Title: Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds
Venue: NeurIPS 2023
Accepted: yes
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

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.

1 Introduction

Analysis of point clouds (finite point sets) has been increasing its importance, with many applications in various fields such as 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 et al., 2017b), and PointMLP (Ma et al., 2022). 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 to incorporate topological information into machine learning is to use persistent homology, which is a central tool in topological data analysis and attracting much attention recently. By defining an increasing sequence of spaces, called a filtration , from a point cloud X ⊂ R d , 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., 2022), surface classification (Zeppelzauer et al., 2016), and fingerprint classification (Giansiracusa et al., 2019).

Yuichi Ike

Persistence diagram

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The standard pipeline to use persistent homology in combination with machine learning is split into the following:

(i) define a filtration for data and compute the persistent homology;

(ii) vectorize the persistent homology;

(iii) input it into a machine learning model.

In (i), one needs to construct appropriate filtration depending on a given dataset and a task. A standard way to define a filtration is to consider the union of balls with centers at points in X ⊂ R d with a uniform radius, i.e., take the union S t = ⋃ x ∈ X B ( x ; t ) , B ( x ; t ) := { y ∈ R d | ‖ y -x ‖ ≤ t } and consider the increasing sequence ( S t ) t . Regarding (ii), one also has to choose a vectorization method depending on the dataset and the task. For example, one can use typical vectorization methods such as persistence landscape and persistence image. As for (iii), we can use any machine learning models such as linear models and neural networks, and the models can be tuned with the loss function depending on a specific task.

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. (2020), for example, have proposed methods to learn vectorization of persistence homology based on data and tasks. Furthermore, for (i), Hofer et al. (2020) have proposed a learnable filtration architecture for graphs, which gives a filtration adaptive to a given dataset. However, this method heavily depends on properties specific to graph data. To establish a method to learn filtration for point clouds, we need to develop a learnable filtration architecture for a 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 a 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 on points. Choosing a weight function w : X → R , we can define

a filtration ( S t ) t by

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 et al., 2020; Nakamura et al., 2015; Hiraoka et al., 2016; Obayashi et 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 ⊂ R d .

The desirable property for network f is isometry-invariance , i.e., f ( TX,Tx ) = f ( X,x ) for any isometric transformation T of R d , so that the associated weighted filtration is isometry-invariant. To address this issue, we implement a function f with a network architecture based on distance matrices. Then, the resulting weight function satisfies the isometry-invariance rigorously . Additionally, we show a theoretical result about the approximation ability for a wide class of weight functions, which supports the validity of our architecture.

Our main contributions are summarized as follows:

We propose a novel framework to obtain adaptive topological features for point clouds based on persistent homology. To this end, we introduce an isometry-invariant network architecture for a weight function and propose a way to learn a weighted filtration.

We theoretically show that any continuous weight function can be approximated by the composite of two continuous functions which factor through a finite-dimensional space. This theoretical result motivates our architecture.

We conducted experiments on public datasets and show that the topological features obtained by our method improved accuracy in classification tasks.

Figure 1 presents one typical use of our framework for a 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 1, our method can learn a filtration appropriately for a given task and data in 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 to the Rips or DTM filtration in several experiments.

1.2 Related work

A neural network architecture for finite point sets was first proposed by Zaheer et al. (2017), as DeepSets. Later, there appeared several geometric neural networks for point clouds, such as PointNet (Qi et al., 2017a), PointNet++ (Qi et al., 2017b), and PointMLP (Ma et al., 2022). These studies addressed the rotation invariance by approximating rotation matrices by a neural network. (Xu et al., 2021) used Gram matrices to implement an isometry-invariant neural network for point clouds. We instead use distance matrices for our isometry-invariant 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 (2007); Chung et al. (2009); Bubenik et al. (2015); Adams et al. (2017); Kusano et al. (2017). However, recent studies such as Hofer et al. (2017) and Carrière et al. (2020) have proposed end-to-end vectorization methods using supervised learning. In particular, Carrière et al. (2020) used the DeepSets architecture Zaheer et al. (2017), 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, Hofer 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. (2021) and Zhang et al. (2022). They

introduced a parametrized filtration with the use of graph neural networks and learned it according to a 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 as Bendich et al. (2007); Cang and Wei (2017); Cang et al. (2018); Meng et al. (2020), have tried to design special filtration in an unsupervised way based on data properties, we aim to learn a filtration in a 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 a neural network ( Zhou et al. (2022) and de Surrel et al. (2022)). Note that similar methods are proposed for image data (Som et al., 2020) and graph data (Yan 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 (Zaheer et al., 2017), which we will use to implement our isometry-invariant structure. A more recent architecture, PointNet (Qi et al., 2017a) and PointMLP (Ma et al., 2022), will also be used for the comparison to our method in the experimental section.

The DeepSets architecture takes a finite (multi)set X = { x 1 , . . . , x N } ⊂ R d of possibly varying size as input. It consists of the composition of two fully-connected neural networks φ 1 : R d → R d ′ and φ 2 : R d ′ → R d ′′ with a permutation invariant operator op such as max , min and summation:

For each x i ∈ X , the map φ 1 gives a representation φ 1 ( x i ) . These pointwise representations are aggregated via the permutation invariant operator op . Finally, the map φ 2 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 of the network as a set. The parameters characterizing φ 1 and φ 2 are tuned through the training depending on 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 : R d → R be a function on R d . For t ∈ R , the t -sublevel set of S is defined as S t := { p ∈ R d | S ( p ) ≤ t } . Increasing t from -∞ to ∞ gives an increasing sequence of sublevel sets of R d , called a 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 α , one can find the value b α < d α such that the feature α exists in S t for b α ≤ t < d α . The value b α (resp. d α ) is called the birth (resp. death) time of the topological feature α . The collection ( b α , d α ) for topological features α is called the persistence diagram (PD) of f , which is a multiset of the half-plane { ( b, d ) | d > b } . The information of connected components, loops, and cavities are stored in the 0th, 1st, and 2nd persistence diagrams, respectively.

Given a point cloud X ⊂ R d , i.e., a finite point set, one can take S to be the distance to point cloud, defined by

Then, the sublevel set S t is equal to the union of balls with radius t centered at points of X : ⋃ x ∈ X B ( x ; t ) , where B ( x ; t ) := { y ∈ R d | ‖ x -y ‖ ≤ t } . In this way, one can extract the topological features of a point cloud X with the persistence diagram. The persistent homology of this filtration can be captured by the filtration called ˇ Cech or Vietoris-Rips 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 a 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 et al. (2016); Nakamura et al. (2015); 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 or solid, use such a filtration. Given a point cloud X ⊂ R d , one can define the radius value r x ( t ) at time t for x ∈ X as follows. We first choose a function w : X → R , which is called weight , and define

The associated weighted Rips filtration is denoted by R [ X,w ] . One choice as a weight function is the distance-to-measure (DTM) function. The resulting filtration is called a DTM-filtration, for which several theoretical results are shown in Anai et al. (2020). One of those results shows that a persistent homology calculated by the DTM-filtrations is 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. (2020) 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 a vectorization method. Note that PersLay can be replaced with any vectorization method in our architecture.

Given a function transformation map φ c with a learnable parameter c and permutation invariant operator op , we can vectorize a persistence diagram D by PersLay with

If we choose appropriate parametrized function φ c , we can make PersLay similar to classical vectorization methods such as persistence landscape and persistence image. In this paper, we utilized the following map as φ c , which makes PersLay similar to persistence image:

3 Proposed Framework

In this section, we describe our framework to obtain topological representations of given point clouds. Our idea is to use persistence homology for extracting topological information and model a weight function w in §2.2 by a 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 = { x 1 , . . . , x N } ⊂ R d , i.e., w is of the form f ( X, · ): R d → R . Designing such a function by a neural network has the following three difficulties:

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 .

g2(x1)

DeepSets

DeepSets

+

g2(22)

g2(c3)

DistanceMatrix

D(X)

g2(cN)

h(X)

g1(c1)

DeepSets

Fully-connected network

f(X,c1)

The resulting function should be isometry-invariant , i.e, for any isometric transformation T , any point cloud X , and x ∈ X , it holds that f ( TX,Tx ) = f ( X,x ) .

The output of the f ( X,x ) should have both of global information X and pointwise information of x .

To address the problem 1, we use the DeepSets architecture (§2.1). To satisfy invariance for isometric transformations as in problem 2, we utilize the distance matrix D ( X ) = ( d ( x i , x j )) N i,j =1 ∈ R N × N and the relative distances d ( X,x ) = ( ‖ x -x j ‖ ) N j =1 instead of coordinates of points, since D ( TX ) = D ( X ) and d ( TX,Tx ) = d ( X,x ) hold for any isometric transformation T . Finally, to deal with problem 3, we regard f as a function of the feature of D ( X ) and the feature of d ( X,x ) . Below, we will explain our implementation in detail.

First, we implement a network to obtain a pointwise feature vector of a point x ∈ R d by

where φ (1) and φ (2) are fully-connected neural networks. This structure (1) can be regarded as the DeepSets architecture whose input is a set of one-dimensional vectors, hence it is permutation invariant.

Next, we use the DeepSets architecture to obtain a feature vector for the entire point cloud X as follows. We first implement pointwise feature vectors with the same architecture as g 1 , i.e.,

where φ (3) and φ (4) are fully-connected neural networks. Then, we get a feature vector h ( X ) for the entire point cloud X by applying the DeepSets architecture again:

where φ (5) is a fully-connected neural network. Here we omit the first inner network for the DeepSets architecture since it can be combined with φ (4) in g 2 .

Finally, we implement f as a neural network whose input is the concatenated vector of h ( X ) and g 1 ( x ) . Specifically, we concatenate h ( X ) with the feature vector g 1 ( x ) for x ∈ R to get a single vector, and then input it into a fully-connected neural network φ (6) :

where θ := ( θ k ) 6 k =1 is the set of parameters that appear in network φ ( k ) ( k = 1 , . . . , 6) . The overall neural network architecture is shown in Figure 2.

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 a dataset given in distance matrix format. In fact, we will apply our framework to such a dataset in Section 5.

Remark 3.2 . If we want to consider features other than the position information assigned to each point in the calculation of the filtration weights, as in the silica glass studies, we vectorize those features and concatenate them together.

By using the weight function f θ ( X, · ) , we can calculate the persistent homology of filtration R [ X,f θ ( X, · )] . Then, we can obtain the vectorization using PersLay parametrized by c . Then, we can input this vector into a machine learning model F θ ′ with parameter θ ′ . Given the dataset { ( X k , y k ) } M k =1 , the objective function to be minimized can be written as

where /lscript is an appropriate loss function depending on tasks. Typically, if the task is regression, /lscript is the mean square loss, and if the task is classification, /lscript is the cross entropy loss. The optimization problem

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. (2021).

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 in 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 to learn (a) and (b) separately.

Let us describe the whole training procedure briefly here. Suppose X is a input point cloud. We write Ψ topo ( X ) ∈ R L 1 for the topological feature which comes from our method, and Ψ DNN ( X ) ∈ R L 2 for the feature that comes from a DNN-based method. Let /lscript be a loss function for the classification task and n be a number of classes. We propose the following two-phase training procedure:

1st Phase. Let C 1 : R L 2 → R n be a classifier that receives feature from Ψ DNN . Then, C 1 and Ψ DNN are learned through the classification task, which is achieved by minimizing ∑ M k =1 /lscript ( C 1 ( Ψ DNN ( X k )) , y k ) .

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 § 5.

2nd Phase. Let C 2 : R L 1 + L 2 → R n be a classifier. We discard the classifier C 1 and fix Ψ DNN . Then, C 2 and Ψ topo are learned through the classification task, which is achieved by minimizing ∑ M k =1 /lscript ( C 2 ([ Ψ DNN ( X k ) /latticetop , Ψ topo ( X k ) /latticetop ] /latticetop ) , y k ) .

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 can be realized by our idea to concatenate two finitedimensional feature vectors.

Consider the space 2 [0 , 1] m of subsets in [0 , 1] m equipped with the Hausdorff distance. For a fixed N ∈ N , we define the following subspace of 2 [0 , 1] m :

The following theorem states that we can approximate any continuous function on X × [0 , 1] m by concatenating the finite dimensional feature vectors of a point cloud and a point. The proof of the theorem is given in Appendix B.

Theorem 4.1. Let f : X × [0 , 1] m → R be a continuous function. Then for any /epsilon1 > 0 , there exist K ∈ N and continuous maps ψ 1 : X → R K , ψ 2 : R K + m → R such that

5 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 a experiment on a 3D CAD dataset. The classification accuracy was used as the evaluation metric.

The first experiment 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 §3.2.

All the code was implemented in Python 3.9.13 with PyTorch 1.10.2. 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 GUDHI 1 . All of the source codes we used for experiments is publically available 2 . More details of the experiments can be found in Appendix C.

5.1 Protein classification

Dataset. In this section, we utilize the protein dataset in Kovacev-Nikolic et al. (2016) consisting of dense configuration data of 14 types of proteins. The authors of the paper analyzed the maltosebinding protein (MBP), whose topological structure is important for investigating its biological role. They constructed a dynamic model of 370 essential points each for these 14 types of proteins and calculated the cross-correlation matrix C for each type 3 . They then define the distance matrix, called the dynamical distance , by D ij = 1 -| C ij | , and they use them to 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 60 × 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 a standard deviation of 0.1.

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 a 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 et al., 2020), the latter of which is robust to outliers. The hyperparameter for DTM filtration was set to maximize the classification accuracy. Additionally, we replaced our topological feature by the output of (2), which we call DistMatrixNet, and compared the classification accuracy. Note that we used DistMatrixNet not for the computation of filtration weights but for the direct computation of point clouds' feature. Since the protein dataset is 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 1. 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 our method learned a weighted filtration adaptively to data and tasks and could provide more informative topological features than the classical ones. Secondly, our method achieved higher accuracy compared to DistMatrixNet, which has a similar

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., 2015). This is a 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 of 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 a topological feature. As DNN-based methods, which were trained through the 1st Phase, we utilized DeepSets (Zaheer et al., 2017), PointNet (Qi et al., 2017a), and PointMLP (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 a classifier based on a 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 DistMatrixNet (2) 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 one 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 a 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 is 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 PointMLP is not higher than that of PointMLP. This would mean that PointMLP has already captured enough information including topological features during the 1st 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, we utilize a weighted filtration and train a neural network that generates a weight for each point.

Since the model is desired to be invariant with respect to isometric transformation, we proposed a network architecture that satisfies such invariance. We theoretically showed a finite-dimensional approximation property for a 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, we can also consider other kinds of filtrations. For example, we can 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 in our method for each cross-validation. While we believe our method is meaningful since it sometimes improves classification accuracy, this is one of the largest limitations of our study. Addressing this issue is one of our future work. Third, since our method adaptively chooses a filtration, our framework is expected to work well even if a subsequent machine learning model has a 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-Aid for Transformative Research Areas (A) (22H05107) and JST KAKENHI JP19H01114.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn 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!\n\nSTRENGTHS:\nIt is a novel approach to use machine learning to learn a suitable filtration function! It has also demonstrated the advantage over traditional approaches.\n\nWEAKNESSES:\nMissing important references for related works. The test examples have limited data points. The model may suffer from over-fitting issues.\n\nQUESTIONS:\n1.\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.\n\nLIMITATIONS:\nIt will be great if the authors can use more realistic examples and compare with state-of-the-art models.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nQuestions 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.\n\nQUESTIONS:\nAre 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?\n\nLIMITATIONS:\nThe 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?", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nA 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.\n\nSTRENGTHS:\n(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).\n\nWEAKNESSES:\n(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\nQUESTIONS:\n(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\nLIMITATIONS:\nSome 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).", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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\nSTRENGTHS:\n1. [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\nWEAKNESSES:\n1. 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\nQUESTIONS:\n1. 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\nLIMITATIONS:\nAuthors 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.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nIt 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\nQUESTIONS:\n1. 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?\n\nLIMITATIONS:\nThe work can be further improved by use more realistic examples and compare with state-of-the-art models.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-08"}