A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions
Authors: David Loiseaux
Mathieu Carrière
Andrew Blumberg
Venue: NeurIPS 2023 poster (2023)
Keywords: Topological Data Analysis
Multiparameter Persistent Homology
Kernel Methods
Convergence Rate
Statistical Learning

Abstract: Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is persistent homology, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case.  A central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on decompositions of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions
David Loiseaux DataShape
Mathieu Carrière
DataShape
Centre Inria d' d'Université Côte d' Azur Biot, France
Centre Inria d' d'Unveersiéé Côte d' Azur Biot, France
Andrew J. Blumberg
Irving Institute for Cancer Dynamics Columbia University New-York, NY, USA
Abstract
Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets, such as graphs and point clouds. One of the most important such descriptors is persistent homology, which encodes the change in shape as a filtration parameter changes; typical parameter the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. particular, a central question is the problem of representing multiparameter persistent homology by elements of vector space for integration with standard machine learning algorithms. Existing approaches
this problem either ignore most of the multiparameter information to reduce to the one-parameter case are heuristic and potentially unstable the face of noise. In this article, we introduce new general representation framework that leverages recent results on decompositions of multiparameter persistent homology. This framework rich information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.
Introduction
Topological Data Analysis (TDA) [8] is a methodology for analyzing data sets using multiscale shape descriptors coming from algebraic topology. There has been intense interest in the field in the last decade, since topological features promise to allow practitioners to compute and encode information that classical approaches do not capture. Moreover, TDA rests on solid theoretical grounds, with guarantees accompanying many of its methods and descriptors. TDA has proved useful in wide variety of application areas, including computer graphics [13, 33], computational biology [34], and material science [6, 35), among many others.
The main tool of TDA is persistent homology. In its most standard form, one is given finite metric space X (e.g., a finite set of points and their pairwise distances) and a continuous function
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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f: + R. This function usually represents a parameter of interest (such as, e.g., scale or density for point clouds, marker genes for single-cell data, etc), and the goal of persistent homology is to characterize the topological variations of this function on the data at all possible scales. Of course, the idea of considering multiscale representations of geometric data is not new [14, 32, 41]; the contribution of persistent homology is to obtain novel and theoretically tractable multiscale shape descriptor. More formally, persistent homology is achieved by computing the so-called persistence barcode of f. which is obtained by looking at all sublevel sets of the form {(- (o),,a))) ) also called filtration induced by f, and by computing a decomposition of this filtration, that is, by recording the appearances and disappearances of topological features (connected components, loops, enclosed spheres, etc) in these sets. When such a feature appears (resp. disappears), e.g., in a sublevel set fetf ((-o0, oo, agl), we call the corresponding threshold ag (resp. a.)) the birth time (resp. death time) of the topological feature, and we summarize this information in a set of intervals, or bars, called the persistence barcode D(f)= == {(as, (2), A C R R U {oo}. Moreover, the bar length Od Og often serves as proxy for the statistical significance of the corresponding feature.
However, an inherent limitation of the formulation of persistent homology is that it can handle only single filtration parameter f. However, in practice it is common that one has to deal with multiple parameters. This translates into multiple filtration functions: standard example is when one aims at obtaining meaningful topological representation of a noisy point cloud. In this case, both feature scale and density functions are necessary (see Appendix A). An extension of persistent homology to several filtration functions is called multiparameter persistent homology [3. 9]. and studies the topologiaal variations of continuous multiparametter function f X + R" with n E N This setting is notoriously difficult to analyze theoretically as there is no result ensuring the existence of an analogue of persistence barcodes, i.e., decomposition into subsets of R" each representing the lifespan of a topological feature.
Still, it remains possible to define weaker topological invariants in this setting, The most common one is the so-called rank invariant (as well as its variations, such as the generalized rank invariant [24], and its decompositions, such the signed barcodes [4]), which describes how the topological features associated to any pair of sublevel sets {ar E X : f(z) I a} and {ar E X f(z) I 3) such that S 3 (w.rt. the partial order in R"), are connected. The rank invariant is construction in abstract algebra, and so the task of finding appropriate representations of this invariant, i.c., embeddings into Hilbert spaces, is critical. Hence, number of such representations have been defined, which first approximate the rank invariant by computing persistence barcodes from several linear combinations of filtrations, a procedure often referenced as the fibered barcode (see Appendix E), and then aggregate known single-parameter representations for them [17, 18, 39]. Adequate representations of the generalized rank invariant have also been investigated recently for n 2 [42].
However, the rank invariant, and its associated representations, are known to be much less informative than decompositions (when they exist): many functions have different decompositions yet the same rank invariants. Therefore, the aforementioned representations can encode only limited multiparam- eter topological information. Instead, in this work, focus candidate decompositions of the function, in order to create descriptors that are strictly more powerful than the rank invariant. Indeed, while there is no general decomposition theorem, there is recent work that constructs candidate decompositions in terms of simple pieces [1, 7, 29] that always exist but do not necessarily suffice to reconstruct all of the multiparameter information. Nonetheless, they are strictly more informative than the rank invariant under mild conditions, are stable, and approximate the true decomposition when it exists!. For instance, in Figure 2, we present a bifiltration of noisy point cloud with scale and density (left), and a corresponding candidate decomposition comprised of subsets of R², each representing topological feature (middle). One can see that there large green subset in the decomposition that represents the circle formed by the points that are not outliers (also highlighted in green the bifiltration).
AAthhough multiparameter persistent homology can always be decomposed as sum of indecomposable pieces ([3, Theorem 4.2], [20]), these decompositions are prohibitively difficult to interpret and work with.
2

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Rank-based
Representations in Hilbert space
Multiparameter function
Multiparameter
invariants
persistent
f:X g
homology
Candidate
Representations Hilbert space
decompositions
Figure 1: Common pipelines for the use of multiparameter persistent homology in data science work provides new contributions to the arrow highlighted in red.
Unfortunately, while more informative, candidate decompositions suffer from the problem than the rank invariant; they also need appropriate representations in order to be processed by standard data science methods. In this work, bridge this gap by providing representations designed for candidate decompositions. See Figure for a summarizing figure.
Putpe Deompodise
1.2
025
codensity
0.5-
030
015
0.0
0.2
0.5
0.00
radius
Figure 2: (left) Bi-filtration of a noisy point cloud induced by both feature scale (using unions of balls with increasing radii) and sublevel sets of codensity. The cycle highlighted in the green zone can be detected as a large subset the corresponding candidate decomposition computed by the MMA method [29] (middle), and in our representation of it (right).
Contributions. Our contributions in this work are listed below:
We provide general framework that parametrizes representations of multiparameter persis- tent homology decompositions (Definition 1) and which encompasses previous approaches in the literature. These representations take the form of parametrized family of continuous functions on IR" that can be binned into images for visualization and data science.
We identify parameters in this framework that result in representations that have stability guarantees while still encoding more information than the rank invariant (see Theorem 1).
We illustrate the performance of our framework with numerical experiments: (1) We demonstrate the practical consequences of the stability theorem by measuring the statistical convergence our representations. (2) achieve the best performance with the lowest runtime several classification tasks on public data sets (see Sections 4.1 and 4.2).
Related work. Closely related to our method is the recent contribution [10], which also proposes representation for decompositions. However, their approach, while being efficient in practice, is heuristic with no corresponding mathematical guarantees. In particular, it is known to be unstable: similar decompositions can lead very different representations, as shown in Appendix B. Our approach can be understood subsequent generalization of the work of [10], with new mathematical guarantees that allow to derive, e.g., statistical rates of convergence.
Outline. Our work organized as follows. In Section 2, we recall the basics of multiparameter persistent homology. Next, in Section 3 we present our general framework and state our associated stability result. Finally, we showease the numerical performances of our representations in Section and conclude in Section

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2 Background
In this section, we briefly recall the basics of single and multiparameter persistent homology. and refer the reader to Appendix C, Appendix D, and [31, 34] for more complete treatment.
Persistent homology. The basic brick of persistent homology is a filtered topological space X, by which we mean topological space X together with a function f: X - R (for instance, in Figure 5, X = IR and f = fr). Then, given a > 0, we call F(a) == == f- a)) C X the sublevel set of f at level a. Given levels 01 S.... S ON, the corresponding sublevel sets nested w.r.t. inclusion, i.e., one has F(a1) C F(a2) C S F(a:) S _... C F(ax). This system is an example of filtration of X, where a filtration is generally defined a sequence of S X. Then, the core idea of persistent homology is
nested subspaces Xi C .... S X; S to apply the kth homology functor Hş on each F(a:). We do not define the homology functor ) is a vector space, whose basis elements represent the kth dimensional topological features of F(a:) (connected components for k loops for k = 1. spheres for l = 2, etc). Moreover, the inclusions F(a,) S F(a,+1) translate into linear maps H& HF (ax)) - H (F(+a))))),, which connect the features of F(o:) and F(a,+1) together. This allows to keep track of the topological features in the filtration, and record their levels, often called times, of appearance and disappearance, More formally, such a sequence of vector spaces connected with linear maps M = (F(a)) + - H. (F(((a))) is called persistence module, and the standard decomposition theorem [15, Theorem 2.8] states that this module can always be decomposed as M = @ [/ab,,aa,, where l[eva, (va,) stands for a module of dimension (i.e., that represents a single topological feature) between ay, and ad, and dimension O (i.e., that represents no feature) elsewhere. It is thus convenient to summarize such module with its persistence barcode D(M) = {[as, a J.sism Note that in practice, one is only given a sampling of the topological space X, which is usually unknown. In that case, persistence barcodes are computed using combinatorial models of X computed from the data, called simplicial complexes. See Appendix C.
explicitly here, but simply recall that each HF(F(aa))
Multiparameter persistent homology. The persistence modules defined above extend straight- forwardly when there multiple filtration functions. An n-filtration, or multifiltration in- duced by a function f X - RR,, is the family of sublevel sets F = (F(o)) F(a:) : {i X : f(x) S a) and S denotes the partial order of R Again, applying the homology functor H₃ on the multifiltration induces a multiparameter persistence module M. How- ever, contrary to the single-parameter case, the algebraic structure of such module is very intricate, and there is no general decomposition into modules of dimension at most 1, and thus no analogue of the persistence barcode. Instead, the rank invariant has been introduced as weaker invariant: is defined, for a module M, as the function RI (a, (3) + rank(M(a) + M(5)) for any a S 8, but is also known miss lot of structural properties of 1M. To remedy this, several methods have been developed to compute candidate decompositions for M [1, 7, 29], where candidate decomposition is a module i' that can be decomposed as M ~ t j==1 M,, where each M, is an interval module, i.c., its dimension at most 1, and its support supp (M,) := fa e R" dim (M,(M(a)) =} an interval of R" (see Appendix D). In particular, when M does decompose into intervals, candidate decompositions must agree with the true decomposition. One also often asks candidate decompositions to preserve the rank invariant.
where
Distances. Finally, multiparameter persistence modules can be compared with two standard dis- tances: the interleaving and bottleneck (or (x)) distances. Their explicit definitions are technical and not necessary for our main exposition, so we refer the reader to, e.g., [3, Sections 6.1, 6.4) and Appendix D for more details. The stability theorem [27, Theorem 5.3] states that multiparameter per- sistence modules are stable: (M, M') < - f'll where f and f' are continuous multiparameter functions associated to IM and M' respectively.
3 T-CDR: a template for representations of candidate decompositions
Even though candidate decompositions of multiparameter persistence modules are known to encode useful data information, their algebraic definitions make them not suitable for subsequent data science and machine learning purposes. Hence, in this section, we introduce the Template Candidate

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Decomposition Representation (T-CDR); general framework and template system for representations of candidate decompositions, i.e., maps defined on the space of candidate decompositions and taking values in an (implicit or explicit) Hilbert space.
3.1 T-CDR definition
Notations. In this article, by slight abuse of notation, will make difference in the notations between an interval module and its support, and we will denote the restriction of an interval support M to given line as M
Definition 1. Let M = M be candidate decomposition, and let M be the space of interval modules. The Template Candidate Decomposition Representation (T-CDR) of M is:
o op(fw(M.) (a0.:..((((MMM)))))) ((M))
(I)
where op is a permutation invariant operation (sum, max, min, mean, etc), w -M - R is a weight function, and Φ .M- + H sends any interval module to a vector in a Hilbert space H.
The general definition of T-CDR is inspired from similar framework that was introduced for single-parameter persistence with automatic representation method PersLay [11].
Relation to previous work. Interestingly, whenever applied on candidate decompositions that preserve the rank invariant, specific choices of op, w and @ reproduce previous representations:
R" + R
Using w M, - 1, M1
+ A(x, M: and op = kth maximum, where
x
le is the diagonal line crossing x, and AC, E) denotes the tent function associated to any segment C R", induces the kth multiparameter persistence landscape (MPL) [39].
Ra p.q op op', where op', w' and ø' are the parameters of any persistence diagram representation from Perslay, induces the multiparameter persistence kernel (MPK) [17].
Using : Mr 1, Mr {
x Rm
IRd
+ w'(M n (p. O'(M, n (p, and
{
R"
+ R
+ exp(-mingg,d(z, Molp)%/o²) and op E. where L is set of (pre-defined) diagonal lines, induces the multiparameter persistence image (MPI) [10].
Using w : M4 + vol(M;), M,
Recall that the first two approaches are built from fibered barcodes and rank invariants, and that it is easy to find persistence modules that are different yet share the same rank invariant (see [38, Figure 3I). On the other hand, the third approach uses more information about the candidate decomposition, but is known to be unstable (see Appendix B). Hence, the next section, we focus on specific choices for the T-CDR parameters that induce stable yet informative representations.
3.2 Metric properties
In this section, study specific parameters for T-CDR (see Definition that induce representations with associated robustness properties. We call this subset of representations Stable Candidate Decomposition Representations (S-CDR), and define them below.
Definition 2. The S-CDR parameters are:
1. the weight function M + supfs 0: By E R" s.t. Eye C supp (M))) by.e is the segment between - = -
[1...., 1 and e. 11,...,
2. the individual interval representations $s : RM + R: (a) (M) = wwussup (M)) n Ry.5), (b) oz (z) (28) vol (supp (M) n Ra,5),
(c) de (M) (267m sup {vol 8) Rw supp (M) n RRr,s)
where Rz.8 the hypersquare {y E RD x S S x + 8) C R". = 8. 1.,,..... 1] RR for any > O, and vol denotes the volume of a set in Ra
5

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3. the permutation invariant operators op Σ and op = sup.
Intuitively, the S-CDR weight function the length of the largest diagonal segment one can fit inside supp (M), and the S-CDR interval representations (a). (b) and (c) are the largest normalized diagonal length, volume, and hypersquare volume that one can fit inside supp (M) R respectively. These S-CDR interval representations allow for some trade-off between computational cost and the amount of information that is kept: (a) and (c) are very easy to compute, but (b) encodes more information about interval shapes. See Figure 2 (right) for visualizations.
Equipped with these S-CDR parameters, we can now define the two following S-CDRs, that can be applied on any candidate decomposition M = M4;
(M,) (M, 1ド= w(M,) (M,),
(M,,((M)= = sup (M.).
(2)
m
1gigm
Stability. The main motivation for introducing S-CDR parameters is that the corresponding S-CDRs are stable in the interleaving and bottleneck distances, as stated in the following theorem.
Theorem 1. Ler M = @ Mi and M' i Mr be two candidate decompositions. Assume that have Σ, wMM), E3 (M,) 2 C. for some C > O. Then for any 8 O, one has (M) Vo s 2(, 8)/5.
(4)
(M)) Vl,s(M))ll s [++ C (dB(M, 8)/8,
(5)
(M) -M..s(M)-VssMMM) & (M)M,
(6)
where A stands for minimum.
proof of Theorem 1 can be found in Appendix F.
These results are the main theoretical contribution in this work, as the only other decomposition-based representation in the literature [10] has such guarantees. The other representations [17, 18, 39, 42] enjoy similar guarantees than ours, but are computed from the rank invariant and do not exploit the information contained in decompositions. Theorem 1 shows that S- S-CDRs bring the best of both worlds: these representations are richer than the rank invariant and stable at the same time. We also provide additional stability result with similar, yet more complicated representation in Appendix G, whose upper bound does not involve taking minimum.
Remark S-CDRs are injective representations: the support of two interval modules are different, then their corresponding S-CDRs (evaluated on a point that belongs to the support of one interval but not on the support of the other) will differ, provided that is sufficiently small.
4 Numerical Experiments
In this section, we illustrate the efficiency of our S-CDRs with numerical experiments. First, we explore the stability theorem in Section 4.1 by studying the convergence rates, both theoretically and empirically, of S-CDRs on various data sets. Then, showcase the efficiency of S-CDRs on classification tasks in Section 4.2, and we investigate their running times in Section 4.3. Our code for computing S-CDRs is based on the MMA [29] and Gudhi [36] libraries for computing candidate decompositions" It is publicly available at https: /att::///tttttt com/ com/Davidee DavidLapous //m/mmutt tiperss and will be merged as a module of the Gudhi library. We also provide pseudo-code in Appendix H.
4.1 Convergence rates
In this section, we study the convergence rate of S-CDRs with respect to the number of sampled points, when computed from specific bifiltrations. Similar the single parameter persistence setting [16], these rates are derived from Theorem 1. Indeed, since concentration inequalities for multiparameter
Several different approaches can be used for computing decompositions [1. 7J. In our experiments, we used MMA [29] (with family of 1000 diagonal lines) because of its simplicity and rapidity.

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persistence modules have already been described in the literature. these concentration inequalities can transfer our representations. Note that while Equations (7) and (8), which provide such rates, are stated for the S-CDR in (3), they also hold for the S-CDR in (2).
Measure bifiltration. Let μ be a compactly supported probability measure of RD, and let ye be the discrete measure associated to sampling of n points from The measure bifiltration associated to μ and μa, is defined as F : {z E R² (x((rr r)) S t), where ((x,)) denotes the Euclidean ball centered on with radius Now, let M and M,, be the multiparameter persistence modules obtained from applying the homology functor on top of the measure bifiltrations F and Jkn. These modules are known to enjoy the following stability result [2, Theorem 3.1, Proposition 2.23 (i)]: di (M, M,,) S (1.
min(dey (n)) 2 dy (1,, pr.) )ff),, where di and dp, stand for the y- o-Wasserstein and Prokhorov distances between probability measures. Combining these inequalities with Theorem I, then taking expectations and applying the concentration inequalities of the Wasserstein distance (see [26, Theorem 3.1] and [21, Theorem 1J) lead to:
SE [ Voo. - Cp.4 E (|x19) n
- (xiba ) 1/ga/4 n
(7)
where V stands for maximum, a 2 if 2p = q = a 1 if d + 2p and q = dp/ (d p) A 2p
> d = 2p and a = 0 otherwise, Cp.g is constant that depends on and q, and X is random variable of law ps.
Čech complex and density. A limitation of the measure bifiltration is that can be difficult to compute. Hence, we now focus on another, easier to compute bifiltration. Let X be smooth compact d-submanifold of RD (d < D), and μ be a measure on X with density f with respect to the uniform measure on X. We now define the bifiltration FC, with:
JC, 3.p = Čech(u) n f ((,,
= fa E RD d(x, S u, f(x) v).
Moreover, given a set Xn, of points sampled from μr, we also consider the approximate bifiltration J.C.", where fu: X + R is an estimation of I (such as, e.g. a kernel density estimator). Let M and Mn be the multiparameter persistence modules associated JCS and FC. Then, the stability of the interleaving distance [27, Theorem 5.3] ensures:
di (M, M,) S Ilf full V dn XX,,,,,
where dı stands for the Hausdorff distance. Moreover, concentration inequalities for the Hausdorff distance and kernel density estimators are also available in the literature (see [16, Theorem 4] and [23, Corollary 15]). More precisely, when the density f is L-Lipschitz and bounded from above and from below, i.e., when 0 < Smin & f S fmax < 0o, and when Jn. is kernel density estimator of with associated kernel k. one has:
loo $
log(1/hm)
E(dn(X <
and E(Ils fnlllo) Lh, V nhe
n
where hn is the (adaptive) bandwidth of the kernel k. In particular, if μ is a measure comparable to the uniform measure of d 2-manifold, then for any stationary sequence hn : h > 0, and considering a Gaussian kernel k one has:
log + Lh.
8E [IlV Voo,ll S
(8)
Empirical convergence rates. Now that we have established the theoretical convergence rates of S-CDRs, we estimate and validate them empirically on data sets. We will first study a synthetic data set and then real data set of point clouds obtained with immunohistochemistr We also illustrate how the stability of S-CDRs (stated in Theorem 1) is critical for obtaining such convergence in Appendix B, where show that main competitor, the multiparameter persistence image [11], is unstable and thus cannot achieve convergence, both theoretically and numerically.
Annulus with non non-unifore non-uunform density. In this synthetic example, we generate an annulus of 25,000 points in R² with non-uniform density, displayed Figure 3a. Then, we compute the bifiltration J. corresponding to the Alpha filtration and the sublevel set filtration of kernel density estimator, with
7

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bandwidth parameter h = 0.1, on the complete Alpha simplicial complex. Finally, we compute the candidate decompositions and associated S-CDRs the associated multiparameter module (in homology dimension 1), and their normalized distances to the target representation, using either ||: or |1-1l The corresponding distances for various number of sample points are displayed in log-log plots in Figure 3b. One can see that the empirical rate roughly consistent with the theoretical one (-1/2 for oll and -1 for l-ll2), even when p + oc (in which case our S-CDRs are stable for dy but theoretically not for d1).
12
(a) Seatter plot synthetic (b) (left) 11-1 and (right) I1 between the target representation and data set colored by a kernel den- the empirical one w.r.t. sity estimator.
Figure 3: Convergence rate of synthetic data set.
Inmmnohisstochemissn data. In our second experiment, we consider a point cloud representing cells, taken from [40]. see Figure 4a. These cells are given with biological markers, which are typically used to assess, c.g., cell types and functions. In this experiment, we first triangulate the point cloud by placing 100 x 100 grid on top it. Then, we filter this grid using the sublevel set filtrations of kernel density estimators (with Gaussian kernel and bandwidth h 1) associated to the CD8 and CD68 biological markers for immune cells. Finally, we compute the associated candidate decompositions of the multiparameter modules in homology dimensions 0 and 1, and we compute and concatenate theoretical convergence rate of
their corresponding S-CDRs. Similar to the previous experiment, our representations is upper bounded by the one for kernel density estimators with the co-norm. The convergence rates are displayed in Figure 4b. Again, one can see that the observed and theoretical convergence rates are consistent.
180
2017
(a) Point cloud of cells colored by (b) (left) 11.112 and (right) II-Il distances between the target representa- CD8 (red) and CD68 (black). tion and the empirical one w.r.t n.
Figure 4: Convergence rate of inmunohistochemissty data set.
4.2 Classification
In this section, we illustrate the efficiency of S-CDRs by using them for classification purposes. We show that they perform comparably or better than existing topological representations as well as standard baselines on several UCR benchmark data sets, graph data sets, and on the immunohisto- chemistry data set. Concerning UCR, we work with point clouds obtained from time delay embedding applied on the UCR time series, following the procedure of [10], and we produce S-CDRs with bifiltrations coming from combining either the Rips filtration with sublevel sets of a kernel density estimator (as in Section 4.1), the Alpha filtration with the sublevel sets of the distance- taaeceetto-msssn

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with parameter = 0.1 (as in [10] and the baselines therein). Concerning graph datasets, we produce S-CDRs by filtering the graphs themselves directly using Heat Kernel Signature with parameter
t = 10, Ricci curvature and node degree (similarly what is used in the literature [11, 22, 45]). In all tasks, every point cloud or graph has a label (corresponding to the type of its cells in the immunohistochemisty data set, and to pre-defined labels in the UCR and graph data sets), and our goal is to check whether we can predict these labels by training classifiers on the corresponding S-CDRs.
For point clouds (immunohistochemissry and UCR), we compare the performances of our S-CDRs (evaluated on 50 x 50 grid) to the one of the multiparameeer persistence landscape (MPL) [39], kernel (MPK) [17] and images (MPI) [10], as well as their single-paramete counterparts (P-L, P-I and PSS-K) K) 3. We also compare to some con-lopological baselines: we used the standard Ripley function evaluated on 100 evenly spaced samples [0, for the immunohistochemistry data set, and k-NN classifiers with three difference distances for the UCR time series (denoted by B1, B2, B3), as suggested in [19]. For graps, we compare S- S-CDRs to the Euler characteristic based multiparameter persistence methods ECP, RT, and HTn, introduced in [22]. In order to also include non topological baselines, we also compare against the state-of-the-an graph classification methods RetGK [44], FGSD [37], and GIN [43].
All scores on the immunohistochemistr data set were computed after cross- validating few classifiers (random forests, support vector machines and xgboost, with their default Scikit-Leans parameters) with folds. For the time series data, our accuracy scores were obtained after also cross-vaud vols-vaaidating the following S-CDR parameters; E {0. 1), op E {sum, mean). 5 {0.01, 0, 0.5, 1). h (0.1, 0.5, 1, 1.5] with homology dimensions and 1, and the following bandwidth parameters for kernel density esimation: b E (0.1%. 1%, 10%, 20%, 30%}, which are percentages of the diameters of the point clouds. with 5 folds. Parameters and results on graph data sets were cross-validated and averaged over 10 folds, following the pipelines of [22]. All results can be found in Table I (immunohistochemistry and UCR-UCK acronyms are provided Appendix I) and Table 2. Bold indicates best accuracy and underline indicates best accuracy among topological methods. Note that there are no variances for UCR data sets since pre-defined train/test splits were provided. One can that S-CDR almost always outperform topological basclines and are comparable to the standard baselines on the UCR benchmarks. Most notably. S-CDRs radically outperform the standard baseline and competing topological measures the immunohistochemistry data set. For graph data sets, results are competitive with both topological and non-topological baselines; S-CDRs perform even slightly better COX2.
Dataset
B1
B3
PSS-K
P-
P-L
MPK
MPL
MPI
S-CDR (Rips
S-CDR
(Alpha
DPOAG
62.6
62.6
77.0
76.9
69.8
70.5
67.6
70.5
71.9
71.9
71.9
DPOC
71.7
72.5
71.7
47.5
67.4
66.3
74.6
69.6
71.7
73.8
74.6
PPOAG
78.5
78.5
80.5
75.9
82.0
78.0
78.0
78.5
81.0
81.9
PPOC
80.8
79.0
78.4
78.4
72.2
72.5
78.7
78.7
81.8
79.4
83.2
PPTW
70.7
75.6
75.6
61.4
72.2
73.7
79.5
73.2
76.1
75.6
75.1
IPD
95.5
95.5
95.0
64.7
61.1
80.7
78.6
71.9
81.2
77.2
GP
91.3
91.3
90.7
90.6
84.7
80.0
88.7
94.0
90.7
96.3
92.7
GPAS
89.9
96.5
91.8
84.5
87.0
93.0
85.1
90.5
88.0
93.7
GPMVF
97.5
97.5
99.7
88.3
87.3
96.8
88.3
95.9
95.3
95.9
PC
93.3
87.8
83.4
76.7
85.6
84.4
86.7
93.1
90.0
Ripley
P
MPL
S-CDR
Immuno
67.2(2.3)
60.7(4.2)
65.3(3.)
91.4((.6)
Table 1: Accuracy scores for UCR and immunohistochemistty data sets.
'Note that the sizes of the point clouds in the immunotistochemiistyy data set were too large for MPK and wwweerreeeeeeerreeeeeeeeeeeeeeee and that all three
MPI using the code provided in https: : //github hoom/Matiieee eucarriereeeeeeeeee kingle-parmmed representations had roughly the same performance, so only display one, denoted

---

Dataset
RetOK
FGSD
GIN
ECP
RT
HT mD
S-CDR
COX2
81.4(0.6)
80.3(0.4)
79.7(0.4)
80.6(0.4)
82.0(0.2)
DHFR
81.5(0.9)
82.0(0.4)
81.3(.44)
83.1(0.5)
81.6(0.2)
IMDB-B BB
71.9(1.))
73.6
(5.1(5.1)
73.3(0.4)
74.0(0.5)
74.7(0.5)
(3.5(0.2)
IMDB-8 M
47.7(0.3)
52.4
52.3(288)
48.7(0.4)
50.2(0.4)
49.9(0.4)
(9.5(0.2)
MUTAG
90.3(1.1)
92.1
90(8.8)
90.0(0.8)
87.3(0.6)
89.4(0.7)
88.4(U.3)
PROTEINS
78.0(0.3)
73.4
76.2(2.6)
75.0(0.3)
75.4(0.4)
75.4(0.4)
73.9(0.2)
Table 2: Accuracy scores on graph datasets.
4.3 Running time comparisons
In this section, provide running time comparisons between S-CDRs and the MPI and MPL representations, in which we measured the time needed to compute all the train and test S-CDRs and baselines of the previous data sets, averaged over the folds (again, note that since UCR data sets already provide the train/test splits, there is no variance in the corresponding results). All representations are evaluated on grids of sizes 50 50 and 100 100, and we provide the maximum running time over {0, 1, 20 All computations were done using a Ryzen 4800 laptop CPU, with 16GB of RAM. We provide results in Table 3, where it can be seen that S S-CDRs (computed on the pinched annulus and immunohistochemisti representations, by a factor of at least 25. As for UCR data sets, which contain only small time series and corresponding point clouds, it can still be observed that S-CDRs can be computed faster than the baselines. Interestingly, this sparse and fast implementation based on corners can also be used to improve on the running time of the multiparameter persistence landscapes (MPL), as one can see from Algorithm 4 in Appendix H (which retrieves the persistence barcode of a multiparameter persistence module along given line; this is enough compute the MPL) and from Table 3.
data sets) can be computed much faster than the other
Ampulus
Immuno
PPTW
GP
Ours (S CDR)
250ms (mms))
275ms (9. Smas)
33.0mm
(3.9hh
45.6mm 5.mm)))
Ours (MPL)
36.9ma/0..8mm.
65.9m.s(0.9mma.
22.4ma
15ml)
31 31.8mm (2.95m.))
(50)
6. 1.13e (23me)
5.67 (23.3mm))
66.2me (12 Smm)
208ms 16.3ma)
MPL (50)
17e(39ma)
(1mmm)
154ma (ms))
630ms (3.00m))
(PT(100)
[125ms)
11. 11.658 (668((.99mm))) 9ims)
289mss (5.0ms))
L699 77.77)
MPI.(100)
(1xm1))m))
31.3x(23.3mmx)
843ma (200ms))
4.43s( (66mm))
Table 3: Running times for S-CDRs and competitors.
5 Conclusion
In this article, we study the general question of representing decompositions of multiparameter persis tence modules in Topological Data Analysis. We first introduce T-CDR: a general template framework including specific representations (called S-CDR) that are provably stable. Our experiments show that S-CDR is superior to the state of the art.
Limitations. (1) Our current T-CDR parameter selection currently done through -ross-validation, which can be very time consuming and limits the number of parameters to choose from. (2) Our classification experiments were mostly illustrative. In particular, it would useful to investigate more thoroughly on the influence of the T-CDR and S-CDR parameters, as well as the number of filtrations, on the classification scores. (3) In order to generate mnite-dimensional vectors, we evaluated T-CDR and S-CDR on finite grids, which limited their discriminative powers when fine grids were too costly to compute.
Future work. (1) Since T-CDR is similar to the PersLay framework of single parameter persis- tence [11] and since, in this work, each of the framework parameter was optimized by neural network, it is thus natural to investigate whether one can optimize T-CDR parameters in data-driven (2) In our numerical applications, we focused
way as well, so as to be able to avoid gross-validation on representations computed off of MMA decompositions [29]. In the future, we plan to investigate whether working with other decomposition methods [1, 7] lead better numerical performance when combined with our representation framework.
10

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Acknowledgments. The authors would like to thank the area chair and anonymous reviewers for their insightful comments and constructive suggestions. The authors would also like thank Hamnah Schreiber for her great help with the implementation of our method. The authors 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 TopModel and NSF grant DMS-2311338
MC was supported by ANR grant AJB was partially supported by ONR grant N00014-22-1-2679
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A Limitation of single-parameter persistent homology
A standard example of single-parameter filtration for point clouds, called the Čech filtration, is to consider f : z + dp(x) := min || -pll- where P is a point cloud. The sublevel sets this function are balls centered
the points in P with growing radii, and the corresponding persistence barcode contains the topological features formed by P. See Figure 5a. When the radius of the balls is small, they form three connected components (upper left), identified the three long red bars in the barcode (lower). When this radius is moderate, a cycle is formed (upper middle), identified as the long blue bar in the barcode (lower). Finally, when the radius large, the balls the whole Euclidean plane, and no topological features are left, except for one connected component, that never dies (upper right).
(a) Persistence barcode obtained from growing balls clean point cloud.
(b) Persistence barcode obtained from growing balls noisy point cloud.
Figure 5: Example of persistence barcode construction for point clouds using Čech filtration. In the middle sublevel sets, we highlight the topological cycles in blue.
The limitation of the Čech filtration alone for noisy point cloud is illustrated in Figure 5b, where only feature scale is used, and the presence of three outlier points messes up the persistence barcode entirely, since they induce the appearance of two cycles instead of one. In order to remedy this, one could remove points whose density is smaller than given threshold, and then process the trimmed point cloud as before, but this requires using an arbitrary threshold choice to decide which points should be considered outliers not.
B Unstability of MPI
In this section, we provide theoretical and experimental evidence that the multiparameter persistence image (MPI) [10], which is another accomposition-base/ representation from the literature, suffers from lack of stability as it does not enjoy guarantees such as the ones of S-CDRs (see Theorem 1). There two main sources of instability in the MPI. The first is due to the discretization induced by the lines: since it is obtained by placing Gaussian functions on top of slices of the intervals the decompositions, if the Gaussian bandwidth (see previous work, third item in Section 3.1) is too small w.r.t. the distance between consecutive lines in L, discretization effects can arise, as illustrated in Figure 6, in which the intervals do not appear continuous shapes.
The second problem comes from the weight function: it is easy to build decompositions that are close in the interleaving distance, yet whose interval volumes very different. See [5, Figure 15], and Figure 7, in which we show a family of modules M (left) made a single interval built from connecting two squares with bridge of diameter > 0. We also show the distances between M and the limit module IM made of two squares with no bridge, through their MPI and S-CDRs (right). Even though the interleaving distance between M and M goes to zero goes to zero, the distances between MPI representations converge to a positive constant, whereas the distances between S-CDRs goes effectively to zero.
We also show that this lack of stability can even prevent convergence. In Figure 8, we repeated the setup of Section 4.1 on synthetic data set sampled from noisy annulus in the Euclidean plane. As
14

---

40.0
24
24
02
030
25
15
1J5 aãe
635
235 L9
175
535
85o
035
135
L90 175
Figure 6: Examples of numerical artifacts occurring when the number of lines too small w.r.t. the Gaussian bandwidth, for two different families of lines.
Ouis, sdvorti
Ourc, shouttin. dt bandyy
N
++1 Figure Example of unstability of MPI.
one can clearly see, convergence does happen for MPI the Euclidean distance the limit MPI representation associated to the maximal number of subsample points suddenly drops to zero after an erratic behavior, while S-CDRs exhibit a gradual decrease agreement with Theorem 1.
3.2
1300 po/tts
Curs, with PCC L80% -M, and E.0.33 with banddeidtte 20775- -2.11)
Number
Figure 8: Example of lack convergence for MPI.
C Simplicial homology
In this section, we recall the basics of simplicial homology with coefficients in Z/2Z, which use for practical computations. A more detailed presentation can found [30, Chapter 1]. The basic bricks of simplicial (persistent) homology are simplicial complexes, which are combinatorial models of topological spaces that can be stored and processed numerically.
Definition Given set of points X, = {I1,..., 2n) sampled in topological space X, an abstract simplicial complex built from X,, is family S(X,) of subsets of X,, such that:
15

---

if r s(X,) and a S then E a)),), and
if o E S(x), then either o n ((x.) or Ø.
Each element ((,,,) is called simplex of s(),). and the dimension of simplex is defined as dim(a) = card(o) ()) - 1. Simplices of dimension are called vertices.
An important linear operator on simplices is the so-called boundary operator. Roughly speaking, turns simplex into the chain of its faces, where chain is formal of simplices. The set of chains has a group structure, denoted by (S()))))
Definition 4. Given a simplex == [Zi, ..... Tigh the boundary operator 0 is defined as
( = Σ D4 Lt,
Fip..
In other words, it is the chain constructed from by removing one vertex at time. This operator à can then be extended straightforwardly to chains by linearity.
Given a simplicial complex, a topological feature is defined as a cycle, i.e., a chain such that each simplex in its boundary appears an even number of times. In order to formalize this property, we remove simplex in chain every time it appears twice, and we let a cycle
chain e s.t. a(c) =
Now, one can easily check that a a 0 for any chain i.e. the boundary of cycle is always cycle. Hence, one wants exclude cycles that are boundaries, since they correspond somehow to trivial cycles. Again, such boundaries form group that is denoted by B. (S(,)).
Definition 5. The homology group in dimension l: is the quotient group
(S(,))
Hs(((X,))) = (S(s)))))
In other words, it is the group (one can actually show it is vector space) of cycles made of simplices of dimension & that are not boundaries.
See Figure 9 for an illustration of these definitions. Finally, given a filtered simplicial complex (with filtration defined as in Section 2), computing its associated persistence barcode using the simplicial homology functor can be done with several softwares, such as, e.g., Gudhi [36].
1/2
U/1
1/
V/
Wo 176 2/4
075
Figure 9: Example of simplicial complex S built from eight points, and made of eight vertices (simplices of dimension 0), fourteen edges (simplices of dimension 1) and six triangles (simplices of dimension 2). The purple path is a cycle; indeed 0[ u7] + [17, vn] + [F4 Uol + [p6. 21)) [vr]+ [17] + [e>] + [ca] + [v4] + [vo] + [eg] + [en] = O since every vertex appears twice. Similarly. the blue path is a cycle as well. However, both paths represent the same topological feature, in the sense that they belong to the same equivalence class of H₃ (S), since their sum is exactly the boundary of the 2-chain comprised of the six triangles of the complex, i.e., they differ only by a trivial cycle.
Hence, the dimension of H (S) is 1.
16

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D Modules, interleaving and bottleneck distances
In this section, we provide a more formalized version of multiparameter persistence modules and their associated distances. Strictly speaking, multiparameter persistence modules are nothing but parametrized family of vector spaces obtained from applying the homology functor to a multifiltration, explained in Section 2.
Definition 6. A multiparameter persistence module M a family of vector spaces (M(a): : E ] together with linear transformations, also called transition maps, sp M(a)) - (3) for any a S B (where I denotes the partial order of IR"), that satisfy p2 pp pl for any a S S.
Of particular interest are interval modules, since they are easier to work with. Definition 7. An interval module M is multiparameter persistence module such that:
its dimension is at most 1: dim(((()) < for any a E R", and
its support supp(M) == {a E IR" : dim(M(aa) = 1} is an interval of R",
where interval of RR is subset of / C R that satisfy:
(convexity) if p,g / and p S S q then I, and
(connectivity) if p.g I, then there exists finite sequence T1, ....... I'm I, for some m E N, such that p ~ T1 ~ 12 ~..... Tm g. where ~ can be either S or 2.
In the main body of this article, we study representations for candidate decompositions of modules, ie.. direct sums* of interval modules that approximate the original modules.
Multiparameter persistence modules can be compared with the interleaving distance [27].
Definition 8 (Interleaving distance). Given > O, two multiparameter persistence modules IM and M' are e-interleaved if there exist two morphisms f: IM + My and g: M' - M2 such that g.46 o f. +2 and fi+e o gg. v where Me the shifted module {M(x + =EEEE (E....,e) 8) E R" and and v are the transition maps of M and M' respectively. The interleaving distance between two multiparameter persistence modules and MY is then defined as di (M, == inf {& 2 0 IM and M'' are e-interleaved)
The main property of this distance is that it is stable for sublevel sets of functions. More precisely, given two continuous functions f. g S + R" defined on simplicial complex S, let (M()), M(g) denote the multiparameter persistence modules obtained from the corresponding multifiltrations { := {or E S: f(o) I =}}ER and { = fo E S g(o) < x) JERR. Then, one has [27, Theorem 5.3]:
that obtained from the
di(M(), Mg)) S || 99l0o-
Another usual distance is the bottleneck distance [5, Section 2.3]. Intuitively, relies on decompo- sitions of the modules into direct sums of indecomposable summands' (which are not necessarily intervals), and is defined as the largest interleaving distance between summands that are matched under some matching.
Definition 9 (Bottleneck distance). Given two multisets A and B, ju: A A B is called a matching if there exist A' C A and B' C B such that μ:: A' B' is a bijection. The subset A' == coim (resp. B' = im(p)) is called the coimage (resp. image) of
Let M = Otez M; and M' = OJE MG be two multiparameter persistence modules. Given 2 0, the modules IM and MY are e-matched if there exists matching μr: I A I such that M, and M M are e-interleaved for all i E cp)m and M; (resp. M2) is e-interleaved with the null module 0 for all i [)coi(([)) (resp. j E J (p)).
The bottleneck distance between two multiparameter persistence modules IM and M is then defined as dg (M, h²) == inf (s 2 M and M' are e e-matched).
'Sinee multiparameter persistence modules are essentially families vector spaces connected by transition maps, they admit direct sum decompositions, pretty much like usual vector spaces do. Recall that module M is indecomposable if M A
Ar
17

---

Since a matching between the decompositions of two multiparameter persistence modules induces an interleaving between the modules themselves, follows that d₁ S dB- Note also that d3 can actually be arbitrarily larger than d1, as showcased in [5, Section 9].
E The fibered barcode and its properties
Definition 10. Let n E N and F = { F(1)} be a multifiltration on a topological space X. e, R" and le.b IR + R be the line in R defined with (e,(()) f-e-b, tha Le,b is the line of direction passing through b. Let F where H. denotes the i-th coordinate. Then, each Fe,b is single-paramete filtration and has corresponding persistence barcode Be,b. The set ((F) { e, (R*) } is called the fibered barcode of F.
P(X) defined with F (t,)(() = N F() (If.,
The two following lemmas from [25] describe two useful properties of the fibered barcode.
Lemma 1 (Lemma in [25]) Let e, b E RR" and d be the corresponding line. Let é min, (e]: Let F, F² be two multi-filtaations, M, M' be the corresponding persistence modules and Be.b E B ((F)) and By B(F") be the corresponding barcodes in the fibered barcodes of F and F". Then, the following stability property holds:
dp (Bo,b, Be,,) S (M(
(10)
Lemma 2 (Lemma's 2 in [25]). Let e', b, RD and Ea, b. &e' 3 be the corresponding lines. Let é mingle), and é' min, [e'];- Let be a mulii-filtration, M be the corresponding persistence module and Be,, Be' y E ((F) be the corresponding barcodes in the fibered barcode of F. Assume M is
decomposable M 1 and let K >> O such that M, C Boo (0, K) == { [ R" ||=|| o0 I K}
for all [1, m.. Then, the following stability property holds:
(K max 1b61 110111 }) lle e'll
+ b 1
d Berr))
(11)
Proof of Theorem
Our proof is based on several lemmas. the first one, we focus on the S-CDR weight function w as defined in Definition 2.
Lemma 3. Let M and M' be two interval modules with compact support. Then, one has
dM 0) =
sup b.desupp(M)
min(d, -by)+ w(M))
(12)
Furthermore, one has the equality
[w(M) - w(M)) <d(,
(13)
Proof. We first show Equation (12) with two inequalities.
First inequality: < Let M be an interval module. If di (M,0) = 0, then the inequality is trivial, now assume that dr(M (M, 0) O. Let 8 > 0 such that å < d,(M, 0). By definition of di, the identity morphism M - M28 cannot be factorized by 0. This implies the existence of some b E IR such that rank(M (b) - MM + 28)) > O; particular, b, 6 28 E supp (M). Making d converge to (M, 0) yields the desired inequality.
Second inequality: > Let (K,))eN be compact interval exhaustion of supp (M), and 6n dn E Kn be two points that achieve the maximum
1 sup min(d; by)+.
2 b
Now, by functoriality of persistence modules, we can assume without loss of generality that b, and dp are the same diagonal line (indeed, if they not, is possible to transform d, into
18

---

d such that 6n and d, are on the same diagonal line and also achieve the supremum). Thus, fank(M((,)) M(d))) > O, and dM(M, 0) I| b ball Taking the limit over leads to the desired inequality.
Inequality (13) follows directly from the triangle inequality applied d1.
In the following lemma, we rewrite volumes of interval module supports using interleaving distances.
Lemma 4. Let M be an interval module, and R C IR be a compact rectangle, with n 2 2. Then, one has:
vol (supp (M) n R) = [[VE==y,.00) 00
di (Ml 0) dx)
where by is the diagonal line crossing and An-1 denotes the Lebesgue measure in R=-1
Using the change of variables yi = J4 Fn and t 2n (which has trivial Jacobian) yields the following inequalities:
Proof.
vol(supp (M) n n))
dx"(x)
Supp((M)n
[yERMUM -0 J JIER Isupp(srionly + dx
[IEER=y.=00 diam (supp (M) 0 ly n R dd dy"-+(y)
where by is the diagonal line passing through Now, since M is an interval module, one has diamm)-ll (supp (M) nly n R) = 2d 1 0), which concludes the proof.
(M)
L
In the following proposition, we provide stability bounds for single interval modules. Proposition 1, If M and M' are two interval modules, then for any > O and S-CDR parameter da in Definition one has:
1. 0 S (s(M)(() < (MM) 5 A for any x E IR"
2. [MM (s(Ill 2(d; (M, M') 6)/8.
Proof. Claim 1. is a simple consequence of Equation (12).
Claim 2. for S-CDR parameter (a) is a simple consequence of the triangle inequality.
Let us prove Claim 2. for (b). Let R" and 8 > 0. One has:
Jos (M)())) - (M)) < (25)m (omm |di (M) 1) di (M'l 1)
S (28)"
di (in M' Ml lon dd>-1(y)
1w:n=00
(M, M') 5)55
where the first inequality comes from Lemma 4, the second incquality is an application of the triangle inequality, and the third inequality comes from Lemmal 1.
Finally, let us prove Claim for (c). Let R" and 8 > 0. Let < d supp (M) n R Let also 0. Then, using Lemma 4, one has:
vol(supp (M) n Ro.a)
dM [ no 0) d)
(25)m
(28) wv.R":y.=0)
< Y (26)
dM Bot7.e--ni, ni, (0 d) (v).
19

---

using the convention Ra.b Ø if 1 b. Now, set == d1(M :
MM nn.)). If + or
d =- y & supp (M') then dM R,,"" M ) = d1 which is impossible. Thus,
M' (20) vol(Bb,)) S 2d1(M Re,s' n 1/8
2
dr(M Ro, 0)
sup a,cER, ofsupp (M)) (25) (vER
2d4 (M R M' n,., 1 (M)
Finally, taking the supremum on S d e supp (M) n Rg yields
(M)))(z) - 0s (M) S 2d1(M R.,"" M R., 21 S (d 16)
The desired inequality follows by symmetry on M and M
Equipped with these results, we can finally prove Theorem 1.
Proof. Theorem 1.
Let M- = 2hh M, and M' = 6y'i Mj be two modules that are decomposable into interval modules and x RM.
Inequality 5. To simplify notations, we define the following: wy == (M:)), oū == (M)(M) and w// := w=](M & == Φ5(2,, Mj). Let also assume without loss of generality that the indices are consistent with a matching achieving the bottleneck distance. In other words, the bottleneck distance is achieved for a matching that matches M, with Mí for every i (up to adding modules in the decompositions of M and M' so that m = m'). Finally, assume without loss of generality that Σ, wt 2 Σ. wy Then, one has:
(V,s(M)(z) (xT)
Σ
wiPs
Euid
Wls
Σ, w/
1
wishi
Σ w/
+E wi E; n Σ
+ E.
wishi
Σ,
Now, for any index i, since di (Mr, Mt) S dn (M, M') and l - w/ll ww/ssal di M2) S (M, M') by
Lemma 3, Proposition 1 ensures that:
|occh - wh.1//aol I |wa - 0i + w/llbe - Hhat S 2(w; + (d MT) A6/6
and
Σ w
Wi
mdp mdB(M, D1') w() ws)
E,w; wi E. m
Σ; w
Σe Wi
Finally,
Σ + w
Σ, w
(M)) V1,6
Σ, w
1 [T ()) w()
S + c
(dn(M, M²) 8)/6.
Inequality 4 can be proved using the proof of Inequality 5 by replacing every wi by 1.
20

---

Inequality 6. Let us prove the inequality for (b). Let R == R2-6,x+ō- One has: (M) - sx,s(M)))a
2
di LyR 0) d)
sup (26)²
(VERMOY. =0
dr(M] 5 (/ 0) 0))))~1((x)
sup (26)²
fMER"OY. =0}
(for any index j) S sup (26)m
((VER' di(Mi (
1,, 0) d "1ynR 0) dAn dy)
supinf dr(Ma, 1yr My M (y)
(28)* (VER=:g==))
Now, as the interleaving distance is equal to the bottleneck distance for single parameter persistence [15, Theorem 5.14], one has: supinf di di(Mall,nne Millnnn) S de dB) MY M,,nk) - di(M),,nu M' ((,,) S dr M')
which leads to the desired inequality. The proofs for (a) and (c) follow the same lines (upper bound the suprema in the right hand term with either infima or appropriate choices in order to reduce to the single parameter case).
G An additional stability theorem
In this section, we define a new S-CDR, with a slightly different type of upper bound. It relies on the fibered barcode introduced in Appendix E. We also slightly abuse notations and use M, denote both an interval module and its support.
Proposition 2. Let M T M, be multiparameter persistence module that can be decomposed into interval modules. Let O, and let S ó 8(M), where
5(M) (6 2 0: I'M achieves dı (Be +ou)) for all x, u s.t. ||ull 1, (e u) 0},
where I'm is the partial matching induced by the decomposition of M. Let (zz o)) denote the function Ra
R
o(z
(-
-Ip-=11"))) 2
and let
+ exp
RB
R
(M)
(14)
T
maxisism fEx [IV(z, a) fill
where C(x, 5, M,) stands for the of interval functions from R" {0, 1} whose support is (c n
M, where E a connecting component of im( O M, and CΛ = []....,1] 1] RD, and where
(E) the 6-thickening of the line L()) induced by (T(() = {a E R L())l_, S 8.}.
Then, Va satisfies the following stability property:
CO,),, where C-(-) is a continuous function such that C,,(5) + when 0.
|| (M) - Va.o S (VTo)"
(15)
Proof. Let M 1 M, and M'' - M2 be two persistence modules that are decomposable into intervals, let v E IR and let 0 < < mi(() ((M)))
Notations. We first introduce some notations. Let N (resp. N') be the number of bars in Beg. (resp. Bya,.), and assume without loss of generality that N < N'. Let I' be the partial matching achieving dn (B B'a..). Let N1 (resp. N2) be the number of bars Ba that are matched (resp. not matched) to bar in B' under r, so that M2) and im(Lox.2) n M, + 0). where C stands for the set
N₂ N2. Finally, note that Beg. =
{e Ei such that e E C(mm(ees.") of connected components (and similarly for B' Beg..) a). and let Fr Bes + B' B' be function defined on all bars of Be that coincides with on the N, bars Bog.z that have an associated bar in B' B' and that maps the N₂ remaining bars of Bo a to some arbitrary bars in the (N' No) remaining bars of B' B
21

---

A reformulation of the problem with vectors. We now derive vectors that allow to reformulate the problem in useful way. Let V be the sorted vector of dimension N containing all weights [[W f where f the interval function whose support (e( M, for some M,, where é E Bes.. is a connected component of im( n M, Now, let V be the vector of dimension N' obtained by concatenating the two following vectors:
. the vector Vi of dimension N whose ith coordinate is [[ - where f' is the interval function whose support is Ts((") n MY for some My. and l' E Bea. is the image under I' of the bar l E _ corresponding the ith coordinate of V, i.e., C' Fr(() where [V], = [[ o) fll and f is the interval function whose support is TT(E) n Mi. for some Ma, In other words, Vi is the (not necessarily sorted) vector of weights computed on the bars of B that are images (under the partial matching achieving the bottleneck distance) of the bars of Beg that were used to generate the (sorted) vector V.
the vector V/2 of dimension (N'- - -N) whose jth coordinate is [I, σ): f'll1. where f' is interval function whose support is T&(E") n Mj for some Mj. and l' E B satisfies l' & im (r). In other words, Vi is the vector of weights computed on the bars of B% B&s,, (in arbitrary order) that are not images of bars of Bea under I.
Finally, we let V be the vector of dimension N' obtained by filling V (whose dimension is N S N') with null values until its dimension becomes N', and we let V sort(V') be the vector obtained after sorting the coordinates of V Observe that:
VV.o (x) Va.a (M' (z)) = IV 1/1 I[V IV- 107--oon sort(V"))
(16)
An upper bound. We now upper bound ||v-v Let g E [1, Then, one has [V]
[[, fll where f is an interval function whose support is (e) n M; for some M, with
E Bes. q I N and e = 0 otherwise; and similarly [V',, = [W(z, o) f'll1. where f is an
interval function whose support is Ta(E") n Mi for some Mj with E' E B B Thus, one has:
VV V'
[I[ r) fll1 ()) f'll
[W 0) f -xx(x -'hh by the reverse triangle inequality
(V) (f (f-f'))l, by linearity
IN(x,))) - 2 by Hölder's inequality
(Vio)* Ilf f'112
Since (f f() is interval function whose support is (T&(E) n M))(TT((") n Mf), one has || - 1'12 = / (()((()) n (.))ATT((E)) n MG). Given a segment and a vector u, we let l, denote the segment + e, and we let eu denote the (infinite) line induced by e,- More precisely:
Ilf - 1 =
= -I U((e))a)))))) u
where u ranges over the vectors such that ||u/l.c I 6, (u, = 0
1(E" (((())
(C ((E MT) + 1(0' ((())))((((())())))
2 /. 4dg (Bes.es Bes. + 4dg Bes.Z Bea.z) + 4dg Bee B' Bes (17)
< 4 / ||ull,o + (MM ||u|| du by Lemma 1 and 2
22

---

Inequality (17) comes from the fact that the symmetric difference between two bars (in two different barcodes) that are both matched (or unmatched) by the optimal partial matching is upper bounded by four times the bottleneck distance between the barcodes, and that (by assumption) the partial matchings achieving dB Be Bes and dB (B's B Bes are induced by and M'.
Conclusion. Finally, one has
VV, (M)) Va. (M)) IV V'I1 [V sort(V')), from Equation (16)
S V 1'
- (Vio) 2 (M, - C,)
(18)
with (.) = Ju | du -> when 8 - 0. Inequality (18) comes from the fact that any upper bound for the norm of the difference between given vector V and a sorted vector V, is also an upper bound for the norm the difference between the sorted version V of V and the same vector V (see Lemma 3.9 in [12]).
While the stability constant not upper bounded by 6, V6, is more difficult to compute than the S-CDRs presented in Definition 2.
H Pseudo-code for S-CDRs
In this section, briefly present the pscudo-code that use to compute S-CDRs. Our code is based on implicit descriptions of the candidate decompositions of multiparameter persistence modules (which are the inputs of S-CDRs) through their so-called birth and death corners. These corners can be obtained with, e.g., the public softwares MMA [29] and Rivet [28].
In order to compute our S-CDRs, we implement the following procedures:
Given an interval module M and a rectangle R .compute the restriction M R
2. Given interval module M (or M M| g). compute di (M, 0). This allows compute weight function and first interval representation in Definition 2.
3. Given an interval module restricted a rectangle, compute the volume of the biggest rectan- gle in the support of this module. This allows to compute the third interval representation in Definition 2.
For the first point, Algorithm I works by "pushing" the corners of the interval on the given rectangle in order to obtain the updated corners.
23

---

Algorithm 1: Restriction of an interval module to a rectangle
Data: birth and death corners of an interval module M, rectangle (2 E Rd m S 2 M)
Result: new ttw__iterval_corneee the birth and death corners of M R
for interval = (interval birth corners, interval death Seaath_coneers in M do
new birth list < []:
for b in seeervalbbirhhcooee
do
if b S M then
8 (max(b,, m.) for E [1,n]]:
Append b' to new new_birth end end
new death list + []; for d in interval death corners do if d 2 then
= (min(d;, M,) for i [1,
Append d' to new new_death___ list:
end end
new tttp-inteea__oreeee +
new new_death_liss
end
For the second point, we proved in Lemma 3 that our S-CDR weight function is equal to di and has a closed-form formula with corners, that we implement in Algorithm
Algorithm 2: S-CDR weight function
Data: birth and death corners of an interval module M Result: distance, the interleaving distance dy (M, 0).
distance +- 0;
do for b in __deatM__oommaaaee corneers do
for d in M
distance + max (result, 1 min, - b:)+);
end
end
The third point also has a closed-form formula with corners, leading to Algorithm 3.
Algorithm 3: S-CDR interval representation
Data: birth and death corners of interval module M
Result: volume, the volume of the biggest rectangle fitting in supp (M)
volume <- O:
for b in M Ti_bbrth_oor coreeer__orrrseeee do for d in M L__ccchh_haee @eoeathcccrrrsstts
corners do volume <- max (result, [I, - b;)+);
end end
Finally, show how get the persistence barcodes corresponding slices of interval module solely from the corners of the interval module.
24

---

Algorithm 4: Restriction of an interval module to a line
Data: birth and death corners of interval module M, diagonal line / Result: barcode, the persistence barcode associated to M
barcode - []:
y f arbitrary point in I;
for interval = (interval birth cornerssrrr, interval death corners)rrrss in M do
birth -- y/ + 1 X milleiuttrraal birth corners max; b; yss death < y + 1 x maKdeinterval death corners min; d; - y45
bar [birh,, death];
Append bar to barcode;
end
I
UCR acronyms
Dataset
Acronym
DPOAG
DistalPPallamddtttneeorrrrrttt
DPOC
Pro.chal/Plalaaxxttttieeeggggggu
PPOAG
ProximaaPPhallannnnnnnnnCCCrr
PPOC
ProximalPPallanIIW
PPTW
ItalyPoverDemand
IPD
GunPoint
GP
GunPointAgesSan
GPAS
CunPoinntMaleVVrruuuuFF
GPMVF
PowerCons
PC
25


Reviews:
[{"review_id": "ihH5Nv1SAu", "reviewer": "Reviewer_pDMj", "summary": "The paper addresses topological data analysis in the multiparameter setting and focuses on the problem of representation of multiparameter persistent homology by vector space elements so that standard ML algorithms can be applied.  The key advance in the paper is the leveraging of decomposition ideas in multiparameter settings to devise a representation framework that is shown to be theoretically stable with practical, efficient computation.  The  strengths in statistical convergence, accuracy and computational speed are validated in specific UCR classification datasets. \n", "questions": "I could not follow the notation in table 1 as it is not clear why there are multiple highlighted numbers (I would assume that the red numbers correspond to the best classification results, but in both table 1 and in the appendix on time series classifications there are multiple flagged columns of relevance).  ", "limitations": "None", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Originality:\nThe paper draws inspiration from past work on multiparameter persistent homology by Carriere et al. It offers a theoretical formulation that provides stability guarantees (e.g. statistical rates of convergence) and is a generalization of the past work.\n\nQuality:  The paper is comprehensive including the appendix.  I did not go through the details of the proofs and cannot comment on the theoretical correctness of the paper.\n\nClarity:  The paper is very clear with comprehensive references to the latest in the field. \n\nSignificance:  The paper appears to be of theoretical significance and will be of importance to researchers in Topological Data analysis. ", "weaknesses": "While the experimental results show the power of S-CDR on several datasets,  It will be interesting to see how the method applied to more complex datasets (e.g. from imaging related applications). The experimental validation is limited but sufficient. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "*Experiments on more complex datasets, e.g., imaging related applications. The experimental validation is limited but sufficient*\n\nWe have included a second set of experiments on graph data that demonstrates that our representations can be successfully applied to other types of data sets. Results are available in Table 3 of the rebuttal pdf. See also our general comment to all reviewers. \n\n*Table 1 notations : clarify the highlighted numbers*\n\nWe agree that our notations were confusing. Red and bold numbers correspond to the best scores in each of the three categories (non-topological baselines / single-parameter persistence / multi-parameter persistence) for the UCR data sets, and to the best score overall for the immunohistochemistry data. We will make more clear in the text by putting the best score overall in bold font and decorate (e.g., by underlining) the best in each category."}, {"review_id": "82LgMmpQMu", "reviewer": "Reviewer_MKJT", "summary": "As the title clearly suggests, a general representation of multiparameter persistent homology (MPH) is introduced. Unlike earlier representations that either yield a loss in information or are unstable, the proposed vectorization is strictly more informative, and is shown to be stable for a specific settings of the parameters in the general approach. The approach is evaluated on several real data sets.", "questions": "(Q1) Looking at Figure 2, it does not seem as challenging, but rather straightforward to represent the plot in the middle with the image on the right?\n\n(Q2) You claim that S-CDR is strictly more powerful than MPK, but MPK achieves better scores on DPOC, PPTW, GPAS, GPMVF data sets (Table 1). This deserves at least a brief discussion.\n\n(Q3) Why do you exclude some data sets considered in [CB20] from your experiments (Table 1)? Are the filtrations used to calculate your S-CDR the same as the filtrations to calculate MPK, MPL and MPI in earlier works? Why do you not evaluate the running times also on the Table 1 data set, and compare it against the results in [CB20]?\n\n\n\nOther minor comments:\n\n-\tThe order of Figures 1 and 2 should be reversed, since the latter is referenced first in the text? Similarly, Sections 4.2 and 4.3, or Table 1 and Table 2 should be reversed?\n-\tWhen listing the contributions, add “(Section 3)” after “statistical convergence of our representations”. The Outline paragraph soon below then becomes almost redundant.\n-\tLine 169, Line 177, Line 343: Perslay -> PersLay\n-\tLine 202: “the two following S-CDR”. I would remove “two,” as it might imply a diminished contribution, since you actually define an S-CDR for every p in N.\n-\tI would start every item in a list on Lines 194-195 and Line 204 on a new line, as this is important information that the reader should be able to find and read easily. If there are space limitations, you can sacrifice the Outline paragraph in Section 1, see one of the minor comments above.\n-\tTable 1: What is highlighted in red?\n-\tLine 314: “the same bifiltration as in the previous section”. Subsection? In this case too, the previous subsection does not contain any information on bifiltration.\n-\tTable 1: What does “P” denote for the last row in the table?\n-\tLine 326: “almost always outperform” is an overstatement.\n-\tTable 2: For better readability, only consider or s or ms, remove them from the individual cells and place in brackets in column titles.\n-\tLine 344: parameter -> parameters\n-\tReferences: Be consistent between capital case vs. lower case for the paper titles. Remove # from 2D, and remove double reference [CdGO15], [CdSGO15].\n", "limitations": "The limitations and future research directions are clearly presented in the final section.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "(S1) I do not have an overview of MPH literature, but assuming that the authors provide a complete and honest overview of earlier work, the contributions in this paper are substantial.\n\n(S2) Figure 1 and the discussion on Lines 170 – 185 clearly and nicely position the work in the existing literature.\n", "weaknesses": "The main experimental results in Table 1 show that the proposed S-CDR approach is half of the time significantly outperformed by other existing methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "*The main experimental results in Table 1 show that the proposed S-CDR approach is half of the time significantly outperformed by other existing methods.*\n\nThis is correct.  The table indicates that for some time series and graph tasks, topological methods are less effective than \"classical\" methods; in these cases, the close scores of the topological methods indicate limits of what TDA can achieve.  The real strength of the topological methods (and notably S-CDR) can be seen from the results on the immunohistochemistry data.\n\n*Looking at Figure 2, it does not seem as challenging, but rather straightforward to represent the plot in the middle with the image on the right?*\n\nThanks for the suggestion. We will make the required changes.\n\n *You claim that S-CDR is strictly more powerful than MPK, but MPK achieves better scores on DPOC, PPTW, GPAS, GPMVF data sets (Table 1). This deserves at least a brief discussion.*\n\nYou are right. We believe that S-CDR shines particularly when candidate decompositions of modules matter the most (w.r.t. rank-invariant based methods). We believe this is what happens for the immunohistochemistry data (and what explains the improvement), while UCR data can be classified using only the information encoded in the rank invariant, so that the extra content in the candidate decompositions and our corresponding S-CDRs is not used so much, leading to comparable yet not strongly superior accuracies. We will update the text to make this clear.\n\n*Why do you exclude some data sets considered in [CB20] from your experiments (Table 1)?* \n\nThis is a very good question. Unfortunately, the immunofluorescence data presented in [CB20] is protected and has not been released publicly, so we could not reuse it. Instead, we provided experiments on a similar public data set from immunohistochemistry.\n\n*Are the filtrations used to calculate your S-CDR the same as the filtrations to calculate MPK, MPL and MPI in earlier works?* \n\nThe filtrations we used are slightly different than the ones of earlier work for MPK, MPL and MPI, but they are very similar: earlier work use Alpha + Distance-to-Measure (DTM) filtrations while we use Rips + codensity (computed with kernel density estimators (KDE)) filtrations. For sake of completeness and fairness, we rerun the experiments using also Alpha and DTM filtrations, see Table 1 in the rebuttal pdf; we did not observe drastic changes in the corresponding results, and the main takeaway of the table stays the same: S-CDRs are quite competitive with both topological and non topological baselines.\n\nThe reason we initially used different filtrations is the following: despite being theoretically related, Alpha and Rips filtrations differ in the final complex they filter. Indeed, on a dataset of n points, Rips filters the complete, abstract simplex with n vertices, while Alpha filters (roughly speaking) the Delaunay triangulation of the points. This induces some lack of stability and convergence rates for Alpha + DTM in the multi-parameter setting, as some critical simplices might never appear in the single-parameter filtrations of some lines in the module if their densities are too small, leading to infinite bars, while the complete simplex associated to Rips contains so many faces that every cycle is eventually closed by some simplex at some point.\n\n*Why do you not evaluate the running times also on the Table 1 data set, and compare it against the results in [CB20]?*\n\nThis is a very good question. The running times improvements for UCR data sets (in Table 1) were not particularly striking due to the quite small data set sizes (leading to running times that are quite small for all competitors) so we did not include them. For sake of completeness, we put them back (for a few representative UCR data sets) in Table 2 of the rebuttal pdf.\n\n*What is highlighted in red?*\n\nRed and bold numbers correspond to the best scores in each category (non-topological baselines / single-parameter persistence / multi-parameter persistence) for the UCR data sets, and to the best score overall for the immunohistochemistry data. We will make this clear in the text. See also our answer to reviewer pDMj.\n\n*What does “P” denote for the last row in the table?*\n\n\"P\" stands for single-parameter persistence. Since all three such methods (\"PSS-K\", \"P-I\", \"P-L\") provided the (almost) same result, we only provided one number in order to save space.\n\nThank you very much also for all the minor suggestions and comments concerning the text. We will make the required changes."}, {"review_id": "2JInWEl3q4", "reviewer": "Reviewer_BEmH", "summary": "The paper investigated the persistent homology with multiple filtration parameters. The authors proposed a framework, T-CDR, which generalized the previous studies in multi-parameter persistent homology. They further presented stability and convergence guarantees on S-CDR, which is a special case of T-CDR introduced to ensure robustness. Third, their theoretical claim and contributions were supported by the empirical convergence studies as well as classification tasks on several immunohistochemistry datasets. \n", "questions": "1. It looks to me most of Section 4.1 is a continuation of Theorem 1, is there any specific reason to put this in this section rather than in Section 3 (and make it a Proposition/Corollary)? \n1. Empirically, how sensitive is the algorithm for the larger intrinsic dimension $d$? If I understand the experiments correctly, they all seem to have $d=2.$ I am curious about how the intrinsic dimension $d$ will impact convergence and/or performance.\n1. I am also curious about how the proposed framework can be applied in higher-order homology descriptors (empirically). This is also related to Question #2 above. \n1. TDA has been applied in numerous different domains such as galaxy, proteins, single-cell sequencing, 3D CAD point clouds, medical imaging [A-C] etc. I am curious if the proposed multi-parameters filtraion work can be expanded in fields outside immunohistochemistry.\n1. How optimal/tight is the bound in (8)? Can we get a better convergence result if we choose op, $\\omega$, and/or $\\phi$ differently?\n1. [Minor] Given that T-CDR is a generalization of both the rank-invraint and candidate decomposition method, should the name template “candidate decomposition” representation be modified to better reflect what it can be capable of?\n1. [Minor/Typo?] Should the citation in L477 be [CB20] instead of the PersLay paper?\n\n---\n[A] Wasserman, Larry. “Topological Data Analysis.” Annual Review of Statistics and Its Application 5 (2018): 501–32.\n\n[B] Chen, Yu-Chia, and Marina Meila. “The Decomposition of the Higher-Order Homology Embedding Constructed from the k-Laplacian.” Advances in Neural Information Processing Systems 34 (2021).\n\n[C] Wu, Pengxiang, Chao Chen, Yusu Wang, Shaoting Zhang, Changhe Yuan, Zhen Qian, Dimitris Metaxas, and Leon Axel. “Optimal Topological Cycles and Their Application in Cardiac Trabeculae Restoration.” In International Conference on Information Processing in Medical Imaging, 80–92. Springer, 2017.\n", "limitations": "Authors have addressed the limitations of their work. Negative social impact is not applicable because this work is a theoretical contribution. \n", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. [Originality] The proposed framework, T-CDR in Definition 1 not only generalized previous work in candidate decomposition (e.g., MPI) as well as approaches in rank-invariant (MPL, MPK). I believe that it opens up new research on ensuring different guarantees by varying parameters/operators in (1).\n1. [Significance] The stability and convergence guarantee (Theorem 1 and (8)) of S-CDR, which is a special case of T-CDR, provides a more stable and useful multi-parameter persistent homology framework.\n1. [Quality] The authors supplement the convergence rate claim with informative empirical convergence studies, showing a clear trend of error rate reduces with $~n^{-1/2}$ as $n$ grows matching eq. (8).\n", "weaknesses": "1. In the empirical convergence rate experiment, it was not clear to me what is the “ground truth” representation you are comparing against. For synthetic data in Figure 3, it makes sense that you can get access to the density $f$ (therefore $\\mathcal F_{C, f}$ and $\\mathbb M$) given that you generated the data from some probability distribution. How do you get the $\\mathbb M$ for the immunohistochemistry data (as in Figure 4)? Provide more clarifications on this will be beneficial. \n1. It seems like MPL runtime can be improved 25x-50x with the sparse implementation in Algorithm 4 (see, e.g., Row #2 of Table 2 vs. Rows #4 and #6 of Table 2); this suggests that the runtime win might be due implementation rather than faster algorithmic time complexity. The claim will be more convincing if the authors can provide more insights, justifications, or analyses of the time complexity as to why the proposed algorithm is more efficient than prior work. \n1. What is the bifiltration parameters used for each experiments in Sections 4? Are they radius and co-density for Figure 3, and CD8 and CD68 for the immunohistochemistry data? Adding more explanation there will increase the clarity more. \n1. In Sections 1-3, the function $f$ is used to define the multi-parameter filtration function; however, in Section 4, the notation is defined as the density (e.g., in L254). I would suggest to choose another notation to avoid confusion.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": " *In the empirical convergence rate experiment, [...] How do you get the for the immunohistochemistry data (as in Figure 4)?*\n\nThis is a very good question. As you mentioned, we do not know the true distribution, so the target that we use (and that we measure the distance to) is the S-CDR obtained from the density computed with a KDE that was fit on the whole point cloud. We then generated the curves by drawing several subsamples, and measuring the differences between the S-CDRs of the KDE fit on the subsamples and the target S-CDR. Note that subsampling in this way is heuristically justified by the theory of the bootstrap/ resampling methods.\n\n *the authors can provide more insights, justifications, or analyses of the time complexity as to why the proposed algorithm is more efficient than prior work.*\n\nThis is a very good observation: indeed, concerning MPL, the huge improvement we observe is definitely due to the data structure we use; while the code in `multipers` needs all barcodes be stored in a heavy list of arrays, we use an implicit, sparse representation of candidate decompositions that allows for fast queries of barcodes and thus efficient MPL and MPI computations. We believe this is also what makes our implementation of T-CDRs and S-CDRs quite fast, in addition to their already favorable theoretical complexity (which is roughly equal to the product of the numbers of birth and death corners of our implicit representations of modules). We will make this more clear in the text.\n\n*What is the bifiltration parameters used for each experiments in Sections 4?*\n\nWe always used Rips + codensity (estimated with kernel density estimators (KDE)), which is more stable yet very similar to the filtrations used in earlier works. Concerning the immunohistochemistry data, CD8 and CD68 were used only to create several different point clouds with their own labels (e.g., the point cloud of CD8 cells, with label \"CD8\", and the one of CD68 cells, with label \"CD68\"), and then they were all processed with Rips + codensity filtrations. We will make this clear in the text. See also our answer to question 3 of reviewer MKJT. \n\n*is there any specific reason to put this in this section rather than in Section 3 [...]?*\n\nThank you for this suggestion. We initially wanted to emphasize the fact that S-CDRs and T-CDRs are general representations that can handle any candidate decompositions, while results in Section 4.1 are specific to the Rips + codensity filtrations. \n\n*Empirically, how sensitive is the algorithm for the larger intrinsic dimension $d$ ?*\n\nThis is a very good question. In terms of convergence, we did some new experiments on point clouds in three dimensions, see Figure 1 in the rebuttal pdf, and did observe convergence with specific rates. In terms of classification performance, notice that while immunohistochemistry data is in two dimensions, UCR data was processed with time delay embedding with a length $3$ window, which produced point clouds in dimension $3$. In terms of running time performances, the intrinsic dimension only has an impact on the candidate decomposition methods (e.g., `MMA`, `Rivet`, etc) and/or the filtration values (e.g., density computed with KDE), but no impact on our T-CDRs and S-CDRs, as they only process persistence modules and are oblivious to how these modules were computed.\n\n*I am also curious about how the proposed framework can be applied in higher-order homology descriptors (empirically).*\n\nWe are a bit unsure of what is meant by higher-order homology. \n--- If this refers to the homology dimension (e.g., $H_1$ vs. $H_{2}$), note that S-CDRs and T-CDRs can be applied straightforwardly as they are oblivious to the homology degrees and only needs persistence modules. We also run new convergence experiments to check whether convergence was still occurring in $H_2$ for a synthetic data set in $R^3$ in Figure 1 of the rebuttal pdf; while rates can be different than homology dimensions $0$ and $1$, we did always observe convergence happening. \n--- If this refers to the number of filtrations, the only limitation is the computational complexity, as the definitions of the T-CDRs and S-CDRs apply in any dimension.\n\n *I am curious if the proposed multi-parameters filtraion work can be expanded in fields outside immunohistochemistry.*\n\nWe have added a new experiment on graph data, which shows that our framework can be applied outside of geometric point clouds. See Table 3 in the rebuttal pdf and our general comment to all reviewers. \n  \n*How optimal/tight is the bound in (8)? Can we get a better convergence result [...]?*\n\nWe believe our bound is not necessarily optimal. Roughly speaking, our bound is obtained with:\n       - Stability of interleaving between Prokhorov and Wasserstein (a)\n       - Control of the Haussdorff distance (b) and of density estimation (c)\nNote that (a) is tight by the universality property (see section 5.2 in `The Theory of the Interleaving Distance on Multidimensional Persistence Modules` (Lesnick, 2015)). However, it should be possible to improve the bound by finding a specific class of data sets that would provide a tighter control of either (b) or (c). Whether it is possible to also get a better bound by fine tuning T-CDR parameters is a very interesting suggestion that we plan to study in future work.\n\n*Should the name template “candidate decomposition” representation be modified to better reflect what it can be capable of?*\n\nThanks for the suggestion. Since T-CDR can only work once a decomposition has been provided (wherever it comes from), we believe that it is important that this appears in the representation name.\n\n*Should the citation in L477 be [CB20] instead of the PersLay paper?*\n\nAre you referring to L177 (the paper only has 458 lines)? In this case, we actually meant PersLay, since $w'$ and $\\phi'$ are applied on restrictions of the persistence module to lines, or, in other words, to single-parameter persistence diagrams."}, {"review_id": "qnEkoZ0ME6", "reviewer": "Reviewer_HVag", "summary": "Persistent homology (PH) is the most important method in topological data analysis (TDA), and multiparameter persistence (MPH) is its natural generalization which is expected to significantly boost its performance. However, because of several mathematical roadblocks, MPH could not be effectively used in real life applications. In this work, the authors proposes a general framework to vectorize MPH. Their framework generalizes all the known existing work as special cases. Furthermore, they provide a subset of vectorizations which are stable. Finally, another obstruction to use MPH in real life applications was their computational costs. The authors new framework provides a much faster way to obtain these vectorizations.", "questions": "From theoretical standpoint, stable vectorizations are always preferable, however, from ML side, when enforcing stability, we might be losing too much information and scarifying performance. With this in mind, can you also suggest a practical, computationally efficient T-CDR methods which potentially provide better performance? ", "limitations": "As authors stated, one of the main limitation of the approach is coming from the generality/flexibility of their framework. For ML applications, there are several choices to make (hyperparameter tuning) to define a suitable vectorization for a given situation. ", "rating": 7, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Framework provided is very general and it addresses a critical need in TDA.\n\nStable MPH vectorizations, and their convergence studies are excellent contribution to the theory of TDA.\n\nComputationally feasible MPH vectorizations will finally enable to use MPH effectively in real life applications.", "weaknesses": "Experiments section could be more detailed, and more in-depth analysis could have been provided for the performance of these vectorizations in real life applications. In particular, the authors compare the performance of their vectorizations only with respect to other TDA models, and some simple other models in point cloud setting. Instead, more thorough analysis could have been provided to compare these MPH vectorizations with respect to SOTA models from different domains.\n\nA performance evaluation in graph setting would be nice as it is another very important application area of PH.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": " *Experiments section could be more detailed, and more in-depth analysis could have been provided for the performance of these vectorizations in real life applications. In particular, the authors compare the performance of their vectorizations only with respect to other TDA models, and some simple other models in point cloud setting. Instead, more thorough analysis could have been provided to compare these MPH vectorizations with respect to SOTA models from different domains.*\n\nThank you for this suggestion. Given that our goal was mostly to show improvement over the topological baselines, we put less emphasis on the non topological ones, as advocated also in the cited earlier works. However, we added a new experiment on graph data, for which MPH is known to work well, and that incorporates more SOTA baselines. See general comment and Table 3 in the rebuttal pdf.\n\n*A performance evaluation in graph setting would be nice as it is another very important application area of PH.*\n\nThank you for this suggestion. We have added a new experiment on graph data, which shows that our framework can be applied to more general structured data than geometric point clouds. See our answer above and our general comment to all reviewers.\n\n*From theoretical standpoint, stable vectorizations are always preferable, however, from ML side, when enforcing stability, we might be losing too much information and scarifying performance. With this in mind, can you also suggest a practical, computationally efficient T-CDR methods which potentially provide better performance?*\n\nThank you for this suggestion. Notice that the parameters we use in the experiments ($p=0,1$) are already not stable for the interleaving distance between persistence modules, but only for the bottleneck distance, as per Theorem 1. One future direction is indeed to investigate other types of T-CDR parameters: an interesting step in this path would be to borrow descriptors from the computer vision literature (such as, e.g., spin images from `Using Spin Images for Efficient Object Recognition in Cluttered 3D Scenes` (Johnson, Hebert, 1999)) in order to capture more information (even if it deteriorates stability) about the module summands of the decomposition in the $w$ or $\\phi$ T-CDR parameters. We will make this clear in the text. "}]

Author rebuttal:
We first want to thank all reviewers for their careful and insightful reviews. We have responded to their comments, and will be happy to provide further explanations if needed.

We have also prepared a rebuttal pdf that contains a few more experiments that were rightfully suggested by the reviewers. If the paper is accepted, these new results will be included in the submission, either in the main paper or in the supplementary. In short, 
1. we re-did classification experiments on UCR data sets and made sure we used the same filtrations as topological baseline competitors (Table 1), 
2. we added the running times of our method for UCR data sets (Table 2),
3. we added a new classification experiments on graph data sets (Table 3),
4. we measured convergence rates for a point cloud in $R^3$ in homology degrees $0$ ,$1$ and $2$ (Figure 1)

More details about the new graph classification experiment: the results we provide were obtained after cross validating across several pairs and triplets of filtrations (including HKS, Ricci curvature and node degree), with the remaining parameters being the same than the ones we used for the UCR classification experiment. We compare S-CDRs to the Euler characteristic based multiparameter persistence methods ECP, RT, and HTn, introduced in `Euler characteristic tools for topological data analysis` (Hacquard, Lebovici, 2023). In order to also include non topological baselines, we also compare against the state-of-the-art graph classification methods RetGK introduced in `Retgk: Graph kernels based on return probabilities of random walks` (Zhang et al, NeurIPS 2018), FGSD introduced in `Hunt for the unique, stable, sparse and fast feature learning on graphs` (Verma, Zhang, NeurIPS 2017), and GIN introduced in `How powerful are graph neural networks?` (Xu et al, ICLR 2019).  These SOTA methods performed the best in the analysis of (Hacquard, Lebovici, 2023) so we decided to go with them and use the accuracies reported there. All methods are averaged over 10 folds with 90/10 training/test splits. 

Overall, one can see from Table 3 in the rebuttal pdf that results are competitive with both topological and non-topological baselines; S-CDRs perform even slightly better on COX2. We believe this is yet another indication that our method can be applied successfully in quite general settings involving multiparameter topology. 

Meta-review: All reviewers agreed on the relevance and impact of the proposed work. As an expert in the topic myself, I am also **very excited** about the prospects of new methods that make multiparameter persistent homology available for a wider audience and a larger set of applications. The authors provided a comprehensive rebuttal and addressed some minor issues. I am thus absolutely confident that the paper is ready for publication and trust the authors to integrate the new results discussed during the rebuttal. It is my great pleasure to **endorse this paper for publication**.
Decision: Accept (poster)