Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures
Authors: David Loiseaux
Luis Scoccola
Mathieu Carrière
Magnus Bakke Botnan
Steve Oudot
Venue: NeurIPS 2023 poster (2023)
Keywords: topological data analysis
multiparameter persistent homology
kernel methods
optimal transport

Abstract: Persistent homology (PH) provides  topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case---where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest---and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes---a recent family of MPH descriptors---as signed Radon measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures
David Loiseaux Luis Scoccola Mathieu Carrière', Magnus B. Botnan", Steve Oudot
DataShape, Centre Inria d' Université Côte d' Azur Mathematicc, University of Oxford *GeomeriX, Inria Saclay and École polytechnique
Mathemattcs, Vrije Universiteit Amsterdam
Abstract
Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus the one one- parameter case- where the descriptors summarize the changes in topology of data as is filtered by single quantity of interest- and there is the use of one-pane one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors elements of Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one- one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, aim bring together the best of both worlds by showing how the interpretation of signed barcodes-4 a recent family of MPH descriptors-as signed measures leads to natural extensions vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-bel based methods on various types of data.
wide array of methods enabling
Introduction
1.1
Context
Topological Data Analysis (TDA) [26] is a field of data science that provides descriptors for geo- metric data. These descriptors encode topological structures hidden in the data, as such they are complementary to more common descriptors. And since TDA methods usually require the sole knowledge of a metric or dissimilarity measure on the data, they are widely applicable. For these reasons, TDA has found successful applications in wide range of domains, including, e.g., computer graphics [58), computational biology [62], or material sciences [63], to name a few.
The mathematical definition of TDA's topological descriptors relies on persistent homology, whose input is simplicial complex (a special kind of hypergraph) filtered by an R"-valued function. The choice of simplicial complex and function is application-depeeddent, common choice being the complete hypergraph on the data (or some sparse approximation) filtered by scale and/or by some density estimator. The sublevel-sets of the filter function form what is called filtration, i.e., family h of simplicial complexes with the property that S. C Sz whenever x S x' E R" (where
Equal contribution.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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by definition z S x + S 1 n). Applying standard simplicial homology [45] with coefficients in some fixed field k to ( yields what is called a persistence module, i.e., an R" parametrized family of lk-vector spaces connected by k--inear maps- foonmally, a funcior from RR to the category vec of vevettor spaces. This module encodes algehraically the evolution of the topology through the filtration (St) heRR, but in a way that is neither succinct nor amenable to interpretation.
Coneise representations of persistence modules are well-developed in the one-parameter case where n = 1. Indeed, under mild assumptions, the modules are fully characterized by their barcode [33], which can be represented as a point measure on the extended Euclidean plane [36]. The classical interpretation of this measure is that its point masses masses-also referred to as bars-encode the appearance and disappearance times of the topological structures hidden in the data through the filtration. Various stability theorems [23, 29, 30, 68] ensure that this barcode representation is stable under perturbations of the input filtered simplicial complex, where barcodes are compared using optimal transport-type distances, often referred to as Wasserstein or bottleneck distances. In machine learning contexts, barcodes are turned into vectors in some Hilbert space, for which is possible to rely on the vast literature in geometric measure and optimal transport theories. A variety of stable vectorizations of barcodes have be designed in this way-see [3] for a survey.
There are many applications of TDA where mulliparameter persistence modules (that is, when n 2 2) are more natural than, and lead to improved performance when compared to, one one-aarameter persistence modules. These include, for example, noisy point cloud data [77]. where one parameter accounts for the geometry of the data and another filters the data by density, and multifiltered graphs [34], where different parameters account for different filtering functions.
In the multiparameter however, the concise and stable representation of persistence modules is known to be a substantially more involved problem [19]. The main stumbling block is that, due to some fundamental algebraic reasons, there is no hope for the existence of a concise descriptor like the barcode that can completely characterize multiparameter persistence modules. This is why research in the last decade has focused on proposing and studying incomplete descriptors- e.g.. [10] for a recent survey. Among these, the signed barcodes stand out as natural extensions of the usual one-parameter barcode [4, 11, 12, 49], being also interpretable as point measures. The catch however is that some of their points may have negative weights, so the measures are signed. As consequence, their analysis more delicate than that of one-parameter barcodes, and is only very recently that their optimal transport-type stability has started to be understood [12, 55], while there currently is still no available technique for turning them into vectors in some Hilbert space.
1.2 Contributions
We believe the time is ripe to promote the use of signed barcodes for feature generation in machine learning; for this we propose the following pipeline (illustrated in Fig, l):
signed barcode deseriptor
goomerric datasets
Altrationn
multifiieeeed
simplicial complexes
homology
multiparameter
persistence
signed barcodes
vectorization
Hilbert space
modules
The choice of filtration being application-deepedeent we will mostly follow the choices made in related work, for the sake of comparison- see also [10] for an overview of standard choices. As signed barcode descriptor, we will mainly use the Hilbert decomposition signed measure (Definition 4), and when the simplicial complex is not too large we will also consider the Euler decomposition signed measure (Definition 5). These two descriptors descriptors, so they will serve as a proof of concept for our pipeline. They also offer the advantage of being efficiently computable, with effective implementations already available [42, 48, 65, 71]. With these choices of signed measure descriptors, the only missing step in our pipeline is the last one-the vectorization. Here is the summary of our contributions, the details follow right after:
arguably the simplest existing signed measure
We introduce two general vectorization techniques for signed barcodes (Definitions 6 and 7).
We prove Lipschitz-continuityy results (Theorems 1 to 3) that ensure the robustness of our entire feature generation pipeline.
We illustrate the practical performance of our pipeline compared to other baselines in various supervised and unsupervised learning tasks.
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f:S R
dim Hu(f)) R + 2
PHa E M
K PBA -
(2:1)
12.01
(8:10
o+1
dim
Figure 1: An instance of the pipeline proposed in this article. Left to right: A filtered simplicial complex (S, f) (in this case bi-filtered graph); the Hilbert function of its Orh dimensional homology persistence module :f(f): R² - vec (which in this case simply counts the number of connected components); the Hilbert decomposition signed measure (H)(S) of the persistence module; and the convolution of the signed measure with a Gaussian kernel.
Vectorizations. Viewing signed barcodes as signed point measures enables to rely on the literature in signed measure theory in order to adapt existing vectorization techniques for usual barcodes to the signed barcodes in natural way. Our first vectorization (Definition 6) uses convolution with a kernel function and is an adaptation of the persistence images of [1]; see Fig. 1. Our second vectorization (Definition 7) uses the notion of slicing of measures of [59] and is adaptation of the sliced Wasserstein kernel of [22]. Both vectorizations are easy to implement and run fast in practice;
assess the runtime of our pipeline in Appendix D. We find that pipeline faster than the other topological baselines by one or more orders of magnitude.
Theoretical stability guarantees. We prove that our two vectorizations are Lipschitz-cottinouus with respect to the Settorovich-uubinsttee norm on signed measures and the norm of the corresponding Hilbert space (Theorems 2 and 3). Combining these results with the Lipschitz-stabiliy of the signed barcodes themselves (Theorem 1) ensures the robustness of our entire feature generation pipeline with respect to perturbations of the filtrations.
Experimental validation. Let us emphasize that our choices of signed barcode descriptors are strictly weaker than the usual barcode in the one- mee-paramtter case. In spite of this, our experiments show that the performance of the me-parameterr version our descriptors is comparable to that of the usual barcode. We then demonstrate that this good behavior generalizes to the multiparameter case, where is in fact amplified since our pipeline can outperform multiparameter competitors (which rely on theoretically stronger descriptors and have been shown to perform already better than the usual one- anc-parameter TDA pipeline) and is competitive with other, non-topological baselines, on a variety of data types (including graphs and time series), For proof that our descriptors are indeed weaker than other previously considered descriptors, we refer the reader to Proposition 2 in Appendix A.
1.3 Related work
We give brief overview of related vectorization methods for multiparameter persistence; we refer the reader to Appendix B for more details.
The multiparameter persistence kernel [32] and mulliparameter persistence landscape [76] restrict the multiparameter persistence module to certain families of one-parameter lines and leverage some of the available vectorizations for one-parameter barcodes. The generalized rank invariant landscape [79] computes the generalized rank invariant over a prescribed collection of intervals (called worms) then, instead of decomposing the invariant as we do. it stabilizes using ideas comings from persistence landscapes [13]. The multiparameter persistence image [20] decomposes the multiparameter persistence module into interval summands and vectorizes these summands individually. Since this process is known to be unstable, there is no guarantee on the stability of the corresponding vectorization. The Euler characteristic surfaces [6] and the methods of [42] do not work at the level of persistence modules but rather at the level of filtered simplicial complexes, and are based on the computation of the Fuler characteristic of the filtration each index in the multifiltratin. These methods very efficient when the filtered simplicial complexes are small, but
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can be prohibitively computationally expensive for nigh-dimensional simplicial complexes, such as Vietoris Rips complexes (Example 2).
Background
In this section, we recall the basics on multiparameter persistent homology and signed measures. We let vec denote the collection of finite dimensional vector spaces over fixed field k. Given n 2 1 E N, we consider IR" as poset, with æ S E IR" x; S y for all 1 S is n.
Simplicial complexes. A finite simplicial complex consists of a finite set So together with a set S of non-empty subsets of Sy such that, if s E Sy then {s} E S, and if 72 E S and @ + T2 S T2, then E S. We denote such a simplicial complex by S, refer to the elements of S as the simplices of S, and refer Sy as the underlying set of S. The dimension of simplex E Sis dim(r) = | E I. In particular, the simplices of dimension O correspond precisely to the elements of the underlying set.
Example 1. Let G be a simple, undirected graph. Then G can be encoded as a simplicial complex S with simplices only of dimensions O and 1, by letting the underlying set of S be the vertex set of G, and by having a simplex of dimension 0 (resp. 1) for each vertex (resp, edge) of G.
Definition 1. A (multi)filtered simplicial complex is a pair (S. (f with S simplicial complex and f: S - RR a monotonic map. i.c., function such that f(T)) S fT2) whenever T1 S T2 S. Given filtered simplicial complex (S, f S - R²) and E R" we get subcomplex S == {T e S: f() S x} S S, which we refer to as the xusublevel set of f).
Example 2. Let (P. d be finite metric space. The (Vietoris-)Rips compler of P is the filtered simplicial complex (S, : $ - R), where S' is the simplicial complex with underlying set P and all non-empty subsets of P as simplices, and f(ppo..... pn.) ...))
max,,; dp Py). Then, for example, given any x e R, the connected components of the r-sublevel set Sf coincide with the single-linkage clustering of P at distance scale (see, e.g., [18]). In many applications, it is useful to also filter the Rips complex by an additional function d P -> (such as a density estimator). The finction-Rips complex of (P, dd) is the filtered simplicial complex (S,g:S - R²) with g((po.... pp.) = (f((po..... :P.))........))... Po))). min; d(p:)) ). Thus, for example, the (t, -t)--uubllved set is the Rips complex at distance scale of the subset of P of points with d-value at least u. See [17, 19, 25] for examples.
Homology. For a precise definition of homology with visual examples, see, e.g., [40]; the following will be enough for our purposes. Given i E IN, the homology construction maps any finite simplicial complex S to a k-vector space HS) and any inclusion of simplicial complexes S C R to a k-lleeer map HS C Rt) H((S) - H2 Rt). Homology is functorial, meaning that, given inclusions SC R T, have an equality (R C T) H. R) H.(S C T)).
Multiparameter persistence modules. An n--parmmeter persistence module consists of an assign- ment M Rm - vec together with, for every pair of comparable elements S y R" linear map M(x S y) : (x) - M (2), with the property that ((r S x) is the identity map for all x E R", and that M(y S M(x S v) = M(x S 2) for all x y E R".
Example 3. Let (S : 5 - R") be filtered simplicial complex. Then, we have an inclusion S!. S.S whenever x S y. By functoriality, given any E N, we get an n-parameter persistence module H.(s): R. - vec by letting (z) == (S!)).
Definition 2. Let . RD - vec. The Hilbert function of M, also known as the dimension vector of M, is the function dim(M) R" - 7 given by dim (i))))((z) = dim(M(a))
In practice, often deals with persistence modules defined finite grid of R These persistence modules are finitely presentable (fp), which, informally, means that they can be encoded using finitely many matrices. See Definition 8 in the appendix for formal definition, but note that this article can be read without a precise understanding of this notion.
Signed measures. The Kantorovich iubbonticc--Rubinteein norm was originally defined by Kantorovich and subsequently studied in [46] the context of measures on a metric space; see [43]. We denote by M(R)) the space of finite, signed, Borel measures on R". and by Mn()) C (RR))) the subspace of measures of total mass zero. For μ E (R"), we let μ = nu+ 1u" denote its Jordan decomposition [8, p. 421], so that uet and μ are finite, positive measures on R"

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Definition 3. For p E oo], the Kantorovich iisibooovichRuubinteeei norm of μa E Mo(RR") is defined as ||
{ {
inf
|
ullp (x) is measure on IR" R" with marginals utt
Ri RR"
can then compare any two measures 14, V E (MR))) of the same total mass using || 1/Ip KR Remark 1. Note that, if p 2 9, then || l1p -
KRR For n = 1, the definition of lpp is
independent of and in that case we just denote it by ||KR.
Given R" let 6 denote the Dirac measure M(R") that can be written as a finite sum of signed Dirac measures.
A finite signed point measure is any measure in
The following result says that, for point measures, the computation of the Kantorovich Rubbiioticc-RRiiiiteeeini norm reduces to an assignment problem, and that, the case of point measures on the real line, the optimal assignment has particularly simple form.
Proposition 1. Let μ E Mo((R") be a finite signed point measure with μ -; S2, and μ E; Sy where X (Z11..., IK and Y = (31...., y are lists of points of R". Then,
||ll| k min
Σ Isisk
||= - --()ll y is permutation of {I....,k} {.
Moreover, if and X and Y such that 0; Fi+1 and yr S Vs+1 for all Eist then the above minimum is attained at the identity permutation (()) = i.
Signed barcodes as measures and stable vectorizations
In this section, we introduce the signed barcodes and measures associated to multifiltrrd simplicial complexes, as well as our proposed vectorizations, and we prove their stability properties.
The barcode [41] of a one-parameter persistence module consists of multiset of intervals of R, often thought of as a positive integer linear combination of intervals, referred to as bars. This positive integer linear combination can, in turn, be represented as a point measure in the extended plane by recording the endpoints of the intervals [36]. Recent adaptations of the notion of one-parameter barcode to multiparameter persistence [4, 11, 12, 49] assign. each multiparameter persistence module M R" - vec, two multisets (B+ B-) of bars; each bar typically being some connected subset R" In [11, 55], these pairs of multisets of bars are referred as signed barcodes. several cases, the multisets B and B are disjoint, and can thus be represented without loss of information as integer linear combinations of bars. This the case for the minimal Hilbert decomposition signed barcode of [55], where each bar is of the form {y E RR" y 2 x} S R" for some E RR. The minimal Hilbert decomposition signed barcode of a multiparameter persistence module M can thus be encoded as signed point measure jaar, by identifying the bar {v E R" 2 #) with the Dirac measure o, (see Fig. 1).
In Section 3.1, we give self-contained description of the signed measure associated to the minimal Hilbert decomposition signed barcode, including optimal transport stability results for it. Then, in Section 3.2, we describe our proposed vectorizations of signed measures and their stability. The proofs of of the results in this section can be found in Appendix A.
3.1 Hilbert and Euler decomposition signed measures
We start by interpreting the minimal Hilbert decomposition signed barcode of [55] as a signed measure. We prove that this notion is well-defined in Appendix A.4, and give further motivation in Appendix A.S.
Definition 4.
The Hilbert decomposition signed measure of multiparameter persistence module M. : RD _ vec is the unique signed point measure Pa E _M(R") with the property that
dim(M(x)) = HM {y E R" y S x})) for all x e R".
Given a filtered simplicial complex, one can combine the Hilbert decomposition signed measure of all of its homology modules as follows.
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Definition 5. The Euler decomposition signed measure of a filtered simplicial complex (8, f) is
Σ(-1)' PBA() IE
Px())
The next result follows from [9, 55], and ensures the stability of the signed measures introduced above. In the result, we denote ||h|[1 Eres [[b(T)]]ı when (S, h) is a filtered simplicial complex. Theorem 1. Let N. let be a finite simplicial complex, and let f.g: S - R be monotonic.
1. For n E {1,2} and i IN, ()ин)S) (H.(()) |RR S n || gl|1.
2. For all E N, ||xx() Px - |lf 15--9111 9|1
Extending Theorem (1.) to number of parameters
is open problem.
Computation. The design of efficient algorithms for multiparameter persistence an active area of research [53, 48]. The worst case complexity for the computation of the Hilbert function of the ith persistent homology module typically in ((SS,-) + |S;| + [S;+11))), where |S&] the number of k-dimensional simplices of the filtered simplicial complex [53, Rmk. 4.3J. However, in one-parameter persistence, the computation is known to be almost linear in practice [5]. For this reason, implementation reduces the computation of Hilbert functions of homology multiparameter persistence modules to one- parameter persistence; we show in Appendix D.1 that this scalable in practice. We consider persistence modules and their corresponding Hilbert functions restricted to a grid (O.... m 1}nh By [55, Rmk. 7.4), given the Hilbert function {O..... m 1)"h -> of module M result, the Euler decomposition signed measure computed in linear time in the size of the complex.
grid (O....,
1)", can compute μM in time O(n m²). Using the next
Lemma 1. 1f (S. f' a filtered simplicial complex, then px(s) = =Erss -EEE(-1))dmg 85(7)-
3.2 Vectorizations signed measures
Now that we have established the stability properties of signed barcodes and measures, we turn the focus on finding vectorizations of these representations. We generalize two well-knows vectorizations of single-parameter persistence barcodes, namely the persistence image [1] and the sliced Wasserstein kernel [22]. The former is defined by centering functions around barcode points in the Euclidean plane, while the latter is based computing and sorting the point projections onto a fixed set of lines. Both admits natural extensions to points in R", which we now define. We also state robustness properties in both cases.
3.2.1 Convolution-base vectorization
A kernel function function is map K R - R20 with JOCR (xx)² da < xc. Given such a kernel function
and E R", let Ky RR - Rz0 denote the function Ky(x) K K(x - y)-
Definition 6. Given p >M(R)), define the convolution of the measure μ with the kernel function K as K * (?(RR) as (K (z) = SEER K(r-2)app - 2)du(2).
Theorem Let K R" - RR be a kernel function for which there exists > O such that [K, K2Ila I c. [yy for all y. R". Then, j. E MM(R") have the same total mass,
| μ K u112 S e || 10-vll v KR
In Propositions 4 and 5 of the appendix, we show that Gaussian kernels and kernels that are Lipschitz with compact support satisfy the assumptions of Theorem 2.
Computation. For the experiments, given a signed measure μe on IR" associated to multiparameter persistence module defined over finite grid in R", evaluate K μ on the same finite grid. As kernel K we use Gaussian kernel.
3.2.2 Sliced Wasserstein kernel
Given A E 5n-1 = {o E R" ||>||e- = S R", let L(Ø) denote the line {Ao À E R), and let 8 IR" - denote the composite of the orthogonal projection R" - I(0) with the map L(0) - R sending X0 to X.

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Definition 7. Let a be measure on S Let u. E MM(R)) have the same total mass. Their sliced Wasserstein distance and sliced Wasserstein kernel with respect a are defined by
SW*(K, Jess The
TLV
KR | do(O)
and käw (1, ; exp (w
respectively, where n.. denotes the pushforward of measures along X² IR" - R.
Theorem 3. Let a be measure on S If p, V E M(R") have the same total mass, then SW" v) S a( ll m v1l2R. Moreover, there exists a Hilbert space H and a map Dew Mo(R") - H. such that, for all p.v E Mo(RR"), we have kēw (p, v) = (9 (p), (w)H and [
sw (4) - (-) S 2 SW.(H.
Computation. We discretize the computation of the sliced Wasserstein kernel by choosing directions ........ Ou) S 5 uniformly at random and using as measure the uniform probability measure with support that sample, scaled by parameter 1/a,, as is common in kernel methods. The norm in R is then computed by sorting the point masses, using Proposition 1.
4 Numerical experiments
In this section, we compare the performance of our method against several topological and standard baselines from the literature. We start by describing our methodology (see Appendix C.I for details about hyperparameter choices). An implementation of our vectorization methods is publicly available at https: /t//tttttub com www.//o///aaaaaee Dawiaaaaoosssmmmiiaaaaaaaa tipers, and will be eventually merged as module of the Gudhi library [70].
Notations for descriptors and vectorizations. In the following, we use different acronyms for the different versions of our pipeline. H a shorthand for the Hilbert function, while E stands for the Euler characteristic function. Their corresponding decomposition signed measures are written HSM and ESM, respectively. The sliced Wasserstein kernel and the convolution- by SW and C. respectively. IP refers to one- ane-parameter. and MP multiparameter. Thus, for instance, MP-HSM-SW stands for the multi-parameter version of our pipeline based on the Hilbert signed measure vectorized using the sliced Wasserstein kernel. The notations for the methods we compare against are detailed in each experiment, and we refer the reader to Section 1.3 for their description.
vectorization are denoted
Discretization of persistence modules. In all datasets, samples consist of filtered simplicial com- plexes. Given such a filtered simplicial complex f S - R" we consider its homology persistence modules H.) with coefficients in finite field, for i E {0, 13. We fix grid size & and / E (0, 1). r S R uniformly spaced, with ró and r/- the 3 and 1 3 percentiles of (So) S R, respectively. We then restrict each module M to the grid (r) . r x .... {rū,..., rk-, rR) where r/ 1.1 (rk - rd) + ró- setting = whenever zj- = ry for some in order to ensure that μum has total mass zero
For each 1 j S n, we take ro S
M (I)....,,
Classifiers. We use an XGBoost classifier [27] with default parameters, except for the kernel methods, for which we use a kernel SVM with regularization parameter in {0.001, 0.01, 1, 10, 100, 1000).
4.1 Hilbert decomposition signed measure vs barcode in one-parameter persistence
As proven in Proposition 2, in Appendix A, the Hilbert decomposition signed measure is theoretically weaker descriptor than the barcode. Nevertheless, we show in this experiment that, in the one- parameter case, our pipeline performs well the following well known vectorization methods based on the barcode: the persistence image (PI) [1], the persistence landscape (PL) [13], and the sliced Wasserstein kernel (SWK) [22]. We also compare against the method pervec (PV) of [15], which consists of a histogram constructed with all the endpoints of the barcode. It is pointed out in [15] that methods like pervec, which do not use the full information of the barcode, can perform very well in practice. This observation was one of the initial motivations for our work. The results of running these pipelines on some of the point cloud and graph data detailed below are Table 1. See Appendix C.2 for the details about this experiment. As one can see from the results, signed barcode scores are always located between PV and (standard) barcode scores, which makes sense given that they encode more information than PV, but less than barcodes. The most interesting part, however, is
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that the Hilbert decomposition signed barcode scores are always of the same order of magnitude (and sometimes even better) than barcode scores.
4.2 Classification of point clouds from time series
We perform time series classification on datasets from the UCR archive [39] of moderate sizes; use the train/test splits which are given in the archive. These datasets have been used to assess the performance of various ropology-based methods in [20]; in particular, they show that multiparameter persistence descriptors outperform one- paraaeeeer descriptors in almost all cases. For this reason, compare only against other multiparameter persistence descriptors: multiparameter persistence landscapes (MP-L., multiparameter persistence images (MP-I), multiparameter persistence kernel (MP-K), K), and the Hilbert function directly used as vectorization (MP-H). Note that, in this example, we are not interested in performing point cloud classification up to isometry due to the presence of outliers. We use the numbers reported in [20, Table IJ. We also compare against non-topological. state-of-the-art baselines: Euclidean nearest neighbor (B dynamic time warping with optimized warping window width (B2), and constant window width (B3), reporting their accuracies from [39]. Following [20], we use delay embedding with target dimension 3, so that each time series results in a point cloud in R Also following [20]. filtered complex we use an alpha complex filtered by sistance-to-mmssure (DTM) [24] with bandwidth 0.1. As one can see from the results in Table 2, MP-HSM-C is quite effective, as is almost always better than the other topological baselines, and quite competitive with standard baselines.
Interestingly, MP-HSM-SW does not perform too well in this application. We believe that this is due to the fact that the sliced Wasserstein kernel can give too much importance to point masses that are very far away from other point masses; indeed, the cost of transporting a point mass is proportional to the distance it is transported, which can be very large. This seems to be particularly problematic for alpha complexes filtered by density estimates (such as DTM), since some of the simplices of the alpha complex can be adjacent to vertices with very small density, making this simplices appear very late the filtration, creating point masses in the Hilbert signed measure that are very far away from the bulk of the point masses. This should not be a problem for Rips complexes, due to the redundancy of simplices in Rips complexes: for any set of points in the point cloud, there will eventually be simplex between them in the Rips complex. In order to test this hypothesis, we run the same experiment but using density to filter a Rips complex instead an alpha complex (Table in Appendix C.4). We see that, in this case, the sliced Wasserstein kernel does much better, being very competitive with the non-topological baselines.
4.3 Classification of graphs
We evaluate our methods standard graph classification datasets (see [54] and references therein) containing social graphs as well as graphs coming from medical and biological contexts.
Since we want to compare against topological baselines, here we use the accuracies reported in [20] and [79], which use train/test splits to compute accuracy. We compare against multiparameter persistence landscapes (MP-L), multiparameter persistence images (MP-I), multiparameter persis- tence kernel (MP-K), the generalized rank invariant landscape (GRIL), and the Hilbert and Euler characteristic functions used directly as vectorizations (MP-H and MP-E). We use the same filtrations as reported in [20], so the simplicial complex is the graph (Example 1), which we filter with two parameters: the heat kernel signature [69] with time parameter 10, and the Ricci curvature [64]. As one can see from the results in Table 3, our pipeline compares favorably to topological baselines. Further experiments on the same data but using 10 train/test splits are given in Appendix C.3: they show that we are also competitive with the topological methods of [42] and the state-of-the-arr graph classification methods of [75, 80, 81].
4.4 Unsupervised virtual screening
In this experiment we show that the distance between our feature vectors in Hilbert space can be used effectively to identify similar compounds (which are in essence multifiltered graphs) in an unsupervised virtual screening task. Virtual screening (VS) is computational approach to drug discovery in which library of molecules is searched for structures that are most likely to bind to a given drug target [61]. VS methods typically take as input query ligand q and set L of test ligands,
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and they return a linear ordering (L)) of L, ranking the elements from more to less similar to g. VS methods can be supervised or unsupervised; unsupervised methods [67] order the elements of I. according to the output of fixed dissimilarity function between & and each element of I,, while supervised methods learn a dissimilarity function and require training data [2, 78). We remark that, in this experiment, we are only interested in the direct metric comparison of multifiltered graphs using our feature vectors and thus only in unsupervised methods.
We interpret molecules as graphs and filter them using the given bond lengths, atomic masses, and bond types. In order to have the methods be unsupervised, we normalize each filtering function using its standard deviation, instead of cross validating different rescalings as in supervised tasks. We use a grid size of 1000 for all methods, and fix for the sliced Wasserstein kernel and the bandwidth of the Gaussian kernel for convolution to 1. We assess the performance of virtual screening methods on the Cleves-Jain dataset [28] using the enrichment factor (FF);; details are in Appendix C.5. We report the best results of [67] (in their Table 4), which are used as baseline in the state-of-the-an methods of [34]. We also report the results of [34], but we point out that those are supervised methods. As one can see from the results in Table 4, MP-ESM-C clearly outperforms unsupervised baselines and approaches the performances of supervised ones (despite being unsupervised itself).
5 Conclusions
We introduced a provably robust pipeline for processing geometric datasets based on the vectorization of signed barcode descriptors of multiparameter persistent homology modules, with an arbitrary number of parameters. We demonstrated that signed barcodes and their vectorizations are efficient representations of the multiscale topology of data, which often perform better than other featurizations based on multiparameter persistence, despite here only focusing on strictly weaker descriptors. We believe that this is due to the fact that using the Hilbert and Euler signed measures allows us to leverage well-developed vectorization techniques shown to have good performance in one- mee-parmettrr persistence. We conclude from this that the way topological descriptors are encoded as important as the discriminative power of the descriptors themselves.
Limitations. (1) Our pipelines, including the choice of hyperparameters for our vectorizations, rely on the cross validation of several parameters, which limits the number of possible choices to consider. (2) The convolution-bass a fine grid, but the performance degrades with coarser grids. This is a limitation of the current version of the method, since convolution in very fine grids does not scale well in the number of dimensions (i.e., in the number of parameters of the persistence module). (3) The sliced Wasserstein vectorization method for signed measures a kernel method, and thus does not scale well to very large datasets.
vectorization method works well when the signed measure is defined over
Future work. There is recent interest in TDA in the differentiation of topological descriptors and their vectorizations. Understanding the differentiabilityy of signed barcodes and of our vectorizations could be used to address limitation (I) by optimizing various hyperparameters using a gradient instead of cross validation. Relatedly, (2) could be addressed by developing a neural network layer taking as input signed point measures, which is able to learn a suitable, relatively small data-dependent grid on which to perform the convolutions. For clarity, our choices of signed barcode descriptor and vectorization of signed measures are among the simplest available options, and, although with these basic choices we saw notable improvements when comparing our methodology to oftte--f-the-eee ropology-based methods, it will be interesting to see how our proposed pipeline performs when applied with stronger signed barcode descriptors, such as the minimal rank decomposition of [1]], or to the decomposition of invariants such as the one of [79]. Relatedly, our work opens up the way for the generalization of other vectorizations from one- parameter persistence to signed barcodes, and for the study of their performance and statistical properties.
A Acknowledgements. The authors thank the area chair and anonymous reviewers for their insightful comments and constructive suggestions. They also thank Hannah Schreiber for her great help in the implementation of our method. They are grateful to the OPAL infrastructure from Université Côte d' Azur for providing resources and support. DL was supported by ANR grant 31A Côte d' Azur (ANR-19. SPRR-19-P311 P3IA--002). LS was partially supported by the National Science Foundation through grants CCF-2006661 and CAREER award DMS-1943755 1943758. MC was supported by ANR grant TopModel (ANR-23-CF23-0004)) SO was partially supported by Inria Action Exploratoire PREMEDIT. This work was carried out in part when MBB and SO were at the Centre for Advanced Study (CAS), Oslo.

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Dataset
SWK
Pi
PL
PV
IP-HSM--WWW
IP-HSM-C
73.4
66.9
68.3
66.9
70.5
70.5
DistaPhallncCOutiinCCrraa
73.6
62.3
68.5
64.5
74.6
75.4
D
62.6
63.3
COX2
78.8(4.0)
78. (8.6(1.0)
79.2(3.7)
78.2(0.8)
79.2(2.6)
80.1(1.3)
DHFR
80.866.5)
76.2(3.5)
78 6(4.6
73.5(4.7)
76.7(5.7)
76.7(344
IMDB-B
69.6(4.4)
66.6(3.3)
63.4(3.3)
63.8(4.1)
64.3(4.6)
63.7(3.6)
Table 1: Accuracy scores of ooo--anameeeee persistence vectorizations on some of the datasets from Tables 2 and 3. The paraamemeeer version of our signed barcode vectorizations, the right, performs as well as other topological methods, despite using less topological information.
Dataset
B1
B2
B3
MP-K
MP-L
MP.]
MP-
MP-HSM-SW
MP-HSM-C
DistaiPhalams Cptrn
DistaPPhalannCutinnee
71.7
72.5
71.7
74.6
69.6
71.7
73.9
71.4
75.4
DistaIPhalanxTW
63.3
59.0
61.2
61.9
ProximalPaalanXOutinn SaalllaxxOtlinAAeeCcoo Ggrropp
78.5
78.5
80.5
78.0
78.5
81.0
82.4
82.9
82.4
Proximal Phalanx OutlineComec
80.8
79.0
78.4
78.7
78.7
81.8
82.1
77.7
82.5
PhalaxiPa
75.6
75.6
73.2
76.1
77.1
77.6
77.6
ECG200
88.0
88.0
77.0
77.0
74.0
83.0
73
71.1
84,1
ItalyPowert PowerrDemadd
MedicalImages
68.4
74.7
73.7
55.4
55.7
60.0
56.6
53.3
56.2
Plane
100.0
100.0
92.4
84.8
97.1
99
91.4
SwedishLeaf
78.9
84.6
79.2
78.2
64.6
83.8
79
66.2
79.8
GunPoint
91.3
91.3
90.7
88.7
94.0
9X7.7
89.3
88.7
94.1
GunPointAgeSpan
96.5
GunPointMalle VersusFenmale
97.5
97.5
99.7
96.8
88.3
95.9
93.7
83.5
96.8
GunnointCllaeessus Young
83.8
99.7
PowerCons
93.3
92.2
87.8
85.6
84.4
86.7
88.3
83.9
88.7
SyathetieControl
88.0
98.3
99.3
50.7
60.3
60.0
55.3
49.7
61
Table 2: Accuracy scores of baselines and multiparameter persistence methods on time series datasets.
Boldface indicates best accuracy and underline indicates best accuracy among topological methods. vectorization with the Hilbert decomposition signed measure (MP-HSM- C) performs very well in comparison to other multiparameter persistence vectorizations, and is competitive with the non-topological baselines.
The convolution-basee
Dataset
MP-K
MP-ESM-SW
MP-FSM-C
COX2
79.9(1.8)
79.0(3.3)
77.92277
79.8(2.9)
78.2(2)
78.42.22 2)
78.4(0.7)
78.2(0.4)
77.1(3)
78.2(1.3)
81.7(1.9)
79.5(2.3)
77.6(2.5)
79.6(1.9)
80.5(3.1)
IMDR-B
68.2(1.2)
71.2(2.0)
71.1(2.1)
65.2(2.6)
72.3(2.4)
71.3(1.))
72.9(2.1
74.7(1.6)
74.8(2.5)
74.4(2.4)
IMDB-M
46.9(2.6)
46.2(2.3)
46.7(2.7)
47(2.7)
47.3(1.9)
47.2(2.5)
47.7(2.4)
47.9(3.2)
47.3(3.2)
MUTAG
86.1(5)2
84.0(6.8)
85.6(7.3)
87.8(4.2)
86.7(5.5)
88.8(4.2)
87.3(5)
87.2(2.6)
85.6(5.3)
88.3(5.8)
PROTEINS
67.5(3.1)
65.8(3.3)
67.3(3.5)
70.9(3.1)
67.4(2.2)
70.2(5.7)
72(3.1)
68.8(2.7)
74.6/2.1)
70.9(0.8)
Table 3: Accuracy and standard deviation scores of topological methods (averaged over 5-fold train/test splits) on graph datasets. Bold indicates best accuracy. Again, the convolution- based vectorization with the Hilbert decomposition signed measure (MP-HSM-C) performs very well when compared to other multiparameter persistence vectorizations.
(max. 50)
F.F
20)
to (max. 10)
USR GZD
13.7
4.7
USR
USR ROCS
17.1
9.1
5.4
GZD ROCS
20.3
10.8
5,3
PSA ROCS
20,5
10.7
6.4
MP II
16.0
5,8
MP-F
21.3
11.8
7.0
MP-ESM-SW
MP-HSM-SW MP-HSM-C
22.5
11.1
6.4
25.4
13.3
7.6
MP ESM-C
28.2
13.7
8.0
ToDD-RF (supervised)
35,2(2.3)
15.6(1.0)
8.1(0.4)
ToDD VT (supervised)
39.611.4)
18.6(0.4)
9.9(0.1)
Table 4: Enrichment factor (EF) for E {2,5, 10) of virtual screening methods on Cleves-Jaie data. Bold indicates best EF among unsupervised methods. The convolution-base vectorization of the Euler decomposition signed measure (MP- (MP-ESM-C) performs significantly better than unsupervised baselines. Moreover, the pipelines based on the vectorization signed barcodes perform better than those using the raw Hilbert and Euler functions directly.
10

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15

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Appendix to Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures
David Loiseaux Luis Scoccola 2* Mathieu Carrière', Magnus B. Botnan", Steve Oudot
DataShape, Centre Inria d' U'iiversité Côte d' Azur
"Mathematics, University of Oxford 'GeomeriX, Inria Saclay and École polytechnique
Mathematiccs, Vrije Universiteit Amsterdam
Table of Contents
A Missing definitions and proofs
16
A.1 Finitely presentable modules
16
A.2 Discriminative power of descriptors
17
A.3 References for Proposition of the main article
17
A.4 Definition of the Hilhert decomposition signed measure
17
A.5 The Hilbert decomposition signed measure as signed barcode
18
A.6 Proof of Theorem of the main article
19
A.7 Proof of Lemma of the main article
19
A.8 Lipschitz-stability of vectorizations
20
B Review of numerical descriptors of persistence modules
22
C Further details about experiments
23
C.I Hyperparameter choices
23
C.2 One parameter experiments
23
C,3 Further graph experiments
23
C.4 Pointeloud classification filtering Rips by density
23
The enrichment factor
25
Runtime experiments
25
Runtime of computation Hilbert decomposition signed measure D.2 Runtime of whole pipeline
25
26
E Stability experiments
26
A
Missing definitions and proofs
For an introduction to multiparameter persistence, see, e.g., [10]. We mostly follow the conventions from [12, 55].
A.1 Finitely presentable modules
Let x E R" We denote by Ps R" + vec the persistence module such that (v(v) = ki 2 x and
P (v) 0 otherwise, with all structure morphisms that are not forced to be zero being the identity
k.
Equal contribution.
16

---

Definition 8. A persistence module is finitely generated projective if it is a finite direct sum of modules of the form Pa. A persistence module is finitely presentable if isomorphic to the cokernel of morphism between two finitely generated projective modules.
The intuition behind the notion of finitely presentable persistence module is the following. By definition, any such module is isomorphic to the cokernel of a morphism Je Pss + Dirr Ps.. The elements 1 in TDA they correspond to topological features being born-whin the elements 1 k Py (y₂) are referred to as the relations and in TDA they correspond to topological features dying or merging.
Pa. (2:4) are often referred to as the generators of the persistence module-anc and
It is worth mentioning that most construction related to TDA produce finitely presentable modules:
Lemma 2. (S, : - RR") is finite filtered simplicial complex, then H;(f) is finitely presentable for all i N.
Proof. Let Sp denote the set of k-dimensional simplices of S. By definition of homology. the persistence module Hz(l) is the cokernel of morphism +rSS Ps(r) K, where K is the kernel of morphism DrES; P)()) - Dres; P(()). Since K is the kernel of a morphism between finitely presentable module, is itself finitely presentable (see, e.g., [9, Lemma 3.14]), follows that Hf) is finitely presentable, since it is the cokernel of morphism between finitely presentable modules (see, e.g., [51, Exercise 4.8]).
Example 4. Let < < bE R. Let (a.b) RR vec be the one- one-parameter persistence module such that b)a, (3r) = k if x E |a, b) and (a.b) (Ir 0 if & [a, b); and, if a < x S y < b, then baa, S y) is the identity linear map k - lk, and is the zero linear map otherwise. The module kja,6) is often referred to as the interval module with support (a, b). This interval module is finitely presentable, since it is isomorphic to the cokernel of any non-zero map P. - P..
An example of a persistence module that finitely presentable is any interval module supported over an open interval, such as ((ab))-
A.2 Discriminative power of descriptors
It is well-known [19, Theorem 12] that the one- parametere barcode is equivalent to the rank invariant, which is descriptor that readily generalizes to the multiparameter case, as follows.
Definition 9 ([19). The rank invariant of persistence module M RR" - vec is the function which maps each pair of comparable elements x S y E IR" to the rank the linear map M(x) - M M
Proposition 2. The Hilbert function is a strictly weaker descriptor than the rank invariant. In particular, the descriptors of [20, 76, 79] are more discriminative than the Hilbert decomposition signed measure.
Proof. The persistence modules M = 0.,))) and N = 1k00,) @ k(), have the same Hilbert function IR -> Z, since in both cases the Hilbert function is just the indicator function of the set [0, oo). However, the rank of the linear map (0 < 2) AM (0) M2) is one, while the rank of
the linear map NOO < 2) (0) - N22) zero, proving the first claim. Since the invariants of
[20, 76, 79] determine the rank invariant of (at least some) one-parameter slices, it follows that they
are strictly more discriminative than the Hilbert function.
A.3 References for Proposition of the main article
Both statements are well known. For the first statement, see, e.g., (57, Proposition 2.1], and, for the second one, see, e.g. [57, Remark 2.30].
A.4 Definition of the Hilbert decomposition signed measure
For intuition about the concepts this section, and their connection to [55], see Appendix A.5. Here, to be self-contained, we unfold the definitions of [55].
17

---

Lemma 3 (cf. [55, Proposition 5.2]). Let M : RR - vec be finitely presentable. There exists pair
finitely generated projective modules (P: Q) such that, as functions R Z. we have
dim (M) = dim ()) - dQm(Q).
Note that the above decomposition of the Hilbert function not unique; in fact, there are infinitely many of these decompositions, given by different pairs (P.Q). Nevertheless, as we will show below, they all yield the same, unique, Hilbert decomposition signed measure. For this we will use the following well known fact from geometric measure theory.
Lemma 4. To prove that μ M(R)), it is enough to show that, for every R we have
M((EER" y S x))) = (U = ))
Proof. Since sets of the form E E Rm : S generate the Borel sigma-algebra, and the measures in M(R") are necessarily finite, the result follows from standard application of the π- theorem [37. Theorem A. 1.4.). For instance, one can follow the proof of [37, Theorem A.1.5], by noting that positivity of the measures is not required for the proof to work.
The Hilbert decomposition signed measure is built as sum signed Dirac measures, one for each summand of P and Q in the decomposition of Lemma 3. The following lemma justifies this insight. Lemma Let æ E R", let Pa R" - vec be the corresponding finitely generated projective module, and let 8 be the corresponding Dirac measure. Then dim (P=) = 6=((w R"
v)).
Proof. Note that, by definition of Pa. we have dim(P:) = æ S y and (Pm().) otherwise. Similarly, (x b}) 1 if S y and (f S v}) 0 otherwise. The result follows.
We now have the required ingredients to define the Hilbert decomposition signed measure formally. While the statement itself only claims existence and uniqueness, the proof actually builds the measure explicitly finite sum of signed Dirac measures (see Eq. (1)), as explained above.
Proposition 3. If M R - vec is fp, there exists a unique point measure μM E M(R") with
dim (a)) par {v E R" : )))),, for all x E RR.
Proof. Let (P.Q) be as in Lemma 3. Let {ailier r be such that P = Dser Ps. and let {ys hee be such that Q ① jes Pe.- To show existence, define the measure
PM Σ 8m: Σ
8yg
(1)
JEJ
Then
dim(M(2) dim P(z)) - dim(((2))
lfi I x2; S 21 [Íj E J y 2
#M<(w E R" w S 2)),
where in the second equality we used Lemma 5. Uniqueness follows from Lemma 4.
A.5 The Hilbert decomposition signed measure as signed barcode
We clarify the connection between the Hilbert decomposition signed measure and the concept of signed barcode of [55].
Let M : IR" - vec be finitely presentable. A pair of finitely generated projective modules (P, Q) as in Lemma 3 is what is called a Hilbert decomposition of dim(M in [55]. Since the modules P and Q are, by assumption, finitely generated projective, there must exist multisets (æi and { hies of elements R" such that P DD.r Pz: and Q = θ jeu Pa. - A pair of multisets of elements of Ra is what is called a signed burcode [55]. The intuition that each element its upset (y E R" S y). which coincides with the support of the module P. (by Lemma 5). Thus, the supports the summands of P (resp. Q) are interpreted as the positive (resp. negative) bars of the signed barcode ((x,)e) {8, be.). See Figs. 2 and for illustrations.
IR" determines
18

---

Holf)
U8
Z
dim(Ho(f)) R +
k
lk
dim
k
IK-
1k
2
1
dim
k
Figure Left to right: The filtered simplicial complex Fig. 1 of the main article; its Orh dimensional homology persistence module Ho()); the Hilbert function of Hf(s); and decomposition of the
Hilbert function of Hf( as a linear combination Hilbert functions of finitely generated projective
persistence modules: dim (f))) dim (0.1) @ P(1,0) P(222) dim ((221) P(12)).
Hllber decomposition of Hlolf)
Hilbert decompocition signed measure Hol))
Figure 3: A Hilbert decomposition (in the sense of [55]) of the module of Fig. 2, and the corresponding Hilbert decomposition signed measure. In the Hilbert decomposition, the supports the persistence modules in yellow are interpreted as positive bars, while the supports of the persistence modules in blue are interpreted as negative bars. Since bars corresponding to finitely generated projective modules are of the form {y E R" y 2 J for some E R",, they are uniquely characterized by their corresponding R"; this why bars in [55] are just taken be points of R"..
A.6 Proof of Theorem of the main article
Claim (I.) follows by combining the stability of sublevel homology in the presentation distance [9, Theorem 1.9 (i) for = 1 with the stability of bigraded Betti numbers with respect to the presentation distance [55, Theorem 1.5], and using the fact that the signed I-Wasserstein distance between the signed barcodes of [55] is equal to the distance induced by the Kantorovich Rubinsttiinnnnn norm between the signed measures associated the signed barcodes, which is proven in [55, Proposition 6.1
We now prove claim (2.), using definitions from [55]. Consider the following two signed barcodes (in the sense of [55]):
dim(r) (B,BB)= (J(T))EESs) (TS)) (-), = (o(T)) TES)), {g(T)) TES dim(7)
dim(r)
dim(7)
is even
is odd
is even
is odd
It is clear that the signed measures associated to these signed barcodes, in the sense of [55, Section 6.4]. are equal Mx(f) and (x(a). respectively. By [55, Proposition 6.11] and [55, Definition 6.1], then enough to prove that there exists a bijection F B, UC- - C4 U B_ such that
Σ -i-F(l I
Σ E1f(r) 9(7)
UE__UC_
To construct such a bijection, we simply map /(r) E B_ to g(T) E C+ when dim(() is even, and g(T) C_ to f(T) B_ when dim(r) is odd.
We remark that the content of claim (2.) is essentially the same that of [42, Lemma 12], although they use slightly different terminology.
A.7 Proof of Lemma the main article
It is well-known [72, Theorem 12.4.1] that, if S is finite simplicial complex, then
1y$ dim dim(H,
Σ (-1)dii(r)) dim(r))
TES
19

---

It follows from this and Proposition 3 that, if (S, f) is filtered simplicial complex, with f: S - R and z R", then
pxin (tV E ER" : ( = Σ
(-1)dim(7)
TES
5(r/s>
Now note that the right-hand side is also equal the measure of the {s R² y S x) with respect the signed measure (-es(=1)dim()) oイ()). The result then follows from Lemma
A.8 Lipschitz-stability of vectorizations
Proof of Theurem of the main article. Let μ and let yi be a coupling between X and A=
Then, for every x. E Ra, we have
(K p. - K * vx) (K * - - (x) / K(x v) du(y) - /
K(x 2) dv(z),
Rm
which is equal fR- xRn (K(x y) K(x 2)) dv z), since K( y) does not depend and (x - 2) does not depend on y. Thus,
1/2
K * 4 K * v12 = (1 ( (K(x- - y)- - K(x - 2)) dé(())))
dx
J XR= (Je (K(x- - y) K(x 2))² de)
1/2
db(v,2)
[ JRHRR" [| - Kyll2 (y,
c lly =|12 del (u,
R
where in the first inequality we used Minkowski's 's integral inequality [44, Theorem 202], and in the second inequality we used the hypothesis about K. Since the coupling v is arbitrary, we have K vil2 S e [AllE e||ı- vllkR, as required.
Proposition Let K Ra - R be Gaussian kernel with zero mean and covariance Σ. For all E R", we have Ky - K2ll2 I e clyy 2|12, with
$2 7/4 dct
where [5-1/ denotes the operator norm of D-1 associated to the Euclidean norm.
Proof. By definition,
K(x) = exp (- | ( /2TT'" " (2)
where || ||- -|x denotes the norm associated to the quadratic form Σ-1
We start by noting that | K2l2 = || K( K2). Now, a standard computation shows that
K 123 = / Jm (2TT)" (E)
exp (111)) dx nn/2 det (E)
20

---

The term (Ky. K2) can be computed similarly, using the formula for the product of Gaussian densities [56, Section 8.1.8]:
(Ky y. K₂)
Ky(z) (x) dx
RRn
exp
(-1|3r -
- 113 1) dx
(2 " (2)(2)
exp(- || 2 11
y/
exp
dx
2 1ー-1
Rd
(2x)² det(I))
2 1s
exp
T//2
det(22)1/2
Thus.
1 exp (- (-1 -ー E-
K 2 Kkg K2) =2 nn/2 det(E) 1/2
2 T1/2 det (Σ) 1/2
where, for the inequality, we use the fact that the function 2 + 1 - exp( (-p(-2) 2) is 1-Lipschitz when restricted to Rzo- The result now follows from the fact that yy =11 S | lll ly 212. by standard property of the operator norm.
Proposition 5. Let K : RR" + R be a-Lipschitz and of compact support. For all y, z E R", we have
IK, Kslla S c ll - 2112. with = a e (2 (u)spp((K))) 1/2 where [supp(K)] denotes the Lebesgue measure of the support of K.
Proof. Note that K2 K2 (x)) = (K(x v) K - 2) S a [| y) - -ax 2) a 11 2/12. by assumption. Now
[K, K2113
(K)(x) (x(x))² dr
Rm
(Ky(x) (x))))² dx
supp((K)) Usupp(k))
a² l1y - 2112 de
(upp(K,) Usupp(K.)
= a² Ill ---2] 2112 [ Jaupp(K,) dx
2 [supp((K]] [ll - =11 S a² .2.2
as required.
Lemma 6. Let be measure on the (n 1) d-ddimensional sphere Sn-1 The function SW is conditionally negative semi- definit on the set (R")...
Proof. Let ................................ . ax E R such that E, a 0 and let pI...., M4 (R").. Let 0 e S let Vi2 p4. and let X; ntt Eoye 1/- Note that x; is a positive measure on R for all i, and that (R) = 1it (R) + Eejs U/ (R) = uzt (R) + Eejr ut (R) = Et = m is independent of i.
Then
KR
||4 1z KR =
1tt + + Σ vi
( + Σ Vi ift$
= 1| - 2z ||
KR
It follows that Σn, a,aj ||4 リグ KR En, 0;0; || - > < 0 since the Wasserstein distance on positive measures of fixed total mass on the real line is conditionally negative semi-definite [22, Proposition 2.1 (i)), as it is isometric to an dii-dintance [57, Remark 2.30]. The result then
follows by integrating over 0 sn-
D
21

---

Proof of Theorem 3. We start with the stability of the sliced Wasserstein distance. By linearity of integration, it is enough to prove that || [| +
KR S l| = | for every s which
follows directly from the fact that orthogonal projection onto L(0) is 1-Lipschitz map when using Euclidean distances.
To prove the existence of Hilbert space H and a map & (R") - H such that, for all μ. V E Mo(R"), we have ksw (1, v/ = ( (4), & (v)) we apply [7. Theorem 3.2.2, p.74]; this requires
prove that SWa is conditionally negative semi-definite distance, which do in Lemma 6. To conclude, we must show that |
"Sw(v)ll & 2 SW" v). This 2Sw (V)ll = 2 2 kgw (1,22) and that the function 2 + 1 exp(-z) is I-Lipschitz when restricted to R20-
follows from the fact that |[
B Review of numerical descriptors of persistence modules
Hilbert function. In the one-parameter case, the Hilbert function is known the Betti curve; see, e.g., e.g., and [73] for an extension beyond the one- parameetreer case. The Hilbert function is itself a numerical descriptor,
it can be used as feature train vector-based machine learning models; it has been used in the multiparameter case in [34], where it performs favorably when compared to application-specific Hate-of-the-art methods for drug discovery tasks.
The (pointwise) Euler characteristic of a (multi)filtered simplicial complex is also readily numerical invariant. It has been used to train machine leaning models in the multiparameter case in [6, 42].
Euler characteristic.
Barcode-based. Many numerical invariants of persistence modules based on the one one- me--arameter barcode have been proposed (see, e.g.. [3] for a survey). Since these methods rely on the one- parameter barcode, they do not immediately generalize to the multiparameter case.
Barcode endpoints and filtration values. It has been shown that the full discriminating power of barcodes is not needed to achieve good performance in topology-based classifications tasks [15, 3]. Indeed, [15] argues that, in many cases, features which only use the endpoints of barcodes, and thus forget the pairing between endpoints, perform as well as features that do use the pairings. The work [3] reaches somewhat similar conclusions, although their descriptors do keep some of the information encoded by the pairing between endpoints given by the barcode (in particular making their descriptors not immediately generalizable to the multiparameter case). The analysis of [15] is particularly relevant to our work: our Hilbert decomposition signed measure can be interpreted as a signed version of their pervec descriptor, and our Euler characteristic signed measure can be interpreted signed version of their filvec descriptor.
Rank invariant. The rank invariant (Definition 9) can be encoded as a function IR Rd Z, by declaring the function to be zero on pairs I y. However, to our knowledge, the rank invariant has not been used directly as a numerical descriptor train vector-based machine learning models. Vectorizations of the rank invariant, and of some of its generalized versions [49, 11, 4], have been introduced building on the notion of persistence landscape.
Persistence landscape. The persistence landscape [13] is a numerical descriptor of one- mne-paraeeter persistence modules which completely encodes the rank invariant of the module. The persistence landscape was extended to a descriptor of multiparameter persistence modules in [76] by stacking persistence landscapes associated to the restriction of the multiparameter persistence modules to all lines of slope 1. Another extension of the persistence landscape, this time to the 2-parameter case, is the generalized rank invariant landscape [79], which relies on the generalized rank invariant restricted to certain convex shapes in IR?.
Multiparameter persistence kernel. Any kernel method for one- one- perammereeer persistence modules, such as [60, 50, 22, 52], gives rise to a kernel method for multiparameter persistence modules, using the methodology of [32], which relies on "slicing", that is, restricting multiparameter persistence modules to monotonic lines in their parameter space.
Multiparameter persistence image. Another vectorization method which relies on slicing multipa- rameter persistence modules in [20]. Their method uses the notion of vineyard [31] to relate the barcodes of different slices and outputs descriptor which encodes these relationships.
22

---

C Further details about experiments
C.1 Hyperparameter choices
We fix _ = 0.01 in all experiments. We also use d = 50 slicing lines and a grid size of k: = 1000 for MP-SW. In supervised learning tasks, all parameters are chosen using 10-fold cross validation (ev). Beside the kernel SVM regularization parameter, these include: the size of the grid l is chosen to be in (20, 50, 100) for MP-SM; we use homology in dimensions O and 1 (except for the Cleves-Jain data, for which we use only dimension 0) with coefficients in the field k Z 2/11Z, which we just concatenate for MP-C, or combine linearly with coefficients chosen in {1, 5. 10) for MP-SW. In general, the parameters of multiparameter persistence module M IR" - vec represent quantities expressed in incomparable units, and are thus a priori incomparable; in order to account for this, we rescale each direction of the persistence module as follows: for MP-SW we choose constant e in {0.5, 1, ].5] for each direction, and scale by c/a, with a E {0.001, 0.01, 1, 10, 100, 1000]; for MP-SM we choose constant in {0.01, 125, 0.25, 0.375, 3.5} for each direction.
C.2 One-parameter experiments
We use alpha filtration [35, Section 2.3.1] 1] for the UCR data and, for the graph data, filter graphs by the node degrees. For UCR data we use 0 and 1 dimensional homology; we take the sum the kernels evaluated on 0 and 1 dimensional features for the kernel methods and otherwise concatenate the features for the other vectorizations. For graph data we use extended persistence as in, e.g., [21].
The parameters of all vectorization methods are chosen by 10-fold cross validation. As classifier, we use support vector machine with parameter y E {0.01, 0.1, 1, 10, 100) and an RBF kernel, except for the kernel methods for which we use precomputed kernel. The regularization pa- rameter is chosen in {0.01, 1, 10, 100, 1000). For sliced Wasserstein kernel (SWK) [21] we use {0.001, 0.01, 1, 10, 100, 1000]; for persistence images (PI) [1] we use the kernel bandwidth in {0.01, 1, 10, 100} and a resolution in {20, 30); for persistence landscape (PL) [13] we let the number of landscapes to be in {3, 4, 5, 6, 7, 8). and the resolution in (50, 100, 200, 300}; and for pervec (PV) we use a histogram with number of bins in {100, 200, 300).
The reported accuracy is averaged over 10 train/test splits in the case of graphs; for UCR we use the given train/test split.
C.3 Further graph experiments
We compare our methods the multiparameter persistence methods based on the Euler characteristic ECP, RT, and HTn of [42]. We also compare against the state-of-the-ar graph classification methods RetGK [81], FGSD [75], and GIN [80]. We choose these because they are the ones that performed the best in the analysis of [42]; in particular, we use the accuracies reported there. Parameters are as Appendix C.I, except that we compute accuracy with 10-fold train/test split. The methods RetGK, FGSD, GIN and those from [42] also use 10-fold train/test splits for accuracy. Note that, for simplicity, and as opposed to [42], we do not cross-validate different choices of filtration, and instead use the following three filtrations: the degree of the nodes normalized by the number of nodes in the graph, the closeness centrality, and the heat kernel signature [69] with time parameter 10. We believe it is possible that better scores can be obtained by considering various other combinations of filtrations. We also remark that choosing three or more filtrations is usually not possible with other persistence based methods ([42) being notable exception).
All scores can be found in Table 5. topological baselines, which shows the benefits of using signed measure and barcode decompositions over raw Euler functions. It is also worth noting that topological methods achieve scores that are comparable with state-of-the-arr, non-topological baselines.
One can see that our methods are competitive with the other
C.4 Pointcloud classification filtering Rips by density
In order to check that the performance of the sliced Wasserstein kernel in Section 4.2 can be improved by using more robust filtration, as explained there, we use here a function-Rips filtration (Example 2) with Gaussian kernel density estimate with bandwidth in {0.001 r, 0.01 r,, 0.1 r, 0.2 0.3 rr.
23

---

Dataset
RetGK
FGSD
GIN
ECP
RT
HT
HSM-MP-SW
FSM-MP-SW
HSM-MP-C
ESM-MP-C
COX2
81.4(U.6]
80.3(0.4)
79.7(0.4)
80.60044)
77.901.3)
78.200.8)
78.1(2.4)
78.4(1.8)
DHIR
81.5(0.9)
82.00.4)
81.30.4)
(3.1
82,8(5)
83.7(5.7)
81.5(311)
81.3(3.2)
IMDR-R
71.9(1.))
73.6
(5.1(5.1)
73.3(0.4)
74.0(0.5)
74.700.5)
74.7(5)
74(3.9)
72.9(2.9)
74.2(3.9)
47.7(0.3)
52.4
52.3(2.8)
48.7(0.))
40.2(0.4)
49.9)0.4)
50.3(3.5)
50.6(3.5)
50.8(3.8)
MUTAG
90.3(1.1)
92.1
90(8.8)
90.0(0.8)
87.30.6)
89.7
86.8(7.))
87.3(9.1)
85.
87.3())
PROTEINS
78.0(0.3)
73.4
76.2(2.6)
75.0(0.3)
75.4(0.4)
75.400.4
74.1
73.6(2.3)
71.4(4)
70(3.5)
Table 5: Accuracy scores (averaged 10-fold train/test splits) on graph datasets. Bold indicates best accuracy and underline indicates best accuracy among topological methods. For our methods and GIN, report standard deviation, while RetGK reports standard error.
Datasel
num. point clouds (train / test]
num. simplices per point cloud
HSM
MP-SW
MP C
DistaiPlillanctuttiinnggggo
1397400
76153
7.69
24.87
4.50
DistalPhalanx Outline
276/ 600
76153
11.61
56.25
6.64
DistalPhalanx
139 /40
76153
7.78
24.39
4.75
400 205
76153
8.12
25.67
5.43
CooiinaiPaaaanxOutinnCCoooo
600
76153
7.77
ProximalPhalanx IW
205 400
76153
8.20
25.25
5.35
ECG200
MelyPoweDDemand
67 1029
1561
0.32
0.49
0.31
Medicallmages
381 760
147536
9.67
18.96
2.49
Plane
105 105
467321
4.52
3,03
2.42
SwedishLeaf
500 625
325625
18.34
53.83
8.81
GunPoint
529543
GunPointAgee
135 316
$29543
5.58
2.53
0.06
GunPointMale Versuuu.eeeeaaaee
/.11
529543
GunPointtOld Versus Young
136 315
529543
5.60
2,50
0.06
PowerCons SymtheticControl
180 180
467321
7.91
6.05
2.16
300 300
30913
4.54
13.33
3.82
Table 6: Time (in seconds) to go from point clouds to the Hilbert decomposition signed measure (column HSM), as well as time (in seconds) to go from the signed measure to the output of our proposed vectorizations (columns HSM-MP-SW and HSM-MP-C).
Datasel
DistalPPallanx Catpaaallannttltnnaaaaa. Agstrou
9227.1
1038.9
217.1
32.36
12.20
DistaPPhalanxTT TW
9396.4
S77.7
138.4
32 18
12.53
Proximal AgeGropp
ProximalPhallnxOOO OutlineCorooc
30822.7
2169.5
497.6
69.22
20.16
ProximalPhalanxTWW
11641.7
375.4
93.4
33.45
13.56
ECG200
1615.3
1355.6
269.0
4.42
3.88
Italy Dewerand
41918.1
1939.0
417.5
0.82
0.64
Medicallmages
147668.
Plane
2036.0
1065.0
249.2
7.55
6.94
Swedishl Leaf
38045.7
72.17
27.15
GunPoint
1977.0
1685,7
422.1
2.91
2.96
CuniPoinAAgeSpun
14013.9
3945.6
1078.6
8.11
5.64
GunPointMale Versuseeale
14069.9
4058.8
1097.0
8.10
5.62
GonPointtOld Versus Young
16668.1
5400.9
1388.5
8.10
5.67
PowerCons
8808.3
SyntheticConrrol
13340.0
595.1
161.9
17.88
8.37
Table 7: Time (in seconds) taken by different vectorization methods for multifiltered simplicial complexes.
Dataset
num, graphs
avg(std) simplices per graph
HSM
MP SW
MP
COX2
467
84.6(8.2
0.91
6.21
2.24
DHFR
756
86.9(18.2)
1.55
16.85
4.09
IMDB BINARY
1000
116.3(113.3)
81.49
14.28
3.97
IMDR-MULTT
1500
78.9(1
91.55
29.74
3.33
MUTAG
188
37.7(10.2)
0.27
0.74
0.52
PROTEINS
1113
111.8(1298)
8.57
26.60
11.96
Table 8: Time (in seconds) to go from graphs to the Hilbert decomposition signed measure (column HSM), well as time (in seconds) to go from the signed measure to the output of our proposed vectorizations (columns HSM-MP-SW and HSM-MP-C).
24

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where r is the radius of the dataset, chosen by ev. As one can see from the results in Table 9, MP-HSM-SW is indeed quite effective with this choice.
Dataset
MP-I.
MP-HSM-SW
DistaiPPhalanx Outhinnuttttteee Aeetiinnnnneruuuu
62.6
62.6
77.0
67.6
70.5
7L,9
71,9
DistalPhalanx CattaaalaanotliieeeeC
71.7
71.7
75.4
DistalPhalanxT%
63.3
63.3
59.0
61.2
56.1
61.9
66.9
CatimmalPaalanxOttiin AgoGroup
78.5
78.5
80.5
78.0
78.5
81.0
34.4
ProximalPlalanxt OutlineCConCo
80.8
79.0
78.4
78.7
78.7
81.8
84.5
ProxinalPhalamm TW
70.7
75.6
75.6
79.5
73.2
76.1
78.0
ECG200
88.0
77.0
74.0
83.0
87.0
IellyPoweDDemand
95.5
95.5
95.0
80.7
78.6
79.8
83.1
Medicallmages
55.7
67.9
Plane
96.2
100.0
100.0
92.4
84,8
97.1
99.0
SwedishLeaf
78.9
84.6
79.2
78.2
64.6
83.8
88.6
GunPoint
91.3
91.3
90.7
88.7
94.0
90.7
97.3
GunPoint Agenn
89.9
96.5
91.8
93.0
85.1
90.5
97.8
GunPointMalle Fersus
97.5
88.3
95.9
58.4
95.2
96.5
83.8
99.0
97.1
100.0
99.7
PowerCons
93_1
84.4
91.7
SyntheticContrel
88.0
98.3
99.3
50.7
60.3
60.0
06.3
Table 9: Accuracy scores of baselines and multiparameter persistence methods on time series datasets. Boldface indicates best accuracy and underline indicates best accuracy among topological methods.
C.5 The enrichment factor
common approach for quantifying the performance of virtual screening methods [67] uses the enrichment factor F.F, defined as follows. Given a query and a test of ligands I., with each ligand labeled as either active decoy with respect to the query, the virtual screening method is run (g, L) producing a linear order (L). Given a E (0, 100),
(F, |active molecules in first (a/100) x | elements of O(L) / ((a/100) x LI)
active molecules in L |4]
We use the Cleves-Jain dataset [28], and follow the methodology of [67]. The dataset consists of common set of 850 decoys D, and, for each of 22 targets {a...., v), two to three compounds {g } and a set of 4 to 30 actives Li. To quantify the performance the method on the target æ, one averages EF. (uf" La U D) over the compounds {ạe J. The overall performance, which is what is reported Table of the main article for different choices of a, is computed by averaging these quantities over the 22 targets.
For topology- methods for virtual screening which use the EF to assess performance, see [47, 16, 34].
D Runtime experiments
We run these experiments in computer with a Ryzen 4800 CPU, and with 16GB of RAM.
D.1 Runtime of computation of Hilbert decomposition signed measure
Hilbert function by reducing multiparameter to one-parameter persistence. Let f 5 R²) be filtered simplicial complex. and let E N. Suppose want to compute the Hilbert function of the homology persistence module Hf( : RT - vec restricted to a grid, which, without loss of 13" for some E N. Given E {O..... 17 1 and b E {O...., m - 1), denote ba:b) = (a), a..1. b). In particular, given a E {O..... m 1pa-1, we get 1-parameter persistence module indexed by b E {0. ... H,(f) b) E vec; we denote this persistence module by (f) : {0. m - 1} - vec.
generality, may assume to be {0, (O....,
1) by mapping b to
We proceed as follows. For each E {O....,
1]n-1, we use the one-parameter persistence algorithm [38] to compute the Hilbert function of the module (f). Thus, we perform m- runs of the me--aaraetter persistence algorithm. The worst case complexity of the one-paraeteer
25

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+ [(+1l))), where |Sg] is the number of k--dimensional simplices of the simplicial complex, but known to be almost linear practice [5].
persistence algorithm is O [[1s-1|+|16) +
In the UCR examples, since we are dealing with Vietoris-Rips complexes which can have lots of simplices, rely the edge-collapse optimization of [82] to speed up computations. Since the point clouds are not too large, we do not perform any further optimizations, but we mention that optimizations like the ones of [14, 66] are available for dealing with large point clouds.
Runtimes. In the fourth column (HSM) of Tables 6 and 8, we report the time taken for computing the Hilbert decomposition signed measure with resolution & 200 starting from filtered simplicial complex. See the experiments section in the article for a description of these filtered simplicial complexes. Then, in the last two columns of Tables 6 and (MP-SW and MP-C). report the time taken for computing our proposed vectorizations starting from the Hilbert decomposition signed measure.
One can see that, in both tables, the bottleneck is usually either the HSM or the MP-SW computation, as MP-C is quite fast to compute. Overall, the whole pipeline (decomposition + vectorization) can be achieved in a quite reasonable amount of time, as the running times never go beyond 102 seconds, which is quite efficient in the realm of topological methods (see also next section).
D.2 Runtime of whole pipeline
In Table we compare the runtime of our full pipeline (from the point clouds obtained from the UCR datasets to the output of our vectorizations) to that of other pipelines based on multiparameter persistence. For other pipelines, we use the numbers in [20].
It is quite clear that our pipeline is much faster than the other topological baselines, by several orders of magnitude. This is generally due to the fact that Hilbert decomposition signed measures can be computed in the same amount of time than fibered barcodes (which are needed by the baselines), and can be turned into vectors in single step with one of our proposed vectorizations, while other baselines require vectorizing all elements of the fibered barcodes.
E Stability experiments
In this experiment, we test our main stability results, Theorems to 3. We fix a simplicial complex K and consider filtrations f : K -> R² using the lower-star filtration associated to functions K₂ -> Ra on the vertices of K. We construct functions on the vertices of K as follows: we treat the function as a vector of dimension [K)), start with a constant vector, and iteratively add uniform random noise to each component. This is effectively a random walk that, at each step. produces function filtering K. Thus, each random walk produces a set of filtering functions (
For each random walk (shown with a different color), we consider the -is-titances be- tween functions IIfi 1, the Kantorovich-Rubbnn Rubinstein distances between Hilbert signed mea- sures [|и MHooff) ) ||kR, the sliced Wasserstein kernel distances between vectorizations (ESM-SW(Hna(s))))
and the L2-distances between convolution vectoriza- HSMMC(()n(tt))ll> See Fig.
tions (HSSMC((H,(t))
The fact that the points in the plot lie below a line with positive slope passing through the origin is a consequence of our stability results for the Hilbert signed measure and for the vectorizations. The fact that the points in the plot lie above a line passing through the origin (at least for points with sufficiently small s-coordinate), shows that, for this kind of data, our proposed vectorizations are strong invariant, meaning that it is able to distinguish different filtering functions.
26

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Kantorovich distance between signed measures
distance between functions
L2 distance between convolutions with Gaussian
Wasserstein kernal distance
measures
Figure Top: I distancc between filtering functions and Seantorovicc-Rubisseee distance between the associated Hilbert signed measures. Bottom: measures and distance between the vectors produced by our vectorizations. Different colors indicate different runs of the random walk used to construct the filtering functions. Axis do not have scale since scale depends on choice of norms and of vectorization parameters.
distance between signed
27


Reviews:
[{"review_id": "LJ5LdbULYI", "reviewer": "Reviewer_61ca", "summary": "This work promote the use of signed barcodes for feature generation, and proposed the feature generation pipeline based on the signed bar codes, \na. the work introduces two general vectorization techniques for signed barcodes;\nb. the authors prove Lipschitz-continuity results that ensure the robustness of the proposed entire feature generation pipeline\nc. the practical performance of the proposed pipeline is compared to other baselines in various supervised and unsupervised learning tasks.\n\nAll of my questions are fully addressed by the rebuttal. I am satisfied with the explanations. I think the work is solid and promising.", "questions": "1. Whether single-parameter persistent homology is equivalent to multi-parameter persistent homology ? What extra-information can multi-parameter PH contributes?\n2. The Kantorovich-Robinstein norm in Definition 3 is inconsistent with that in proposition 1, under what conditions are they equivalent in general ?\n3. The generalization of theorem 1 to higher dimension is open, is the difficulty intrinsic or just technical ?", "limitations": "The authors have adequately addressed the limitations, which are helpful for practical applications.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "This work generalizes the single parameter bar codes to multi-parameter bar codes, introduces general vectorization techniques for signed barcodes, and prove Lipschitz-continuity results to ensure the robustness of the proposed pipeline. The theoretic results are important and convincing. \nThe paper is well written, the key concepts are explained thoroughly, the proofs of the main theorems are clear and in details. The experimental results are analyzed in details.", "weaknesses": "The motivation for multi-parameter persistent homology could be further emphasized. From theoretical point of view, it is unclear what extra-information can multi-parameter PH bring compared to single-parameter PH. \n\nIt will be helpful to add some experiments to validate the stability theorems, and estimate the Lipschitz-constant.\n\nThe definition of the Kantorovich-Rubinstein norm for point measures in proposition 1 is inconsistent with that in definition 3. The definition 3 allows point mass splitting, namely one source point can be mapped into multiple target points, hence \\psi is a transportation scheme; but in proposition 1, each source point is mapped to a single target point, \\gamma is a transportation map. This needs to be clarified.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and feedback.\n\n- *Motivation for multiparameter persistence and extra info compared to one-parameter.*\nThank you very much for pointing out the lack of references motivating multiparameter persistence and its relationship to one-parameter persistence.\nTo address this, we will add the following short paragraph right after line 47:\n\n>There are many applications of TDA where multiparameter persistence modules are more natural than, and lead to improved performance when compared to, one-parameter persistence modules.\n>These include\n>noisy point cloud data\n>[Vipond, Bull, Macklin, Tillmann, Pugh, Byrne, Harrington. Proc Natl Acad Sci USA 2021],\n>where one parameter accounts for the geometry of the data and another filters the data by density;\n>multifiltered graphs\n>[Demir, Coskunuzer, Segovia-Dominguez, Chen, Gel, Kiziltan. NeurIPS 2022],\n>where different parameters account for different filtering functions;\n>and time-varying data\n>[Chen, Segovia-Dominguez, Coskunuzer, Gel. ICLR 2022],\n>where an extra parameter accounts for time-dependence.\n\n- *Definition 3 vs Proposition 1.*\nProposition 1 is stating a property of the Kantorovich-Rubinstein norm, namely that, when evaluated on a point measure (i.e., a measure given by an integer linear combination of Dirac deltas), there always exists an optimal coupling that does not split masses.\nWe gave details in the originally submitted appendix, but we agree that the fact that this is a property and not a different definition should be further emphasized to avoid confusions.\nWe will add the following sentence right before Proposition 1 with that goal:\n\n> The following result says that, for point measures, the computation of the Kantorovich--Rubinstein norm reduces to an assignment problem, and that, in the case of point measures on the real line, the optimal assignment has a particularly simple form.\n\n- *Theorem 1 for higher $n$.*\nThis is a very interesting question.\nWe know of an extension for arbitrary $n$ but with a Lipschitz constant that depends on the simplicial complex $K$; although not as strong as the cases $n = 1,2$, this result does provide justification for the method for higher parameters.\nHowever, the proof is quite technical and, unlike the cases $n=1,2$, it does not follow easily from previous work and requires several new definitions; for this reason we decided to omit it.\nAn extension for arbitrary $n$ with a Lipschitz constant that is independent of the simplicial complex $K$ seems plausible and is the subject of current work.\n\n- *Experimental validation of stability.*\nThank you for suggesting this.\nPlease see the rebuttal PDF we have submitted where such an analysis is carried out for the pipeline (multifiltered complex) -> (Hilbert signed measure) -> (convolution vectorization), showing that our vectors are indeed stable and also provide a quite strong lower bound.\nIf the paper is accepted, in the final version we will include an analysis for the three constructions: (multifiltered comples) -> (Hilbert signed measure), (Hilbert signed measure) -> (convolution vectorization), (Hilbert signed measure) -> (sliced Wasserstein kernel).\n"}, {"review_id": "DyqEYYefbH", "reviewer": "Reviewer_cn3Z", "summary": "The authors introduce first vectorizations of multiparameter persistent homology (MPH) via signed barcodes, that are easy to compute and shown to be stable. The two proposed vectorizations often outperform the state of the art MPH methods on a variety of data sets.", "questions": "(Q1) The proposed approach is theoretically weaker than some other (Appendix A, Proposition 1), but it outperforms them. This should be at least briefly commented/discussed.\n\n(Q2) It seems that the two and three filtrations used respectively in Sections 4.2 and 4.3 for the calculation of proposed MPH featurizations are not the same as the other considered MPH approaches from the literature? How is the comparison fair/reasonable then? \n\n(Q3) Can you comment on why MP-HSM-SW is much better for some, whereas MP-ESM-C performs significantly better for other data sets?\n\nOther minor comments:\n\n-\tDefinition 6: Name the introduced notion K * \\mu.\n-\tLine 81: Briefly summarize the conclusions of the runtime assessment, i.e. that your pipeline is much faster than the other topological baselines, by several orders of magnitude (Appendix D.2, Table 3).\n-\tTheorems and Propositions: For better readability, it would be useful to name the theoretical results.\n-\tSection 4.1: Describe the data and filtration.\n-\tLine 301: “As one can see from the results in Table 2, MP-HSM-SW is quite efficient”. Rephrase, since Table 2 does not show the computation times, but accuracy. \n-\tTable 4 caption: MP-HSM-C -> MP-ESM-C?\n-\tReferences: Be consistent between capital case vs. lower case for journal names, and also between full and shortened names, e.g. Journal of Machine Learning Research or J. Mach. Learn. Res. (similarly, Discrete Comput. Geom, Found. Comput. Math, …). Correct 2d, Xgboost, ucr, betti, Tudaset.\n", "limitations": "The limitations and future work are discussed in detail.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "(S1) The paper is clearly organized and written, it reads really well.\n\n(S2) The experiments are rather extensive and the results seem convincing.", "weaknesses": "I did not identify major weaknesses in this work, besides some confusion about the experimental settings addressed in question (Q2) below. I am open to raising my score if this issue is properly addressed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and feedback.\n\n- *Comment on/discuss the fact that the proposed approach is theoretically weaker than other approaches (Appendix A, Proposition 1), but it outperforms them.*\nWe believe this is due to the fact that, despite being weaker than the rank invariant, using the Hilbert/Euler functions allows us to leverage and use the vectorization techniques that have worked the best for one-parameter persistence (in our case, Gaussian convolution and sliced Wasserstein kernel).\nIn other words, while the rank invariant is richer, the ability to use it in machine learning pipelines is (currently, and to the best of our knowledge) limited in the sense that it forces one to use more complicated vectorizations.\nIn the original submission, we have commented on this briefly in line 349 of the Conclusions.\nWe agree that this is too succinct, and we will add the following more detailed comment:\n\n>[...] strictly weaker descriptors.\n>We believe that this is due to the fact that using the Hilbert and Euler signed measures allows us to leverage well-developed vectorization techniques shown to have good performance in one-parameter persistence.\n>We conclude from this that the way topological descriptors are encoded is as important as the discriminative power of the descriptors themselves.\n\n- *Filtrations of Sections 4.2 and 4.3 are not the same as the other considered MPH approaches from literature.*\nWe agree and thank you for pointing this out.\nWe have rerun the experiments of sections 4.2 and 4.3.\nFor section 4.2, we now used alpha complex + DTM filtration as in [Carrière, Blumberg. NeurIPS 2022].\nWe are including the new results for the first 6 UCR time series datasets of the original submission in Table 1 of the rebuttal PDF we submitted.\nFor section 4.3, we now used Heat Kernel Signature (with time 10) + Ricci curvature as in [Carrière, Blumberg. NeurIPS 2022].\nWe are including the new results for the graph datasets of the original submission in Table 2 of the rebuttal PDF we submitted.\nThe takeaway for both tables is the same: our methods are outperforming previous MPH vectorization methods for these tasks.\nWe note two interesting things: SWK is doing slightly worse than using Rips+KDE as in the original submission (but still better than previous MPH vectorizations), while the convolution-based vectorization is doing better; we believe that the improved performance of the convolution-based vectorization could be due to some implementation improvements having to do with how measures are snapped to a grid.\nIf the paper is accepted, we will include the full table as well as the Hilbert and Euler function columns (not part of previous work, but included in the original submission for the sake of comparison), which are now missing due to time constraints.\n\n- *Comment on why MP-HSM-SW is much better for some, whereas MP-ESM-C performs better for other data sets.*\nBoth vectorizations have advantages and drawbacks.\nThe convolution-based vectorization requires the choice of a relatively small grid in order to make training practical, whereas SWK can take a much finer grid since it does not result in a more costly vectorization.\nBut, the SWK can be sensitive to outliers in the form of far-away point masses, since transporting them can be costly in terms of Wasserstein distance; meanwhile, for the convolution-based vectorization a far-away point mass just results in a bump whose distance to zero is bounded regardless of how far it is.\nWe believe that the best method depends highly on the data type.\n\n- *Other minor comments.*\nThank you for spotting these; we will address them.\nOne question: when you say \"For better readability, it would be useful to name the theoretical results\", do you mean giving a short description of the result like \"Theorem 3 (Stability of sliced Wasserstein)\"?\n"}, {"review_id": "wDE4SOz2wi", "reviewer": "Reviewer_VRYZ", "summary": "The paper vectorizes data descriptors coming from multiparameter persistent homology for classifying point clouds and measuring similarity between graphs extracted from databases of times series and molecules. ", "questions": "Lines 14-15 claim that \"The resulting feature vectors are easy to define and to compute\". If this is true, could the authors write a full easy definition of these vectors in a few lines?\n\nConcerning the computation, line 245-256 seem more honest: \"We discretize the computation of the sliced Wasserstein kernel by choosing directions {θ1, . . . , θd} ⊆ Sn−1 uniformly at random\". Are there any theoretical guarantees for this approximate computation and what is the asymptotic cost in the size of the input?\n\nThe authors correctly highlight the importance of stability (Lipschitz continuity) under noise. However, the practical experiments in section 4.2 use discrete grids, which makes all features discontinuous under perturbations of times series or point clouds. Is it possible to avoid this discretization?\n\nSection 4.3 discusses datasets of graphs coming from social networks, biology, and medicine. How can edges be justified in these experimental graphs? \n\nIn social networks, friendships or other links between people have different strengths and are always subjective. In molecules, even the strongest covalent bonds between atoms have no strict definitions and only abstractly represent inter-atomic interactions, so there are physical sticks or strings between them. That is why most databases of molecules and materials such as QM9 contain only atomic positions but no links between them.\n\nWhich of the invariants in Appendix B are provably Lipschitz continuous under perturbations of points, sat in the bottleneck distance on the persistence diagram?\n\nWhich of these invariants are faster or stronger than the isometry invariants from Widdowson et al (CVPR 2023)?\n\nWhich (dis)similarity functions mentioned in the paper satisfy all metric axioms? The importance of metric axioms including the triangle inequality was theoretically justified in the paper arXiv:2211.03674, which should be cited in every work on clustering and metric distances because any clustering based on non-metrics is not trustworthy.", "limitations": "The main limitation of the 1-parameter persistence is its weakness as an isometry invariant, which should have been clear to all experts in computational geometry many years ago but was demonstrated only recently. The paper by Smith et al (arxiv:2202.00577) explicitly constructs generic families of point clouds in Euclidean and metric spaces that are indistinguishable by persistence and even have empty persistence in dimension 1. These examples can be easily extended to more than one parameter at least for some filtrations.\n\nThough Topological Data Analysis was largely developed by mathematicians, the huge effort over many years was invested into speeding up computations, rather surprisingly, instead of trying to understand the strengths and weaknesses of persistent homology, especially in comparison with the much simpler, faster, and stronger invariants of clouds under isometry. \n\nPersistence in dimension 0 was extended to a strictly stronger invariant mergegram by Elkin et al in MFCS 2020 and Mathematics 2021, which has the same asymptotic time as the classical 0D persistence and is also stable under perturbations of points.\n\nA SoCG 2022 workshop included a frank discussion concluding that there was no high-level problem that persistent homology solves. Is there such an ultimate problem for multi-parameter persistence? In fact, persistence as an isometry invariant essentially tries to distinguish clouds of unlabeled points up to isometry, not up to continuous deformations since even non-uniform scaling changes persistence in a non-controllable way. \n  \nOn the other hand, the isometry classification problem for point clouds was nearly solved by Boutin and Kemper (2004), who proved that the total distribution of pairwise distances is a generically complete invariant of point clouds in any Euclidean space. The remaining singular cases were recently covered by Widdowson et al in NeurIPS 2022 and CVPR 2023. ", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper includes rigorous definitions and proves (in the appendices) three theorems from section 4.", "weaknesses": "The paper seems to over-advertise the strengths of persistence.\n\nLines 1-2 claim that \"Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs\". If the authors really meant a topological classification of graphs, this classification has been known to Euler in the 18th century, so there is no need to re-invent topological invariants of graphs by using persistence.\n\nLines 22-23 claim that \"TDA methods usually require the sole knowledge of a metric or dissimilarity measure on the data\". In fact, the definition already of persistence requires a choice of filtration of simplicial complexes on given data points, which seriously affects the resulting persistence whose further analysis requires many more extra parameters, see lines 351-352: \" Our pipelines, including the choice of hyperparameters for our vectorizations, rely on the cross validation of several parameters, which limits the number of possible choices\"\n\nExamples 1-3 are definitions or comments, not illustrative examples that could help the readability. \n\nThe questions and limitations below include further concerns about experiments and comparisons with past work. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and feedback.\n\nWe start with two clarifications: Our applications go well beyond classification of point clouds up to isometry, and even in the point cloud application, we do not seek a strong isometry invariant since such invariants are necessarily highly sensitive to, e.g., outliers.\nIt is also beyond the scope of this rebuttal to discuss the efficacy of PH, which is well-established given the vast amount of existing applications in ML ([Hensel, Moor, Rieck, 2021] and references).\n\n- *\"Lines 1-2 [...] using persistence\"* We do not understand this comment: the quoted sentence does not mention \"topological classification of graphs\". All we are saying here is that persistence allows for the use of topology to produce descriptors of geometric data. A weighted graph can be taken as a single graph, but persistence allows one to incorporate much richer information into topological descriptors by using filtrations derived from the weights.\n\n- *\"Lines 22-23 [...] possible choices\"* The fact that our method has hyperparameters does not prevent it from being applicable in arbitrary metric spaces or weighted graphs. One can resort to default choices as in Section 4.4.\n\n- *\"Examples 1-3 [...] readability\"* Ex 1 provides a classical example of 1-dimensional simplicial complex; Ex 2 describes a classical family of multifiltered complexes; Ex 3 gives a usual way of getting a persistence module from a filtration. We do not see why these examples should not be helpful to the reader.\nDoes the reviewer have other examples in mind?\n\n- *\"Could the authors [...] few lines\"* The Hilbert function of a module is its pointwise dimension (def 2), and the Hilbert signed measure is computed by Mobius inversion of the Hilbert function (amounting to a convolution with a kernel of small support; this is reference to [41, Rmk. 3] in line 213). The formulas for vectorizations are explicitly given in defs 6 and 7; and the computation is described in lines 232 and 245, respectively. We will be happy to make this comment into pseudocode, if the reviewer believes this will improve readability.\n\n- *\"Are there any [...] input\"* There is no approximation: our definition of the Wasserstein kernel takes any measure on the $(n-1)$-sphere, and our stability result holds for any such measure.\nThis allows us to use a discrete measure so that the integral can be evaluated exactly. One could consider the approximation of SWK with uniform measure on the sphere by a SWK with discrete measures, as in [Carriere et al. 2017] and the analysis would be almost identical. For space concerns, we decided to have the theory of the paper focus on stability results and not on injectivity/inverse results.\nThe time complexity of the computation of the SWK is linear in the number of point masses of the discrete measure on the $(n-1)$-sphere.\n\n- *\"Is it possible to avoid this discretization?\"* Discretizations make the vectors discontinuous, but, crucially, Lipschitz stability theorems still hold up to the discretization size (on grids of step size $\\epsilon$, stability holds up to additive term of $O(\\epsilon)$), so near-by datasets still get near-by vectors, and approximation can be controlled by the user.\nIt is possible to avoid discretizations: the computation of the Hilbert and Euler signed measures can be done exactly since fp MP modules can be described exactly over some data-dependent finite grid.\nSo the SWK can be computed exactly on any finite point measure.\nThe convolution vectorization does require a grid if one seeks a vectorization (i.e., a Euclidean vector for each sample); however, if a kernel method is sufficient, one can dispense with discretizations because the $L^2$ inner product between Gaussian mixtures can be computed exactly (see, e.g., the vectorization of [Reininghaus et al. 2015]).\n\n- *\"How can edges [...] graphs\"* We seek to understand the performance of our vectorizations for graph classification, so we take the input datasets at face value like previous papers.\nWhile interesting, the discussion about the validity of edgers is outside the scope of the paper.\n\n- *\"Which of [...] persistence diagram?\"* We are not sure to understand this question: since the invariants being referred to are invariants of multiparameter persistence modules, there is, a priori, no persistence diagram. Could you please clarify this question?\n\n- *\"Which of these [...] (CVPR 2023)?\"* In Section 4.2, we do not seek to use strong isometric invariants for point clouds such as the ones in (CVPR 2023) because we want stability under, e.g., outliers.\nWe rather see the point clouds as empirical measures, and use density-geometric bifiltrations whose one-parameter slices are stable under Wasserstein perturbations [Chazal, Cohen-Steiner, Mérigot (2011)].\n\n- *\"Which (dis)similarity [...] axioms?\"* Our vectorizations have the space of finite signed measures with total mass zero as domain, equipped with the Kantorovitch norm, which is an honest norm; thus, its induced metric satisfies all metric axioms. The codomain of our vectorizations are Hilbert spaces, where the metric is derived from the inner product and therefore is an honest metric.\n\n- *\"About an ultimate problem/goal for multiparameter persistence\"* Besides recent applications of multiparameter persistence (see motivating paragraph for MPH, for now in the response to reviewer 61ca), various theoretical papers show that TDA with several parameters/directions can address difficult practical problems: [Blumberg, Lesnick. Found Comp Math, 2022] shows that two-parameter persistence provides Gromov-Prokhorov Lischitz-stable invariants (i.e., robust to outliers) of metric probability spaces that do not require user chosen parameters, such as a KDE bandwidth. [Curry, Mukherjee, Turner. Trans Amer Math Soc Ser B, 2022] shows that several one-parameter filtrations, taken together, can uniquely characterize simplicial complexes in Euclidean space up to action of the orthogonal group.\n"}, {"review_id": "ciN05xjyhM", "reviewer": "Reviewer_csyn", "summary": "In this paper, the authors are addressing a very critical need in topological data analysis (TDA), vectorization of multiparameter persistence (MPH). Persistent homology (PH) is the key method in TDA, but in its current form, it allows only a single function to use in its key process, filtration. By enabling multiple functions, MPH is a natural generalization of (PH) with much finer information, however, there are several mathematical obstructions to use them effectively in applications. \n\nIn this paper, the authors propose an effective way to vectorize the barcode information in the MP module by sacrificing some to keep the computation feasible and process practical. They offer two versions of MP vectorizations where one being kernel and the other being direct vectorization, both can be effective in different settings. The authors made extensive experiments in several settings, including point cloud, graph classification, and virtual screening to compare their model with other MP vectorizations and SOTA models. Their model consistently outperforms existing MP vectorizations.\n\n", "questions": "This is more of a question/remark. A low-hanging fruit might be to combine (concatenate) Hilbert functions (MP-H) with MP-HSM-C (or MP-ESM-C) as you already compute MP-H during the process if I understand the process right. Considering the closeness of the performances in graph classification and virtual screening, this combined vectorization might perform better.", "limitations": "As mentioned in weaknesses, the main limitations are scalability and hyperparameter tuning which could be very tricky, especially choosing the right grid size. Hence, performance vs. computational feasibility can be a problem in large datasets.", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The authors use a recent idea of signed barcodes cleverly to obtain practical vectorizations. There is a significant need to employ MPH idea with good and feasible vectorizations, and this paper makes a valuable contribution in this direction.\n\nThey propose two versions of their vectorizations, a kernel (MP-SW) and a direct vectorization (MP-C). In the past years, depending on the domain, we see that one can perform better than the other from our experience with the applications of single persistence. So, this versatility of outputs is very valuable for ML applications.\n\nComputational time is significantly better than existing MP vectorizations. Combining with its good performance, this might be the most important contribution of the paper from ML perspective.\n\nTheir extensive experiments in various settings show that their model consistently outperforms the existing MPH vectorizations.", "weaknesses": "The main weakness is that the paper might be too technical for non-experts and ML audience. However, considering the depth of the problem, I can see that the authors did an enormous effort to make this important subject accessible to a wider audience.\n\nWhile new vectorizations consistently outperform the existing MP models in point cloud settings, the performance is not as strong in graph classification and specifically, virtual screening cases (TODD also uses a version of MPH).  \n\nAs MPH is an involved process, hyperparameter tuning might need serious expertise in TDA in real life applications.\n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and feedback.\n\n- *Performance on graph data.*\nWe agree that the results in section 4.3 (graph data) are not as good as those in section 4.2 (point cloud data).\nWe note, however, that the performance on the virtual screening task of our method is very good; ToDD is a supervised method, whereas in this task we are using our vectorizations in a completely unsupervised way (we included ToDD to give an idea of how well TDA-based tools can work on this type of data).\nWith respect to section 4.3, we are using an invariant (the Hilbert function or, equivalently, the Hilbert signed measure) that is strictly weaker than that used by other topological methods (such as the rank invariant), which could be the reason.\nIn particular, we believe that the issue is not with the vectorizations themselves, and that the gap in performance could be bridged by using signed barcodes/measures coming from the rank invariant.\n\n- *Combine/concatenate MP-H and MP-HSM-C (or MP-ESM-C) to get better performance.*\nWe think this is an interesting idea which should be experimented with.\nIn this paper, we have decided to keep the approach as straightforward as possible to try to isolate the performance of the vectorization from the performance of the machine learning model and other application specific choices.\nWe address the tuning of the approach, including combination of different vectorizations, in ongoing work.\n"}]

Author rebuttal:
We sincerely thank all the reviewers for their time and feedback.
We have responded to their questions, and we will be happy to provide further clarifications, if required.

Only a couple of short paragraphs (included in the responses to specific reviewers, below) are required in the main body of the paper to address the reviewers' feedback.
With respect to the appendix, an experiment will be added following a suggestion by reviewer 61ca (a short version of the experiment is in Fig. 1 of the rebuttal PDF).

Meta-review: Reviewers unanimously agreed on the relevance of the proposed work. Making multiparameter persistence amenable to more efficient calculations promises novel integrations into modern machine learning techniques and novel insights. With most reviewers already opting for acceptance of the paper, it was a pleasure to see an enthusiastic and in-depth rebuttal by the authors. While some concerns could not be alleviated, they mostly pertain to the overall question of the use of persistent homology in machine learning—answering those is beyond the scope of the paper, but the authors are encouraged to think about more ways to contextualise their work vis-à-vis the machine learning community at large.

I am thus very happy to **endorse this work for publication and presentation at the conference**. I encourage the authors to use the time to revise their paper in terms of accessibility. While the overall efforts and the initial state were already lauded by reviewers, I also agree with reviewer `csyn` that additional work concerning accessibility is encouraged since it makes this exciting topic available to a larger audience.

The authors are also encouraged to consider additional baselines, as mentioned by reviewer `VRYZ`.
Decision: Accept (poster)