Fuzzy c-Means Clustering for Persistence Diagrams
Authors: Thomas Davies
Jack Aspinall
Bryan Wilder
Long Tran-Thanh
Venue:  (2021)
Keywords: Topological data analysis
fuzzy clustering

Abstract: Persistence diagrams concisely represent the topology of a point cloud whilst having strong theoretical guarantees. Most current approaches to integrating topological information into machine learning implicitly map persistence diagrams to a Hilbert space, resulting in deformation of the underlying metric structure whilst also generally requiring prior knowledge about the true topology of the space. In this paper we give an algorithm for Fuzzy c-Means (FCM) clustering directly on the space of persistence diagrams, enabling unsupervised learning that automatically captures the topological structure of data, with no prior knowledge or additional processing of persistence diagrams. We prove the same convergence guarantees as traditional FCM clustering: every convergent subsequence of iterates tends to a local minimum or saddle point. We end by presenting experiments where our fuzzy topological clustering algorithm allows for unsupervised top-$k$ candidate selection in settings where (i) the properties of persistence diagrams make them the natural choice over geometric equivalents, and (ii) the probabilistic membership values let us rank candidates in settings where verifying candidate suitability is expensive: lattice structure classification in materials science and pre-trained model selection in machine learning.

Under review as a conference paper at ICLR 2021
FUZZY C-MEANS CLUSTERING FOR PERSISTENCE DIAGRAMS
Anonymous authors Paper under double-blind review
ABSTRACT
Persistence diagrams concisely represent the topology of point cloud whilst having strong theoretical guarantees. Most current approaches to integrating topological information into machine learning implicitly map persistence diagrams to a Hilbert space, resulting in deformation of the underlying metric structure whilst also generally requiring prior knowledge about the true topology of the space, this paper we give an algorithm for Fuzzy c-Means (FCM) clustering directly on the space of persistence diagrams,22 cnabling unsupervised learning that automatically captures the topological structure of data, with prior knowledge additional processing of persistence diagrams. We prove the same convergence guarantees traditional FCM clustering: every convergent subsequence of iterates tends local minimum or saddle point. We end by presenting experiments where our fuzzy topological clustering algorithm allows for unsupervised top-k candidate selection in settings where (i) the properties of persistence diagrams make them the natural choice over geometric equivalents, and (ii) the probabilistic membership values let us rank candidates in settings where verifying candidate suitability is expensive: lattice structure classification materials science and pre-trained model selection in machine learning.
INTRODUCTION
Persistence diagrams, a concise representation of the topology of a point cloud with strong theoretical guarantees, have emerged as new tool in the field of data analysis (Edelsbrunner Harer, 2010). Persistence diagrams have been successfully used to analyse problems ranging from financial crashes (Gidaa & Katz 2018) to protein binding (Kovacev-Nikolic et al. 2014). but the oon-Hilbertian nature of the space of persistence diagrams means it is difficult to directly use persistence diagrams for machine learning. In order better integrate diagrams into machine learning workflows, efforts have been made to map them into more manageable form; primarily through embeddings into finite feature vectors, functional summaries, or by defining a positive-definite kernel on diagram space. In all cases, this explicitly or implicitly embeds diagrams into a Hilbert space which deforms the metric structure, potentially losing important information. With the exception of Topological Autoencoders, techniques integrate these persistence-bant based summaries as topological regularisers and loss functions currently require prior knowledge about the correct topology of the dataset, which clearly not feasible in most scenarios.
Against this background, we give an algorithm to perform Fuzzy c-Means (FCM) clustering (Bezdek 1980) directly on collections of persistence diagrams, giving an important unsupervised learning algorithm and enabling learning from persistence diagrams without deforming the metric structure. We perform the convergence analysis for our algorithm, giving the same guarantees as traditional FCM clustering: that every convergent subsequence of itcrates tends to local minimum or saddle point. We demonstrate the value of our fuzzy clustering algorithm by using it to cluster datasets that benefit from both the topological and fuzzy nature of our algorithm. We apply our technique in two settings: lattice structures in materials science and the decision boundaries of CNNs. A key property for machine learning in materials science has been identified as "invariance to the basis symmetries of physics [...] rotation, reflection, translation" (Schmidt et al. 2019). Geometric clustering algorithms do not have this invariance, but persistence diagrams do, making them idcally suited for this application; we can cluster transformed lattice structure datasets where geometric
equivalents fail. In addition this, our probabilistic membership values allow to rank the top-k

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most likely lattices assigned to cluster. This is particularly important in materials science, as further investigation requires expensive laboratory time and expertise. Our second application is inspired by Ramamurthy et al. (2019). who show that models perform better on tasks if they have topologically similar decision boundaries. We use our algorithm to cluster models and tasks by the persistence diagrams of their decision boundaries. Not only is our algorithm able successfully cluster models to the correct task, based just on the topology of its decision boundary, but we show that higher membership values imply better performance on unseen tasks.
1.1 RELATED WORK
Means of persistence diagrams. Our work relies on the existence of statistics in the space of persistence diagrams. Mileyko et al. (201T) first showed that means and expectations are well-defined in the space of persistence diagrams. Specifically, they showed that the Fréchet mean, extension of means onto metric spaces, is well-defined under weak assumptions on the space of persistence diagrams. Turner et al. (2012) then developed an algorithm
compute the Fréchet mean. We adapt the algorithm by Turner et al. to the weighted case, but the combinatoric nature of their algorithm makes it computationally intense. [acombe al. (2018) framed the computation of means and barycentres in the space of persistence diagram as an optimal transport problem, allowing them to use the Sinkhorn algorithm (Cuturi & Doucet 2014) for fast computation of approximate solutions. The vectorisation of the diagram required by the algorithm by Lacombe et al. makes it unsuitable for integration into our work, as we remain in the space of persistence diagrams. Techniques to speed up the matching problem fundamental to our computation have also been proposed by Vidal et al. (2020)
and Kerber et al. (017))
Learning with persistence-based summaries. Integrating diagrams into machine learning work- flows remained challenging even with well-defined means, as the space is non non-Hiibertian Turner Spreemann 2019). As such, efforts have been made to map diagrams into Hilbert space; primarily
either by embedding into finite feature vectors (Kališnik] 2018. Fabio & Ferri 2015 Chepusstanoval
et al. 2013) or functional summaries (Bubenik) 2015 Rieck al. (019)) or by defining positive-
definite kernel on diagram space (Reininghaus et al. 2015] C Carrière et al. 2017 Le & Yamada 2018))
These vectorisations have been integrated into deep learning either by learning parameters for the
embedding (Hoter et al.) 2017 Carrière et al. 2020 Kim et al. |020 /hhao & Wang 2019 Zielinski
al.
2019), or part of topological loss regulariser (Chen et al.] [2018] Gabrielsson al.
2020
Clough et al. 2020 Moor et al. 2019). However, the embeddings used in these techniques deform the metric structure of persistence diagram space (Bubenik & Wagner 2019) Wagner 2019
Carrière & Bauer 2019, potentially leading to the loss of important information. Furthermore, these techniques generally require prior knowledge of 'correct' target topology which cannot plausibly be known in most scenarios. In comparison, our algorithm acts in the space of persistence diagrams so it does not deform the structure of diagram space via embeddings, and is entirely unsupervised, requiring no prior knowledge about the topology.
Hard clustering. Maroulas et al. (017)) gave an algorithm for hard clustering persistence diagrams based on the algorithm by Turner et al. Lacombe et al. (2018) gave an alternate implementation of hard clustering based on their algorithm for barycentre computation, providing a computational speed-up up over previous the work by Maroulas et al. The primary advantages of our work over previous work on hard clustering are as follows.
The probabilistic membership values allow rank datasets in the cluster, enabling top- candidate selection in settings where verifying correctness is expensive. The value provided by this fuzzy information is demonstrated in the experiments.
(ii) The fuzzy membership values provide information about proximity to all clusters, whereas hard labelling loses most of that information. In our experiments we demonstrate that this additional information can be utilised in practice.
(iii) The weighted cost function makes the convergence analysis (which we provide) entirely non- trivial in comparison to the non-fuzzy case. We consider this convergence analysis a primary contribution of our paper.
(iv) Fuzzy membership values have been shown to be more robust to noise than discrete labels Klawonn 2004).
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(v) Unlike hard clustering, fuzzy clustering is analytically differentiable, allowing integration of the fuzzy clustering step into deep learning methods (Wilder et al. 2019
Geometric equivalents. The most similar unsupervised learning technique to our algorithm is Wasserstein Barycentre Clustering (WBC). It clusters datasets of point clouds by the Wasserstein distance between the point clouds, rather than the Wasserstein distance between their persistence diagrams, We compare our algorithm experimentally to WBC using ADMM (Ye & Li 2014).
Bregman ADMM (Ye et al., 2017), Subgradient Descent (Cuturi & Doucet, 2014), Iterative Bregman Projection (Benamou et al. 2013), and full linear programming (Li & Wang 2008), Each these algorithms computes approximates the Wasserstein barycentre in different ways. Theoretically, fuzzy discrete distribution clustering (d. A. T. de Carvalho et al. 2013) is similar to our algorithm, but the addition of the diagonal in the persistence diagram makes our work distinct.
1.2 OUR CONTRIBUTIONS
1. Our main contribution is an algorithm for Fuzzy c-Means clustering of persistence diagrams, along with the convergence analysis. Given a collection of persistence diagrams 1_,..., ........D.i Dn we alternatively calculate cluster centres Mi,...,, Ml, and membership values r,kk E [0, 1] which denote the degree to which diagram J3 is associated with cluster Mi.. We prove Theorem I showing that every convergent subsequence of these alternative update steps tends local minimum or saddle point of the cost function. This is the same convergence guarantee provided by traditional FCM clustering (Bezdek et al., 1987), but requires additional work as the space of persistence diagrams with the Wasserstein distance has far weaker theoretical properties than Euclidean space.
Updating the cluster centres requires computing the weighted Fréchet mean. We extend the algorithm given by Turner et al. (2012) to the weighted case. case. justifying our addition of weights by extending their proof to show that the updated algorithm converges.
3. We implement our algorithm in Python, available in the supplementary materials. It works with persistence diagrams from commonly used open-source libraries for Topological Data Analysis (TDA) so is available for easy integration into current workflows, offering a powerful unsupervised learning algorithm to data science practitioners using TDA.
4. We demonstrate the application of our algorithm to settings where (i) the properties of persistence diagrams makes clustering them the natural choice over geometric equivalents and (ii) the probabilistic membership values be used to rank candidates for top-k selection. Our algorithm classifies transformed lattice structures from materials science where geometric equivalents fail, whilst giving probabilistic rankings help prioritise expensive further investigation. We also cluster the persistence diagrams of decision boundaries and labelled datasets, showing that our fuzzy clustering captures information about model performance on unseen tasks.
TOPOLOGICAL PRELIMINARIES
Topological Data Analysis emerged from the study of algebraic topology, providing a toolkit to fully describe the topology of dataset. We offer quick summary below; for more comprehensive details see Edelsbrunner & Harer (2010). A set of points in Rd are indicative of the shape of the distribution they are sampled from. By connecting points that are pairwise within e > 0 distance of each other, we can create an approximation of the distribution called the Vietoris-Rips complex [Vietoris) 1927). Specifically, we add the convex hull of any collection of points that are pairwise at most apart to the c- c-Vietoris-Re Rips complex. However, choosing an remains problematic; too low value and key points can remain disconnected, too high a value and the points become fully connected. To overcome this we use persistence: consider the approximation over all values of simultancously. and study how the topology of that approximation evolves as grows large. We call the collection of complexes for all filtration.
For each e, we compute the p-homology group. This tells us the topology of the e- c-Vietoris-Ripp complex: the O-homology counts the number of connected components, the 1-homology counts the number of holes, the 2-homology counts the number voids, etc. (Edelsbrunner et al. 2000).
Dionysus and Ripser Bauer 2019
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Death
6 0
0.71
1.42
O-PH
o: 1-PH
0.71
1.42
0.71 1
1.42
Birth
(a) Vietoris-Rips filtration
(b) Persistence diagram
Figure 1: An example Vietoris-Ripe Rips filtration with its persistence diagram. We only add convex hulls when points are pairwise apart, so there is hole at e whilst the diagonals of the square are not close enough fill the hole in. This hole can be seen point 1.42) in the 1-PH persistence diagram.
The p-persistent homology (p-PH) group is created by summing the p-homology groups over all e. This results in p-PH group that summarises information about the topology of the dataset at all granularities. If a topological feature, such as connected component or hole, persists throughout large range of granularities, then it's more likely to be a feature of the distribution. If only persists for short amount of time, then it's more likely to be noise (Cohen-Steined et al.) (207)). We can stably map p-PH group into multiset in the extended plane called persistence diagram (Chazal et al.. 2012). Each topological feature has birth and death: feature is born when it enters the complex, and dies when the complex grows enough to destroy it. For example, in Figure I (a), feature born at = when four lines form a hole. This feature dies at 1.42 when the hole is filled in. This is shown in the persistence diagram in Figure 1(b) as point at (1, 1.42) in I-PH. By computing get complete picture of the
the birth/death points for each topological feature in the filtration, topology of the point cloud at all granularities (omorodiaa & Carlsson 2005). The persistence diagram is the collection of birth/death points, along with the diagonal A iaa a) a E RR) with infinite multiplicity, added in order to make the space of persistence diagrams complete (Mileyko
al. 2011
3
ALGORITHMIC DESIGN
3.1 CLUSTERING PERSISTENCE DIAGRAMS
In order cluster need distance the space of persistence diagrams. We use the 2- Wasserstein L2 metric as it is stable on persistence diagrams of finite point clouds (Chazal et al.] 2012). The Wasserstein distance is optimal transport metric that has found applications across machine learning. In the Euclidean case, it quantifies the smallest distance between optimally matched points. Given diagrams D1 ID2 the distance is
1/2
(D,,,,Da)
inf Σ || [|x-~((a -1)=-(<0113 7:D1-DD zED,
where the infimum is taken over all bijections 7 D1 D3, Note that added the diagonal with
infinite multiplicity to each diagram, these bijections exist. If an off-diagonal point is matched to the diagonal the transportation cost is simply the shortest distance the diagonal. In fact, the closer point is to the diagonal, the more likely it is to be noise (Cohen-Stenna) Steiner et al., 2007). so this ensures distance is not overly affected by noise.
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We work in the space D a (D W2 A) < H this leads geodesic space with known structure (Turner et al. 2012 Given a collection of persistence diagrams {Dy}}=1 C D1,2 a and a fixed number of clusters c, we wish to find cluster centres {M+} C D12, along with membership values ryk [0, that denote the extent to which m is associated with cluster Mg. We follow probabilisticc fuzzy clustering, so that >x "jk- = for each j.
We extend the FCM algorithm originally proposed by Bezdek 1980). Our Tyk is the same as traditional FCM clustering, adapted with the Wasserstein distance. That
Wa(MA, 1B)) WM D,)
Tyk
To update M4, we compute the weighted Fréchet mean Di of the persistence diagrams {D)}}=1 with the weights [= Specifically,
M.
arg mii
3. fe WDD, for l I....,c.
As the weighted Fréchet mean extends weighted centroids to general metric spaces, this gives fuzzy cluster centres. The computation of the weighted Fréchet mean is covered in Section 3.2. By alternatively updating (I) and (2) we get sequence of iterates. Theorem 1, proven in Appendix A. provides the same convergence guarantees as traditional FCM clustering.
Theorem 1. Every convergent subsequence of the sequence of iterates obtained by alternatively updating membership values and cluster centres with (1) and (2) tends a local minimum saddle point of the cost function J(R, = Ei=Ei= r (2,,,,,)))
Observe that we only guarantee the convergence of subsequences of iterates. This is the same as traditional FCM clustering, so follow the same approach to stopping condition and run our algorithm for a fixed number of iterations. The entire algorithm is displayed in Algorithm 1.
Algorithm FPDCluster
Input Diagrams D {D, number of clusters maximum iterations MAXITER
Output Cluster centres MI (M, M membership values R = {r3R}
D ADDDIAGONALS(I)
end for 8: end for
2: MI = INITCENTRES(D)
3: for count in 1.MAXITER do
for l: in 1..c
4: for j in 1.n do
10: M2g < WFRECHETMEAN(D, Rs)
5: for in L..c do
11: end for
W.M, Tyı + (2i=) W2(,.,))
12: end for
6:
13: return M, R
3.2 COMPUTING THE WEIGHTED FRÉCHET MEAN
Turner al. 2012 give algorithm for the computation of Fréchet means. In this section extend their algorithm and proof of convergence to the weighted case. Algorithm we add copies of the diagonal to ensure that each diagram has the same cardinality; denote this cardinality m. To compute the weighted Fréchet mean, need find M, (x) ) that minimises the Fréchet function in (2). Implicit to the Wasserstein distance is a bijection 7i ( + 2j for
To ensure that our persistence diagrams are all in this space, we map points infinity a hyperparameter that much larger than other death values in the diagram. This hyperparameter ensures that the one point at infinity will always be matched to the corresponding point at infinity when computing the Wasserstein distance between diagrams. This can be avoided entirely by computing the diagrams with extended persistence (Cohen-Steine al. 2009), which removes points infinity.
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(a) Synthetic datasets
I-PH persistence diagrams
(c) Cluster centres
Figure 2: Our algorithm successfully clustered the persistence diagrams, finding cluster centres that represent the topology of the datasets. The cluster centres have zero, one, or two significant off-diagonal points, representing zero, one, or two holes in the datasets.
each j. Supposing we know these bijections, we can rearrange the Fréchet function into the form F(M[a) ブ Jt W2 (M&, D,)² > Σy " (1 x 2000
In this form, the summand minimised for ( by the weighted Euclidean centroid of points (xr (0) y Therefore to compute the weighted Fréchet mean, we need to find the correct bijections. We start by using the Hungarian algorithm to find an optimal matching between MI, and each Dj. Given a J. for each point ( E M2,, the Hungarian algorithm will assign optimally matched point oy E Dj- Specifically, we find matched points
(0) 5 j-=
(lungarian(MI, DBj), for each j 1, 1,.-.,
Now, for each ( E M2 we need to find the weighted average of the matched points 60)"
f=1
However, some of these points could be copies of the diagonal, we need to consider three distinct cases: that each matched point is off-diagonal, that each one a copy the diagonal, or
that the points are mixture of both. We start by partitioning I,..., into the indices of the off-
diagonal points g = {i 1 * } and the indices of the diagonal points 2 = { z }
for each L..... Now, if JOD
then (i) is copy of the diagonal. If not, let E Σjegfe r "2 be the weighted mean of the off-diagonal points. If o
[Eeafe 2 [k
(0) = w. Otherwise, let WA the point on the diagonal closest to w. Then our update
( Ejeat 23,0)) (1)
+ +E 7 "A
for 1,..., m.
(4)
プラ jk
2
We alternate between (3) and (4) until the matching remains the same. Theorem 2, proving that this algorithm converges to local minimum of the Fréchet function, proven in Appendix Also in Appendix B is Algorithm 2, giving an overview of the entire computation.
Theorem 2. Given diagrams D,, membership values Tzk- and the Fréchet function F(D) =
Ej=1 " (To, 2, then M2 {u(() mi is a local minimum of F if and only if there is unique optimal pairing from Ml, to each of the MD and each o is updated via (4).

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Rerations For FCM
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Figure Heatmaps showing average number of iterations for fuzzy clustering of persistence diagrams (left) and the weighted Fréchet mean computation (right) to converge. Convergence of the FCM algorithm determined when the cost function is stable within +0.5%. Convergence experiments were carried out randomly generated persistence diagrams with three repeats.
EXPERIMENTS
4.1 SYNTHETIC DATA
Exemplar clustering. stat by demonstrating our algorithm on simple synthetic dataset designed to highlight its ability to cluster based the topology of the underlying datasets. We produce three datasets of noise, three datasets of a ring, and three datasets figure-of-eights, all shown in Figure 2(a). In Figure 2(b) show the corresponding 1-PH persistence diagrams. Note that the persistence diagrams have either zero, one, or two significant off-diagonal points, corresponding to zero, one, or two holes in the datasets. We then three clusters. Having only been given the list of diagrams, the number of clusters, and the maximum number of iterations, 2(c) shows that the cluster centres have zero, one, or two off-diagonal points: our algorithm has found cluster centres that reflect the topological features of the datasets. Because are reducing the cardinality and dimensionality of datasets by mapping into persistence diagrams, we also demonstrate a speed-up of least an order of magnitude over Wasserstein barycentre clustering methods. Details of these timing experiments are in Appendix C.1
our algorithm to cluster the nine persistence diagrams into
algorithm successfully clusters the diagrams based their topology. Figure
Empirical behaviour. Figure shows the results of experiments run to determine the empirical performance of our algorithm. We give theoretical guarantees that every convergent subsequence will tend to a local minimum, but in practice it remains important that our algorithm will converge, and within reasonable timeframe. To this end we experiments on total of 825 randomly generated persistence diagrams, recording the number iterations and cost functions for both the FCM clustering and the weighted Fréchet mean (WFM) computation. We considered the FCM to have converged when the cost function remained within +0.5%. As explained in Section 3.2, the WFM converges when the matching stays the same. Our experiments showed that the FCM clustering consistently converges within <5 iterations, regardless of the number of diagrams and points per diagram (note that the time per iteration increases as the number of points/diagrams increases, even if the number of iterations remains stable). We had no experiments in which the algorithm did not converge. The WFM computation requires more iterations as both number of diagrams and number of points per diagram increases, but we once again experienced no failures to converge in each of our experiments. In general, running the algorithm offered no difficulties on laptop, and we believe the algorithm is ready for use by practitioners in the TDA community.
Further details of all experiments available in Appendix
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(a) Face-centred cubic structure
(b) Body- cubic structure
(c) Diamond
(d) Cis- Cis-hinger polydiacctylene
Figure Cubic structures (top) and carbon allotropes (bottom). Our algorithm can cluster transformed lattice structures where comparable geometric algorithms fail.
4.2 LATTICE STRUCTURES
key property for machine learning materials science has been identified as "invariance to the basis symmetries of physics [...] rotation, reflection, translation" (Schmidt et al., 2019). Removing the need for a standardised coordinate system allows machine learning methods to be applied to broader range of existing coordinate datasets generated by experimental methods (c.g.. x-ray diffraction and scanning electron microscope imaging) and computational methods (e.g.. density functional theory). Persistence diagrams, which capture affine transformation- invarrinn properties of datasets, are ideally suited for application in this domain. Our algorithm enables successful unsupervised learning on these datasets for the first time. Additionally, the fuzzy membership values allow top-k ranking of candidates suggested by our algorithm. This is particularly important in materials science, where further investigation of materials can be extremely costly.
The large majority of solids are comprised of lattices:
regularly repeating unit cells of atoms. This lattice structure directly determines the properties material (Hull Bacon 201 I and it has been predicted that machine learning will reveal presently unknown links between structure and property by identifying new trends across materials (Meredig 2019 Wei et al. 2019). We apply our algorithm to two examples of lattice structures from materials science: cubic structures and carbon allotropes. Cubie structures are important because they are ubiquitous. The most common lattice structures, particularly amongst pure metals, are face-centred cubic (FCC) structures and body-centred cubic (BCC) structures (Putms) 1992). shown in Figures 4(a) and 4(b). Carbon allotropes, such as graphene and diamond, are widely anticipated revolutionise electronics and optoelectronics (Wang et al. 2016). We focus on the carbon allotropes diamond and cis-hinged polydiacetylene, shown in Figures 4(c) and 4(d).
We use atomic positions for the unit-cells of iron mp-150 and iron mp-13 from the Materials Project (Jain al. 2013). representing BCC and FCC structures respectively, for our first experiment. For our second experiment we use diamond and cis-hinged polydiacetylene unit-cell atomic positions from the Samara Carbon Allotrope Database (Hoffmann et al. (016)). We simulate distinct collections of lattices by transforming the atomic coordinates, with no information about bonds given to the algorithms. The properties of persistence diagrams mean that we can successfully cluster the atomic coordinates derived from the same base unit-cell regardless of the transformations applied to the coordinate system, fulfilling the key property identified above (we consider a clustering successful when all datasets from the same lattice structure have their highest membership values for the same cluster). In comparison, we run Wasserstein barycentre clustering on the same datasets using several state-of- the-art algorithms for barycentre computation and approximation. Each can only successfully cluster the cubic structures after reflection, and none of them successfully cluster the carbon allotropes after any transformations. We give specific values for these results in Appendix C.2.
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4.3 DECISION BOUNDARIES
Learnt models have been shown to perform better on datasets which have a similar persistence diagram to the model's decision boundary (Ramamurthy et al. 2019). In fact, topological complexity has been shown to correlate with generalisation ability (Guss & Salakhutdmov 2018 Gabrielsson & Carlsson 2019 Rieck et al. 2018). We utilise our algorithm to cluster the topology of models and tasks, showing that high membership values imply better performance on tasks. Specifically, given a dataset with n classes, we fix one class to define
1 tasks: binary classification of the fixed class vs each of the remaining classes. On each of these tasks, we train a model. We compute the decision boundary of the model f. defined as (I1, .......,m........... fx)) where f(z) is the model's prediction for a = ) and the decision boundary of the tasks, defined via the labelled dataset as (æI,.... Zm. y) where y is the true label. We compute the 1-persistence diagrams of the tasks' and models' decision boundaries and cluster them to obtain membership values and cluster centres. To view task and model proximity through our clustering, we find the cluster centre with the highest membership value for each task, and consider the models closest to that cluster centre. Note that in general you cannot do this with hard clustering: most of the time path will not exist from task cluster centre to model, because each task/model is only associated with a single cluster. This contrasts with fuzzy clustering, where you have information about how close each model/task is to each cluster centre. We further discuss why this does not work for hard clustering in Appendix C.3.
To assess the ability of our model/task clustering, we performed the above experiment on three different datasets: MNIST (LeCun et al.) (2010), FashionMNIST (Xiao et al. (017). and Kuzushiji- MNIST Clanuwat et al. 2018). We repeat each experiment three times. Our goal is to evaluate whether or not the clustering is capturing information about model performance on tasks, so baseline we use the average performance of all models on a fixed task, averaged over all tasks. Recall that we are just using the topolegy of the decision boundary: no information about model performance on the task was used until we assessed the ability of the clustering to capture information about the tasks. We start by verifying what happens if we use the model closest to the cluster centre associated with the task (i.e., top-1), 1). We see significant increase in performance, indicating that the topological fuzzy clustering has selected the model trained on the task: this means the clustering is working. We also compute the top-3 and top-2 performance change over average. We still see a statistically sigmificant increase in performance over average performance, indicating that our membership values are capturing information about model performance on unseen tasks. These results are shown in Table By successfully matching models to tasks using fuzzy clustering, we offer new technique for future model marketplaces, demonstrating the value of topological clustering when analysing decision boundaries.
Table I: Performance increase/decrease over average task performance when using learnt fuzzy membership values for model selection. The increase in performance demonstrates that our fuzzy clustering automatically clusters models near tasks they perform well on.
Performance change random model selection (%)
Top-3
Top-2
Top-1
MNIST
+2.1742.18
+10.81+188
+20.88
FashionMNIST
4.0000044.00
+21.94+4.73
+23.30=8.17 72
Kuzushiji
+6.61 11.78
+11.18+2.45
+21.89=5.55
CONCLUSIONS
We have developed FCM clustering on the space of persistence diagrams, adding an important class of unsupervised learning Topological Data Analysis' toolkit. Our topological fuzzy clustering paves the way for applications in materials science and pre-trained model marketplaces, where we envisage our algorithm being used as part of automated materials discovery and model selection.

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Reviews:
[{"review_id": "WEdl0dkAmP", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper aims to apply Fuzzy c-Means (FCM) clustering to persistence diagrams and prove convergent subsequence of iterates tends to a local minimum or saddle point. The motivation of the work is direct and clear. This paper addresses the problem of persistence diagram clustering via using  weighted Frechet mean . \n\nThis is a incremental work though replacing Euclidean distance in FCM by Wasserstein distance. The contributions in this work are not quite promising. For instance, the weighted Frechet mean in Sec3.2 is the well-known Wasserstein Barycenter whose behavior is well studied in optimal transport works. Thus Theorem2 can be not be considered as the contribution of this work. \n\nAnother drawback is that experiment in this work is very weak. There are only three dataset tested in this work. It's not quite convincing. It would be promising if the proposed work valid in other shape datasets such as SHREC2010 or SHREC2014. These datasets were frequently used for testing algorithms of topological data analysis .\n\nI cannot recognize the merits of this work compared with previous papers, such as: Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport.\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "CWomj3cJi_2", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "# Synopsis of the paper\n\nThis paper develops a novel algorithm for performing fuzzy clustering\n(i.e. non-hard assignment of points to cluster centres) on persistence\ndiagrams, i.e. topological data descriptors. This is a highly relevant\ncontribution because persistence diagrams 'live' in a space that makes\nmetric analyses somewhat cumbersome. By contrast, the proposed method,\neven though this is not strictly highlighted in the paper, can be used\nas a principled way to obtain 'representatives' of a data set.\n\nThe critical algorithmic insight of the paper lies in developing a new\nway to calculate Fréchet means; this makes it possible to adapt FCM to\nthis domain.\n\nA set of experiments demonstrates the utility of the proposed approach.\n\n# Summary of the review\n\nThis is a well-written paper with a highly relevant contribution for\nthe TDA community. I am excited to see such a clustering algorithm\nfinally emerge for persistence diagrams, and I envision that this paper\nwill be a very useful contribution to the field.\n\nThat being said, there are some issues in the current write-up that\nprevent me from fully endorsing this work for now, namely:\n\n1. Presentation: the paper will be somewhat confusing for non-experts in\n   TDA. While this is to be expected to some extent, there are several\n   places in which the paper could be improved to provide some more\n   intuition, making it possible that even non-experts can appreciate\n   the contribution.\n\n2. The experiments appear to be somewhat preliminary. The experiment\n   with synthetic data, for example, only comprises few samples that\n   are clustered; a large portion of this section is spent on discussing\n   an application in materials science instead (which is of course\n   important, but I feel that it is hard to both appreciate the\n   application domain and the algorithm at the same time). In addition,\n   some details about the empirical behaviour of the method are not\n   discussed.\n\nIf these two points were to be rectified in a revision of the paper,\nit would help the contribution to shine more. Please see below for more\ndetailed comments.\n\n# Detailed comments\n\n- In terms of exposition, I would suggest to cite the Vietoris--Rips\n  complex construction (and also refer to the complex by this\n  alternative name; I think VR is much more common that just 'Rips')\n\n- Would it not be easier to show a Vietoris--Rips filtration instead of\n  the Čech one? I would suggest updating Figure 1 to account for this.\n  To provide a more intuitive view to TDA, the caption of the figure\n  could also be extended to describe the creation of cycles in the\n  point cloud, for example. This is the first figure that readers will\n  see, so any updates are well worth the effort, in my opinion. \n\n- Footnote 2 needs some clarification: is the paper suggesting to\n  compute the full VR complex (until the full simplex of $n$ vertices\n  has been reached)? If not, there *will* be multiple points at\n  infinity.\n\n- Would it not be possible to sidestep the issue of infinite points\n  entirely by using, say, extended persistence? It is my understanding\n  that the method does not 'care' about the way the persistence diagrams\n  are calculated, right? Hence, there are no structural assumptions\n  being made here.\n\n- The Fréchet mean is not necessary unique. Does this pose any problems\n  for the algorithm? I would assume that the calculated clustering might\n  also not be unique (or one of multiple equivalent solutions), but\n  I lack the intuition here. This should be briefly discussed.\n\n- Before equation 2, it should be 'Fréchet mean $\\widehat{\\mathbb{D}}$',\n  i.e. the variable indicating the mean should be used here. \n\n- How is the convergence behaviour of the algorithm? How many\n  iterations does it usually take until clusters start to stabilise? \n\n- The section on 'Computing the Fréchet mean' could be improved in terms\n  of the flow. Would it be possible to provide an overview algorithm as\n  well?\n\n- In terms of the limitations of the method, it is my understanding that\n  it heavily depends on the Wasserstein distance. Is this correct? I am\n  asking because this distance is known to be computationally\n  challenging to compute, so I was wondering while reading the paper\n  whether a similar algorithm could be derived for *kernels* between\n  persistence diagrams. While the experiments discuss runtime already in\n  the supplements, I am not convinced about the overall scalability of\n  the algorithm. (this is not a fault of the paper, I merely want the\n  limitations to be discussed in more details)\n\n- As already mentioned above, I would suggest updating the discussion of\n  the lattice structure data if possible. I feel that this is an\n  interesting topic, but I would rather see more experiments in the\n  paper and a shortened description of the background (it could be put\n  into the supplemental section).\n\n- As an additional suggestion for improving the experiments, I would\n  suggest running the synthetic test data experiments with more\n  diagrams. This *can* be an excellent introductory experiment to\n  showcase the capabilities of the algorithm, but at present, it falls\n  slightly short of that.\n\n- To me, the 'decision boundaries' section could be extended. This is\n  a really exciting application; the ability to find representatives \n  of diagrams opens up all kinds of new avenues! Is it possible to link\n  this more to previous results, i.e. Ramamurthy et al.?\n\n- As for the discussion on generalisation capabilities, I would suggest\n  citing prior work (Rieck et al., 'Neural Persistence: A Complexity\n  Measure for Deep Neural Networks Using Algebraic Topology'), which\n  mentions relationships between topology-based measures and\n  generalisation capabilities.\n\nAll in all, I am convinced that this has the potential to be a strong\naddition to the TDA community!\n\n# Style & clarity\n\nThe paper is well-written; there are a few sentences that I failed to\nparse correctly, though:\n\n- 'invariance to the basis symmetries': should this be 'basic\n  symmetries' instead? Moreover, why is there an ellipsis (...) after\n  'physics'? Should this be '[...]' to indicate that parts of the\n  quotation were left out?\n\n- As a matter of personal style preference, I would prefer to say 'The\n  algorithm by Turner et al.' rather than \"Turner et al.'s algorithm\".\n  The latter strikes me as somewhat confusing.\n\n- I would suggest to use no contractions in formal writing, hence\n  \"cannot\" instead of \"can't\" etc.; this is a minor point, but it since\n  the remainder of the paper is written so neatly, I cannot help but\n  point out ways to improve it even more.\n\n- 'so does not' --> 'so it does not'\n\n# Update after rebuttal & discussions\n\nI thank the authors for their thorough rebuttal. While the technical details are acknowledged and addressed for the most part, the experimental setup could still be improved. R1 mentioned that the work by Lacombe et al. might also be applicable as a comparison partner. Investing in a more thorough scenario would strengthen the paper by a lot.\n\n# Further update after discussions\n\nThe primary subject of our discussions concerned the experimental setup. While I still see this paper favourably, it would be strengthened by a more in-depth comparison with the work by Lacombe et al. The core of the paper would be more convincing if the utility of the fuzzy clustering could be highlighted better in a set of scenarios that are more comparable with existing TDA literature.", "extra_scores": {}, "extra_text": {}}, {"review_id": "6wTpzgLuFL2", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a new clustering algorithm for persistence diagrams. They use fuzzy c means clustering. The motivation for fuzzy clustering is that it allows each datum (persistence diagram) to have weighted (soft) membership in different clusters. The partial membership value is the ratio of the distance to that cluster center to sum of all memberships to other clusters. Empirical results on synthetic and real data shows that clustering of persistence diagrams outperforms others that depends on geometry. A convergence theorem is provided, based on previous fuzzy k-means convergence proof.\n\nThis paper is well written. The clustering of persistence diagrams is an important practical problem and the proposed algorithm does seem to work.\n\nHowever, overall the benefit of the new soft-membership clustering is not clear. The authors claim that (Lacombe et al. NeurIPS 2018) is hard clustering. But indeed (Lacombe et al. 18) relaxed the diagram to a continuous function. So it is unclear why it cannot have some of the benefits of the proposed method. Also the following paper needs to be discussed.\n\nPersistence Bag-of-Words for Topological Data Analysis - IJCAI'19\n\nEmpirically, some important tables should be included in the main paper. It is difficult to find the results in the supplemental material. The comparison to hard membership is hard to find. Also maybe Lacombe et al should be compared with? \n\nFurthermore, the real world data experiments is very specially chosen to show that when geometric transformation is involved, topology-based method is better than geometry-dependent baselines. But this is kind of obvious. And there are definitely better adaptations of these baselines to be more robust to these geometric transformations (e.g., using ICP algorithm to compute distance for WBC). I suggest the authors look for better dataset (ideally datasets used by supervised methods) and show that PH based clustering algorithm is useful in these datasets.\n\nI think the bottomline to me is that the key ideas are really not quite surprising. \n\n------------\n\nI have read the authors' response. I do not think the answers addressed my concerns.\n\nOn the positive side, I do think the paper is written well and the idea is clear. It is true that a soft-clustering of PDs on the true Wasserstein distance has not been done. The paper also provided proof of the convergence of the algorithm.\n\nMy main concerns are the following:\n\nFirst and most importantly, the method is not particularly surprising. It basically combines the most classic soft-clustering algorithm and the Frechet mean computation algorithm of Turner et al. Theoretically, the proof is extending the convergence result of the soft-clustering algorithm to the Wasserstein distance of PDs. I would not say the proof is trivial. But it is not that surprising, as we already knew that the Frechet mean of PDs is computable. I would be much more excited if the theoretical result is about the optimality rather than just convergence.\n\nEmpirically, the paper did not provide comparison with (Latombe et al. 18). Just saying that they did not do it exactly in the PD space is not good enough. A lot of practically powerful methods (persistence image, various kernels, etc) are approximations/relaxations outside the PD space. These approximations/relaxations can bring computational advantage, and sometimes better learning efficiency. Therefore, we need to know how this method is compared with (Latombe et al. 18) in efficiency and clustering performance. (Latombe et al. 18) can naturally have both hard- and soft-clustering versions. A thorough comparison with the different versions can show how important it is to stick with the PD space rather than the relaxation. My guess is that in practice sticking with PD space is not that important, or maybe even worse due to bad local optima of the Frechet mean. But I would be very happy to be proven wrong.\n\nAnother issue is the limited experiments. The material data does seem to be a good fit. But the authors could use some of the classic topology-friendly data (shape, dynamic data, graph) from existing supervised methods. Any labeled binary/multiclass data can be used to evaluate clustering. An even more ambitious goal is to prove the usefulness of clustering in the supervised task. For example, the authors can show that a bag-of-words approach (using the clustering result) can improve classification performance.\n\nOverall, I feel that the methodology is not very exciting to me, and the experiments are insufficient. If the main argument is the algorithm computes on the PD space and the proof of convergence, this paper may better fit a theoretical conference.\n\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "6iofO9Cr1e", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors propose a novel algorithm for the fuzzy clustering of persistence diagrams. To determine cluster centroids, the Wasserstein-2 distance is used to minimize the weighted Fr\\’echet mean between a potential cluster center and all PDs considered for clustering. The authors proof convergence of the clustering algorithm and conduct experiments on 1) synthetic data, 2) lattice structures, and 3) decision boundaries of neural networks. In the last experiment is was shown that models whose PDs cluster close to the PDs of a given task lead to higher classification performance than random classifiers, demonstrating the merit of PDs as a useful tool for model selection. \n\nThe paper is very well written and clearly presented. Even though I did not check all proofs in the appendix, it seems like a mathematically sound approach that leads to good clustering results. The contributions and the relevance of the topic are apparent; however, I think the experimental section is weaker than it could be. While I like the experiments on the synthetic data set and the decision boundaries, I think the weakest part of the paper are the experiments on the lattice structures. Respective paragraphs are packed with technical details which are not crucial to understand the problem at hand. The results are very qualitative and outsourced to the appendix. Furthermore, Figure 3 is interesting to look at and to get a better understand of the structural differences of the data; however, it would be nice to have visual representations of the results of 4.2. For example, would it be possible to choose $c=4$ and find the four different categories that you show in Figure 3 (by selecting 10-20 structures from each category and run your algorithm on them)? You could visualize a representative of each category (like in the current figure) and their respective PDs (just plotted on top of each other for each category). In a third panel, it would be interesting to see a representation of the centroids and all PDs using Multidimensional Scaling based on the W2 distance. I would also recommend using a more quantitative way of evaluating cluster quality (e.g. [1]).\nA second weakness I see in the experiments is that in the introduction, you mention previous work on fuzzy discrete distribution clustering (by de Carvalho et al.). Still, you do not compare your results to their method (does adding the diagonal make such a big difference?).  \n\nMinor comments and questions:\n\n•\tRelated work: When you mention the correlation of topological complexity with generalization ability, you miss the work by Rieck et al. [2].\n\n•\tI am a fan of citing software libraries when they provide how to in their website. Hence I would recommend citing Ripser [3] (see https://github.com/Ripser/ripser for BibTeX entry) and potentially other Software you used in the main paper.\n\n•\tYou say “[…] the vectorization […] required by Lacombe et al.’s algorithm makes it unsuitable for integration into our work”. Could you elaborate on this? Can you not even choose this as a comparison partner?\n\n•\tWhat do you mean when you say other persistence-based learning strategies required prior knowledge of a ‘correct’ target topology which can’t plausibly be known? The work by Moor et al. for example, extracts persistence directly from the input, which is “trivial” to know since you can compute it. \n\n•\tTopological Preliminaries: Why do you introduce the \\v{C}ech complex when you use Vietoris-Rips for your experiments? \n\n•\tOne feat of your approach is that you add tuples on the diagonal to make the cardinality of all compared PDs match. I wonder how susceptible your approach is to increasingly different cardinality (that has to be filled). Maybe you can conduct a small ablation study on your synthetic data set and evaluate how “good” your mean PDs are when cardinalities are increasingly different.\n \n[1]: RJGB Campello, ER Hruschka. A fuzzy extension of the silhouette width criterion for cluster analysis. Fuzzy Sets and Systems, 2006 – Elsevier\n\n[2]: Bastian Rieck, Matteo Togninalli, Christian Bock, Michael Moor, Max Horn, Thomas Gumbsch, Karsten Borgwardt. Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology. ICLR 2019. \n\n[3]: Ulrich Bauer. Ripser: efficient computation of Vietoris-Rips persistence barcodes. Preprint.\n\n======================\nUpdate: Thank you for your rebuttal, I still think this is a promising paper and lean towards acceptance. However, after the discussion and the update of the manuscript, my score will remain the same.  \n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper is overall clearly written, and the proposed approach of performing clustering on the space of persistence diagrams can be a significant contribution.

However, during the discussion, the reviewers share the concern about insufficient empirical evaluation. In particular, datasets are limited and (Lacombe et al. 2018) is not included as a comparison partner, although the authors had a chance to include it in the author response phase. Since this point is crucial, I will reject the paper.

Addressing these points will largely improve the paper, and also reviewers put a great effort to give detailed reviews for the paper. So I hope the authors take the reviews into consideration for further revision of the paper.
Decision: Reject