PERSISTENCE SPHERES: BI-CONTINUOUS REPRESENTATIONS OF PERSISTENCE DIAGRAMS.
Authors: Matteo Pegoraro
Venue: ICLR 2026 Poster (2026)
Keywords: topological data analysis
topological machine learning
linearization
lift zonoid
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: We propose a novel functional representation of persistence diagrams that is bi-continuous: it is Lipschitz continuous with respect to the 1-Wasserstein distance and admits a continuous inverse on its image.

Abstract: Persistence spheres are a new functional representation of persistence diagrams. In contrast to existing embeddings such as persistence images, landscapes, or kernel-based methods, persistence spheres define a bi-continuous mapping: they are Lipschitz continuous with respect to the 1-Wasserstein distance and admit a continuous inverse on their image. This provides both stability and geometric fidelity, placing persistence spheres among the few representations of persistence diagrams that offer an inverse-continuity guarantee.
We derive explicit formulas for persistence spheres and show that they can be computed efficiently with minimal parallelization overhead. Empirically, we evaluate them on clustering, regression, and classification tasks involving functional data, time series, graphs, meshes, and point clouds. Across these benchmarks, persistence spheres are competitive with, and often improve upon, standard baselines including persistence images, persistence landscapes, persistence splines, and the sliced Wasserstein kernel. Additional simulations in the appendices further support the method and provide practical guidance for tuning its parameters.

Published as a conference paper at ICLR 2026
PERSISTENCE SPHERES: BI-CONTINUOUS REPRESEN- TATIONS OF PERSISTENCE DIAGRAMS
Matteo Pegoraro
Department of Informatics Università della Svizzera Italiana Lugano, Switzerland {pegorm) @usi.. ch
ABSTRACT
Persistence spheres new functional representation of persistence diagrams. In contrast to existing embeddings such as persistence images, landscapes, or kernel-based methods, persistence spheres define bi-continuous mapping: they are Lipschitz continuous with respect to the 1- 1-Wasserstein distance and admit a continuous inverse on their image. This provides both stability and geometric fi- delity, placing persistence spheres among the few representations of persistence diagrams that offer an inverse-continuity guarantee. We derive explicit formu- las for persistence spheres and show that they can be computed efficiently with minimal parallelization overhead. Empirically, we evaluate them on clustering, regression, and classification tasks involving functional data, time series, graphs, meshes, and point clouds. Across these benchmarks, persistence spheres are com- petitive with, and often improve upon, standard baselines including persistence images, persistence landscapes, persistence splines, and the sliced Wasserstein kernel. Additional simulations in the appendices further support the method and provide practical guidance for tuning its parameters.
INTRODUCTION
Topological Data Analysis (TDA) is emerging field that leverages concepts from algebraic topol- ogy to study the shape of data, offering coordinate-free and noise-robust methods for extracting meaningful patterns. At the core of TDA lies persistent homology, a framework that captures multi- scale topological features of dataset. By recording the scales at which features such as connected components, loops, and voids appear (birth) and disappear (death), persistent homology produces compact descriptors of data shape. These descriptors are commonly represented as persistence dia- grams (PDs) or barcodes, which provide stable and interpretable summaries amenable to qualitative exploration and (limited) quantitative analysis (Edelsbrunner & Harer 2010 Oudoi 2015
Data Analysis with Persistence Diagrams. To integrate topological information into data analy- sis pipelines, PDs often compared using Wasserstein distances defined through partial optimal transport (POT) (Divol & Lacombe 202T). These distances play a crucial role in ensuring robust- to perturbations, but they also impose highly non-linear geometry non-linearity significantly limits the range of statistical tools that can be directly applied to PDs. For instance, even basic operations such computing averages are non-trivial: they are usually formulated in terms of Wasserstein barycenters (Mileyko et al. 2011), which are computationally intensive to approximate and may fail to yield unique solutions.
the space of PDs. This
Topological Machine Learning: Vectorizations and Kernel Methods. To overcome these lim- itations, numerous vectorization methods have been developed to embed PDs into linear spaces, enabling the use of classical statistical and machine learning techniques. Such embeddings underpin the field of topological machine learning (Papamarkou et al. 2024). where topological features and topological loss functions have proven effective in both predictive and representation learning tasks (Moor et al. 2020; Wayland et al., 2024). For comprehensive surveys we refer to Pun et al. (2022); Ali al. 2023); Papamarkou al. (2024) ) here we only recall the main approaches.

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Broadly, these methods fall into two main categories. The first consists of explicit embeddings of PDs into linear spaces, while the second comprises kernel methods (Reininghaus et al.] 2015]
Kusano al., 2018: Carriere et al. 2017). which employ the kernel trick to define feature maps implicitly. Within the class of explicit embeddings, one can further distinguish between approaches based on dexcriptive statistics (Asaad al. [022). algebraic representations exploiting polynomial rings or tropical coordinates (Kališnik| | |Monod et 2019 Fabio & Ferri 2 functional
representations. which associate each diagram scalar field over chosen domain (Bubenik)
2013
Adams et al. 2017 Biscio & Møller 2019 Dong et al. 2024 Gotovac Dogaš & Mandarić
2025
and other approaches (Mitra & Virk 2024).
Main Contributions. In this work, we build the framework of Gotovac Dogas & Mandarić (see Remark 2) and introduce a new functional representation of PDs, mapping each diagram D to function yp S + R. We prove that this map is Lipschitz continuous with respect to the 1- I-Wasserstein distance between diagrams, and that inverse, on its domain definition, is also continuous. The continuity of the forward map guarantees stability, in the sense that similar diagrams produce similar functions, while continuity of the inverse ensures that functional similarity always reflects similarity at the level of diagrams. A key consequence, which we plan explore in future work, is that this inverse- contini- makes persistence spheres natural candidate for loss design on persistence diagrams: convergence in function space implies convergence at the diagram level. To the best of our knowledge, similar or stronger guarantees have so far been obtained by restricting to diagrams with at most n points, as Bate & Garcia Pulido (2024); Mitra & Virk (0024).
2025
PRELIMINARIES
2.1 CONVEX SETS AND SUPPORT FUNCTIONS
We briefly review the notation and concepts from convex analysis and geometry that will be used throughout. Standard references include
(197)) & Wets (1979).
Definition 1. Given two convex sets A, B R$, their Minkowski sum and their multiplication with non-negative scalar À 2 0. defined
A + B {a + E E AA= (Aa a E A}.
Definition 2. Given a compact convex sel A C IR3, its support function is la IR3 - R, defined as ha(x) maxag A a).
One can check that 1) any support function A is completely determined by its restriction on S2: the operator A + hA is linear: A @ A₂ + A + X2hB.
To compare different convex sets will use the Hausdorff distance.
Definition 3. Given two compact subsets A, B C 7, with (7, dz) being metric space, their
Hausdorff distance defined as d (A, B) = max(max dz (a, B), maxtEB dz (b, A)). With (z B) == infre dz (a, b).
Now we can state the following classical result.
Proposition 1. Given two compact convex sets A, B C R23, the following holds:
max
(v) hn(v)) = (A,
VES?
where du is the Hausdorff distance induced by the Euclidean metric on IR3, In particular, the operator A + hA is injective.
2.2 INTEGRABLE MEASURES ON 2²
For any Borel measure on RR, and any f R2 + R m-measurable, we set (n. f) = Jgp fp)dd((p) Moreover, for any 2 O, we B, (p R2
||plla S r), and B; R2 B,.
In the following we will use integrable measures and uniformly integrable sequences of measures. See Hendrych & Nagy (022) for more details such topics.
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Definition 4. A positive finite Borel measure on R2, μ., is called integrable if:
(p 1 = J (Ielldd((p) oc.
Similarly, sequence of integrable measures {Pa hee is uniformly integrable if:
lim sup [plladup)()) p-000 nEN
BE
To compare measures, need weak and vague convergence of measures, which standard no- tions in measure theory. See, for instance, Kallenberg (1997).
Definition
sequence of integrable measures (JaIneN converges weakly p if (phn f +
(1,, l' for every J IR2 + I continuous and bounded. Instead, if (Jn, - (1, for every
f:R² R22 Th continuous and compactly supported, we say that [PnIneN converges vaguely to p
We write
= for weak convergence and Uin S for vague convergence.
2.3 PERSISTENCE DIAGRAMS
For general overview on PDs and their relevance in TDA, refer Appendix A Here, we adopt measure-theoretie perspective to define PDs. First, we introduce the following notation:
R < = {(x, v) R x <). A== E R2 x x
Definition 6. A PD positive finite measure μD EpED apōp. with 6p being the Dirac delta
centered E RR, D C R2cy being a fnnte set, and ag E N. We refer to the set D as the support of the diagram.
Following Divol & Lacombe 2021)
we give the following definition.
Definition 7, For any measure μ and for any subset C R R2 we define:
Persz(n) = J (y x z)dp((z, y)).
When Z = IR2 w<y we simply write Pers(u).
As proven in Skraba & Turner 2020), in the context of stability for linear operators defined on spaces of measures, we forced to work with the 1-Wasserstein metric. To introduce such metric with a notation convenient for the proofs that follow, we define the following terms.
Definition 8. Consider two diagrams Un and uny partial matching between μn EpED Apop
and HD = EpED: 8p8p a triplet (D,, D', D, + D',) such that:
D, C D and D', D';
Tr + D', is a bijection.
We may indicate partial matching just with 7, for the sake of brevity.
Given partial matching y between PDD EpED apop and pD EpeD 6póp. for every p E D,,, we set "pp := min(ap, b Similarly, for every E D' we set 7i := mbn ay-1())) The cost of y can then be defined as follows:
<(n)
Σ PED
% lp 7p))l +
Σ (ap - Alloo
Σ (b, - --)ll--44 Alloct REDY
DED,
(1)
apllp Allxo +
byll9
Alloo
DED
DED",D
Definition 9. The 1-Wasserstein distance between PDs is defined as:
Wn (HDD inf(c(7) partial matching between pn and pD

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The definition of the 1- 1-Wasserstein distance adopted here is equivalent other formu- lations in the literature. In particular, the bijection & D, - D' can be interpreted as a transport map between the measures EpED, "/p 6p and LgED; "t öq. while the associated cost c(r) corre- sponds to the transportation cost, incluling the cost of sending the remaining mass of both diagrams to the diagonal A (see Divol Lacombe (2
Remark 1.
We recall the following key result from Divol & Lacombe (021)
Theorem 1. (i(D.PD.) + 0 if. and only if. PD.. 4 aD and Pers(up.) + Pers(up).
2.4 LIFT ZONOIDS OF DISCRETE MEASURES
We now introduce the final components needed to define our topological summaries. Throughout, we adopt the following notation: for point p (z, y) R²a, we set (I, == (1, y) R& The reader may refer to Figure[ a visual illustration of the constructions introduced below. For conciseness, Figure T also anticipates material from the following sections and therefore includes some notation that will be formally introduced later on.
As a preliminary step, we recall the construction of the lift zonoid associated with an integrable measure, as presented in Koshevoy & Mosler (1998); Hendrych & Nagyl (2022). For simplicity and coherence with our setting, we restrict attention to discrete measures.
Definition 10. Given a discrete measure μ Ei= a.Spo Ps IR? and a, > 0, the lift zonoid of μ is the following convex set (zonotope);
4,
+ a; (0, C CR,
with 10, (1.p.)) being the segment joining the origin 0 E R3 and the point (1,p)). In particular, the lift zonoid of the zero measure is the set (0} E R.3
Note that the lift zonoid construction is linear: \Mly + d2p2 A1 + A2Z Moreover, the support function of [0, (I] amounts to + ReLU((e, maximum of the inner product is attained on one of the extremes the segment. Hence, by linearity, lift zonoids of discrete measures can conveniently be expressed as sums of rectified linear units.
== max(0, (u, (1, as the
Koshevoy & Mosler [1998); Hendrych & Nagy (2022) prove the following result.
Proposition Given an integrable measure μ and a sequence of integrable measures {Mmn JnEN. the following hold:
du (Zus Zun.) + if, and only if. nn. "> y and fnen uniformly integrable.
3 PERSISTENCE SPHERES
We are now ready to define persistence spheres as the support functions (see Definition 2 of lift zonoids associated with (weighted) PDs, restricted to $2, A running example of this construction is provided in Figure I
As for other functional representations of PDs, see Adams al. (2017). we need to re-weight di- agrams with function w IR2 + (0, 1] so that the weight assigned to points goes to zero as we approach A. Given a diagram puD Eped Sp3p and function R (0, 1), we set 4* := EpED w(p)ayōp-
Definition 11. Given a PD un and function R22 (0, 1). the persistence sphere (PS) of pD with weighting w is defined as v == (hzzgg )1s2.
For any function w R22 + (0, we set [p(p) == (L. We want to control the decay of w as points approach the diagonal. We do so with the following technical conditions.
Definition 12. A function
R22 (0, 1] is called stable lift weighting if:
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Ho totple
(c, Nirely, Dladdh)
Death
odg 1/2
(1-4)
(1.p)
Birts
Birt
Death
4xg220 (1,0)
Birth
Birth
2,5
20 (l. (.
Dealh
RetUI <.(.p)))) >
Brrhh
Figure 1: A detailed example illustrating the construction of the lift zonoid of a discrete measure. In the upper left panel, we start from the measure associated with PD D, endowed with a weighting w (see Section 3). From the point p we obtain the segment wp)][0, (1. p)]. and analogously for q, embedded in Iº with coordinates (z, Birth, Death). We also provide a 2D representation given by the projection onto the plane Death = 0. The lift zonoid Z is then obtained as the Minkowski sum of these two segments. Finally, report the explicit expression of its support function, which, by linearity, is obtained as the sum of the support functions of each segment.
Tw is C-Lipschitz w.t.. || lla for some C >
the following inequality satisfied for every p = (x.y) R2 and some fixed C' > 0:
4 = C'IIp Alloc (²
[TT()) SC
we used the term "lift weighting" emphasize its role in the context of lift zonoids. Since no ambiguity arises in this work, we will omit the qualifier "lift" from now on for brevity. Definition 13. A continuous function w Rp2 > (0, 1] is called an effective (lift) weighting if for any sequence of diagrams (HD.) FueN.
In Definition 12
lim sup w(p) lipll dp (p) = 0 1-000
lim sup Pers (PD..) = 0. 1-0
By
Definition 13 controls the behavior of l' at infinity. To see that, note that, for every > O, there is R such that for every
R and p BC we have: |plla 2 1 1- [[(1,)]]a
For any such
(1 - =) sup [ w())l((.,p)ladpp... (p) I sup / w(p))llladdpp (p).
B:
Which means that, in the context of the definition, we have:
lim sup
[[Eu())lldpoo.)) (p) 0.
(2)
We now provide examples of stable and effective weightings.

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Proposition 3. Ser X(p) 21I(1,))lla The following stable weightings:
2 π
((p) Ka
(p) X(p),
w
arctan
for any K and a 2 1. They are also effective weightings for a = 1.
In what follows, in case a 1, we write wk for simplicity. The weighting w is our preferred choice for constructing PS, because it performs very well in practice (see Section 4). In addition, the parameters and a are highly interpretable and effective for handling noise in PDs, to the point that they potentially be qualitatively selected (see Appendix B
We conclude this section highlighting that, by linearity, the PS of a PD DD = EpED apop. with weighting function we can be explicitly written as:
Pib (v) hizn (v) Σ w()a (c,,U(e
PED
3.1 CONTINUITY THEOREMS
We now state our main results, which contain the continuity properties anticipated in the introduc- tion. First we state and prove them in terms of lift zonoids and Hausdorff distances, which simplifies the proofs, and then, using Proposition
!
we derive the bi-continuity of PSs.
Theorem 2. Let PD: I'D be PDs and let w R2 (0, 1] be a stable weighting. We have:
du(Zug Znz,) S /2 max C'} (HD.
with C, C' > O being the stability constants of w (see Definition 12 Theorem 3. Let (MD. hae be sequence of PDs such that du (Zwz Zuzz - 0, with R² (0, 1] being an effective weighting. Then, W, Wi MD) -
Summarizing the statements Theorem 2 and Theorem and writing them replacing lift zonoids with PSs, we obtain the following result.
Corollary 1. Within the setting of the previous results, we have:
for every E [1, c0] there exist Cp > O such that, for every pair of diagrams μ.D. AD' we have ll Ppp S CyWi (HD,PD) PD:);
if if PHD 1l + O, then WI(UD., HDD 0.
As anticipated in the introduction, Corollary D establishes a particularly strong link between the Wasserstein geometry of persistence diagrams and their functional representation. For instance, for general embeddings into Hilbert spaces, bi-Lipschitz embedding of the Wasserstein space is known to be impossible (Carrière & Bauer 2019). Consistent with this perspective, in Appendix B.2 we present a simulation showing that PS and SW achieve the highest fidelity with respect to the I- Wasserstein distance.
Remark Gotovac Dogaš Mandarić 2025 define their functional representation as qi with (p)) y x. This weighting is stable, since for any C 0. [[T'())]| C'lp Alboo whenever [|pl|2 is sufficiently large. In particular, the map PD + PpD fails to be stable. For instance, let D,, {p.) [[nn2, n² + })}. Then ZD 10, (1, If 0 obtain
Wi(Dn, D) = n_00 O, dz(Zn ZDz) 2 v2 n n= oo.
On the other hand. reasoning as above one can verify that wp) = y x is effective. Consequently, Theorem BanremmBholh holds for the functional representation in Gotovac Dogaš & Mandarid (2025), although this fact not established in that work. Moreover, in Appendix b we test this weight function showing how it leads inferior performances.

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4 EXPERIMENTS
We evaluated PSs on a range of clustering, regression, and classification case studies, comparing their performance with persistence images (PIs), persistence landscapes (PI.s,, persistence splines (PSpl) (Dong et al.] 2024). the sliced Wasserstein kernel (SWK), and PersLay architectures (for supervised problems with sufficient sample size). For PSs, Pls, PSpl. and PLs we used random forest classifiers and regressors, while SWK was coupled with SVMs. Performance was measured using R2 for regression and accuracy for classification, averaged over independent runs for the "Eyeglasses" case study and 10 runs for all other supervised tasks. Clustering performance was evaluated via the Rand index, averaged over 200 independent repetitions. Computational aspects and runtime simulations are discussed in Appendix C
4.1 DATASETS
Clustering Case Study We consider an unsupervised simulation based on a standard functional data analysis (FDA) generative model (Ramsay & Silverman (005). We first construct two smooth random functions ff and g by cubic-spline interpolation of points in [0, 1] x R: on a regular grid = 1, independently, 2). For fixed noise level
0
2200 = 1 we sample y ~ (0, 502), i = ,,..., 1) and g we generate 50 noisy realizations of each function by sampling uniformly {a² 1,000 i.idd in [0, 1] and setting by = f(aa) + eq, e/ ~ ~N,0 o²), and analogously for g. The noise level takes values O-dimensional PD, obtained from the sublevel-
and interpolate {(z,, ((),,v()) Pm0 to obtain f
0,
{10, 15, 30). Each noisy curve is encoded set filtration of the linear interpolation of the sampled points. For each topological summary, as well for the 1-Wasserstein and sliced Wasserstein distances, we compute the pairwise distance matrix, perform hierarchical clustering with average linkage, cut the dendrogram into two clusters, and evaluate the partition via the Rand index. The whole pipeline repeated 200 times for each and the best performing algorithm is selected via grid-search.
"Eyeglasses" Case Study The "Eyeglasses" dataset is a regression case study we designed using the eyeglasses generative model from the scikit-tee This model takes two radii as parameters, and a noise variable which was kept equal to 1. The first radius was always set equal to 20, while the second was sampled according a normal distribution with mean 10 and standard deviation 2.5. We sampled 2000 point clouds and 1-dimensional PDs were obtained from the Vietoris-Rip Rips filtration. For 5 independent runs, we used a 30% 70% split between training and test data and threefold cross-validation was used to select
python package (Saul & Tralie 2019).
Functional datasets from the scikit-fda Package For the following functional datasets, we used zero-dimensional persistent homology derived from the sublevel set filtration of the functions. Data were split into training and test in a 70% 30% ratio, and hyperparameters were selected via threefold cross- cross-validates validation. All datasets are freely available in the scikit- toikiitforr Python package (Ramos- Caaneoo et al. 2024). The datasets "Growth" and "NO," were smoothed using Nadaraya- Watson kernel smoother with bandwidth 3, chosen by visual inspection.
The "Tecator" dataset (https://lib) stat. cmu . edu edd/ asstsstststtttttssseest publicly available measurements collected using the "Tecator Infratec Food and Feed Analyzer". Building on the derivatives of these curves, we explore the same regression problem as in Ferraty & Vieu (2006), trying to regress the fat content of the food samples.
consists of
The "NO," " dataset (Febrero et al., 2008) contains hourly measurements of daily nitrogen oxides (NO,)) emissions in the Barcelona area. The data is labeled based on whether the emission curve was recorded on a weekday or a weekend, and our goal is thus to reconstruct this labeling through supervised classification.
The "Growth" dataset (Tuddenham & Snyder 1954]. also known as "The Berkeley Growth Study", contains height measurements of girls and boys, recorded yearly between ages and 18. A com- mon approach is to analyze the first derivative of the growth curves to distinguish growth dynamics between boys and girls (Vitelli et al. 2010).
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Datasets from Bandiziol & De Marchi (2024) The classification case studies involving the datasets "DYN SYS", "ENZYMES JACC". "POWER". and "SHREC14" were taken from Ban- diziol & De Marchi 2024). As in the previous setting, used 70%-30% train-test split, with hyperparameters selected via threefold cross-validation. For these datasets, we could di- rectly rely on the PDs associated with the classification tasks, which are publicly available at https://git com/ http:///c.aaaaaaiooo/eprrreeeeaa Lettrennceeeeeeeee
In selecting the problems, we prioritized classification tasks with balanced classes and diversity in data type, including point clouds, graphs, time series, and 3D meshes. We now summarize the considered datasets; further details can be found Bandiziol & De Marchi (2024).
The dataset "DYN SYS". first introduced in Adams et al. (017)) and referred to as "Orbit Recog- nition" in Bandiziol & De Marchi (2024). consists of point clouds generated by a one-parameter discrete dynamical system, with the parameter ranging in f2.5, 3.5, 4.4.1, 4.3). The classification task, considered in [Adams et al. (2017): Bandiziol & De Marchi (024) as well as in our work, is to predict the parameter value from the associated point cloud, problem also studied in Carriere et al.) (0177). For each parameter value, 50 independent point clouds were generated, each containing 1000 points with starting positions chosen uniformly at random, yielding dataset of 250 elements. The PDs computed by Bandiziol & De Marchi (2024) contain only one-dimensional features.
The dataset "ENZYMES JACC" addresses graph classification problem. Graphs represent protein tertiary structures obtained from the BRENDA enzyme database (https://wwl www. brenda-enzymes: org/), org//) and the task is to classify each of the 600 graphs into one of six enzyme classes. Edges were weighted by their Jaccard index, and PDs were computed from the resulting sublevel set filtration, combining both zero- and one-dimensional features.
The dataset "POWER", from the UCR Time Series Classification Archive (https://wwy
Cs.Ucr.edus -emmnnnnt teteim__seee me__srrees___e lel_ddaaaa21a___ 87), consists of 1096 time series. The pipeline in this case applied the sliding window embedding (Ravishanker & Chen [202]), followed by the extraction of zero-, one-, and two-dimensional features, which were then merged into single diagram for each time series.
Finally, the dataset "SHREC14" (Pickup al. 2014 is benchmark for non- non-rigid 3D shape classi- fication. It contains meshes of human models across 20 poses and 15 body types (e.g.. man, woman, child), resulting in 300 total meshes. In BBandizoll De Marchi (2024). the Heat Kernel Signature (HKS) (Sun et al., 2009; BBoonstein & Kokkinos 2010) was used to extract one-dimensional PDs from the corresponding sublevel set filtrations.
Datasets "Human Poses" and "MeGill 3D Shapes" The remaining datasets. "Human Poses" and "McGill 3D Shapes". were obtained from https: /ttts::/giituz. com/ ct ctraiie/ TDALabs/ 3DS 3Dshapes................pp...e.b i ipynb The corresponding classification pipelines are documented in the referenced notebook: for the human pose task, a sublevel set filtration of the height function was used, while for the McGill shape classification task, a sublevel set filtration of the HKS was ap- plied. We note that the "McGill 3D Shapes" dataset used here is subsample of the original version, which is no longer fully accessible online. In both case studies, the train-tess split (80%~20%) 20%) was imposed by the dataset limitation of having only 10 samples per class.
4.2 PARAMETERS DETAILS
Now we describe the parameters used for the vectorization and kernel methods, while we defer the discussion of the employed PersLay architectures to Appendix D
Random Forests and SVM Parameters We used SVM pipelines with regularization parameter
C, chosen from (10-3, 10-2, 10 1, 10, 102 103, 104} and precomputed kernel (SWK). For the Random Forests models the number of estimators trained by each forest was chosen in {100, 200). Both were implemented using the
Python package (Pedregosa et al. 201T
Linearization Methods Parameters We now summarize the hyperparameters used for each lin- carization method. For Pls, PSpl and PLs, the range/support parameters were selected by inspecting the full dataset, independently of the training/test split; this introduces minor inconsistency. which
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(c) The PS of the PD in
Figure 2b from the "Eye-
(a) Plot of three point clouds in the "Byeglasses" case study.
(b) A PD from the "Eye- glasses" case study.
glasses study, in po- lar coordinates.
Subject Fose
1.23
Class Pase
1.00
3.5
86-
3.2
65-
00.2
0.50
(d) The (derivatives of the) "Growth" dataset, with the two classes highlighted.
-0.75
(1) sample from the
sample from the "McGill 3D Shapes"
dataset.
"Human Poses" dataset.
Figure 2: Data, PDs, and PSs from some of the experiments in Section
could be avoided in practice by choosing sufficiently generous bounds based only on the training data. For PS the domain is fixed and compact, so this issue does not arise.
PS: used the weighting function wi with K {0, 10-4, 10-3,
10- 10-3,10-3 10-10011000 1..001.0.20 0.25, 0.5),
and a E (1,3,5] where = 0 denotes (with a slight abuse notation) the constant weight p 1. PSs are functions on 5 in spherical coordinates and were expanded in spherical harmonics (Müller 2006) using pyshtools /Wieczorek & Meschede 2018), yielding orthonormal feature representation for sci scikit-less
With Driscoll- Healy grid (Driscoll& & Healy] 1994) having 2N latitudinal and 4.N longitudinal nodes, the feature dimension N we cross-validated 2.N E {30, 40, 50, 60, 70). For the clus-
tering pipeline we used 2N = 100, K E (10-3, 30- 2 10-11-11100 1, 0.3, 0.6) and (1,
Only for the McGill 3D Shapes dataset, we considered 2.N, 14.
Pl: with seikit-tde persim, we set pixel size by enclosing all PDs in a birth firt--persisenne rectangle and dividing its shortest side by Npi. E {100, 500), then rounding to the nearest power of 10. Using the default Gaussian kernel, we took o pixel_size se/N.. No E {0.1, 102 10 10 10 1000, and the persistence ex- ponent n E {1, 1,,44488] 4] in weight-params (100, 500), N₂, E (10-3, 10-2, 10-1, 10²), and n E (2, 4,8), 8), and for McGill 3D Shapes, we restricted Npix E {5, 10..
For clustering we restricted to Npx E
PL: all persistence landscapes were evaluated on a common grid of 5000 points and con- catenated (no hyperparameters). For clustering we used grid of 1000 points.
PSpl: following Dong et al., 2024). we used spline grid of size h2 with {5, 10, 20, 40, 50} and iterations in {5, 10, 50, 100}. As eminence function, we adopted the persistence-bue
one from the original mat code, ported python from "emi- nencef.m". For clustering we restricted to h E {10, 20, 40] and iterations in (10, 50, 100).
SWK: used the sliced Wasserstein kernel from gudhi (Project 2025). fixing M 100 directions and tuning a E (10-5, 10-1, 10-3 10-2 10-1 1,, 10 for the Gram matrix.

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Table 1: Results of the case studies: we report average R2 for regression and average accuracy for classification, across 5 runs for Eyeglasses and 10 runs for the remaining supervised tasks. Unsuper- vised clustering (FDA rows) is evaluated via Rand index over 200 runs, We report mean standard deviation. Bold entries denote the best-perfeer rest-performing method in each row; dagger marks methods whose 95% confidence interval overlaps with that of the best method.
PL
PSpl
PersLay
Unsupervised
FDA 10 (0.800 0.221)
0.845 + 0.158
0.786 + 00165
0.753 + 0.213
0.556 + 0.00T
0.762 + 0.220
FDA 15 (0.717 1 0.223
0.806 A 0.167
0.730 4: 0.159
0.676 4 0.200
0.538 a: 0.062
0.696 4: 0.207
FDA 30 (0.548 4 0.107
0.688 + 0.144
0.621 + 0.103
0.542 = 0.085
0.518 =: 0.014
0.578 = 0.120
Regression
Ryeglasses
0.466 0.003"
0,922
0.000
0.018 0.964
0.965
0.971 0.01
0,248 0.031
0.997 + 0033
0.895
0.010
Classification
Growth
0,850 1 0,052
0.743 0.135!
0.768
0.060
0.807 0.033'
0.507 0.043f
0,768 0.058
NOX
0.869 + 0.041
0,780
0.000
0.789
0.062
0.823 0.133
0.717 0.078
0.840 + 0.055
DYS
0.829 a: 0.028
0.419 4: 0.015
0.840
0.024
0.829 d: 0.032
0.696 4 0.044
0.828 4: 0.028
ENZYMES JACC
0.319 = 0.036'
0.312 + 0.036
0.377 0: 032)
0.373 4: 0.014
0,243 4: 0.023
0,283 = 0.055
POWER
0.769 1: 0,021
0.653 1: 0.066
0.756 2: 0.0015
0.745 1: 0.022
0.725 1 0.038
0.767 1: 1501
SHREC14
0.931 =: 0.022
0.894 +: 0.071
0.943 =: 0.0241
0.949 +: 0.028
0.879 0.018
0,886 0.0W2!
0.640 0.077
0.081
0.405 0.108
0.102
0.082
McGill Shapes
0.541 0.085'
0.151
0.102
0.561 0.104
0.00
4.3 RESULTS
As reported in Table PSs consistently matched or outperformed established topological represen- tations across all the considered tasks. PSplines also performed very well in all supervised settings, suggesting a robust and remarkably low-dimensional representation that particularly convenient for fitting supervised models; however, in line with Appendix [B.2] they performed markedly worse than the other methods in the unsupervised scenario. It is also worth noting that PersLay was likely penalized by the relatively small sample sizes (often between 100 and 1000 observations) and, in the case of Eyeglasses, by the fact that we were unable to identify network architecture yielding competitive performance (see also Appendix[D)
In keeping with the fact that all considered methods have been successfully used in the literature, none of them was dramatically inferior overall. The main practical difficulty arose with Pls, for which identifying suitable parameter ranges proved more delicate and occasionally led to very long runtimes due to slow training of random forests (and other supervised models we tried). Finally, for the McGill 3D Shapes dataset we observed that PSs and, in particular, PIs were more unstable and harder to optimize, with higher variability in accuracy compared to the other methods; in response, substantially reduced the dimensionality of their vectorizations. We did not observe similar behavior on any of the other datasets, as further illustrated in Appendix B
5 CONCLUSION AND BROADER IMPACT
We introduced PSs, novel functional representation of persistence diagrams that is both Lipschitz continuous and admits a continuous inverse on its image, yielding a bi-continuous correspondence with respect to the I-Wasserstein geometry. This combination of stability and geometric fidelity sets PSs apart from existing vectorization methods. Empirically, we find that PSs are not only competitive with, but frequently outperform, widely used alternatives such as PIs, PLs, and SWK.
Several avenues for future work remain. Alternative weighting schemes may yield more expressive summaries. Tools from FDA could support advanced statistical methodologies, such as confidence sets, hypothesis testing, and limit theorems for point processes Biscio et al. (2020). Reconstruction techniques for recovering PDs from scalar fields on the sphere are under development, exploiting existing optimization frameworks for ReLU neural networks. Visualization strategies could enhance interpretability. PSs may also be integrated into modern representation-leamminn pipelines through differentiable topological layers, for instance in their applicability. Finally, extending the construction to signed measures could provide a natural vectorization for bi-parameter persistence Loiseaux al. (003)).
fashion, thereby broadening
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REPRODUCIBILITY STATEMENT
Section 4.2 and Appendix C provide the main details required to reproduce our results. All datasets used are publicly available, and the explicit formulation of our method in Section ensures repro- ducibility. We also submit the code necessary to run the experiments as supplementary material (excluding the data).
THE USE OF LARGE LANGUAGE MODELS
Large Language Models were occasionally employed to refine and polish the writing.
REFERENCES
Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18 (8):1-35, 2017.
Dashti Ali, Aras Asaad, Maria-Jose Jimenez, Vidit Nanda, Eduardo Paluzo-Hidalgee and Manuel Soriano- Trigueros. survey of vectorization methods in topological data analysis. IEEE Trans- actions Pattern Analysis and Machine Intelligence, 45(12):14008) 100440009000,,0 2023.
Aras Asaad, Dashti Ali, Taban Majeed, and Rasber Rashid. Persistent homology for breast tumor classification using mammogram scans. Mathematics, 10(21):4039, 2022..
Cinzia Bandiziol and Stefano De Marchi. Persistence symmetric kernels for classification: A com- parative study. Symmetry, 16(9):1236, 2024.
David Bate and Ana Lucía Garcia Pulido. Bi-lipschitz embeddings the space of unordered m- tuples with partial transportation metric. Mathematische Annalen, (90(2):3109-3133) 2024.
Christophe AN Biscio and Jesper Møller. The accumulated persistence functioo, a new useful func- tional summary statistic for topological data analysis, with view to brain artery trees and spatial point process applications. Journal of Computational and Graphical Statistics, 28(3):671-681, 2019.
Christophe AN Biscio, Nicolas Chenavier, Christian Hirsch, and Anne Marie Svane. Testing good- of fit for point processes via topological data analysis. Electronic Journal Statistics, 14 (1):1024-1074, 2020.
Michael M Bronstein and lasonas Kokkinos. Scale-invariant heat kernel signatures for non non-rigid shape recognition. In 2010 IEEE computer society conference on computer vision and pattern recognition, pp. 1704-1711, IEEE, 2010.
Peter Bubenik. Statistical topologiaal data analysis using persistence landscapes. Journal of Machine Learning Research, 16:77-102, 2015.
Peter Bubenik and Pawet Dłotko. persistence landscapes toolbox for topological statistics. Journal of Symbolic Computation, 78:91-114, 2017.
Mathieu Carrière and Ulrich Bauer. On the metric distortion of embedding persistence diagrams into separable hilbert spaces. In 35th International Symposium Computational Geometry (SoCG 20/9), pp. 21-1. Schloss Dagstuhl-Leeinniz-Zennnn für Informatik, 2019.
Mathieu Carriere, Marco Cuturi, and Steve Oudot. Sliced wasserstein kernel for persistence dia- grams. In International conference on machine learning, pp. 664-673. PMLR, 2017.
Mathieu Carrière, Frédéric Chazal, Yuichi Ike. Théo Lacombe, Martin Royer, and Yuhei Umeda. Perslay: A neural network layer for persistence diagrams and new graph topological signatures. In International Conference Artificial Intelligence and Statistics, pp. 2786-2796 PMLR, 2020.

---

Published as a conference paper at ICLR 2026
Barbara Di Fabio and Massimo Ferri. Comparing persistence diagrams through complex vectors. In International conference on image analysis and processing, pp. 294-305. Springer, 2015.
Vincent Divol and Théo Lacombe. Understanding the topology and the geometry the space of persistence diagrams via optimal partial transport. Journal of Applied and Computational Topology, 5:1-53, 2021.
Zhetong Dong, Hongwei Lin, Chi Zhou, Ben Zhang, and Gengchen Li. Persistence b-spline grids: stable vector representation of persistence diagrams based on data fitting. Machine Learning, 113 (3):1373-1420, 2024.
James R Driscoll and Dennis M Healy. Computing fourier transforms and convolutions on the 2-sphere. Advances in applied mathematics, 15(2):202-250, 1994.
Herbert Edelsbrunner and John Harer. Persistent homology-a survey. In Surveys on discrete and computational geometry, volume 453 of Contemporary Mathematics, pp. 257-282. American Mathematical Society, Providence, RI, 2008.
Herbert Edelsbrunner and John L Harer. Computational topology: an introduction.
American
Mathematical Society, 2010.
Manuel Febrero, Pedro Galeano, and Wenceslao González-Manteigg Outlier detection in functional data by depth measures, with application identify abnormal nox levels. Environmetrics, 19(4): 331-345, 2008.
Frédéric Ferraty and Philippe Vieu. Nonparametric functional data amalysis: theory and practice. Springer Verlag, NY, 2006. URL
Vesna Gotovac Dogaš and Marcela Mandarić. Topological data analysis for random sets and its application in detecting outliers and goodness of fit testing. Statistical Methods & Applications, pp. 1-45, 2025.
Allen Hatcher. Algebraic topology. Cambridge University Press, Cambrige, 2000.
František Hendrych and Stanislav Nagy. note on the convergence of lift zonoids of measures. Stat, 11(1):c453, 2022.
Sara Kališnik. Tropical coordinates on the space of persistence barcodes. Foundations of Computa- tional Mathematics, 19(1):101-129 2019.
Olav Kallenberg. Foundations of modern probability, Springer, 1997.
Gleb Koshevoy and Karl Mosler. Lift zonoids, random convex hulls and the variability of random vectors. 1998.
Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Kernel method for persistence diagrams via kernel embedding and weight factor. Journal of Machine Learning Research, 18(189):1-41, 2018.
David Loiseaux, Luis Scoccola, Mathieu Carrière, Magnus Bakke Botnan, and Steve Oudot. Stable vectorization of multiparameter persistent homology using signed barcodes as measures. Ad- vances in Neural Information Processing Systems, 36:68316-68342. 2023.
Yuriy Mileyko, Sayan Mukherjee, and John Harer. Probability measures the space of persistence
diagrams. Inverse Problems, 1240077 2011.
Atish Mitra and Ziga Virk. Geometric embeddings of spaces of persistence diagrams with explicit distortions. arXiv preprint arXiv:2401.052 205.09298 2024.
Anthea Monod, Sara Kalisnik, Juan Ángel Patino-Galindo and Lorin Crawford. Tropical sufficient statistics for persistent homology. SIAM Journal Applied Algebra and Geometry, 3(2):337- 371, 2019.
Michael Moor, Max Horn, Bastian Rieck, and Karsten Borgwardt. Topological autoencoders. In International conference machine learning, pp. 7045-7054, PMLR, 2020.
12

---

Published as a conference paper at ICLR 2026
Claus Müller. Spherical harmonics, volume 17. Springer, 2006.
Steve Y Oudot. Persistence theory: from quiver representations to data analysis, volume 209. American Mathematical Society Providence, 2015.
Theodore Papamarkou, Tolga Birdal, Michael Bronstein, Gunnar E Carlsson, Justin Curry, Yue Gao, Mustafa Hajij, Roland Kwitt, Pictro Lio. Paolo Di Lorenzo, al. Position: Topological deep learning is the new frontier for relational learning. In International Conference on Machine Learning, pp.
PMLR, 2024.
F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Pretten- hofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-leam: Machine learning in Python. Journal of Machine Learning Research, 12:2825-2836) 2011
D. Pickup, X. Sun, P. L. Rosin, R. R. Martin, Z. Cheng, Z. Lian, M. Aono, A. Ben Hamza, A. Bron- stein, M. Bronstein, S. Bu, U. Castellani, S. Cheng, V. Garro, A. Giachetti, A. Godil, J. Han, H. Johan, I.. Lai, B. Li, C. Li, H. Li, R. Litman, X. Liu, Z. Liu, Y. Lu, A. Tatsuma, and J. Ye. SHREC' 14 track: Shape retrieval of non-rigid 3d human models. In Proceedings of the 7th Eu- rographics workshop on 3D Object Retrieval, EG 3DOR' 14. Eurographics Association, 2014.
The GUDHI Project. GUDHI User and Reference Manual. GUDHI Editorial Board, 3.11.0 edition, 2025. URL httph:tts/:/uutt
iinriai.......... wtt.f/////....ooo 11.0/
Chi Seng Pun, Si Xian Lee, and Kelin Xia. Persistent-hooologyybbee and a comparative study, Artificial Intelligence Review, 55(7):5169-5211 2022.
machine learning: survey
Carlos Ramos-Carreñna José L. Torrecilla, Miguel Carbajo Berrocal, Pablo Marcos Manchón, and
Alberto Suárez. scikit-fda: A Python Package for Functional Data Analysis. Journal of Statistical Software, 109(2):1-37, May 2024.
James O. Ramsay and Bernard W. Silverman. Functional Data Analysis. Springer, New York, NY, USA, 2005.
Nalini Ravishanker and Renjie Chen. An introduction to persistent homology for time series. Wiley Interdisciplinary Reviews: Computational Statistics, 13(3) 548, 2021.
Jan Reininghaus, Stefan Huber, Ulrich Bauer, and Roland Kwitt. A stable multi-scale kernel for topological machine learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4741-4748, 2015.
Tyrrell Rockafellar. Convex Analysis, volume 28. Princeton University Press, 1997.
Gabriella Salinetti and Roger J.-B. Wets. On the convergence of sequences convex sets in finite dimensions. SIAM Review, 21(1):18-33, 1979.
Nathaniel Saul and Chris Tralie. Scikit-tda: Topological data analysis for python, 2019. URL htttps://do..orgg http://.rrrrr...../e/eeee/e 5281/22nod.. 2533369
Primoz Skraba and Katharine Turner. Wasserstein stability for persistence diagrams. arXiv preprint
arXiv:2006. 16824, 2020.
Jian Sun, Maks Ovsjanikov, and Leonidas Guibas. A concise and provably informative multi-scale signature based on heat diffusion. In Computer forum, volume 28, pp. 1383-1392 Wiley Online Library, 2009.
RD Tuddenham and MM Snyder. Physical growth of california boys and girlsfrom birth to age 18. Calif. Publ. Child Develop, :183-366, 1954.
Valeria Vitelli, Laura Maria Sangalli, Piercesare Secchi, and Simone Vantini. Functional clustering and alignment methods with applications. Communications in Applied and Industrial Mathemat- ics, 1(1): 1(1):205-224 2244, 2010. htLhttts::///// htts:://doo..rg/.... 685 / 1100AI1486
Jeremy Wayland, Corinna Coupette, and Bastian Rieck. Mapping the multiverse of latent represen- tations. In International Conference Machine Learning. PMLR, 2024.
13

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Published as a conference paper at ICLR 2026
Mark A Wieczorek and Matthias Meschede. Shtools: Tools for working with spherical harmonies. Geochemistry, Geophysics, Geosystems, 1948) 2018.
A FILTRATIONS AND PERSISTENCE DIAGRAMS
As mentioned the introduction, TDA provides wide range of techniques to extract features with desirable invariance properties. It does by leveraging algebraic topology, which offers natural framework for identifying structures that remain unchanged under continuous deformations of the domain (Hatcher 2000). A central tool in TDA persistent homology, which builds these ideas to track the evolution of homological features, such as path-connected components (O-dimensional holes) and loops (I-dimensional holes), across filtration, that is, nested family of topological spaces.
Filtrations generated by real-valued functions and point clouds are among the most general and widely used. Given a topological space X and a function f X - IR, one considers the sublevel sets X2 (( ((-oc, fl). The changing topology of the family {Xt)tes encodes information about the structure of f.
Similarly, for a finite set X R", one can consider the filtration
X, U ( Rd | zEX
pll < <1).
visual representation of this filtration, known the Čech filtration of X, is provided in Figure
To extract topological information from a filtration, one typically applies homology functors Ha, HI..... with coefficients in field. The resulting families of vector spaces, usually referred to as persistence modules, track the birth and death of features such as path-connected components and loops.
To make this information amenable to data analysis, persistence modules are encoded by topological summaries. Among these, persistence diagrams are arguably the most widely used in TDA; for detailed survey, see, for instance, (Edelsbrunner & Harer 2008).
Loosely speaking, a persistence diagram is multiset of points (Cz, Cy) in the upper half of the plane, with cy > Cz. where ca denotes the value of the parameter at which homology class in X, first appears (its birth), and Gy is the value of / which the same class either disappears or merges with previously born class (its death).
B THE ROLE OF THE WEIGHTING FUNCTION AND THE ASSOCIATED PARAMETERS IN PSs
In this section, we examine the weighting function
arctan A(p) Ka
wx (p)
which is the one used in our simulations and case study, and we analyze the effect of its parameters K and a. Observe that studying this function Eysy is, by symmetry, equivalent to extending it to IR² by replacing y with ly zr. We will adopt this viewpoint throughout the section.
We begin with an informal, qualitative description, aimed at providing an intuitive overview of the roles played by wx. K, and and then move on to rigorous mathematical justification of the claims. The reader may find helpful to refer Figure 4 and Figure 5 throughout the discussion.
At high level, the function wx is smooth step function on the plane: it starts from 0 along the diagonal y = x and, as the direction of p rotates away from the diagonal y = æ toward y -r, it transitions monotonically to higher values, bounded by 1. The parameter K controls both the location and the width of this transition region: determines where the weight assigned to points begins to decrease from towards smaller values their persistence approaches 0, and how spread out this transition is. By contrast, the parameter a primarily affects the steepness of the step, without shifting its location (see Figure
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Death
Death
F7
Ho
H1
14
r6 Birth
Fo
Birth
Tg
16
T7
Figure schematic illustration of point cloud in R.? its Čech filtration, and the associated persistence diagrams in homology dimensions O and 1. Path-connected components consisting of singletons are highlighted with circles, while I-cycles are emphasized using colored polygons. The colors of the polygons are consistent with those of the corresponding points in the diagram associated with H₃ and with the horizontal bars representing the life-spans of topological features. Red bars path connected components.
Recall that increasing a beyond 1 generally breaks the bi-continuity guarantees of the PS represen- tation. However, as will see in Appendix [.I]] choosing 1 can still be beneficial in practice when working with particularly noisy diagrams. For the rest of this section, we focus on the case 1.
We now make the above qualitative picture more precise, trying to derive an explicit characterization of how K affects the step function. In doing so, central role played by the quantity
x
Xp)
(1,
which, for each point p = (x, v). measures the ratio between its persistence and the Euclidean norm of the corresponding vector (1, the lift-zonoid representation (in the unweighted case). Alternatively, one can see:
--N 1(00,)]l2
A(p) = (0. (21((1,
The term M- is the cosine of the angle between the vectors (z, 3) and (-1, while the second V2llpll
term is constant if we keep (p) constant. As consequence, À vanishes on the diagonal y x, its values increase with the angle between (z, y) and the vector (1, 1). and they are maximized along y -r, while remaining strictly smaller than 1. Feeding this function into sigmoidal nonlinearity such as the arctan produces a step-like function that is 0 on y = æ, increases as we move away from the diagonal, and gradually flattens
we approach y -r, as in Figure 4
Multiplying À by 1/K rescales its Lipschitz constant by 1/K. Since arctan (() < 1 for all 4, follows that, for a
Lip(wR) < Lip(A/K) = Lip(A)
2
π
π K
Hence larger values of K produce a flatter transition (smaller Lipschitz constant), while smaller K yield sharper step. In other words, K controls the steepness of the resulting step function and
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shrinks enlarges the transition region between the diagonal and the plateau where the weight is close to (see Figure 4).
As shown in Figure the level sets of the weighting not parallel to the diagonal, which would be the case if wk depended only on the persistence of the points. Instead, they exhibit a radial behavior with respect to the origin. This feature is implicitly enforced by the definition of the lift zonoid, which built from segments of the form [0. [1.)]] and is somewhat analogous to what happens with the sliced Wasserstein kernel, where points projected onto straight lines passing through the origin. In fact, both representations induce metrics which are not translation invariant on PDs.
Building this picture, now add further layer of mathematical rigor by deriving explicit expressions for the (asymptotes of the) level sets of A/ K. These formulas underpin the visualizations reported in Figure 4 and Figure and the practical considerations we discuss in Remark
Since 2 x, for a fixed value we can write
=5: X(p)
--z
(y-x)² (2Kz)² (1 +)²).
K
2K (1,,,,l,
This identity leads to
((2K2)² - 1) + x²² ((2K2)² 1) 2ry + (2K2)² -.
(4)
Suppose momentarily that (2Kz)² - + 0. Since À is maximized along the direction y -I, from [I(1, v)ll2 > [1 122 we obtain
(2Kz)² 2.
V2K
As a consequence, ((2K2)² - 1)2 S 0. Using this, one can show that Equation describes hyperbola, centered the origin, whose focal points lie the line y - (indeed, y/ corresponds to z = 0 and thus cannot be intersected by the hyperbola when > 0). This means that we can interpret the level sets of X/ via the asymptotes of this hyperbola.
Introducing the notation A = 2K and B = 1 - -A3, the slope of the asymptotes of this hyperbola can be written as
1 1 V1 B
M1,2
(5)
Recall that A E [0.2] and B E [-1.1]
Moreover, B is a monotone decreasing function w.r.t (and so 1 - B is monotone increasing) and changes sign K
To get an even more interpretable view on this, let 6 be the angle between the vectors (1, m.) (representing the asymptotes) and (1, 1) (representing y x). That
1 1+ m
0, cos
2
2(1 +
Figure displays the functions (K)), whose behavior we now briefly discuss. As K grows, the lines identified by each angle 0 move from the line y x to the line -r, with the line of slope mi rotating counterclockwwse and that of slope m2 rotating clockwise. At first sight, Figure 5a may seem to contradict this interpretation for larger values of K, as 0 suddenly decreases, but a closer inspection shows that this is not the case.
When B changes sign, also m1, change sign. Moreover, when K Zz mi and so, when mi changes sign, it goes from pointing upward almost vertically, to pointing downward almost vertically, and so the angle f jumps, as represented in Figure [a but the line it represents still moves counterclockwise, and our interpretation remains consistent. Instead, since m2 -+ 0, it does not jump when K goes across z shown in Figure 5b
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To summarize, K governs the geometry of the level sets (x(p) 2, i... the loci of points in the diagram that are assigned a fixed weight 2. For instance, as K - O, we have m1,2 + 1, meaning that the corresponding level sets move toward the diagonal y z. Conversely, as K increases, the level sets associated with a fixed value z move farther away from the diagonal. The case = (and B 0) yields singularity, which now admits clear interpretation: the asymptote with coefficient my becomes the vertical line I = O. Indeed, substituting 2K= = 1 into Equation (4) gives 2xy + O.
Figure also provides additional insight, with practical implications, into the behavior of ( which quantifies how quickly the asymptotes are displaced as K varies. The region most relevant for applications is typically the one close to the diagonal (i.e., for smaller values of K), where the dependence on K' is essentially linear.
For example, and as illustrated in Figure 5b setting K = 0.5 roughly corresponds to assigning score of 0.5 z arctan(1) to points lying at an angle of x x/4 with respect to y 2, that is, at an angle of π 7/2 with the æ-axis (namely, on the y-axis, halfway between y x and -z). By approximate linearity, choosing = 0.25 instead assigns the same score 0.5 = arctan(1) to points near an angle of /8 with y x, and so on.
In other words, for K S 0.5, the relationship between K and the angle can be bounded as T K. particular, v22K is first-order approximation of (1))(K) for K V2=K S 0, I 2
close to zero Before proving this, we summarize these considerations and their practical implica- tions in the following remark.
Remark 3. Taken together, these observations show that any regular grid in (0,0.5) will move the level set (p( = 0.5 from arbitrarily close to the diagonal y = x towards the y-axis in roughly uniform angular steps. Since noise in persistence diagrams typically lies near y x, is often preferable, as we do in practice, to instead use an irregular grid that is denser near the lower end of this interval. Additional values larger than 0.5 can also be considered, especially for unsupervised analyses. if the data are concentrated in the quadrant x < 0,, >> O (see Figure 4b) and are strongly affected by noise; in that case, values of a >> 1 should be explored as well, as illustrated in Appendix B.I trading bicontinuity guarantees for increased stability. Note, however, -a. We
that K = 1/V22 x 0.7 corresponds to the level set wx (p) = 0.5 lying on the diagonal emphasize that this entire analysis is driven solety by the structure of A K and does not depend on the specific choice of the sigmoidal nonlineariry. Consequently, the qualitative behavior of the level sets and their dependence on K would carry over to other sigmoid functions beyond arctan(z).
Lastly we derive the first order approximation of (22((K) for K + 0.
By Taylor:
0² 0 for A 0.
X cos(f)
2
Thus, can write 0 2 2X for X 1. Set:
+ m;
cos(O,) = X
2(1 + m?)
For K + 0, we have X; + 1. Again by Taylor:
2 2-2X; (mg - 1)2
O(mm: 1)2) for m; +1 1.
Lastly, consider:
2
22 A²
(m: 1)2
+ V B
1 + Av2 AZ
1) +
4
B
A²
for A + O. Remember that A 2K2, so if K + 0 then +
Putting the pieces together, we have:
03 ~ (2Kz)²
for K ~ 0.
2
Since know that 0 2 obtain: 0, R v2zK for K O.
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K-0.6,
K-0.1, a
100
100
Mk(p1 = 0.5
Hk(p) 0.3
0.5
75
75
08
25
25
0.4
-25
-25
-50
-30
0.1
-100
0.0
-100
0.0
100
(a) Heatmap of wx for = 0.1, a = 1. High- (b) Heatmap of wk for K = 0.6, = 1. High-
lighted (the asymptotes of) the level set wx (p) (2/x) arctan (1) 0.5.
lighted (the asymptotes of) the level set wk (p)
(2/x) arctan 0.5.
K=0.3,
k=0.3,a=4
Me(p)
0.7
Me((p)
08
50
50
0.6
0.4
0.3
-25
-25
02
-75
-75
-100
-100
-55
-50
-25
25
50
75
100
-100
-75
-50
-25
50
75
100
(c) Heatmap of wx for 0.3,
High- (d) Heatmap of wk for K 0.3, 4. High-
lighted (the asymptotes of) level set (p( (2/z) arctan(1) = 0.5.
lighted (the asymptotes of) the level set wx (p)
(2/x) arctan 0.5.
Figure 4: Comparison of the step function wk for different values of K and In the first row, decreasing K shrinks toward the diagonal y = x the region where the weight is significantly reduced for low-persistence points, thereby enlarging the area where the weighting flattens out near 1. The asymptotes, shown for the level set wK(p) = 0.5, highlight how the location of a fixed score shifts under different values of K. Moreover, the transition from lower to higher scores becomes more gradual for larger K, reflecting smaller Lipschitz constant. In the second row, the role of in handling highly noisy diagrams is illustrated: increasing a leaves the level set wx unchanged in position, but sharply increases the steepness of the step. This produces extended region with low scores near the diagonal, and a correspondingly large flat region with high scores farther away from it.
0.5
B. THE ROLE OF a
To illustrate the effect of a and its potential relevance in practice, we propose the following simula- tion study. The guiding idea is that, when persistence diagrams contain an overwhelming number of points close to the diagonal that can be regarded noise (i.e. do not carry meaningful variability), the 1- I-Wasserstein distance can be severely affected. Indeed, among the p- Wasserstein distances, the case p = 1 is the least robust, since the cost of sending points to the diagonal scales linearly with persistence. On the other hand, precisely this linear sensitivity can make the I-Wasserstein distance particularly effective for discrimination tasks.
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m//
6- 2K
/2X
B Zor
Zox
3m/4
3m/4
by m41
10,
Sm/t
Bra3kon
002
#
Km
Km
100
m04
-09
0.0
0.1
0.2 0.4 0.5
1.6
5.7
0.0
0.3
0.2
0.3
0.4
0.5
1.6
67
(a) The function (K) for 1, measured in ra- (b) The function 02(K) for dians. The highlighted points correspond the dians. The highlighted points correspond to the two diagonals: y
measured in ra-
(red) and y = = (purple). The diagonals: y x (red) and y = (purple). The black dashed lines emphasize the approximately lin- black dashed lines emphasize the approximately lin- ear behavior of 0, (K') for small values of K, while ear behavior of 012(K) for small values of K, while the vertical orange dashed line marks K 1/(2)). the vertical orange dashed line marks K 1/(2),, where my becomes singular and the corresponding where my becomes singular and the corresponding level set acquires a vertical asymptote.
level set acquires vertical asymptote.
Figure 5: Behavior of the angle (in radians) of the asymptotes the level sets function of K. Near the diagonal y = x (slope π7 /4), the angular coefficient varies approximately linearly with K, and then starts to increase more rapidly as K approaches 1/ 1/(v2z). The region close to (i.e. near the diagonal) is the most relevant for handling noise, and the essentially linear dependence K there implies that a simple uniform grid in K explores uniformly the corresponding range of angles.
To mitigate the influence of near-diagonal noise, other linearization methods such as Pls and persis- tence splines adopt more rapidly decaying weighting schemes. For instance, in the case of PIs one may use higher powers of persistence as weights. This is exactly the role played by a for PSs: as shown in Figure 4d larger values of induce sharper decay of the weighting as one approaches the diagonal y T.
We now present simple simulation to demonstrate this effect in practice. We generate persistence diagrams belonging to two classes i = 1, each of cardinality 50, such that within each class there is comparable "core" of points away from the diagonal, together with highly variable number of points near y = x. Formally, for each class i E 2} and each replicate k E (1....,50), the diagram has the form
= = ((b)h d )3
{(br 22 33
and is sampled as follows:
7e 5%,,5%)
my u(1,, -1). 1)).
mi = [ma].
11 uid N(M, (M/110)3),
81. und N(M, (M,110)))
j 1, Tkk
mik
Pj Lid NMM (Mr/10)3),
PI. ltd (MM/100 (M,/10)3),
= =b + 1p
= 10t
The first subset (()) d 1T represents the informative points in the diagram, while
((b)* ) captures the noisy points near the diagonal. The parameter N controls the expected cardinality of the informative part, whereas N' governs the noisy part. Note that the variables {m+} sampled uniformly in [1, N' 11. and therefore have much higher variance than )
In our experiments, we set = 50, Mi = 100, and M2 = 70, and consider N' E (10, 1000, 5000, 10000).
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Disc
202
202
36
25
63
325
34
520
(a) A plot of all diagrams from (b) A plot of all diagrams from (c) A plot of all diagrams sam-
class I, for N' 5000, col- class 1, for N' 5000, col- pled when N' = 5000, colored ored using wk with K 0.2 ored using wk with K 0.2 according to their classes. and
We display the data and the effect of increasing to sharpen the transition between low and high weight points. The parameter 0.2 was chosen by visual inspection to separate the noisy points from the others.
Figure 6: Figures related to Appendix B.1
For each value of N', we compute PSs from the sampled diagrams and then perform hierarchical clustering with average linkage on the resulting PS distance matrix. Cutting the dendrogram to obtain two clusters, whose partition we evaluate with the Rand index.
We report the results obtained with PS using K = 0.2 (chosen by visual inspection; see Figure 6 6. For reference, we also compare with Pis (with grid maximize clustering accuracy), using persistence the weighting
and two values of a, namely a and parameters selected function, first with q = and then with = 6. The results in Table indicate that higher levels of stability, corresponding larger values of q and a, lead improved clustering performance.
Table 2: Rand index for different methods and noise levels N'.
Method
N'
10
N'
1000
N' = 5000
N
10000
PS 1
0.97
0.66
0.59
0.60
PS 6
0.97
0.97
0.92
0.96
PI
1.00
1.00
0.71
0.81
Pl q
1.00
1.00
1.00
1.00
B.2 THE ROLE OF K: AN UNSUPERVISED SIMULATION
In this section, we present simulation study illustrating that the main effect of the parameter K is to modulate the contribution of points near the diagonal, thereby altering the geometry of the vectorizations only when many noisy points are present. To demonstrate this, we consider setting where points are sampled uniformly on large bounded subset of {(I, y) E R22 x < y). i.c. without an explicit concentration of points near the diagonal, and examine how varying K affects pairwise relationships between diagrams. We extend this experiment to all vectorization methods considered the paper, assessing how well each topological summary reflects the Wasserstein geometry the underlying persistence diagrams.
To this end, we generate independent random pairs of persistence diagrams and compare their 1- Wasserstein distance with the distances induced by the corresponding vectorizations (or the sliced Wasserstein distance in the case of SWK). Specifically, we sample 500 independent pairs of dia- grams, compute their 1-Wasserstein distances, and correlate these values with the distances between their vectorized representations.
Each diagram is generated as follows: first draw an integer N uniformly from [1. 10 then sample an N x 2 matrix with entries independently and uniformly drawn from [1, 10+]. Finally, we add the first coordinate to the second to enforce the constraint y.
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Table Correlation with 1-Wasserstein distance for different parameter settings (values rounded to 10 10-*).
SW (s)
PS (K)
PI N,.
PL
PSpl (h,iter)
: 0.9989
0: 0.9993
(50, 0.001) 0.9283
0.9650
(10,5) 0.1831
10: 0.9989
0.001 0.9993
(50,0.1): : 0.9220
(10, 10) 0.1752
20 0.9989
0.01 0.9993
(50,1)
:
0.1 0.9994
(50, 0.9214
(20, 10) 0.2433
0.5 0.9996
(100, :.000..0)))0090
(40,5) 0.3805
(.10,
(10, 10) : 0.3316
(1:
(100, 10) 0.9214
The results, reported in Table for different parameter choices across methods, show the impact of these hyperparameters an unsupervised setting. The ranges explored for PI and PSpl are chosen based on the best-performing configurations in the case studies, while for PS we fix Ng = 100 and 1.
Table 3 also provides practical validation of Corollary for this generative process, PS achieves the highest correlation with the original Wasserstein distances, independently of the choice of K. The next best performance, with almost identical results, is obtained by the sliced Wasserstein dis- tance, which the other vectorization method for which a form of inverse continuity has been established Carriere Carriier et al. (2017). By contrast, PSpl exhibits by far the weakest agreement with the Wasserstein geometry, suggesting that it may be ill-suited for unsupervised analyses, as confirmed by Table I
C COMPUTATIONAL ASPECTS, RUNTIMES, AND ADDITIONAL IMPLEMENTATION DETAILS
The computation of PS scales linearly with both the number of points in the diagram and the size of the evaluation grid. As shown Equation (3). it reduces to evaluating standard mathematical functions in one or two variables. Since these evaluations are independent across points, the pro- cess can efficiently parallelized. As a result, PSs are potentially cheaper than PIs, which require binning and integration. and PLs, whose fastest known algorithm has complexity O(n log + n.) (Bubenik Diotko (017)), where n = #D and is the number of nonzero landscapes. Approx- imating SWK (and treating the number of slices as constant) incurs computational cost of On log n) (Carriere et al. (017). When evaluated on a grid, PSs have the same dimensionality as PIs on a comparable grid, since both are scalar fields on 2D manifolds.
For PSpls we ported in python the matlab code found in ht htttp::/// //t::://gttttb/t// com/ 2C119/PP since the grid size of PSpls was never exceeding 502, even without optimizing the code, were able to run all the needed experiments. Still, since our code is not optimized, the upcoming runtimes comparison will not feature PSpls.
C.I RUNTIME SIMULATION
In this simulation, we aim illustrate the linear computational cost PSs. To this end, we randomly sample PDs of varying sizes and compute the corresponding PS, PI, and PI representations. To keep the comparison fair, we choose grids so that the resulting vectorizations have comparable sizes: PSs computed
grid of shape (100, 200) (so their spherical harmonics representation has 1250 coefficients), PIs on a grid of shape (200, 100), and PLs on a grid of 500 points, yielding a representation whose size depends on the number of landscapes. We also report runtimes for PSs including the decomposition into spherical harmonics, so as to compare against the runtimes of the actually employed vectorization.
The generative process for the random PDs is straightforward: for each N E (5, 10, 10 103 10 2- 10a). sample matrix of shape (N, 2) with entries drawn independently and uniformly from
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baasata M--20
tie pira Aatavsi Sw1
04/54
1427
04/04
(a) Boxplots of runtimes for ran- (b) Boxplots of runtimes for ran-
(c) Boxplots of runtimes for ran-
dom diagrams with N = 10 dom diagrams with N 100 points.
dom diagrams with N points.
10
points.
Burtires
05-
6.50
1.004800
(d) Boxplots of runtimes for ran-
(e) Boxplots of runtimes for ran-
(1) Median runtimes as func- tion of the number of point in diagram, in log-log scale.
dom diagrams with N 10 dom diagrams with N 2- 10* points.
points.
Figure 7: Figures associated with Appendix C The boxplots indicate that PSs achieve runtimes comparable established implementations of PIs and PLss, while Figure 7 illustrates almost perfectly linear growth of the median PS runtime with respect to the number of points in diagram.
10%] We then add the first coordinate to the second to enforce x < y. For each value of N, we generate 1000 random diagrams.
The results, reported in Figure clearly display a linear relationship between runtime and the num- ber of points in the diagrams (see in partieular Figure 7f.. and show that the anterical-harmmniie based vectorization is very efficient. At the same time, Figure 7 indicates that PIs and PLs are often computed more quickly. Recall that, for PIs and PLs, relied on the highly optimized scikit -aad persim module, We therefore expect that our current PS implementation could be further optimized and potentially integrated into this module.
D PERSLAY ARCHITECTURES
In our experiments with PersLay, each persistence diagram is first rescaled to lie in [0, 1] x [0, 2] and then processed by multi-branch architecture combining up four topological chan- nels with a small dense head. Between NGasss = and NGaus = 3 branches employ different image resolutions, namely 10 x 10, 20 20, or 50 50, on the fixed birth-death box [0, 1] x [0, 2]; the corresponding variance parameter is learned during training. The remaining branch uses TentPers PerslayPhi evaluated a grid of Nieut samples in [0, 2), with Ntent ranging from 100 to 1000, and the sample locations treated as trainable parame- ters. In all branches we adopt wwwessinmiixtuueeeee. components the weighted features are aggregated over points by sum pooling and passed through a wranch-speciffy mutch-normalization layer; the Gaussian-image outputs and the tent output are then flattened and concatenated into a single feature vector.
K layWeigghh with mixture plane, where KGauss varies between 5 and 15. Within each branch,
On top of this representation we place a small fully connected head: first dense layer with NR&LU ReLU units, where NRELU E {16..... 483, followed by a dropout layer with rate lärop E [0, 0.4], and second dense layer with NRELU 2 ReLU units. This is followed by task-specific output layer: for classification we use a softmax layer with C' (number classes) units and optimize categorical cross-entropy, while for regression
single linear unit optimized with mean squared emror.
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Dataset
NGaws
Munt
KGawa
NBLLU
Ydrop
Eyeglasses
Eyeglasses
10
D
Tecator
500
15
0.15
Growth
500
13
16
0.25
10
DYN.SYS
500
13
0.40
22
ENZYMES JACC
POWER
500
15
48
0.40
SHRECI4
0.40
-10-
Table 4: PersLay hyperparameters used for each case study,
-150
-100
-50
50
100
130
200
PCL
Figure 8: PS-based PCA of the Eye- glasses dataset.
In all cases, the network is trained with the Adam optimizer. During our analyses, we explored different configurations within the parameter ranges described above, monitoring training rums of 20 to 50 epochs (depending network size) with batch size 32, and then selected the final architectures reported in Table 4 All other experimental choices (e.g. train-test test splits) were kept consistent with the remaining pipelines.
As already mentioned in the main text, we were unable to identify a PersLay architecture that achieved competitive performance the Eyeglasses dataset, even after adjusting the train-test ratio to increase the amount of training data. We therefore focused on very shallow networks, motivated by the low intrinsic dimensionality of this case study: PCA on the spherical harmonics representation of PS shows that the first principal component alone explains 0.985 of the variance (see Figure [8. In the end, the berf-perg architecture used single Gaussi anPers layer with resolu- tion 10x x 10, no Tent PerslayPhi branch, no dropout, and Gaussian mixture with KGaus = 10 components.
Lastly, we note that, by choosing as point transformation CCrrrèér et al. 2020
p ReLU (1)),
52
one essentially recovers the basic building block of PSs. This suggests that PSs could, in principle, be integrated directly into PersLay architecture as an additional topological channel, for instance by treating a grid on S as a set of trainable features, along with the weights, in close analogy with how the landscape-based transformation is handled.
E ABLATION STUDIES
Lastly, we present of ablation studies designed isolate the practical impact of individual PS parameters on performance. To this end, we considered the four datasets from Bandiziol & De Marchi (2024) and, in each experiment, varied a single parameter while keeping all others fixed. These datasets were chosen because, for a fixed method, they exhibit substantial variability in clas- sification accuracy: some case studies are considerably more challenging than others.
The default parameter configuration, i.e., the values used whenever a parameter was not under abla- tion, was set to 2N,, = 40, 10-2, and a = The number of estimators random forest was fixed to 100 throughout.
We first varied 2N, E (30, 40, 50, 60). We then explored K E 0-1,0,0.00 1-010,0001,0.11003.. where K -1, K 0 respectively denote, with abuse of notation, the weighting function (y x2/2 (used by Gotovac Dogaš & Mandarić 2025)), and the constant weighting function equal to 1. Finally, examined 11,2,4,8.
The results, reported Table 5 show that PSs are very robust to parameter choices in supervised case studies. This is largely because, in supervised settings, the learning algorithm can compensate for suboptimal weighting functions and still effectively suppress noise in the data (which is precisely the role of K and a). In unsupervised situations, the choice should be guided by Appendix B
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Table Ablation Studies. Mean + standard deviation (over 10 runs) for each dataset and parameter setting.
POWER
DYN_SYS
SHRECI4
ENZYMES
2N 30
0..002000026
0.779 = 0.035
0.909 = 0.020
0.364
2No 40
0.764
0.783 0.018
0.887 + 0.035
0.354 + 0.028
2No 50
0.768++0000
0.817 + 0.021
0.881 0.037
0.378 0.022
2No 60
0.704
0.809 = 0.026
0.893 = 0.032
0.386 0.026
K = -1
0.481 0.001
0.155 + 0.012
0.031 0.011
0.134 0.010
K
0.809 = 0.029
0.847 0.023
0.349 0.027
K = 0.01
0.790 = 0.029
0.884 = 0.023
0.362
K 0.1
0.766
0.806 = 0.022
0.912 0.031
0.371 0.027
K = 0.5
0.801 = 0.015
0.907 = 0.033
0.357 + 0.025
Q =
0.770 0.022
0.783 = 0.037
0.893 = 0.028
0.364 0.031
2 a =
0.769 0.020
0.805 = 0.030
0.877 = 0.026
0.368 + 0.035
0.755 0.019
0.799 = 0.017
0.886 0.030
0.347 = 0.038
0.014
0.803 + 0.022
0.908 0.028
0.371 + 0.024
Remarkably, the only choice which yielded considerably worse results is K = -1, which used to indicate the weighting function used in Gotovac Dogaš Mandarid (2025).
Performance is also highly stable with respect the choice of grid (and thus the dimension of the vectorization), despite the different sizes and difficulty levels the considered case studies. We attribute this to the fact that each PS is Lipschitz function the sphere, and therefore well-behaved and not overly difficult to approximate via spherical harmonics expansions.
As noted in Section 4.3 the McGill 3D Shapes dataset is the only exception this pattern: despite its modest size, substantially increasing the dimensionality of the PS vectorization led to clearly improved results.
PROOFS OF THE RESULTS
Proposition Ser A(p) 211(1,))ll=
The following are stable weightings:
( = arctan ((p))a
K(p) = X(p),,
Ka
for any K 0 and a 2 They are also effective weightings for 1.
Proof. The functions T. have the following forms:
(y x))a
Ta( 20 x. 1 (1,
(y x)
wx (x. y) arctan
(1,
2 K [| x. v)
Lipschitzianity is obtained because the components of the functions Io and Tax are differentiable and have bounded partial derivatives on IR2 1
To check the norm condition for stability, we write down the expressions of T
(y x
TT v)ll2
2 (1, v...v
(y x)aa
[T (4, arctan
111,,,,
20 barKa ( x. 5)
At this point, we observe that:
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(y x)a-1
(1 I. 2)12 0- [0, 1);
2
and that:
(y x
x)a
arclan
[I a. v) S k
2a K (1, y>1
a (11
H. 8)112
The first observation is enough to prove stability for 2, while the second and the first observations, combined, prove for wk.
Now we prove that both weightings are effective for a 1, exploiting Equation functions have become:
Note that the
x) 1lp Allxo
[|T (2. f)1l
2
(y (y-z) x)
[T (I, 2)ll2 arctan
[[(1, x, 21)l2
|1 I, 4)1l2
To that 7. is effective, it suffices to observe that, plugging the expression of ||T fl12 in Equa- tion ( we directly obtain the thesis.
Now deal with Tag- Set JDD, EpEd, an.ppp
We rewrite Equation 2
as:
lim sup
Σ
WT (p)ll2 S lim sup
Σ
an. wxx (p)ll2 0.
0-000
P PeDallllllas
p.Da..
Thus, for every O, there is R such that, for every R the following holds:
sup pEDmilpplla7
Σ
Twx(())l <E
(6)
Hence, for every n, we have:
Σ
aretan
(y x)
TE
(7)
21,,,)l
2r
p= (=(=,) D 2el
In particular, Equation implies 2 2K](,,,0)
(y-w)
+ 0 for + co. Equivalently, for every C > 0,
there is ro such that 21...))) C.
The key observation now, is that, due to the concavity of + arctan(2), which easy see due to the strict monotonicity of its derivative I+ we have:
arctan
< arctan(2) <
(8)
for every [0, and every fixed E 2 In fact, + arctan(s) is the straight line joining (0, 0) (y=z)
and (s, ar))an())) Thus, for every C > 0 and for every ra such that 2 (1.(1.))) for every
< C, we have
Σ
an x)
20n aretan peDasllpllaSree
(y x) 2K][ v. 12
[[(1, xr 1)
PE Drc. 1
with C" 2arctan(C) xCK
In particular, find O' > 0 such that:
C' lim sup Persge (HD.) S lim sup
Σ
an.pl Tay (p)ll2 + 0.
y-000
p pla
Which proves the thesis.
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Theorem 4. Let P.D P'D be PDs and let w :R² [0, be stable weighting. We have:
du dn(Zpg 2nz, S v2 max C*} Wi Wi(DD.HD),
with C, C' 0 being the stability constants of w.
Proof. Consider μD = EpED Appp. = EgED bydg and partial matching between them.
Without loss of generality, suppose C = max ( CC,C",
generic point Zuz has the form:
P=
Σ PED
apSp apppTu(p) E Zuz,
with 8p E [0, 1]. We start by considering p E D.y and P E Zug with the following form:
P (p)))_(p)
Note that, by definition, 7/p E N and 7/p 2 1.
Consider the point:
Q Q->T[u(7(D))
Since 1/p by6) there is [0, such that sby(p) Yp- Thus, Q Zaz.
We have:
P Qlla S (p) [=((()))]22 S 7,O'lp -
7(2)
For P in the form of Equation 9 we define ((P) =
Consider now a generic P E Zuz=
P Σ pp (w)) Σ
7p 7p +
Σ (ap (w)))((())) Σ
ap8pT
PED
PED,
PED,
PED-D,
We build Q follows:
Q= Σ
sp Sp
PED,
We have:
11- Σ TT( - -T(()()))22 Σ
splap -
PED,
PED,
(10)
(p))l1Tup))112
PED-D. D.
Plugging into Equation 10) the following facts:
1. [p) I Clp - Alloc
2. || 11255 1 22 1ooi
3. sp < 1 for every p D,
we obtain:
||P - -Q12
C V2 Σ
70l1p op) + Σ (ap 7p)llp - Alloo +
Σ
apllı Alloo
PED,
PED,
PED DED-D, D2
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Since can do this construction for any partial matching 7, for every P Zuy, we found Q such that:
Reversing the role of μD and μD we obtain the thesis.
Theorem 5. Let {uD. be sequence of PDs such that du (Zwz. 2uz) - 0, with R22 (0, 1] being effective weighting. Then Wi (HD., PD) + O.
Proof. Combining Proposition 2 with the assumption du dz(Z,z 2μz) O, we obtain that JD " 1 and that (18 INE uniformly integrable.
Let / E Ca(RR and set E == supp(f) C 22 Since E is compact and En A = Ø3, by continuity and strict positivity of w we have me == minper w()) > Hence = f/w is continuous on E and bounded by llffloo/mg Extending h by 0 outside E (note that f is on dE), we obtain a bounded continuous function on I² and therefore, by weak convergence,
= (JLD f) f Idpp. -
- -s
f dup. = fIndvö, (1) (WB, (uD: f).
Thus J.D,, 3 J.p- If we prove Pers(uD,,) + Pers(un)) then we can conclude by Theorem By uniform integrability we have:
lim sup w(p) ll P lle duD, (p) lim sup P10000 nEN Bę
P| ll dpD. (p) O.
1-000 neN. B
Since is an effective weighting, this implies:
lim sup Perso; (JDD.) - O, r-000 neN
For any r 2 we can write:
f R2 (y =))pp,(((z,)))
/ (y ((,,) / (y - (--ddd, (Zx, v)). B.
2 PPrrs(uD.) =
Similarly, we can write:
2 Pers(up)) = { RR
- (0-2)dpp((z,u))) v)) / (y- (===))p(((Ex))) (y (0-r)dpp((=;)))
B3
B,
I is big enough, being D finite, we have supp(D) C. Br, and so S (v - r)dup(((; u)) = 0 and J.. (y z)duD((x, y)) 2 Pers(uDD.
Fix some big enough so that the above holds. Since B, compact write a positive test function R2 w + [0, 1] such that:
g is continuous;
g on B,;
supp(g) is compact.
For such a g. we obtain:
0 < / (y- x)dup, (x.,
()v, z)duD, ((x,9)).
13
Moreover, using vague convergence, we get:
/ 1 g(x,v)(y r)dun ((x, v)) 3 2 Pers((p))
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But:
1 g(a,y)(y- ()
g(a, z)duu. + / g(a, u)ly g(z,8)(v - 2)duD, ((1, ())
B
B.
f (y- - x) (p + /
g( 20 - r)ddp, y)).
Br
B;
Thus:
0 s { g(r,y)(y x)duD, ((x,0)) - / (y- xddu ((x,)) B.
/ g( wy r)dun, (1x, 2 / (v-z)d xddun ((r,
B
Putting the pieces together, for every & > 0 there exist T' and N₂ such that, for every n 2 No:
/ (u (u- b)) Ise,
2 Pers(un,.)
Brr
/
2 Pers(uD)
g(2, v) - ((du IS E,
g(2, 4) - x xddpD. ((x, y)) -J (y (y-x)da z)dpu, ((z, s E, RR2N entailing Pers((u,,) - Pers(uu) is 38, concluding the proof.
5
Brc
28


Reviews:
[{"review_id": "KCO7PhsHhz", "reviewer": "Reviewer_FZZ6", "summary": "This paper proposes persistence spheres (PS), a new functional representation of persistence diagrams (PDs) for topological data analysis. Unlike existing embeddings, persistence spheres yield a bi-continuous embedding, which guarantees both stability and geometric fidelity. The paper defines persistence spheres via the support functions of lift zonoids of re-weighted persistence diagrams, proves continuity theorems, and provides explicit, efficiently computable formulas. Experiments across several benchmark datasets and show that persistence spheres are comeptitive or superior to established methods in regression and classification.", "questions": "1. How sensitive are the results to the specific hyperparameter choices—particularly the weighting function $\\omega$, the number of basis functions — and could the authors provide an ablation or sensitivity analysis to clarify their impact on performance and stability?\n2. Runtime comparisons: the paper claims linear complexity, but actual wall-clock runtimes versus PIs/PLs are not reported—can authors provide them?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper demonstrates originality by introducing a mathematically rigorous yet computable embedding that enhances geometric fidelity compared with existing topological data analysis (TDA) vectorizations. Its quality is supported by solid theoretical foundations, complete proofs, and carefully reproducible experiments. The empirical validation spans diverse application domains, including functional, graph, mesh, and point-cloud data, underscoring the versatility of the proposed method. In terms of scalability, the approach achieves linear complexity with respect to the number of persistence diagram points and allows straightforward parallelization.", "weaknesses": "The paper lacks an ablation or sensitivity analysis on key hyperparameters such as the weighting function $\\omega$ or the number of basis functions. These parameters likely influence both the expressiveness and stability of the representation, yet their impact is neither theoretically discussed nor empirically evaluated.\nGrammar errors:\nLine 465: “due the high variability” → should be “due to the high variability.”", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RVFHKMEvyQ", "reviewer": "Reviewer_D5QX", "summary": "This paper presents persistence spheres, a new way to represent persistence diagrams as functions. The method provides a bi-continuous mapping that is Lipschitz continuous with respect to the 1-Wasserstein distance and has a continuous inverse on its image. These properties ensure both stability, meaning similar diagrams produce similar functions, and geometric fidelity, meaning that similar functions reflect similar diagrams. Together, they offer an optimal balance between robustness and faithfulness to the original geometry of persistence diagrams.", "questions": "How do persistence spheres advance the broader understanding of how topological information can be integrated into machine learning, beyond simply providing another competitive embedding for persistence diagrams?", "limitations": "", "rating": null, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "The experiments demonstrate that persistence spheres perform consistently well across a wide range of data types.", "weaknesses": "The paper as a whole reads more like a sequence of definitions and technical explanations than a cohesive narrative, which makes it difficult for readers to grasp the overarching motivation and significance of the work. While the theoretical foundations are clearly laid out, the presentation lacks a guiding storyline that connects these formal definitions to the broader research goals and practical implications. Nearly two pages are devoted solely to dataset descriptions, and almost another full page focuses on hyperparameter settings and implementation details. While such information supports reproducibility, it disrupts the flow of the paper and could be moved to an appendix or supplementary material. This space could instead be used to strengthen the narrative by clarifying research goals, providing interpretive discussion, and highlighting the significance of the findings. Moreover, several analytical gaps limit the depth of the study. There are no direct runtime benchmarks, so although computational efficiency is discussed qualitatively, explicit timing comparisons with existing topological representations such as persistence images, persistence landscapes, or the sliced Wasserstein kernel are missing. Hyperparameter sensitivity is not analyzed, leaving unclear how robust the method is to parameter changes, and no ablation studies are provided to isolate the effect of specific design choices such as the weighting function or spline grid size. Scalability testing is also limited to moderate-sized datasets, offering no insight into how the approach performs on large or high-dimensional data. Finally, the reported results rely on averages over runs but include no statistical significance tests or confidence intervals, making it difficult to assess whether performance differences are meaningful.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Q29sVqdUOl", "reviewer": "Reviewer_4NTj", "summary": "This paper proposes a novel functional representation for Topological Data\nAnalysis, specifically for Persistence Diagrams (PDs).\n\nThe approach builds upon the theory connecting persistence diagrams with\nmeasure theory and optimal transport. The functional associated with a\npersistence diagram (viewed as a discrete measure) is then derived from the\nsupport function of a zonoid (a convex set) constructed from this measure.\n\nAfter the necessary background, the authors present the novel functionnal :\nPersistence Sphere, and show that it satisfied multiple desirable properties\nfor a TDA representation, namely: stability, and separability.\n\nThe paper concludes with an experimental section, showcasing the performances\nof this approach on various datasets.", "questions": "- See weaknesses.\n - When comparing with the SWK, two question naturally arise : 1) is it\n possible to have a better separability bound ? e.g., f(W_1(...)) <= || PS...\n ||_p  for a non-trivial non-decreasing function f ? and 2) The sliced\n wasserstein distance can be exactly computed on small enough diagrams, since\n the combinatorix is finite, and can thus be decomposed into a finite number\n of cells. What about the || PS_1 - PS_2 ||_p of two diagrams ?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Well written. \n - Proofs seem to be sound \n -  The proposed construction sounds reasonable : stability (w.r.t.\n Wasserstein 1) is a must have for persistence diagram representations, and\n the separability property is very nice\n -  Computational complexity is linear\n - The code is available, and the code is reproducible in notebooks, with\n clear parameter selection.", "weaknesses": "- This paper is hard to read for someone who is not familiar with measure\n theory / TDA. For instance, there is no real intuition on what is a\n persistence diagram, how to interpret the matching cost, etc. As the paper\n presents a representation for PDs though, I think this approach can be argued\n to be fair. \n - The experiment section is weak. IIUC the SOTA is roughly from 2017 (sliced\n waserstein kernel).\n - l60-70 : Unless I misunderstood something, SWK is a stable and bicontinuous\n representations as well, with stronger separability properties (cf [SWK for\n PD, theorem 3.3]). The comparison to SWK is still on your favor though, since\n SWK is not an unsupervied vectorization method.\n - Def. 10 : a little picture/explanation could help\n - l193 : uniformly integrable is not def\n - l225 : \"and p \\in B_r^c\" \n - l232 : link with uniform integrability?\n - l240 : \"and α ≥ 1. They are also effective weightings for α = 1\" what is\n the point of the last sentence ?\n - l244 : This is a bit fast here. I suppose the ReLU comes from the fact that\n the segments of the zonoid are based on 0.\n - l257 : the [Carrière Bauer] paper mentionned in the \"Contribution\"\n paragraph fits here ?\n - l273 : this is a bit fast as well here. I assume that $Z_\\varnothing =\n \\{0\\}$?\n - l477-483 : some references should be added\n - l647 : K\n - l740 : I'm not sure the notation with \\Phi help a lot\n - l777 : By prop. 2, [...] to conclude with thm 1\n - l787 : effective weighting\n - l793, 798: a factor 2 is missing", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SRYGMSNcTN", "reviewer": "Reviewer_9Ldx", "summary": "The paper proposes a new way of vectorizing persistence diagrams (PD). The idea of the vectorization procedure is the following: for each point $(b, d)$ of a PD with multiplicity $k$, consider a line segment from the origin to the point $(k, kb, kd)$. Take the Minkowski sum of all these segments, it is a convex polytope in $\\mathbb{R}^3$. Now map the PD to the support function of this polytope (restricted onto the unit sphere). In fact, one has to apply weighting (points that are closer to the diagonal must contribute less). The paper establishes conditions for the weighting function that guarantee that the mapping PD -> support function is a Lipschitz embedding with contiuous inverse. The experimental part demonstrates the efficiency of this vectorization technique on various classification problems, by comparing it to many other known methods.\nNB: the unweighted vectorization scheme was proposed in the literature, and the paper contains the appropriate references and attribution; the actual contribution of the paper is weighted case, its theoretical properties and experimental evaluation.", "questions": "Why not use spherical harmonics? I mean, to represent the vectorization instead of splines one could use the Fourier decomposition. Or has it been tried and the splines perform better?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- The paper is rather well-written and very clear, being also practically self-contained.\n- The properties that the embedding needs are certainly important, not just from a theoretical viewpoint, but also practically. Therefore the criteria on the weighting established in the paper are a solid contribution.\n- The experiments cover various problems and look convincing.\n- The proposed vectorization method is embarassingly parallel and can be expressed by a very simple formula.", "weaknesses": "- functions on a sphere are hard to work with, unlike, say, persistent images which are honest-to-goodness 2D numerical arrays. This fact is acknowledged in the paper.\n- the paper could include the results for some of the experiments that the unweighted case paper made.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wXsBPbGUE8", "reviewer": "Reviewer_YUj9", "summary": "This paper proposes a bi-continuous representation (with respect to the 1-Wasserstein distance) of Persistence Diagrams (PDs) via the restricted support of lift zonoids, i.e. Persistence Sphere (PS). The property of being bi-continuous is theoretically proved. The experiments focus on regression and classification tasks.", "questions": "See Weaknesses.", "limitations": "", "rating": 2, "confidence": 5, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "Strengths:   \n\n- The bi-continuity w.r.t. Wasserstein distance is very interesting and important for ensuring PD correspondence, the key component in Topological Machine Learning.\n- The background definitions and proofs are through and self-contained.\n- The bi-continuity holds for different weight functions ws. This provides flexibility for the representation.\n- The theoretical proofs for continuity theorems are sound.\n- The datasets in experiments are extensive and the results are convincing.", "weaknesses": "On the presentation aspect:\n- The discretization method of the Persistence Sphere, a restricted support function of lift zonoids is unclear.\n- The paper focuses too much on the math notations. Only one visualization (Fig c) on the proposed method is provided. The lack of visual examples in $\\mathbb{R}^3$ of the key components such as lift zonoid of a PD (definition 10) and PS (definition 11) makes these concepts difficult to understand intuitively and reduces readability.\n- The background of Persistence Diagram, a multiset on the open half plane, is very limited. Maybe a detailed one including the filtration process could be included in the Appendix.\n\nOn the experiments aspect:\n- The baselines (PI, PL, SWK) are very limited. More recent methods such as Persistence B-spline grids [1], an unsupervised method, should be included.\n- The baselines are all unsupervised vectorization methods. Since the paper focus on supervised task like classification, I think it is necessary to compare PS with supervised vectorization like PersLay [2] to show whether PS can outperform supervised baselines. \n- The paper gives computational analysis without showing the actual time cost of PS. A scale-up test would make the analysis more convincing.\n- Although it is nice that the bi-continuity holds for different weight functions $w$s, when it comes to experiments, it would be necessary to show how the choice of weight function affects the results. An ablation study on the choice of weight function would be nice.\n\n[1] Dong Z, Lin H, Zhou C, et al. Persistence B-spline grids: stable vector representation of persistence diagrams based on data fitting[J]. Machine Learning, 2024, 113(3): 1373-1420.  \n\n[2] Carrière M, Chazal F, Ike Y, et al. Perslay: A neural network layer for persistence diagrams and new graph topological signatures[C]//International Conference on Artificial Intelligence and Statistics. PMLR, 2020: 2786-2796.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)