many numerical invariants of persistence modules based on the one-parameter barcode have been proposed (see, e.g., [3] for a survey). Since these methods rely on the one-parameter barcode, they do not immediately generalize to the multiparameter case.
Barcode endpoints and filtration values. It has been shown that the full discriminating power of barcodes is not needed to achieve good performance in topology-based classifications tasks [15, 3]. Indeed, [15] argues that, in many cases, features which only use the endpoints of barcodes, and thus forget the pairing between endpoints, perform as well as features that do use the pairings. The work [3] reaches somewhat similar conclusions, although their descriptors do keep some of the information encoded by the pairing between endpoints given by the barcode (in particular making their descriptors not immediately generalizable to the multiparameter case). The analysis of [15] is particularly relevant to our work: our Hilbert decomposition signed measure can be interpreted as a signed version of their pervec descriptor, and our Euler characteristic signed measure can be interpreted as a signed version of their filvec descriptor.
Rank invariant. The rank invariant (Definition 9) can be encoded as a function  $\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{Z}$ , by declaring the function to be zero on pairs  $x\not\leq y$ . However, to our knowledge, the rank invariant has not been used directly as a numerical descriptor to train vector-based machine learning models. Vectorizations of the rank invariant, and of some of its generalized versions [49, 11, 4], have been introduced building on the notion of persistence landscape.
Persistence landscape. The persistence landscape [13] is a numerical descriptor of one-parameter persistence modules which completely encodes the rank invariant of the module. The persistence landscape was extended to a descriptor of multiparameter persistence modules in [76] by stacking persistence landscapes associated to the restriction of the multiparameter persistence modules to all lines of slope 1. Another extension of the persistence landscape, this time to the 2-parameter case, is the generalized rank invariant landscape [79], which relies on the generalized rank invariant restricted to certain convex shapes in  $\mathbb{R}^2$ .
Multiparameter persistence kernel. Any kernel method for one-parameter persistence modules, such as [60, 50, 22, 52], gives rise to a kernel method for multiparameter persistence modules, using the methodology of [32], which relies on "slicing", that is, on restricting multiparameter persistence modules to monotonic lines in their parameter space.
Multiparameter persistence image. Another vectorization method which relies on slicing multiparameter persistence modules is in [20]. Their method uses the notion of vineyard [31] to relate the barcodes of different slices and outputs a descriptor which encodes these relationships.
C Further details about experiments
C.1 Hyperparameter choices
We fix  $\beta = 0.01$  in all experiments. We also use  $d = 50$  slicing lines and a grid size of  $k = 1000$  for MP-SW. In supervised learning tasks, all parameters are chosen using 10-fold cross validation (cv). Beside the kernel SVM regularization parameter, these include: the size of the grid  $k$  is chosen to be in  $\{20, 50, 100\}$  for MP-SM; we use homology in dimensions 0 and 1 (except for the Cleves-Jain data, for which we use only dimension 0) with coefficients in the field  $\mathbb{k} = \mathbb{Z} / 11\mathbb{Z}$ , which we just concatenate for MP-C, or combine linearly with coefficients chosen in  $\{1, 5, 10\}$  for MP-SW. In general, the  $n$  parameters of a multiparameter persistence module  $M: \mathbb{R}^n \longrightarrow \mathrm{vec}$  represent quantities expressed in incomparable units, and are thus a priori incomparable; in order to account for this, we rescale each direction of the persistence module as follows: for MP-SW we choose a constant  $c$  in  $\{0.5, 1, 1.5\}$  for each direction, and scale by  $c / \sigma$ , with  $\sigma \in \{0.001, 0.01, 1, 10, 100, 1000\}$ ; for MP-SW we choose a constant  $c$  in  $\{0.01, 0.125, 0.25, 0.375, 0.5\}$  for each direction.
C.2 One-parameter experiments
We use an alpha filtration [35, Section 2.3.1] for the UCR data and, for the graph data, we filter graphs by the node degrees. For UCR data we use 0 and 1 dimensional homology; we take the sum the kernels evaluated on 0 and 1 dimensional features for the kernel methods and otherwise concatenate the features for the other vectorizations. For graph data we use extended persistence as in, e.g., [21].
The parameters of all vectorization methods are chosen by 10-fold cross validation. As classifier, we use a support vector machine with parameter  $\gamma \in \{0.01, 0.1, 1, 10, 100\}$  and an RBF kernel, except for the kernel methods for which we use a precomputed kernel. The regularization parameter is chosen in  $\{0.01, 1, 10, 100, 1000\}$ . For sliced Wasserstein kernel (SWK) [21] we use  $\sigma \in \{0.001, 0.01, 1, 10, 100, 1000\}$ ; for persistence images (PI) [1] we use the kernel bandwidth in  $\{0.01, 0.1, 1, 10, 100\}$  and a resolution in  $\{20, 30\}$ ; for persistence landscape (PL) [13] we let the number of landscapes to be in  $\{3, 4, 5, 6, 7, 8\}$ , and the resolution in  $\{50, 100, 200, 300\}$ ; and for pervec (PV) we use a histogram with number of bins in  $\{100, 200, 300\}$ .
The reported accuracy is averaged over 10 train/test splits in the case of graphs; for UCR we use the given train/test split.
C.3 Further graph experiments
We compare our methods to the multiparameter persistence methods based on the Euler characteristic ECP, RT, and HTn of [42]. We also compare against the state-of-the-art graph classification methods RetGK [81], FGSD [75], and GIN [80]. We choose these because they are the ones that performed the best in the analysis of [42]; in particular, we use the accuracies reported there. Parameters are as in Appendix C.1, except that we compute accuracy with 10-fold train/test split. The methods RetGK, FGSD, GIN and those from [42] also use 10-fold train/test splits for accuracy. Note that, for simplicity, and as opposed to [42], we do not cross-validate different choices of filtration, and instead we use the following three filtrations: the degree of the nodes normalized by the number of nodes in the graph, the closeness centrality, and the heat kernel signature [69] with time parameter 10. We believe it is possible that better scores can be obtained by considering various other combinations of filtrations. We also remark that choosing three or more filtrations is usually not possible with other persistence based methods ([42] being a notable exception).
All scores can be found in Table 5. One can see that our methods are competitive with the other topological baselines, which shows the benefits of using signed measure and barcode decompositions over raw Euler functions. It is also worth noting that topological methods achieve scores that are comparable with state-of-the-art, non-topological baselines.
C.4 Pointcloud classification filtering Rips by density
In order to check that the performance of the sliced Wasserstein kernel in Section 4.2 can be improved by using a more robust filtration, as explained there, we use here a function-Rips filtration (Example 2) with a Gaussian kernel density estimate with bandwidth in  $\{0.001\cdot r,0.01\cdot r,0.1\cdot r,0.2\cdot r,0.3\cdot r\}$ ,
Dataset	RetGK	FGSD	GIN	ECP	RT	HT nD	HSM-MP-SW	ESM-MP-SW	HSM-MP-C	ESM-MP-C
COX2	81.4(0.6)	-	-	80.3(0.4)	79.7(0.4)	80.6(0.4)	77.9(1.3)	78.2(0.8)	78.1(2.4)	78.4(1.8)
DHFR	81.5(0.9)	-	-	82.0(0.4)	81.3(0.4)	83.1(0.5)	82.8(5)	83.7(5.7)	81.5(3.1)	81.3(3.2)
IMDB-B	71.9(1.0)	73.6	75.1(5.1)	73.3(0.4)	74.0(0.5)	74.7(0.5)	74.7(5)	74(3.9)	72.9(2.9)	74.2(3.9)
IMDB-M	47.7(0.3)	52.4	52.3(2.8)	48.7(0.4)	50.2(0.4)	49.9(0.4)	50.3(3.5)	50.6(3.5)	48.5(4.2)	50.8(3.8)
MUTAG	90.3(1.1)	92.1	90(8.8)	90.0(0.8)	87.3(0.6)	89.4(0.7)	86.8(7.1)	87.3(9.1)	85.1(9)	87.3(9)
PROTEINS	78.0(0.3)	73.4	76.2(2.6)	75.0(0.3)	75.4(0.4)	75.4(0.4)	74.1(2)	73.6(2.3)	71.4(4)	70(3.5)
Table 5: Accuracy scores (averaged over 10-fold train/test splits) on graph datasets. Bold indicates best accuracy and underline indicates best accuracy among topological methods. For our methods and GIN, we report standard deviation, while RetGK reports standard error.
Dataset	num. point clouds (train / test)	num. simplices per point cloud	HSM	MP-SW	MP-C
DistalPhalanxOutlineAgeGroup	139 / 400	76153	7.69	24.87	4.50
DistalPhalanxOutlineCorrect	276 / 600	76153	11.61	56.25	6.64
DistalPhalanxTW	139 / 400	76153	7.78	24.39	4.75
ProximalPhalanxOutlineAgeGroup	400 / 205	76153	8.12	25.67	5.43
ProximalPhalanxOutlineCorrect	600 / 291	76153	12.39	56.83	7.77
ProximalPhalanxTW	205 / 400	76153	8.20	25.25	5.35
ECG200	100 / 100	134137	2.66	1.76	1.22
ItalyPowerDemand	67 / 1029	1561	0.32	0.49	0.31
MedicalImages	381 / 760	147536	9.67	18.96	2.49
Plane	105 / 105	467321	4.52	3.03	2.42
SwedishLeaf	500 / 625	325625	18.34	53.83	8.81
GunPoint	50 / 150	529543	2.44	0.47	0.52
GunPointAgeSpan	135 / 316	529543	5.58	2.53	0.06
GunPointMaleVersusFemale	135 / 316	529543	5.56	2.53	0.05
GunPointOldVersusYoung	136 / 315	529543	5.60	2.50	0.06
PowerCons	180 / 180	467321	7.91	6.05	2.16
SyntheticControl	300 / 300	30913	4.54	13.33	3.82
Table 6: Time (in seconds) to go from point clouds to the Hilbert decomposition signed measure (column HSM), as well as time (in seconds) to go from the signed measure to the output of our proposed vectorizations (columns HSM-MP-SW and HSM-MP-C).
Dataset	MP-K	MP-L	MP-I	HSM-MP-SW	HSM-MP-C
DistalPhalanxOutlineAgeGroup	9227.1	1038.9	217.1	32.56	12.20
DistalPhalanxOutlineCorrect	36734.6	3492.6	833.7	67.87	18.25
DistalPhalanxTW	9396.4	577.7	138.4	32.18	12.53
ProximalPhalanxOutlineAgeGroup	11573.1	759.5	244.5	33.80	13.56
ProximalPhalanxOutlineCorrect	30822.7	2169.5	497.6	69.22	20.16
ProximalPhalanxTW	11641.7	375.4	93.4	33.45	13.56
ECG200	1615.3	1355.6	269.0	4.42	3.88
ItalyPowerDemand	41918.1	1939.0	417.5	0.82	0.64
MedicalImages	147668.1	2404.7	599.5	28.63	12.16
Plane	2036.0	1065.0	249.2	7.55	6.94
SwedishLeaf	38045.7	3329.3	693.5	72.17	27.15
GunPoint	1977.0	1685.7	422.1	2.91	2.96
GunPointAgeSpan	14013.9	3945.6	1078.6	8.11	5.64
GunPointMaleVersusFemale	14069.9	4058.8	1097.0	8.10	5.62
GunPointOldVersusYoung	16668.1	5400.9	1388.5	8.10	5.67
PowerCons	8808.3	3234.8	811.4	13.97	10.08
SyntheticControl	13340.0	595.1	161.9	17.88	8.37
Table 7: Time (in seconds) taken by different vectorization methods for multifiltered simplicial complexes.
Dataset	num. graphs	avg(std) simplices per graph	HSM	MP-SW	MP-C
COX2	467	84.6(8.2)	0.91	6.21	2.24
DHFR	756	86.9(18.2)	1.55	16.85	4.09
IMDB-BINARY	1000	116.3(113.3)	81.49	14.28	3.97
IMDB-MULTI	1500	78.9(117.9)	91.55	29.74	3.33
MUTAG	188	37.7(10.2)	0.27	0.74	0.52
PROTEINS	1113	111.8(129.8)	8.57	26.60	11.96
Table 8: Time (in seconds) to go from graphs to the Hilbert decomposition signed measure (column HSM), as well as time (in seconds) to go from the signed measure to the output of our proposed vectorizations (columns HSM-MP-SW and HSM-MP-C).
where  $r$  is the radius of the dataset, chosen by cv. As one can see from the results in Table 9, MP-HSM-SW is indeed quite effective with this choice.
Dataset	B1	B2	B3	MP-K	MP-L	MP-I	MP-HSM-SW
DistalPhalanxOutlineAgeGroup	62.6	62.6	77.0	67.6	70.5	71.9	71.9
DistalPhalanxOutlineCorrect	71.7	72.5	71.7	74.6	69.6	71.7	75.4
DistalPhalanxTW	63.3	63.3	59.0	61.2	56.1	61.9	66.9
ProximalPhalanxOutlineAgeGroup	78.5	78.5	80.5	78.0	78.5	81.0	84.4
ProximalPhalanxOutlineCorrect	80.8	79.0	78.4	78.7	78.7	81.8	84.5
ProximalPhalanxTW	70.7	75.6	75.6	79.5	73.2	76.1	78.0
ECG200	88.0	88.0	77.0	77.0	74.0	83.0	87.0
ItalyPowerDemand	95.5	95.5	95.0	80.7	78.6	79.8	83.1
MedicalImages	68.4	74.7	73.7	55.4	55.7	60.0	67.9
Plane	96.2	100.0	100.0	92.4	84.8	97.1	99.0
SwedishLeaf	78.9	84.6	79.2	78.2	64.6	83.8	88.6
GunPoint	91.3	91.3	90.7	88.7	94.0	90.7	97.3
GunPointAgeSpan	89.9	96.5	91.8	93.0	85.1	90.5	97.8
GunPointMaleVersusFemale	97.5	97.5	99.7	96.8	88.3	95.9	98.4
GunPointOldVersusYoung	95.2	96.5	83.8	99.0	97.1	100.0	99.7
PowerCons	93.3	92.2	87.8	85.6	84.4	86.7	91.7
SyntheticControl	88.0	98.3	99.3	50.7	60.3	60.0	66.3
Table 9: Accuracy scores of baselines and multiparameter persistence methods on time series datasets. Boldface indicates best accuracy and underline indicates best accuracy among topological methods.
C.5 The enrichment factor
A common approach for quantifying the performance of virtual screening methods [67] uses the enrichment factor  $EF$ , defined as follows. Given a query  $q$  and a test of ligands  $L$ , with each ligand labeled as either active or a decoy with respect to the query, the virtual screening method is run on  $(q, L)$  producing a linear order  $O(L)$ . Given  $\alpha \in (0, 100)$ ,
$$
E F _ {\alpha} (q, L) := \frac {\left| \text {a c t i v e m o l e c u l e s i n f i r s t} (\alpha / 1 0 0) \times | L | \text {e l e m e n t s o f} O (L) \right| / \big ((\alpha / 1 0 0) \times | L | \big)}{\left| \text {a c t i v e m o l e c u l e s i n} L \right| / | L |}.
$$
We use the Cleves-Jain dataset [28], and follow the methodology of [67]. The dataset consists of a common set of 850 decoys  $D$ , and, for each of 22 targets  $x \in \{a, \dots, v\}$ , two to three compounds  $\{q_i^x\}$  and a set of 4 to 30 actives  $L_x$ . To quantify the performance of the method on the target  $x$ , one averages  $EF_{\alpha}(q_i^x, L_x \cup D)$  over the compounds  $\{q_i^x\}$ . The overall performance, which is what is reported in Table 4 of the main article for different choices of  $\alpha$ , is computed by averaging these quantities over the 22 targets.
For topology-based methods for virtual screening which use the EF to assess performance, see [47, 16, 34].
D Runtime experiments
We run these experiments in a computer with a Ryzen 4800 CPU, and with 16GB of RAM.
D.1 Runtime of computation of Hilbert decomposition signed measure
Hilbert function by reducing multiparameter to one-parameter persistence. Let  $(S,f:S\to \mathbb{R}^n)$  be a filtered simplicial complex, and let  $i\in \mathbb{N}$ . Suppose we want to compute the Hilbert function of the homology persistence module  $H_{i}(f):\mathbb{R}^{n}\longrightarrow \mathrm{vec}$  restricted to a grid, which, without loss of generality, we may assume to be  $\{0,\ldots ,m - 1\} ^n$  for some  $m\in \mathbb{N}$ . Given  $a\in \{0,\dots,m - 1\}^{n - 1}$  and  $b\in \{0,\dots,m - 1\}$ , denote  $(a;b) = (a_1,\dots,a_{n - 1},b)$ . In particular, given  $a\in \{0,\dots,m - 1\}^{n - 1}$ , we get a 1-parameter persistence module indexed by  $b\in \{0,\dots,m - 1\}$  by mapping  $b$  to  $H_{i}(f)(a;b)\in \mathrm{vec}$ ; we denote this persistence module by  $H_{i}^{a}(f):\{0,\dots,m - 1\}\longrightarrow \mathrm{vec}$ .
We proceed as follows. For each  $a \in \{0, \dots, m-1\}^{n-1}$ , we use the one-parameter persistence algorithm [38] to compute the Hilbert function of the module  $H_i^a(f)$ . Thus, we perform  $m^{n-1}$  runs of the one-parameter persistence algorithm. The worst case complexity of the one-parameter
The persistence algorithm is  $O\left((|S_{i - 1}| + |S_i| + |S_{i + 1}|)^3\right)$ , where  $|S_k|$  is the number of  $k$ -dimensional simplices of the simplicial complex, but it is known to be almost linear in practice [5].
In the UCR examples, since we are dealing with Vietoris-Rips complexes which can have lots of simplices, we rely on the edge-collapse optimization of [82] to speed up computations. Since the point clouds are not too large, we do not perform any further optimizations, but we mention that optimizations like the ones of [14, 66] are available for dealing with large point clouds.
Runtimes. In the fourth column (HSM) of Tables 6 and 8, we report the time taken for computing the Hilbert decomposition signed measure with resolution  $k = 200$  starting from a filtered simplicial complex. See the experiments section in the article for a description of these filtered simplicial complexes. Then, in the last two columns of Tables 6 and 8 (MP-SW and MP-C), we report the time taken for computing our proposed vectorizations starting from the Hilbert decomposition signed measure.
One can see that, in both tables, the bottleneck is usually either the HSM or the MP-SW computation, as MP-C is quite fast to compute. Overall, the whole pipeline (decomposition + vectorization) can be achieved in a quite reasonable amount of time, as the running times never go beyond  $10^{2}$  seconds, which is quite efficient in the realm of topological methods (see also next section).
D.2 Runtime of whole pipeline
In Table 7, we compare the runtime of our full pipeline (from the point clouds obtained from the UCR datasets to the output of our vectorizations) to that of other pipelines based on multiparameter persistence. For other pipelines, we use the numbers in [20].
It is quite clear that our pipeline is much faster than the other topological baselines, by several orders of magnitude. This is generally due to the fact that Hilbert decomposition signed measures can be computed in the same amount of time than fibered barcodes (which are needed by the baselines), and can be turned into vectors in a single step with one of our proposed vectorizations, while other baselines require vectorizing all elements of the fibered barcodes.
E Stability experiments
In this experiment, we test our main stability results, Theorems 1 to 3. We fix a simplicial complex  $K$  and consider filtrations  $f: K \longrightarrow \mathbb{R}^2$  using the lower-star filtration associated to functions  $K_0 \longrightarrow \mathbb{R}^2$  on the vertices of  $K$ . We construct functions on the vertices of  $K$  as follows: we treat the function as a vector of dimension  $|K_0|$ , start with a constant vector, and iteratively add uniform random noise to each component. This is effectively a random walk that, at each step, produces a function filtering  $K$ . Thus, each random walk produces a set of filtering functions  $\{f_i\}_{1 \leq i \leq k}$ .
For each random walk (shown with a different color), we consider the  $L^1$ -distances between functions  $\| f_i - f_j\| _1$ , the Kantorovich-Rubinstein distances between Hilbert signed measures  $\| \mu_{H_0(f_i)} - \mu_{H_0(f_j)}\| _2^{KR}$ , the sliced Wasserstein kernel distances between vectorizations  $\| \mathrm{HSM - SW}(\mu_{H_0(f_i)}) - \mathrm{HSM - SW}(\mu_{H_0(f_j)})\|_{\mathcal{H}}$ , and the  $L^2$ -distances between convolution vectorizations  $\| \mathrm{HSM - C}(\mu_{H_0(f_i)}) - \mathrm{HSM - C}(\mu_{H_0(f_j)})\| _2$ . See Fig. 4.
The fact that the points in the plot lie below a line with positive slope passing through the origin is a consequence of our stability results for the Hilbert signed measure and for the vectorizations. The fact that the points in the plot lie above a line passing through the origin (at least for points with sufficiently small  $x$ -coordinate), shows that, for this kind of data, our proposed vectorizations are a strong invariant, meaning that it is able to distinguish different filtering functions.
Figure 4: Top:  $L^1$ -distance between filtering functions and Kantorovich-Rubinstein distance between the associated Hilbert signed measures. Bottom: Kantorovich-Rubistein distance between signed measures and distance between the vectors produced by our vectorizations. Different colors indicate different runs of the random walk used to construct the filtering functions. Axis do not have scale since scale depends on the choice of norms and of vectorization parameters.