Title: Node-Level Topological Representation Learning on Point Clouds
Venue: NeurIPS 2024
Accepted: no
Keywords: Topological Data Analysis

TDA

Hodge Laplacian

Higher-Order Networks

Simplicial Complexes

Algebraic Topology

Differential Geometry

Point Clouds

Persistent Homology
Abstract: Topological Data Analysis (TDA) allows us to extract powerful topological, and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description of the global structure of the point cloud. However, common machine learning applications like classification require point-level information and features to be available. In this paper, we bridge this gap and propose a novel method to extract node-level topological features from complex point clouds using discrete variants of concepts from algebraic topology and differential geometry. We verify the effectiveness of these topological point features (TOPF) on both synthetic and real-world data and study their robustness under noise.

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Abstract

Topological Data Analysis (TDA) allows us to extract powerful topological, and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description of the global structure of the point cloud. However, common machine learning applications like classification require point-level information and features to be available. In this paper, we bridge this gap and propose a novel method to extract node-level topological features from complex point clouds using discrete variants of concepts from algebraic topology and differential geometry. We verify the effectiveness of these topological point features (TOPF) on both synthetic and real-world data and study their robustness under noise.

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1 Introduction

In modern machine learning [39], objects are described by feature vectors within a high-dimensional space. However, the coordinates of a single vector can often only be understood in relation to the entire data set: if the value x is small, average, large, or even an outlier depends on the remaining data. In a 1 -dimensional (or low-dimensional) case this issue can be addressed simply by normalising the data points according to the global mean and standard deviation or similar procedures. We can interpret this as the most straight-forward way to construct local features informed by the global structure of the data set.

In the case where not all data dimensions are equally relevant, or contain correlated and redundant information, we can apply (sparse) PCA to project the data points to a lower dimensional space using information about the global structure of the point cloud [51]. For even more complex data, we may first have to learn the encoded structure itself: indeed, a typical assumption underpinning many unsupervised learning methods is the so-called 'manifold hypothesis' which posits that real world data can be described well via submanifolds of n -dimensional space [36, 21]. Using eigenvectors of some Laplacian, we can then obtain a coordinate system intrinsic to the point cloud (see e.g. [47, 4, 15]). Common to all these above examples is the goal is to construct locally interpretable point-level features that encode globally meaningful positional information robust to local perturbations of the data. However, none of these approaches is able to represent higher-order topological information, making point clouds with these kind of structure inaccessible to point-level machine learning algorithms.

Instead of focussing on the interpretation of individual points, topological data analysis (TDA), [9], follows a different approach. TDA extracts a global description of the shape of data, which is typically considered in the form of a high-dimensional point cloud. This is done measuring topological features like persistence homology, which counts the number of generalised 'holes' in the point cloud on multiple scales. Due to their flexibility and robustness these global topological features have been shown to contain relevant information in a broad range of application scenarios: In medicine, TDA has provided methods to analyse cancer progression [33]. In biology, persistent homology has been used to analyse knotted protein structures [5], and the spectrum of the Hodge Laplacian has been used for predicting protein behaviour [50].

This success of topological data analysis is a testament to the fact that relevant information is encoded in the global topological structure of point cloud data. Such higher-order topological information is however invisible to standard tools of data analysis like PCA or k -means clustering, and can also not be captured by graph models of the point cloud. We are now faced by a situation where (i) important parts of the global structure of a complex point cloud can only be described by the language of applied topology, however (ii) most standard methods to obtain positional point-level information are not sensitive to the higher-order topology of the point cloud.

Contributions We introduce TOPF (Figure 1), a novel method to compute node-level topological features relating individual points to global topological structures of point clouds. TOPF (i) outperforms other methods and embeddings for clustering downstream tasks on topologically structured data, returns (ii) provably meaningful representations , and is (iii) robust to noise . Finally, we introduce the topological clustering benchmark suite, the first benchmark for topological clustering.

Related Work The intersection of topological data analysis, topological signal processing and geometry processing has many interesting related developments in the past few years. On the side of homology and TDA, the authors in [16] and [41] use harmonic co homology representatives to reparametrise point clouds based on circular coordinates. This implicitly assumes that the underlying structure of the point cloud is amenable to such a characterization. In [2, 26], the authors develop and use harmonic persistent homology for data analysis. However, among other differences their focus is not on providing robust topological point features. [24] uses the harmonic space of the Hodge Laplacians to cluster point clouds respecting topology, but is unstable against some form of noise, has no possibility for features selection across scales and is computationally far more expensive than TOPF. For a more in-depth review of related work, see Appendix A

Organisation of the paper In Section 2, we give an overview over the main ideas and concepts 62 behind of TOPF. In Section 3, we describe how to compute TOPF. In Section 4, we give a theoretical 63

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result guaranteeing the correctness of TOPF. Finally, we will apply TOPF on synthetic and real-world data in Section 5. Furthermore, Appendix A contains a brief history of topology and a detailed discussion of related work. Appendix B contains additional theoretical considerations, Appendix C describes the novel topological clustering benchmark suite, Appendix D contains details on the implementation and the choice of hyperparameters, Appendix E gives a detailed treatment of feature selection, Appendix F discusses simplicial weights, and Appendix G discusses limitations in detail.

2 Main Ideas of TOPF

A main goal of algebraic topology is to capture the shape of spaces. Techniques from topology describe globally meaningful structures that are indifferent to local perturbations and deformations. This robustness of topological features to local perturbations is particularly useful for the analysis of large-scale noisy datasets. To apply the ideas of algebraic topology in our TOPF pipeline, we need to formalise and explain the notion of topological features . An important observation for this is that high-dimensional point clouds and data may be seen as being sampled from topological spaces - most of the time, even low-dimensional submanifolds of R n [21].

In this section we provide a broad overview over the most important concepts of topology and TDA for our context, prioritising intuition over technical formalities. The interested reader is referred to [7, 27, 49] for a complete technical account of topology and [38] for an overview over TDA.

Simplicial Complexes Spaces in topology are continuous , consist of infinitely many points, and often live in abstract space . Our input data sets however consist of finitely many points embedded in real space R n . In order to bridge this gap and open up topology to computational methods, we need a notion of discretised topological spaces consisting of finitely many base points with finite description length. A Simplicial Complex is the simplest discrete model that can still approximate any topological space occuring in practice [43]:

Definition 2.1 (Simplicial complexes) . A simplicial complex (SC) S consists of a set of vertices V and a set of finite non-empty subsets (simplices, S ) of V closed under taking non-empty subsets, such that the union over all simplices ⋃ σ ∈ S σ is V . In the following, we will often identify S with its set of simplicies S and denote by S k the set of simplices σ ∈ S with | σ | = k +1 , called k -simplices . We say that S is n -dimensional, where n is the largest k such that the set of k -simplices S k is non-empty. The k -skeleton of SC contains the simplices of dimension at most k . If the vertices V lie in real space R n , we call the convex hull in R n of a simplex σ its geometric realisation | σ | . When doing this for every simplex of S , we call this the geometric realisation of S , |S| ⊂ R n .

Concretely, we can construct an n -dimensional SC S in n + 1 steps: First, we start with a set of vertices V which we can identify with the 0 -simplices S 0 . Second, we connect certain pairs of vertices with edges, which constitute the set of 1 -simplices. We can then choose to fill in some triples of vertices which are fully connected by 1 -simplices with triangles, i.e. 2 -simplices. More generally, in the k th step, we can add a k -simplex for every set σ k of k +1 vertices such that every k -element subset σ k -1 of σ k is already a ( k -1) -simplex.

Vietoris-Rips and α -complexes We now need a way to construct a simplicial complex that approximates the topological structure inherent in our data set X ⊂ R n . Such a construction will always depend on the scale of the structures we are interested in. When looking from a very large distance, the point cloud will appear as a singular connected blob in the otherwise empty and infinite real space, on the other hand when we continue to zoom in, the point cloud will at some point appear as a collection of individual points separated by empty continuous space; all interesting information can be found in-between these two extreme scales where some vertices are joined by simplices and others are not. Instead of having to pick a single scale, the Vietoris-Rips (VR) filtration and the α -filtration take as input a point cloud and return a nested sequence of simplicial complexes indexed by a scale parameter ε approximating the topology of the data across all possible scales.

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Definition 2.2 (VR complex) . Given a finite point cloud X in a metric space ( M , d ) and a nonnegative real number ε ∈ R ≥ 0 , the associated VR complex V R ε ( X ) is given by the vertex set X and the set of simplices S = { σ ⊂ X | σ = ∅ , ∀ x, y ∈ σ : d ( x, y ) ≤ ε }

Intuitively, a VR complex with parameter ε consists of all simplices σ where all vertices x ∈ σ have a pair-wise distance of at most ε . For r ≤ r ′ , we obtain the canonical inclusions i r,r ′ ( X ): V R r ( X ) ↪ →

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V R r ′ ( X ) . The set of VR complexes on X for all possible r ∈ R ≥ 0 together with the inclusions then form the VR filtration on X . For large point clouds, using the VR complex for computations becomes expensive due to its large number of simplices. In contrast, the more sophisticated α -complex approximates the topology of a point cloud using far fewer simplices and thus we will make use of it. For a complete account and definition of α -complexes and our reason to use them, see Appendix B.

Boundary matrices So far, we have discussed a discretised version of topological spaces in the form of SCs and a way to turn point clouds into a sequence of SCs indexed by a scale parameter. However, we still need an algebraic representation of simplicial complexes that is capable of encoding the structure of the SC and enables extraction of the topological features : The boundary matrices B k associated to an SC S store all structural information of SC. The rows of B k are indexed by the k -simplices of S and the columns are indexed by the ( k +1) -simplices.

Definition 2.3 (Boundary matrices) . Let S be a simplicial complex and ⪯ a total order on its vertices V . Then, the i -th face map in dimension n f n i : S n →S n -1 is given by

with v 0 ⪯ v 1 ⪯ · · · ⪯ v n and ̂ v i denoting the omission of v i . Now, the n -th boundary operator B n : R [ S n +1 ] → R [ S n ] with R [ S n ] being the real vector space over the basis S n is given by

When lexicographically ordering the simplex basis, we can view B n as a matrix . We call R [ S n ] the space of n -chains. Now, B 0 is the vertex-edge incidence matrix of the associated graph consisting of the 0 - and 1 -simplices of S and B 1 is the edge-triangle incidence matrix of S

Betti Numbers and Persistent Homology We now turn to the notion of topological features and how to extract them. Homology is one of the main algebraic invariants to capture the shape of topological spaces and SC. From a technical point of view, the k -th homology module H k ( S ) of an SC S with boundary operators B k is defined as H k ( S ) := ker B k -1 / Im B k . The generator or representative of a homology class is an element of the kernel ker B k -1 . In dimension 1 , these are given by formal sums of 1 -simplices forming closed loops in the SC. Importantly, the rank rk H k ( S ) is called the k -th Betti number B k of S . In dimension 0 , B 0 counts the number of connected components, B 1 counts the number of loops around 'holes' of the space, B 2 counts the number of 3 -dimensional voids with 2 -dimensional boundary, and so on.

If we are now given a filtration of simplicial complexes instead of a single SC, we can track how the homology modules evolve as the simplicial complex grows. The mathematical formalisation, persistent homology , thus

turns a point cloud via a simplicial filtration into an algebraic object summarising the topological feature of the point cloud. For better computational performance, the computations are usually done in one of the small finite fields Z /p Z . Because we will later be interested in the sign of numbers to distinguish different simplex orientations, we will use Z / 3 Z -coefficients, with Z / 3 Z being the smallest field being able to distinguish 1 and -1 .

The Hodge Laplacian and the Harmonic Space In the previous part, we have introduced a language to characterise the global shape of spaces and point clouds. However, we still need to find a way to relate these global characterisations back to local properties of the point cloud. We will do so by using ideas and concepts from differential geometry and topology: The simplicial Hodge Laplacian is a discretisation of the Hodge-Laplace operator acting on differential forms of manifolds:

Definition 2.4 (Hodge Laplacian) . Given a simplicial complex S with boundary operators B k , we define the n -th Hodge Laplacian L n : R [ S n ] → R [ S n ] by setting

The Hodge Laplacian gives rise to the Hodge decomposition theorem: 161

Algorithm 1 Topological Point Features (TOPF)

Input: Point cloud X ∈ R n , maximum homology dimension d ∈ N , interpolation coeff. λ .

Compute persistent homology with generators in dimension k ≤ d .

Select set of significant features ( b i , d i , g i ) with birth, death, and generator in F 3 coordinates.

Embed g i into real space and project into harmonic subspace of SC at step t = λb i +(1 -λ ) d i .

Normalise projections to e k i and compute F i k ( x ) := avg x ∈ σ ( e k i l ( σ )) for all points x ∈ X . Output: Features of x ∈ X

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Theorem 2.5 (Hodge Decomposition [34, 46, 44]) . For an SC S with boundary matrices ( B i ) and 162 Hodge Laplacians ( L i ) , we have in every dimension k 163

This, together with the fact that the k -th harmonic space is isomorphic to the k -th real-valued 164 homology group ker L k ∼ = H k ( R ) means that we can associate a unique harmonic representative 165 to every homology class. The harmonic space encodes higher-order generalisations of smooth flow 166 around the holes of the simplicial complex. Intuitively, this means that for every abstract global 167 homology class of persistent homology from above we can now compute one unique harmonic 168 representative in ker L k that assigns every simplex a value based on how much it contributes to the 169 homology class. Thus, the Hodge Laplacian is a gateway between the global topological features 170 and the local properties of our SC. It is easy to show that the kernel of the Hodge Laplacian is the 171 intersection of the kernel of the boundary and the coboundary map ker L k = ker B n -1 ∩ ker B ⊤ n . 172 Because we have finite SCs we can identify the spaces of chains and cochains. This leads to another 173 characterisation of the harmonic space: The space of chains that are simultaneously homology and 174 cohomology representatives. 175

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3 How to Compute Topological Point Features

In this section, we will combine the ideas and insights of the previous section to give a complete account of how to compute Topological point features (TOPF). A pseudo-code version can be found in Algorithm 1 and an overview in Figure 1. We start with a finite point cloud X ⊂ R n .

Step 1: Computing the persistent homology First, we need to determine the most significant 180 persistent homology classes which determine the shape of the point cloud. By doing this, we can 181 also extract the 'interesting' scales of the data set. We will later use this to construct SCs to derive 182 local variants of the global homology features. Thus we first compute the persistent k -homology 183 modules P k including a set of homology representatives R k of X using an α -filtration for n ≤ 3 and 184 a VR filtration for n > 3 . We use Z / 3 Z coefficients to be sensitive to simplex orientations. In case we 185 have prior knowledge on the data set, we can choose a real number R ∈ R > 0 and only compute the 186 filtration and persistent homology connecting points up to a distance of at most R . In data sets like 187 protein atom coordinates, this might be useful as we have prior knowledge on what constitutes the 188

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'interesting' scale, reducing computational complexity. See Figure 3 left for a persistent homology diagram.

Step 2: Selecting the relevant topological features We now need to select the relevant homology classes which carry the most important global information . The persistent homology P k module in dimension k is given to us as a list of pairs of birth and death times ( b k i , d k i ) . We can assume these pairs are ordered in non-increasing order of the durations l k i = d k i -b k i . This list is typically very long and consists to a large part of noisy homological features which vanish right after they appear. In contrast, we are interested in connected components, loops, cavities, etc. that persist over a long time, indicating that they are important for the shape of the point cloud. Distinguishing between the relevant and the irrelevant features is in general difficult and may depend on additional insights on the domain of application. In order to provide a heuristic which does not depend on any a-priori assumptions on the number of relevant features we pick the smallest quotient q k i := l k i +1 /l k i > 0 as the point of cut-off N k := arg min i q k i . The only underlying assumption of this approach is that the band of 'relevant' features is separated from the 'noisy' homological features by a drop in persistence. If this assumption is violated, the only possible way to do meaningful feature selection depends on application-specific domain knowledge. We found that our proposed heuristics work well across a large scale of applications. See Figure 3 left and centre for an illustration and Appendix E for more technical details and ways to improve and adapt the feature selection module of TOPF. We call the chosen k -homology classes including k -homology generators in dimension f i k .

Step 3: Projecting the features into harmonic space and normalising In this step, we need to relate the global topology extracted in the previous step to the simplices which we will use to compute the local topological point feature. Every selected feature f i k of the previous step comes with a birth time b i,k and a death time d i,k . This means that the homology class f i k is present in every SC of the filtration between step ε = b i,k and ε = d i,k and we could choose any of the SCs for the next step. Picking a small ε will lead to fewer simplices in the SC and thus to a very localised harmonic representative. Picking a large ε will lead to many simplices in the SC and thus to a very smooth and 'blurry' harmonic representative with large support. Finding a middle ground between these regimes returns optimal results. For the interpolation parameter γ ∈ (0 , 1) , we will thus consider the simplicial complex S t i,k ( X ) at step t i,k := b 1 -γ i,k d γ i,k for k > 0 and at step t i,k := γd i,k for k = 0 of the simplicial filtration. At this point, the homology class f i k is still alive. We then consider the real vector space R [ S t i,k k ( X )] with formal basis consisting of the k -simplices of the SC S t i,k . From the persistent homology computation of the first step, we also obtain a generator of the feature f i k , consisting of a list Σ i k of simplices ˆ σ j ∈ S b i,k k and coefficients c j ∈ Z / 3 Z . We need to turn this formal sum of simplices with Z / 3 Z -coefficients into a vector in the real vector space R [ S t i,k k ( X )] : Let ι : Z / 3 Z be the map induced by the canonical inclusion of {-1 , 0 , 1 } ↪ → R . We can now define an indicator vector e i k ∈ R [ S t i,k k ( X )] associated to the feature f i k .

While this homology representative lives in a real vector space, it is not unique, has a small support, and can differ largely between close simplices. All of these problems can be solved by projecting the homology representative to the harmonic subspace ker L k of R [ S t i,k k ( X )] . Rather than directly projecting e i k to the harmonic subspace, we make use of the Hodge decomposition theorem (Theorem 2.5) which allows us to compute the gradient and curl projections solving computationally efficient least square problems:

and then setting ˆ e i k := e i k -e i k, grad -e i k, curl . (Cf. Figure 3 right for a visualisation.) Because homology representatives are gradient-free, we only need to consider the projection of e i k into the curl space.

Step 4: Processing and aggregation at a point level In the previous step, we have computed a set of simplex-valued harmonic representatives of homology classes. However, these simplices likely have no real-world meaning and the underlying simplicical complexes differ depending on the birth and death times of the homology classes. Hence in this step, we will collect the

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features on the point-level after performing some necessary preprocessing. Given a simplex-valued vector ˆ e i k and a hyperparameter δ , we now construct e i k : S t i,k k ( X ) → [0 , 1] by setting e i k : σ ↦→∈ {| ˆ e i k ( σ ) | / ( δ max σ ′ ∈S t i,k k ( X ) | ˆ e i k ( σ ′ ) | ) , 1 } such that ˆ e i k is normalised to [0 , 1] , the values of [0 , δ ] are mapped linearly to [0 , 1] and everything above is sent to 1 . We found empirically that a thresholding parameter of δ = 0 . 07 works best across at the range of applications considered below. However, TOPF is not sensitive to small changes to δ because entries of ˆ e i k are concentrated around 0 .

For every feature f i k in dimension k with processed simplicial feature vector e i k and simplicial complex S t i,k , we define the point-level feature map F k i : X → R mapping from the initial point cloud X to R by setting

For every point v , we can thus view the vector ( F k i ( v ): f k i ∈ F ) as a feature vector for v . We call this collection of features Topological Point Features (TOPF). (Cf. Figure 4 for an example).

Choosing Simplicial Weights By default, the simplicial complexes of α - and VR filtrations are unweighted. However, the weights determine the entries of the harmonic representatives, increasing and decreasing the influence of certain simplices and parts of the simplicial complex. We can use this observation to increase the robustness of TOPF against the influence of heterogeneous point cloud structure, which is present in virtually all real-world data sets. For a complete technical account of how and why we do this, see Appendix F.

4 Theoretical guarantees

In this section, we prove the relationship between TOPF and actual topological structure in datasets:

Theorem 4.1 (Topological Point Features of Spheres) . Let X consist of at least ( n + 2) points (denoted by S ) sampled uniformly at random from a unit n -sphere in R n +1 and an arbitrary number of points with distance of at least 2 to S . When we now consider the α -filtration on this point cloud, with probability 1 we have that (i) there exists an n -th persistent homology class generated by the 2-simplices on the convex hull hull of S , (ii) the associated unweighted harmonic homology representative takes values in { 0 , ± 1 } where the 2 -simplices on the boundary of the convex hull are assigned a value of ± 1 , and (iii) the support of the associated topological point feature (TOPF) F ∗ n is precisely S : supp( F ∗ n ) = S . (iv) The same holds true for point clouds sampled from multiple n i -spheres if the above conditions are met on each individual sphere.

We will give a proof of this theorem in Appendix B.

Remark 4.2 . In practice, datasets with topological structure consist in a majority of cases of points sampled with noise from deformed n -spheres. The theorem thus guarantees that TOPF will recover these structural information in an idealised setting. Experimental evidence suggests that this holds under the addition of noise as well which is plausible as harmonic persistent homology is robust against some noise [2].

5 Experiments

In this section, we conduct experiments on real world and synthetic data, compare the clustering results with clustering by TPCC, other classical clustering algorithms, and other point features, and demonstrate the robustness of TOPF against noise.

Topological Point Cloud Clustering Benchmark We introduce the topological clustering benchmark suite (Appendix C) and report running times and the accuracies of clustering based on TOPF and other methods and point embeddings, see Table 1. We see that TOPF outperforms all classical clustering algorithms on all but one dataset by a wide margin. We also see that TOPF closely matches the performance of the only other higher-order topological clustering algorithm, TPCC on two datasets with clear topological features, whereas TOPF outperforms TPCC on datasets with more complex structure. In addition, TOPF has a consistently lower running time with better scaling for the more

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complex datasets, while also not requiring prior knowledge on the best topological scale. As for the other point embeddings, Node2Vec is not able to capture any meaningful topological information, whereas the performance of clustering using geometric features depends on the data set.

Feature Generation In Figure 4, we show qualitatively that TOPF constructs meaningful topological features on data sets from Biology and Physics, and synthetic data, corresponding to for example rings and pockets in proteins or trajectories around different attractors in dynamical systems. (For individual heatmaps see Figure 8)

Robustness against noise We have evaluated the robustness of TOPF against Gaussian noise on the dataset introduced in [24] and compared the results against TPCC, Spectral Clustering, Graph Spectral Clustering on the graph constructed by TPCC, and against k -means in Figure 5 Left . We have also analysed the robustness of TOPF against the addition of outliers in Figure 5 Right . We see that TOPF performs well in both cases, underlining our claim of robustness.

6 Discussion

Limitations TOPF can -by design - only produce meaningful output on point clouds with a topological structure quantifiable by persistent homology. In practice it is thus desirable to combine TOPF with some geometric or other point-level feature extractor. As TOPF relies on the computation of persistent homology, its runtime increases on very large point clouds, especially in higher dimensions where α -filtrations are computationally infeasible. However, subsampling, either randomly or using landmarks, usually preserves relevant topological features while improving run time [41]. Finally, selection of the relevant features is a very hard problem. While our proposed heuristics work well across a variety of domains and application scenarios, only domain- and problem-specific knowledge makes correct feature selection feasible.

Future Work The integration of higher-order TOPF features into ML pipelines that require pointlevel features potentially leads to many new interesting insights across the domains of biology, drug design, graph learning and computer vision. Furthermore, efficient computation of simplicial weights leading to the provably most faithful topological point features is an exciting open problem.

Conclusion We introduced point-level features TOPF founded on algebraic topology relating global structural features to local information. We gave theoretical guarantees for the correctness of their construction and evaluated them quantitatively and qualitatively on synthetic and real-world data sets. Finally, we introduced the novel topological clustering benchmark suite and showed that clustering using TOPF outperforms other available clustering methods and features extractors.

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A Extended Background

A brief history of topology and machine learning Algebraic topology is a discipline of Mathematics dating back roughly to the late 19 th century [42]. Starting with Henri Poincaré and continuing in the early 20 th century, the mathematical community became interested in developing a framework to capture the global shapes of manifolds and topological spaces in concise algebraic terms. This development was partly made possible by the push towards a formalisation of mathematics and analysis, in particular, which took place inside the mathematical community in the 1800's and early 1900's (e.g. [17, 30, 28]). The axiomatisation of analysis in the early 20 th century is an important result of this process. These abstract ideas made it possible for Topologists to talk about the now common notions of Euler characteristics, Betti number, simplicial homology of manifolds, topological spaces, and simplicial and CW complexes. Over the course of the last 100 years, branching into many sub-areas like low-dimensional topology, differential topology, K-theory or homotopy theory [1, 31], algebraic topology has resolved many of the important questions and provides a comprehensive tool-box for the study of topological spaces. These achievements were tied to an abstraction and generalisation of concepts: topological spaces turned into spectra, diffeomorphism to homotopy equvialences and later weak equivalences, and Topologists turned to category theory [20], model categories [6] and recently ∞ -categories [35] as the language of choice.

The 21 st century saw the advent and rise of topological data analysis (TDA, [8, 12]). In short, mathematicians realised that the same notions of shape and topology that their predecessors carefully defined a century earlier were now characterising the difference between healthy and unhealthy tissue, between normal and abnormal behaviour protein behaviour, or more general between different categories in their complex data sets.

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Related Work The intersection of topological data analysis, topological signal processing and geometry processing has many interesting related developments in the past few years. On the side of homology and TDA, the authors in [16] and [41] use harmonic co homology representatives to reparametrise point clouds based on circular coordinates. This implicitly assumes that the underlying structure of the point cloud is amenable to such a characterization. Although circular coordinates are orthogonal to the core goal of TOPF, the approaches share many key ideas and insights. In [2, 26], the authors develop and use harmonic persistent homology and provide a way to pool features to the point-level. However, their focus is not on providing robust topological point features and their approach includes no tunable homology feature selection across dimensions, no support for weighted simplicial complexes, and they only construct the simplicial complex at birth. In their paper on topological mode analysis, [11] use persistent homology to cluster point clouds. However, they only consider 0 -dimensional homology to base the clustering on densities and there is no clear way to generalise this to higher dimensions.

Onthe more geometric-centred side, [19] already provide a notion of harmonic clustering on simplices, [13, 14] analyse the notion of geometry and topology encoded in the Hodge Laplacian and its relation to homology decompositions, [45] study the normalised and weighted Hodge Laplacian in the context of random walks, and [24] use the harmonic space of the Hodge Laplacians to cluster point clouds respecting topology. Finally, a persistent variant of the Hodge Laplacian is used to study filtrations of simplicial complexes [37].

In [24], the authors have introduced TPCC, the first method to cluster a point cloud based on the higher-order topological features encoded in the data set. However, TPCC is (i) computationally expensive due to extensive eigenvector computations, (ii) depending on high-dimensional subspace clustering algorithms, which are prone to instabilities and errors, (iii) sensitive to the correct choice of hyperparameters, (iv) requiring the topological true features and noise to occur in different steps of the simplicial filtration, and it (v) solely focussed on clustering the points rather than extracting relevant node-level features. This paper solves all the above by completely revamping the TPCC pipeline, introducing several new ideas from applied algebraic topology and differential geometry. The core insight is: When you have the time to compute persistent homology with generators on a data set, you get the topological node features with similar computational effort.

B Theoretical Considerations

More details on VR and α -filtrations Vietoris-Rips complexes are easy to define, approximate the topological properties of a point cloud across all scales and computationally easy to implement. However for moderately large r , the associated VR complex contains a large number of simplices up to ( | X | n ) n -simplices for large enough r -leading to poor computational performance for any downstream task on some large point clouds. One way to see this is the following: After adding the first edge that connects two components or the final simplex that fills a hole in the simplicial complex the VR complex keeps adding more and more simplices in the same area that keep the topology unchanged. One way to mitigate this problem is to pre-compute a set of simplices that are able to express the entire topology of the point cloud. For a point cloud X ⊂ R n , the α -filtration consists of the intersection of the simplicial complexes of the VR filtration on X with the (higher-dimensional) Delaunay triangulation of X in R . Due to algorithmic reasons, the filtration value of a simplex is then the radius of the circumscribed sphere instead of the maximum pair-wise distance of vertices. This reduces the number of required simplices across all dimensions to O ( | X | ⌈ n/ 2 ⌉ ) . However, the Delaunay triangulation becomes computationally infeasible for larger n .

Definition B.1 ( n -dimensional Delaunay triangulation) . Given a set of vertices V ∈ R n , a Delaunay triangulation DT ( V ) is a triangulation of V such that for any n -simplex σ n ∈ DT ( V ) the interior of the circum-hypersphere of σ n contains no point of DT ( V ) . A triangulation of V is a SC S with vertex set V such that its geometric realisation covers the convex hull of V hull( V ) = |S| and we have for any two simplices σ , σ ′ that the intersection of geometric realisations | σ | ∩ | σ ′ | is either empty or the geometric realisation | ˆ σ | of a common sub-simplex ˆ σ ⊂ σ, σ ′ .

If V is in general position, the Delaunay triangulation is unique and guaranteed to exist [18].

Definition B.2 ( α -complex of a point cloud) . Given a finite point cloud X in real space R n , the α -complex α ε ( X ) is the subset of the n -dimensional Delaunay triangulation DT ( X ) consisting of all σ ∈ DT ( X ) with a radius r of its circumscribed sphere with r ≤ ε .

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Proof of the main theorem We will now give the proof of the theorem that guarantees that TOPF works. First, let us recall Theorem 4.1:

Theorem 4.1 (Topological Point Features of Spheres) . Let X consist of at least ( n + 2) points (denoted by S ) sampled uniformly at random from a unit n -sphere in R n +1 and an arbitrary number of points with distance of at least 2 to S . When we now consider the α -filtration on this point cloud, with probability 1 we have that (i) there exists an n -th persistent homology class generated by the 2-simplices on the convex hull hull of S , (ii) the associated unweighted harmonic homology representative takes values in { 0 , ± 1 } where the 2 -simplices on the boundary of the convex hull are assigned a value of ± 1 , and (iii) the support of the associated topological point feature (TOPF) F ∗ n is precisely S : supp( F ∗ n ) = S . (iv) The same holds true for point clouds sampled from multiple n i -spheres if the above conditions are met on each individual sphere.

Proof. Assume that we are in the scenario of the theorem. Now because the n -volume of ( n -1) -submanifolds is zero, we have that with probability 1 the points of S don't lie on a single ( n -1) sphere inside the n -sphere. Let us now look at the α -filtration of the simplices in S : Recall that the filtration values of a k -simplex is given by the radius of the ( k -1) -sphere determined by its vertices. Because all of the ( n +1) -simplices σ n +1 with vertices V ⊂ S in S lie on the same unit n -sphere S n , they all share the filtration value of α ( σ n +1 ) = 1 . By the same argument as above, with probability 1 there are no ( n +1) points in S that lie on an unit ( n -1) -sphere. Thus all of the n -simplices σ n lie on ( n -1) -spheres S n with a radius r < 1 smaller than 1 and hence have a filtration value α ( σ n ) smaller than 1 . Let

be the maximum filtration value of an n -simplex on the boundary of the convex hull of S . Then, then a linear combination g of the n -simplices of the boundary of the convex hull of S with coefficients in ± 1 is a generator of a persistent homology class with life time ( b, 1) (this follows from the fact that n -spheres and their triangulations are orientable). This proves claim (i) .

Because of the assumption that all points not contained in S have a distance of at least 2 to the points in S , all ( n +1) -simplices σ n +1 with vertices both in S and its complement in X will have a filtration value α ( σ n +1 ) ≥ 1 of at least 1 . Recall that all ( n +1) -simplices σ n +1 ⊂ S with vertices inside S have a filtration value of α ( σ n +1 ) = 1 . Thus the adjoint of the n -th boundary operator B ⊤ n is trivial on the homology generator g . Thus, we have that for the n -th Hodge Laplacian

and hence g is a harmonic generator for the entire filtration range of ( b, 1) , which proves claim (ii) . Claim (iii) and (iv) then follow from the construction of the TOPF values. □

C Topological Clustering Benchmark Suite

We introduce seven point clouds for topological point cloud clustering in the topological clustering benchmark suite (TCBS). The ground truth and the point clouds are depicted in Figure 6. The point clouds represent a mix between 0 -, 1 - and 2 -dimensional topological structures in noiseless and noisy settings in ambient 2 -dimensional and 3 -dimensional space. The results of clustering according to TOPF can be found in Figure 7.

D Implementation

We will release an implementation of TOPF and the code and data required to reproduce the experimental results of this paper under https://anonymous.4open.science/r/topf_ submission-5C40/ . In particular, we will release the topological clustering benchmark suite.

All experiments were run on a Apple M1 Pro chipset with 10 cores and 32 GB memory. TOPF and the experiments are implemented in Python and Julia. For persistent homology computations, we used GUDHI [48] (© The GUDHI developers, MIT license) and Ripserer [52] (© mtsch, MIT license), which is a modified Julia implementation of [3]. For the least square problems, we used the LSMR implementation of SciPy [22]. We used the Node2Vec python implementation https: //github.com/eliorc/node2vec (© Elior Cohen, MIT License) based on the Node2Vec Paper [25]. We used the pgeof Python package for computation of geometric features https://github.

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com/drprojects/point_geometric_features (© Damien Robert, Loic Landrieu, Romain Janvier, MIT license). We use parts of the implementation of TPCC https://git.rwth-aachen. de/netsci/publication-2023-topological-point-cloud-clustering (© Computational Network Science Group, RWTH Aachen University, MIT license).

D.1 Hyperparameters

All the relevant hyperparameters are already mentioned in their respective sections. However, for convenience we gather and briefly discuss them in this section. We note that TOPF is robust and applicable in most scenarios when using the default parameters without tuning hyperparameters. The hyperparameters should more be thought of as an additional way where detailed domain-knowledge can enter the TOPF pipeline.

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Maximum Homology Dimension d The maximum homology dimension determines the dimensions of persistent homology the algorithm computes.

For the choice of the maximum homology degree d to be considered there are mainly three heuristics which we will list in decreasing importance (Cf. [24]):

In applications, we usually know which kind of topological features we are interested in, which will then determine d . This means that 1 -dimensional homology and d = 1 suffices when we are looking at loops of protein chains. On the other hand, if we are working with voids and cavities in 3d histological data, we need d = 2 and thus compute 2 -dimensional homology.

Algebraic topology tells us that there are no closed n -dimensional submanifolds of R n . Hence their top-homology will always vanish and all interesting homological activity will appear for d < n .

In the vast majority of cases, the choice will be between d = 1 or d = 2 because empirically there are virtually no higher-dimensional topological features in practice.

In our quantitative experiments, we have always chosen d = n -1 .

Thresholding parameter δ In step 4 of the algorithm, we normalise and threshold the harmonic representatives. After normalising, the entries of the vectors lie in the interval of [0 , 1] . The thresholding parameter δ now essentially determines an interval of [0 , δ ] which we will linearly map to [0 , 1] , while mapping all entries above δ to 1 as well. This is necessary as most of the entries in the vector e i k are very close to 0 with a very small number of entries being close to 1 . Without this thresholding, TOPF would now be almost entirely determined by these few large values. Thus this step limits the maximum possible influence of a single entry. However, because most of the entries of e i k are concentrated around 0 , small changes in δ will not have a large effect and we chose δ = 0 . 07 in all our experiments.

Interpolation coefficient λ The interpolation coefficient λ ∈ [0 , 1) determines whether we build our simplicial complexes close to the birth or the death of the relevant homological features at time t = b 1 -λ d . This then in turns controls how localised or smooth the harmonic representative will be. In general, the noisier the ground data is the higher we should choose λ . However, TOPFis not sensitive to small changes in λ . We have picked λ = 0 . 3 for all the quantitative experiments, which empirically represents a good choice for a broad range of applications.

Feature selection factor β Increasing β leads to TOPF preferring to pick a larger number of relevant topological features. Without specific domain-knowledge, β = 0 represents a good choice.

Feature selection quotients max_total_quot , min_rel_quot , and min_0_ratio These are technical hyperparameters controlling the feature selection module of TOPF. For a technical account of them, see Appendix E. In most of the cases without domain knowledge, they do not have an effect on the performance of TOPF and should be kept at their default values.

Simplicial Complex Weights Although the simplicial weights are not technically a hyperparameter, there are many potential ways to weigh the considers SCs that can highlight or suppress different topological and geometric properties. In all our experiments, we use w ∆ weights discussed in Appendix F.

E How to pick the most relevant topological features

Simplified heuristic The persistent homology P k module in dimension k is given to us as a list of pairs of birth and death times ( b k i , d k i ) . We can assume these pairs are ordered in non-increasing order of the durations l k i = d k i -b k i . This list is typically very long and consists to a large part of noisy homological features which vanish right after they appear. In contrast, we are interested in connected components, loops, cavities, etc. that persist over a long time, indicating that they are important for the shape of the point cloud. Distinguishing between the relevant and the irrelevant features is in general difficult and may depend on additional insights on the domain of application. In order to provide a heuristic which does not depend on any a-priori assumptions on the number of relevant

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features we pick the smallest quotient q k i := l k i +1 /l k i > 0 as the point of cut-off N k := arg min i q k i . The only underlying assumption of this approach is that the band of 'relevant' features is separated from the 'noisy' homological features by a drop in persistence.

Advanced Heuristic However, certain applications have a single very prominent feature, followed by a range of still relevant features with significantly smaller life times, that are then followed by the noisy features after another drop-off. This then could potentially lead the heuristic to find the wrong drop-off. We propose to mitigate this issue by introducing a hyperparameter β ∈ R > 0 . We then define the i -th importance-drop-off quotient q k i by

The basic idea is now to consider the most significant N k homology classes in dimension k when setting N k to be

Increasing β leads the heuristic to prefer selections with more features than with fewer features. Empirically, we still found β = 0 to work well in a broad range of application scenarios and used it throughout all experiments. There are only a few cases where domain-specific knowledge could suggest picking a larger β .

To catch edge cases with multiple steep drops or a continuous transition between real features and 626 noise, we introduce two more checks: We allow a minimal q k i of min _ rel _ quot = 0 . 1 and a 627 maximal quotient q h 1 / q k i of max _ total _ quot = 10 between any homology dimensions. Because 628 features in 0 -dimensional homology are often more noisy than features in higher dimensions, we add 629 a minimum zero-dimensional homology ratio of min _ 0 _ ratio = 5 , i.e. every chosen 0 -dimensional 630 feature needs to be at least min _ 0 _ ratio more persistent then the minimum persistence of the 631 higher-dimensional features. Because these hyperparameters only deal with the edge cases of 632 feature selection, TOPF is not very sensitive to them. For all our experiments, we used the above 633 hyperparameters. We advise to change them only in cases where one has in-depth domain knowledge 634 about the nature of relevant topological features. 635

VR complex alpha complex

Point cloud

Unweight. harm. rep.

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1/Num tris

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F Simplicial Weights 636

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In an ideal world, the harmonic eigenvectors in dimension k would be vectors assigning ± 1 to all k -simplices contributing to k -dimensional homological feature, a 0 to all k -simplices not contributing or orthogonal to the feature, and a value in ( -1 , 1) for all simplices based on the alignment of the simplex with the boundary of the void. However, this is not the case: In dimension 1 , we can for example imagine a total flow of 1 circling around the hole. This flow is then split up between all parallel edges which means two things: I Edges where the loop has a larger diameter have smaller harmonic values than edges in thin areas and II in VR complexes, which are the most frequently used simplicial complexes in TDA, edges in areas with a high point density have smaller harmonic values than edges in low-density areas. Point II is another advantage of α -complexes: The expected number of simplices per point does not scale with the point density in the same way as it does in the VR complex, because only the simplices of the Delaunay triangulation can appear in the complex.

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We address this problem by weighing the k -simplices of the simplicial complex. The idea behind this is to weigh the simplicial complex in such a way that it increases and decreases the harmonic values of some simplices in an effort to make the harmonic eigenvectors more homogeneous. For weights w ∈ R S k , W = diag( w ) , the symmetric weighted Hodge Laplacian [45] takes the form of

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Because we want the homology representative to lie in the weighted gradient space, we have to scale its entries with the weight and set e i k,w := W -1 / 2 e i k . With this, we have that

We propose two options to weigh the simplicial complex. The first option is to weigh a k -simplex by the square of the number of k +1 -simplices the simplex is contained in:

where the +1 is to enforce good behaviour at simplices that are not contained in any higher-order simplices. One of the advantages of the α -complex is that we don't have large concentrations of simplices in well-connected areas. The proposed weighting w ∆ is computationally straightforward, as it can be obtained as the column sums of the absolute value of the boundary matrix |B k | . The weights also deal with the previously mentioned problem II : As the homology representative is scaled inversely to the weight vector w , the simplices in high-density regions will be assigned a low weight and thus their weighted homology representative will have a larger entry. By the projection to the orthogonal complement of the curl space, this large entry is then diffused among the high-density region of the SC with many simplices, whereas the lower entries of the simplices in low-density regions are only diffused among fewer adjacent simplices.

However, the first weight is not able to incorporate the number of parallel simplices into the weighting. This is why we propose a second simplicial weight function based on generalised effective resistance.

Definition F.1 (Effective Hodge resistance weights) . For a simplicial complex S with boundary matrices ( B k ) , we define the effective Hodge resistance weights w R on k -simplices to be:

where diag( -) denotes the vector of diagonal entries and ( -) + denotes taking the Moore-Penrose inverse.

Intuitively for k = 1 , we can assume that every edge has a resistance of 1 and then the effective resistance coincides with the notion from Physics. Thus simplices with many parallel simplices are assigned a small effective resistance, whereas simplices with few parallel simplices are assigned an effective resistance close to 1 . However, computing the Moore-Penrose inverse is computationally expensive and only feasible for small simplicial complexes.

In Figure 9, we show that the weights w ∆ are a good approximation of the effective resistance in terms of the resulting harmonic representative. The standard form of TOPF used in all experiments uses w ∆ -weights.

G Limitations

Topological features are not everywhere The proposed topological point features take relevant persistent homology generators and turn these into point-level features. As such, applying TOPF only produces meaningful results on point clouds that have a topological structure. On these point clouds, TOPF can extract structural information unobtainable by non-topological methods. Although TDA has been successful in a wide range of applications, a large number of data sets does not possess a meaningful topological structure. Applying TOPF in these cases will produce no additional information. Other data sets require pre-processing before containing topological features. In Figure 4 left , the 2 d topological features characterising protein pockets of Cys123 only appear after artificially adding points sampled on the convex hull of the point cloud (Cf [40]).

Computing persistent homology can be computationally expensive As TOPF relies on the computation of persistent homology including homology generators, its runtime increases on very large point clouds. This is especially true when using VR instead of α -filtrations, which become computationally infeasible for higher-dimensional point clouds. Persistent homology computations for dimensions above 2 are only feasible for very small point clouds. Because virtually all discovered relevant homological features in applications appear in dimension 0 , 1 , or 2 , this does not present a large problem. Despite these computational challenges, subsampling, either randomly or using landmarks, usually preserves relevant topological features and thus extends the applicability of TDA in general and TOPF even to very large point clouds.

Automatic feature selection is difficult without domain knowledge While the proposed heuristics 699 works well across a variety of domains and application scenarios, only domain- and problem-specific 700 knowledge makes truthful feature selection feasible. 701

Experimental Evaluation There are no benchmark sets for topological point features in the 702 literature, which makes benchmarking TOPF not straightforward. On the level of clustering, we 703 introduced the topological clustering benchmark suite to make quantitative comparisons of TOPF 704 possible, and benchmarked TOPF on some of the point clouds of [24]. On both the level of point 705 features and real-world data sets, it is however hard to establish what a ground truth of topological 706 features would mean. Instead we chose to qualitatively report the results of TOPF on proteins and 707 real-world data, see Figure 4. 708

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1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The claims about TOPF are supported by the theoretical background in Section 2 and Section 3, quantitatively and qualitatively validated and benchmarked in Section 5. Furthermore, a theoretical guarantee can be found in Section 4.

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It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We believe that being open about limitations is crucial for the practice of doing good Science. We briefly discuss the main limitations in ?? , and talk in detail about limitations in Appendix G. Finally, we are open about limitations when talking about the theoretical background and the algorithm in Section 2 and Section 3 and the remark in Section 4.

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Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

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The answer NA means that the paper does not include theoretical results.

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Answer: [Yes]

Justification: We list all the steps necessary to reproduce TOPF in Section 3, Appendix E, Appendix F and talk in detail about the hyperparameter choices in Appendix D.1. Furthermore, we will both release the Topological Clustering Benchmark Suite and the code necessary to reproduce all experiments in this paper.

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Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nLarge scale topological descriptors of data are leveraged to compute point/node-level descriptors, which encode to which large scale topological feature each point belongs to. For this, a combination of applied algebraic topology and applied harmonic analysis is used. More specifically, large scale homological features are computed using persistent homology, then represented with harmonic cocyles, and then averaged locally to obtain a point-level descriptors. The problem of topological clustering (already introduced in the literature) is addressed, whose objective is to determine to which large scale topological feature a certain data point belongs to. A set of benchmark datasets are introduced for topological clustering. The pipeline is applied to these datasets as well as real world datasets.\n\nSTRENGTHS:\n- Large scale topology of data is leveraged to assign point/node-level features to data. This gives concrete meaning to what it means for a data point to belong to a certain large scale topological feature.\n- The method is based on well-established mathematical concepts.\n- The concept of topological clustering is interesting and has potential.\n- A suite of synthetic datasets is introduced.\n\nWEAKNESSES:\nRegarding unjustified claims:\n\n- Existing approaches are undersold. Specifically, in the introduction it is said that \"none of these approaches is able to represent higher-order topological information\" and that \"such higher-order topological information is however invisible to standard tools of data analysis like PCA of k-means clustering\". However, cluster structure is topological structure. Does \"higher order\" mean homology in dimensions 1 and above?\n- Remark 4.2 says that \"datasets with topological structure consist in a majority of cases of points sampled with noise from deformed n-spheres\". This seems like a really strong claim. Is there any evidence of this?\n\nRegarding theory:\n\n- Theorem 4.1 applies in a very restricted scenario. Moreover, I do not understand why the harmonic representative takes values in {0, -1, 1}. This seems very surprising since harmonic cycles/cocycles almost always take fractional values (in order to minimize energy). I did not understand the proof of this fact; specifically, why $g$ being a harmonic generator for the entire filtration range of $(b,1)$ implies this claim.\n\nRegarding the methodology:\n\n- The method, specifically line 225, seems to assume that a cycle with coefficients in $Z/3Z$ will also be a cycle when interpreting those coefficients (0,-1, or 1) as real numbers. However, this need not be the case. To see this it suffices to consider a simplicial complex given by a triangle with no interior. Thus, step 3 of Algorithm 1 (and the method more generally) seems to be heuristic.\n- The setup up Table 1 is unclear to me. How can one compare TOPF, which produces feature vectors, with, say, DBSCAN, which produces a clustering?\n- Figure 4 is hard to interpret. For example, how should one assess the effectiveness of the algorithm in Fig 4(a)?\n- The methodology has many hyperparameters. Some choices, like delta=0.07 in line 241, seem arbitrary.\n\nQUESTIONS:\n1. Does your method have an interpretation in the case of a Riemannian manifold? That is, suppose that I take a Riemannian manifold and a harmonic (smooth) cocycle. Is there a corresponding point-level feature function? Does it have an interpretation? Perhaps it is related to the pointwise norm of the harmonic cocycle?\n2. How were the parameters of the other algorithms in table 1 selected?\n3. Which further applications do you have in mind? Figure 4 is interesting, but does not hint at real life applications.\n\nLIMITATIONS:\n- The experimental evaluation is limited.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces TOPF, a topological feature extraction mechanism on point cloud data. The authors consider Vietoris-Rips/$\\alpha$ filtrations over point clouds and compute the persistent homology. They propose a heuristic to select the “top” features from the barcodes. They consider the corresponding representatives for these features and project them onto the harmonic space of the simplices. These projected vectors are then normalized and used to construct a point-level feature vector. The authors use this framework for clustering. Towards this end, the authors introduce a topological point cloud clustering benchmark and report the experimental results on this benchmark.\n\nSTRENGTHS:\nThe authors propose to use Hodge Laplacian and Hodge decomposition to compute feature vectors over points in point-cloud data, which is a novel idea.\n\nWEAKNESSES:\n1. I do not fully understand the “learning” the representation here, because the representation is not particularly being learnt. It is being computed by using the persistent homology of the point cloud. \n\n2. Experimental evaluation is limited to clustering. And even in clustering, it is primarily limited to shapes which are partially/fully topologically spherical.  \n\n3. The robustness of the approach is due to the robustness of harmonic persistent homology known in the literature. \n\n4. The paper uses well-known notions in the TDA literature in the context of point-clouds, which amounts to an incremental progress in this direction. \n\n5. It would strengthen the paper if the authors include a small paragraph explaining why projecting onto the harmonic subspace solves the problems that exist in using the homology representatives directly. \n\nMinor: \n\nPage 2, Line 81: ‘Spaces in topology are “continuous”’. Continuity is a notion defined for functions on topological spaces and not for topological spaces themselves. Spaces are connected.\n\nQUESTIONS:\nCan this approach be used for other point-cloud related machine learning tasks?\n\nLIMITATIONS:\nYes, the authors have discussed limitations.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-12"}

comments: SUMMARY:
The paper introduces an approach to select and compute some point-level topological features for point cloud or general data set analysis.
The main ideas is to define a multi-scale simplicial complex representation, thus we can track how the homology modules change along the filtration and then select the homologies that persist for a long range of scales.

STRENGTHS:
- Topological features are usually not localized, the idea of being able to bring back the topological descriptor to the relevant points is quite novel and impactful.
- The approach is theoretically sound and well analyzed.
-The experimental evaluation is limited but convincing.

WEAKNESSES:
- the feature selection is very heuristic.
- The evaluation is only on point cloud clustering. Since we are evaluating effectiveness and robustness of localized features, feature/point correspondence problems would have been interesting.

QUESTIONS:
Is there scope to learn how to select the topological features

LIMITATIONS:
The main limitation, i.e. the selection of the features, has been briefly addressed.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a novel method for extracting per point topological features - TOPF. The method builds on previous results in topological data analysis which described a shape or a point cloud with a single global feature, by generating per-point topologically-aware features. The paper presents a quantitative evaluation and comparison of the proposed method with prior art on a new benchmark consisting of several synthetic examples, evaluates the robustness of the proposed method under noise, as well as presents qualitative examples of its performance on synthetic and real work data.\n\nSTRENGTHS:\n* The paper is well written and easy to follow. Prior art and the proposed algorithm description is detailed and comprehensive.\n* To my understanding, the paper describes a novel method for per-point feature extraction based on topological information contained in a point cloud, and describes theoretical guarantees for its correctness on point clouds sampled from multiple n-spheres.\n* The paper describes a new topological point clustering benchmark dataset consisting of seven synthetic point clouds with up to 5 labels, and evaluate the proposed and existing methods on this dataset showing that the proposed method outperforms existing methods in most cases.\n\nWEAKNESSES:\n* The paper lists common machine learning applications requiring point level features as a motivation for the proposed method. However, only quantitative experiments for point cloud clustering on a set of synthetic examples, and anecdotal evidence of performance on real world data, were presented. In order to fully understand the potential of the proposed approach to be applied beyond synthetic data, it would be beneficial to include additional evaluation, qualitative and quantitative, on real-world data and additional applications, e.g. as described in lines 304-307.\n* Specifically, it would be interesting to see experiments on non-synthetic datasets with topological structure mentioned in line 266.\n* Additionally, comparison with other well performing modern machine learning methods, such as graph neural networks for point cloud clustering, needs to be discussed, for completeness.\n\nQUESTIONS:\n* What is the runtime of the proposed method? How does it change with the point cloud size and does it have limitations on the size of point cloud that can be processed with it?\n\nLIMITATIONS:\nThe authors adequately addressed the limitations and impact of the proposed approach.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-28"}