Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds
Authors: Naoki Nishikawa
Yuichi Ike
Kenji Yamanishi
Venue: NeurIPS 2023 poster (2023)
Keywords: point cloud
persistence homology
isometry-invariant networks
filtration learning

Abstract: Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.

Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds
Naoki Nishikawa
Yuichi Ike Kyushu University
The University of Tokyo kishikawa-naoki ecc. u-tokyo.ac. ac. jp
ike@imi kyushu- ac jp
Kenji Yamanishi
The University of Tokyo y yamanishi Seiiiiimisttt.. w..-t..yo...cc.o jc.jp
Abstract
Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective incorporate global topological features, which are typically extracted by persis- tent homology. In the calculation of persistent homology for point cloud, we choose filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose framework that learns filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invarrint, we develop a neural network architecture with such invariance. Additionally, we show a thcoretical result on shite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experi- mental results demonstrated the efficacy of our framework in several classification tasks.
Introduction
Analysis point clouds (finite point sets) has been increasing its importance, with many applications in various fields such shape recognition, material science, and pharmacology. Despite their importance, point cloud data were difficult to deal with by machine learning, in particular, neural networks. However, thanks to the recent development, there have appeared several neural network architectures for point cloud data, such as DeepSet (Zaheer et al.) 2017). PointNet (Qi et al.) 2017a), PointNet++ (Qi al. 20176), and PointMLP Ma et al. (022). These architectures have shown high accuracy in tasks such as shape classification and segmentation.
In point cloud analysis, topological global information, such as connectivity, the existence of holes and cavities, is known to be beneficial for many tasks (see, for example, (Hiraoka et al, 2016 Kovacev-Nikolic et al. 2016)). One way incorporate topological information into machine learning is to use persistent homology, which a central tool in topological data analysis and attracting much attention recently. By defining increasing sequence of spaces, called filtration, from point cloud X C Rdd we can compute its persistent homology, which reflects the topology of X. Topological features obtained through persistent homology are then combined with neural network methods and enhance the accuracy of many types of tasks, such as point cloud segmentation (Liu et al., (022). surface classification /Zepellzauer et al. 2016), and fingerprint classification (Giansiracusa al. 2019).
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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4.87
Rips
4.13
-4.00
(a)
2.51
Procedure to utilize our framework to a classification task and its comparison to when we use Rips filtration. For given set of points, define a value called "weight" for each point in (a), using neural network method. In (b), a ball centered at each point is gradually expanded. The balls centered at points with larger weights expand later than the balls centered at points with smaller weights. In (c), we examine the topology of the sum of the balls as they grow larger, and aggregate the information into a diagram called persistence diagram. to vector, which can be used as input of machine learning models. When we use standard Rips filtration, the balls expand uniformly from all of the points, so that we cannot correctly capture the large global hole. On the other hand, when classification task chosen, and the weights on the outliers get larger. As a result, we can capture the large global hole in the point cloud. Note that the resulting weights assigned to each point not pre-specified, but are fully learned from the data.
Figure 1:
(d), the persistence diagram is converted
use our method, the suitable filtration to solve
The standard pipeline to use persistent homology in combination with machine learning is split into the following:
ii) define filtration for data and compute the persistent homology;
(ii) vectorize the persistent homology;
(iii) input it into machine learning model.
In (i), one needs to construct appropriate filtration depending on a given dataset and task. A standard way to define filtration is to consider the union of balls with centers at points in X c Rd with uniform radius, i.e., take the union S, = Urex (xx; t), B(x; t) = {y E Rd |l - x s t} and consider the increasing sequence (S:).. Regarding (ii), one also has to choose a vectorization method depending the dataset and the task. For example, such as persistence landscape and persistence image. As for (iii), we can use any machine learning models such linear models and neural networks, and the models can be tuned with the loss function depending on a specific task.
use typical vectorization methods
Recently, several studies have proposed ideas to learn not only (iii) but also (i) and (ii). Regarding (ii), Hofer et al. (2017) and Carrière et al. (020),, for example, have proposed methods learn vectorization of persistence homology based on data and tasks. Furthermore, for (i), Hofer er al. (2020) have proposed a learnable filtration architecture for graphs, which gives a filtration adaptive to given dataset. However, this method heavily depends properties specific to graph data. To establish a method to learn filtration for point clouds, we need to develop a learnable filtration architecture for point cloud incorporating the pairwise distance information. In this paper, we tackle this challenging problem to adaptively choose a filtration for point clouds.
1.1 Contributions
In this paper, we propose novel framework to obtain topological features of point clouds based on persistent homology. To this end, we employ a weighted filtration, whose idea is to take the union of
balls with different radii depending points. Choosing weight function X + R. we can define

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filtration (St), by
S = U B(55--u0 - ZEX
This type of filtration often extracts more informative topological features compared to non-weighted filtrations. Indeed, a weighted filtration plays an important role in various practical applications (Anai
al. 2020 Nakamura et al. 2015 Hiraoka et al., 2016 Obayashi al. 2022). Then, we introduce a neural network architecture to learn a map that associates a weight function f(X,-) with a point
cloud X CR.
The desirable property for network f' is isometry-invarincee, i.e., J(TX,Ta) = f(X, .) for any isometric transformation T of IR. so that the associated weighted filtration is isometry-invarian To address this issue, we implement function f with network architecture based on distance matrices. Then, the resulting weight function satisfies the show theoretical result about the approximation ability for wide class of weight functions, which supports the validity of our architecture.
rigorously. Additionally, we
Our main contributions are summarized as follows:
1. We propose a novel framework to obtain adaptive topological features for point clouds based on persistent homology. To this end, introduce an isometry-invarian network architecture for weight function and propose a way to learn a weighted filtration.
2. We theoretically show that any continuous weight function can be approximated by the composite of two continuous functions which factor through finit-ddimensional space. This theoretical result motivates our architecture.
3. We conducted experiments on public datasets and show that the topological features obtained by our method improved accuracy in classification tasks.
Figure T presents one typical use of our framework for classification task. Our method first calculates the weight of each point using a neural network model. The colored point clouds in the second row depict the assigned weights for each point. In the bottom one, our architecture assigns large weights on the outliers in the point cloud, which implies that our method can choose the filtration adaptively to data and tasks. Although the DTM filtration is also effective in the example shown in Figure | our method can learn a filtration appropriately for a given task and data a supervised way. Therefore, our method would be effective for data and tasks with difficulties other than outliers. In $5) we will show that our method improves the classification accuracy compared the Rips or DTM filtration in several experiments.
1.2 Related work
neural network architecture for finite point sets was first proposed by Zaheer et al. (001 7). as DeepScts. Later, there appeared several geometric neural networks for point clouds, such as Point-
Net (Qi et al. 2017a), PointNet++ (Qi et al. 20176), and PointMLP (Ma et al. 2022). These studies addressed the rotation invariance by approximating rotation matrices by neural network. (Xu et al. 202T) used Gram matrices to implement an isometry-inver invariant neural network for point clouds. We instead use distance matrices for isometry- -invaraat network architecture since the pairwise distance information is important in persistent homology.
Regarding vectorization of persistent homology, in the past, it has usually been chosen in unsupervised situations, as seen in Bubenik and Kim (007); Chung et al. (2009); Bubenik et al. (2015) Adams et al. 2017); Kusano al. (201)). However, recent studies such Hofer et al. (2017) and Carrière
ct al. (0200 have proposed end-to-end vectorization methods using supervised learning. In particular,
Carriere et al. (2020) used the DeepSets architecture Zaheer et al. (207). whose input is a finite (multi)set, not a vector, for vectorization of persistent homology to propose PersLay. Although our method also depends on DeepSets, it should be distinguished from PersLay; our method uses it for learning a weight for computing persistent homology while PersLay learns a vectorization method of persistence homology.
For graph data, Hoter et al. 2020) proposed the idea of filtration learning in supervised way, i.e., to learn the process in (i), and this idea was followed by Horn et al. (202I) and Zhang et al. (0222) They

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introduced parametrized filtration with the use of graph neural networks and learned it according to task. The persistent homology obtained by the learned filtration could capture the global structure of the graph, and in fact, they achieved high performance on the graph classification problem. However, filtrations for point clouds cannot be defined in the same way, as they are not equipped with any adjacency structure initially, but equipped with pairwise distances. Moreover, the filtration should be invariant with respect to isometric transformations of point clouds, and we need a different network architecture to that for graphs. While many previous studies, such Bendich et al. (200)): Cang and Wei 22017); Cang et al. (2018); Meng et al. (2020), have tried to design special filtration in an unsupervised way based
data properties, we aim to leam filtration in data-driven and supervised way. To the best of our knowledge, this study is the first attempt to learn a filtration for persistent homology on point cloud data in a supervised way.
Another approach to obtaining topological features is to approximate the entire process from (i) to (iii) for computing vectorization of persistent homology by neural network ([Zhou et al. (2022) and de Surrel al. (0022). Note that similar methods are proposed for image data (Som et al.) 2020)
and graph data (an et al., 2022). Since these networks only approximate topological features by classical methods, they cannot extract information more than features by classical methods or extract topological information adaptively to data.
2 Background
2.1 Network Architectures for Point Clouds
There have been proposed several neural network architectures for dealing with point clouds for their input. Here, we briefly explain DeepSets (aheer et al.] (017)), which we will use to imple- ment our isometry-invarinn
structure. more recent architecture, PointNet (Qi et al., 2017a and PointML (Ma et al. 2022), will also be used for the comparison to our method in the experimental section.
The DeepSets architecture takes finite (multi)set X = { .xn C RRd of possibly varying size as input. It consists of the composition of two fully-connected neural networks Φ1 Rd + Rd and Φ2: Rdd + Rdd with permutation invariant operator op such as max, min and summation:
DeepSets(X) = (2(0p(((((=))))
For each 2; E X, the map ④1 gives a representation Φ (x;). These pointwise representations are aggregated via the permutation invariant operator op. Finally, the map Φ2 is applied to provide the network output. The output of DeepSets is permutation invariant thanks to the operator op, from which we can regard the input the network as a set. The parameters characterizing Φ1 and de are tuned through the training depending the objective function.
2.2 Persistent Homology
Here, we briefly explain persistence diagrams, weight filtrations for point clouds, and vectorization methods of persistence diagrams.
Persistence Diagrams Let S: Rd + R be a function on Rd. For E R, the t-sublevel set of is defined Se := {p E Rd S(p) tt. Increasing from -00 to oc gives increasing sequence of sublevel sets of Rd, called filtration. Persistent homology keeps track of the value of t when topological features, such as connected components, loops, and cavities, appear or vanish in this sequence. For each topological feature a, one can find the value b, < d, such that the feature exists in S, for b, S < da. The value b, (resp. d,) is called the birth (resp. death) time of the topological feature diagram (PD) of f. which is a multiset of the half- {b, d) d > 6}. The information of connected components, loops, and cavities stored in the Oth, 1st, and 2nd persistence diagrams, respectively.
The collection (ba, da)) for topological features is called the persistence
Given point cloud X C IRd, i.e., finite point set, one can take S to be the distance to point cloud, defined by
S(z)
min ||
x2|1

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Then, the sublevel set St is equal to the union of balls with radius t centered at points of X: Urex B(x; t), where t) == {y E R" || - yll I t}. In this way, one can extract the topological features of point cloud X with the persistence diagram. The persistent homology of this filtration can be captured by the filtration called Cech or Victoris-Ripp filtration (Rips filtration for short). In this paper, we use Rips filtration for computational efficiency. See Appendix A for details. The Rips filtration can be computed only by the pairwise distance information.
Weighted Filtrations for Point Clouds While the Rips filtration considers ball with the same radius for each point, one can consider the setting where the radius value depends on each point
(see, for example, Edelsbrunner (1992)) For instance. Hiraoka al. (2016); Nakamura al. (0155) ; Obayashi et al. (2022). which use persistent homology to find the relationship between the topological nature of atomic arrangements in silica glass and whether the atomic arrangement is liquid solid, use such a filtration. Given point cloud X C R. one can define the radius value (xx(t) at time for X as follows. We first choose a function w: X I. which called weight, and define
i w(x).
(t)())
w(x) otherwise.
The associated weighted Rips filtration is denoted by R,X, w). One choice as weight function is the distance-to- taateee-tt-mmsuur (DTM) function. The resulting filtration is called a ITM-iiltaaiion, for which several theoretical results are shown in Anai et al. (2020). One of those results shows that persistent homology calculated by the DTM-filtrations robust to outliers in point clouds, which does not hold for the Rips filtrations.
Vectorization Methods of Persistence Diagrams Persistence diagrams are difficult to handle by machine learning since they are a multiset on a half-plane. For this reason, several vectorization methods of persistence diagrams have been proposed, such as persistence landscape (Bubenik et al. 2015) and persistence image (Adams et al. 2017). Carrière et al. (2220) proposed the method called PersLay, which vectorizes persistence diagrams in a data-driven way using the structure based on DeepSets. In this paper, we utilize this method as vectorization method. Note that PersLay can be replaced with any vectorization method in our architecture.
Given a function transformation map de with a learnable parameter and permutation invariant operator op, can vectorize persistence diagram D by PersLay with
PersLay (D) = op op(()o(()))E))
we choose appropriate parametrized function de. we can make PersLay similar to classical vectorization methods such as persistence landscape and persistence image. In this paper, utilized the following map as de. which makes Persl La similar to persistence image:
exp (-
|1 c1
exp (-
111. CM 11
]
(c(g)
2
2
3 Proposed Framework
In this section, we describe our framework to obtain topological representations of given point clouds. Our idea to use persistence homology for extracting topological information and model a weight function w in 2.2 by neural network, which can be learned adaptively to data. Moreover, we propose a way to combine the topological features with deep neural networks.
3.1 Filtration Learning for Point Clouds
First, we introduce a structure to learn a filtration on point clouds end-to-end from data. For that purpose, we learn a weight function w for a weighted filtration, by modeling it by a neural network. We assume that the weight function depends on an input point cloud X (Ii,..., &N) C Rd i.e., wi is of the form f(X,-) :): Id + IR. Designing such a function by a neural network has the following three difficulties:
1. The weight function f(X, should be determined by the whole point cloud set X and does not depend on the order of the points in X.
5

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DeepSets
02
Distance Matrix
(
((X)
91 (21
DeepSets
Fulty-connected network
fo a1)
Figure 2: The architecture of the network. This figure is for the case of getting the weight of point D1 in the point cloud X. Note that d,),,;,) equal to (the transpose of) the i-th row of D(X), ie., (d(x,, 21)))
2. The resulting function should be isometry-invaricn i.e, for any isometric transformation T, any point cloud X, and x E X, it holds that f(TX, Tx) 2).
3. The output of the a)) should have both of global information X and pointwise information of x.
To address the problem transformations as problem 2 we utilize the distance matrix D(X) = (d(z,, 2z)) E R NxN
the DeepSets architecture ( ($2. To satisfy invariance for isometric
and the relative distances d( X x)) = (I|=: zy 1)}² instead of coordinates of points, since D(TX)= = D(X) and d(T', X, Tx) d(X, xx) hold for any isometric transformation T. Finally, to deal with problem 3 we regard / as a function of the feature D(X) and the feature of (() Below, we
will explain implementation in detail.
First, we implement a network to obtain pointwise feature vector of a point x e Rd by
91 (x) (22) (op(fo (||= 2 2,11)> )),
where (()) and 2f are fully-connee culnecconeected neural networks. This structure can be regarded as the DeepSets architecture whose input of mi-dimensional vectors, hence it permutation invariant.
Next, the DeepSets architecture obtain feature vector for the entire point cloud X as follows. We first implement pointwise feature vectors with the same architecture as i.c..
g2 (x;) = of (4) (op(fo
(3) (d(x,.;))))
where (3) and () are fully-connected neural networks. Then, we get feature vector h(X) for the entire point cloud X by applying the DeepSets architecture again:
h X の (op( (=)((_
where (55) fully-connected neural network. Here omit the first inner network for the DeepSets architecture since it can be combined with (((1) in g2-
Finally, we implement f as a neural network whose input is the concatenated vector of h(X) and 9i (x). Specifically, we concatenate h(X) with the feature vector (I) for x E IR to get a single vector, and then input it into fully-connected neural network (6();
= (1h()).
where 0 : (0x) is the of parameters that appear in network (1)) (k.....,,6) The overall neural network architecture is shown in Figure 2
,6.
Remark 3.1. Since our implementation of the weight function depends on distance matrices, not on the coordinates of each point, we can apply our method to dataset given in distance matrix format. In fact, we will apply our framework to such dataset in Section

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Remark 3.2. If we want to consider features other than the position information assigned to each point in the calculation the filtration weights, as in the silica glass studies, we vectorize those features and concatenate them together.
By using the weight function fo(X,,), we can calculate the persistent homology of filtration R[X, Jo(X,.) Then, we can obtain the vectorization using PersLay parametrized by c. Then, we can input this vector into a machine learning model For with parameter 0'. Given the dataset {(XR, y1)) ) the objective function be minimized can be written as
M 8 (For (PerssPayyy()X fo (x)
0(),0',c)
1 M
Σ
where l is appropriate loss function depending tasks. Typically, if the task is regression, é is the mean square loss, and if the task is classification, é is the cross entropy loss. The optimization problem
min C(O,07 0',0 0,8".c
is solved by stochastic gradient descent. Note that the differentiability of the objective function and the convergence of the optimization algorithm is guaranteed by results in Carrière et al. (202I)
3.2 Combining Topological Features with a DNN-based method
While we can use the topological features alone, we can also concatenate them with the features computed by a deep neural network (DNN). In order to do this, we need to learn not only (a) networks in our framework but also (b) DNN. Although (a) and (b) can be learned simultaneously since the output of the resulting feature is differentiable with all parameters. the networks cannot be successfully optimized our experiments. This would be because the loss function is not smooth with respect to the parameters in (a), which can also make the optimization of (b) unstable. To deal with this issue, we propose learn (a) and (b) separately.
Let us describe the whole training procedure briefly here. Suppose is a input point cloud. We write L (opp((() R& for the topological feature which comes from method, and DNN (X) E R for the feature that comes from a DNN-based method. Let be a loss function for the classification task and be number of classes. We propose the following two-phase training procedure:
1st Phase. Let Ci: RLa - Rn classifier that receives feature from VDNN. Then, C and DNN are learned through the classification task, which is achieved by minimizing DM (( (MDNN(XX)))
2nd Phase. Let C2: RL - R" be classifier. We discard the classifier C1 and fix VDNN. Then, C2 and Vtopo are learned through the classification task, which achieved by minimizing IM ((C2((PDDN) (X&)T, M Vtopo (X+) TTT), yk).
This training procedure is observed to make the optimization stable and to help the architecture to achieve higher accuracy. We show the experimental result in $
4 Approximation Ability of Weight Function
In this section, we theoretically show that our architecture has a good expression power. We prove that approximation of any continuous map be realized by idea concatenate finite- dimensional feature vectors.
Consider the space 2/0.11 of subsets in [0, 1J equipped with the Hausdorff distance. For fixed N N, we define the following subspace of 2
X = (X C 10
|X| S NJ.
The following theorem states that we can approximate any continuous function on X' [0, 1)mh by concatenating the finite dimensional feature vectors point cloud and a point. The proof of the theorem is given in Appendix B
7

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Theorem 4.1. Let f: X x 10. - R. be continuous function. Then for any >> 0, there exist
K and continuous maps 41 X 1/2 R RK+m mm
+ such that
(f(X,x) (x((vi(X) , TT) dx <e.
sup XEX [0.1jm
Experiments
To investigate the performance of our method in classification tasks, we conducted numerical experiments on two types of public real-world datasets. In $5.1 we present a simple experiment on a protein dataset to demonstrate the efficacy of our method. Next, in $5.2 we show the result of experiment on 3D CAD dataset. The classification accuracy was used as the evaluation metric.
The first experiment was conducted to show the validity of method where only used the topological feature, without DNN features. In the second experiment, on the other hand, we combine the topological features and features from DNNs using the method described in $3.2
All the code was implemented in Python 3.9.13 with PyTorch 1....2. The experiments were conducted on CentOS 8.1 with AMD EPYC 7713 2.0 GHz CPU and 512 GB memory. Persistent homology was calculated by Python interface of GUDH All of the source codes we used for experiments is publically available More details of the experiments can be found in Appendix
5.1 Protein classification
Dataset. In this section, we utilize the protein dataset in Kovacev-Nikolie et al. (2016) consisting of dense configuration data of 14 types of proteins. The authors the paper analyzed the maltose- binding protein (MBP), whose topological structure is important for investigating its biological role. They constructed dynamic model of 370 essential points each for these 14 types of proteins and calculated the cross-correlation matrix C' for each type They then define the distance matrix, called the dynamical distance, by Dij
|C,yl. and they use them classify the proteins into two classes, "open" and "close" based on their topological structure. In this paper, we subsampled 60 points and used the distance matrices with the shape x 60 for each instance. We subsampled 1,000 times, half of which were from open and the rest were from closed. The off-diagonal elements of the distance matrix were perturbed by adding noise from a normal distribution with standard deviation of O.
Architectures and comparison baselines. Given a distance matrix, we computed the proposed adaptive filtration and calculate the persistent homology. Then, the 1st persistence diagram of that filtration was vectorized by PersLay. The obtained feature was input into a classifier using linear model, and all the models were trained with cross entropy loss. For comparison, we evaluate the accuracy when we replace our learnable filtration with a fixed filtration. As a fixed filtration, we used Rips or DTM filtration (Anai al.) 2020), the latter of which robust to outliers. The hyperparameter for DTM filtration was set to maximize the classification accuracy. Additionally, we replaced our topological feature by the output D. which we call DistMatriNNet, and compared the classification accuracy. Note that we used MistMatriXNet not for the computation of filtration weights but for the direct computation of point clouds' feature. Since the protein dataset given in the format of distance matrices, we did not utilize standard neural network methods for point clouds.
Results. We present the results of the binary classification task of proteins in Table We can see that our framework overperformed the other methods in terms of the accuracy of the classification.
We describe some observations below: Firstly, the result of our method is better than those of Rips and DTM filtrations. This would be because method learned a weighted filtration adaptively to data and tasks and could provide more informative topological features than the classical ones. Secondly, method achieved higher accuracy compared to DistMatriXNet, which has similar
https://gudhi... inria.
hattp:://iitub... com/ www..ot.tttt.eeeeeo http:///wwotottooottcccctttttana
'The correlation matrix C can be found in
301543862_corz
8

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Table 1:
The accuracy of the binary classification task of protein structure. We compared our methods with NistMatrixNet and persistent homology with Rips/DTM filtration. We can see that our method performs better than other three methods.
DistMatrixNet
Rips
DTM
Ours
65.0 = 12.0
79.9 3.0 78.0 1.6 81.9 2.1
number of parameters to the proposed method. This suggests that persistent homology is essential in the proposed method rather than the expressive power of DistMatrixNet.
5.2 3D CAD data classification
Dataset. In this section, we deal with the classification task on ModelNet10 (Wu et al. 2013). This is dataset consisting of 3D-CAD data given by collections of polygons of 10 different kinds of objects, such as beds, chairs, and desks. We choose 1,000 point clouds from this dataset so that the resulting dataset includes 100 point clouds for each class. The corresponding point cloud was created with the sampling from each polygon with probability proportional to its area. Moreover, to evaluate the performance in the case where the number of points in each point cloud is small, we subsampled 128 points from each point cloud. The number points is relatively small compared with the previous studies, but this setting would be natural from the viewpoint of practical applications. We added noise to the coordinates of each sampled point using a normal distribution with a standard deviation of 0.1.
Architecture and comparison baselines. We utilized the two-phase training procedure described in $3.2. where we concatenated a feature computed by DNN-based method and topological feature. As --Nse- methods, which were trained through the 1st Phase, we utilized DeepSets (Zaheer) et al. 2017). PointNet (Qi et al.) (017a). and PointMIP (Ma et al. 2022). A given point cloud was input into our adaptive filtration architecture, and the resulting 1st persistence diagram was vectorized by PersLay. The concatenated features were input into classifier based on linear model, and the
parameters are tuned with the cross entropy loss.
We compared the result of 1st Phase, which is the accuracy of a DNN alone, and that of 2nd Phase to validate the efficacy of our learnable filtration. We also compared our method with a fixed filtration, Rips or DTM, instead of our learnable filtration. Furthermore, we replaced our topological feature replaced by the output of NistMatrixNet and compared the classification accuracy.
Results.
The results are shown in Table 2 Below, we present some observations for the results.
Firstly, our method outperformed Rips filtration, which is of the most common filtrations. Moreover, the accuracy of our method is almost the same as or higher than that of DTM filtration. While DTM filtration has hyperparameters that need to be tuned, our method achieved such result by learning an adaptive filtration automatically through the training. This result implies that our method is useful in efficiently choosing filtrations.
Secondly, our method again overperformed DistMatrixNet. This indicates that the combination of persistent homology and DistMatrixNet was crucial similarly to the protein dataset.
Thirdly, and most importantly, it can be observed that the classification accuracy better when our method was concatenated with DeepSets/PointNet compared to using DeepSets/PointNet alone. Additionally, the accuracy when we combine our method and DeepSets/PointNet is competitive when using PointMLP alone. These observations indicate that concatenating the topological features obtained by our method yields positive effects. The accuracy when we combine our method and PointMI.P is not higher than that of PointMI.P. This would mean that PointMI.P has already captured enough information including topological features during the phase, so that the information obtained by persistent homology may be redundant, potentially negatively impacting the classification.
6 Conclusion
In this paper, we tackled the problem to obtain adaptive topological features of point clouds. To this end, utilize weighted filtration and train a neural network that generates weight for each point.

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Table 2: Results for the classification task of 3D CAD data. The row named "Ist Phase" shows the results when we used the feature calculated by DNN only, and the rows named "2nd Phase" shows the results when we concatenated them with the feature calculated by DistMatrixNet or topological feature obtained by persistent homology. Compared can achieve higher accuracy when we concatenate the topological feature obtained by our method. Additionally, regardless which DNN method we use, using topological feature computed by our method improves accuracy better than using topological feature computed by Rips filtration and feature computed by DistMatrixNet. Moreover, the accuracy when we combine our method with DeepSets/PointNet is competitive with using PointMLP alone.
using the DeepSets/PointNet alone, we
DeepSets
PointNet
PointMLP
# of parameters
813,488
3,472,500
3,524,386
1st Phase
65.7 + 1.4
64.3 4.4
68.8 + 6.3
2nd Phase
DistMatrixNet
65.7 = 4.8
55.7 = 13.9
53.8 = 7.4
Rips
67.0 + 2.6
68.4 2.4
57.8 12.4
DTM
68.0 + 2.5
68.7
57.2 + 6.8
Ours
67.5 + 2.5
68.8 2.0
60.0 = 6.3
Since the model desired to invariant with respect to isometric transformation, we proposed a network architecture that satisfies such invariance. We theoretically showed fiite-dimennsionl approximation property for wide class of weight functions, which supports the validity of our architecture. Our numerical experiments demonstrated the effectiveness of our method in comparison with the existing methods.
Limitations and future work There are several limitations and remained future work for this study, First, while we concentrated on determining the weight function of the weighted Rips filtration in this paper, can also consider other kinds of filtrations. For example, we expand the balls from each point at different speeds. Second, we did not applied our method to larger datasets as we needed to compute persistent homology repeatedly in the training, which make the computational cost higher. In fact, it took about seven hours to train the neural networks our method for each cross- wross-validation. method is meaningful since it sometimes improves classification accuracy, this
While believe one of the largest limitations of our study. Addressing this issue is one our future work. Third, since method adaptively chooses filtration, framework is expected to work well even if subsequent machine learning model has simple architecture. We plan to validate the hypothesis experimentally and theoretically.
Acknowledgments
We appreciate Marc Glisse and Théo Lacombe for the helpful discussion. This work was partially supported by Grant-in-Ain for Transformative Research Areas (A) (22H05107) and JST KAKENHI JP19H01114.
10

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Reviews:
[{"review_id": "OaW79b8hm3", "reviewer": "Reviewer_xfZT", "summary": "This work proposes a novel framework to obtain adaptive topological features for point clouds based on persistent homology by introducing an isometry-invariant network architecture for a weight function and proposes a way to learn a weighted filtration. The work theoretically proves that any continuous weight function can be approximated by the composite of two continuous functions which factor through a finite-dimensional space. The experiments on public datasets shows the proposed method improves the accuracy in classification tasks.\n\nAll of my questions are carefully addressed in the rebuttal. The work is theoretically solid and has potential.", "questions": "1. What are the criteria for good weight functions ? What are the theoretical explanations for them ? \n2. For the purpose of classification, the extracted topological features improves the results. It is unclear how the loss functions are differentiable with respect to the topological features and in turn the weight functions. The homology is intrinsically discrete, the differentiability is not obvious. Some explanation will be helpful. \n3. Are the weight functions affected by the point cloud quality ?  For example, if the scanning quality is improved, how does that affect the weight function?", "limitations": "The work can be further improved by use more realistic examples and compare with state-of-the-art models.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The idea of learning filtration by the weight function is novel. The approximation ability theorem is important. The architecture design is based on this theorem and the isometry-invariance, which is rigorous and interpretable. The experimental results are convincing.  The paper is well written, all the concepts, math symbols, theorems are thoroughly explained. The deductions are clear and easy to follow. The experimental results are convincing.", "weaknesses": "It is not clear why different weight functions affect the qualities of the topological features, and what are the criteria for the persistent diagrams. Some theoretical explanations and numerical demonstration will be helpful.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the valuable feedback for improving our paper. We will reflect all the comments and suggestions in our final version. In the following, we respond to specific concerns and questions raised by the reviewer.\n\n> It is not clear why different weight functions affect the qualities of the topological features, and what are the criteria for the persistent diagrams. Some theoretical explanations and numerical demonstration will be helpful.\n\n> Question 1. What are the criteria for good weight functions? What are the theoretical explanations for them?\n\nThe persistent homology depends on the weight function; Rips filtration is the case when the weights are all zero, and some studies(, for example, Anai et al. (2020)) have used the DTM function to deal with outliers. In this study, we took the approach of learning these weights in a data-driven and supervised manner in order to increase classification accuracy. Therefore, we can say that the classification accuracy is a criterion for the weights in this case.\n\n> Question 2. For the purpose of classification, the extracted topological features improves the results. It is unclear how the loss functions are differentiable with respect to the topological features and in turn the weight functions. The homology is intrinsically discrete, the differentiability is not obvious. Some explanation will be helpful.\n\nThank you for your important suggestion. Although the homology is intrinsically discrete, the differentiability of persistent homology has already been discussed in previous studies such as [1], [2], [3], and [4]. The differentiability of the loss function in our method can be directly derived from these results. We believe that such a differentiability argument is now standard and it would not be needed in the main text. We would add such an argument in the appendix if necessary. \n\n[1] M. Gameiro et al. A topological measurement of protein compressibility. *Japan Journal of Industrial and Applied Mathematics*, 32:1–17, 2015.\n\n[2] C. Chen et al. A topological regularizer for classifiers via persistent homology. In *The 22nd International Conference on Artificial Intelligence and Statistics*, pages 2573–2582. PMLR, 2019.\n\n[3] M. Carrière et al. Optimizing persistent homology based functions. In *International Conference on Machine Learning*, pages 1294–1303. PMLR, 2021.\n\n[4] J. Leygonie et al. A framework for differential calculus on persistence barcodes. *Foundations of Computational Mathematics*, pages 1–63, 2021.\n\n> Question 3. Are the weight functions affected by the point cloud quality? For example, if the scanning quality is improved, how does that affect the weight function?\n\nWe are grateful for your intriguing question. We believe that the weight functions would be affected by the quality of the point clouds. However, we are currently not sure how the affection will be. To clarify this, in our future work, we will conduct the experiment to investigate how our trained filtration will change if the scale of the noise (for example, the variance of Gaussian noise) in the dataset is changed. \n\nWe hope that we addressed all your questions and concerns adequately. In light of our clarifications, please consider increasing your score to accept. Please let us know if \nwe can provide any further details and/or clarifications."}, {"review_id": "yyZcw3zpyP", "reviewer": "Reviewer_kFjr", "summary": "This paper investigates the extraction of global topological features using the framework of persistent homology. The authors have proposed a neural network architecture to learn the filtration weights for each point in an end-to-end and data-driven manner, which is later supported by an approximability theorem. Additionally, a two-phase training procedure is introduced to further improve the performance of the proposed architecture. The proposed framework is then applied to different tasks such as protein 3D CAD classifications. \n", "questions": "1. TDA is also used as data-analysis/unsupervised purposes (e.g., in finding enclosing holes) [A-C], I am curious whether the learned filtration weight $f(X, \\cdot)$ can reveal the true topological structures or if one can design some sort of loss function and learn the weight function accordingly?\n1. How to correctly understand the connection between the architecture (in Figure 2) and Theorem 4.1? The weighting function $f_\\theta(X, x_1)$ is a concatenation of $h(X)$ and $g_1(x)$, but in Theorem 4.1, the $f(X, x)$ can be approximated by a concatenation of $\\varphi_1(X)$ and $x$ itself. \n1. [Minor language issue] When I first read the paper, it is not clear what the “architecture” and “approximation result” in L12-13 meaning (original sentence: “Additionally, we theoretically show a finite-dimensional approximation result that justifies our architecture.”). Consider adding some details there to improve clarification. For instance, you might want to change it to something like this: “Additionally, we theoretically show a finite-dimensional approximation \\textbf{of any filtration function}, which \\textbf{justifies (or motivates) the proposed neural network architecture}.”\n\n---\n[A] Wasserman, Larry. “Topological Data Analysis.” Annual Review of Statistics and Its Application 5 (2018): 501–32.\n\n[B] Chen, Yu-Chia, and Marina Meila. “The Decomposition of the Higher-Order Homology Embedding Constructed from the k-Laplacian.” Advances in Neural Information Processing Systems 34 (2021).\n\n[C] Wu, Pengxiang, Chao Chen, Yusu Wang, Shaoting Zhang, Changhe Yuan, Zhen Qian, Dimitris Metaxas, and Leon Axel. “Optimal Topological Cycles and Their Application in Cardiac Trabeculae Restoration.” In International Conference on Information Processing in Medical Imaging, 80–92. Springer, 2017.\n\n", "limitations": "Authors have addressed the limitations of their work. Negative social impact statement is not necessary in this work, as the primary focus of this manuscript lies in its theoretical contribution.\n", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. [Originality] The filtration weight is usually constructed without considering the label information, e.g., a constant for VR complex and k-NN info for DTM. The authors proposed a framework to learn the weights for each point from the label using a neural network.\n1. [Quality] Being able to provide an approximability theorem to support/motivate the choice of the neural network architecture is nice. \n1. [Clarity] Great overview on the persistent homology as well as the limitations of the current framework. The author laid out the contribution in the very beginning of the introduction which can help present the novelty of this work. Overall, a well-written paper.\n1. [Significance] TDA framework has long been suffered from the topological noise issue. Being able to propose a way to automatically learn a weight function to suppress the noises is an important contribution to the field. \n", "weaknesses": "1. It looks like there are two contributions in this paper: 1) propose a way to learn the weighted filtration as per Section 3.1, with theoretical guarantee in Section 4; and 2) Section 3.2 in the ability to approximation with composite function. The contribution in 1) is more related to filtration learning as it learns a weighting function, but 2) is a more general use case. However, by looking at the experimental results, it is not clear which contribution is more significant, i.e., the performance gain in Table 1 comes from 1), but the gain in Table 2 comes from 2). It would be nice to consolidate the comparisons between different experiments and provide some discussions. That way, the readers can get a better understanding of the interplay between different contributions. \n1. It is not clear to me from reading the main text on how you choose the hyper-parameters for DTM. From reading the appendix, it looks like the parameters are not chosen by cross-validation (Section C.2). Is there any specific reason to not choose this parameter end-to-end using cross-validation? How will a different $q$ and $k$ affect the prediction performance in the protein datasets?\n1. It looks like Theorem 4.1 suggests the function can be approximated, but it does not mention whether it can be recovered. Will it be possible to get some results regarding the convergence (i.e., with $n \\to \\infty$, will the $\\epsilon$ shrink?) and/or whether $\\psi_1$ and $\\psi_2$ can be estimated by the proposed architecture (i.e., how close $\\psi_1$ and $\\psi_2$ to $h$ and $\\phi^{(6)}$, respectively).\n1. Related to #4, can we support Theorem 4.1 by running a synthetic example? Specifically, can we show that the true weighting function $f$ can be recovered by the $f_\\theta$ in Figure 2?\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for the detailed comments and suggestions for improving our paper. We will reflect all the comments and suggestions in our final version. In the following, we respond to specific concerns and questions raised by the reviewer.\n\n> Weakness 1. It looks like there are two contributions in this paper: 1) propose a way to learn the weighted filtration as per Section 3.1, with theoretical guarantee in Section 4; and 2) Section 3.2 in the ability to approximation with composite function. The contribution in 1) is more related to filtration learning as it learns a weighting function, but 2) is a more general use case. However, by looking at the experimental results, it is not clear which contribution is more significant, ...\n\nSection 3.2 describes how to combine topological features with more general features obtained by DNNs to solve classification tasks, and it does not describe the ability to approximation with composite function. The first experiment in Section 5 was conducted to show the validity of our method where we only used the topological feature, without DNN features. In the second experiment, on the other hand, we combine the topological features and features from DNNs using the method described in Section 3.2. In the final version, we will clearly state the difference between the two experiments at the beginning of Section 5.\n\n> Weakness 2. It is not clear to me from reading the main text on how you choose the hyper-parameters for DTM. From reading the appendix, it looks like the parameters are not chosen by cross-validation (Section C.2). ...\n\nThank you for your important remark. As you pointed out, the experimental results in the main text were not ones selected in cross-validation, while we conducted experiments with some hyper-parameters as we showed in Appendix. In the final version, we would like to fix them as we showed in the global response. \n\n> Weakness 3. It looks like Theorem 4.1 suggests the function can be approximated, but it does not mention whether it can be recovered. Will it be possible to get some results regarding the convergence ... and/or whether $\\psi_1$ and $\\psi_2$ can be estimated by the proposed architecture...\n\n> Question 2. How to correctly understand the connection between the architecture (in Figure 2) and Theorem 4.1? ...\n\nThank you for your essential remark. Since we proved Theorem 4.1 to demonstrate the validity of our idea to concatenate the (finite-dimensional) global feature $\\psi_1(X)$ (which corresponds to $h(X)$) and local feature $x$ (which corresponds to $g(x)$), it does not clarify whether  $\\psi_1$ or $\\psi_2$ can be recovered by the network architecture proposed in this paper indeed. We leave it as future work to provide stronger theoretical results for our network or to propose a new architecture with a complete approximation guarantee.\n\n> Weakness 4. Related to #4, can we support Theorem 4.1 by running a synthetic example? Specifically, can we show that the true weighting function $f$ can be recovered by the $f_\\theta$ in Figure 2?\n\nWe appreciate your constructive suggestion that we should support Theorem 4.1 by running a synthetic example. \n\nIn this setting, we do not have a “true weight function”, but we just find the filtration that can achieve high classification accuracy. \n\nWe have the experimental result to support Theorem 4.1 that shows our network can recover the DTM function, as shown in the table below. This table shows the error when our network is trained by the regression task to approximate the DTM functions. This result means that our method can choose filtrations from the space including Rips and DTM filtration if trained appropriately. We will add this result in the final version. \n\n| value of k |      error      |\n|:----------:|:---------------:|\n|  0 (Rips)  | 0.0000 ± 0.0000 |\n|      2     | 0.0015 ± 0.0000 |\n|      3     | 0.0016 ± 0.0000 |\n|      4     | 0.0017 ± 0.0000 |\n|      5     | 0.0018 ± 0.0000 |\n|     10     | 0.0022 ± 0.0001 |\n\n> Question 1. TDA is also used as data-analysis/unsupervised purposes (e.g., in finding enclosing holes) [A-C], I am curious whether the learned filtration weight $f_\\theta(X, \\cdot)$ can reveal the true topological structures or ...\n\nThank you for your interesting comments. We believe that we can sometimes get some insights on the topological structure of the point cloud data $X$ from the learned filtration weight $f_\\theta(X, \\cdot)$. For instance, for the point clouds shown in Figure A (in the PDF of the global response), one can find that the points with large weights are outliers. We remark that these weights were automatically learned in a data-driven way without any prior information. \n\nAs for the loss function, we are currently using the classification loss as a loss function, but we can also consider other types of loss functions. \nFor example, if we consider the reconstruction task and its loss, one might obtain the filtration weight that is effective in extracting all of the topological information in the point clouds. \nIn fact, we have conducted such an experiment and obtained appropriate weights in some cases. We leave it as future work to do much investigation on how using other types of loss functions affects the filtration weights. \n\n> Question 3. [Minor language issue] When I first read the paper, it is not clear what the “architecture” and “approximation result” in L12-13 meaning … Consider adding some details there to improve clarification. For instance, …\n\nThank you for pointing out this issue and suggesting an alternative sentence. In the final version, we will change the expression to clarify how our theoretical result contributes. \n\nWe hope that we addressed all your questions and concerns adequately. In light of our clarifications, please consider increasing your score to accept. Please let us know if we can provide any further details and/or clarifications."}, {"review_id": "r3KtcuzZR0", "reviewer": "Reviewer_L8GX", "summary": "A neural network that learns the filtration for persistent homology on given point cloud data is introduced, theoretically justified, and evaluated experimentally on 2 data sets.", "questions": "(Q1) PH wrt some filtrations (e.g., height, useful to distinguish MNIST digit 6 from digit 9) is not isometry-invariant, so why do you impose this condition? \n\n(Q2) Figure 1: Comment more why this approach is meaningful, since 2 holes are not recognized with 1-dim PD for any of the two point clouds? Consider plotting (next to) more reasonable weights.\n\n(Q3) You never mention simplicial complexes, so that it remains unclear why the filtration discussed on lines 143-144 is Rips, and not Čech?\n\n(Q4) How does your weighted Rips filtration compare to the weighted Rips filtration discussed in [3] (Proposition 3.5), where point cloud point x appears according to its filtration function value (weight) f(x), and an edge (x, y) appears when f(x), f(y) and distance d(x, y) satisfy certain properties?\n\n(Q4) Related to (Q3) and (Q4), do all point cloud points immediately appear in your filtration? In particular, if a point has a really large weight, the balls centered at this point will expand very late in the filtration, but will the point (ball with radius 0) be there from the beginning? This is important e.g. if the point is an outlier, since we commonly want to ignore such a point.\n\n(Q5) “Although (a) and (b) can be learned together since the output of the resulting feature is differentiable with all parameters, it would make the optimization unstable.” Why, can you explain more?\n\n(Q6) In Section 5, can you provide more intuition on what is captured with DistMatrixNet? \n\n\nOther minor comments:\n\n-\tThe homology is persistent (not persistence homology), but we talk about persistence landscapes and images (not persistent landscape or image), rephrase throughout the paper.\n-\tLine 86: Explicitly mention DeepSets.\n-\tLine 142: “one can take function S defined by” -> “one can take S to be the distance to point cloud, defined by”\n-\tLine 156:  “which can also [add: be] computed only by distance matrices”. Do you not need the weights too?\n-\tLine 168: What is function u, where is it used?\n-\tLine 169: “we can vectorize persistent a persistent diagram”. Rephrase.\n-\tLine 234: Do topological features and DNN features have to have the same dimension L?\n-\tLine 251: “we can approximate any continuous function on X x [0,1]^m can be approximated”. Rephrase.\n-\tLine 253: Cite specific Appendix.\n-\tLine 283: “we replaced our topological featured replaced”. Rephrase.\n-\tLine 285: “Note that we use DistMatrixNet not for the computation of filtration weights.” This sentence seems weird, what do you mean?\n-\tTable 2 caption: “concatenated the feature” -> “concatenated with the feature”?\n-\tMention explicitly that the code is made publicly available.\n-\tReferences: Check capitalization of acronyms in paper titles (e.g., Dtm, Perslay, Ripsnet, Toposeg, Homcloud, Pointnet, 3d, Pointet++, Pi-net, 3d shapenets, Sgmnet, …), and be consistent between capital case vs. lower case for journal names.\n\n[3] Anai, Hirokazu, et al. \"DTM-based filtrations.\" Topological Data Analysis: The Abel Symposium 2018. Springer International Publishing, 2020.\n", "limitations": "Some limitations and corresponding future research directions are mentioned, but they do not address the lack of insights on when a learned filtration can be expected to be beneficial (compared to e.g. Rips and DTM filtrations).", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "(S1) If this is indeed the first work that considers learning filtrations on point clouds, I find the idea very relevant.\n\n(S2) The filtration learning approach is very nicely motivated and described (Lines 185 – 196).", "weaknesses": "(W1) The need for learned filtrations should be better motivated. Think of an example point cloud where some other learnable filtration is more meaningful than Rips or DTM, visualize all three filtrations and their PDs. For example, we know that DTM is more suitable than Rips in the presence of outliers, but when is another filtration better than Rips and DTM? This would provide guidance to readers when it would make sense to use your approach, instead of relying simply on the Rips and DTM filtration. From Table 1 results, it seems that some answers might lie in the protein data set you consider. \n\n(W2) [1] and [2] seem to be related work, but are not referenced?\n\n(W3) The improvement with a learned filtration is good for protein classification, but it is for sure not convincing for the 3D CAD data (whereas it is much more complicated and computationally difficult). It would therefore be useful to provide more information about the data (including visualizations), and more detailed insights. I took a look at Appendix C, but this does not provide answers to these questions. Negative insights are also meaningful, i.e., that learning a filtration might not be useful for a lot of problems, and that relying on Rips or DTM filtration is good enough.\n\n(W4) You write: “… the classification accuracy is better when our method was concatenated with DeepSets/PointNet compared to using DeepSets/PointNet alone. The accuracy when we combine our method and PointMLP is not higher than that of PointMLP. This would mean that concatenating the topological feature is effective when the number of parameters of a DNN-based method is relatively small.” I find this argument extremely flawed, since the number of parameters for PointNet and PointMLP is very comparable?\n\n\n[1] Zhang, Simon, Soham Mukherjee, and Tamal K. Dey. \"GEFL: Extended Filtration Learning for Graph Classification.\" Learning on Graphs Conference. PMLR, 2022.  \n[2] Horn, Max, et al. \"Topological graph neural networks.\" arXiv preprint arXiv:2102.07835 (2021).\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the detailed comments and suggestions for improving our paper. We will reflect all the comments and suggestions in our final version. In the following, we respond to specific concerns and questions raised by the reviewer.\n\n> (W1) The need for learned …\n\nThank you for your important suggestion. To clarify the validity of learning filtration using our method, in Section 1, we will add a sentence “We will show some experimental results which show that our method improves the classification accuracy compared to when using Rips or DTM filtration\". \n\nOne example that motivates learning filtration is the point cloud in Figure A in the global response. In this example, the trained weight function gives large values to the outliers. Although this is similar to the DTM function, we remark that such a weight function was obtained in a data-driven and supervised way, without any information (other than data) in advance. We believe that the fact that our method can learn suitable filtrations for classification without any prior specification, as in this example, allows us to motivate the use of our method.\n\n> (W2) [1] and [2] seem …\n\nWe are grateful to you for pointing out the lack of references. We will add these references in the final version. \n\n> (W3) The improvement with …\n\nThank you for your constructive comments. We attached some visualization of the learned weights for the 3D CAD data (Figure B). We can observe that the points in some of the point clouds would have appropriate weight to be classified correctly, while there exist point clouds such that all points are assigned a weight of 0. This shows that learning filtration with our method would contribute to improving classification accuracy for some of the data, while is not effective for some data. \n\nOn the other hand, as you pointed out, the accuracy improvement by the proposed method for 3D CAD data is not remarkable. Based on this, in the final version, we will describe that Rips or DTM filtration is effective enough for some data (such as the furniture surface data) so that our method does not lead to further improvements in accuracy while learning filtration with our method is beneficial for some data (such as protein data). \n\n> (W4) You write: “…” I find this …\n\nThank you for your essential remark. After reviewing the results, we now believe that the lack of accuracy is due to incompatibility with PointMLP, rather than to the large number of parameters. We hypothesize that PointMLP has already captured enough information, including topological features, during the 1st phase. If so, the information obtained by persistent homology may be redundant, potentially negatively impacting the classification. We will appropriately replace the current observation with the hypothesis above in the final version.\n\n> (Q1) PH wrt some …\n\nIn this paper, we focus on the classification of point clouds. In this setting, it would be natural to impose the isometry-invariance.\n\n> (Q2) Figure 1: Comment more …\n\nThank you for your great idea on the image explaining the procedure of our method. We will replace Figure 1 with Figure A in the PDF as we stated in the global response. \n\n> (Q3) You never mention …\n\n> (Q4) How does your …\n\n> Related to (Q3) and (Q4), …\n\nWe appreciate your comments that we should add some explanation about simplicial complex and Vietoris-Rips filtration. Due to the page constraints, we could not include them in the main text. We instead described the increasing family of balls, which gives an intrinsic understanding, but it was not an honest way. We will modify the sentence from line 145 and create a new section to give detailed explanations of simplicial complexes, Cech, and Rips filtrations in the appendix. More concretely, we will change the sentences from line 145 as follows: “The persistent homology of this filtration can be captured by the filtration called Cech or Vietoris-Rips filtration (Rips filtration for short). In this paper, we use Rips filtration for computational efficiency. See Appendix … for details”. In the same section in the appendix, we will also describe the weighted Rips filtration we used, which is totally the same as the one defined in Section 3.3 of Anai et al. (2020). \n\n> (Q5) “Although …” Why, can you explain more?\n\nThe loss function can be differentiable with respect to all of the parameters in the network (a) and (b), but is not smooth with respect to the parameters in (a)*.* Because of this, if we optimize both of the networks in (a) and (b), not only the resulting networks of (a) but also that of (b) are (empirically) not optimized well. This is why we separately optimized the networks in (a) and (b). We will append these details in the final version. \n\n> (Q6) In Section 5,  …\n\nWe believe that DistMatrixNet can extract the information of point clouds using the relative distance information. The experimental results show that DistMatrixNet is effective when used as a weight function in the proposed method, but not when used directly for classification. This might mean that DistMatrixNet can be effectively used to obtain information about the role of each point in the point cloud, but is not suitable for getting information to distinguish its global shape.  Further investigation on the role of the DistMatrixNet is future work. \n\n> Other minor comments:\n\nWe appreciate your detailed comments and giving us the suggested fixes. We will fix every mistake in the final version.\n\n> **Limitations:** … they do not address …\n\nThank you for your important remarks that the description in the limitation section is not enough. In the final version, we will add some insights on the learned filtration compared to Rips and DTM filtration, including negative ones. \n\nWe hope that we addressed all your questions and concerns adequately. In light of our clarifications, please consider increasing your score to accept. Please let us know if we can provide any further details and/or clarifications."}, {"review_id": "E67wIAlAZy", "reviewer": "Reviewer_k7w9", "summary": "The paper develops a neural network to learn weights of given points in addition to other internal parameters to classify 3D clouds of unlabeled points on several public datasets. \n", "questions": "Are there any theoretical guarantees that the output of the proposed neural network distinguishes infinitely many (almost all?) point clouds as pairwise distances do?\n\nWhat theoretical results in the paper are stronger than the past work by Widdowson et al in CVPR 2023?\n\nIn subsection 5.1, what is the dataset size relative to the Protein Data Bank? \n\nIn subsection 5.2, how representatives are \"subsampled 128 points\" (line 302) for objects such as \"beds, chairs, and desks\" (line 298). For example, can humans distinguish between a bed and a desk by looking at 128 randomly sampled points? \n\nWhat is the asymptotic complexity (in the number of points in a given cloud) of the algorithm described in section 3? What was the actual running time for training and testing, and what are the technical specifications of the used machine? ", "limitations": "The very last paragraph on limitations only discusses talks about future work, for example about \"generalizing our framework to include a wider class of weighted filtrations\" (line 342). Yes, there are numerous papers that go even further and include a given point cloud as a raw input without computing any justified invariants. \n\nSince the authors have written detailed and accurate mathematical proofs in the appendix, they could probably agree that *examples prove nothing* because counter-examples can still exist, especially when all possible data (as for point clouds) fill a continuous space.\n\nFor continuous data, a continuous parametrization or metric can be more suitable than a discrete classification, which practically cuts a continuous space into disjoint pieces.\n\nSince all tables of experimental results include accuracies of at most 84% in table 1 (maximum 75% in table 2), the best and certainly publishable contribution seems to be Theorem 4.1. Can a mathematical venue be more suitable for this important result? \n\nTo help the authors with future submissions, the key insight from \"AlphaFold2 one year on\" https://www.nature.com/articles/s41592-021-01365-3 exposes the major limitation of all brute-force predictions, not only for AlphaFold. What resources (money, people, even electricity and water) are needed not only to get the first predictions but to annually update predictions with new training data? Is it really sustainable in the long term? \n", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The authors should be highly praised for a rigorous approach to an important problem of point cloud classification by using isometry invariants from persistent homology. \n\nThe paper is generally well-written.", "weaknesses": "Questions arise already when reading the abstract in line 11: \"to make the resulting persistent homology isometry-invariant, we develop a neural network\". For standard filtrations such as Vietoris-Rips, Cech, or Delaunay complexes on a point cloud, the persistent homology is already an isometry invariant of a given cloud of unlabeled points because constructions of all complexes above depend only on inter-point distances.\n\nThe main drawback of persistent homology is its weakness as an isometry invariant, which should have been clear to all experts in computational geometry many years ago but was demonstrated only recently. The paper by Smith et al (arxiv:2202.00577) explicitly constructs generic families of point clouds in Euclidean and metric spaces that are indistinguishable by persistence and even have empty persistence in dimension 1. \n\nThough Topological Data Analysis was largely developed by mathematicians, the huge effort over many years was invested into speeding up computations, rather surprisingly, instead of trying to understand the strengths and weaknesses of persistent homology, especially in comparison with the much simpler, faster, and stronger invariants of clouds under isometry. \n\nPersistence in dimension 0 was actually extended to a strictly stronger invariant mergegram by Elkin et al in MFCS 2020 and Mathematics 2021, which has the same asymptotic time as the classical 0D persistence and is also stable under perturbations of points.\n\nA SoCG 2022 workshop included a frank discussion concluding that there was no high-level problem that persistent homology solves. In fact, persistence as an isometry invariant essentially tries to distinguish clouds up to isometry, not up to continuous deformations since even non-uniform scaling changes persistence in a non-controllable way. \n  \nOn the other hand, the isometry classification problem for point clouds was nearly solved by Boutin and Kemper (2004), who proved that the total distribution of pairwise distances is a generically complete invariant of point clouds in any Euclidean space. The remaining singular cases were recently covered by Widdowson et al in NeurIPS 2022 and CVPR 2023. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for the detailed comments and suggestions for improving our paper. First, let us explain the problem setting and our motivation for this study to resolve your misunderstanding.\n\nIn this paper, we deal with the classification of *labeled point clouds.* In this setting, *point clouds are labeled beforehand* (for example, in the experiment on protein dataset, each point cloud has a label open or close), and two point clouds with the same label are not necessarily isometric. Our aim is NOT to distinguish the point clouds up to isometry.\n\nIn solving such classification tasks, topological features extracted by persistent homology (PH) would be useful. In fact, PH has been shown to be *effective for point cloud analysis and classification* for material science [1, 2], biology [3, 4], and medical science [5, 6]. Although it is isometry-invariant if the filtration is chosen to be isometry-invariant, the use of PH is NOT limited to distinguishing the point clouds up to isometry.\n\n[1] T. Nakamura et al. Persistent homology and many-body atomic structure for medium-range order in the glass. *Nanotechnology*, 26(30):304001, 2015.\n\n[2] A. Hirata et al. Structural changes during glass formation extracted by computational homology with machine learning. *Communications Materials*, 1(1):98, 2020.\n\n[3] V. Kovacev-Nikolic et al. Using persistent homology and dynamical distances to analyze protein binding. *Statistical Applications in Genetics and Molecular Biology,* 15(1):19–38, 2016.\n\n[4] Z. Cang and G. Wei. \"TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions.\" *PLoS computational biology* 13(7), e1005690, 2017\n\n[5] X. Zhu et al. Stochastic Multiresolution Persistent Homology Kernel. In *IJCAI 2016* (pp. 2449-2457).\n\n[6] N. Singh et al. Topological descriptors of histology images. In *Machine Learning in Medical Imaging: 5th International Workshop, MLMI 2014. Proceedings 5* (pp. 231-239).\n\n> the persistent homology is already an isometry invariant\n\nWe consider weighted Rips filtration whose weight function is implemented by the neural network. If we do not force this network to be isometry-invariant, the resulting PH can be non-isometry-invariant. This is why we make the network to be isometry-invariant.\n\n> What theoretical results in the paper are stronger than the past work by Widdowson et al … ?\n\nThe result by Widdowson et al (2023) is not a competitor of our study but rather one that could potentially improve our results by incorporating it into our study. Currently, we do not have strict guarantees about the approximation ability of our network based on distance matrices. Widdowson et al (2023) showed that Simplexwise Centered Distribution (SCD) is complete isometry invariant and continuous. This helps us to obtain theoretical guarantees of our network or to propose a new network architecture with stronger theoretical guarantees. That is our future work, and we are grateful for your insight. We will mention this in the final version. \n\n> In subsection 5.1, what is the dataset size relative to the Protein Data Bank?\n\nAs explained in lines 266—276, we created the protein dataset composed of 1,000 data by subsampling from 14 types of proteins. This is much smaller than the datasets in Protein Data Bank. We did not applied our method to larger datasets since we need to compute PH repeatedly in the training, which makes the computational cost higher. We will add this limitation in the final version. \n\n> In subsection 5.2, how representatives are \"subsampled 128 points\" …, can humans distinguish between a bed and a desk … ?\n\nThank you for your important comment. Subsampling 128 points sometimes makes it hard for humans to judge the label for each point cloud. We subsampled a relatively small number of points due to the high computational cost, as we described above. It is our future work to conduct experiments for point clouds with a large number of points. \n\n> What is the asymptotic complexity … ?\n\nThe computational complexity can be upper-bounded by $O(N^9)$, where $N$ is the number of points. However, actual computational complexity is expected to be smaller since computational cost of PH is known to be empirically less than cubic complexity with respect to the number of simplices.\n\nRegarding the actual running time, it takes about seven hours to train the neural networks in our method. The bottleneck of computational time would be the computation of PH. While this is one of the limitations of our study, we believe our method is meaningful since it sometimes improves classification accuracy.\n\nWe will include this information related to the computational cost in the final version.\n\nThe technical specifications of the used machine are described in lines 261—264.\n\n> The very last paragraph on limitations only discusses talks about future work\n\nThank you for pointing out the lack of a description in the limitation. We will add some explanation about the limitation of our method such as the high computational cost.\n\n> Are there any theoretical guarantees …\n\n> Since the authors have written …\n\nOur method aims to estimate labels for meaningful labeled point clouds, rather than distinguishing between all possible point configurations up to isometry. Our theory supports the validity of the architecture to solve such a classification task. We believe our method is effective for this task.\n\n> Since all tables …\n\nDue to the high computational cost of our method, we could not apply our method to large point clouds, which makes the accuracy lower. If we improve our method so that we can apply it to larger point clouds, the accuracy will be much higher. We will state this fact in the limitation section of the final version.\n\nWe hope that we addressed your questions and concerns. In light of our clarifications, please consider increasing your score to accept. Please let us know if we can provide any further details and/or clarifications."}, {"review_id": "sqiNiZ3uEi", "reviewer": "Reviewer_q2Rp", "summary": "In this manuscript, the authors develop a special module that can adaptively learn a suitable filtration function for persistent homology (PH) and its downstream tasks. In particular, their learning module is specially designed, so that resulting persistent homology is isometry invariant. Their model has been tested on two datasets for protein classification and CAD data classification. The model is novel and very interesting! ", "questions": "1.\tOne of the key point for the submission is to \"propose a novel framework to obtain adaptive topological features for point clouds\", as mentioned in Page 3, line 71. However, the submission seems to only focus on learning \"weight function\" using \"different radii\"! In fact, many other works have been done for designing special filtration process based on data properties, such as local homology, weighted homology, element-specific homology, cohomology (incorporated with special weights), etc. These works can all be viewed as learning (unsupervisedly) \"adpative topological features\".  For instance,\n\n(a) local homology:  Bendich P, Cohen-Steiner D, Edelsbrunner H, Harer J, Morozov D (2007) Inferring local homology from sampled stratifed spaces. In foundations of computer science, 2007. FOCS’07. 48th Annual IEEE symposium on, IEEE, pp. 536–546\n\n(b) element-specific homology & special distance matrices: Cang ZX, Mu L, Wei GW (2018) Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening. PLoS Comput Biol 14:e1005929\n\nCang ZX, Wei GW (2017) TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions. PLoS Comput Biol 13:e1005690\n\n(c) Weighted simplicial complexes: Zhenyu Meng, D Vijay Anand, Yunpeng Lu, Jie Wu, and Kelin Xia, \"Weighted persistent homology for biomolecular data analysis.\" Scientific Report, 10 (1), 1-15 (2020)\n\n(d) Topological antoencoders: Moor, Michael, Max Horn, Bastian Rieck, and Karsten Borgwardt. \"Topological autoencoders.\" In International conference on machine learning, pp. 7045-7054. PMLR, 2020.\n\nFurther, \"adaptive topological features for point clouds\" can also been achieved through different types of simplicial complexes or hypergraphs. For instance, the topological features of Rips complexes are dramatically different from Dowker complexes, Neighborhood complexes, Hom-complexes, etc. \n\nXiang Liu, Huitao Feng, Jie Wu, and Kelin Xia, \"Dowker complex based machine learning (DCML) models for protein-ligand binding affinity prediction.\" PLOS Computational Biology, 18(4), e1009943\n\nXiang Liu, and Kelin Xia, \"Neighborhood complex based machine learning (NCML) models for drug design.\" In Interpretability of Machine Intelligence in Medical Image Computing, and Topological Data Analysis and Its Applications for Medical Data, pp. 87-97. Springer, Cham (2021).\n\nXiang Liu, Huitao Feng, Jie Wu, and Kelin Xia, \"Hom-complex-based machine learning (HCML) for the prediction of protein–protein binding affinity changes upon mutation\", Journal of Chemical Information and Modeling, 62 (17), 3961-3969 (2022)\n\n2.\tThe discussions of vectorization of PH are not proper. In particular, “The idea to learn vectorization of persistent homology was pioneered by Hofer et al. (2017” is incorrect. To learn statistic or combinatorial properties (unsupervised) from persistent diagram or persistent barcodes is a common approach. For instance,\n\nBubenik, Peter, and Peter T. Kim. \"A statistical approach to persistent homology.\" Homology, homotopy and Applications 9, no. 2 (2007): 337-362.\n\nChung, Moo K., Peter Bubenik, and Peter T. Kim. \"Persistence diagrams of cortical surface data.\" In Information Processing in Medical Imaging: 21st International Conference, IPMI 2009, Williamsburg, VA, USA, July 5-10, 2009. Proceedings 21, pp. 386-397. Springer Berlin Heidelberg, 2009. \n\nBubenik P (2015) Statistical topological data analysis using persistence landscapes. J Mach Learn Res 16:77–102\nChi Seng Pun, Si Xian Lee, and Kelin Xia, \"Persistent-homology-based machine learning: a survey and a comparative study.\" Artificial Intelligence Review, (2022) \n\nDey, Tamal Krishna, and Yusu Wang. Computational topology for data analysis. Cambridge University Press, 2022.\n\n3.\tIn the test examples, the data sizes seem to be relatively small, i.e., less than 2000 datapoints. But the learning module has multiple fully connected layers. In this way, overfitting can easily be a problem. The authors are suggested to do some ablation studies to address the issue. Further, some more details for PH analysis are needed. For instance, filtration sizes for the data, vectorization of persistent images, etc. ", "limitations": "It will be great if the authors can use more realistic examples and compare with state-of-the-art models. \n", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "It is a novel approach to use machine learning to learn a suitable filtration function! It has also demonstrated the advantage over traditional approaches. ", "weaknesses": "Missing important references for related works. The test examples have limited data points. The model may suffer from over-fitting issues. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the detailed comments and suggestions for improving our paper. We will reflect all the comments and suggestions in our final version. In the following, we respond to specific concerns and questions raised by the reviewer.\n\n> 1. One of the key point for the submission is to \"propose a novel framework to obtain adaptive topological features for point clouds\", as mentioned in Page 3, line 71. However, the submission seems to only focus on learning \"weight function\" using \"different radii\"! In fact, many other works have been done for designing special filtration process based on data properties, such as local homology, weighted homology, element-specific homology, cohomology (incorporated with special weights), etc. These works can all be viewed as learning (unsupervisedly) \"adpative topological features\".\n\nThank you for pointing out that the expression “adaptive topological feature” might be confusing for readers, because \"adaptive topological features\" does not necessarily mean that the extraction of topological features is done in a supervised way. \n\nIn this paper, we tried to learn a filtration in a data-driven and supervised way, which is our main contribution. This is not completely on the same line with previous studies designing special filtration in an unsupervised way based on data properties. We missed some of the important references you cited and will add them in the final version. \n\n> Further, \"adaptive topological features for point clouds\" can also been achieved through different types of simplicial complexes or hypergraphs. For instance, the topological features of Rips complexes are dramatically different from Dowker complexes, Neighborhood complexes, Hom-complexes, etc.\n\nWhile we concentrated on determining the weight function of the weighted Rips filtration in this paper, we can also consider other kinds of filtrations as you pointed out. We will clearly state this remark in the limitation section in the final version. \n\n> 2. The discussions of vectorization of PH are not proper. In particular, “The idea to learn vectorization of persistent homology was pioneered by Hofer et al. (2017” is incorrect. To learn statistic or combinatorial properties (unsupervised) from persistent diagram or persistent barcodes is a common approach.\n\nWhile we intended to state that Hofer et al. (2017) first proposed to determine vectorization in an end-to-end/data-driven/supervised/automatic way, the sentence you pointed out is inappropriate. In the final version, we will modify this expression and add references related to the vectorization method. \n\n> 3. In the test examples, the data sizes seem to be relatively small, i.e., less than 2000 datapoints. But the learning module has multiple fully connected layers. In this way, overfitting can easily be a problem. The authors are suggested to do some ablation studies to address the issue.\n\nThank you for your important suggestion. We will conduct additional experiments to investigate whether we can reduce the number of network parameters, and add the results in the appendix.\n\n> Further, some more details for PH analysis are needed. For instance, filtration sizes for the data, vectorization of persistent images, etc.\n\nSince we used (weighted) Rips filtration, the number of simplices can be computed from the number of points in the point clouds, which is stated in lines 273 and 302. To make it clear, we will add this in the appendix in the final version.\n\nWe used PersLay (Carriere et al., 2019) as a vectorization method, and the detailed settings for this are described in Appendix B.3. \n\n\nWe hope that we addressed all your questions and concerns adequately. In light of our clarifications, please consider increasing your score to accept. Please let us know if we can provide any further details and/or clarifications."}]

Author rebuttal:
We appreciate the detailed comments and suggestions for improving our paper. We will reflect all the comments and suggestions in our final version. In the following, we show you some information that we would like to share with all of the reviewers. 

1. A reviewer pointed out that Figure 1 is confusing. To address this issue, we would like to replace Figure 1 with Figure A in the attached PDF. 
2. A reviewer pointed out that the hyper-parameter $k$ and $q$ in DTM filtration should be chosen to maximize the average classification accuracy with cross-validation. While the experimental results for DTM with different parameters were included in the Appendix, the results in the main text were not for optimal parameters, so we will replace this with the following table in the final version.
    
    For protein data:
    
    | DistMatrixNet | Rips | DTM | Ours |
    | --- | --- | --- | --- |
    | 65.0 ± 12.0 | 79.9 ± 3.0 | 78.0 ± 1.6 | 81.9 ± 2.1 |

    For 3D CAD data: 
    
    |  |  | DeepSets | PointNet | PointMLP |
    | --- | --- | --- | --- | --- |
    | 1st Phase |  | 65.7 ± 1.4 | 64.3 ± 4.4 | 68.8 ± 6.3 |
    | 2nd Phase | DistMatrixNet | 65.7 ± 4.8 | 55.7 ± 13.9 | 53.8 ± 7.4 |
    |  | Rips | 67.0 ± 2.6 | 68.4 ± 2.4 | 57.8 ± 12.4 |
    |  | DTM | 68.0 ± 2.5 | 68.7 ± 2.3 | 57.2 ± 6.8 |
    |  | Ours | 67.5 ± 2.5 | 68.8 ± 2.0 | 60.0 ± 6.3 |
3. A reviewer pointed out we should visualize the resulting weight function learned by our method. We showed some examples of the weight function for 3D CAD data in Figure B in the attached PDF. We will add this figure in the appendix. 

We also respond to the comments from each reviewer. If there are further questions/comments/suggestions, we would be happy to address them in the discussion period.

Meta-review: Reviewers agreed that the current submission constitutes a relevant and timely addition to the ever-growing literature of persistent homology in general, and topological machine learning in particular. While some concerns were raised about some methodological details, the authors provided a strong rebuttal and discussed all aspects satisfactorily.

I am thus happy to endorse this paper for publication, primarily because the learning of filtration for point clouds proves to be a **substantial algorithmic improvement** for computational topology and topological machine learning. While some works already learn [one](https://proceedings.mlr.press/v119/hofer20b) or [more](https://openreview.net/forum?id=oxxUMeFwEHd) filtrations in the graph context, current work for point clouds is mostly concerned with defining new filtrations, without having a way to adjust them to the task at hand. This critical need is now addressed by the paper at hand. Using an elegant and extensible architecture, the proposed work constitutes an important framework that will serve as the basis for many future extensions. A particularly relevant aspect for learning filtrations, also missing from previous work, constitutes an analysis of the expressive power; the authors close this gap and show that their architecture can approximate all relevant weight functions.

 The concerns by reviewer `k7w9` are acknowledged, but pertain to the wider area of persistent homology, not this specific paper. The authors are **strongly encouraged** to incorporate the comments made by the reviewer, in particular those pertaining to the experimental setup—additional data sets that showcase the power of the proposed method would strengthen the paper more. Likewise, additional work concerning clarity improvements would help to make the better overall more accessible and increase its well-deserved visibility.
Decision: Accept (poster)