Graphcode: Learning from multiparameter persistent homology using graph neural networks
Authors: Florian Russold
Michael Kerber
Venue: NeurIPS 2024 poster (2024)
Keywords: Topological Data Analysis
Multiparameter Persistent Homology
Machine Learning
Geometric Deep Learning
Graph Neural Networks
Primary area: graph_neural_networks
TL;DR: We introduce a novel method to learn from two-parameter persistent homology using graph neural networks.

Abstract: We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale parameters. Such multi-parameter topological summaries are usually based on complicated theoretical foundations and difficult to compute; in contrast, graphcodes yield an informative and interpretable summary and can be computed as efficient as one-parameter summaries. Moreover, a graphcode is simply an embedded graph and can therefore be readily integrated in machine learning pipelines using graph neural networks. We describe such a pipeline and demonstrate that graphcodes achieve better classification accuracy than state-of-the-art approaches on various datasets.

Graphcode: Learning from multiparameter persistent homology using graph neural networks
Michael Kerber"
Florian Russold
Institute of Geometry
Institute of Geometry
Graz University Technology kerber@tugraz
Graz University of Technology
russold@tugraz.co
Abstract
We introduce graphcodes, a novel multi-scale summary of the topological properties of dataset that is based on the well-established theory of persistent homology- Graphcodes handle datasets that are filtered along two real-valued scale parameters. Such multi-parameter topological summaries usually based complicated theoretical foundations and difficult to compute; in contrast, graphcodes yield informative and interpretable summary and can be computed as efficient one-parameter summaries. Moreover, a graphcode is simply an embedded graph and can therefore be readily integrated in machine learning pipelines using graph neural networks. We describe such a pipeline and demonstrate that graphcodes achieve better classification accuracy than state-of-the-as approaches on various datasets.
1 Introduction
quote attributed to Gunnar Carlsson says "Data has shape and shape has meaning". Topological data analysis (TDA) is concerned with studying the shape, or more precisely the topological and geometric properties of data. One of the most prominent tools to quantify and extract topological and geometric information from dataset is persistent homology. The idea is to represent dataset multiple scales through a nested sequence of spaces, usually simplicial complexes for computations, and measure how topological features like connected components, holes or voids appear and disappear when traversing that nested sequence. This information can succinctly be represented through barcode, or equivalently a persistence diagram, which capture for every topological feature its lifetime along the scale axis. Persistent homology has been successfully applied in wealth of application areas [4 23 30 36), often combination with Machine Learning methods see the recent survey 22 for comprehensive overview.
shortcoming of classical persistent homology that it is bound single parameter, whereas data often is represented along several independent scale axes (e.g., think of RGB images which have three color channels along which the image can be considered). To get a barcode, one is forced to chose fixed scales for all but one scale parameters. The extension to multi-parameter avoids to make such choices. Similar to the one-parameter setup, the data is represented in a nested multi-dimennsonal grid of spaces and the evolution of topological features in this grid is analyzed. Unfortunately, a succinet representation extension, which makes the theory and algorithmic treatment more involved. Nevertheless, the quest of how to use informative summaries in multi- malli--arameter persistence an active field of contemporary research. A common theme in this context is vectorization, meaning that some (partial) topological information extracted from the dataset and transformed into high-dimensional vector suitable for machine learning pipelines.
persistent homology 7 ?
barcode is not possible this
Equaa contribution.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Penistent Homology
Induced homology map
Trchared homology meap
Persistent
Homology
Induced homoology map
Persistens Homology
Bifiltration
Graphcode
Figure 1: Schematic overview of our approach.
Contribution. We introduce graphcodes, a novel representation the homology a dataset that filtered along two scale parameters. The idea, depicted in Figure I is to consider one-parameter slices of the dataset by fixing one parameter, obtaining a stack persistence diagrams. We define map between consecutive diagrams in this stack, resulting in bipartite graph connecting these diagrams. The graphcode is the union of these bipartite graphs over all consecutive pairs.
Since the maps connecting diagrams depend on choice of basis for each persistence diagram, the graphcode is not a topological invariant. Nevertheless, the graphcode is a combinatorial description whose features are easy interpret and which permits, for fixed bases per diagram, complete reconstruction the persistence module induced by the bifiltration. Moreover, the structure as an (embedded) graph in R3 admits direct integration of graphcodes into machine learning pipelines via graph neural network architectures. In that way, graphcodes avoid the vectorization step of other homology-enhame sometimes slow in practice. In contrast, we describe an efficient algorithm to compute the graphcode of bifiltered simplicial complex which essentially computes all required information through single out-of-order matrix reduction of the boundary matrix of the entire complex. While the worst-case complexity is cubic, the practical performance is closer linear for realistic datasets (5) 29
learning pipelines which often require more parameter choices and are
We demonstrate how graphcodes facilitate classification tasks. For that, we implemented a machine learning pipeline that feeds the graphcodes into a simple graph neural network pipeline. On graph datasets used in related works on multi-paramenn persistent learning, our approach shows compara- ble classification quality. As proof of concept, also created synthesized dataset of point clouds in 2R2 that contain a number of densely sampled disks and annuli plus some uniform noise. Clearly, topological classifiers related methods on this type of data terms of accuracy. At the same time, graphcodes are faster computed than all alternative topological descriptors, sometimes by several orders of magnitude. We also demonstrate that graphcodes perform better than other topological methods on two further datasets, established in related work on TDA, which consist of samples from different random point processes and orbits generated by dynamical system, respectively.
well-suited for such data. In experiments, graphcodes outperform
Related work. Our method can be viewed as generalization of PersLay | to the two-parameter case. PersLay is neural network layer that enables vectorization-fin learning from mne-parameter persistent homology. uses a deep set [77] architecture to directly take persistence diagrams as input. A conceptually simpler generalization of PersLay would consist of only using the union of persistence diagrams of the one- mee-prraettrr slices, that is, the graphcode without connecting edges. We show in our experiments, however, that the edges improve the accuracy in the two-parameter case.
Most of the previous methods used in applications are based on transforming persistence diagrams into vectors in Euclidian space, other data structures suitable for machine learning, Examples

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persistence landscapes 8 persistence images [ or scale-space kernels [32]. These vectorization methods for one-parameter persistence modules have been generalized in various forms to the two- parameter case [10
35 The difference in the two-parameter case is that the vectorizations are not based on a complete invariant like the persistence diagram but on weaker invariants like the rank-invariant, generalized rank-invariant or the signed barcode. Hence, these vectorizations capture the persistent homology only partially. Moreover, even this partial information is often times computationally expensive. In contrast, method avoids to compute direct vectorization, although we point out that a vectorization is implicitly computed eventually within the graph neural network architecture. To our knowledge there exists other method that allows to feed complete representation of two-parameter persistent homology into a machine learning pipeline.
16
18
26
Our approach also resembles persistent vineyards 17 in the sense that two-ttoooet parameter process is considered as dynamic 1 1-parameter process and the evolution of persistence diagrams is analyzed. Indeed, vineyards produce layered graph of persistence diagrams, just as graphcodes (see Fig VIII.6 in [1), but they operate in different setting where the simplicial complex is fixed throughout the process and only the order of simplices changes, whereas graphcodes rely on bifiltered simplicial complex data that only increases along both axis. Most standard constructions of multi-parameter persistence yield such bifiltered complex and graphcodes are more applicable in this situation.
Generating bifiltered simplicial complexes out of point cloud data is computationally expensive and an active area of research. In the context of the aforementioned two-dimente two-dimensional point clouds that we analyze with graphcodes, we heavily rely on sublevel Delaunay bifiltrations which were introduced very recently by Alonso et al. [2]. That algorithm (and its implementation) render the two-parameter analysis of such point clouds possible in machine learning contexts, partially explaining why previous methods have only tested their approaches on very small point cloud data, if at all.
Outline. We review basic notions of persistent homology in Section 2 and define graphcodes in Section NattionBbasoe based on these definitions. We decided for a "down-to-earth" approach, defining graphcodes in terms of cycle bases in simplicial complexes to keep concepts concrete and relatable geometric constructions for the benefit of readers that are not too familiar with the algebraic foundations of persistent homology. Moreover, this treatment simplifies the algorithmic description to compute graphcodes in Section 4 We explain the machine learning architecture based on graphcodes in Section 5 and report on experimental results in Section 6 We conclude in Section 7
2 Persistent homology
We will use the following standard notions from simplicial homology. For readers not familar with these concepts, we provide a summary in Appendix For a more in-depth treatment see, for instance, the textbook by Edelsbrunner and Harer [19].
For an (abstract) simplicial complex and p 2 0 an integer, let Cp K)) denote its p-th chain group with Zz coefficients (which is, in fact, a vector space) and 0 : Cp(K) -> C Cp-1 (K) the boundary map (see also Figure F Let (K) be the kernel of ap. Elements of Zp( are called p-cycles. (K) is the image of 0,+1, and its elements are called p-boundaries. The quotient Z, (,)) called the p-th homology group (K(K), whose elements are denoted by [al] with a a p-cycle, or just [a] if the underlying complex is clear from context. See Figure 2 (right) for an illustration of these concepts.
We call a basis (21...., 2m) of (P( (boundary-Jconsiisee (ZI...., 22) a basis of (K). In this case, [=4+]]...., [2m]] is a basis of HK) ). A consistent basis can be obtained by first determining basis of Bp (K) and completing it to basis of 7pp ()). Clearly, (K) usually has many consistent bases.
if there exists some E 2 0 such that
Homology maps. Let L C K be a subcomplex. The inclusion map from to K maps p-cycles of L to p-cycles of K and bouuddasies of L to p-boundaries of K. It follows that for every p. the map i. : HL) + M,(K), (a)r. + [a]x is a well-defined map between homology groups. This map is major object of study in topological data analysis, as it contains the information how the topological features (i.e., the homology classes) evolve when we embed them on a larger scale (i.e., larger complex).
3

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Figure 2: Left: A simplicial complex with 11 0-simplices, 19 1-simplices and 7 2-simplices. A 2-chain consisting of three 2-simplices is marked with darker color, and its boundary, a collection of 7 1-simplices is displayed in thick.
Right: The 1-cycle marked in thick on the left is also a 1-boundary, since it is the image of the boundary operator under the 4 marked 2-simplices. On the right, the 1-cycle going along the path ABCDE is not a I-boundary; therefore it is a generator of an homology class [a] of H1 (K). Likewise, the 1-cycle a' going along ABCFGH is not 1-boundary neither. Furthermore, [a'] [a] since the sum a a'' is the 1-cycle given by the path AEDC GH, which is a 1-boundary because of the 5 marked 2-simplices. Hence, and a' represent the same homology class which is characterized by looping aroung the same hole in K.
Being a linear map, is can be represented as matrix with Zz-coefficients. It is instructive to think about how to create this matrix: Assuming having fixed consistent bases A, for Zp(L) and Ax for K') (P))
and considering basis element [o] of Hp([.) we write the p-cycle a as linear combination with respect to the basis Ag. We can ignore summands that correspond to basis elements of Bp (P),, (K), and the remaining entries yield the image of [a)], in (K),, and thus one column of the matrix of i..
Alternativly,, is takes a diagonal form when picking suitable consistent bases A and Ak. We can do that as follows: We start with basis A' of (L) n (,),K,, the set of p-cycles in L that become "trivial" when included in K. This subspace contains B, and we can choose A' such that it starts with basis of (L). followed by other vectors. Now we extend A' to a basis Ag of (L) which is consistent by the choice of A'. Since Zp(L) is subspace of (K),), we can extend AL to basis AK of Z,(K), Importantly, we can do so such that Ag is consistent, because the sub-basit A' maps to Bp (') and As \ A' does not, so we can first extend A' to a basis of (K). ensuring consistency, and then extend a full bais of (K). this way, considering a homology class (a) with a basis element of AL, a is also basis element of Ax, and the map i. indeed takes diagonal form.
().
Filtrations and barcodes. A filtration of simplicial complex K is nested sequence K2 - K₂ ) - K, = of subcomplexes of K. Applying homology, we obtain vector spaces (Kg) and linear maps (,,))) + H,(K,) whenever S j. This data is called persistence module. For simplicity, we assume that H, is the trivial vector space, implying that all p-cycles eventually become p-boundaries.
is possible to iterate the above construction for single inclusion to obtain consistent bases A, for Zp (K,) such that all maps (,,,)) - H,(K)) have diagonal form. Equivalently, there is one basis of (K) that contains consistent bases for all subcomplexes Ki,..., . Kn. To make this precise, we Zp(K), there is minimal such that a E Zp (Ke) for all l 2 i. This is called the birth index of a. Moreover, there is a minimal j such that a E B, (K2) for all 2 j. This is called the death index of a. By our simplifying assumptions that Hp(K) is trivial, every indeed has a well-defined finite death index. The half-open interval ( J) consisting of birth and death index is called the bar of We say that is already born at index e if its birth index is most f. We call a alive at l if alpha is already born at l and E is strictly smaller than the death index of a; otherwise we call it dead at l.
first observe that for every
Definition 2.1. For a filtration of K as above, a barcode basis is basis A of (p(KK) where each basis element is attached its bar such that the following property holds: For every i, the elements of A dead at form a basis of B, K,), and the elements of A already born at i form a basis of Zpp K,). In particular, these cycles form a consistent basis of Z,, (K:).
See Figure 3 (left) for an illustration. The collection of bars of a barcode basis is called the barcode of K. Indeed, while there is unique barcode basis for K, they all give rise to the same barcode.

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K1
K2
Kg
D.31
n.31
The bases AG and Ag together with their graphcode sufficient recover the homology maps 6; (L,) + Hp (Ki), induced by the inclusion L; S K₃: Since basis of Z,: is contained in hard see that this map equals the map d B
d b
L
L1
L3
Figure 3:
Left, lower row: (L) is generated by the cycles abed and abd. They form a barcode basis, with attached bars (1, and [2, 2), respectively. Note that also abd and bed form basis of Z1 (L), but that is not a barcode basis as none of these cycles is already born at L1. so they do not induce a basis of 21 (L1). Left, upper row: Here, abd and bed form barcode basis with attached bars [0, 2) and [1, (). respectively, and abd and abed as well (with identical barcode).
Right: Choosing the basis abcd, abd for Z1 (L) and abd and bed for Z1 (K), have obed abd + bed, hence the cycle abed has two outgoing edges, to both basis elements in K. We ignore the basis vector abd of L in the figure, since its birth and death index coincide, the corresponding feature has persistence zero.
This barcode (or the equivalent persistence diagram, where a bar (i.j) interpreted as a point (i, f) in the plane) yields topological summary of the filtration, revealing what topological features are active on which ranges of scale. Therefore, barcodes are a suitable (discrete) proxy for dataset and are heavily used in applications.
The concept of a barcode basis enhances the barcode with consistent choice of representative cycle for every bar. In practice, this extra information is obtained with no additional computation costs because the standard algorithm compute barcodes computes barcode basis by by-product.
3 Graphcodes
We now consider the case of two filtered complexes L, K such that L; S K2 for all
Ki
K2
Kn = K Inn
(I)
L Ln
Assume we have fixed barcode bases Ag for (L) and Ax for Zp(K). The inclusion L C K induces a linear map á Zp(L) + (K),, mapping each element of Ag to a linear combination of Ax with Ez-coefficients, or equivalently, to subset of Ax. The map can be represented as bipartite graph over A,, L Ak. By furthermore replacing elements of A,, and Ax with their attached bars, we can interpret this graph See Figure 3 (right) for an illustration. We emphasize that the graphcode depends on the chosen barcode bases for L and K, thus the graphcode is not unique, and not topological invariant.
graph between barcodes, which we call the graphcode of (
AL. Ø restricts to map Zp(L, (L.) - Zp (K,). and
induced by the inclusion (L.). + Zp (K,). Moreover, the basis of (L.) within A₁ is simply determined by those elements in A,. that are already born at i, by definition of the barcode basis. Within this basis, the homology class represented by cycles alive at i form a basis of H,,,.)) The image these classes under yields linear combination of cycles in Z,(K)) which are all already born, and removing the summands corresponding to dead cycles yields the image of o in H, (K,).
5

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Bifiltrations.
Assume our data is now bifiltered simplicial complex written as
Km,
Km,
Kmn K
I
3...
]
(2)
K2.1
K2,2
Ka.
]
]
K1,1
K1,2
Ki,
Such structure often times appears in applications where dataset is analyzed through two dif- ferent scales. An example is hicrarchical clustering where the points are additionally filtered by an independent importance value.
We can iterate the idea from the last paragraph to a bifiltration in a straight- manner: Let A, be a barcode basis for the horizontal filtration K₁, > Ki,2 - - K K₁, With that bases fixed, there is graphcode between the i-th and (i + 1)-th horizonal filtration, and we define the union of these graphs as the graphcode the bifiltration. The vertices the graphcode are bars of the form [6, d) that are attached to basis A,, and we can naturally draw the graphcode in I23 by mapping the vertex (b, d, i). This yields a layered graph in I3 with respect to the 3rd coordinate with edges only occurring between two consecutive layers. As discussed in Appendix [D] graphcodes can be defined for arbitary two-parameter persistence modules. They can also be defined for arbitrary fields, in which case we obtain graph that has not only node but also edge attributes.
4 Computation
The vertices and edges of graphcode in homology dimension p can be computed efficiently in Onn²) time where n is the total number of simplices of dimension
+ 1. We expose the full algorithm in Appendix B in self-contained way and only sketch the main ideas here for brevity.
First of all, it can be readily observed that the standard algorithm to compute persistence diagrams via matrix reduction yields barcode basis in (1r²) time (see [1]]. Doing for every horizontal slice in (2) yields the vertices of the graphcode, and computing the edges between two consecutive slices can be reduced solving linear system via matrix reduction as well, resulting ((n²) time as well for any two consecutive slices. This is not optimal though as it results in a total running time of (sn3) with s the number of horizontal slices.
To reduce further to cubic time, we perform an out-of-order matrix reduction, where the (p + 1)- simplices sorted with respect their horizontal filtration value, but added to the boundary matrix in the order of their vertical value. This reduction process, which still results cubic runtime, yields sequence of snapshots of reduced matrices that correspond to the barcode basis on every horizontal slice, and thus yields all vertices of the graphcode. The final observation is that with additional book-keeping when going from snapshot the next, we can track how the basis elements transform from one horizontal slice to the next and these changes encode which edges are present in the graphcode.
Finally, the practical performance can be further improved by reducing the size of the graphcode, by keeping small, by ignoring bars whose persistence is below a certain threshold, and by precomputing a minimal presentation instead of working with the simplicial input. See Appendix for details.
Learning from graphcodes using graph neural networks
We describe our pipeline that exemplifies how graphcodes can be used in combination with graph neural networks (GNN's). The inputs are layered graphs with vertex attributes [bb, d), i]. with [6, d) a bar of the barcode at the i-th layer. We can add further meaningful attributes like the additive d - b and/or multiplicative persistence to the nodes suggest the GNN that these might be important. Any graph neural network architecture can be used to learn from these topological summaries. We propose the architecture depicted in Figure 4 It starts with sequence of graph attention (GAT) layers

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Max Pool
KR
Input
GAT Layers
Dense Layers
Output
Figure 4: Neural network architecture for graphcodes.
34] taking the graphcodes as input. The idea is that the network should learn to pay more attention to adjacent features with high persistence which are commonly interpreted the topological signal. These layers are followed by a local max-pooling layer that performs max-pooling over all vertices in common slice. Then concatenate the vectors obtained from the local max-pooling over all slices and feed the resulting vector into a standard feed-forver forward neural network (Dense Layers).
If we remove all the edges from the graphcodes, this model can be viewed as combination of multiple Perslay architectures [I]] one for each slice of the bifiltration. In such case, the model would implicitly learn for each barcode individually which bars are important for classification. Adding the edges, in turn, enhances this model as propagation between neighboring layers is possible: a bar that is connected to important bars in adjacent layers is more likely to be significant itself.
We also point out that the separate pooling by slices crucial our approach. It takes advantage of the additional information provided by the position of a slice in the graphcode. If we simply embed the entire graphcode in the plane by superimposing all persistence diagrams and do one global pooling, the outcome gets significantly worse.
6 Experiments
We have implemented the computation of grapheodes in a dedicated C++ library and the machine learning pipeline in Python. All the code for our experiments is available in the supplementary materials. The experiments were performed on an Ubuntu 23.04 workstation with NVIDIA Geffrcee RTX 3060 GPU and Intel Core i5-6600K CPU.
Graph datasets. We perform series of experiments on graph classification, using a sample of TUDatasets, collection of graph instances (27]. Following the approach in [10], we produce a bifiltration of graphs using the Heat Kernel Signature- Curvature bifiltration. From these bifiltrations, we compute the graphcodes (GC) and train a graph neural network as described in Section to classify them. More details on these experiments can be found in Appendix C.T and the supplementary materials. We compare the accuracy with
persistence images (MP-I) [0., multi-parameter persistence kernels (MP-K) [8, marti-parammter persistence land- sacapes (MP-L) [35]. generalized rank invariant landscapes (GRIL) [16] and multi parameter Hilbert signed measure convolutions (MP-HSM-C) [26). All these approaches produce a vector and use XGBoost 15 to train classifier.
The results Table indicate that graphcodes are competitive on most of these datasets in terms of accuracy. In terms of runtime performance, the instances are rather small and all approaches terminate within few seconds (with the exception of GRIL that took longer). Also, while the numbers in Table I for the previous approaches are taken from (26). we have partially rerun the classification using convolutional neural networks instead of XGBoost. Since the results were comparable, we decided to use the numbers from the previous work.
Graphcodes do not outperform other methods on these datasets but one can observe that there is no descriptor that consistently outperforms the other descriptors. We also observe that the performance of a certain descriptor on a certain dataset seems a little bit arbitrary. For example, (MP-HSM-CC has arguably the best overall performance but has the worst performance on COX2. A possible explanation could be that there not enough topological signal in these datasets. This might be unfavorable for graphcodes as they capture more information at the cost of invariance. We also note that the different formats of the topological descriptors require different classifiers and make a direct comparison of the results difficult. This test was included primarily because it is the standard test in related work. Still, seems unclear that topological descriptors are well suited for these datasets as, for example, on the PROTEINS dataset GNN architecuusss reach up to 85% accuracy [38]
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Table I: Graph classification results. The table shows average test set prediction accuracy in %. The numbers in all columns except the last one are taken from Table 3 in [26].
Dataset MP-1
MP-K
MP-L
GRIL
MP-HSM-C
GC
PROTEINS
67.3
67.5#3.1
65.8
70.9=3.1
74.6=2.1
73,6+2.6
DHER
80.2+-2.2
81.7++19
79.5+22.3
77.6-2.5
21.9
76.4+
COX2
77.9+-2.7
7..9+1.8
79.0+
79.8-2.9
77.1+3.0
78.7+44.9
MUTAG 85.6+77.3
86.1+5.22
84.0++6.8
87.8 4.2
85.6+5.3
86,4+6.1
IMBD-BINARY
71.122.1
68.21112
71.212.0
65.2-2.6
74.8
65.412.77
Table 2: Average test prediction accuracy in % over 20 train/test runs with random 80/20 train/test
split on the point cloud dataset and computation time in seconds of the topological descriptors. We note that GRIL could only be computed with low resolution.
MP-1
MP-L
P-1
GRIL
MP-HSM-C
GC
GC-NE
Accuracy
64.114.77
37,2
43.6.12.2
74.9.12.7
57.012. 282
86.9 1.4
82.8.1.9
Time
9176
3519
1090
333187
95
Shape dataset. To demonstrate that graphcodes are powerful topologiaal descriptors. we ap- ply them on synthetic shape dataset with a strong topological signal. We construct 5 classes of shapes co....,
......,C, C4 as follows: Class o consists of annuli and centers and radii are sampled uniformly such that the shapes do not overlap. This implies that the homology of class G degree one has rank Now we uniformly sample points from these shapes and add uniform noise. The Figure on the left shows an example of class ca-
disks in the plane. The
We generate 1000 random shape configurations and point sam- ples per class to obtain a dataset of 5000 point clouds. The point clouds are labeled with the homology of the underlying shape configuration. The goal is to classify the point clouds according to their homology in degree one. To filter the homology signal from the noise, we first compute local density estimate at each point in point cloud and compute the Delaunay-bifin -iiflttrtiini [2 with respect to relative density scores. This yields dataset of 5000 labeled bifiltrations. From these bifiltrations, we compute the graphcodes well the topological descriptors for the same related approaches as for the graph case. Additionally we also compute one- mee-prrmeter persistence images (P-I) based on a one one-parameter alpha filtration.
The time to compute these topological descriptors is reported in Table Graphcode is faster than every other method, in some cases by orders of magnitude. On the other hand, training graph neural networks more time-consuming than for convolutional neural networks, and thus graphcodes require more time in the subsequent training phase. In our experiments, the training took around minutes for graphcodes and around minute for other methods.
We now split the various datasets of topological descriptors and class labels 80/20 into training set and test set without labels, train neural networks on the training sets and test their ability to make predictions
the test sets. Further details on these experiments can be found in Appendix C.2 and the supplementary materials. The results in Table[]show that on this inherently topological classification task, graphcodes outperform every other method by significant margin. To demonstrate that the graphcode edges that connect consccutive layers add significant information, we run the graphcodes with edges removed (GC-NE). The results show even without edges, graphcodes yield a better accuracy compared to related approaches, but also that the edge information further improves accuracy.
same experiment
Random point process dataset. The following variation of an experiment proposed in [ they consider 4 types of random point processes, namely Poisson, Matérn, Strauss and Baddeley- Silverman process, and try to discriminate the Poisson null model from the other processes using a hypothesis test based multiparameter persistent Betti numbers. The latter 3 processes are prototyp- ical models for attractive behaviour, repulsive behaviour and complex interactions, respectively. We instead use multiparameter topological descriptors and neural networks to classify these processes.
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Table 3: Average test set prediction accuracy in % over 20 train/test runs with random 80/20 train/test split on the point-procest dataset.
Dataset
MP-1
MP-L
P-1
GRIL
MP-HSM-C
GC
GC-NE
Processes
66.0#2.5
50.2+3.0
35.5
61.111.6
70.7:4.9
83.4
83.1
Table 4: Orbit classification results. The table shows average test set prediction accuracy in %. The numbers in all columns except the last two are taken from Table in [.].
Dataset
PSS-K
PWG-K
SW-K
PF-K
PersLAY
GC
GC-NE
Orbit5k
72.4+2.4
76.6+0.7
83.6+0.9
85.9+0.8
87.7+1100
88.5-1.1
88,4++1.5
Orbit100k
89.2+H..3
92.3-0.3
91.5+0.3
We create dataset Processes consisting of 4 classes, each of which consisting of 1000 point clouds sampled from the above processes and use the topological descriptors and neural networks, discussed for the shape dataset above, for the classification. More details can be found in The results reported in SableB]show again that graphcodes outperform other topological descriptors. They also show that for these random point processes the influence of the edges of the graphcodes is much smaller than for the shape dataset. This is expected since prominent persistent features along the density direction are very unlikely in random point processes.
Orbit dataset. Finally we test our pipeline on another dataset which has been established in topological data analysis benchmark in the one- mne-parmetter setting. The purpose of this experiment is twofold: On the one hand it demonstrates that grapohcodes can be applied to very big datasets. On the other hand it compares the voo-parameter graphcode pipeline to its one- aee-paraetter analog PersLay The dataset consists of orbits generated by a dynamical system defined by the following rule:
Fn+1 F. + ryn (1 Vn)
mod! 1
Un+1 = yn + r (1 - xn+1) mod! 1
where the starting point (zo. yo) sampled uniformly in [0. 1]2. The behaviour of this dynamical system heavily depends on the parameter r > O. Following [T]. we create two datasets consisting of 5 classes of orbits of 1000 points generated by this dynamical system, where the 5 classes correspond to the following five choices of the parameter r = 2.5, 3.5, 4.0, 4.1 and 4.3. The datasets Orbit5k and Orbit 00k consist of 1000 and 20000 orbits per class, respectively. We again use our graphcode pipeline, discussed for the shape dataset, to classify them. The computation of the graphcodes of the 100000 point clouds took just 27 minutes demonstrating the efficiency of our algorithm. The results are reported in Table Where where we compare them to the results achieved by Persistence Scale Space Kernel (PSS-K) 332 Persistence Weighted Gaussian Kernel (PWG-K) [24], Sliced Wasserstein
Kernel (SW-K) [1), Persistence Fisher Kernel (PF-K) (25) and (PersLAY) [II reported in Table
1 of 11] The results demonstrate that graphcodes perform better than the one-parameter methods and underpin conjecture that the performance of the graphcode-GNN pipeline relative to other methods gets better as the size of the dataset increases. More details can be found in C.4
Graphcodes with different bases. The edges of the graphcode of a wwo-parameter persistence module depend on a choice of bases. So far, in all experiments, we used graphcodes with bases produced by our graphcode-algoriine next discuss the performance graphcodes on the shape classification task introduced above with respect to different choices of bases. We compute the graphcodes using the dataset from the previous paragraph "Shape dataset". At first we construct graphcode dataset (GC-ER) using an exhaustive column reduction
which is based on the standard reduction algorithm. We
[ instead of the standard reduction. Next we construct graphcode dataset (GC-RS) where we randomly shuffle the bases. This is done by performing valid column additions on the input presentation with a 5% probability. Finally, we produce graphcode dataset (GC-BC) containing 20 graphcodes constructed with random base shuffles for each input instance.
We observe that the bases chosen by the standard reduction and exhaustive reduction algorithm are far from random. The input presentation arising from a simplicial bifiltration is usually sparse. We find that this initial sparseness is preserved by the standard and exhaustive reduction algorithm in the sense that both lead to sparse graphcodes. we do random base changes the input presentation we reduce the sparseness of the input which also leads to a loss of sparseness in the output graphcodes.

---

Table 5: Average test prediction accuracy in % over 20 train/test runs of Graphcodes with different choices of bases on the shape dataset.
GC
GC-NE
GC-ER
GC-RS
GC-BC
100 Epochs
1.
12
86.7
86.611.7 85.0+1..7
84,5=2,4
200 Epochs
1.61+1.6
83.7+1.0
87.0+
88.1+
The result is an increase in the average number of edges of the produced graphs. Average number of edges with standard reduction: ~ 826, exhaustive reduction: 841, random shuffle: 1977.
We train the same graph neural network as in the previous experiments on these alternative graphcode datasets and report the results in Table For the (GC-BC) dataset modify the training process in the following way: In the i-th epoch of the training process we use the (i mod 20)-th graphcode for each instance. This approximates change of basis of each graphcode after each training epoch by picking one 20 available bases. We find that this training procedure disproportionally benefits from larger number of training epochs. Therefore, we run the same experiments with twice the number of training epochs.
The results in Table 5 show that the exhaustive reduction (GC-ER) does not significantly change the result compared to the standard bases (GC). The random basis shuffle (GC-RS) leads to slightly worse performance and slight increase in variability of the results but we note that the performance still better than without edges (GC-NE). If we use randomly shuffled bases but provide the network 20 different bases for each instance we match, and with more training epochs, even exceed the performance of (GC) and (GC-ER), These results indicate that changing the graphcode bases during the training process increase the performance.
7 Conclusion
Our shape experiment shows that current implementations of topological classifiers struggle with simple datasets that contain clear topological signal but also lot of noise. A possible explanation is that the vectorization step in these methods blurs the features too much or relies on invariants which might be too weak for a classifier to pick up delicate details. Graphcodes, on the other hand, provide an essentially complete description of the (persistent) topological properties of the data and delegates finding the relevant signal the graph neural network. As additional benefit, some vectorizations are challenging to compute, whereas graphcodes can be computed efficiently.
The biggest drawback of graphcodes is certainly that they dependent are not uniquely defined for a given dataset. The bases chosen by the standard reduction algorithm special in the sense that they lead to sparse graphs. Doing random basis changes on the graphcode dataset leads to denser graphs and a slightly worse performance on the shape classification task. But doing multiple random basis changes during the training process and, thus, providing the neural network different bases for the same instance, increases the performance even beyond the performance of the sparse graphs.
choice of basis and therefore
goal for future work is to combine sparse graphcodes and random basis changes during the training process. A possible direction could be to decompose the two-pat parameter modules into indecomposable summands, compute the graphcodes for the indecomposables and perform random basis changes on the individual components during the training process. By working only with graphcodes of decompositions we could reduce the number of possible edges.
We speculate that the combination of computational efficiency and discriminatory power will make graphcodes a valuable tool in data analysis. With the advent of more efficient techniques to gen- erate bifiltrations for large datasets, we foresce that the potential of graphcodes will be a study of investigation in the coming years.
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Acknowledgments and Disclosure of Funding
This research has been supported by the Austrian Science Fund (FWF), grant numbers W and 33765-N.
The authors thank David Loiseaux for his help with using the multipers- priiipeer--pckaeeee and Shreyas Samaga and Soham Mukherjee for their help with using the GRIL-package.
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A Basic topological notions
An (abstract) simplicial complex K with vertex set V is collection of subsets of V. called simplices, with the property that whenever a E K and S 0, then 7 E K as well. In that case, is called a face of a. A subcomplex L of K is a subset of K that is itself a simplicial complex. The dimension of simplex is its number of vertices minus 1 1- this corresponds to the interpretation that the vertices are embedded in Euclidean space and a simplex identified with the convex hull of the embedded vertices that define the simplex. Simplices in dimension 0, 1, 2 are called vertices, edges, and triangles, respectively. A facet of a p-simplex a is a face of a that has dimension p - 1. A p-simplex has exactly (p + 1)ffctt..
Let Zg the field with two elements, p 2 an integer and K be a simplicial complex. The set of p-simplices forms the basis of a -e-vector space CP) and we call elements of this vector space p-chains. Put differently, chain is formal linear combination of p-simplices with coefficients in {0, 1) and can thus be interpreted as subset of p-simplices. The boundary of a y-simplex o, written do, is the (p- - 1)-chain formed by all facets of a. For instance, the boundary of triangle the sum of the three edges that (geometrically) form the boundary of the triangle. The boundary extends to a linear map Ø (K) - Cp-1(K) (K) because it is defined on a basis of (K)
A p-chain e is called p-cycle if de = O. The p-cycles are the kernel elements of the map Ô and hence form vector space which we denote by 2,(K) The aforementioned three edges bounding a triangle form a 1-cycle because when taking their boundary, every vertex appears twice, and hence vanishes because we work with -zocofffcceenss.
A p-chain c called a p-boundary if there exists a (p + 1)-chain d such that ad = c. In other words, the y-boundaries are the images of the map & and hence, form vector space that we denote by Bp (K). A fundamental property is that for any chain c, we have that ((ec) = 0. As an example, for e triangle, the above examples exemplify that indeed, the boundary of the boundary of the triangle is trivial. As a consequence, p-boundaries are p-cycles and hence (B) is a subspace of (p))
The p-th homology group (K) is the quotient (p( (K). We will use the standard notion of "homology group" even though it is even a vector space in our case. The rank of (,) can be interpreted as the number of p-dimensional holes in K: indeed, "hole" in K must have p-cycle (i.e., an element of Zp (K) that encloses the hole, but we need to disregard such cycles that loop around space that is filled by (p + 1) simplices, hence
divide out (K). We write [a]x for the elements of Hp (K), called homology classes, where a is a p-cycle, called a representative cycle of x if and only if a + a' E (K). When the complex K is clear from context, we also just write [a].
[o) K- By definition of quotients, [a]ı = [a']
B Computation
Computing barcode bases. We start by reviewing the classical matrix reduction algorithm for computing persistent homology. For fixed filtration of simplicial complex K and p 2 0, we put the (p + 1) ---mmpliieess of K in a total order that respects the filtration, meaning that if a (p + 1) 1)-simplex of K enters the filtration at index and a (p + 1) 1)-simplex 7 of K enters at index j with i < j. then precedes in the order. Writing Up+1 1 for the number of (p + 1) 1)-simplices of K, we can assign to each (p + 1)-simplex an index in {1..... [1pt1) } that reflects this order. Doing the same for the p-simplices of K (with total number n,)), the boundary matrix 2 of K in dimension p is defined as the (ni x npp+1)-matri
mp+13
over Ze where each row corresponds to a p-simplex and each column to a (p + 1) -iimple,,, and the entry at position (i,j) equals 1 if and only if the i-th p-simplex is facet of the j-th (p + -)-simplex.
The columns of 0 span (K).) Recall our simplifying assumption that Hp (K) is trivial, implying that the columns also span ZK).). However, they are not necessarily a basis because of linear dependance. To obtain a basis, we apply matrix reduction on Ô: for non-zero column, denote by its pivol the highest index whose coefficient is not zero. Then, traverse the columns from left to right, writing ej for the currently considered column. As long as Cj is not zero and has the same pivot as previous column e with < j. add e, to cz. Every column addition will make the pivot entry eo disappear since we work with Zg- cocofffiisss
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This process results in a reduced matrix R in which no two columns have the same pivot. The remaining non-zero columns of R are linearly independent and thus form basis of (K) The bar attached to each cycle can easily be read off the matrix R: fixing cycle represented by column j. let a be the (p + ))-smpllex that corresponds to column j in 0. Then the death index is the minimal d such that a is contained in Ka. Furthermore, let i denote the pivot of column and let be the p-simplex that corresponds to row in Then, the birth index is the smallest such that T is contained in Ko. Exploiting the fact that the matrix reduction only performs left-to-right column additions and that the resulting basis has pairwise-disjoin pivots, one can show that this basis forms barcode basis of (p).
Moreover, the reduction process has the additional property that any column that is ever added to another column is already reduced, that is, will not be modified further in the process. We record this for later:
If a matrix A gets reduced to R as above, and a column addition cj + c + Cj happens during the reduction process, then G. is a column of R.
Proposition B.1.
Being variant of Gaussian elimination, the algorithm matrix à requires O(n²) in the worst case (with n = max ma }. However, the initial sparseness of O results in a close-to-linear performance in many applied scenarios 5 29
compute a barcode basis out of the boundary
The assumption that H)) is trivial might seem crucial for the construction, but one can lift this assumption without much computational overhead. In that case, one also needs to perform matrix reduction on the boundary matrix spanned by (p - 1)- and p-simplices and do some additional book-keeping to obtain basis elements for p-cycles that do not die in K. We omit details for brevity.
Efficient grapheodes through batched matrix reduction. We consider the algorithmic problem to compute the graphcode of a bifiltration of a simplicial complex K. We assume that the bifiltration 1-critical which means that for every simplex o, there a index pair (io.jo), such that o E K if and only if in < i and jo < j. In other words, every simplex has a unique entrance time into the bifiltration; while this assumption does not apply for every bifiltration, it is still satisfied for many instances that occur in practice; furthermore, there are techniques to transform other types of bifiltrations to the 1-critical case [3]
Our input is a list of simplices of K together with a critical index pair jo) per simplex, defining a 1-critical bifiltration as in ( The output is the graphcode of the bifiltration.
The straight- forward approach to compute the graphcode is to first compute a barcode basis for each horizontal slice é independently, obtaining the vertices of the graphcode. Then, express every basis element at level using the barcode basis at level E + 1 to determine the edges between levels l and + 1. This requires solve one linear system per basis element. With s the total number of horizontal slices and n the number of simplices, this approach requires O(sn²) to compute the barcodes bases and also (sn²)) to get the edges, since every linear system can be solved in (nn) time using the reduced matrices.
The straightfforward approach is non-optimal because it computes the barcode bases on each level from scratch. Since Ke+1 contains Ke, we can devise a more efficient strategy to update a barcode basis for Ke to a barcode basis for Ke41. In this way, we obtain the barcode bases for all horizontal filtrations in (n²) time.
First, we sort all (p + 1) sisiiplliess of K with respect to their second critical index, and refine to a total order, assigning to each (p + 1)-sipple an integer in (1,.... 1p+1). Initialize A to be an empty matrix, whose number of columns equals 7p+1- Precompute for every level E the set of (-simiiissl that are contained in K₃, but not in K₃-1 (these are precisely those simplices whose first critical value equals f). calling them the l-th batch. Now assume that A contains barcode basis for the horizontal level e - 1. Add the columns of the b-th batch to A at the appropriate place with respect to the chosen total order and apply the matrix reduction from the previous paragraph on A. The resulting reduced matrix yields a barcode basis for level f.
This above algorithm computes all vertices of the graphcode in worst-case ((2²) time, and also efficiently in practice, as it basically performs a single reduction of the boundary matrix of K, in some order that is determined by the bifiltration. What is perhaps remarkable is that with some extra bookkeeping, the algorithm also computes the edges of the graphcode on the fly. To see that, consider non-zero column of the boundary matrix before the l-th batch gets added, representing a basis
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element of (1( If the column does not change during the reductions caused by the batch, also basis element in the barcode basis for K2, and there a single edge connecting the two copies of a in the basis for K and Kg. If the column changes, this is caused by column addition with column from the left. Because of Proposition B. the added column is a basis element 21 of the f-th barcode basis, and we know that a 21 + " where (2) is the cycle represented by the column after the addition. Now if (1 gets modified by another basis element 22 of the 8-th barcode basis, we get a 21 + 22 + and so on. The process either stops when the column becomes 0, in which case (A) = 0 for some k, and we have obtained the linear combination of or the process stops because some a (*) is reduced and will not be modified further in the reduction. In that case, (*)) is itself a basis element 2 of the f-th barcode basis. and we obtain that a 21 22 + ... + 2k-1- Storing the linear combination during the process does not affect the running time, hence the graphcode can be computed in (n²) time.
Speed-ups. When implementing the above algorithm, we observed that the resulting graphcodes can become rather large and the practical bottleneck in the computation is to merely create the graphcode data structure. We suggest two ways to reduce the size, resulting in a much better performance: first of all, standard constructions for bifiltrations result in a large number of horizontal slices to consider. Instead, we propose to fix an integer > 0, and to equidistantly split the parameter range of the first critical value equidistantly of individual graphcodes, we propose to only consider relevant bars, where relevance means that the persistence of the bar (i.e., the distance between death and birth) is above some threshold. In particular, this removes bars of zero length from consideration, which are often the majority of all bars in barcode. We then only return the subgraph of the graphcode induced by the relevant bars. In Figure 3 (right), we have applied this filter, for instance.
positions, obtaining $ slices in the graphcode. To reduce the size
Finally, we observed significant performance gains by first computing a minimal presentation [20] of the input bifiltration, and computing the graphcode of this minimal presentation. A minimal presentation consists of generators and relations capturing the homology of bifiltration. The entire approach works in an analogous way, with generators taking the role of p-simplices and relations the role of (p 1)-simpliess. We skip details for brevity.
C Details on experiments
In all our experiments, we compute the graphcodes from minimal presentations using our C++ library. The graphcode software requires parameters specifying the homology degree, the number of slices and the direction in which to take the slices. If the minimal presentation is already computed for specific homology degree (an option provided by MPFREE bhtbbukkt.org/mmkebbr///g.. FUNCTION_ DELAUNAY < bitbucket.rg///kerrrreeeeeeee http://funcciooo_dlaaaa... the degree parameter can be omitted. One can also specify a relevance threshold to obtain the subgraph induced by all nodes (b, d) such that d - bb=bb b>6 I.
and
C.1 Graph experiments
For the datasets listed in Table T we first compute the Heat Kernel Signature-Ricci Curvature bi- filtration using a function provided by github.com/ Caam/TDA-Jae "DA-Jyymiti/GRIL and then compute a minimal presentation of this filtration using the software MPFREE 20). In all the graph experiments we compute the graphcodes using homology degree one, 20 slices and relevance-threshold t = O. We tested both possible slicing directions controlled by the option crimary-paramedd software and found that primary parameter 1 is slightly better. The computation of the graphcodes of these datasets takes between and seconds. We additionally augment the vertices [b, d), i] of the raw graphcodes with their multiplicative and additive d b persistence. The vertex attributes [bb, d, d
in the graphcode
b d b)) a] yield slightly better results. The slices index i is only used for the local max-pooling and not as part of the vector attributes for the neural network.
We then train a graph neural network classifier on these graphcode datasets where we use the architecture specified in Section 5 The details of the chosen parameters for the GNN architectures can be found in the code provided as supplementary material. We randomly shuffle the dataset and split it 80/ /20 into a labeled training set and test set without labels, train the GNN on the training
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set and evaluate it on the test set. We run this procedure 20 times and average the achieved test set prediction accuracy over this 20 train/test runs. The results are reported in Table T
C.2 Shape experiments
The point cloud dataset is generated as follows. For class we have to put annuli and 5 - disks on an empty canvas in a way such that they don't It overlap. We put the shapes on the canvas one by one by uniformly sampling radii and centers in such way that newly added shape would have at least separation e > 0 from all shapes that are already there. At the end we put uniform noise with a uniformly sampled density over the whole canvas. For each of the 5 classes we generate 1000 of these random shape configurations and take a point sample from each of them. This leads a dataset of 5000 point clouds in the plane. The details of all the chosen parameters can be found in the code in the supplementary materials.
The point clouds have a lot of randomness to them. The only thing that two point clouds in the same class have in common is that, before adding the noise, they both sampled from space with the same homology in degree one. A human could probably still predict with high accuracy how many annuli are in a picture (ef. the example figure in Section 6). This is because the underlying regions of the shapes have much higher density. To enable a classifier based on topological descriptors to do a similar kind of inference we have introduce some measure of density. Hence, compute local density estimate at every point of point cloud based on the number of neighbours in a circle with a given radius. We then score the points with respect to these local density estimates and use these score values as function values for the Delaunay-bifiltattion bifiltraatinn computed with FUNCTION DELAUNAY (2)
For the graphcode computation use homology degree one, 10 slices and slices the bifiltration in direction of fixed density. On these datasets we have to set a positive seeeaance-thresheee of 0.1 because the resulting graphcodes would be too big for the available GPU memory in the GNN training process. The density scores and the slicing along fixed density will yield the following slices: The first slice contains those 5% of the points with the lowest density. The second slice contains those 10% of the points with the lowest density, etc. The computation time reported in Table is the time needed compute the graphcodes of the whole dataset from the minimal presentations.
We then train a graph neural network classifier on these graphcode datasets where we use the architecture specified in Section 5 The details of the chosen parameters for the GNN architectures can be found in the code provided as supplementary material. We randomly shuffle the dataset and split it 80/ /20 into a labeled training set and test set without labels, train the GNN on the training set and evaluate it on the test set. We run this procedure 20 times and average the achieved test set prediction accuracy over this 20 train/test runs. We repeat this experiment for the graphcodes after removing all the edges. The results are reported in Table
As comparison we test various other topological descriptors on this classification task. We start with one-parameter persistence images obtained from one-parameter alpha- filtrtiitt in homology degree one the point clouds. For the computation use the Gudhi package [3) The computation time reported in Table 2 is the time needed to compute the persistence images of the whole dataset from the alpha filtrations.
Next we compute the multiparameter persistence images, landscapes and the signed measure convo- lutions using the multipers package computed from minimal presentations, for homology degree one, of the Delaunay-bifi all these vectorizations we use a resolution of 100 x 100, i.e., the output are 100 x 100 images. For the landscapes we use the first 5 landscapes. The computation time reported in Table is the time needed to compute the vectorizations of the whole dataset from the minimal presentations.
aapiiiaaapus/muuiiiiessss These vectorizations are For
Finally we compute the generalized rank invariant landscapes (GRILs), for homology degree one, using the GRIL package wwwggggithoonnoo www.b..omm/iie Wwoo//DD-Jyaaaat VDDA-Jyamit//GIII Since the GRIL package takes bifiltrations as input we first have to compute the Delaunay- bifilttatioissi in non non- oon-presentation form and convert them to inputs suitable for GRIL. Since the computation of the GRIL's is costly we were forced to choose a rather large step size to make the computation of the landscapes feasible. We found that enlarging the step size lead to better results than reducing the resolution. The resulting images are of size 17 x 17. The computation time reported in Table 2 is the time needed to compute the landscapes of the whole dataset from the precomputed GRIL inputs.
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For the computation of all multiparameter vectorizations we first scale the bidegrees of the bifiltration, i... the parameter of the alpha complex and the density scores, to one make the two parameters comparable.
We then train convolutional neural network classifiers on these image datasets. As in the GNN case, we randomly split the datasets into training and test sets using 80/20 split, train the network on the training set and test it on the test set. We run this procedure 20 times and take the average test set prediction accuracy. The results are reported in T
We note that despite the big step size in the GRILs, leading to rather coarse images, the performance is quite good compared to other vectorization methods. We believe that, given the computational resources to use smaller step size, the GRILs would perform significantly better.
C.3 Random Point-Process Experiments
Following [6] we consider four classes of point processes and create the dataset Processes by simulating random samples of these processes. The four classes of our dataset correspond to the four different processes and consist of 1000 random samples per class. We simulate all processes in two-dimensional Euclidian space where we restrict the sampling window [0, 1j2. The first process is a standard homogeneous Poisson process which, in our case, corresponds to uniformly sampling a Poisson distributed number of points with given intensity in [0, 1]2. The second process is a Matérn cluster process which is based on a parent Poisson process, whose points can be viewed as cluster centers, where each parent point creates a Poisson distributed number of child points uniformly sampled in a sphere centered at the parent. The third process is a Strauss process which models repulsive behaviour. In the Strauss process there a penalty on points sampled within a given distance of each other based on an interaction parameter. To sample these processes, we use MatérmPointProceee and process where we subdivide the sampling window [0, 1J2 into a grid of boxes and sample 0, or 2 points each box with probabilities 0.45, 0.1 and 0.45, respectively. Since we could not find implementation of this process in MATHEMATICA we implemented this process in PYTHON. The details of all parameter choices can be found in the supplementary materials. We choose the parameters in such way that sample of any of the above processes contains about 200 points on average.
the functions
in MATHEMATICA.
The last process we consider is Meddeley-Silvermen
As in the previous experiments we compute a Delaunay- bifiltaatoii based on local density estimates and then compute graphcodes in homology degree one using 10 slices along fixed density values without threshold. The results are reported in Table
We then train graph neural network classifier on this graphcode dataset where we use the architecture specified in Section 5 The details of the chosen parameters for the GNN architectures can be found in the code provided as supplementary material. We randomly shuffle the dataset and split it 80/20 into a labeled training set and test set without labels, train the GNN on the training set and evaluate it on the test set. We run this procedure 20 times and average the achieved test set prediction accuracy over this 20 train/test runs. We repeat this experiment for the graphcodes after removing all the edges. The results are reported in Table 3
As a comparison we also compute persistence images based on a one-parameter alpha filtration and multiparameter persistence images, landsacapes and signed measure convolutions as well as generalized rank invariant landscapes from the bifiltrations and classify them using convolutional neural networks. For all these experiments we use the same settings as for the shape datasets. The results can be found in Table 3
C.4 Orbit Experiments
The orbit datasets Orbit5k and Orbit100k are created as follows. For each of the five parameter values r = 2.5, 3.5, 4.0, 4.1 and 4.3 we uniformly sample 1000 and 20000 points (Zu, yo) in [0, 1]2, respectively, and run the dynamical system 3 for 1000 steps. this way we obtain the two datasets Orbit5k and Orbi consisting of 5 classes of 1000 and 20000 point clouds, respectively, where each point cloud consists of 1000 points in R22. The class labels are the values of r used generate a point cloud.
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After constructing the datasets we compute local density estimates at every point, score the points with respect these density estimates and compute a Delaunay-bifilration with respect these density scores. In contrast to the point clouds of the shape dataset, the point clouds from the orbit datasets do not have particularly prominent dense regions which explains why the the difference between the methods based shape dataset. From these bifiltrations we compute graphcodes in homology degree one, using 10 slices along fixed density values and use persistence threshold of 0.002.
-errammeeter persistence and graphcodes is smaller than for the
We then train graph neural network classifier on these graphcode datasets where we use the architecture specified in Section [5] The details of the chosen parameters for the GNN architectures can be found in the code provided as supplementary material. We randomly shuffle the dataset and split it 70/30 (to be consistent with (TT) into labeled training set and a test set without labels, train the GNN on the training set and evaluate and 10 times for O0k and average the achieved test set prediction accuracy over this train/test runs. We note that in [1 they average over 100 train/test runs. For time reasons, especially on the relatively large Orbit100k dataset,
the test set. We run this procedure 20 times for Orbit5k
avoided such large number of runs but, since the results have low variability, this does not make a significant difference. We repeat this experiment for the graphcodes after removing all the edges. The results are reported in Table We note that in [ they use persistent homology in degree zero and degree one for the classification. Thus we achieve the reported accuracy with less information. We can observe that the performance of grapheodes relative to perslay and the other methods as well as the influence of the graphcode- edgess increases as the size of the dataset increases. This demonstrates that the true power of the combination of graphcodes and graph neural networks really starts to manifest itself on larger datasets.
D Graphcodes of general two-parameter persistence modules
In Section we defined graphcodes of two-parameter persistence modules arising from bifiltered simplicial complexes. In this section, we show that graphcodes can be defined for arbitrary two- parameter persistence modules. For the graphcode construction, consider two-parameter persistence module as a sequence of one-parameter persistence modules connected by morphisms. A one-parameter persistence module M is diagram
M?
M
M
Mn-1,
Mi
M
Mn-1
Mn.
where Mj is a finite-dimensional vector space and M Mi + M.+1 is a linear map. The elementary building blocks of one-parameter persistence modules the so-called interval modules ((a,)) defined by
k if <i<b
(I(.b))s
0 else
id if < b-
(I(a,)))
0 else
A morphism of one-parameter persistence modules 6: M + N is a collection of linear maps Φ, M, N; such that the following diagram commutes:
N?
N
N
No
N
N₂
N
N
@1
@2
M
M3
M
M
M
M2
M
M
The theorems of Reuul-Remaa--oor Rewa--Schaiddddd 3 Theorem 1] and Gabriel [21] Chapter 2.2] imply that every mne-paramtter persistence module M is isomorphic to a unique direct sum of interval modules. i.c.. M 2 D H(az,b)) We define by Dgm(M) (M) == {(ag, by) E 18330 S S g} the persistence diagram of M. The points or intervals (aj, by) E Dgm(M) uniquely determine M up to isomorphism. We call isomorphisms μ:: .M _ O g Ioyy,,) barcode basis of M. This the abstract analog of the barcode basis of Definition 2.1 Note that there might be many choices for such an isomorphism.
The results discussed above can be interpreted on an elementary level in the following way: for persistence module M there exists choice of bases of the vector spaces M, such that all the matrices
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M are in diagonal form, i.e., every basis element in M, is either mapped to unique basis element in Ma+1 is mapped zero. Since there no unique way of transforming arbitrary bases of M into a barcode basis there is no unique isomorphism.
Since every persistence module is isomorphic morphisms of persistence modules, it is enough to understand morphisms between interval modules. Given two interval modules (a,b) and ((,d) the vector space Hom (Ja.)) (c..)) of morphisms [(a,6) + di..d) has the following simply structure:
direct of interval modules, understand
[k iffcssadds
Hom (Jia)) I
(0 else
(4)
This means that, if the intervals overlap as described in (4) then, up a scalar factor k. there is unique morphism (a.b) A [(c,d)- Otherwise the only possible morphism is the mero-morphism.
For choice of barcode bases ju: M
1 and N S + [(nd))) morphism
s: + induces morphism e/
M
N
2 j=1 (ag.by)
defined by vh n- Such morphism between direct sums is completely determined by the morphisms between individual summands, i.c.
Hom(M (M, N)
Hom Hom(Iiaj,b))) (ard))
j=1 1=1
The morphisms between summands are given by composition with the inclusion and projection to these summands
[(a.be))
de)
+ ==1 I(y.by)
1=1
Hence, by 2h == ग. the o in is either zero or determined by a scalar Ale E la and we can represent the morphism ve by a matrix M(Ø) of the form
[a1.b] b1)
[ag, b2)
lag by)
[ci, di) [c2, d2)
X
Ag
A
2g
[ons dn)
Anz
Aug
where X is the scalar determining the morphism v Note the analogy to matrices representing maps between vector spaces with respect a choice of basis. Example D. Consider the following morphism of persistence modules
(1 0)
N:
k
k
k
0
(D)
(1)
(0)
(1)
M
0
4
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---

In this case, we have M = (2.4) and N = [1.4) @ ((23), i... M and N are already in barcode form. The morphism o: M - N given by the vertical maps sends (2244) to both summands I[,4) and (2.3)-
Therefore, we obtain
[2. 4)
M(6)
(1,4)
(
1
(5)
(2,3) 1
As in the case of matrix representations of linear maps, representing with respect different bases leads to different coefficients. Therefore, the matrix M(() is not unique. It depends on the choice of barcode bases jl and v.
Example D.2. The morphisms of persistence modules @teen and Oback given by the front- and back-face of the following diagram are isomorphic
k²
(1)
(1 i
(b
k
(1)
1 0
k2
(1)
k
2
but they induce the following different matrices
[1,3)
(1,3)
M(Oriat)
(1,3)
(1,3)
M(Orack)
(1,3)
(
1
[1,3)
To get rid of the dependence on scalar factors, from now on we assume that & Ez. This implies that the entries of .M()) are either 0 and allows us to view
(mio)
M(O) 0
as the adjacency-matrix of a bipartite graph with vertex set Dgm( M) U Dgm(N).
Definition D.3 (Graphcode general). A graphcode ((Ø) [V (Ø), B(d) of a morphism 6: . + N of one- paramerreetrr persistence modules with respect to choice of barcode bases is the bipartite graph defined by
V(ő) = DDgm (M) U Dgm(N)
E(d) = (v. w) Dgm(M) x Dgm(N)).(O)... =1
By construction, the graphcode describes the morphism 6: -+ N up to isomorphism. In some sense a graphcode is just representation of the morphism with respect specific (barcode) bases.
We can now extend graphcodes from single morphism one-parameter persistence modules to two-parameter persistence modules which can be viewed as sequence of one-parameter persistence
21

---

modules:
Mom
Mim.
M Ing
M mm
Mom
M Mim
M nm
Mr
Mam-1
M 1m-1
mm
M1
M2
M
M
M
M
M
M2
Mn
M1
Mi
M
M
MiL
M
M
M
M1
M21
M
There is a graphcode I for every morphism M slices. Thus, we can define the graphcode of the ivo-parameter persistence module as the union of the graphcodes for all morphsims M Example D.4.
M.: + M between horizontal
(1)
(0)
M3
(1)
(1
0)
(0)
(1
1)
M2
k
k
0
(v)
(1)
(0)
(1)
M1
k
k 0
4
In Example D.I we already determined the matrix corresponding to the morphism from the first to second slice. Thus, M(Ø1) M(Ø) for M(Ø) as in (5. Similarly we obtain the matrix from the second third slice
[1, (2.3) M(r2)= = (1,3)
1 1)
(6)
By combining the matrices M(Ø1) and (( we obtain the following graphcode G
(1,3)
(1,4)
[2,3)
[2, 4)
The graphcode is complete deseription of wo-parameter persistence module in the following sense: Given the graphcode we can reconstruct the persistence module up to isomorphism. This follows directly from the construction.
Finally we note that we don't 't necessarily have to restrict to Z, coefficients. If is an arbitary field we can define the graphcode in similar fashion as graph with labeled edges, where the label of an edge records the scalar factor determining the morphism between the two corresponding interval summands. We can still use graphoodes defined in this way for arbitrary fields k as inputs for graph neural networks by using an architecture that allows edge weights.
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Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: We explain the construction of Graphcodes Section 3 and [ describe an efficient algorithm to compute them in Section 4 and and demonstrate their performance
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The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
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Reviews:
[{"review_id": "TffCSzc8GU", "reviewer": "Reviewer_k1Z5", "summary": "This paper proposes a computationally fast method to extract information from a 2-parameter persistence module. The authors consider a 2-parameter persistence module as slices of 1-parameter persistence modules. Each 1-parameter persistence module can be represented as barcodes. The authors consider these barcodes as basis and represent the maps between the barcodes, induced by the bifiltration, as an attributed graph. This attributed graph is used as an input to a GNN architecture. The information extracted by this method is not a topological invariant, however, empirical results show that the method gives comparable results on TU datasets and better results on some synthetic datasets as compared to existing multiparameter methods.", "questions": "1. Was there a specific reason to choose GAT as the GNN architecture? How does GC perform with other GNN architectures? \n\n2. Was there a specific reason to choose max-pooling? How does the model perform with sum/average pooling? \n\n3. It is interesting to see from Table 3 and Table 4 that adding edges in GC amounts to a marginal increase in performance. Does this mean that the information carried by the maps between barcodes is not as important? \n\n4. How does the proposed method stand with respect to the expressivity as compared to other multiparameter persistent homology methods?", "limitations": "Yes, the authors have discussed limitations.", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Overall, the paper is well-written and organized. \n\n2. The problem that the authors are trying to tackle is a hard one of capturing relevant information in 2-parameter persistence in the absence of a complete invariant. \n\n3. The method proposed in the paper has substantial theoretical backing and an algorithmic component. \n\n4. The proposed method is computationally fast, as indicated by results in Table 2.", "weaknesses": "1. The authors have shown experiments on TU Datasets and some synthetic datasets. GC seems to perform well on the synthetic datasets, however, the performance on TU datasets is not particularly impressive. The authors claim that it might be because there are not enough topological signals to capture in those datasets, which is not fully convincing.  \n\n2. The exact experimental setup (hyperparameter choices, number of GAT layers etc.) is not described in the paper, not even in the appendix.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for his comments.\n\n**Question 1 and 2:** The idea of choosing a GAT network is that topological (homological) features that are persistent across both scale parameters are reasonable candidates for the topological signal and that the network should learn to pay more attention to the features of high persistence in consecutive slices that are connected by an edge. We also tested standard graph convolution networks and average pooling which both performed worse. We want to point out that the proposed architecture is meant to serve as a proof of concept and we don't want to claim that it is close to optimal. There is certainly a lot of room for improvement and we hope that especially the expertise of the machine learning community will help to advance this architecture.\n\n**Question 3:** The graphcodes are very strong topological descriptors but this expressivity comes at the cost of basis dependence. We believe that this tradeoff is more favourable for graphcodes if there is more topological signal present in the data. The random point process data does not really have a significant topological signal and also many of the Orbit point clouds look very much like random noise. Therefore, the edges of the graphcode don't contribute as much as they do in the shape dataset. But for the Orbit100k dataset they add another 0.8\\% which might be significant given the already very high classification accuracy. Also on the TUDatasets, given the small size of these datasets and the at least questionable topological signal, we think that the tradeoff between expressivity and (basis) consistency becomes unfavourable for graphcodes. But we want to point out that, although graphcodes are not dominant on the TUDatsets, on some of those datasets graphcodes still outperform some of the other methods. \n\n**Question 4:** Please see \"Expressivity and non-invariance of graphcodes\" in the overall rebuttal above.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BKnzZ8dFPx", "reviewer": "Reviewer_NxrC", "summary": "The paper introduces \"graphcodes,\" a novel multi-scale summary of the topological properties of datasets using graph neural networks. Unlike traditional persistent homology, which uses a single parameter, graphcodes handle datasets filtered along two real-valued scale parameters, resulting in a more informative and interpretable summary. The paper outlines how graphcodes can be efficiently computed and integrated into machine learning pipelines, demonstrating improved classification accuracy over state-of-the-art approaches.", "questions": "1.\tHow is the graph constructed in graphcode? Is it a directed graph?\n2.\tFrom Table 1, MP-HSM-C performs much better than GC. Moreover, as stated in Line 251, all approaches terminate within a few seconds even on the devices mentioned in Line 240. So do we really need a method that runs a few seconds faster at the cost of losing considerable accuracies?\n3.\tAs stated in Line 227, GC-NE can be viewed as Perslay. It seems that Perslay is already accurate and efficient enough. From my perspective, it’s more practical to use Perslay rather than graphcode considering the experimental results provided in this paper. Besides, how does GC-NE perform on graph classification tasks?\n4.\tAs mentioned in line 172, the graphcode depends on the chosen barcode bases. I wonder if there is a risk that different choices of barcode bases could impact the robustness and consistency of GC?", "limitations": "See Questions", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "The paper introduces graphcodes, extending persistent homology to two-parameter scales, offering a novel and efficient method that integrates seamlessly into machine learning pipelines via graph neural networks. It demonstrates superior classification accuracy on various datasets compared to existing methods and provides an interpretable summary of topological features. The approach claims efficient computation comparable to one-parameter summaries, adding value through its innovation and practical performance.", "weaknesses": "From the experimental results, it’s hard to find too much practical meanings of using graphcode.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for his comments.\n\n**Question 1:** The construction of the graph is explained in Section 3, Appendix B and D. The vertices are the points of the persistence diagrams (topological cycles) of the individual slices where two points in consecutive slices are connected by an edge if the representative cycle of one point maps to the representative cycle of the other point. See Figure 3 for a schematic depiction. We only use undirected edges but using directed edges pointing in the direction of the filtration is certainly something that can be explored in the future.\n\n**Question 2:** Multi-parameter persistent homology especially in combination with machine learning is still in its infancy. Only recent advances in algorithms made it possible to apply these methods on larger datasets. Therefore, there is still a lack of good benchmark datasets for multi-parameter topological descriptors. We used the TUDataset for comparison because it was the most common dataset in the related literature on combining multi-parameter persistent homology and machine learning but we don't see any evidence that topological descriptors are particularly well suited for these datasets. We believe that these datasets where chosen mainly because of their small size. It is true that all approaches terminate within a few seconds but for example the MUTAG dataset has only 188 samples. As shown in Table 2, on larger datasets other methods took significantly longer than graphcode.\n\n**Question 3:** The original Perslay architecture is constructed for one-parameter persistent homology. So GC-NE is not exactly the same as Perslay. To our knowledge the adaption of Perlay to two-parameter persistent homology has also not been considered before. A stronger topological descriptor is only beneficial in the presence of enough topological signal. On some datasets weaker topological descriptors will be sufficient and the edges will not contribute much to the accuracy. But for example on the shape dataset in Table 2 the edges improve the accuracy by 4.1\\% at basically the same computational cost. \n\n**Question 4:** Please see \"Expressivity and non-invariance of graphcodes\" in the overall rebuttal above.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "m5GPYEe5RX", "reviewer": "Reviewer_1X25", "summary": "This paper introduces \"Graphcode\", a new representation for summarizing the topological properties of datasets filtered along two parameters. Graphcodes are based on persistent homology but aim to provide a more interpretable and efficient summary than existing multi-parameter topological descriptors. The key idea is to collect one-parameter sliced barcodes with basis-choice dependent mappings between consecutive persistence diagrams, which results in a bipartite graph structure, as the so-called Graphcode summary. \n\nThe authors present an efficient algorithm to compute Graphcodes and demonstrate how they can be directly used as input to graph neural networks for machine learning tasks. Experiments on several datasets show that Graphcodes have competitive or superior performance compared to other existing topological descriptors of multi-parameter persistence, especially on tasks with clear topological signals.", "questions": "- How sensitive are the results to the choice of basis used in constructing the Graphcode? Is there a way to make this choice optimal or at least consistent/stable locally?\n- Since the Graphcode is not topological invariant, one concern might be that if the GNN based on such method is still permutational invariant / equivalent? If yes, more discussion or justification should be included in the main text. If not, the potential issue or constraints in application should be discussed. \n- The theoretical foundations and analysis of what information Graphcode capture compared to other descriptors is somewhat limited. Can this be expanded?", "limitations": "As the author already mentioned in the paper, one main concern is raised from the issue that the Graphcode, as a topological descriptor, is not invariant. It might also break the stability property of persistence modules. A GNN based on Graphcode might no longer be permutation invariant or equivalent. More discussion about such limitations in both theory and application will buy the paper more benefits.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- A new topological representation that captures two-parameter topological information. \n- Efficient computation algorithm that scales well to large datasets.", "weaknesses": "- Graphcode are not topologically invariant, depending on choice of basis. Impact of this is not fully explored.\n- Theoretical analysis of expressiveness/information captured by Graphcode is limited.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for his comments.\n\n**Question 1:** The standard graphcode algorithm chooses a basis based on the reduction performed by the slicewise persistence algorithm. So in some sense the algorithm chooses a specific basis. In the uploaded software we also added an option  (do-exhaustive-reduction) to compute a basis based on minimal lexicographic cycles and an option (do-random-shuffle) to make random changes to the basis. The minimal lexicographic cycles didn't change the classification accuracy while the random bases lead to slightly worse results. The problem with the way we perform random basis changes is that it strongly increases the number of edges. This can be explained by the way the reduction in the persistence algorithm works which leads to a relatively small graph. But we think that there is room for improvement in shuffling the bases without massively increasing the number of edges. But overall changing the bases did not dramatically affect the results.\n\n**Question 2:** The permutation invariance of graph neural networks is a feature of the architecture itself and is true independent of the input graph. But if the question is if two different graphs representing the same persistence module can lead to different classification results then theoretically this can happen. One way to deal with this problem is to randomly change the basis of the graphcode in every iteration of the training process to teach the neural network to extract the ismorphism type of the persistence module independent of the basis. As explained in the previous paragraph the current implementation of this random basis change increases the number of edges too much which makes the whole process slow. But this is something that can be worked out and we are optimistic that especially the expertise of the machine learning community can help a lot to improve the architecture.\n\n**Question 3:** Please see \"Expressivity and non-invariance of graphcodes\" in the overall rebuttal above.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ry1N82B6rH", "reviewer": "Reviewer_j5Nq", "summary": "This paper introduces graphcodes, biparametric persistence summaries that are empirically efficient to compute yet not topologically invariant. It also provides a dedicated C++ library for calculating graphcodes. Graphcodes are structured as layered graphs; each layer contains vertices representing points in persistence diagrams derived from \"horizontal slices\" of a bifiltration. Edges connecting these points are induced by the inclusion maps between homology groups across the vertical slices.  The authors propose the use of graphcodes as effective representations of data topology for graph neural networks, which facilitates the learning from topological features in datasets through deep learning approaches.  The performance of graphcodes is empirically tested and benchmarked against other (multi)persistent summaries across three distinct tasks: graph classification, shape classification, and binary classification of samples derived from various random point processes. While graphcodes demonstrate superior performance in shape and binary classification tasks, they do not outperform state-of-the-art methods in graph classification.", "questions": "- Vertices of the graph are given by persistence diagrams computed from \"horizontal\" slices, and edges from vertical morphisms. Does the construction depend on which parameter you select to be \"horizontal\" and which parameter you select to be \"vertical\"? \n\n- Can computations be parallelized? If so, what is the impact on the computation time? I saw that you compute the whole graph reducing one big matrix. However, it seems to me that parallelization, like reducing several matrices at a time, could be worth in this context.\n\n- Could this be extended to $n$-parametric persistent homology? \n\n- Are there differentiability results for these computations? You compare your method with PersLay. One of the good points of PersLay is precisely that backpropagation can be computed using PersLay as in intermediate layer due to differentiability results of one-dimensional persistence barcodes. Can something be said about this?", "limitations": "Authors have addressed adequately the limitations of their paper through the text.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "**Significance**:\n\nThe paper tackles an important problem in applied topological data analysis: the development of topological summaries or representations that \"characterize\" (or provide critical information about) multiparameter persistence modules in such a way that they are easy to use/study, as persistence diagrams and their vectorizations do in one-dimensional persistent homology. More than that, the proposed summaries, called graphcodes, are not only good representations in the sense that they are easy to understand (stacked and connected persistence diagrams) as one-dimensional persistence diagrams, but are also easy to represent and manipulate in machine learning tasks, as are diagram vectorizations in the one-dimensional case. The graph nature of the representation makes graphcodes ideal for machine learning tasks, where topological information can be combined with graph learning architectures to automatically extract topological information about the input. This last point may be really relevant in topological deep learning as an automatic topological feature extractor from topological domains.\n\n**Originality**:\n\nAlthough I am not an expert on the specific topic of multiparameter persistent homology, I know that its integration with machine learning is still in its beginning, and there are not a lot of papers exploring how to create efficient and ML-suitable representations of the topological information of data. This one does, and it does so in a very original and understandable way. Seeing biparametric filtrations as graphs is something clever that I have not seen before and opens the door to automatically extracting topological information from the input data using graph learning methods. \n\n**Quality and Clarity**:\n\nRegarding quality, I find the main text (and the parts of the appendix I read) to be of very good quality. For an expert in topological data analysis, the paper can be followed smoothly, and the main concepts are illustrated with figures. For a non-expert, the theoretical part will be probably hard, but I cannot think of any way of making it clearer with the limited number of pages. Regarding the experiments, the number of tasks is enough for a theoretical paper.", "weaknesses": "The main weakness of the paper is that the topological summary is not a topological invariant and depends on a choice of basis. This likely means that, in a real scenario, neural networks fed with graphcodes would require numerous examples to learn how to overcome this limitation. In a context where equivariance and invariance are becoming increasingly important in neural networks to avoid issues like this one, this may prevent graphcodes from being a suitable choice in deep learning pipelines. However, something similar occurs with graph eigenvector positional encodings, yet they are still used in state-of-the-art graph neural networks. The suitability of graphcodes may depend on how sensitive the performance of neural networks trained with graphcodes is to the choice of basis. Given that this conference is oriented towards machine learning, I believe that experiments assessing this sensitivity are crucial for the paper's acceptance.\n\nRegarding the entire set of experiments, I feel the discussion on the graph classification problem is not fair. For instance, it is mentioned that the performance of their methods is affected by small training sets for GNN architectures, but for the proteins dataset, a GNN achieves an accuracy of 84.91%, which is significantly higher than the methods reported, according to PapersWithCode: https://paperswithcode.com/sota/graph-classification-on-proteins. Although it is a drawback that the results with graphcodes are inferior to their counterparts, I do not think this is a critical weakness, and I believe it is preferable to have a more critical discussion of the method. Similarly, I think that the comparison with other methods in graph classification is unfair because they do not report the accuracies of neural networks trained with the other summaries. Although you mention partial experimentation, it is not specified adequately how these experiments were conducted, how many parameters they used, etc., to dismiss the experiments and present results from other tables.\n\nRegarding the text, it is unusual for me to work with half-open intervals [a, a). As stated in Figure 2, some cycles can be born at the same time they die. While it is useful for computations to have these bars, it is not standard to consider persistence diagrams with only some diagonal points. Even the notation is counterintuitive (what does an interval like this signify?). Perhaps it would be worth adding a comment about this fact. \n\nThere is a typo in line 134 (bais -> basis).\n\nI believe there is a typo in line 143: I think you meant to say that it \"contains consistent bases for all chain vector spaces $Z_p(K_1)$, ..., $Z_p(K_n)$\"; subcomplexes are not vector spaces.\n\nFinally, regarding literature review, I feel like the Mapper paper [1], and [2] and [3] could be good additions to the section, The first one is used usually to generate graph representations of datasets that \"preserves\" topology and can also be used as inputs to graph neural networks. The second one analyzes expressivity of topology for graph learning (coincides very well with the topic). The third one is a topological layer based on persistent homology for graphs.\n\n[1] Singh, Gurjeet, Facundo Mémoli, and Gunnar E. Carlsson. \"Topological methods for the analysis of high dimensional data sets and 3d object recognition.\" PBG@ Eurographics 2 (2007): 091-100.\n[2] Rieck, Bastian. \"On the expressivity of persistent homology in graph learning.\" arXiv preprint arXiv:2302.09826 (2023).\n[3] Horn, Max, et al. \"Topological graph neural networks.\" arXiv preprint arXiv:2102.07835 (2021).\n\nPS: Some basic experiments (really basic) on sensitivity and a slight modification of the discussion for graph classification being more critique would increase my score to weak accept. More critical experiments on sensitivity, or/and a good discussion about this section may likely make me to increase the score even higher.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for his comments.\n\n**Weaknesses:** Please also see \"Expressivity and non-invariance of graphcodes\" in the overall rebuttal above. \n\nGiven a fixed size dataset one can create multiple graphcodes corresponding to different choices of bases for each instance. This is a way to teach the neural network to learn relevant information independent of the chosen basis without the need of a bigger initial dataset.\n\nWe only included the TUDatsets as they seem to be the standard benchmark for multi-parameter topological descriptors. We are aware that GNN's outperform topological descriptors on these datasets and we don't see any evidence that these datasets are particularly well suited for topological descriptors. But we want to point out that graphcodes are not inferior to other methods on all of these datasets. They are certainly not dominant but on some of the datasets they outperform some of the other methods. And non of the methods outperforms every other method on every dataset. One can also not directly compare the data required for a GNN taking the graphs directly as an input to a GNN taking the graphcodes as an input as the graphcodes can be much bigger than the initial graphs.  \n\nWe actually did experiments testing the sensitivity to different choices of basis. In the uploaded software we implement an option (do-exhaustive-reduction) to compute a basis based on minimal lexicographic cycles and an option (do-random-shuffle) to make random changes to the basis. The minimal lexicographic cycles didn't change the classification accuracy while the random bases lead to slightly worse results. We are happy to include these experiments in a final version. We also propose to change the basis of each graphcode in every iteration of the training process to teach the neural network to extract the relevant information independent of the bases. The problem is that randomly changing the basis is not trivial. The simple method we implemented in (random shuffle) increases the number of edges a lot. Doing this kind of basis changes in every training iteration would make the training process slow. But this is certainly something that can be improved in future work.\n\n**Question 1:** Graphcodes can be constructed by taking slices vertically or horizontally which  we have also implemented in the software. Both choices of direction lead to differtent graphs but both of them contain the same complete information of the two-parameter persistence module. One could also consider taking diagonal slices. \n\n**Question 2:** Yes one could compute the subgraph for any two consecutive slices independent of each other in parallel. For computing graphcodes from arbitrary two-parameter persistence modules that would probably be the most efficient option. But in the case of persistence modules coming from bifiltered simplicial complexes every simplicial slice is contained in the following one. What makes the graphcode algorithm so efficient is that we don't have to compute the persistence of all slices successively but we can compute the graphcode of all slices at once in the same asymptotic time as computing the persistence of the top slice.\n\n**Question 3:** This could theoretically by extended to persistence modules over any poset of the form $P\\times \\mathbb{R}$ where the $\\mathbb{R}$ part is compressed by taking barcodes and maps between bars over different points in $P$ are connected by edges in a similar fashion.\n\n**Question 4:** This is an interesting question but unfortunately we don't have an answer at the moment.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
At first, we want to thank all reviewers for their comments. As some points came up in more than one review we want to address them here in a general rebuttal and address more specific questions in individual rebuttals below.

**Expressivity and non-invariance of graphcodes:** The most important point we want to address is the critique of non-invariance of graphcodes which came up in all of the reviews. It is a well-known theorem in the field of topological data analysis that there can not exist a complete (capturing all the information) and discrete invariant of multi-parameter persistence modules. Some reviewers asked about the expressivity or the information captured by graphcodes. The graphcode of a two-parameter persistence module is a complete descriptor which means that it captures all the information of the two-parameter persistent homology up to isomorphism. It is easy to reconstruct the initial module up to isomorphism from the graphcode. It is just an alternative compact representation of the module that is suitable as input for graph neural networks. Other established methods construct a discrete invariant from a module which comes at the cost of significant loss of information. The theorem mentioned before implies that one is forced to make this tradeoff between invariance and loss of information and the lack of invariance of graphcodes has to be put into this context. From this perspective, on appropriate datasets (like the shape dataset), the lack of invariance is rather a strength than a shortcoming since it is the only way we can capture all the information. A two-parameter persistence module can be equivalently represented by two graphs with different edges (the vertices and vertex labels are actually invariant) but both graphs contain exactly the same information.

**Graphcodes as inputs for neural networks:** Graphcodes are a novel approach to combine multi-parameter persistent homology and machine learning by giving up on invariance for the benefit of providing complete two-parameter persistence information to a neural network. Since this has not been considered before and as the reviewers have pointed out there are a lot of open questions like: How well can a graph neural network architecture learn to extract the isomorphism type of the underlying module from the graphcode and use this information to make predictions? What is the best architecture to learn from graphcodes? How to efficiently change the bases of the graphcodes in the learning process to teach the neural network to extract the isomorphism type of a module independent of the basis?  As acknowledged by all the reviewers, we built the theoretical foundation of graphcodes, an efficient algorithm to compute graphcodes and proposed an architecture that works well on some datasets which provides a proof of concept. Hence, we think that these open questions are not a reason to reject the paper. Every line of research has to start at some point. Since most of these open questions are on the machine learning side of the pipeline we believe that it would be very beneficial to bring this work to the attention of the machine learning community.

Meta-review: This paper presents a new multi-scale summary of topological features of graphs, called a *graphcode*. Graphcodes are obtained from bifiltrations, i.e. a filtration process along two (independent) variables on a graph. Leveraging ideas from computational topology, in particular persistent homology, the paper calculates graphcodes as a representation of maps between consecutive slices in a stack of persistence diagrams, i.e. topological descriptors. The resulting features can be easily integrated into a deep-learning model, leading to a new way of solving graph-learning tasks.

Initial support of the paper was not very pronounced; reviewers agreed on the utility, novelty, and relevance of the idea, but cited concerns about the experimental setup as well as some of the theoretical properties. For the latter point, the fact that graphcodes—unlike many other methods in topology-based graph learning—are *not* a topological invariant was raised multiple times. The authors acknowledge this fact openly in their submission, though, and the AC believes that this is not a shortcoming *per se*, since graphcodes can nevertheless be gainfully employed empirically (and most, if not all, graph-learning methods do not even discuss properties like invariance). That being said, the other shortcoming, raised by multiple reviewers, concerns the experimental results and their framing. For a new method like this, which the AC construes to be primarily about showing that multi-parameter persistent-homology methods can be computationally efficient, a paper cannot be expected to report state-of-the-art results. Nevertheless, the AC essentially echoes the feedback by reviewer `j5Nq`, who made additional suggestions about (a) discussiong **additional related work**, and (b) an **improved contextualisation of the results**.

Since the authors engaged openly and actively in their rebuttal, several of the concerns could already be alleviated. Thus, the authors are trusted to implement the aforementioned suggestions, as well as other suggestions raised by reviewers (mostly concerning the accessibility of the work), in the camera-ready version of their paper.
Decision: Accept (poster)