A sparse, fast, and stable representation for multiparameter topological data analysis
Authors: David Loiseaux
Mathieu Carrière
Andrew Blumberg
Venue: Submitted to ICLR 2023 (2023)
Keywords: Topological Data Analysis
Algebraic Topology
Persistent Homology
Kernel Methods

Abstract: Topological data analysis (TDA) is a new area of geometric data analysis that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for point clouds.  One of the most important shape descriptors is persistent homology, which studies the topological variations as a filtration parameter changes; a typical parameter is the feature scale.

For many data sets, it is useful to consider varying multiple filtration parameters at once, for example scale and density.  While the theoretical properties of one-parameter persistent homology are well understood, less is known about the multiparameter case.  Of particular interest is the problem of representing multiparameter persistent homology by elements of a vector space for integration with traditional machine learning.

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 general representation framework for multiparameter persistent homology that encompasses previous approaches. 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 TDA practitioners. 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. 

Under review as a conference paper at ICLR 2023
A SPARSE, FAST, AND STABLE REPRESENTATION FOR MULTIPARAMETER TOPOLOGICAL DATA ANALYSIS
Anonymous authors Paper under double-blind review
ABSTRACT
Topological data analysis (TDA) is a new area of geometric data analysis that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for point clouds. One of the most important shape descriptors is persis- tent homology, which studies the topological variations as a filtration parameter changes; typical parameter is the feature scale.
For many data sets, it is useful to consider varying multiple filtration parameters at once, for example scale and density. While the theoretical properties of single parameter persistent homology are well understood, less known about the multi- parameter case. Of particular interest is the problem of representing multiparameter persistent homology by elements of a vector space for integration with traditional machine learning.
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 general representation framework for multiparameter persistent homology that encompasses previous approaches. We establish theoretical stability guarantees under this framework well efficient algorithms for practical computation, making this framework an applicable and versatile tool for TDA practitioners. 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) Carlssso, 2009) is 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 have allowed 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 (Carrière et al. 2015a: Poulenard let al. 2018), computational biology (Rabadán & Blumberg. 2019), and material science (Buchet et al.
2018 Saadatfar et al. 2017). among many others.
The main tool of TDA is persistent homology. In its most standard form, one is given finite metric space (c.g.. finite set of points and their pairwise distances) and a continuous function f X - IR. This function usually represents a parameter of interest (such as, e.g., scale 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. Of course, the idea of considering multiscale representations of geometric data is not new (Chapelle et al.] 2002] Ozer 2019 Witkin 1987); the contribution of persistent homology to obtain novel and theoretically tractable multiscale shape descriptor. More formally, this is achieved by computing the so-called persistence diagram of f. which is obtained by looking at all sublevel sets of the form ((-oc, al)) and by computing a decomposition of these sets, 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 sublevel set f "[(-oo, asl), call the corresponding threshold ag (resp. a.)) the birth time (resp. death time) of the topological feature, and we summarize this information in the persistence diagram D(J)= = {(as, a.)) Jaed C IR2 Moreover, it is also

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usual to consider the distance to the diagonal Odd at proxy for the statistical significance of the corresponding feature. The most important foundational result in the subject establishes that persistence diagrams stable they will not change much when computed on a small perturbation of the function f Cohhn--teener al. 2007
However, an inherent limitation of the formulation of persistent homology is that it can handle only a single filtration parameter f. However, in practice it is common that one has to deal with multiple parameters. This translates into multiple filtration functions: a standard example is given in Figure | where both feature scale and density functions topologiaal representation of a noisy point cloud. An extension of persistent homology to this more general setting is called multiparameter persistent homology (Botnan & Lesnick| [022] Carisson & Zomorodian (009), since roughly speaking it amounts to studying the topological variation of a continuous multiparameter function f: X + R" with n E N". This setting is notoriously difficult to analyze theoretically, there is no general decomposition theorem and hence no analogue of persistence diagrams.
necessary obtain meaningful
Still, it remains possible to define topological invariants, even in this setting. The most common one the so-called rank invariant, which describes how the topological features associated to any pair of sublevel sets {z E X f(x) S. a and {ir E X f(z) S }} such that S 3 (w.r.t the partial order in IR"), are connected. Since it is defined as an algebraic construction, and, as such, not suitable as input for subsequent data science purposes, the task of finding appropriate representations of this invariant, ie., embeddings into Hilbert spaces, is critical. Hence, a number of such representations
have been defined recently (Corbet al. 2019 Coskunuzer al. 2021
Vipond 2020).
However, while the rank invariant is equivalent to the persistence diagram in the single-parameter case, it is known to be much less informative in the multiparameter case, when the function admits a decomposition: many functions have different decompositions yet same rank invariants. Therefore, all aforementioned representations can encode only limited multiparameter topological information. Instead, in this work, we define representations based on 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 now whole body of work that allows to define stable, candidate decompositions (Asashiba et al. 2019 Botnan et al. 2021 Cai et al. 2020: Dey & Xin 202T Loiseaux et al. 2022). that are more informative than the rank invariant and that agree with the true decomposition when it cxiste] Our representation is motivated from this series of recent contributions, and expects one of such candidate decompositions as input. Closely related to our method is the recent contribution (Carrière & Blumberg 2020), which also proposes a representation built from given decomposition. However, their approach, while being efficient in practice, is a heuristic with no corresponding mathematical guarantees. In particular, it is known to be unstable: similar decompositions can lead to very different representations. Our approach can be understood as a subsequent generalization of the work of (Carrière & Blumberg 2020), with new mathematical guarantees that allow to derive, e.g., statistical rates of convergence.
codensity
03
Yarlins
Figure 1: (left) Example of point cloud X filtered by both feature scale (computed from unions of balls centered on X with increasing radii) and (co)density. A point will be in the bifiltration its density is high enough, and if it is close enough to X. Note that the circle is present in the red zone, and can be detected as a large summand in the decomposition of the corresponding module (middle), or in a representation of it (right).
SStricty speaking, multiparameter persistent homology can always be decomposed in an abstract way (Botnan & Lesnick [2022] Theorem 4.2). However, such abstract decompositions are hard to interpret and work with, hence the motivation for finding candidate decompositions that are more visual and intuitive.
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Contributions. Our contributions in this work are listed below:
We provide a general framework that parametrizes representations of multiparameter persis- tent homology (Definition 3.1 and which encompasses previous approaches in the literature.
We identify parameters this framework that result in representations that have stability guarantees while still encoding more information than the rank invariant (see Theorem 3.4).
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 of our representations. (2) We achieve the best performance with the lowest runtime on several classification tasks on public data sets (see Sections 4. and 4.2
we recall the basics of multiparameter persistent homology. Next, in Section 3 we present our general framework and state our associated stability result. Finally, showcase the numerical performances of our representations in Section and we conclude in Section 5
Outline. Our work is organized as follows. In Section 2
BACKGROUND
In this section, we briefly recall the basics of single and multiparameter persistent homology, and refer the reader to (Oudot 2015 Rabadán & Blumberg 2019) for thorough treatment.
Persistent homology. standard procedure for studying a data set provided as finite metric space, i.e., set X with a distance dx: X x X - IR, is to form graph with vertex set X and an each edge connecting oy and 2, when dx (xx, 2Ty) j] < c. Persistent homology a simultaneous generalization of this procedure, replacing the graph with simplicial complex (a combinatorial model of topological
space) and aggregating information across different scale parameters e.
More generally, given filtered topological space X, by which we mean topological space X together with a function f: X - R. the system of inverse images F(a) == f [ al) fit together produce system F(a1) F(a2) C ... F(a.) S .... where 04 S 0++1- This system is an example of filtration of X, where a filtration is generally defined as a sequence of nested subspaces X1 C ... X, S ... C X. Then, one can apply the kth homology functor Hş on each F(a.). We do not define the homology functor explicitly here, but simply recall that each Ha(((a;)) is vector space, whose basis elements represent the keth dimensional topological features of F(a;) (connected components for k = O, loops for A: 1, spheres for k = 2, etc). Moreover the inclusions F(a:) S F(a(+1) translate into linear maps H& (F(,)) - H& (F ((++1)))) )), which connect the features of F(ax) and F(a:+1) together. Such a sequence of vector spaces connected with linear maps a persistence module, which is generally defined as a functor from (R, S) (regarded as a category) to Vec, the category of vector spaces.
Multiparameter persistent homology. The persistence modules defined above extend straight- forwardly when there are multiple filtration directions. An n-filtration, or multifiltration, is a family of subspaces J (Xa S X JaER- such that X. S Xxx whenever S a' (in the partial order of IR". Moreover, we say that a multifiltration is 1-critical if there exists a function f X + I such that X, is homeomorphic to the sublevel set (a E X f<x S a} (note that single parameter filtrations are always l-critical). Again, applying the homology functor Hę on a multifiltration J induces a multiparameter persistence module M M(F). The support of M is then defined as the set supp (M) := (ar E T dim(M((a:) > 0}. Any module which is of dimension everywhere on its support is called an indicator module, and, in addition, when its support is an interval of R" (Botnan& & Lesnick 2022 Definition 2. it is called an interval module.
multiparameter persistence module M can always be decomposed in a unique way: M, (Botnan & Lesnick 2022 Theorem 4.2), where the M; are indecomposable modules that are called the sunmands of M. See Figure [ (middle) for an example of decomposition. For single parameter filtrations. the summands of the decomposition are known to be interval modules MI = @ a.g... and can be summarized in a persistence barcode D(M) = {(a,, ad,, ) h hisigm This result does not extend to the multiparameter setting however since summands can have dimension
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larger than 1 on their support. Hence, the rank invariant has been introduced weaker invariant for such modules: is defined as the function RI : (a, () + rank(M[(a) - (55 for any a I B.
Finally, multiparameter persistence modules can be compared with two standard distances: the interleaving and bottleneck (or ()) distances. Their explicit definitions are technical and not necessary for main exposition,
refer the reader to, e.g.- (Botnan & Lesnick 2022 Sections
6.1, 6.4) for more details. The stability theorem (Lesnick) 2013
Theorem 5.3) states that l-critical
multiparameter persistence modules are stable: d1 M') S
|| -f'l oo where f and f'
functions associated to the 1-critical modules MI and MI' respectively.
3 A FRAMEWORK FOR PERSISTENCE MODULE REPRESENTATION
Even though 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 propose a general framework for studying representations of multiparameter persistence modules, i.c., maps defined the space of multiparameter persistence modules and taking values in an (implicit or explicit) Hilbert space. Defining such maps crucial for subsequent data science tasks, most of the standard methods in statistical machine learning expect the input to be Euclidean vectors, or at least to be equipped with scalar product.
3.1 PERSISTENCE MODULE REPRESENTATION
Framework. In this article, we study the following general representation framework:
Definition 3.1. Let M be multiparameter persistence module. Let M 1 M, be its decomposi- tion (note that the M, need not be intervals), and let M be the space of indecomposable persistence modules. A multiparameter representation of M. is:
Vop.n. (MI) = (p((w(M)) ( (M))).))
where op is a permutation invariant operation (sum, max, min, mean, etc), w M - IR is a weight function, and Φ .M- + H. sends any indecomposable module to a vector in a Hilbert space H.
This general definition is inspired from similar framework that was introduced for dingle-paraman persistence with the automatic representation method Perslay (Carrière et al., 2020).
Remark 3.2. Definition 3.1 requires being able to decompose multiparameter persistence modules. Several approaches have been proposed for this problem in the literature, that provide either exact (Dey & Xin 2022, Lesnick & Wright, 2015) or approximate (Loiseaux et al., 2022) decompositions, or decompositions that are restricted to specific homology dimensions (Cai al. 2020).
Relation to previous work. Interestingly, specific choices of op, w and á can reproduce the previous work from the literature. For sake of simplicity, assume MI = e M, be decomposed into indicator modules (we can thus identify any summand Mc with its support), then:
R"
R
Using w : M, + 1, 1. M1
+ M and op = kth maximum, where
lz the diagonal line crossing x, and A(,,) denotes the tent function associated any segment E C R", induces the kth multiparameter persistence landscape (Vipond 2020).
Ra x Rd
Rd
Using w M, + 1, M,
+ w'(M, / n (p.4)) ('MM, n [p, q)) and
p.q
op = op', where op', w' and ø' are the parameters any persistence diagram representation from Perslay, induces the multiparameter persistence kernel (Corbet al. 2019).
TR + R
eeep(-mingezdee Ma //2 and op = Σ. where is a set of (pre-defined) diagonal lines, induces the multiparameter
Using w M; + vol(M)) M, T
persistence image (Carrière & Blumberg 2020
Concerning the first two approaches, it is worth noting that they do not depend on the decompositions and the global geometric shapes of the support of the summands since they equivalent to the rank
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invariant. In particular, it is easy to build different modules with same rank invariants and thus same landscapes and kernels (see (Loiseaux et al. 2022 Section 7.1) for examples). On the other hand, the third approach does use the module decomposttion but is known to be unstable (Figure 11 in (Botnan & Lesnick] 2018)) provides two modules that are close in the interleaving distance but that also have multiparameter persistence images that far away from each other). In the next section, we study representations that allow to encode the summand shapes while also enjoying stability guarantees.
3.2 METRIC PROPERTIES
In this section, we study specific parameters within our framework in Definition 3.1 and prove associated theoretical properties about their associated representations.
Definition. We begin by defining parameters that we call geometric and stable (GS) in the sense that they capture shape information about the summands, and that they also lead to robust representations. Definition 3.3. The geometric and stable (GS) representation parameters are:
1. the weight function .M + sup 0 : By E Rº s.t. ly C supp supp(MM)..wherr lg is the segment between [I...., 1| and + E ,I,
2. the individual summand representations (M) R= - R: (a) os (M) = wsuup (M) n R2,6), (b) Φs (a) (285 vol (supp (M) n Ri,,),
(c) ds (M) (28ym sup {vol(R 8.) R supp (M) n R,,,),
where R the hypersquare {y IR x 8 S y S x + 8) S R", = 8. =0. [I...... 1]
TR for any 8 >> O, and vol denotes the volume of set in RR".
3. the permutation invariant operators op- Σ and op sup.
In other words, our weight function is the length of the largest diagonal segment one can fit inside supp (M), and summand representations (a), (b) and (c) are largest diagonal length, volume, and largest hypersquare volume one can fit locally inside supp (M) n Rg,5 respectively.
Equipped with these GS parameters, we can now define the two following GS representations, that can be applied any multiparameter persistence module M + My:
(M)) == Σ ((M))
プ j=1 (M,)) Ss (M:), for some p E R
(2)
iml
V Voo,((M)) == sup (M).
(3)
15
These GS parameters allow for trade-off between computational cost and the amount of information that kept: (a) and (c) are very easy to compute on interval modules, but (b) encodes more information about summand shapes. See Figures [ (right) and 2b for visualizations.
Stability. The main motivation for introducing our GS representations is that one can show that
they are stable in the interleaving and bottleneck distances, as stated in the following theorem. We assume that persistence modules have support included in compact set K C R for simplicity (but our results extend straightforwardly to modules restricted to K if their supports are infinite).
Theorem 3.4. Let M M and M' M be two multiparameter persistence modules that can be decomposed with indicator modules. Assume that we have m Σ, w(M.), the E, w(M) 2 C. for some C > O. Then for any parameter 8 O, one has
(MI) Vo.s(M')ll (MI A 8)/8,
2 C (MT) Vi,a(M)) 1+
(dn(M, A 8)/6.
| (MI) Voc. (di(M, MM) ^ A)55
(6)
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These results are the main theoretical contribution in this work, the only other decomposition-basee -aase representation in the literature (Carrière & Blumberg 2020) has no such guarantees. The other representations (Corbet et al. 2019 than ours, but are computed from the rank invariant and do not exploit the information contained in decompositions. Theorem shoss that our GS representations bring the best of both worlds: they are richer than the rank invariant and stable at the same time.
et al. 2021 [ 2020) enjoy similar guarantees
proof of Theorem 3.4 can be found in Appendix C One might worry that the minimum with d encountered in the upper bounds in Theorem 3.4 show that our GS representations are too stable and not discriminative enough for practical purposes; while we did not observe such behaviour in our numerical experiments, we also provide an additional stability result with a similar, yet slightly different representation Appendix B with an associated upper bound without minimum.
Remark 3.5. Our GS representations are injective on indicator modules: if the support of two modules are different, then the GS representations (evaluated on point that belongs to the support of one summand but not on the support of the other) will differ, provided that is sufficiently small.
NUMERICAL EXPERIMENTS
In this section, illustrate our GS representations with numerical experiments. First, we explore the stability theorem in Section 4.1 by studying the convergence rates of our GS representations on various data sets. Then, we showcase the efficiency of our GS representations on classification tasks Our code for computing GS representations is based on the MMA (Loiseaux et al. 2022) and Gudhi (The GUDHI Project 2022) libraries and is publicly available and described in
in Section 4.2
Appendix
D
We also provide a few (favorable) running time comparisons in Appendix
Decomposition.
Our GS representations require to compute a persistence module decomposition first. Many different approaches can be used for this task (Loiseaux et al.) 2022, Dey & Xin| 2022
Cai et al. 2020 Lesnick Wright 2015). In our experiments, we always use the MMA approximate
decomposition Loiseaux et al. 2022 because of its simplicity and rapidity.
4.1
CONVERGENCE RATE
In this section, we study the convergence rate of our representation with respect to the number of sampled points. Similar to the single parameter persistence setting (Chazal et al.) (015)) these rates are derived for free from our stability theorem. Indeed, since concentration inequalities for multiparameter persistence modules have already been showed in the literature, these concentration inequalities apply directly to our representations. Note that while Equations (7) and (8), which provide such rates, are stated for GS representation 3 they also hold for GS representation (
Measure bifiltration. Let be a compactly supported probability measure of R.D, and let μn be the discrete measure associated to a sampling of n points from μ.. The measure bifiltration associated to and Mn, is defined as F" == {x E IR' (ax r)) < t}, where B r) denotes the Euclidean ball centered on with radius r. Now, let M. and M,, the multiparameter persistence modules obtained from applying the homology functor on top of the measure bifiltrations, F² and F These modules are known to enjoy the following stability result (Blumberg & Lesnick, 2021 Theorem 3.1, Proposition 2.23 (i)):
di (M,M,) S dpr(j, pn) S min(dy (1, Pz), dly (1. Mn) )+))
where dry and dp, stand for the p- W-Wassesstein and Prokhorov distances between probability mea- sures. Combining these inequalities with Theorem 3.4 then taking expectations and applying the concentration inequalities of the Wasserstein distance (see (Lei 2020 Theorem 3.1) and (Fournier &
Guillin 2015 Theorem 1)) lead to:
8E 5 V Vo0.6 (M, < (cp [XX])
apiva J loga/9
F+T
where a = 2 2p = q = d, a = 1 & 2p and q = dp/(d - p) A 2p or > d = 2p and a = otherwise, Cp.g is a constant that depends on p and q, and X is random variable of law μ.
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Čech complex and density. A limitation of the measure bifiltration is that it can be difficult to compute. Hence, we now focus on another, easier to compute bifiltration. Let X be compact d-submanifold of R (d S D), and be a measure X with density f with respect to the uniform measure on X. We now define the 1-critical bifiltration F.C. with:
FC. = Cech(u) 0 (f E ERR d(x, X) S u, [()
Moreover, given a set X,, of n points sampled from j4, we also consider the approximate bifiltration JCCf,, where fn: X IR an estimation of f (such as, e.g., kernel density estimator). Let M and Mr, be the multiparameter persistence modules associated to F= and JCCff., Then, the stability of the interleaving distance (Lesnick 2013 Theorem 5.3) ensures:
(M, s llf fulloo V (x,
where dæ stands for the Hausdorff distance. Moreover, concentration inequalities for the Hausdorff distance and kernel density estimators are also available in the literature under some conditions (see (Chazal al. 2015 Theorem 4) and (Kim et al., 2019, Corollary 15)). More precisely, when the density / is I.-Lipschitz and bounded from above and from below, i.e., when 0 < fmin I f fmax= < co, and when fn a kernel density estimator of f with associated kernel k;, one has:
log n y
à
log(1/h,)) nhg
(X,
S
and B(II - fxlloo) S LI
where fn is the (adaptive) bandwidth of the kernel k and K is random variable of density k. In particular, p is a measure comparable the uniform measure of a 2-manifold, then for any stationary sequence h, == h > O, and considering a Gaussian kernel h; one has:
log n + LV logg()).
SI (MM) Voo, (M
(8)
Empirical convergence rates. Now that have established the theoretical convergence rates of our GS representations, we estimate and validate them on data sets. We will study a synthetic data set and real data set of point clouds obtained with immunohistochemistry
Anmulus with non-umiform density. In this synthetic example, we generate an annulus in IR2 with a non- uniform density, displayed in Figure 2a Then, we compute the bifiltration FC.,, corresponding the Alpha filtration and the sublevel set filtration of a kernel density estimator, with bandwidth parameter 0.1, on the complete Alpha simplicial complex. Finally, we compute the representations (see Figure 2bb of the associated multiparameter module (in homology dimension 1), and their normalized
distances to the target representation, using either I1-112 or |1-1loo: The corresponding distances for various number of sample points are displayed in log-log plots in Figure 2c One can see that empirical rate is roughly consistent with the theoretical one (-1/2 for ||- and -1 for l1-112), even when p * do (in which case our GS representations are stable for dg but not for di).
Inmunohistochemistry data. In our second experiment, consider point cloud representing cells, taken from (Vipond et al. 2021), see Figure 3a These cells are given with biological markers, which typically used to assess, e.g., cell types and functions. In this experiment, the point cloud by placing a 100 x 100 grid on top of it. Then, we filter this grid using the sublevel 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 multiparameter modules homology dimensions and 1, and we compute and concatenate their corresponding GS representations. The convergence rates are displayed in Figure (3b] Similar to the previous experiment, the theoretical convergence rate of our representations is upper bounded by the one for kernel density estimators with the co-norm, which is of order with respect to the number
first triangulate
of sampled points. Again, the observed convergence rate consistent with the theoretical ones.
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-14
-12
0.0
-05
114 boeniner Paa
-23
-1.0 -0.5
0.0
0.5
1.0
(a) Scatter plot of the synthetic data set, comprised of 25,000 points, and colored by a kernel density estimator with bandwidth = 0.5.
(b) GS representations with parameters, p (upper left), = 1 (upper right), = oo (bottom).
10"
2.555
10P
10-
1.30
2.336+
scaled 9uD norm
10
307
307
307
107
107
307
22*
Mumoze poirs
Numser poinzs
(c) (left) 11:112 and (right) I| (right) between the target representation and the empirical one w.r.t.
Figure 2: Convergence rate of synthetic data set.
300
1.09
2000
2008
4003
30K
(a) Point cloud of cells colored by CDS (red) and CD68 (black).
(b) (left) ||:1| and (right) ||-1| distances between the target repre- sentation and the empirical one
Figure 3: Convergence rate of immunofluorescence data set.
4.2 CLASSIFICATION
We illustrate the efficiency of our GS representations by using them for classification purposes. We show that they perform comparably or better than existing topological representations as well as standard baselines on UCR benchmark data sets. We work with point clouds obtained from time delay embedding on time series from the UCR data sets, following the procedure of (Carrière & show that on the inmunohistochemistr data set of point clouds from the previous section, where each point cloud represents single cells of given type (CD8, CD68, etc), our method is significantly better than the competition. In both tasks, every point cloud has label (corresponding, e.g., to the type of its cells in the immunohistochemistry data), and our goal is check whether can predict these labels using the bifiltration defined in the previous section.
Blumberg 2020). Furthermore,
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We also compare the performances of our GS representations (evaluated on a 50 50 grid) to the one of the multiparameter persistence landscape (MPL) (Vipond 2020), kernel (MPK) (Corbet et al., 2019) and images (MPI) (Carrière & Blumberg 2020) We also compare
topological baselines: we used the standard Ripley function evaluated on 100 evenly spaced samples in [0, 1] for immunohistochemistry data, and k-NN classifiers with three difference distances for the UCR time series (denoted by B1, B2, B3), as suggested in (Dau et al. 2018). All scores on the immunohistochemiitty data were computed with folds after cross-validating few classifiers (random forests, support vector machines and xgboost). We also evaluated the performances of few different GS representation parameters p and ō separately. For the time series data, our accuracy scores were obtained after also cross-validating the following GS parameters; E {0, }}. op E {sum, mean), 8 E {0.01, 0.1, 0.5, 1). | (immunohistochemistrry) and (time series-noke note that there are no variances since train/test splits were provided in the data sets). One can see that our GS representations almost always outperform topological baselines, and are comparable to the standard baselines on the UCR benchmarks. Most notably, all tested GS parameters showed improvement over the baselines for the immunohistochemisty data.
0.5, 1, 1.5] with homology dimensions 0 and
1. All results can be found in Tables
MPL.
MPT
Ours
DPGRG
62.6
62.6
77.0
67.6
70.5
71.9
71.9
DPO2
71.7
71.7
73.8
Ours
p
Accuracy (%)
DPTH
63.3
63.3
59.0
61.2
56.1
61.9
62.6
PROAG
80.5
78.5
81.9
10
74.1 = 2.9
P202
80.8
79.0
78.4
78.7
78.7
818
79.4
1
10 10-3
75.6 3.4
70.7
75.6
73.2
76.1
75.6
506200
88.0
88.0
77.0
77.0
74.0
83.0
83.0
10-1
68.7 + 3.0
95.5
95.5
95.0
80.7
78.6
71.9
81.2
10 10-2
77.81 2.0
MT
68.4 4
74.7
73.7
55.4
55.7
60.0
64.3
96.2
100.0
100.0
92.4
84.8
97.1
99.0
10
76.0
SL
78.9
84.6
79.2
78.2
64.6
83,8
83.8
10 10-4
70.5 3.0
GP
91.3
91.3
90.7
88.7
94.0
90.7
96.3
GPAS
89.9
91.8
93.0
85.1
90.5
88.0
MPL Ripley
60.5 - 3.5
GPMUP
97.5
97.5
99.7
96.8
88.3
95.9
95.3
67.2 2.3
GPOVY
95.2
83.8
99.0
97.1
100.0
100.0
PC
93.3
92.2
87.8
85.6
84.4
86.7
9..11
SC
60.3
61.3
Table 1: Accuracy scores for immunohisto- chemistry data.
Table 2: Accuracy scores for time series.
5 CONCLUSION
In this article, we study the general question of representing multiparameter persistence modules in Topological Data Analysis. We first introduce general theoretical framework that allows to recover existing approaches in the literature, as well as to define new ones. Then, we identify and prove the stability of some specific representations in that framework, that call geometric and stable (GS) representations since they are able to encode the shapes of the summands comprised in the modules, making these representations more powerful than already existing ones. We finally showcase the stability and efficiency of these GS representations in various theoretical and numerical applications about statistical convergence and classification accuracies.
Several questions remain open for future works:
The framework that introduced in Definition 3. is similar the Perslay framework of single parameter persistence (Carrière et al. 2020). In this work, each of the framework parameter was optimized by a neural network. It thus natural to investigate whether one can optimize the module representation within our framework in data-driven way well.
In our numerical applications, we focused on representations computed off of MMA decom- positions (Loiseaux et al. 2022). In the future, we plan to investigate whether working with other decomposition methods (Cai et al. 2020 Dey & Xin 2022) lead to better numerical performances when combined with representation framework.
Even though our experiments were done on data sets equipped with bifiltrations, our code and framework is well-defined for any number of filtrations. Investivating the dependences and performances of our framework on more than two filtrations is thus of particular interest.
Note that the sizes of the point clouds obtained from immunohistnchemissty were too large for MPK and MPT using the code provided in wttthtt::/// http:///hhtb.bccccmmm/acaaaarrra
cipers

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A ADDITIONAL BACKGROUND
Definition A.1. Let n E N and F (.. be a multi-filtraton on a topological space X. Let e, b IR, and Le,s : IR - R" be the line in R" defined with (t, t e b, that is, le.6 is the line of direction e passing through b. Let Fe,b R + P(X) defined with (1) = - FO ((f,,())))) where [:]; denotes the i-th coordinate. Then, each Fo.b is single- parameter filtration and has corresponding persistence diagram, barcode Ba.b- The set B(F)= (Beb e, E } is called the fibered barcode of F.
The two following lemmas from Landi 2018) describe two useful properties the fibered barcode.
Lemma A.2 (Lemma in (Landi 2018)). Let e, Rh and le.b be the corresponding line. Let è = min; [e]; Let F, F be two filtifiltratioon,, M. M' be the corresponding persistence modules and Be, E B(F) and B', B(F') be the corresponding barcodes the fibered barcodes of F and F.. Then, the following stability property holds:
di d(M,M)) M')
dg Bo. B²,,) S
Lemma A.3 (Lemma 2 in (Landi 2018 Let L' E IR" and La,, lot be the corresponding
lines. Let é min;[e]s and e¹ min; e'| Let F be multi-iiltration, M be the corresponding persistence module and Be,,, Be E B(F) be the corresponding barcodes in the fibered barcode of
F. Assume M41 is decomposable MI i and let > O such that M, C Boc ()) = {x-
R | S K} for all i E [1.m]. Then, the following stability property holds:
(K max{ 116'11 }) ||e e'll.o + bbb 1l
dp d ())
(10)
E'
B AN ADDITIONAL. STABILITY THEOREM
In this section, we define new GS representation, with slightly different type of upper bound. It relies on the notations introduced in Appendix [A] We also slightly abuse notations and use M, to denote both an indicator module and its support.
Proposition B.1. Let a > O, and let 0 S 5 < 8(MI), where
5(MI) == in 2 0: T'e achieves dg (BeA.Z: Bes.zt5u) for all x, u.s. ||u|l = 1, (cs, u) = 0}, where TM is the partial matching induced by the decomposition of ML. Let N(xx 0) denote the function (xx
R
(-
11p-a/l 2
and let
exp
R"
+ R
V8.o (MI
(11)
maxisism max [[(x, -flh
where C(x, 5, M,) stands for the set of indicator functions from R" to {0, 13 whose support is Ts(E) n Ms where is connecting component of (m((los.a)) 0 Mj and es = =[I,...,1] R", and
where Ta (l) is the 6-thickening of the line T.(E) induced by l: (2(E) (x E R T(f]l & 5).
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Then, Vo.o satisfies the following stability property:
22t (MM (o)) where Cn(-) is a continuous function such that (,) + 0 when 6 + 0.
I|V (M) Vo.o (M S (ViTo)"
(12)
Proof. Let M M and M''' e be two persistence modules that are decomposable into indicators, let x E R&& and let 0 < 5 5 min( d(M'))
Notations. We first introduce some notations. Let N (resp. N" be the number of bars in Beg." (resp. B. ..), and assume without loss of generality that N S N'. Let I' be the partial matching achieving dn (B Bes..)) .) Let N1 (resp. N2) be the number of bars in Beg that are matched (resp. not matched) to a bar in Beg.a under r, so that N = N1 + N2. Finally, note that Be = {e Ei such that & E (fim(fe..:)) n M,) and im(Les, n M, * 0}. where C stands for the set of connected components (and similarly for BeA,.). and let Fr Bea." + Bes. be function defined on all bars of Be that coincides with I on the N1 bars of Bea.z that have an associated bar in _e,... and that maps the N2 remaining bars of Bea. to some arbitrary bars in the (N' NN) remaining bars of B B.....
A reformulation of the problem with vectors. We now derive vectors that allow to reformulate the problem in a useful way. Let V be the sorted vector of dimension N containing all weights [[, fll where is the indicator function whose support is (C) n M, for some M,, where l E Be 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 [[V - where f' is the indicator function whose support is (e') n Mj for some Mj. and E' E B's. is the image under I of the bar l E Bea. corresponding to the ith coordinate of V, i.e., l' (f) where [V], = [1,(,, fllı and f is the indicator function whose support is n for some Mig. In other words, V is the (not necessarily sorted) vector of weights computed on the bars of B'a,* that are images (under the partial matching achieving the bottleneck distance) of the bars of Bea, that were used to generate the (sorted) vector V.
the vector 1/2 of dimension (N' -N) whose jth coordinate is [[, a)) f'll where f' is an indicator function whose support is (e") n M", for some My. and E' E B'a.. satisfies l' & im (.r)). In other words, V/2 is the vector of weights computed the bars of B (in an arbitrary order) that are not images of bars of Be under I.
Finally, we let V be the vector of dimension N' obtained by filling V (whose dimension is N < 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:
VA.o (z) Va. (M.)) - IV V/ = IV - sort(("'),
(13)
An upper bound. We now upper bound ||v-v/ Let q E [1.N'] Then, one has [V],
(1 fll where f is indicator function whose support is Ts( n M, for some M, with E Bea. if q S N and 0 otherwise; and similarly [V', = [W σ) -''ll: where f is an indicator function whose support is T2((") n Mi for some M2 with e' E B Thus, one has:
(V- 14
|l)(x fll1 (2 -f'l1
< [A o): f -N o) f'll, by the reverse triangle inequality
[L o) ffll") by linearity
I [1((2,)))l ||f f'll2 by Hölder's inequality
=(Voo)* -flllz
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Since (f - f*) is an indicator function whose support is (T&(E) (e) Mj), has Ilf fl1 = v / (T)(()) M.)A(T((e() MM)) Given a segment and vector u, let Lu denote the segment f, and we let en denote the (infinite) line induced by lu More precisely:
Ilf 1 = =lJennaAUerrnn
where ranges over the vectors such that ||ull I 5, (1, =0 (((MM))(())")))))))
I(C" O M.) +Ieen Mi) ((0 C M
J, 4dg (Bea. Bes Bes + 4d (Bes.z B' =x + 4dg (B' B B'
C Be
(14)
< / ||u||b0 + (M, + ||ulloc du by Lemma A.2 and A.3
Inequality (14) 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 dg (BeA. ees. and
By Be are induced by MI and M''
Conclusion. Finally, one has
V.so Va,o (M)) = IV - V-] = -Vo-sor((V'))
sorl(V")), from Equation 13
< ||V - V'
< (VTT)** 1d
Co,(5),
(15)
with (o) = 8 fn du + when 8 - 0. Inequality 15 comes from the fact that any upper
bound for the norm the difference between a given vector V and a sorted vector V, is also an upper bound for the norm the difference between the sorted version V V and the same vector V (see Lemma 3.9 in (Carrière et al. 2015b)).
While the stability constant is not upper bounded by 6, Va,a is more difficult to compute than the GS representations presented in Definition 3.3
C PROOF OF THEOREM 3.4
Our proof is based several lemmas. In the first one, focus on the GS weight function as defined in Definition 3.3
Lemma C.1. Let M and M' be two indicator modules with compact support. Then, one has
di d =
sup min(d, by)+ w(M).
(16)
b.desupp(M)
Furthermore, one has the equality
Iw(M) - w(M') S d1
(17)
Proof. We first show Equation (66 with two inequalities.
First inequality: < Let M be an indicator module. If dr(M, 0) then the inequality is trivial, so we now assume that di (M, 0) > 0. Let > 0 such that ó < dj(M, 0). By definition of dj, the identity morphism M M (28) cannot be factorized by O. This implies the existence of some b E R" such that rank (b) - (b 26)) > 0; in particular, b, + 28 E supp (M). Making converge to di (M, 0) yields the desired inequality.
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Second inequality: Let neE) be compact interval exhaustion of supp (M), and ons dn E Kn be two points that achieve the maximum
1 sup min(d;- - by)+. 2 b,dek,
Now, by functoriality of persistence modules, we can assume without loss of generality that by and dy are on the same diagonal line (indeed, if they are not, possible to transform d, into d, such that bn and d, are on the same diagonal line and also achieve the maximum). Thus, rank(M/(b,) + M(d,)) > O, and d1(M,0) 2 ||d, - ballow Taking the limit over N leads to the desired inequality.
Inequality (17 follows directly from the triangle inequality applied on dj.
In the following lemma, we rewrite volumes of indicator module supports using interleaving distances. Lemma C.2. Let M be an indicator module, and R C IR" be a compact rectangle, with n 2 Then, one has:
vol (supp (M) n R) = (VEER:..=0) di (M (M)nRR))) dxx-1(y)
where by is the diagonal line crossing y. and X- denotes the Lebesgue measure in R=-!
Proof. Using change variables y, = -; Un and t = 2n (which has a trivial Jacobian) yields the following inequalities:
[
vol(supp (M) n R)
dx"(x)
Suu
1app(M))n + dt (v)
(NER" 00 JIER
(VER*aYM diam (supp (M) nly n R) dx"-1(y) (v)
where ly is the diagonal line passing through y. Now, since M an indicator module, one has diamnjMl (supp (M) n R) = 2d[(M I,n 0), which concludes the proof.
In the following proposition, we provide stability bounds for single indicator modules.
Proposition C.3. If M and M' are two indicator modules, then for any 0 and GS parameter di
in Definition 3.3 one has:
1. 0 S (M)( w(M) A for any x E R".
2. [ - (s(M')l & (ddI(M, A 6)/6.
Proof. Claim 1. is simple consequence of Equation (16).
Claim 2. for GS parameter (a) is simple consequence of the triangle inequality.
Let us prove Claim 2. for (b). Let E "R" and > 0. One has:
lds (M)()) - (x))) N (28)" 10vn=00 dd (M),n,,,s di (M' 1 /dd"-1(y)
10mn==0 di (M Mirno M' l ddn
(28)m
((d M') 5)/88
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where the first inequality comes from Lemma C.2 the second inequality is application of the triangle inequality, and the third inequality comes from Lemma A.2
Finally, let us prove Claim 2. for (c). Let R" and 8 > 0. Let S d supp (Mf n R Let
also y > 0. Then, using Lemma C.2 one has:
1 (28)
vol(supp (M) n Ro.a)
ddMM
Ro,an, 0)d1"~1(v)
(26)* (VER"
2
2
_ 8 (25)m (VERRM
d
Rat Botra n .) dd" (y),
using the convention Ra,b = 2 if a o b. Now, set == di(M Re,s" M' R )- If b + or
- y & supp (M') then di(M] R.. Ml Res =>> di M') which is impossible. Thus,
(20)n Rol( s (M R
M' Rs,8 )/6
2
dr(M] Rn, 0) (vER"
R.cER,,s M'') (26)m
sup
(M R..s" MM R.. )/6 + @ss
Finally, taking the supremum on b S E supp (M) n R yields
(z) (s()) - (M . M1 Rs, 18 $ (M)
The desired inequality follows by symmetry on M and M'.
Equipped with these results, can finally prove Theorem 3.4
Proof. Theorem 3.4
Let M- = 20 M2 and M' = - Mj be two modules that are decomposable into indicator modules and R
Inequality B To simplify notations, we define the following: w; := w(M,,, وτ == Φs (2) and w// = :=((M) M,)) H/ == Φ1 M2). 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 Mt for every (up to adding modules in the decompositions of M and M'' so that m = m²). Finally, assume without loss of generality that E. w/ 2 E, w. Then, one has:
I (M))(())( (=M))=
Σ
1
Σ
E, w;
E.ur w/
Σ wi
Σ Euich...
+ E,w; W, Σ, w/ Σ wrsra
1
Σ, w!!
Now, for any index since dM,,, Mf) du (M, Ml) and |wo - w/ll S dM, M₁) S du (M, MY)
by Lemma C.I. Proposition C.3 ensures that:
b w.H.aal S |w, Hi, x + w/l Tlaal 2(w, + w ) (dg (M, M') ^ 61/5
and
1
Σ, w
wi
mdg (M, M')
Eww; w/ Ee m
E, w
Σe Wi
E ( w.)
Finally,
Σ, wy + W/
Σx wi
V (M))( VM.s
2(d 0)/6
Σ, me!
I wi
E, w!)
2
4 + c
(d (M, ^ 8)/5.
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Inequality 4 can be proved using the proof of Inequality|5]] by replacing every wy by 1.
Inequality Let us prove the inequality for (b). Let == Rz- One has:
(M) Voo.s
2
di(M: ,n 0) d)d"-1(w)
sup
(26)"
(yER":y=0)
2
di (1(M 1, 0) d)
-sup
3 (28)m
(VERMINN 0)
2
(for any index ÿ) S ssup
di di 1, 0) di di(M s,mme 0) (1)=)(w)
i (26)n
(DERm:B.=0)
2
sup inf d1 (M)
M, M Milenn dd"
(28)"
(vER"
Now, the interleaving distance is equal the bottleneck distance for single parameter persistence (Chazal et al. 2016 Theorem 5.14), one has:
sup J
inf dM i, Mile,nR I du (M ( MY Minlnnnnn ( M. Mlll,nnn d1,MM) A
{
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 the single parameter case).
D SEEUDO-CODE FOR OUR REPRESENTATIONS
In this section, we briefly present the pseudo-code that we use to compute our GS representations. Our code is based on implicit descriptions of the decompositions of multiparameter persistence modules (which are the inputs of our GS representations) through their so-called birth and death corners. These corners can be obtained with, e.g., the public softwares MMA (Loiseaux al., 2022) and RIVET [Lesnick& & Wright 2022).
In order to compute our GS representations, we implement the following procedures:
1. Given an indicator module M and rectangle R C R compute the restriction M
2. Given module M (or M 1)) compute di(M, 0). This allows (combined with the first algorithm above) to compute our weight function and first summand representation in Definition 3.3
3. Given a module restricted to rectangle, compute the volume of the biggest rectangle in the support of this module. This allows (combined with the first algorithm above) to compute the third summand representation in Definition 3.3
For the first point, Algorithm D corresponds to "pushing" the corners of the module on the given rectangle in order to obtain the updated corners.
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Algorithm I: Restriction of a module to a rectangle
Data: birth and death corners of a module M, rectangle R = {z E R² m S 2 $ M]
Result: new now_summands. corners, the birth and death corners of M| for summand = (sununand birth
M wewammaaaddeehhhhoooco
new.birth.list 4 []:
for b in surmaad birth corners do if b < M then
= [max(bs, m²) for E [1,n]];
Append b' new_birth_ao
end
end
new death list < []:
for d in summand death corners do if d 2 then
d' {minnd,, M,) for i [1,n]}: Append d' new -ewtth end
end
summands corners [new birth list, new death list];
end
For the second point, we proved in Lemma C.I that our GS weight function equal to d1(M, 0) and has closed-form formula with corners, that we implement in Algorithm 2
Algorithm 2: GS weight function
Data: birth and death corners of a module Result: distance, the interleaving distance dy (M,0).
distance +- 0;
for b in M www_birttcorreeee
do for d in M death corners do
distance + max (rcsult, 1 2 min, (d; b:)+);
end end
The third point also has a closed-form formula with corners, leading to Algorithm
Algorithm 3: GS summand representation
Data: birth and death corners of module M
Result: volume, the volume of the biggest rectangle fitting in supp (M)
volume +-
for b in M birth corners do for d in M death corners do volume <- max (result, [d; - bb)))::
end
end
E RUNNING TIME COMPARISONS
In this section, we provide running time comparisons between our GS representations and the MPI and MPL representations from the literature.
worth noting that our implementation relies on an implicit description of the summands in the module decomposition, which are characterized with their corners (Loiseaux et al.) 2022). This is in contrast with the implementations of the other representations that are available in the mult multipers library, which require slicing the modules with many lines first. This shows from Table 3, where can be seen that our representations
18

---

Under review as a conference paper at ICLR 2023
(computed on the pinched annulus and immunohistochemiisty data sets defined in Section 4 be computed much faster than the others, by a factor of at least 25 (all representations are evaluated on grid of size 100 x 100, and provide the maximum running time All computations were done using Ryzen 4800 laptop CPU, with 16GB of RAM. Interestingly, this sparse and fast implementation based corners also be used to improve the running time of the multiparameter persistence landscapes (MPL), as one can see from our pseudo-code in Algorithm 4 (which retrieves the persistence barcode of multiparameter persistence module along given line; this is enough to compute the MPL) and from the second line of Table
{0, (o)))
Algorithm& 4: Restriction of a module to a line
Data: birth and death corners of module M, diagonal line / Result: barcode, the persistence barcode associated to M z
barcode < []:
y +- an arbitrary point in for summand = /summand. wittthccrrehrrrsseeeeeeeeeeeeeee corners, wew.:maannddetthhcooeeeee in M do
birth + y + 1 x minbesummand birth corners max; death < y + 1 x maxdemmmdnndd death.corner. min; d; - bar <- [birth, death];
Append bar to barcode;
end
Annulus
Immuno
Ours (GS)
250ms + 2ms
275ms = 99.ms
Ours (MPL)
36.9ms + 0.8ms
65.9ms = 0.9ms
MPI (50)
6.43s = 25ms
5.67s = 23.3m
MPL (50)
17s = 39ms
15.6s = 14ms
MPI (100)
13.1s + 125ms
11.65s 7.9ms
MPL (100)
35s + 193ms
31.3s = 23.3ms
Table 3: Running times for GS representations and competitors.
19


Reviews:
[{"review_id": "k196CqV-dc", "reviewer": "Reviewer_WBCA", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes a framework to use multiple parameters to compute persistent homology for topological data analysis. The proposed sparse implementation reduces memory usage and running time. Experiments show the proposed method converges fast and achieves the best performance on several classification tasks.", "summary_of_the_review": "The paper proposes a general framework for multiparameter persistent homology with geometric and stable parameters. ", "strength_and_weaknesses": "Strength:\n- The proposed framework unifies previous work by selecting different operations and functions. The proposed parameters consider geometry and stability, which are effective in classification.\n- Experiments show the proposed method has fast convergence rate, and better quality than others.\n\nWeakness\n- The paper has heavy theory, and it’s not easy to read for the audience without a background in topological data analysis and persistent homology. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "gEQCBuaJ0aC", "reviewer": "Reviewer_su2T", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper presents a new topological data analysis framework based on persistent homology. The main contributions are first the general framework with multiparameter persistency and a sparse implementation. The experiments show interesting results on point cloud classification. ", "summary_of_the_review": "This article proposes a framework for topological data analysis. It combines multiparameter persistency with persistent homology to produce robust feature to describe data. The features are shown to produce a controlled approximation of the data robust to noise. Experiments show interesting results for classification of complex data. This paper is easy to read and may reach a larger public than just its community.", "strength_and_weaknesses": "## Strength\n\nThe proposed framework is interesting on several points.\n1. The combination of multiparameter persistency with persistent homology allows to catch complex features inside the data. The multiparameter is then crucial to deal with data with more than one dimension.\n2. The representation is robust (using geometric and stable parameters) so usable in context where noise may be important.\n3. There is an analysis of both the approximation quality of the feature and the convergence toward a complete representation of the data.\n\n## Weaknesses\n\nSuch framework have several weakness,\n1. High dimensional data are still an issue with such framework.\n2. The scope of such methods is unclear and should be discuss. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "MInCLi0f_DU", "reviewer": "Reviewer_a971", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper discusses a framework of multiparameter persistence, which has been a long-standing problem in the field of topological data analysis with persistence homology, and proposes three types of specific representations (GS representations) with stability guarantee.   The numerical results demonstrate the convergence rates with respect to the sample size and the effectiveness in classification accuracies with real-world data.  \n", "summary_of_the_review": "I think that the paper contains significant contributions to the problem of multiparameter persistent modules, which is one of the important issues in topological data analysis.  However, the presentation of the paper has much room for improvement, as detailed in Strength and Weaknesses.  \n", "strength_and_weaknesses": "Strength:\n- A framework for representing multiparameter persistence modules is proposed, covering some previous methods. \n- The proposed methods with specific representations have a theoretical guarantee of stability, which is an important property of a topological feature in applications. \n- In the two multiparameter cases of the measure bifiltration and Cech complex with density, the theoretical analysis gives convergence rates of the proposed GS representations, and the rates are also confirmed numerically. \n- The experimental results with real-world data sets demonstrate better classification accuracies than the existing multiparameter representations.\n\nWeakness:\n- The significance of the proposed framework and method is not clearly presented in the paper.  \n    - While the multiparameter persistence is a more general approach, the practical advantage in data analysis over 1-parameter persistence is not necessarily clear in the paper.  For the real-world data, more basic comparisons with 1-parameter persistent homology as well as other relevant methods should be included.  \n    - The advantage over the existing multiparameter methods is not sufficiently explained, although it is somehow discussed.  One of the advantages should be stability.  In comparison with Carriere & Blumberg (2022),  the authors just cite Botnan & Lesnick (2018) without details.  As an important contribution, the reason for having stability, unlike the others, should be explained more clearly and demonstrated with some basic numerical examples.  \n- The significance of the theoretical analysis in Section 4.1 is unclear.  \n    - As discussed in the paper, the dependence of constant $\\delta$ is not clear enough.  While the authors say that the empirical results do not show bad behavior by choice of \\delta, more detailed discussions are necessary about whether this dependence on $\\delta$ appears as an artifact by the theoretical derivation or an inevitable factor by the method. \n    - In the upper bound (8), the bandwidth parameter $h$ remains constant, and the bound involves a constant factor.  Can we say that the first term $1/\\sqrt{n}$ is the convergence rate?  \n- Some parts of the paper are tough to understand for standard readers in the machine learning field.  Some examples are:\n    - In the last paragraph of page 3, \"indecomposable\" module is undefined. \n    - In Figure 1, it is not clearly explained what is depicted in the middle and right figures.  More explanations are needed. \n- The paper may not be easy to read for most researchers in the machine learning field.   While I understand the difficulty of explaining persistent homology to non-experts, the authors could demonstrate better the importance of the problem to be solved in this paper.  Also, I recommend the authors include a brief introduction about the basic notions of persistence homology in Supplementary Materials.  \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "6ugfuAfR6x", "reviewer": "Reviewer_Bro2", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper mentions introduces a multiparameter persistent homology based solution to the \"finding shape descriptors problem\" in point clouds.  The authors also introduce their solution for topological data analysis as a generic tool. The authors introduce some upper limits for the stability for their work.\n\n", "summary_of_the_review": "This paper mentions introduces a multiparameter persistent homology based solution to the \"finding shape descriptors problem\" in point clouds.  The authors also introduce their solution for topological data analysis as a generic tool. The authors introduce some upper limits for the stability for their work.\n\nOverall, due to the vast amount of Math usage, instead of words, this paper remains hard to read and requires way more time to understand fully (as a reviewer, I could not give that much time to check the correctness of each equation, unfortunately). The used ideas could be found in the literature. As such, the motivation and the novelty of this paper remains unclear to me. The experimental results include only a few baseline algorithms and they run on toy datasets. There are a vast amount of classifiers out there, and the authors could include a large amount of classifiers to support their claims in their experiments.", "strength_and_weaknesses": "Strengths: \n\n- The paper introduces a multiparameter based approach for modeling. The authors provide proof for their theorem (which I did not check for correctness).\n\n- The run time of the algorithm seems faster when compared to the run time of the used baseline algorithms as shown in Table 2.\n\n- The algorithm shows more than 10 % improvement on accuracy, when compared to MPL and Ripley. (However, more baseline algorithms could also be used there for a better judgment).\n\n\nWeaknesses: \n- The idea of using multiple parameters for geometric modeling and for filtering is not new, and there are other multiparameter based solutions exist in the literature. The authors can do a better taxonomy of those works from the literature by including them in their work and in their experiments (for example, the authors should also include the following 3 works as they also are highly related to this work: (i) O. Chapelle, V. Vapnik, O. Bousquet, and S. Mukherjee, \"Choosing multiple parameters for support vector machines\", Machine learning, 46(1-3):131–159, 2002; and (ii) Ozer, S. (2018), \"Similarity domains machine for scale-invariant and sparse shape modeling\", IEEE Transactions on Image Processing, 28(2), 534-545; (iii) Witkin, Andrew P. \"Scale-space filtering.\" In Readings in Computer Vision, pp. 329-332. Morgan Kaufmann, 1987)\n\n- The paper uses heavy Math based representation, which is hard to follow. Furthermore, the proposed \"algorithm\" is not clear to understand. The structure of this paper could be improved by focusing the flow of the paper to explain the steps of the proposed approach first. Currently the proposed algorithm is given as Appendix and it is not citing the any particular equation from the main text. That could be simplified. \n\n- Experimental results show marjinal improvement on toy datasets and the approach is compared to k-NN classifier along with 3 additional random algorithms in Table 3.  The baseline algorithms seem limited. For classification, the authors could use many classification algorithms, when their performance metric is accuracy. \n\nIn Table 3, the caption states that \"classification results for time series\". It is not well explained how the time series problem is considered as a geometry based multi-scale classification problem there. That could be clarified.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject