Point-Level Topological Representation Learning on Point Clouds
Authors: Vincent Peter Grande
Michael T Schaub
Venue: ICML 2025 poster (2025)
Keywords: Topological Data Analysis
Differential Geometry
Representation Learning
Hodge Laplacian
Simplicial Complex
Persistent Homology
Point Clouds
Primary area: general_machine_learning->unsupervised_and_semisupervised_learning
TL;DR: We use topological data analysis to relate global topology to local point features on point clouds.

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 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.
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 and heterogeneous sampling.

Point-Level Topological Representation Learning on Point Clouds
Vincent P. Grande
Michael T. Schaub
Abstract
typical assumption underpinning many unsupervised learn- ing methods is the so-called "manifold hypothesis" which posits that real world data can be described well via subman- ifolds of n-dimensional space (Ma & Fu, 2012; Fefferman et al., 2016). Using eigenvectors of some Laplacian, we can then obtain a coordinate system intrinsic to the point cloud (see e.g. (Shi & Malik, 2000; Belkin & Niyogi, 2003; Coif- man & Lafon, 2006)). Common to all these above examples is the goal to construct locally interpretable point-level features that encode globally meaningful positional infor- mation robust local perturbations of the data.
Topological Data Analysis (TDA) allows us to ex- tract powerful topological and higher-order infor- mation on the global shape of a data set or point cloud. Tools like Persistent Homology give a sin- gle complex description of the global structure of the point cloud. However, common machine learning applications like classification require point-level information and features. In this paper, we bridge this gap and propose a novel method to extract point-level topological features from complex point clouds using discrete variants of concepts from algebraic topology and differential geometry. We verify the effectiveness of these topologiaal point features (TOPF) on both syn- thetic and real-world data and study their robust- ness under noise and heterogeneous sampling.
Instead of focussing on the interpretation of individ- ual points, topological data analysis (TDA), (Carlsson & Vejdemoo-ohannsoo, 2021), follows a different approach. TDA extracts a global description of the shape of data, which is typically considered in the form of a nigh-dimensional point cloud. This is done measuring topological features like persistent homology, which counts the number of gen- eralised "holes" 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 ap- plication scenarios: In medicine, TDA has provided methods to analyse cancer progression (Lawson et al., 2019). In biol- ogy, persistent homology has been used to analyse knotted protein structures (Benjamin et al., 2023), and the spectrum of the Hodge Laplacian has been used for predicting protein behaviour (Wee et al., 2024).
1. Introduction
In modern machine learning (Murphy, 2022), objects are described by feature vectors. However, the coordinates of 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 low-dimensional case this issue can be addressed simply by normalising the data points according to the global mean and standard deviation similar procedures. We can interpret this as the most ttraihht-forard way to construct local features informed by the global structure of the data set.
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 the tools of data analysis discussed above like PCA diffusion maps. We are now faced by the problem that (i) important parts of the global structure of 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.
In the case where not all data dimensions are equally rele- vant, 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 struc- ture of the point cloud, For even more complex data, we may first have to learn the encoded structure itself: indeed,
Department of Computer Science, RWTH Aachen Uni- versity, Germany. Correspondence to: Vincent Grande <grande @ www.--ac.c.........
Contributions We introduce TOPF (Figure 1), novel method to compute point-level topological features re- lating individual points to global topological structures of point clouds. TOPF (i) outperforms other methods and em- beddings for clustering downstream tasks on topologically
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Point-Level Topological Representation Learning on Point Clouds
Applications
Input
vectora
Output
Featura
Topological Node Embeddings in Biology, Machine Learning. Computer Vision, etc.
Bars correspond to loops in the paint cloud
1 Compute persistent homology
2
Pick significant generators
Weighted harmonic 3
4
Average over incident simplices
representatives
Figure 1: Computing Topological Point Features (TOPF). Input. A point cloud X in n-dimensional space. Step 1. To extract global topological information, the persistent homology computed on an a/VR-filtration. The most significant topological features across all specified dimensions are selected. Step 2. k-homology generators associated all features step of the filtration where Sı,a is alive. Step 3. The homology generators are projected to the harmonic space. Step 4. The vectors normalised obtain vectors ek indexed over the k-simplices. For every point and feature f E F, we compute the mean of the entries of ej corresponding to simplices containing æ. The output is a X x [F matrix which can be used for downstream ML tasks. Optional. We weigh the simplicial complexes resulting in a topologically more faithful harmonic representative in Step
fi,A F are computed. For every feature, a simplicial complex is built
structured data, returns (ii) meaningful representations, and (iifi) robust to moderate noise and heterogeneous sampling. Finally, we introduce the topological clustering benchmark suite, the first benchmark for topological clustering.
possibility for features selection across scales and compu- tationally far more expensive than TOPF. An overview over the thriving field of topological data analysis can be found in (Wasserman, 2018; 8; Munch, 2017). For more in-depth review of related work, see Appendix A. Because there are different views on what constitutes representation learning, we note that the learnt representations of TOPF are the result of homology and linear algebra computations, rather than trained features of an autoencoder or other neural network.
Related Work The intersection of topological data analy- sis, 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 2009) and (Perea, 2020)
(De Silva & Vejdemo-Johansee use harmonic cohomology representatives reparametrise point clouds based on circular coordinates. This implicitly assumes that the underlying structure of the point cloud is amenable to such characterization.
Organisation of the paper In Section 2, we give an overview over the main ideas and concepts behind TOPF, In Section describe how compute TOPF, Finally, will apply TOPF on synthetic and real-world data in Sec- tion 4. Furthermore, Appendix A contains brief history of topology and a detailed discussion of related work. Ap- pendix contains additional theoretical considerations. In Appendix C, we give a theoretical result guaranteeing the correctness of TOPF in an idealised setting. Appendix D describes the novel topological clustering benchmark suite, Appendix E contains details the implementation, the choice of hyperparameters and a discussion of runtime com- plexity, Appendix F contains more details on the experi- ments, Appendix G gives a detailed treatment of feature selection, Appendix H discusses simplicial weights, and Appendix discusses limitations in detail.
In (Basu & Cox, 2022; Gurari et al., 2023), 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. (Car- rière al., 2015) construct point features using Topology well. However, their signatures only summarise the local topology of the neighbourhood of the point rather than the relation between the point and the global topological fea- tures. (Grande & Schaub, 2023a) uses the harmonic space of the Hodge Laplacians to cluster point clouds respecting topology, but is unstable against some form of noise, has no
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Point-Level Topological Representation Learning on Point Clouds
Code We provide TOPF as an easy-to-use python pack- age with example notebooks at https: / /ttpbbb....uu..gg//g//// com/ vincent- grande/ of installable via pip.
by scale parameter approximating the topology of the data across all possible scales.
Definition 2.2 (VR complex), Given a finite point cloud X in metric space (M, d) and non-negative real num- ber E E Rz0, the associated VR complex VR;(X) is given by the vertex set X and the set of simplices S = {o C X a + Va, o d(z, v) S
2. Main Ideas of TOPF
A main goal of algebraic topology is to capture the shape of spaces. Techniques from topology describe globally mean- ingful 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.
A VR complex at e consists of all simplices a where all ver- tices x E a have distance of at most e. For r S r', we obtain the canonical inclusions iy.y (X): (X) + V R2 (X). For a simplex a and a filtration F, we denote by its filtration value F(o) the smallest E such that o E Fe. The set of VR complexes X for all possible E Rg0 together with the inclusions then form the VR filtration on X. For large point clouds, the VR filtration becomes computationally ex- pensive due to its large number of simplices. In contrast, the a-filtration approximates the topology of a point cloud using far fewer simplices. Thus we will make use of a- complexes in settings of ambient dimension lower than 4, where their construction is computationally feasible. For complete discussion and definition of a-complexes, Appendix B.
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. For more de- tails, the reader is referred to (Bredon et al., 1993) (algebraic topology) and (Munch, 2017) (TDA).
Simplicial Complexes Spaces in topology are continuous (connected), consist of infinitely many points, and often live in abstract spuce. Our input data sets however consist of finitely many points embedded in real space I=. In order to bridge this gap and open up topology to computational methods, need notion of discretised topological spaces consisting of finitely many base points with finite descrip- tion length. A simplicial complex is the simplest discrete model that can still approximate any topological space oc- curing in practice (Quillen, 1967):
Boundary matrices We still need an algebraic represen- tation of simplicial complexes that is capable of encoding the structure of the SC and enables extraction of the topo- logical features: The boundury matrices Bn associated to an SC S store all structural information of the sc. The rows of B& are indexed by the k-simplices of S and the columns are indexed by the (k- -)-simplices.
Definition 2.1 (Simplicial complexes). A simplicial com- plex (SC)& S consists of a set of vertices V and set of finite non-empty subsets (simplices, 5) of V closed un- der taking non-empty subsets, such that the union over all simplices Ures a is V. We will often identify S with its set of simplicies S and denote by Sy the set of simplices S with |a| h 1, called k-simplices. We say that S is n-dimensional, where is the largest & such that S5 is non-empty. The k-skeleton of S contains the simplices of dimension at most k. If the vertices V lie in R2" we call the convex hull in T of a simplex a its geometric realisation lol. When doing this for every simplex of S, we call this the geometric realisation of S. S R".
Definition 2.3 (Boundary matrices). Let S be a simplicial complex and < total order its vertices V. Then, the i-th face map in dimension f. S., + Sa-1 is given by
f (wo, UI,..., Un + feo, U1..... 0 Un
with up I vn .... I v. and o denoting the omission of oe. Now, the n-th boundary operator B, [[[S++]] + R[S,] with [[[S.] being the real vector space over the basis S,, is given by
12-11
B
Σ(1))
1=0
We can construct an n-dimensional SC S in n + 1 steps: We start with a set of vertices V (Sy). We then connect certain pairs of vertices with edges (S1). Afterwards, we fill in some fully connected triangles (S2), and so on.
When lexicographicallyy ordering the simplex basis, we can view B, as a matrix. We call K[S,] the space of n-chains. Bo is the vertex-edge incidence matrix of the associated graph consisting of the 0- and 1-simplices of S and B, is the edge-triangle incidence matrix of S
Vietoris-Rips and a-complexes We now need way to construct a simplicial complex that approximates the topo- logical structure inherent in our data set X C. R". Instead of having to pick a single scale, the Vietoris-Rips (VR), fil- tration and the a-filtration take as input a point cloud and return a nested sequence of simplicial complexes indexed
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. The k-th homol- ogy module HS) of an SC S with boundary operators By
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Point-Level Topological Representation Learning on Point Clouds
Persistence diagram
14
selected features
Death
10
8 e
Persistance
Birth
Figure 2: PH sketch and TOPF pipeline applied to NALCN channelosome, a membrane protein (Kschonsak et al., 2022). Left: Bars represent life times of features (Grande & Schaub, 2023b). Centre left: Steps 1&2a, when computing persistent 1-homology, three classes are more prominent than the rest. Centre right: Step 2b: The selected homology generators. Right: Step 3: The projections of the generators into harmonic space are now each supported on one of the rings.
is defined as (S) = ker 1 Im Im Bx. The generator or representative kernel ker - In dimension 1, these are given by formal sums of 1-simplices forming closed loops in the sc. Impor- tantly, the rank rk (S) is called the k-th Betti number By of S. In dimension O, Bo counts the number of connected components, B1 counts the number of loops around 'holes' of the space, B2 counts the number of 3-dimensional voids with 2- 2-dimensional boundary, and so on.
Algorithm Topological Point Features (TOPF)
homology class is an element of the
Input: Point cloud X E R", maximum homology di- mension d N, interpolation coeff. A.
1. Compute persistent homology with generators in dim. k S d. (Sec. 2: Betti Numbers & Persistent Homology)
2. Select set of significant features (b,, dis 9.) with birth, death, and generator in F3 coordinates (See Step 2).
3. Embed gi into real space (1), and project into harmonic subspace (2) of SC at step d>b!-> or Ad; + (1 A)6,- 4. Normalise projections to e and compute F+(x) =
Given a filtration of SCs, we can track how the homology modules evolve as the simplicial complex grows. The math- ematical formalisation, persistent homology, thus turns point cloud via simplicial filtration into an algebraic ob- ject summarising the topological features of the point cloud (Edelsbrunner & Harer, 2008).
avgred (e)(()) for all points x E X (3).
Output: Features of x E X
ker L 2 (R) means that we associate unique hurmonic representative to every homology class. Intu- itively, this means that for every abstract global homology class of persistent homology at filtration step from above we can now compute one unique harmonic representative in ker Lg that assigns every simplex value based on how much contributes to the homology class. Thus, the Hodge Laplacian is a gateway between the global topological fea- tures and the local properties of our SC. We discuss how this relates to the theory of differential forms and Hodge theory in the continuous case in Appendix B.2.
The Hodge Laplacian and the Harmonic Space We need to relate the global characterisations of PH back to local properties of the point cloud. We will do so by using ideas and concepts from differential geometry and topol- ogy: The simplicial Hodge Laplacian is discretisation and generalisation of the Hodge-Laplace operator acting differential forms of manifolds:
Definition 2.4 (Hodge Laplacian). Given a simplicial com- plex S with boundary operators Bx, we define the n-th Hodge Laplacian In [[(s.] + R(S)) by setting
3. How to Compute Topological Point Features
In B B B
In this section, we will combine the ideas and insights of the previous section to give complete account of how to compute topological point features (TOPF). A pscudo-code version can be found in Algorithm 1 and an overview in Figure 1, We start with a finite point cloud X RR"
Theorem 2.5 (Hodge Decomposition (Lim, 2020; Schaub et al., 2021; Roddenberry et al., 2021)). For an SC S with boundary matrices (B,) and Hodge Laplacians (Lg), we huve in every dimension
TR [Rx]
Im B
ker La
Im Lm curl space
Step 1: Computing the persistent homology First, we determine the most significant persistent homology classes which determine the shape of the point cloud. Doing this, we also extract the "interesting" scales the data set. We will later use this to construct SCs localise the global
gradiont space
harmonic space
This, together with the fact that the k-th harmonic space isomorphic to the k-thh real-valued homology group
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Point-Level Topological Representation Learning on Point Clouds
homology features. Thus we first compute the persistent k-homology modules Pk including set of homology rep- resentatives Rg of X using an a-filtration for < 3 and a VR filtration for n > 3. We use 2/3Z coefficients to be sensitive to simplex orientations. In case we have prior knowledge on the data set, we can choose a real number R E R50 and only compute the filtration up to the parame- ter R. In data sets like protein atom coordinates, this might be useful we have prior knowledge on what constitutes the "interesting" scale, reducing computational complexity. See Figure 2 centre left for PH diagram.
between these regimes returns optimal results. Given inter- polation hyperparameter "y E (0, 1), we will thus consider the simplicial complex 8th, (X) at step ti, == 6/,5² di x for k > 0 and at step t4 == .yd_, for & = 0 of the simplicial fil- tration. At this point, the homology class f is still alive. We then consider the real vector space IR [S fr basis consisting of the k-simplices of the Sc Shi,t From the persistent homology computation of the first step. we also obtain a generator of the feature f%.- consisting of list Ej of simplices oj E S and coefficients oj E 22/32. We need to turn this formal sum of simplices with coffecoeffscientss into a vector in the real vector space R[S (X)]: Let :: Z/3Z be the map induced by the canonical inclusion of 0-1,0,11 + R. We can now define an indicator vector e E RlS & Vejdemo-Johansson, 2009)).
& (X)] with formal
Step 2: Selecting the relevant topoloiiall features We now need to select the relevant persistent homology classes which carry the most important global information. The persistent homology Ps module in dimension k is given to us as a list of pairs of birth and death times (6f ).). We can assume these pairs are ordered in non-increasing order of the durations Eht = d - D 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 of ;= //-1/ > O as the point of cut-off N₂ := arg min, q. 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 ieplicatiio-speciffe domain knowledge. We found that our proposed heuristics work well across a large scale of applications. See Figure 2 left and centre for an illustration and Appendix G for more technical details and ways to improve and adapt the feature selection module of TOPF. We call the chosen k- k-homology classes with k- homology generators fi.
S (X) associated to the feature fV (CI. (De Silva
{o ((())) 3d E : 0z 10
<b(o)= =
(I)
else
Empirically, ei is homology generator for real coefficients as well in the vast majority of cases, although our construc- tion only guarantees for B5-1 = 0 mod We discuss how to fix the rare case where this does not work in Ap- pendix B.3 and now assume to work with real homology representative e While this homology representative lives in a real vector space, it is not unique, has a small sup- port, and its value can differ largely even between close simplices. All of these problems can be solved by project- ing the homology representative to the harmonic subspace ker LA of [ (X)] Rather than directly projecting ex to the harmonic subspace, we make use of the Hodge decom- position theorem (Theorem 2.5) which allows us to solve computationally efficient least square problems:
ejkccul := BB arg min |1e ex.grad [
Baall
(2)
and get the harmonic representative et = e
ei (CF. Figure 2 right for a visualisation.) Because homology representatives are gradient-free, ef ek. 0 and we only need to compute ek.cdd
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 features. Every selected feature ft of the previous step comes with a birth time by. and a death time di,.. This means that the homology class fü is present in every SC of the filtration between step (i.e. filtration value) by and di, k and we could choose any of the SCs for the next step. Pick- ing small filtration value will lead to fewer simplices in the SC and thus to a very localised harmonic representative. Picking a large filtration value 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
Step 4: Processing and aggregation at point level In the previous step, we have computed set of harmonic representatives of homology classes. Each harmonic repre- sentative assigns each simplex in the correspondings SCs a value. 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 features the levet-level after performing some necessary prepro- cessing. Given vector ěj indexed over simplices and a hyperparameter ó, we now construct et: S. (X) - [0, 1]
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Point-Level Topological Representation Learning on Point Clouds
by setting
Thus, using the canonical isomorphism to the dual vec- tor space, all elements of ker B; are homology represen- tatives and of ker B are cohomology representatives. The kernel of the Hodge Laplacian ker L; is the inter- section of ker B, and ker B, Hence every element in ker L, = ker B,nker B automatically is a homology and cohomology representative which is unique by matching dimensions. This gives us an explicit isomorphism between ker L, and the (co-Jhomology.
a (1é((o)) o'ES (X)
())..
max
such that ej is normalised to [0, 1], then the values of [0, 6] are mapped linearly to [0, 1] and everything above is sent to 1. We found empirically that a thresholding parameter of 5 = 0.07 works best across at the range of applications considered below. However, TOPF is not sensitive to small changes to 8 because entries of éj are concentrated around 0 (cf. Appendix 1.
In step 3, we start with some arbitrary homology representa- tive r in ker B, However, we want a harmonic representa- tive h in ker Lg- = ker B, ker B for the same homology class, Le. with h = r - c for c E Im Bi+1 the curl space quotiened out in homology- By the orthogonality of the Hodge decomposition, this c is given by the projection of to the curl space.
For every feature fü in dimension & with processed sim- plicial feature vector cj and simplicial complex Sti*,, we define the point-level feature map FR: X + IK mapping from the initial point cloud X to R by setting
Eorestin DEON ((x))) F. : U + max Hor E St ug:uE o+)))* v
(3)
Comparison with Topological Point Cloud Clustering Given point cloud, TPCC ((Grande & Schaub, 2023a)) selects an e-radius and builds the associated VR simplicial complex. It then computes the associated simplicial Hodge Laplacians and determines a basis of their O-eigenvectors. TPCC embeds the simplices bases on the eigenvector coordi- nates, performs subspace clustering on the embeddings and aggregates this information back to the point level deter- mine a final clustering. However, TPCC is (i) computation ally expensive due to extensive eigenvector computations, (ii) depending on high-dimensional subspace clustering al- gorithms, which are prone to instabilities and errors, (iii) sensitive to the correct choice of hyperparameters, (iv) re- quiring 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 rele- vant point-level features. TOPF solves (i) by using homology representatives from persistent homology and projecting them into the harmonic space, which requires solving a computationally efficient least-squares problem instead of an eigenvector problem. Problem (ii) and (iv) are solved by selecting relevant persistent homology classes by the TOPF heuristic instead of considering and clustering the entire harmonic space of given simplicial complex. Finally, chal- lenge (iii) is partially addressed by the heuristics of step 3 and 4 and the general architecture being more robust to noise.
For every point 2. can thus view the vector (w): 1 E F) as a feature vector for v. We call this collection of features Topological Point Features (TOPF). (Cr. Figure 3 for an example).
Choosing Simplicial Weights The above discussed the- ory works analogously for weighted SCs. We discuss how TOPF employs weighted scs to mitigate the influence of noise and heterogeneous sampling in Appendix H.
Theoretical Guarantees In the appendix in Theorem C.1 we give theoretical guarantees for when TOPF provably works on idealised point cloud.
Homology, Cohomology, and Hodge Laplacian kernels Many methods of TDA employ cohomology instead of ho- mology because of computational benefits. The TOPF would work with persistent cohomology instead of persistent ho- mology, however the cohomology representatives will be associated to different harmonic representatives which we found to be less faithful of topological features. We will briefly elaborate on this relationship. Formally, the cochain space where cohomology lives is the dual of the chain space of homology. For finte-dimennional vector spaces with the standard inner product, there is a canonical isomorphism between the two spaces induced by sending a vector v to the linear extension of the map sending v to 1. Thus, we can identify the chains and cochains. We have the boundary maps B; and the coboundary maps BT, By the universal coefficient theorem real-valued homology and cohomology are isomorphic, with
4. Experiments
In this section, we conduct experiments on real world and synthetic data, compare the clustering results with clus- tering by TPCC, other classical clustering algorithms, and other point features, and demonstrate the robustness of TOPF against noise. In Table 2, we perform an ablation study with respect to the harmonic projections of step 3 of TOPF.
H, = ker B, Im B, 2 H² ker BJ BTT Im B.T
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Point-Level Topological Representation Learning on Point Clouds
S (e)
(c)
(a)
(b)
(d)
TOPF 3d real-world and synthetic point clouds. For every point, we highlight the largest corre- sponding topological feature, where colour stands for the different features and saturation for the value of the fea- ture. (a): Atoms of mutated Cys123 of E. coli (Hidber al., 2007). We added auxiliary points on the convex hull and considered 2-homology, to detect the protein pockets which are crucial for protein-environmect interactions, ef. (Oda et al., 2024). (b): Atoms of NALCN channelosome (Kschonsak et al., 2022) display three distinet loops. (c): Points sampled in the state space of a Lorentz attractor. The two features correspond to the two lobes of the attractor. (d): Point cloud spaceship
Figure 3:
Figure 4: TOPF on high-dimensional point cloud. We used TOPF on 6500 points sampled from the 24-dimensional conformation space of cyclooctane (Martin & Watson, 2011). We show the ISOMAP projection from 24 dimen- sions. Top: Three features automatically selected by TOPF. Bottom: TOPF-Clustering for 3 and clusters. TOPF can correctly cluster similar points according to their topolog- ical function. Four clusters, TOPF can even identify the anomalous points (blue) violating the manifold structure.
The performance differences of TPCC and TOPF on differ- ent point clouds of the TCBS comes mainly down to how "difficult" and "robust" the topological structure encoded in the datasets is. In particular, 2Spheres2Civ Sphereennircle are directly sampled from unions of manifolds without any noise and sufficient sampling den- sity and TPCC and TOPF perform similarly. In compari- son, in Ellipses and Spaceship, the topological fea- tures have shorter life times and live different scales. There is single scale containing all features, and for most holes/voids, there is not even scale whithout noisy holes with small persistence present in the filtration. TOPF can deal with these situations, whereas TPCC cannot. For the difference on HalvedCircle, we posit that this is due to the better feature aggregation of TOPF. TPCC requires to per- form subspace clustering
our newly introduced topo- logical clustering benchmark suite (See Appendix D). (e): Latent space of VAE trained image patches showing topological structure (See Figure 12 for details).
and
Topological Point Cloud Clustering Benchmark We introduce the topological clustering benchmark suite (Ap- pendix D) and report running times and the accuracies of clustering based on TOPF and other methods and point em- beddings, see Table 1. We see that spectral clustering on TOPF vectors (listed as TOPF) outperforms all classical clus- tering 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 algo- rithm, TPCC two datasets with clear topological features, whereas TOPF outperforms TPCC on datasets with more complex structure. In addition, TOPF has consistently lower running time with better scaling for the more complex datasets, while not requiring prior knowledge on the best topological scale. Furthermore, TOPF outperforms WSDesc (Li al., 2022) pretrained on the 3DMatch dataset (Zeng al., 2017) and DGCNN (Wang al., 2019) pretrained on the ShapeNetPart data set. This highlights that neural network architectures like DGCNN need specific application- specific training data for good performance that is simply not available in all cases. TOPF is an unsupervised method for extracting interpretable topological features founded in algebraic topology and differential geometry.
harmonic edge embedding, which is both unstable and sensitive to parameters, which we posit TPCC has no information to choose correctly in this setting.
On 4Ci reles+Grid, TOPF performs worst out of the point clouds of TCBS. Figure and Figure show the ground truth and TOPFclustering on 4Circles+Grid. In short, TOPF chooses suboptimal scales which are not well-enough connected. Figure 10 supports this interpretation, increasing the interpolation hyperparameter À improves TOPF perfor- mance on this dataset. We have used a fixed set of hyperpa- rameters for all experiments on the TCBS for transparency.
Feature Generation In Figure 3, we show qualitatively that TOPF constructs meaningful topological features on
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Point-Level Topological Representation Learning on Point Clouds
Table I: Quantitative performance comparison of clustering with TOPF and other features/clustering algorithms. Four 2d and three 3d data sets of the topological clustering benchmark suite (Appendix D, cf. Figure 8 for ground truth and Figure 9 for labels by TOPF). We ran each algorithm 20 times and list the mean adjusted rand index (ARI) with standard deviation a and mean running time. We omit for algorithms with σ = 0. Spectral clustering on TOPF vectors consistently outperforms or almost matches the other algorithms while having significantly better run time than the second best performing algorithm TPCC. Spectral Clustering (SC), DBSCAN (applied the points) are standard clustering algorithms, TOMATO is topological clustering algorithm (Chazal et al., 2013), Geo clusters using 12-dimensional point geometric features extracted by pgeof. "Weakly Supervised 3D Local Descriptor Learning for Point Cloud Registration" (WSD (Li et al., 2022)) is pretrained on 3DMatch data and produces 32-dimensional feature vectors. DGCNN (Wang et al., 2019) is pretrained on ShapeNetPart segmentation, with the parameters determined by hyperparameter search. PAM+ISOMAP denotes k-medoids clustering on the ISOMAP embedding. We highlight all ARI scores within +0.05 of the best ARI score.
TOPT
TPCC
SC
DBSCAN
AgC
ToMATO
Goo
PN
WSD
DGCNN
PAM+ISOMAP
4spheres
ARI
0.81
0.52:10.17
0.37
0.00
0.45
0.32
0.20
0.30
0.13
0.29
0.39
time
13.9
23.3
0.2
<0.1
<0.1
<0.1
0.2
<0.1
<0.1
<0.1
0.2
Ellipses
ARI
0.95
0.47:1:004
0.25
0.19
0.52
0.29
0.81
0.50
0.35
0.70:10.00
0.56
time (6)
13.5
14.4
0.1
<0.1
<0.1
<0.1
0.1
<0.1
<0.1
<0.1
<0.1
4Clrcles+Grid
0.70
0.39=20.04
0.90
0.92
0.89
0.82
0.41
0.55
0.06
0.75:
0.78
time (s)
13.7
28.5
0.5
<0.1
<0.1
<0.1
0.3
<0.1
40.1
<0.1
0.3
Halved Vircle
ARI
0.71
0.18:20.12
0.24
0.00
0.20
0.16
0.08
0.36
0,00
0.39
0.13
time (x)
13.6
14.3
0.1
<0.1
<0.1
<0.1
0.1
<0.1
<0.1
<0.1
<0.1
28pheres2Circle ARI
0.94
0.97=00001
0.70
0.00
0.51
0.87
0.12
0.55
0.95
0.84
0.63
time (x)
20.7
1662.2
1.6
<0.1
0.3
<0.1
0.9
<0.1
<0.1
0.2
6.5
SphereinCirole
ARI
0.97
0.98.E-c0.1
0.34
0.00
0.29
0.06
0.69
0.39
0.46
0.76-1-0006
0.94
time (s)
14.5
8.0
<0.1
<0.1
<0.1
<0.1
0.08
<0.1
<0.1
<0.1
<0.1
Spaceship
ARI
0.92
0.56=#0.03
0.28
0.26
0.47
0.30
0.87
0.41
0.76
0.79+00.00
0.58
time (s)
341.8
16.7
<0.1
0.2
<0.1
<0.1
<0.1
0.1
mean
ARI
0.86
0.58
0.44
0.16
0.48
0.40
0.45
0.44
0.39
0.64
0.58
time
15.1
298.9
<0.1
0.3
<0.1
<0.1
<0.1
1.0
synthetic data and data sets from Biology and Physics, cor- responding to for example rings and pockets in proteins or trajectories around different attractors in dynamical systems, (for individual heatmaps see Figure 14). In Figure 4, we show that TOPF works for the high-dimensional conforma- tion space of cyclooctane.
Embedding Space of Variational Autoencoders and High- dimensional spaces Variational autoencoders (VAR) are unsupervised neural networks that learn to extract a low- dimensional embedding of a high-dimensional data set. We have trained 2 VAES on 518 image patches which were sam- pled along a topological structure on a larger image with a latent space dimension of 3 and of 16. We show that run- ning TOPF on the two latent spaces as well as directly on the 874 (748())dimmnniio) base space can recover the topological structure underlying the capturing process of the training set. We visualise the feature vectors on the latent space in Figure 3 (c) and Figure 13 and provide additional details in Figure 12 and Appendix F.
Robustness Against Noise, Downsampling, and Sam- pling Heterogeneity We have evaluated the robustness of TOPF against Gaussian noise on the dataset introduced in (Grande& & Schaub, 2023a) and compared the results against TPCC, Spectral Clustering, Graph Spectral Clustering on the graph constructed by TPCC, and against k- m-manss in Figure We have also analysed the robustness of TOPF against the addition of outliers in Figure 6. We see that TOPF performs well in both cases, underlining our claim of robustness. Fi- nally. in Figure 5, we show that TOPF performs well under sampling heterogeneity and under downsampling to sparse point clouds, while comparing TOPF to the other baselines. We posit that this is due to the robustness of a-complexes, the chosen simplicial weights, and the general robustness of features extracted by persistent homology. In our eyes, the above result showcases the strength of our approach incorporating ideas from topological data analysis and dif- ferential geometry, given that heterogeneous sampling and sparse point clouds constitute a major challenge for classical geometry learning.
In Figure 4, we use TOPF on high-dimensional input point cloud representing the conformation space of cyclooctane (Martin & Watson, 2011). We see in the ISOMAP projections of Figure 4 that TOPF extracts reasonable features represent- ing the topology of the conformation space. This shows the feasability of TOPF on very high-dimensional spaces. Note that TOPF even recovers the anomalous points violating the manifold hypothesis, as done in (Stolz et al., 2020).
5. Discussion
TOPF can by ingful output on point clouds with topological structure
Limitations
by design produce mean-
8

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Point-Level Topological Representation Learning on Point Clouds
Downsampling
Heterogeneous Sampling
Table 2: Ablation study for the harmonic projection in step 3. We see that in the more complex datasets, perfor- mance without the harmonic projection drops significantly. Only 2Spheres reshcireeeiiiie rcles and Sphereing rele are unions of simple manifolds without noise and thus TOPF can recover meaningful features even without harmonic pro- jection.
0.8
0.8
0.7.
0.6
0.6
0.5
0.5
20.4
0.4
0.3
0.3
0.2
0.2
0.1
ARI TOPF
0.0
Dataset
AR1 TOPF
without harmonic projection
100
n_points
downsamplin factor
4apheres Eilipses
0.81
0.08
TOPF
KMeans OPTICS
DBSCAN
0.95
0.53
Spectral Clustering
Agglomerative Clustering Geo
4Circles-Grid
0.70
0.11
Mean Shift
WSDesc
Halved Circle
rapheresSCrrales
0.94
0.95
SphereinCircle
0.97
1.00
Figure 5: Performance of TOPF Clustering under down- sampling and heterogeneous sampling on 4spheres. Left: We perform random downsampling to test the per- formance of TOPF on sparse point clouds. TOPF maintains strong performance until- ~ 100 points, which proves that TOPF performs well on very sparse point clouds. For exper- iments all datasets of TCBS see Figure 15. Right: We investigate the performance under heterogeneous sampling. We divide the point cloud into halves cutting up topological structures and downsample one of the halves with a factor of up to 100. The experiments show that TOPF still outper- forms the other methods under a sampling irregularity of 1 : 10, indicating a strong robustness against heterogeneous sampling. For experiments on more datasets of TCBS see Figure 16.
Spaceship
0.92
0.49
0.86
0.47
quantifiable by persistent homology. practice it is thus desirable to combine TOPF with some geometric or other point-level feature extractor. As TOPF relies the com- putation of persistent homology, its runtime increases very large point clouds, especially in higher dimensions where a-filtrations are computationally infeasible. However, subsampling, either randomly or using landmarks, usually preserves relevant topological features while improving run time (Perea, 2020). Finally, selection of the relevant features is very hard problem. While our proposed heuristics work well across variety of domains and application scenar- ios, only domain- and problem-specific knowledge makes correct feature selection feasible.
Method
TORE untuned
KMconn Spectral Chadweiig
TPGC Spectral on
08
06
Future Work The integration of higher-order TOPF fea- tures into ML pipelines that require point-level features po- tentially leads to many new interesting insights across the do- mains 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.
0.2
100
400 500 E0000 Outliars
D2
0,4
0.6
0.8
Noiie
Figure 6: Performance of TOPF Clustering with noise/outliers on SphereinCircle, 95% CI. The radii are 3 and 1, the mean distance to the closest neighbour is 0.22. Left: We add i.i.d. Gaussian noise to every point with standard deviation indicated by the noise parameter. Even when compared with TPCC on a data set specifically crafted for TPCC, TOPF requires significantly less information and delivers almost equal performance. Tuned for datasets with a high noise, the TOPF outperform TPCC. Right: We add outliers with same standard deviation as the point cloud. We then measure ART restricted to original points. Compared with TPCC on a data set specifically crafted for TPCC, TOPP requires significantly less information and delivers matching to superior performance.
Conclusion We introduced point-level features TOPF founded on algebraic topology relating global structural features to local information. We gave theoretical guaran- tees for the correctness of their construction and evaluated them quantitatively and qualitatively on synthetic and real- world data sets. Finally, introduced the novel topological clustering benchmark suite and showed that clustering us- ing TOPF outperforms other available clustering methods and features extractors. In particular, we showed that TOPF performs well both on very sparse datasets and on datasets under heterogeneous sampling.

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Point-Level Topological Representation Learning on Point Clouds
Acknowledgments
Bousfield, A. K. The localization of spaces with respect
homology. Topology, 14(2):133-150, 1975. doi: https: htt.:o/g/0.1116/00000-9333333333
thank the anonymous referees for the detailed discus- sions and helpful feedback.
VPG acknowledges funding by the German Research Coun- cil DFG within Research Training Group 2236 UNRAVEL. MTS acknowledge wengosssweeesseooo partial funding by the Ministry of Cul- ture and Science (MKW) of the German State of North Rhine- Westphalia ("NRW Rtickkehpprgrrmmm") and the European Union (ERC, HIGH-HOPES, 101039827). Views and opin- ions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
Bredon, G., Ewing. J., Gehring, F., and Halmos, P. Topology and Geometry. Graduate Texts in Mathematics. Springer, New York, 1993.
Bubenik, P. et al. Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res., 16(1):77- 102, 2015.
Carlsson, G. and Sgjdemo-Johanssen M. Topological Data Analysis with Applications. Cambridge University Press, 2021.
Carlsson, G., Zomorodian, A., Collins, A., and Guibas, Persistence barcodes for shapes. In Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing, pp. 124-135, 2004.
Impact Statement
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Carrière, M., Oudot, S. Y., and Ovsjanikov, M. Stable topological signatures for points on 3d shapes. Com- puter graphics forum, volume 34, pp. 1-12. Wiley Online Library, 2015.
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Point-Level Topological Representation Learning on Point Clouds
A. Extended Background
A brief history of topology and machine learning Algebraic topology is a discipline of Mathematics dating back roughly to the late 19th century (Poincaré, 1895). Starting with Henri Poincaré and continuing in the early 20th 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 formalisation of mathematics and analysis, in particular, which took place inside the mathematical community in the 1800's and early 1900's (e.g. (Dedekind, 1888: Hilbert, 1899; Hausdorff, 1908)). The axiomatisation of analysis in the early 20th century is an important result of this process. Another key step towards modern algebraic topology was the development of homological algebra which was led by the insights of Emmy Noether (Noether, 1926). The main idea was to replace the Betti numbers by the more structured and powerful homology groups. Over the course of the last 100 years, branching into many sub-areas like low-dimensional topology, differential topology, K-theory or homotopy theory (Atiyah, 1989; Hu, 1959), 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 abstraction and generalisation of concepts: topological spaces turned into spectra, diffeomorphisms to homotopy equivalences and later weak equivalences, and Topologists turned to category theory (Eilenberg & MacLane, 1945), model categories (Bousfield, 1975) and recently oc-categories (Lurie, 2006) as the language of choice.
The 21x century saw the advent and rise of topological data analysis (TDA, (Bubenik et al., 2015; Chazal & Michel, 2021)). 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.
Topological data analysis is closely connected to distance- and density-based methods for data analysis like DBSCAN or tools from manifold learning like ISOMAP (Tenenbaum et al., 2000) or UMAP (Mclnnes et al., 2018). Persistent homology captures the shape of a point cloud across all scales and can be captured in persistence barcodes (Carlsson et al., 2004), persistent landscapes (Bubenik al., 2015) or persistence images (Adams et al., 2017). In multiparameter persistent homology, an additional filtration parameter like density is included, allowing for stronger representations like graphcodes (Kerber & Russold, 2024) or GRIL (Xin et al., 2023). The persistent homology transform computes persistent homology over a height filtration across different directions in the data set, further uncovering geometric properties of the dataset (Turner et al., 2014). The mapper graph is a tool to visualise high dimensional point clouds with graphs using the (persistent) homology of slices according to some generalised heigh function (Singh et al., 2007).
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 (De Silva & Vejdemo-Johansson 2009) and (Perea, 2020) use harmonic cohomology 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 (Basu& & Cox, 2022; Gurnari et al., 2023), the authors develop and use harmonic persistent homology and provide a way to pool features to the 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 birth. In their paper on topological mode analysis, (Chazal et al., 2013) use persistent homology to cluster point clouds. However, they only consider O-dimensional homology to base the clustering on densities and there is no clear way to generalise this to higher dimensions.
On the more geometric-centred side, (Ebli & Spreemann, 2019) already provide a notion of harmonic clustering on simplices, (Chen & Meilä, 2021; Chen et al., 2021) analyse the notion of geometry and topology encoded in the Hodge Laplacian and its relation to homology decompositions, (Schaub et al., 2020) study the normalised and weighted Hodge Laplacian in the context of random walks, and (Grande & Schaub, 2023a) use the harmonic space of the Hodge Laplacians cluster point clouds respecting topology. Finally, a persistent variant of the Hodge Laplacian is used study filtrations of simplicial complexes (Mémoli et al., 2022).
There are other constructions of isometry-invarian local shape descriptors, as for example proposed in (Mémoli, 2011). However, their aim is not relate the global topology back to the local features.
In (Moor et al., 2020), the authors basically set up a topological loss functions that ensures that topological features of the original point cloud are preserved in the latent space of the autoencoder. This is different from the approach TOPF takes, as
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Point-Level Topological Representation Learning on Point Clouds
TOPF tries detect and extract topological features. Similar things can be said about topological node-2-vec (Hiraoka et al., 2024) and some experiments in (Carriere al., 2021) or some experiments of (Carriere et al., 2024): They all try come up with ways to preserve global topological structure while embedding point cloud, while we on the other hand construct the embedding based on the relationship between the points and the global topological structure.
On graphs, (Arafat et al., 2024) construct topological representations of graphs to study adversarial graph learning. They construct a local topological features for nodes and global topological features for the entire graph. In contrast to our approach, they do not relate the global topological features back to the local level, but rather consider the local topology on the point level and the single global topological summary separately. Furthermore when constructing their features, they use vision transformers on the persistence images of their witness filtrations, which does not allow for the same amount of interpretability as TOPF does.
In (Grande & Schaub, 2023a), 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, (ifi) sensitive to the correct choice hyperparameters, (iv) requiring the topological true features and noise to occur in different steps of the simplicial filtration, and (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 data set, you get the topological node features with similar computational effort.
B. Theoretical Considerations
B.1. More details on VR and a-filtrations
Vietoris- 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 large number of simplices - up to (IxI) 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 to pre-compute a set of simplices that are able to express the entire topology of the point cloud. For a point cloud X C R", the a- a-filtration consists of the intersection of the simplicial complexes of the VR filtration on X with the (higher-dimensional) Delaunay triangulation of X in IR. Due to algorithmic reasons, the filtration value of simplex is then related to the radius of the circumscribed sphere instead of the maximum pair-wise distance of vertices, where the filtration value a (a) of a simplex a is given by the minimum E such that a is contained in ag (X), i.e. (a) = inf{= E R : a E ag (V)). This reduces the number of required simplices across all dimensions to (( However, the Delaunay triangulation becomes computationally infeasible for larger n.
Definition B.1 (n-dimensional Delaunay triangulation). Given a set of vertices V C R", a Delaunay triangulation DT(V) is a triangulation of V such that for any n-simplex on E DVV the interior of the circum-hyperseee point of DT(V). triangulation of V is a sc S with vertex set V such that its geometric realisation covers the convex hull of V hV) |S] and we have for any two simplices a, a' that the intersection of geometrie realisations |a|| n Jo'l is either empty or the geometric realisation [o] of common sub-simplex ē C C,
of on contains no
If Vin is in general position, the Delaunay triangulation is unique and guaranteed to exist (Delaunay et al., 1934). We will now first introduce a slightly simpler version of the a-filtration, the a filitraaio...
Definition B.2 (a" cco-plex of point cloud). Given a finite point cloud X in real space IR",, the a -coopllxx at is the subset of the n-dimenniooaa Delaunay triangulation DX) consisting of all k-simplices d% E DT(X) with a radius r of its circumscrihed (k: sp-see with S for all l: < n.
The a and a-filtration share the same set of simplices and agree on the filtration values of the simplices on all top- dimensional simplices and all other simplices which are Gabriel simplices.
Definition B.3 (Gabriel Simplex). Given a set of vertices V C R" and its Delaunay triangulation DT(V). A simplex E DV( is called Gabriel if there exists no point e E V such that v is contained in the interior of the cireumsphere of a.
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Point-Level Topological Representation Learning on Point Clouds
Definition B.4 (a-complex of point cloud, (Kerber & Edelsbrunner, 2013; Edelsbrunner, 2011)). Given finite point cloud X in real space IR,", the a-complex Og (X) is the a" -complex of V, except that the filtration value a(a) of all non-Gabriel simplices with associated points X in the interior of the circumsphere of a is given by min a(a U {x)).
We note that due to the definition of the Delaunay triangulation, all n--implices are Gabriel simplices and hence this is well-defined. This is an equivalent formulation of the original definition, which was for example shown in (Kerber & Edelsbrunner, 2013). We chose to go with the above formulation, as it the form used in implementations of e-filtrations.
We note two things: (i) The a" -illtraatioo and a-filtration have fewer simplices than the VR filtration and (ii) the filtration values of individual simplices differ between the a', the a and the VR filtration. In partieular, the order of the filtration values can be different across the two types of filtration. For 1-simplicial complexes, the a and VR filtration values on the simplices present in both are equivalent: The VR filtration value of an edge is its length 1, whereas its a -flltratiion value is the radius r = 1/2 of the associated O-sphere consisting only of the two vertices.
B.2. Differential forms and a continuous analogue of TOPF
In this section, we will discuss how a continuous analogue of TOPF on Riemannian manifolds would look like. Simply considering the simplicial homology of the manifold, as we did in the case of a discrete simplicial complex, is however not sufficient: Harmonic cycles and cocycles don't I exist in the simplicial chain complex of the manifold. This is because the space of chains in simplicial homology on manifolds is the space of chains with its dual, the space of cochains. From an intuitive point of view, harmonic chain would need to take infinitesimal small values on every simplex which is not allowed. Rather, the right language to formalise harmonic representatives is Differential Geometry, Hodge Theory, and harmonic forms. Dodziuk proved in (Dodziuk, 1976) that the discrete Hodge Laplacian as used in this paper if weighted correctly converges spectrally to the Hodge Laplacian on integrands over manifolds.
dimensiolll Thus, we cannot canonically identify
differential forms which are eobitrry-dimenssine
There is a beautiful connection between differential geometry and algebraic topology: A theorem by de Rham states that given a Riemannian manifold M, the real-valued homology HA (M; MRR) R)) with coefficients in IR is isomorphic to the de Rham cohomology HM (M; R) on differential forms on M for arbitrary l e Zoo, some form of cohomology defined via the chain complex of vector spaces of differential k-forms on M. We can define an analogue of the discrete Hodge Laplacian on differential forms by the differential induced via exterior derivates d and its adjoint ó:
A = 8d + d8
The kernel ker A of theorem now gives us a natural vector space isomorphism between the space of harmonic k-forms on M, H+ and the ttth real-valued homology group
the Hodge Laplacian is called the space of harmonic forms. Similar to the discrete case, the Hodge
(M; R) Ha (M).
We can rephrase this as follows: the harmonic forms can be seen as unique and natural homology representatives.
Let us now get some intuition for what this means for continuous TOPF features: In dimension 0, 0-forms are functions f: M - R. this case, we can simply take their value f(x) at x E M as the corresponding point feature at x. In dimension 1, there is a correspondence between 1-forms and vector fields on M. In this case, we can take the norm of the vector field that corresponds to given harmonic form a point via the map + |o(z)]-
In general in dimension k, this is more complicated. Luckily, as we are interested in point-features, we can do all computations pisnt-wise. We consider a point æ E M and a harmonic form w. At point x, w determines an element in the exterior algebra on the dual of the tangent space of M at x, i.e wa E A k Th M and we need to define norm on this space. An orthonormal basis on T: M, (z)),.... en (x) gives rise to a basis consisting of elements ei, (x) A... A Ci, (x) of the exterior algebra. We can now evaluate Wa on the elements of this basis wa (ei, (x)) A... A ei (x)) and take the square root of the sum of squares of the individual results
TOPF E
Σ Si
We (e,, (......... Ae,, (z))
As one can show, this result is independent of the choice of ONB of T&M M and gives a point-wise norm of the differential forms.
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Point-Level Topological Representation Learning Point Clouds
In the discrete case discussed in this paper, however, don't evaluate on all basis elements described above, but rather on all k-simplices adjacent to These k-simplices can thus be seen as samples from the exterior algebra of æ. This sampling induces some differences to the continuous case, which address with the post-processing procedure as described in the paper.
We note that some authors prefer to reserve the name of Hodge Laplacian for the continuous case of differential geometry and for the representations of finite element theory and discrete exterior calculus. Thus, what we call Hodge Laplacian on simplicial complexes sometimes known as the combinatorial Laplacian.
B.3. Fixing real lifts of homology generators in 2/32-cocfficients
Figure 7: 2/32-homology generator with "faulty" lift to IR-coefficients. Left: Initial lift produced in 2/32-coofficiietss Right: Fixed lift with light blue edges representing the original lift with coefficients in {-1, 1] and dark blue edges representing the fix with coefficients in 3Z. The underlying data is the trefoil knot embedded in RR, as generated by (Perea et al., 2023).
In step 3. of the TOPF algorithm in Section 3, we attempt to lift homology generator with simplices gj E Σg and coefficients in 2/32-coefficients to a homology generator in RRcoefficients using the embedding
(())) 0
=d, E E 85 else
(0)
where v: 2/3Z is the map induced by the canonical inclusion of (-1, -, TR. Thus, we know that oj is a cycle modulo BB- = 0 mod A-prior however, there is no reason why this equality should hold in as well. As turns out, in virtually all examples in practice, we already have et ker Bx-1 which very convenient. The same phenomenon occurs for cocycles well, see (De Silva & Johanmo-Johansson, 2009). We will now assume to be in case of et ker Ba-1- We can now attempt to fix this using linear integer program finding SSA such that
3B 13 = 16
And then we set fix()), == ej - 3y. If this linear program is not feasible, ei has no integer/real lift and TOPF will skip the associated topological feature. In Figure 7, we give a visual example a faulty lift and fix obtained by the above procedure. This is similar to the implementation used to compute Eilenberg-Macll coordinates DREiMac (Perea et al., 2023).
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C. Theoretical guarantees
In this section, we want to investigate possible theoretical guarantees for TOPF computed on idealised datasets. Instead of directly working with the a-complexes, we will first prove guarantees for the closely related construction of a -complexes. a filtrations (Definition B.2) are closely related to a-filtrations (Definition B.4) used in the implementation and agree on the filtration values most of the simplices, have however nicer properties. Finally, we give conditions on when this result applies to the a-filtrations, which is relevant to the actual implementation of TOPF.
Theorem C.1 (Topological Point Features of Spheres), Let X consist of at least (n+22 + 2) points (denoted by S) sampled uniformly at random from a unit n-sphere in R=+1 and an arbitrary number of points with distance of at least 2 to S. When we now consider the o² -filtration on this point cloud, with probability 1 we have that (i) there exists n-th persistent homology class generated by the n-simplices on the convex hull of S, and (ii) the support of the associated topological point feature feature (TOPF) Ft is precisely S: Fi) = S. (iit) The same holds true for point clouds sampled from multiple n, ns-spheres if the above conditions are met on each individual sphere. (iv) If there are no other points contained in or on the circumspheres of the n-simplices of the convex hull of S, then the same above holds true for TOPF computed using the a-filtration. (v) More generally, for any contractible subset M of the (n + 1)-sipllices such that all n-simplices on the boundary fulfill the above condition, this holds when using the ar-filtration, the discussed n-simplices (i) and the subset of points contained in M (ii).
We will give a proof of this after the following remark.
Remark C.2. The key idea of the proof is to use that a afffittraiions and partially a-filtrations assign the filtration value based on the radius of the circumsphere of the k-simplex. Because all points in S lic on the same n-sphere, with probability 1 we can write down the filtration value of the (n + 1)-sipplices explicitly. This is of course a very idealised setting. However 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 (Basu & Cox, 2022). For the 2D setting, condition (iv) is equivalent to assuming that there is no line through the centre of the circle such that one half plane does not contain any points of S.
We will now give the proof of the theorem that guarantees that TOPF works. We discuss the assumptions of the theorem in Remark C.3.
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 ne-sphere. Let us now look at the a-filtration of the simplices in S: Recall that the filtration values of k--immlee given by the radius of the (A: - 1)-sphere determined by its vertices. Because all of the (n+ 1--simllices 0x+1 with vertices V C S in S lie the same unit n--phere Sn. they all share the filtration value of a(an+1) = 1. By the same argument above, with probability 1 there are no (n + 1) points in S that lie on an unit (n - --phhee Thus all of the n-simplices o, lie on (n - 11) 1)-spheses Sn, with radius r <1 1 smaller than 1 and hence have a filtration value a(aa.) smaller than 1. Let
b = max (fa(an) nn C (hhll((S))
be the maximum filtration value of an n-simplex on the boundary of the convex hull of S. 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 distance of at least the points in S, all (n + 1)- simplices 1++1 with vertices both in S and its complement in X will have a filtration value a(σn+1) > 1 of at least 1. Recall that all (n + 1 s)-simplices .n+1 C S with vertices inside S have a filtration value of a( = 1. Thus the adjoint of the n-th boundary operator BT is trivial on the homology generator g for a simplicial complex constructed during (b, 1). Thus, we have that for the n-th Hodge Laplacian
L.g- BTT B, 9 0 0
and hence g is a harmonic generator for the entire filtration range of (b, 1). Claim (ii) and (iii) then follow from the construction of the TOPF values.
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Point-Level Topological Representation Learning on Point Clouds
For claim (iv), it suffices notice that the n-simplices a the convex hull are precisely non-Gabriel simplices iff there exist point x: E S lying inside the circumsphere of a. Because the assumption of part (iv) excludes this case, all n-simplices are Gabriel simplices and the a-filtration values (Definition B.4) and a* a"-filtration values (Definition B.2) agree on the n-simplices and (n + 1) simpplccss of the Delaunay triangulation. Because these are the only considered filtration values for the proof, the claim follows.
Finally for part (v), we see that the same argument now applies to the simplices on the boundary, consisting of n-simplices of M. consisting of (n + simipllies Because M is contractible, it is connected and has trivial homology. Thus, its boundary homotopy equivalent to an n-sphere. Because in this case only the points X' on the boundary of M are contained in the generating n-simplices of the considered homology class, the associated TOPF feature will have support on X'. This concludes the proof.
Remark C.3. The setting of the theorem is a very special setting. In general on data, harmonic chains have entries with absolute values somewhere in [0, 1]. In the setting of the theorem however, the harmonic representative will take values in 11,10. 0, 1. We will briefly explain why this is the case: Consider a k- k-homology representative r with simplex-valus values in {-1,0, 03 We consider the harmonic projection h of r into the harmonic space. Then, for a k-simplex o, we have that h()) = =1 iff r(o) = 1 and a is not the face of any (k + -)simpliies.
This can for example be seen by representing as the difference of r and its gradient and curl parts h = r T'rad Yeurl- Because r is already cycle, B&r = 0 and thus rgra O. Now, the curl part can be written as stemming from signal on Bk+1Ieurt- However, because is not the face of a k + 1-simplex, (own(or 0 and thus
the the k + 1-simplices, Taur
h(() = rr)).
The setting of the theorem precisely constructs a case where we have k-simplices and no (I: + 1) 1)-simplices in the relevant part of the filtration. This is the case because in an a a"-filtatiinn, the filtration value of a k-simplex is the radius of its cireumscribed (k= - sp-eeee All relevant (A: + 1) s-sipplices in the theorem lie on the k-unit sphere, and thus have filtration value 1. This however an idealised setting. Because we construct the simplicial complex to compute the harmonic representative somewhere in the middle (determined by the interpolation coefficient) between the birth and death time of the homology class, empirically the majority of the k- simplices of the k-homology generators are faces of (k: + 1) )-iimplicss. In this case, the harmonic representative smooths out the feature.
D. Topological Clustering Benchmark Suite
We introduce seven point clouds for topological point cloud clustering in the topological clustering benchmark suite (TCBS). Some of these point clouds have been adapted from (Grande & Schaub, 2023a). The ground truth and the point clouds are depicted in Figure 8. The point clouds represent mix between O-, 1- and 2-dimensional topological structures in noiseless and noisy settings in ambient 2-dimensional and 3-dimensional space. Point clouds E1 1 spsess and spaceshi i p already incorporate heterogeneous sampling/densittes. The results of clustering according to TOPF can be found in Figure 9.
We constructed the benchmark by sampling from topologically different shapes with varying sampling density in different ambient dimensions. For example for the point cloud 28pheres2Circles spheres and two circles. We then divided the point cloud into four ground truth clusters, one for each of the two spheres and two circles and assigned every point the cluster corresponding to the object it was sampled from. Thus, the ground truth labels of point corresponds to the topological structure it was sampled from. We provide functions to resample some of the point clouds with variable point numbers and provide method to provide sampling heterogeneity as described in Figure 16.
we combined points sampled from two
E. Implementation
We have created an easy-to-use python package TOPF which can be found at github (https://gittub) com/ vincent-grande (toff and PyPi (https: (https://pypi. httpss///yp....opppoooeoo wttpo/gggg/c///t///o/ooooo/eoe/t
with many examples from the paper and the topological clustering benchmark suite. The package currently works under macOS and Linux. The package contains both the code to generate the topological point features, as well as the code to reproduce the various visualisation steps in this paper.
All experiments for TOPF 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 (The
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Point-Level Topological Representation Learning Point Clouds
b)
8)
Figure 8: Data sets of the Topological Clustering Benchmark Suite (TCBS) with true labels. Top: 2D data sets. From left to right: a): 4Spheres (656 points), b): Ellipses (158 points), c): Spheres+Grid (866 points), d): Halved Circle (249 points). Bottom: 3D data sets. From left right: e): 2Spheres2Ci reles (4600 points). f): SphereinCircle (267 points), g): spaceship (650 points).
a)
b)
e)
9)
Figure 9: Data sets of the Topological Clustering Benchmark Suite (TCBS) with labels generated by TOPF. Top: 2D data sets. From left to right: a): 4Spheres (0.81 ARI), b): Ellipses (0.95 ARI), c): Spheres+Grid (0.70 ARI), d): Halved Circle (0.71 ART). Bottom: 3D data sets. From left to right: e): 28pheres2circles (0.94 ART). f): SphereinCircle (0.97 ARI), g): spaceship (0.92 ARI).
GUDHI Project, 2015) (Ø The GUDHI developers, MIT license) and Ripserer (Cufar, 2020) (0 mtsch, MIT license), which is modified Julia implementation of (Bauer, 2021). For the least square problems, we used the LSMR im-
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Point-Level Topological Representation Learning on Point Clouds
plementation of SciPy (Fong & Saunders, 2011).
We used the pgeof Python package for computation of geometric features httpps://iithus wttp:/ttthhccccccoooeeeoeooooaee
wwt.cccmddrr/e www.m///rreeetsss///ccooooo http:///wwappptttt/ooooooooeeeee ntt_geeeeerrrceeeeeeeeeeee features (Ø Damien Robert, Loic Lan-
drieu, Romain Janvier, MIT license). We use parts of the implementation of TPCC httttp::/git..wttttt raathaahehhaaahe /
(@ Computational Network Science Group, RWTH Aachen University, MIT license). We use the implementation of WSDese (Li et al., 2022), https://g cwtp//raaigleell//Reeesee.. (Lei Li, Hongbo Fu, Maks Ovsjanikov. CC BY-NC 4.0 License). We use the implementation An Tao, https:/ / /gi thub. commontatteooooooooooooooooccaaa ant 97 10/997/dd/o/o /dgcnn . pytorch of Dynamic Graph CNNS (Wang et al., 2019), MIT license. The idea for the implementation of the fix of the lift of the Z/ r/peeeepresettattve RR-cofficiietss was inspired by DREiMac, (Perea al., 2023).
E.1. Hyperparameters
1.0
09
0.0
0.0
filename
4spheres
Ellipses
ARI
0.7
Spheres+Grid
Halved Circle
2Spheres2Circles
0.6
Spaceship
SphereinCircle
0.5
0.5
0.4
0.1
Interpolation coefficient
0.2
0.3
0.4
0.6
0.025
0.060
0.075
0.100 0.125 D.1.0 a. 0.002
delta
0.95
0.95
0.90
filename
4spheres
0.80
0.85
Ellipses
ARI
ARI
Spheres+Grid
0.75
Halved Circle
0.80
2Spheres2Circles
Spaceship
0.70
SphereinCircle
0.75
0.65
0.60
0.70
Feature Damping Coefficient
0.2
0.8
0.8
1.0
Clustering Method
spoctral
Figure 10: Hyperparameter robustness of TOPF the TCBS. TOPF performs reasonably well across large range around the default parameters (Interpolation coefficient: 0.3, 8: 0.07, damping coefficient (non-negative): 0, Clustering method: splctral). The clustering method is used on the TOPF vectors to determine the cluster-ids.
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-knoww knowledge can enter the TOPF pipeline. We have performed additional experiments to validate the robustness of TOPF to moderate changes parameters, see Figure 10.
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Point-Level Topological Representation Learning on Point Clouds
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. (Grande & Schaub, 2023a)):
I. 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 suffices when we are looking at loops of protein chains. On the other hand, if we working with voids and cavities in 3d histological data, we need d and thus compute 2-dimensional homology.
II. Algebraic topology tells us that there are no closed n-dimensional submanifolds of R", Hence their top-homology will always vanish and all interesting homological activity will appear for d < n.
III. 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 8 In step of the algorithm, we normalise and threshold the harmonic representatives. After normalising, the entries of the vectors lie in the interval of [o, 1]. The thresholding parameter 8 now essentially determines an interval of [0, 6] which we will linearly map to [0, 1]. while mapping all entries above å to 1 as well. This is necessary as most of the entries in the vector ef 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 a single entry. However, because most of the entries of ej. are concentrated around 0, small changes in 5 will not have a large effect and we chose 8 0.07 all our experiments.
Interpolation coefficient The interpolation coefficient À E (0, determines whether we build our simplicial complexes close the birth the death of the relevant homological features time t b'-Àd. This then in turns controls how localised smooth the harmonic representative will be. In general, the noisier the ground data is the higher we should choose X. However, TOPF is not sensitive to small changes in A. We have picked À = 0.3 for all the quantitative experiments, which empirically represents a good choice for a broad range of applications.
Feature selection factor 3 Increasing leads to TOPF preferring to pick a larger number of relevant topological features. Without specific Nomain-knowledge
represents a good choice.
Feature selection quotients max total cotal_quot, mii_mmilllll__aaaa pul__uutt,t and min 0 ratio These technical hyperpa- rameters controlling the feature selection module of TOPF. For technical account of them, Appendix G. In most of the cases without domain knowledge, they do not have 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 wa weights discussed in Appendix H.
Ablation study In Table we performed an ablation study and benchmarked TOPF against TOPF while skipping the harmonic projection step with the Hodge Laplacian (Step 3.). The result show that in general, skipping the step with the Hodge Laplacian decreases the performance of TOPF by wide margin. The exception to this are Sphere InCi rcle e or 2Spheres PeeesCCrccess where the two methods have similar performance. In these two last examples, points are directly sampled from a manifold, and thus the loops and spheres are very "thin". Thus, the homology generator is already good approximation of all the simplices responsible for a topological feature. In general however, this is not the case, necessating the use of Differential Geometry and the Hodge Laplacian.
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Point-Level Topological Representation Learning on Point Clouds
E.2. Runtime
In Figure 11, we analyse the runtime of the current TOPF implementation on two point clouds from the topological clustering benchmark suite. We increase the point density while keeping the structure of the point cloud intact. The runtime of our implementation with pparsification number of points with significant constant term. The significant constant term is probably due to our implementation starting new julia kernel and communicating between python and julia. A python-only or julia-only implementation would thus speed up computations.
x 'ff'' in the regime of Figure 11 appears to scale linearly with the total
Theoretical runtime complexity
1. a). Constructing the complex:
The Vietoris-Rips complex will have 1(n*+1) simplices, where is the number of points and & the maximum homology dimension. Apart from this, there is no significant computational effort needed. Note that we can decrease this if we set maximum VR-radius, done in many works from the literature. Computing the alpha-complex in ambient dimension d is equivalent computing the convex hull in ambient dimension d + 1, which has a complexity of ~ Ond/22). For a fixed dimension d, the number of simplices is linear in n. b). Computing persistent homology and generators: As we need the homology generators, we use the recently introduced involuted persistent homology (Cufar & Virk, 2023). While the authors discuss the runtime performance, they do not give a concrete complexity bound in (Cufar & Virk, 2023). However, they deduce that it has a similar run-time to usual persistent cohomology computations, whose performance is for example discussed (De Silva et al., 2011).
2. Picking the significant generators has negligible impact.
3. Computing the harmonic projections. Given homology representative in dimension k. computing the harmonic representative amounts to computing sparse matrix vector product of BR+1 and a sparse least squares problem Ismr(Ba+1, r), i.e. solving
min ||r WER
B8+12112
where Sk+1 is the number of (k + 1) simmplicss in the simplicial complex and Bx+1 has (k: + 2)Sx+1 non-zere entries. This is sparse least-squares problem, which we solve using the iterative sparse solver smr (Fong & Saunders, 2011). Because this is an iterative solver, the authors do not give runtime complexity, but discuss the computational requirements in (Fong& & Saunders, 2011).
4. Pooling and averaging over the simplicial neighbours has negligible runtime constraints.
F. More Details on the Experiments
figh-dimensional Point Clouds In Figure 4, we TOPF on tigh-dimensional input point cloud representing the conformation space of cyclooctane (Martin & Watson, 2011). Because the ambient dimension is too large for a- a-filtrations, Vietoris-Rips filtration with more simplices is used. The VR filtration depends on the ambient dimension only for computing the distance matrix, which is a negligible part of the runtime. Hence, a similar performance can be expected for higher dimensions than 24 as well. To counteract the increase in computational complexity, TOPF automatically downsamples the point cloud using a mixture of landmark and random sampling. We see in the ISOMAP projections of Figure 4 that TOPF extracts reasonable features representing the topology of the conformation space. Setting Neelsters to 4 even reveals the manifold anomalies in separate cluster (blue), similar to the results of (Stolz et al., 2020). The runtime for this high-dimensional experiment was 189.8 seconds.
Variational Autoencoders We have trained a VAE on image patches sampled with topological structure, see Figure 12. We have sampled the image patches around two loops and connecting line. Using TOPF, could recover this structure in the latent space of the VAE highlighting how topological point features can help in explainable AI. We note, of course, that many latent spaces do not carry any higher-order topological information, making characterisation by TOPF infeasible. Rather we showed that when we expect a topological structure to exist in the data set due to some specifics of the data, we can recover this topological structure even in the latent space of the VAE using TOPF, We have repeated the experiments for a higher-dimensional latent space and received similar results, see Figure 13.
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Figure 11: Runtime on two examples from Topological Clustering Benchmark Suite while increasing point density. Left: Halved Circle, Right: 2 Spheres 2 Circles. Experiments kept the topological structure intact while increasing the number of points sampled. Both the runtime of the vanilla TOPF algorithm, as well of TOPF using automatic subsampling heuristic as implemented in the python package, are listed.
Additional heatmaps on proteins In Figure 14, provide additional experiments of TOPF on protein data. This time, we report every single feature vector produced by TOPF in a separate plot. In the experiment with Cys123, we wanted to show that TOPF can detect so-called protein pockets. Because pockets do not form holes detectable by TDA straightforward, the literature suggests adding (and later removing) these points on the convex hull, which turns the pockets into holes, see for example (Oda et al., 2024).
Performance with decreasing sampling density In Figure 15, we analyse the robustness of TOPF with respect to decrease in sampling density. We have selected 2Spheress res rrleelee rcles as the dataset with the highest original sampling density for this experiment. It shows that TOPF can still produce meaningful results even for significantly downsampled point clouds. We note that the considered point clouds are in particular very sparse when compared to the point clouds usually considered by neural 3D point cloud architectures. We interpret the results as basically repeating the good performance of TOPF, except for the datasets El psses i nF 11 pses and spacesh i pp, where the performance drops rapidly (at least in logarithmic plot). Our interpretation of this that both datasets consist of submanifolds with different sampling density, which intersect/are contained in one another. Thus, the point density is key distinguishing the classes, and a random change in these densities due to downsampling can make the encoded structure hard to detect or even causes them to vanish in sense of topology.
Performance under heterogeneous sampling In Figure 16, we analyse the robustness of TOPF with respect to hetero- geneous sampling. The results show that TOPF performs well even in the presence of inhomogeneous sampling, and the degrading of performance can largely be attributed to decrease in minimum sampling rate, rather than the inhomogeneous nature.
From a theoretical point of view, this should not be surprising: The birth of the topological feature will now depend on the minimum sampling density, while the death time still depends on the size of the feature. The heuristic employed by TOPF will then choose a maximal triangulation radius which lies between these two radii. In the dense part, the triangulation will be denser then in the sparse parts. However, due to the the character of the alpha filtration, the weighting of the simplices discussed in Appendix H and the thresholding this does not cause an issue in the features generated by TOPF.
Details on experiments with WSDesc WSDesc (Weakly Supervised 3D Local Descriptor Learning for Point Cloud Registration), (Li et al., 2022), uses voxel-based representations of point clouds to extract robust 3D local descriptors for point cloud registration. As showcased in the original paper, WSDese showcases a good generalisation for point clouds outside of the training set. We used WSDese pretrained on the 3DMatch data. WSDess was used with 512 keypoints, and
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Point-Level Topological Representation Learning on Point Clouds
3.
Variational Auto Encoder 2.
Latent space with TOPF features
1.
TOPF features overlayed on sampling locations
4.
Sample square image patches around the points marked in red
=> TOPF can extract topological information fron latent spaces of NN and autoencoders
Using TOPF to explore the topological structure in latent/embedding spaces of Variational Auto Encoders (VAE) 1. Given a picture (of cow strolling along the shore of Lake Prags), we sample 578 square patches around the centre points marked in red in the image. The centre points are taken from the sides two squares and a line connecting the two squares. The assumption is that this topological structure (with two holes) is present in the sample space as well. 2. We train a VAE on the set of 578 square patches with latent space dimension of (down from 33 -3.) We run TOPF [0, 2] on the latent space to extract the two most significant point-wise topological
Figure 12:
with f fixed_num_feeee _____n____________ mum_features features. 4. We overlay the topological features from the latent space over the centre points of the corresponding image patches. We see that TOPF has roughly recovered the topological structure inherent in the sample space as described in step 1. We note that the image patches with centre on the middle horizontal line are almost identical, making it virtually impossible to distinguish them. Note that although TOPF thinks they contribute to the left loop, their corresponding feature entries are lot smaller than those the image patches (shown by lighter colours and smaller dots the plot.)
tested with a varying of keypoints up to ~ 5000. We thus tested WSDesc with the maximum number of keypoints possible per input point cloud.
Details on experiments with DGCNN We use the implementation An Tao, htttps://githhe Agenn.pptoorn pytppcc of Dynamic Graph CNNs (Wang et al., 2019). We use the data pretrained on the ShapeNetPart segmentation dataset, where we use the features of the second last layer. This showed the best performance. We determine the optimal hyperparameters (k, object class, clustering method, layer to use) for every dataset of TCBS individually and only present the results the best parameter set. The spectral clustering always outperformed k-means. The authors of the original paper state that DGCNN performs very well on datasets with above 500 points, and still is robust to downsampling beyond this point. Thus we always input the maximum available number of points to DGCNN, As described in (Wasserman, 2018), we scale the input point cloud to into a unit sphere. We pad the 2d data sets with zeroes in the third dimension. TOPF still outperforms DGCNN on the TCBS. We believe that this is due to the heterogeneous sampling, small sampling set size and most importantly the lack of training data for the neural network.
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Figure 13: VAE experiment described in Figure 12 for 16-dimensional latent space of the VAE and the original -in-nimmensional image space. Left: The first three of the 16 dimension of the latent space embeddings, colour represents the fourth dimension. Centre left: The weighted harmonic representatives of the selected topological features. Centre right: The features produced by TOPF overlayed the centre pixels of the associated image patches, done in Figure 12 Step 4., just without the picture of the cow. Right: TOPF features obtained from the original dimenniimnni image space. This shows that TOPF can help analyse topological structure in high-dimensional data.
Figure 14: TOPF heatmaps for three proteins. Top left: NALCN channelosome (Kschonsak et al., 2022) Top right: Mutated
Cys123 of E. coli (Hidher al., 2007), with convex hull added during computation, only 2-dimensional homology features Bottom: GroEL of E. coli (Chaudhry et al., 2004) (Selected features).
G. How to pick the most relevant topological features
Simplified heuristic The persistent homology Pt module in dimension & is given to us as list of pairs birth and death times (b/,at) We can assume these pairs are ordered in non-increasing order of the durations 18 db b/. This list
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Performance of TOPF while decreasing sampling density. The z-axis is scaled logarithmically and TOPF achieves an adjusted rand index (ART, random algorithms achieve ART of over 0.9 for ~ 700 samples, down from 4600 original points on the 2Sphe resperre2Cinco wersseirrrlel. rele relesDataset. The smallest considered sample size was 16 points. The points were sampled at random. We ran each experiment 100 times and report the standard deviation.
Figure 15:
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 long time, indicating that they important for the shape of the point cloud. Distinguishing between the relevant and the irrelevant features is in general difficult and may depend additional insights the domain of application. In order to provide heuristic which does depend any a-priori assumptions on the number of relevant features we pick the smallest quotient of = 1/41/15 > 0 as the point of cut-off N& == arg min, g.. The only underlying assumption of this approach is that the band of "relevant" features separated from the "noisy" homological features by 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 find the wrong drop-off. We propose mitigate this issue by introducing a hyperparameter Rzo- We then define the i-th importance-dro- quotient of by
tht = t (1 8/1).
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Figure 16: Performance TOPF under heterogeneous sampling. Top: We divide four of the point clouds with a symmetry into two halves. We downsample only one of the halves, creating sampling irregularities within the individual topological features. We repeat the experiment 100 times and report the achieved ARI. Note that the lowest value on the r-axis means downsampling by factor of 100, meaning that the smaller point clouds will then almost only consist of one half. We compare the performance of TOPF with the other basclines, noting that TOPF outperforms them for reasonable heterogeneities. Bottom Left: 4spheres with downsampling factor 0.2 with true labels. Bottom Right: 2spheres2circles with downsampling factor 0.1 with true labels.
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The basic idea is now consider the most significant N₁ homology classes in dimension & when setting N₁. to be
N2 arg min
Increasing 3 leads the heuristic to prefer selections with more features than with fewer features. Empirically, we still found 3 = 0 to work well in broad range of application scenarios and used it throughout all experiments. There are only few cases where domain-speciff knowledge could suggest picking larger 3.
To catch edge cases with multiple steep drops or a continuous transition between real features and noise, we introduce two more checks: We allow a minimal vf of min_rel_quot = 0.1 and a maximal quotient <i/ of max_tottal between any homology dimensions. Because features O-dimensional homology are often more noisy than features in higher dimensions, we add minimum zero-dimensional homology ratio of min_ 0 ratio = 5, i.e. every chosen O-dimensional feature needs to be at least min O ratio more persistent then the minimum persistence of the higher-diment gigher-dimeniioal features. Because these hyperparameters only deal with the edge cases of feature selection, TOPF is not very sensitive to them. For all our experiments, we used the above hyperparameters. We advise to change them only in cases where one has in-depth domain knowledge about the nature of relevant topological features.
= 10
Fixed number of topological features Alternatively, it is possible to specify a fixed desired number N₂ of topological features per dimension d. TOPF then automatically returns the Na most relevant features in dimension d. For practical purposes, we can then weigh these features by a function w--) in the life time d; - b, or the life quotient d./ of the features. Possible picks for include an exponential function w<(x) = e", quadratic function w(x) = x²² linear function w(x)= shoet-lived features will be given too much weight in comparison to the few more relevant features. Selection the best weight function is an interesting open problem for future work.
scaled sigmoid function. Our intuition suggests that when picking functions like a linear function, the many
H. Simplicial Weights
In an ideal world, the harmonic eigenvectors in dimension & would be vectors assigning 11 to all k-simplices contributing to k-dimensional homological feature, 0 to all k-simplices not contributing or orthogonal to the feature, and 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 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 a-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.
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 E St W = diag(w), the symmetric weighted Hodge Laplacian (Schaub et al., 2020) takes the form of
W1/2 + W-1/2 L = W W1/2
1-22
Because we want the homology representative to lie in the weighted gradient space, we have to scale its entries with the weight and set , = W e/ With this, we have that
B
W2/2 2/ = BL W 2/2 = B =0
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 + 1-simplices the simplex is contained in:
ws (Ox) = 10:0=11/00 1/(Hou~1 E Stt : os C Co 31 1)²
where the +1 is to enforce good behaviour at simplices that are not contained in any higher- simplices. One of the advantages of the a-complex is that we don't 't have large concentrations of simplices in well-connected areas. The proposed
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Point cloud
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Figure 17: Effect of weighting a simplicial complex on harmonic representatives. Red edges have large absolute values and blue edges have small absolute values of the considered quantity. Top: VR complex. Bottom: a-complex Left: The base point cloud with different densities. 2nd Left: Unweighted harmonic homology representative of the large loop. 3rd Left: Effective resistance of the 1-simplices. 3rd Right: Harmonic homology representative of the complex weighted by effective resistance. 2nd Right: Inverse of number of incident triangles (Definition H.1). Right: Harmonic homology representative of the complex weighted by number of incident triangles. Up to a small threshold, the standard harmonic representative the VR complex is almost exclusively supported in the low-density regions of the simplicial complex. This leads to poor and unpredictable classification performance in downstream tasks. contrast, the harmonic homology representative of the weighted VR complex has more homogenous support along the loop, while still being able discriminate the edges not contributing to the loop. The a-complex suffers less from this phenomenon (at least in dimension 2), and hence reweighing not necessarily required.
weighting wa is computationally straightforward, as it can be obtained as the column sums of the absolute value of the boundary matrix [Bx]- The weights also deal with the previously mentioned problem II: As the homology representative scaled inversely to the weight vector w. the simplices high-density regions will be assigned a low weight and thus their weighted homology representative will have larger entry. By the projection to the orthogonal complement of the curl space, this large entry then diffused among the high-density region of the sc with many simplices, whereas the lower entries of the simplices in low-density regions only diffused among fewer adjacent simplices.
(Paik & Park, 2023) consider the dual problem of weighting the simplicial complex such that a harmonic cohomology representative becomes sampling-densiee independent. They show that the dual to our approach, i.e. by dividing by the sum of lower-incide incident simplices (i.e. nodes) of the two vertices of an edge produces an harmonic representative stable under heterogeneous density.
However, the first weight able incorporate the number of parallel simplices into the weighting. This is why propose second simplicial weight function based generalised effective resistance.
Definition H.1 (Effective Hodge resistance weights). For simplicial complex with boundary matrices (Bt), we define
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the effective Hodge resistance weights wR k-simplices to be:
WR == diag (B k-- B)-1)²
where diag(-) denotes the vector of diagonal entries and
(-) y denotes taking the Moore-Penrose inverse.
assume that every edge has resistance of 1 and then the effective resistance coincides with the notion from Physics. Thus simplices with many parallel simplices are assigned small effective resistance, whereas simplices with few parallel simplices are assigned an effective resistance close to 1. However, computing the Moore-Penross inverse is computationally expensive and only feasible for small simplicial complexes.
Intuitively for k
In Figure 17, we show that the weights wA 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 wa -weightss.
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 meaningful topological structure. Applying TOPF in these cases will produce no additional information. Other data sets require pre-rocessiig before containing topological features. In Figure left, the 2d topological features characterising protein pockets of Cys123 only appear after artificially adding points sampled on the convex hull of the point cloud (Cf. (Oda et al., 2024)).
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 a-filtrations, which become computationally infeasible for higher-dimensional point clouds. Persistent homology computations for dimensions above are only feasible for very small point clouds. Because virtually all discovered relevant homological features in applications appear dimension 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.
or 2, this does not present large problem.
Automatic feature selection is difficult without domain knowledge While the proposed heuristics works well across variety of domains and application scenarios, only domain- and problem-speciffe knowledge makes truthful feature selection feasible.
Experimental Evaluation There are no benchmark sets for topological point features in the literature, which makes benchmarking TOPF not straightforward. On the level of clustering, we introduced the topological clustering benchmark suite make quantitative comparisons of TOPF possible, and benchmarked TOPF on some the point clouds of (Grande Schaub, 2023a). On both the level of point features and real-world data sets, it is however hard to establish what ground truth of topological features would mean. Instead chose qualitatively report the results of TOPF proteins and real-world data, see Figure 3.
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Reviews:
[{"review_id": "IXGTeSnP9N", "reviewer": "Reviewer_7iPh", "summary": "Inspired by TDA, the authors proposed a point-level topological representation learning method for point cloud data analysis. Specifically, they introduced topological point features (TOPF) to extract point-level features from point clouds through discrete algebraic topology and differential geometry. The TOPF allows local feature extraction while retaining high-order geometric information. Experimental results show that TOPF outperforms existing methods in clustering and robustness tests, and is verified on synthetic data and real datasets.\n\n### Update after rebuttal\nI maintain my rating.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "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": "Thank you very much for your feedback! We are happy about the many strengths of TOPF identified by you. We will now address your comments:\n\n> Weakness 1: Although the author analyzed TOPF from a theoretical perspective, I still think that the author's TDA cannot be actually applied in the current field of point cloud analysis, and it is currently only in the theoretical analysis stage. Its high complexity and time consumption make it difficult to apply in practice.\n\nThank you for your comment! We applied TOPF on synthetic and a variety of real-world datasets and conducted experiments on large-scale point clouds in appendix E. Furthermore, persistent homology, which has similar runtimes, has been used successfully in the literature and many applications. We believe that this is convincing evidence that TOPF is a valuable scientific contribution. Because TOPF has the single goal of extracting topological information from homology, we do not claim it to be relevant in all point cloud tasks. \n\n(Indexing continues at weakness 4 in your review.)\n\n4. Please refer to our rebuttal to reviewer yYZ5. Furthermore, we evaluate large-scale scenarios of point clouds with up to **40,000** points in E.2, showing that TOPF, particularly with the proposed landmark downsampling heuristic, is **feasible even on large point clouds**.\n\n5. >This method may not be applicable to very high dimensional point clouds, \n\nThank you for your comment! **TOPF works well on high-dimensional data sets**. In the paragraph “Embedding Space of Variational Autoencoders and High-dimensional spaces” of sec. 4, we apply TOPF successfully to high-dimensional latent spaces and directly to an **8478(!)-dimensional** image space. In Figure 4, we show a projection of the results of TOPF on a 24-dimensional point cloud.\n\n> and the authors should explicitly discuss its limitations.\n\nWe agree with your comment: We already give a summary of the limitations of TOPF in sec. 5. We discuss the limitations in great detail in section I: Limitations in the appendix.\n\n6. > Is it possible to provide a detailed computational complexity analysis of TOPF?\n\nThank you for your comment! We discuss the computational complexity in detail in section E.2 in the appendix, split into the different steps of the algorithm and conduct experiments regarding the scaling behaviour. \n\n7. > Compared with existing neural network-based methods, can TOPF outperform them? Please provide relevant experiments and analysis.\n\nThank you for your comment! **TOPF outperforms neural network architectures** on the task TOPF was designed for, namely extracting topological features from point clouds. We benchmark TOPF’s performance on the TCBS against the neural network-based methods PointNet and WSDesc, see Table 1. Furthermore, we have now added a benchmark against DGCNN (Wang et al.). TOPF outperforms these approaches (pretrained on large-scale part segmentation datasets) with a mean ARI of TOPF of **0.86** to 0.44 (PointNet) 0.39 (WSDesc) 0.64 (DGCNN). We discuss this in more detail in Section 4. We have pretrained the neural architectures on large-scale shape segmentation data sets. For a detailed experimental setup, see Appendix F: More Details on the experiments.\n\n> The legend of Figure 1 is too long, and it is recommended to split it into multiple sentences.\n\nThank you for this suggestion! We hope the caption of Figure 1 to be self-contained and an explanation for the figure, thus we do not know how to further shorten it. Do you have any suggestions? We already broke down the caption into 10 short to medium-length sentences. Does this suffice for you?\n\n> The resolution of Figure 3 needs to be improved to enhance readability.\n\nWe are apologise, but we do not understand this comment. On our end, the figure has sufficient resolution and an associated file size of 1.5 MB. Could you further elaborate on your suggestion?\n\nThank you again for your review! Based on your feedback, we believe the paper is in an even stronger state than before! We believe to have addressed a majority of your concerns and would be interested in hearing back from you!", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. The author introduces some theoretical concepts of topology and persistent homology in detail.", "experimental_designs_or_analyses": "Yes. In Table 1, the authors compared TOPF clustering with other feature/clustering algorithms, showing some effectiveness.", "relation_to_broader_scientific_literature": "The authors use TDA and geometric learning for point cloud analysis. They include current literature on persistent homology, Hodge Laplace operator, and topological clustering methods.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n1. The authors propose a new approach based on persistent homology to link global topological features with point-level information.\n2. This method is robust to noise and non-uniform sampling.\n3. TOPF does not require a large amount of training data and is more interpretable and widely applicable.\nWeakness:\n1.Although the author analyzed TOPF from a theoretical perspective, I still think that the author's TDA cannot be actually applied in the current field of point cloud analysis, and it is currently only in the theoretical analysis stage. Its high complexity and time consumption make it difficult to apply in practice.\n2. At the same time, the authors use simple point cloud datasets. Can you provide the results and time complexity on ModelNet40 and ScanObjectNN, or even experimental results in large-scale scenarios?\n3. This method may not be applicable to very high dimensional point clouds, and the authors should explicitly discuss its limitations.\n4. Is it possible to provide a detailed computational complexity analysis of TOPF?\n5. Compared with existing neural network-based methods, can TOPF outperform them? Please provide relevant experiments and analysis.", "other_comments_or_suggestions": "1. The legend of Figure 1 is too long, and it is recommended to split it into multiple sentences.\n2. The resolution of Figure 3 needs to be improved to enhance readability.", "questions_for_authors": "See weakness.", "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": "", "extra_scores": {}}, {"review_id": "MGkfimJWjR", "reviewer": "Reviewer_WnWi", "summary": "The paper presents a method (TOPF) to extract point-level topological features of point clouds, i.e., to assign to each point in the cloud a feature vector that encodes to which generators of homology it contributes. Topological features are thereby computed across all scales using persistent homology on a (Vietoris-Rips or alpha) filtered simplicial complex built on the point cloud, and then, the 'correct' scale of interest is selected by a heuristic. \nTo identify, which simplices of the complex generate which topological features Hodge theory is used which ensures the existence of unique  harmonic representatives of (co)homology classes. In the last step, the topological features of the simplices in the complex are converted to topological features of the points in the underlying point cloud by an averaging procedure.  \nEmpirically, the generated features are evaluated on a clustering task and their robustness against noise, downsampling and the addition of outliers is studied.\n\n## Update after rebuttal\n\nI thank the authors for their rebuttal and the additional explanations. Unfortunately, the main issue I identified, i.e., a limited empirical evaluation, see Claims and Evidence (i), was not addressed. I will therefore keep my score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thank you very much for your detailed and thorough review. We will reply to the raised issues in as much detail as the character limit permits.\n\n**Claims And Evidence:**\nThank you for your detailed feedback! While we still believe that the listed contributions are technically true, we now realise that parts might be misleading. We omit the “provably” from (ii).\nRegarding point (iii), we interpret this differently: When downscaling and using heterogeneous downsampling, from a certain degree onwards, there is no topological signal left in the data. The other algorithms detect simpler structural features and thus are not affected. However, without topological features, the performance of TOPF degrades, which is some form of sanity check. In Figure 16, Bottom left, we see that even a downsampling factor of 0.2 significantly deteriorates the topological signal.\n\nHowever, we will write \"[...] robust to moderate noise [...]\".\n\n>  I was wondering what performance classic clustering algorithms based on manifold distances would achieve on the datasets,\n\nThis is a good suggestion: PAM using Isomap distances has a mean ARI of 0.58 with 0.39/0.56/0.78/0.13/0.63/0.58/0.94 on the individual datasets. This puts PAM+ISOMAP between the performance of clustering algorithms directly on the points (kMeans mean ARI: 0.44) and TOPF (mean ARI: 0.86). We will add this to table 1.\n\n> a short description of TPCC should be added to the main part together with a list of changes done for TOPF and their effects.\n\nThank you for this good suggestion, we will do this!\n\n**Comments and Suggestions:**\nThank you very much, we have addressed your comments.\n\n**Questions for authors:**\n> [...] Would you elaborate on how you identify homology and the kernel of the Laplacian and how this relates to Step 3 in section 3? [...]\n\nThis is a good question! Formally, the cochain space is the dual of the chain space. For finite-dimensional vector spaces with the standard inner product, there is a canonical isomorphism between the two spaces induced by sending a vector $v$ to the linear extension of the map sending $v$ to $1$. Thus, we can identify the chains and cochains. We have the boundary maps $B_i$ and the coboundary maps $B_i^T$. By the univ. coefficient theorem *real-valued* homology and cohomology are isomorphic, with $H_i=\\ker B_i/Im B_{i+1}\\cong H^i=\\ker B^T_{i+1}/Im B_i^T$. Thus, using the canonical isom. to the dual vector space, $ker B_i$ are homology and $ker B^T_{i+1}$ are cohomology representatives. $\\ker L_i$ is the intersection of $\\ker B_i$ and $\\ker B^T_{i+1}$. Hence every element in $\\ker L_i$ automatically is a homology representative. This gives us an explicit isomorphism between $\\ker L_i$ and the (co-)homology. \n\nIn step 3, we start with some arbitrary homology representative r in $\\ker B_i$. However, we want a harmonic representative h in $\\ker L_i = ker B_i\\cap ker B^T_{i+1}$ for the same homology class, i.e. with h =r - c for c in $im B_{i+1}$, the curl space quotiened out in homology. By the orthogonality of the Hodge decomposition, this c is given by the projection of r to the curl space $im B_{i+1}$.\nWe will discuss this in step 3 of section 3 and in more detail in appendix B.2!\n\n> I am specifically curious why TOPF is able to outperform TPCC on some datasets (HalfCircles, Ellipses, ...) by a large extent but achieves similar performance on SphereInCircle .\n\nAs you mentioned, we already give a brief comparison. The performance differences of TPCC and TOPF on different comes mainly down to how “difficult” and “robust” the topological structure encoded in the datasets is. In particular, 2Spheres2Circles and SphereInCircle are directly sampled from unions of manifolds without any noise and sufficient sampling density. In comparison, in  Ellipses and Spaceships, the topological features have shorter life times and live at different scales. There is no single scale containing all features, and for most holes/voids, there is not even a scale whithout noisy holes with small persistence present in the filtration. TOPF can deal with these situations, whereas TPCC cannot. For the difference on HalvedCircle, we posit that this is due to the better feature aggregation of TOPF. TPCC requires to perform subspace clustering on an harmonic edge embedding, which is both unstable and sensitive to parameters, which we think TPCC has no information to choose correctly in this setting. \n\n> Do you have an explanation or conjecture why TOPF performs rather weak on the 4Circles+Grid data?\n\nFigure 8&9 show the ground truth and TOPF clustering on 4Circles+Grid. In short, TOPF chooses suboptimal scales which are not well-enough connected. Figure 10 supports this interpretation, increasing the interpolation hyperparameter lambda improves TOPF performance on this dataset. We have used a fixed set of hyperparameters for all experiments on the TCBS for transparency. We will add an extended discussion of this to the paper.", "claims_and_evidence": "The paper makes the following claims (in the 'contributions' paragraph)\n1. \"*TOPF (i) outperforms other methods and embeddings for clustering downstream tasks on topologically structured data.*\"\n2. \"*it 'returns provably meaningful representations*\"\n3. \"*it is 'robust to noise and heterogeneous sampling*\"\n\nRegarding 1. This is true for the clustering tasks studied in the empirical evaluation. However, the underlying data is introduced with this work and specifically designed such that the method can shine. That is, the only signal in the data is the topological information, see Figure 8.\nTherefore, the **claim (1) is not wrong, but the bar is set rather low.**\n\nRegarding 2. This likely refers to Theorem C.1 in the appendix which states that the method returns correct results when points are sampled uniformly from a sphere in $\\mathbb R^d$. While it seems plausible that the method is correct under more general settings, ***provably meaningful representations* are showed only in this very simple setting**.\n\nRegarding 3. This is true for the addition of outliers and noise in Figure 6. The downsampling results in Figure 5 are mixed, however. The method outperforms the baselines to a large extent under moderate downsampling (factor 1--10) but fails (much more than the baselines) for larger factors. \n\nHowever, while the author exaggerate when listing the contributions, in the remainder of the paper they are actually very nuanced and careful about the scope, limitations and underlying assumptions of their method. This gives a very positive and honest impression.", "methods_and_evaluation_criteria": "The chosen evaluation criteria do make sense. That being said, the empirical evaluation is of too limited scope to verify the claims from above, as (aside from the qualitative evaluation in Figure 3) the generated features are only evaluated with respect to clustering on one dataset which was specifically designed so that the method can shine (see above).  \nIt is surprising that the generated features are not evaluated for different tasks, particularly as a central motivation to this work is that *\"common machine learning applications like classification require point-level information and features\"*. When reading this sentence in the abstract I was already expecting experiments which verify that the topological features computed by TOPF provide complementary information that is beneficial for these common machine learning tasks the authors had in mind.\n\nRegarding the clustering results in table 1. I was wondering what performance classic clustering algorithms based on manifold distances would achieve on the datasets, e.g., PAM using distance estimates from Isomap (without the MDS step) or something similar.", "theoretical_claims": "Theoretical results (even the statements) are only presented in the supplementary material, which according to the reviewing instructions \"encouraged (but not required) to read\". Due to the high reviewing load I decided not to thoroughly check the these results.", "experimental_designs_or_analyses": "There are no issues apart from the ones already listed in §Methods And Evaluation Criteria.", "relation_to_broader_scientific_literature": "The paper is situated in the domain of topological data analysis and introduces a method to generate point-level features that encode topological information. The connection to machine learning consists in that these features can be used by machine learning models and that this is done in the experiments in this paper. From my perspective, ICML might not be the best fit for the paper, but it still fits. \n\nClosely related is work by Grande & Schaub (Topological point cloud clustering, ICML 2023) who also cluster points based on topological features computed from the Hodge laplacian and proceed in similar steps, i.e., building a complex, computing homology representatives using the laplacian, aggregating information from the complex to the point cloud. In fact, the authors list sevaral limitations of the former (TPCC) and state that they revamp their pipeline in a way that resolves them, but they do so only in the supplementary material.  \nGiven the close relation between these two works, a short description of TPCC should be added  to the main part together with a list of changes done for TOPF and their effects.", "essential_references_not_discussed": "I am unaware of missing related work.", "other_strengths_and_weaknesses": "A major strength of this work (which regrettably did not fit into the previous points) is the proposed method itself. The way the point features are computed  leads to interpretable features, appears to be easy to implement, computationally efficient and overall, a good idea. In addition to the usefulness of the features for downstream learning tasks (which as discussed above should have been analyzed more extensively) they can serve also for visualization purposes, cf. Figures 2 & 3. Moreover, source code is provided.", "other_comments_or_suggestions": "Typos: In the caption of figure 5, there is a missing ':' after 'Left'  \nThere appears to be a typo in Step 3 regarding the multiple usage of $\\epsilon$ around line 261 and one has to guess what is actually meant there.", "questions_for_authors": "- When you describe the main ideas in section 2, you relate the kernel of the Laplacian to the homology groups. However, the Laplacian typically operates on **co**chains and thus, Hodge isomorphy typically holds between **co**homology and the space of harmonic forms. Would you elaborate on how you identify homology and the kernel of the Laplacian and how this relates to Step 3 in section 3? Without this information, I found the latter (which is a key component of the method) a bit confusing. (I am aware of your comparison with de Rham homology in Appendix B.2 but did not find the neccessary information there). \n\n- Would you please compare your method to TPCC by Grande & Schaub. I am specifically curious why TOPF is able to outperform TPCC on some datasets (HalfCircles, Ellipses, ...) by a large extent but achieves similar performance on SphereInCircle . \n\n- Minor: Do you have an explanation or conjecture why TOPF performs rather weak on the 4Circles+Grid data?", "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": "", "extra_scores": {}}, {"review_id": "00JRShOo7h", "reviewer": "Reviewer_u1pt", "summary": "The paper introduces TOPF (Topological Point Features), a method for extracting point-level topological features from point clouds using tools from algebraic topology and differential geometry. The authors propose leveraging persistent homology and harmonic representatives from the Hodge Laplacian to relate global topological structures to local point features. Experiments on synthetic and real-world datasets demonstrate that TOPF outperforms existing methods in clustering tasks, exhibits robustness to noise and heterogeneous sampling, and scales to high-dimensional data. A new topological clustering benchmark suite is introduced to evaluate performance.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "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": "Thank you very much for your careful review and feedback!\n\n> Weakness: It could be better to analyze the runtime of the computational complexity of persistent homology regarding different point cloud sizes.\n\nThank you for this suggestion! We analysed the computational complexity in appendix E.2 in detail, broken up into the different steps of the TOPF pipeline. Furthermore, we provided numerical experiments for runtime while increasing the size of the point cloud up to 40,000 points in Figure 11. Finally, we provide the number of points in the point clouds of the Topological Clustering Benchmark Set in Figure 8 in the appendix and provide the mean runtime of TOPF in Table 1.\n\nWe hope to have addressed your concerns and want to thank you once again for your review!", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "I'm not an expert in this field, which cost me a lot of time to understand the meaning of each theoretical claim. I didn't recognize any obvious proof of error.", "experimental_designs_or_analyses": "The experimental design is sound.", "relation_to_broader_scientific_literature": "The work builds on persistent homology [1] and harmonic representatives [1], extending TPCC (Grande & Schaub, 2023a) by reducing computational cost and improving robustness.\n\n[1] Edelsbrunner H, Harer J. Persistent homology-a survey[J]. Contemporary mathematics, 2008, 453(26): 257-282.\n[2] De Silva V, Vejdemo-Johansson M. Persistent cohomology and circular coordinates[C]//Proceedings of the twenty-fifth annual symposium on Computational geometry. 2009: 227-236.\n[3] Grande V P, Schaub M T. Topological point cloud clustering[C]//Proceedings of the 40th International Conference on Machine Learning. 2023: 11683-11697.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strength:\nThis paper introduces a novel benchmark suite for topological clustering.\nThe paper is well-written, with comprehensive theoretical analysis and visualizations to support the hypothesis.\n\nWeakness:\nIt could be better to analyze the runtime of the computational complexity of persistent homology regarding different point cloud sizes.", "other_comments_or_suggestions": "See the comment above", "questions_for_authors": "See the comment above", "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": "", "extra_scores": {}}, {"review_id": "blqvPZbpcP", "reviewer": "Reviewer_yYZ5", "summary": "The paper proposes to extract point-level features given the global structure of the point cloud, using concepts from algebraic topology and differential geometry.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "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": "Thank you for your review and your valuable feedback!\n\n> The experiments and the visualization are good. I wonder whether it is possible to evaluate on more diverse tasks, like ModelNet40 classification, ShapeNet segmentation, and S3DIS segmentation, like many point cloud papers evaluate.\n\nThank you for your comment! The point clouds in datasets like ModelNet40 or ShapeNet have none to very few topological features, while the neural architectures learn to extract some different form of features. However, TOPF was specifically design to extract the topological features from homology. It might be interesting to extend the ideas from TOPF to geometrical information as well in future work, but this will require many new ideas and is out of scope for the current paper. The above is the reason why we have introduced the novel Topological Clustering Benchmark Suite to benchmark TOPF.", "claims_and_evidence": "The proposed method can compute point-level topological features conditioned on the global topological structures of the point cloud. The proposed method outperforms other methods in down streaming tasks. It also achieves provably meaningful representation, and is robust to noise.\n\nThe design of the module is grounded with strong theoretical foundation and explained clearly. The qualitative results and experiments verify the effectiveness of the proposed method.", "methods_and_evaluation_criteria": "The proposed method is novel and grounded with theoretical foundation. What I found interesting and important is, the proposed module does not require training.", "theoretical_claims": "The theoretical claims make sense to me. But the detailed math is not carefully checked.", "experimental_designs_or_analyses": "The experiments and the visualization are good. I wonder whether it is possible to evaluate on more diverse tasks, like ModelNet40 classification, ShapeNet segmentation, and S3DIS segmentation, like many point cloud papers evaluate.", "relation_to_broader_scientific_literature": "This work seems to be very useful in point cloud segmentation tasks, because it can extract pointwise feature conditioned on the global information. I will be very interested in seeing the performance of the proposed method on a point cloud segmentation task.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "No", "other_comments_or_suggestions": "As mentioned in the previous sections, I found the proposed method interesting and strongly grounded, while showing an experiments on point cloud segmentation will make the paper much stronger.", "questions_for_authors": "No", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The draft proposes the association of a vector of topologically derived features to points in a cloud as a means to enrich the representation for downstream tasks such as segmentation.
I'm somewhat divided in my appreciation of the novelty of the main idea/rationale as some other authors (including myself) have used similar tricks of associating vectors with some topological information to our observations in order to achieve some desired boost to whatever the task at hand. But at the same time, the methodology itself is clearly novel. Further, in line with this year's guidelines for meta reviews, the paper is certainly sound and correct. I suppose this dichotomic feeling is somewhat backed by the original split scores (two 2 and two 4; although one of the 2 was later increased after the rebuttal).
Among the things appreciated by the reviewers are the sensible theoretical claims, the well-explained approach (to which I add it is supportive of the replicability), and the lots of good stuff in the appendices. On the weaknesses, unfortunately, the results shown in Table 1 are disparate and there are no inferential statistics to support empirical claims.
When it comes to the rebuttal it is clear that the authors played a strategy of providing strong answers to those comments they appear to like, but attempted to discredit those they did not like instead of affording a credible answer or simply disregarding them e.g. "we have addressed your comments" yet I can't see any explanation. This results in a mixed rebuttal with lights and shadows.
Decision: Accept (poster)