A $2$-parameter Persistence Layer for Learning
Authors: Cheng Xin
Soham Mukherjee
Shreyas N. Samaga
Tamal K. Dey
Venue: Submitted to ICLR 2023 (2023)
Keywords: topological data analysis
graph representation
persistent homology
2-parameter persistence
graph neural network

Abstract: $1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as cycle basis hidden in data. It has found its application in strengthening the representation power of deep learning models like Graph Neural Networks (GNN). To enrich the representations of topological features,  here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vectorization on $2$-parameter persistence modules called Generalized Rank Invariant Landscape {\textsc{Gril}}. We show that this vector representation is stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vectorization and its gradients. We also test our methods on synthetic graph datasets and compare the results with some popular graph neural networks.

Under review as a conference paper at ICLR 2023
A 2-PARAMETER PERSISTENCE LAYER FOR LEARNING
Anonymous authors
Paper under double-blind review
ABSTRACT
1-parameter persistent homology, cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected compo- nents and cycles hidden in data. It has found its application in strengthening the representation power of deep learning models like Graph Neural Networks (GNN). To enrich the representations of topological features, here we propose to study 2-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we intro- duce novel vectorization on 2-parameter persistence modules called Generalized Rank Invariant Landscape (GRIL). We show that this vector representation is lipschitz (stable) and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vectorization and its gradients. We also test our methods on synthetic graph datasets and benchmark graph datasets, and compare the results with previous vector representations of 1- 1-parameter and 2-parameter persistence modules
INTRODUCTION
Machine learning models such as and Graph Neural Networks (GNNs) (Gori et al. 2005 Scarselli ctt al. 2009; Kipf & Welling 2017: Xu et al.) 2019) are well-known successful tools from the ge- ometric deep learning community. The representation power of such models can be augmented by infusing topological information as some vector representation of persistent homology of the underlying space hidden in data. Many recent works have successfully integrated topological infor- mation with machine learning models. (Carrière et al. 2020: Kim et al. 2020 Gabrielsson et al.)
2020
Hofer et al.] 2020 Horn et al. [022 Swenson et al. 2020, Bouritsas et al. 2022; Corbet al. 2019 Carrière& & Blumberg. 2020] Vipond 2020). In most of these works, the authors use 1-parameter persistence homology as the topologiaal information. However, in (Corbet et al. 2019) Vipond 2020; Carrière & Blumberg 2020). the authors use vector representations of 2-parameter persistence modules. In (Carrière & Blumberg. 2020) and (Corbet et al. 2019), these representa- tions are based on slices of 2-parameter persistence modules along lines, which are first studied and computed by (Lesnick Wright 2015). In (Vipond) 2020). the author generalizes the notion of 1-parameter persistence landscapes (Bubenik, 2015). In this paper we propose a novel vector representation Generalized Rank Invariant Landscape (GRIL) for 2-parameter persistence modules which encodes richer information beyond fibered barcodes alone. The building blocks are based on the idea of generalized rank invariant (Kim & Mêmoli 202T: Dey al. (2022). The construction of GRIL. is generalization of persistence landscape (Bubenik 2015] Vipond| 2020). We will show that the vector representation GRII. is 1-L-pschitz and differentiable with respect to the filtration function f. which allows us to build differentiablee topological layer, PERSGRIL, in machine learning pipeline. We demonstrate its use on synthetic datasets and standard graph datasets i From the perspective of direct use of 2-parameter persistence modules into machine learning models, to the best of our knowledge, this is the first work of its kind.
Persistent homology is a useful tool for characterizing the shape of data. Rooted in the theory of algebraic topology and algorithms, it has spawned the flourishing area of Topological Data Analy- sis(TDA). The classical persistent homology, also known as, 1-parameter persistence module, has
attracted plenty of attention from both theory (Edelsbrunner & Harer 2010 Oudot 2013) Carlsson
& Vejdeemo-Johansson 202 Dey & Wang 2022 Hofer et al. 2017 Li al. 2022
Dey & Wang
the code for full implementation will be available after review process completed.

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2022
Mémoli al. 2022) and applications Yan et al. 202 Zhao al. 2020; Yang et al. 2021b
Banerjee et al. 2020: Wu et al. [2200] Wang et al. 2020 Chen et al. 2021 1021] Hu et al. 2021 Yan et al, 2022). The standard pipeline of 1-parameter persistence module as follows: Given domain of interest X' (e.g. a topological space, point cloud data, a graph, or simplicial complex) with scalar function f X' - R, one filters the domain X' by the sublevel sets X, _ {ir E X f(x) S a} along with continuously increasing threshold forms an increasing sequence of subspaces = X C Xa₃ S .... C X4 = X. Along with the filtration, topological features appear, persist, and disappear over some intervals. We consider p- homology groups Hp(-) (over a field, see (Hatcher (2000) of the subspaces in this filtration, which results into sequence of vector spaces. These vector spaces are connected by mncluiion-induce linear maps forming an algebraic structure 0 =x) - Hp(,,,) + > (x (Hatcher 2000)). This algebraic structure, known as 1parameter persistence module induced by and denoted as M can be uniquely decomposed into a collection of atomic modules called interval modules, which completely characterizes the topological features in regard to the three behaviors- appearance, persistence, and disappearance of all p-dimensional cycles. This unique decomposition of 1- 1-parameter persistence module is commonly summarized as a persistence diagram (Edelsbrun- ner et al., (002) or barcode (Zomorodian & Carlsson, 2005). Figure | (left) shows filtration of a simplicial complex which induces a 1 1-parameter persistence module and its decomposition into bars.
IR. The collection { J. which is called filtration,
K₁
K2
R3
Ks
Ku
Hg 9
2
4
5
o6
Hi
Figure I: (left) 1-parameter filtration and bars; (right) 2-parameter filtration inducing 2- 2-parameter persistence module whose decomposition is not shown.
Some problems in practice may demand tracking the topological information a filtration that is not necessarily linear. For example. in (Adcock al., 2014), 2-parameter persistence module is shown to be better for classifying hepatic lesions compared to 1-parameter persistence. In (Keller et al. 2018), virtual screening system based on 2-parameter persistence modules are shown to be effective for searching new candidate drugs. In such applications, instead of studying a sequential filtration filtered by a scalar function, one may study a grid-filtration induced by a IR2-valued bi- filtration function ʃ X' + Ra with R² equipped with partial order u S V U1 S U1, U2 S Wy: see Figure rightt for an example of 2-parameter filtration. Following a similar pipeline as the 1-parameter persistence module, one will get a collection of vector spaces { {M uERE indexed by vectors u (u1, W2) IR2 and linear maps (MS(u S v) Mf. - M u S v E R2] for all comparable u I v. The entire structure Mr, in analogy to the 1-parameter case, is called 2- 2-parameter persistence module induced from f. Unlike 1 1-parameter case, the algebraic structure of 2- 2-parameter persistence modules is much more complicated. There is complete discrete invariant like persistence diagrams or barcodes for 2-parameter persistence modules (Carlsson & Zomorodian, 2009]. A good non-complete invariant for 2-parameter persistence modules should characterize as many non- non-isomorphic topological features as possible. At the same time it should be stable with respect small perturbations of filtration functions, which guarantees its important properties of continuity and differentiability for machine learning models. Therefore, how to build a good summary for 2- 2-parameter persistence modules which is also applicable to machine learning models is an important problem.
2 2-PARAMETER PERSISTENCE LANDSCAPE
From the perspective of representation learning, persistence module can be viewed special representation of discrete topological space, like point cloud data or graph embedding, which cap- tures geometric and topological information. 1-parameter persistence module captures information about topological features that persist across different scales. Here, we consider a bi-filtration which leads to 2-parameter persistence module. To better utilize the richer information captured by 2-
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parameter persistence modules, here propose GRIL (Generalized Rank Invariant Landscape), stable and differentiable vectorized representation of a 2-parameter persistence module.
Let M MM be 2-parameter persistence module induced by filtration function f. We say connected subset S I2² is an interval if Yu S v < w, [u E 1, w E 1] = [2 E 11. The restriction of M an interval denoted as Ml, J, is the collection of vector spaces {M u E 7] along with linear maps (M(u & <) u, E f)}. One can define generalized rank:
rkm - rank lim M), + lim M r
where lim M + lim Ml2 the unique linear map from the limit of M), to the colimit of M]1- |u S w S t is a rectangle subset in R², lim M), M, and lim M/, = M,. Then rk (1) equals the traditional rank of the linear map M(u S v). We refer the reader LMacLane 1971) for the definitions of limit and colimit in category theory. The basic covering set W on R², which called generalized rank invariant (Kim & Mémoli 2021) over W.
When / [u, v) - (w RR
idea of GRIL is to compute collection of generalized ranks {ik (I) ew
We choose W to be set of Worms defined as follows:
W
p
8
0,421,PER
0,
1.pE
where
P
- {q
3a
R S (E
1)8: b19 --a------ (a, a))l S 8}.
We call the p in the center point of the f-worm and 8 the width of the f-worm. As special case, when 1,
P {q |pp
00 S 8} is a ō-square with side 25 centered p. is the union of all 8-squares 9 centered at some
In general, for any 2 1, > 0, &-worm
P b
point q on the ofa-diagonal line segment p + (1, -1 with la| I (e 1)6. Therefore, we can also equivalently write p 5
Uq=p+(a.-a |a|$(1-1)6
See Figure (left) for an illustration of 2-worm
example. We now define GRIL.
d+p
Figure 2: A 2-worm, discretized 2-worm and expanded discretized 2-worm. p denotes grid resolu- tion. The blue dotted lines show the intermediate staircase with step-size p. The red dotted lines form parts of the squares with size d which replaced by the blue dotted lines in the worm. The last figure shows the expanded 2- 2-worm with red and blue dotted lines. The expanded 2-worm has width p which is the step expansion of the worm with width d.
Definition 2.1 (Generalized Rank Invariant Landscape ). For persistence module M, the Gener- alized Rank Invariant Landscape (GRII.) of M is a function XM R2 x N4 x N4 - R defined
M (p, k) sup(rk" (p kj. 620
(1)
Proposition 2.1. GRIL is equivalent to the generalized rank invariant on W. Here the equivalence means bijective reconstruction from each other (proof in Appendix
In practice, choose center points p from some finite subset P C R². e.g. finite uniform grid in R², and consider l < K, < L for some K, L E N4. Then A can be viewed as a vector of dimension [P] x K x L. See Figure 3 for illustration of the overall pipeline of our construction
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of X starting from filtration function a simplicial complex. Figure shows the discriminating power of GRIL where we see that GRIL can differentiate between shapes that are topologically non-isomorphic.
Srngitcaa Compllar flod - it 100)
Sinpliad Blicatin
(0.3)
(2.20
(1.0)
Proilaco uodile
X
Coupolic uno WeEES ditieent J8
Figure 3: The construction starts from simplicial complex with bi-filtration function as shown on the top right. The simplicial complex consists of two vertices connected by one edge. Based on the bi-filtration, simplicial bi-filtration can be defined as shown on the top left. On the bottom left, 2-parameter persistence module is induced from the above simplicial filtration. If we check the dimensions of the vector spaces on all points of the plane, there are 1-dimensional vector spaces on red, blue and light purple regions. On the L-shaped dark purple region, the vector spaces have dimension 2. Finally, on this 2-parameter persistence module, we calculate A (p. E) for all tuples (p, k) E P x K x L to get our GRII. vector representation. By Defintion 2.1 the value AM (p, k, E) corresponds the width of the supremum ê-worm which the generalized rank is least On the bottom right, the interval red the maximal 2-worm for AM (p, = 1, 1,8- 2). The green interval is the maximal 2-worm for X (g, k 2, 2). The yellow square is the maximal 1-worm for MM (r, = 1, = 1), and the blue interval is the maximal 3-worm for MM (r, k = 1, =
Stability and Differentiability of GRIL. An important property of GRIL is its stability property which makes immune small perturbations of the input bi-filtration while still retaining the ability to characterize topologies. We show GRII. is 1-Lipschitz (stable) with respect to input filtrations.
Proposition 2.2. Given two filtration functions f, f' X R2, |AM (proof in Appendix
AAs""
11o
- Ilʃ l'
Remark 2.1. Note that when X a finite space (c.g. finite simplicial complex (see Definition A.1 point cloud) with |X| = n then, any f: X - IR can be represented as a vector in Ran
We now define PERSGRIL.
Definition 2.2 (PERSGRIL.). For a finite space X' with | and fixed k,l,p. PERSGRIL is function A P IR2n - given by Ab*(f) M² (k, Proposition 2.3. PERSGRU. Lipschitz contimuous with respect to the bi-filtration functions on finite spaces. (proof in Appendix
"y-axiss represents Rips filtration and r-axis represents density. Density value of vertex v, denoted by 7u. defined as exp(Avg. nearest neighbor distance of v). For an edge e := (u, e) and triangle t := (u, 8, w). ry is defined as max("1.. 7.) and max (T., Te, 7w)) respectively, Rips filtration value of all vertices r, 0. For an edge = (u, v). the Rips filtration value Twv is the Euclidean distance between u and v. For a triangle t = (u, v, w). the filtration value max(rue, Tuse Tow) is the maximum over all its edges. The filtration value rounded off the nearest hundredth decimal place for visualization purposes.
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1000 MEX KOW
Figure 4: GRIL topologiaal discriminator: each shows a point cloud P, its density-Rips bi-filtration H GRIL value heatmap for I-dimensional homology and generalized ranks 1 and k ==2 2 named Åι and λ2 respectively. First Betti number (3,) of a circle is which is reflected in A being non-zero. 31 for two circles is 2 which reflected in both x and x being non-zero. Similarly, 91 circle and disk together is 1 which is reflected A1 being non-zero but X2 being zero for this point cloud.
By Rademacher's theorem (Evans & Gariepy 2015), we have PERSGRI,, a Lipschitz continuous function, being differentiable almost everywhere.
Corollary 2.4. PERSGRIL is diffcrentiable almost everywhere.
The differentiability of PERSGRIL in Corollary 2.4 refers to the existence of all directional deriva- tives. But the existence of steepest direction as the "gradient" of PERSGRIL might not be unique. We propose an algorithm to efficiently compute one specific steepest direction based on the follow- ing theorem.
X Theorem 2.5. Consider the space of all filtration functions {f : - R2) on a finite space X with | n, which equivalent
R2n. For fixed k. L.P, there exists a measure-zero subset Z C R2n such that for any f E R22n satisfying the following generic condition: Yx & y E X, fx)1 # (()):J(x)) # f(y)2. there exists assignment X [+1,0, +f]² such that
V.A R. (J) A lima-so (f+(() (5)
maxgex VgAke(f)
a||
The proof of Theorem 2.5 in Appendix B also shows how to find the assignment with the cor- responding set of supporting simplices. This result leads us to update the simplicial filtration with such assignment See the description of enhancing topological features in section
3 ALGORITHM
We present algorithm compute GRII. in this section. High-level idea of the algorithm is as follows: Given bi-filtration function f X + R?, for each (p, E P x K x L, we need to compute M (p. k. E) sups20 (rk M/ width over worms on which the generalized rank is at least k. In order to find the value of this by applying the algorithm
k.. essence, we need compute the maxinum
width, we use binary search. We compute generalized rank Mk proposed in DDy et al. 2022), which uses zigzag persistence boundary path. This zigzag
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persistence is computed efficiently by recent algorithm proposed in (Dey & Hou 2022 We denote the sub-routine to compute generalized rank over a worm by COMPUTERANK in a algorithm mentioned below. COMPUTERANK(] I) takes as input bi-filtration function and an interval I, and outputs generalized rank over that interval. In order to use the algorithm proposed in (Dey et al.) 2022). the worms need have their boundaries aligned with grid structure defined on the range of Thus, we normalize f to be in the range [0, 1] x [0, 1], define a grid structure on [0, 1] x [0, 1] and discretize the worms. Let Grid
{(M M
m n | m, n E , M}} for some M E 2+.
We denote the grid resolution p 1/M. We uniformly sample center points for the worms
Grid from this grid. We consider discrete worms
P
Uq=pt(a.aa)
q for all p E la 11-118
géGrid
See Figure as an illustration of discrete worms. Now all the discrete worms
P intervals
whose boundaries are aligned with the Grid. We apply the procedure COMPUTERANK
P
to compute rka P Let i (p,k,t) srps M
2 kj. One can observe that
AM (p. k) A (p. k. e) & p. Therefore, we compute Ả as an approximation of with the approximation gap controlled by the grid resolution The pseudo-code is given in Algorithm T The algorithm is described in detail in Appendix C
Algorithm COMPUTEGRII.
Input: f -Bi-iltration function, e 0, 2 1, P C Grid, p: grid resolution
Output: A(p, I) Persistence landscape at point for fixed l and dmin t -p, dmax +
2: while dmin S dmaz do
3:
x (dmin dmaa)) 1/2
4:
+ Pl
5:
COMPUTERANK(F.. IF r = THEN
6:
7:
rk d
8:
dmnn +- d+p +
9:
ELSE > K. THEN
10:
dmin <- +
ELSE
THEN
12:
dmpr + d d
RETURN
Time complexity. Assuming grid with ć nodes and bi-filtration of a complex with simplices on it, one can observe that each probe in the binary search takes O((l + n²) time where w < 2.37286 is the matrix multiplication exponent (Alman & Williams) 2021). This is because each probe generates a zigzag filtration of length Ott) with O(n) simplices. Therefore, the binary search takes O((t + n log t) time giving total time complexity of O(t((t n) log t)) that accounts for (t)) worms.
4 EXPERIMENTS
We create a differentiable topological layer based on GRIL named PERSGRIL which is in line with Definition 2.2 In essence, PERSGRIL takes bi-filtration function as input and gives the value of GRIL on the persistence module generated by the filtration function output.
Experiment with HourGlass dataset. We test our model on synthetic dataset (HourGlass) that entails binary graph classification problem over collection of attributed undirected graphs. Note that this synthetic dataset is designed to show that some attributed graphs can easily classfied by 2- 2-parameter persistence modules while not easy for 1-parameter persistence moduels or commonly

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used GNN models. Each graph G from either class is composed with two circulant subgraphs G1, G2 connected by some cross edges. The node attributes are order indices generated by two different traversals T2. The label of classes corresponds to these two different traversals Tú, T2- Therefore, the classification task is that given an attributed graph G, the model needs to predict which traversal is used to generate G. See Figure (left) with same graph structure but node attributes generated by two different traversals. More details can be found Appendix D.I] We denote HourGGass[a.b] as the dataset of graphs generated with node size of each circulant subgraphs in range [a. b). We generate three datasets with different sizes: HourGlass[10,22
example of two attributed graphs
HourGlass(211.0 [21300] HourGlass(31,40, Each dataset contains roughly 400 graphs. We evenly split HourGlass [2,330] into balanced training set and testing set on which we compare PERSGRII with several commonly used GNN models from the literature including: Graph Convolutional Networks (Kip) & Welling, 2017), Graph Isomorphism Networks (GIN) (Xu ctt al., 2019) and a 1- 1-parameter persistent homology vector representation called persistence image (Persimg (Adams et al. 2017). All GNN models contain 3 aggregation layers. All models use 3- layer multilayer perceptron (MLP) as classifiers. More details about model and training settings can be found in Appendix [.I] After that we also test these trained models on HourGlass{ 10,20] and HourGlass[311 ,440 to check if they can generalize well on smaller and larger graphs. The experiment results are shown in Table We can see that this dataset can be easily classified by our model based on 2- 2-parameter persistence modules with good generalization performance. But it is not easy for 1-parameter persistence method like Persimg or some GNN models.
Testing accuracy of models on HourGlass
Model
GCN
GIN
PersImg
PERSGRIL
HourGlass[21.30]
87.2514.0
84.00
74.00:17.4
100.0+0.00
HourGlass| 10,20]
6..3144.6
62.98
+334
50.33+1.66
99.79+-.11
(louGGlas(311400
87.75
79.10
86.9515.0
100.01:0.0
Table 1: Table of testing results from different models. Last two rows show the testing results on HourGlass[10,22] and HourGlass[31,40] of models trained on HourGlass(21,30).
4,3
6,4
14.8
16,9 9
2,2
12,7
18, 10
25,72
1,1
10, 15
11,6
20, 20
3, 1
0. 14
13, 16
19, 19
5. 12
7,13 13
17,17
17,
18
Figure 5: (Left) An example of a graph consisting of two circulant subgraphs. The pair of indices on each node represents the its order the traversals T₁ and T2 respectively. Both traversals start from the left node as the root node. (Right) Cross edges placed across two subgraphs.
Graph experiments. We perform a series of experiments on graph classification to test the proposed model. We use standard datasets such as PROTEINS, DHFR, Cox2 and MUTAG (Morris et al.) 2020). A quantitative summary of these datasets is given in Appendix D.2 On these datasets, we compare the performance of GRIL with other models such as multiparameter persistence landscapes (MP-L) (Vipond 2020), multiparameter persistence images (MP-I) (Carrière & Blumberg 2020). multiparameter persistence kernel (MP-K) (Corbet et al. 2019) and PersLay (Carrière et al. 2020
In (Carriére& & Blumberg 2020). the authors use the heat kernel signature (HKS) and Ricci curvature on the graphs to form bi-filtration. We also use the same bi-filtration and report the result in the column Gril HKS-RC. Since the graphs in all of these datasets have node attributes, we also form Density-Alpha bi-filtration on the node features and compare the performance of GRIL on this bi- filtration (Gril D-Alpha) with other methods. Density-Alpha bi-filtration uses Distance-to-Measure function as the filtration function for one coordinate and an Alpha complex filtration in the other coordinate. We use simple 1-layer MLP classifier in order to test the discriminating power of GRIL features. We can see from Table 2 that GRIL with 1-layer MLP gives better performance as compared to multiparameter persistence image, multiparameter persistence kemne, multiparameter
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persistence landscapes PROTEINS, Cox2 and MUTAG. However it doesn't t seem perform as well on DHFR. On PROTEINS, Cox2 and MUTAG, GRIL has comparable performance with PersLay. (Carriére et al. 2020). Perslay also uses 1-layer MLP as the classifier. However, PersLay uses spectral features of the graph along with the 1-parameter persistence diagrams corresponding to the filtration given by the heat kernel signature the filtration function. The reported results after ten-fold cross validation. The full details of the experiments are given in Appendix D.2
Dataset
MP-1
MP-K
MP-L
PersLay
GRIL HKS-RC
GRIL D-Alpha
PROTEINS
67.3 3.5
67.5 3.1
65.8 - 3.3
74.8 + 0.3
71.6 4.2
72.6 + 4.9
DHFR
8.3
81.7 = 1.9
2.3
80.3 0.8
+.7
61.8 + 2.5
Cox2
77.9
79.9 + 1.8
79.0 3.3
80.9 1 1.0
78.7 + 0.0
80.6 1 2.5
MUTAG
85.6 0.617.7
86.2 = 2.6
85.7 1 2.5
89.8 1 0.9
89.4 + 7.0
88.4 1 8.4
Table 2: 10-fold cross validation test accuracy of different models on graph datasets. The values of the MP-I, MP-K, MP-L columns LersLay Lay column are as reported in (Carrière et al. 2020).
reported in (Carrière & Blumberg 2020 and those in the
We have compared the performance of this model with different values of l in X(p, k. The results are reported in Table 5 in Appendix [ We report the computation times for these datasets in Table 6 in Appendix
D In Table we show the performance of GRII. with different grid resolutions.
Differentiability of PERSGRIL: A proof of concept. In the previous experiments we showed how PERSGRIL can be used to obtain topological signatures from graphs to facilitate a specific downstream task, which in our case is graph classification problem. In that application, we build PERSGRIL a static filtration function. By static, we mean that we computed the topological features and used them as an input to a classifier. In this experiment, we demonstrate, as proof of concept. how PERSGRL can be easily integrated in differentiable framework (with the theoretical foundation laid in See specifically Theorem 2.3 like the standard neural network architectures. We show this by rearranging the positions of input points, i.e. encouraging formation of clusters, holes by choosing suitable loss functions.
As shown in Figure 6 input to PERSGRIL is points sampled non-uniformly from two circles. Re- call that GRIL. is defined over a 2-parameter persistence module induced by some filtration function f ( For every vertex v, assign (u) = 1 exp( _ Σ d(v, ez)), where v; denotes i-th nearest neighbor of the vertex v and d(u, v.) denotes the distance between and e. For our ex- periments we fix a = 5. We set fy(v) O. We compute ALPHACOMPLEX filtration (Edelsbrunner & Harer 2010) of the points and for each edge == (u. v) we assign (e) = max( Jz (+) and [y(C)
To obtain valid bi-filtration function on the simplicial complex we extend the bi-filtration function from 1-simplices to 2 2-simplices, i.e. triangles. We pass f as an input to PERSGRIL, coded with the framework PYTORCH (Paszke et al., 2019), that computes persistence landscapes. PERSGRIL uniformly samples n center points from the grid [0, 1j2. Since GRIL value computation can be done independently for each k and a center point, we take advantage of parallel computation and implement the code in parallel manner. In the forward pass get GRIL values A(p, k; E) for generalized rank 1 = 1,2, worm size i and homology of dimension 1 while varying p over all the sampled center points. After get the GRIL values, we compute the assignment s according to Theorem 2.5 During the backward pass, we utilize this assignment to compute the derivative of PERSGRIL with respect the filtration function and consequently update it. We get values of A(- 1, 2) for n center points, We treat these n values as a vector and denote is as λy. Similarly, we use X2 to denote the vector formed by values X(- 2, 2). We minimize the loss L = - -(I| 1 + A (23 Figure 6 shows the result after running PERSGRIL for 200 epochs. The optimizer we use to optimize the loss function is Adam (Kingma & Ba 2015) with learning rate of 0.01.
exp(d(u,,)
CONCLUSION
This work presents GRIL, a 2-parameter persistence vectorization based on generalized rank invari- ant which show is stable and differentiable with respect to the bi-filtration functions. Further, present an algorithm for computing GRIL which is a synergistic confluence of the recent develop- ments in computing generalized rank invariant of a 2-parameter module and an efficient algorithm
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300 63 60 120 09
133
650
431 300
Figure 6: The figures show the rearrangement of points according to the loss function, which in our case is increasing the nomn of ê and X vectors. We start with two circles containing some noisy points inside. We observe that the points rearrange to form two circles because that increases norm of ê and ò2 vectors.
for computing zigzag persistence. We propose PERSGRIL, a differentiable topological layer, which be used as topological feature extractor in a differentiable manner. As topological feature extractor, PERSGRIL can perform better than Graph Convolutional Networks (GCNs) and Graph Isomorphism Networks (GINs) on some synthetic datasets. It performs better than the existing mul- tiparameter persistence methods on some graph benchmark datasets. Further, we give proof of concept for the differentiability of PERSGRIT. by rearranging the point cloud enhance its topolog- ical features. We believe that the additional topological information that a 2-parameter persistence module encodes, as compared to a 1-parameter persistence module, can be leveraged to learn better representations. Further directions of research include using PERSGRIL with GNNs for filtration learning to learn more powerful representations. We hope that this work motivates further research into exploring this direction.

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A BACKGROUND AND DEFINITIONS
Here, we give the detailed definitions of all the concepts explained in the paper. We begin by defining simplicial complex.
Definition A.1 (Simplicial Complex). An abstract simplicial complex is a pair (V, Σ) where V is a finite set and Σ is a collection of non-empty subsets of V such that if a E Σ and if T S then r E Σ. topological space [V, Σ)] can be associated with the simplicial complex which can be defined using bijection L: V (1,2,... ...... [V]} as the subspace of RI formed by the union U h(o). where h(o) denotes the convex hull of the set fes(s) where denotes the standard OEE
basis vector in R!V)
We shall now define a zigzag filtration and the zigzag persistence module induced by it. Definition A.2. A zigzag filtration is sequence of simplicial complexes where both insertions and deletions of simplices are allowed, the possibility of which we indicate with double arrows:
X, X.
Applying homology functor on such a filtration we get zigzag persistence module that is sequence of vector spaces connected either by forward or backward linear maps:
H. (Xo) ++ HX(X)) + ..... ++ H. (X)).
Now, we give the definition of 2-parameter filtration over IR² and the 2-parameter persistence module induced by it.
Definition A.3 (2 2-parameter simplicial filtration over R²). A 2-parameter simplicial filtration, also called bi-filtration, over I22 a collection of simplicial complexes {XuJuek> with inclusion maps (u1; u2) and (v1,
X, > Xv for u < v. that is, in I U2 and U1 S 1/2 where
Definition A.4 (2- 12-parameter Persistence Module). Given a bi- bi-filtration, {Xu) by considering the homology of the simplicial complexes in the bi-filtration over the finite field 22, we get a collec- tion of vector spaces {Mu u E R²) along with a collection of linear maps (Mn- Mn - Mv S v} Each inclusion map the bi-filtration induces a linear map between the corresponding homology vector spaces.
Having defined 2- 2-parameter filtration and 2-parameter persistence module, we now define the notion of an Interval in R2. In the definition, we shall make use of the standard partial order on R², i.e., S if u1 < and u2 I 02 for (u1, U2) and v = (2),
Definition A.5. An interval in IR2 is subset @ & I C R22 that satisfies the following:
1. If p. E and S r < q. then r E I;
2. If q I, then there exists finite sequence (p Po- Pl......
..r......... Pm. =q) E so that every consecutive points p., Pi+1 are comparable in the partial order for i E (O...., m 1}.
We now give the formal definition of generalized rank invariant over intervals in RR. However, generalized rank invariant can defined over any locally finite connected poset.
Definition A.6 (Generalized Rank (Kim & Mémoli 202I)). Given 2-parameter persistence mod- M and an intervals C R², the generalized rank of M restricted to I, rk (I), defined as
( raak|[iim/ M) + lim M
Here lim M), lim M denote the limit and colimit of the functor M when restricted to I. We refer the reader to (MacLanne 1971) for the definitions of limit and colimit in category theory.
For collection of intervals I, the collection rk2 invariant of M over I.
[rkM (I) [ I} is called generalized rank
We can define a metric on the space of persistence modules based on their generalized rank invariants over all intervals in R2
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Definition A.7 (Erosion Distance (Patel 2018 Kim & Mémoli 2021 Let Int(R²) be the collec- tion all intervals IR?. Let M and be two persistence modules. The erosion distance is defined
de N) inf (VT E Int rk (T) rk (I+s) and rk rk) rk (Tte))
Here It denotes the E-extension of the interval I.
B STABILITY AND DIFFERENTIABILITY PROOFS
In this section, we provide the proof for stability and differentiability of GRIT.. We begin by defining some metrics on the space of persistence modules based on GRII..
Definition B.1. Given two persistence modules M and N, a morphism f M + N is collection of linear maps (fu M,, >> NuJueR UERR such that fa Nu-vv Mu-vo fw: Vu v.
Definition B.2. Given a persistence module M and E R, we define the shift module M through - = Mu+e and M = Mutc-rote- Here E = (u, +,, u2 +
Definition B.3. For a pair of persistence module M and N and some e E R2o. an e-interleaving between M and N pair morphisms Φ M + N and N - M such that Vu R², Mu-put+ = Mutt Ou and Nu-utte = Qute tuu. If such interleaving exists, we say M and N are c-interleaved.
Definition B.4. For two persistence modules M and N, define the interleaving distance as (M,, N) infezo[ (M and N are e-interleaved).
Definition B.5, For persistence module M, N with GRILS AM ,N,, define
de dcMM, N)
J - 1lx
We shall now look at a property of GRIL that will help in proving the stability.
Definition B.6. Given any interval / and & 2 0, let + be the e-extension of defined as:
I+
U P PET
(2)
where P
A {q: ||p - 4l1ox S is the oc-norm e-neighbourhood of x.
Proposition B.1.
[p
+e
C
p
In order to better analyze the stability property of persistence landscape, we define a distance in similar flavour erosion distance for the underlying collection of all worms.
Notation B.7. Denote the collection of all worms as W
P 8 0, R2]
Definition B.8. For W
P
8
O.IENN+
R2
define distance aly as follows:
(M, A inf E P
W,
Mr
rk
P and N
P
2 M
P
(3)
Proposition B.2. de = d < de, where de is the erosion distance.
Proof. da I dg obvious by definition.
To show de < d.y. Given two persistence modules M, N, assume (E, c. For fixed
p. ie, l, let X (M(pF k, on and XN (p, k) = 82. Without loss of generality, assume 62 2 8j. We want to show that 82 5i I c. By the construction of dy we know that for any a > 0,
> N
(x:) 2 rk (P 5iteta
(x)). One can get d1 + + 82 = c+ > 62 51.
ta
By taking 0. we have d2 <.
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To show de I dc. Let dc = 8. For any I= I, want to show that rk
p
rk" and rk" (p.) 2 rk" M (P c40
We prove the first inequality. The second one can
be proved similar way. Let h rk
P45 then X (N k. 2 c 6. By the assumption
de (M, 6, we know that X (p, (p, k. e, which implies rk p k rk c+5
Proposition. 2.1 GRII. is equivalent the generalized rank invariant W. Here equivalence means bijective reconstruction from each other.
Proof. Constructing GRIL from generalized rank invariant on W is immediate from the definition of GRII..
On the other direction, for any p, 5, f, the generalized rank rkw as follows:
can be reconstructed by GRIL.
p
arg max k E) 2 2
(4)
not hard to check that, this construction, combined with the construction of persistence land- scape, gives bijective mapping between (generalized) rank invariants over W and GRILS.
O
By the stability property of crosion distances, we can immediately get the stability of GRIL as follows:
Proposition. 2.2 For two filtration functions f. X IR? I|AM - A 11 S 1lf fooo
Proof. Let M and M be the persistence modules derived by f and f' respectively. Then, have the following chain of inequalities:
M AM 1l de(M', M²" s de(M!, M() S dr(M!, (O(MMMMM))) S 115--'ll
where dj (M', M ) the interleaving distance. The second last inequality has been shown in (Kim & Mémoli 2021).
Recall that when X' is a finite space (e.g. finite simplicial complex, point cloud) then, any f: X R2 can be considered as 2 matrix which can be linearized into vector in R2n Let denote that vector by fyy- Proposition 2.3 PERSGRIL is Lipschitz continuous with respect to bi-filtration functions on finite spaces.
Proof. Given filtration functions f, f' and their corresponding vector representations Uy R it is easy to see that f'llo I 2/|u, uy lloe I 2/|1, uy Combining this with the chain of inequalities in the previous proposition, get that PERSGRIL is Lipschitz continuous with respect to the underlying filtration functions.
Theorem 2.5). Consider the space of all filtration functions {f : X + R²) on a finite space with | n, which is equivalent to R For fixed k. f, p, there exists measure-zero subset Z C R such that for any f R2n Z satisfying the following generic condition: Vr + E X, fx)1 * (11,0, +222 such that
f()) + (y)). there exists assignment
Af + as) (f)
V. V.Ak&())
lim a-0
max V k (f). 9EX
a||s||00
Proof. By Corollary 2.4 we know there exists some measure-zero set R C R such that PERSGRIL differentiable in R 1R2n
R. Let M M be 2-parameter persistence module induced from some generic filtration function f E and p> be an (-worm in R² centered some point p. Let a(I) be the boundary of I excluding the right most vertical edge and bottom most vertical edge shown in (Dey et al. (022) that, over the boundary @(I), a so-called zigzag persistence module can be defined by taking the restricting M to a(T) (in practice
(See Figure 7 as an illustration).
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it is enough to take zigzag path to approximate the smooth off-diagonal boundary) on which the be another ł-worm centered at p for some d' * d. One can observe that, if the zigzag filtrations on a(T) and a(T') have the same order of insertion and deletion of simplices, then the number of full bars on M aun) and M Marrr) are the same, which means rk" (I) rk" (I'). Now let
number of full bars is equal to rk M (T). Let I' = p
XM XM E, p).
P D
P
P Pdtte for some small enough Based on the definition of A we know that rk rk" k
and rk (1+) < k. which means that zigzag filtrations change on some simplices while moving from a(I-) to a(I+). Either the collection of simplices changes or the order of simplices changes. The former case corresponds to the simplices with x or g-coordinate aligned with some vertical or horizontal edges on the (T). The latter case corresponds to those pairs of simplices (a, r) such that f(o) V [(T) - (max(f(o)), i, [T)) max(f(()) f(()2) is on some off-diagonal edges on a(I). By the generic condition of the filtration function f. we can locate those simplices as the set S, which we call support simplices. The assignment function is defined on each a E S by assigning s(or) = +1 or = which is consistent with the moving direction of the edge from a() to ((4)). ). We discuss the assignment values case by case:
We can divide the boundary into four edges: bottom (off-diagonal) edge eb, top (horizontal) edge et, left (vertical) edge e,, right (off-diagonal) edge e,.
1. s(o) = +() if a has y-coordinate the same as Cs2
2. (() = (-f,0) 0) if a has z-cordinnte the same as Et.
3. s0) (0, -0). = (-1.0) f(o) v f(r) is on es and f(o)1 s f(T),. is on e, and f(a) S s(7),.
4. (()) (411 ((o
(+1,0)
See Figure as an illustration. We assume f satisfies the condition that the supporting simplices in S either all belong cases 1 and 2 or all belong to cases 3 and 4, but not combination of them. It is not hard to see that the collection of f for which this condition does not hold is measure zero set in R2n Let us denote the collection of all such f's by F. Then, Z F R is measure zero set in R²n which consists of f's which do not satisfy the condition and those points where PERSGRIL is not differentiable.
Now, check for such generic f & Z so that the directional derivative V v.A(f) is indeed a maximal directional derivative. For the cases 3 and 4, the stability property in Proposition [2. implies that, for any > 0 and any direction vector E R2n with |وو|| xo = 1, we have X(f + ag) X(f) S a. Also is not hard to check that >(ʃ + as) - x() = a for a > small enough since the zigzag persistence of M with J d+o
has the same collection of simplices and orders as M
with I = p f. " which means they have the same rank. Therefore, we have V|[g|l> = 1f X(f) S x(f + as) X(f = v,A(f) < V.A(f). For the case (the case 2 similar), the support simplex is edge e,. Now for any direction vector g E R and > 0 small enough, let Ad A(f + ag) A(f) and let Aye, be the difference between y-coordinates of e, 's from P ( Ad since in order to change A(f) by Ad and P d+ad Note that Ad = l and [M(f+ag) Ay., Aye, one has to at least move edge by Aye.. which correspondingly changes the y-coordinate of s(r) by Aye. From the above argument, we can get the directional derivative V yg.(ff) is bounded from above by the ratio Aie. = _ v()(). The case for a < 0 symmetric.
In summary, V.X(J) indeed maximizes the directional derivative for f.
C ALGORITHM
Here, we describe the algorithm in detail. In practice, we are usually presented with a piecewise linear (PL) approximation f of R²-valued function ff on discretized domain such finite simplicial complex. The Pl.aapproximation f itself is R22 Discretizing the parameter space
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s(o) = (0,2)
0(1) el
13
p
9b
( (0. -1)
07
= 2
5((0)) (-1.0)
s(0a) (-2.0)
Figure 7: Two examples of 2-worm 1, 1'. Blue and red lines boundaries of and I' respectively on which the zigzag persistence modules are constructed for computing ranks. o;, i = 1, 2, 4 are four support simplices Ö(I). s(o;) is the assignment function values on dy.
R² by a grid, we consider a lower star bi-filtration of the simplicial complex. Analogous to the 1-parameter case, lower star bi-filtration is obtained by assigning every simplex the maximum of the values over all of its vertices each of the two co-ordinates. With appropriate scaling, these (finite) values can be mapped that because of the maximization of values over all vertices, we have the property that two simplices
subset of points
uniform finite grid over [0, 1] x [0, 1]. Observe
C have values f(oa) E R² and f(r) E R² where f(a)= < f(r). A partial order of the simplices according to these values provide a bi-filtration over the grid (0, 1] [0. 1].
Choosing center points for worms. Let us denote the chosen grid Grid { m 3) m, n (0.1....,MA M}} } for some M E Z+. We denote the grid resolution as - 1/M. We sample a uniform subgrid P Grid as the collection of center points for the worms to be used to build GRIL.
Discretized 6-worms. We saw the definition of é-worm in the previous section. However, in prac- tice, since work with discrete grid rather than R², use discretized {-worms tion. The approximation gap is determined by the grid resolution A discretized Ê-worm centered p with width d is the union of 2/ squares with centers p (kd, -kd) and p (kd, -kd) 1} along with the intermediate staircases between two squares of step- size equal to grid resolution (p). Figure 2 (middle) shows the discretization of 2-worm. This construction is sensitive to the grid resolution.
approxima-
where k E {0, 1, 10.,,...,
Computing generalized ranks. We need to compute the generalized rank rk
for every
worm P to decide whether to increase its width or not. We use a result of (Dey et al. 2022) to compute MM P It says that rk M
can be computed by considering zigzag module and computing the number of full bars (bars that begin at the start of the zigzag filtration and persist until the end of the filtration) in its decomposition. This zigzag module decomposition can be obtained by restricting the bi-filtration on the boundary of M (pf) and using any of the zigzag persistence algorithms on the resulting zigzag filtration. We use the recently published efficient algorithm and its associated software (Dey & Houl 2022) for computing zigzag persistence.
Computing the value of GRIL using binary search. For worm p and given k 2 1, apply binary search to compute the value of GRIL. Let us denote the grid resolution by p. We do the binary search for d in the range [dinin dmax)] where dmai and dmax length of the grid. In
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each iteration, compute rk
p for d (dmia dmax)/2 and check if rk
P
2 k. We
increase the width of the worm by updating
d
5
10
Figure 8: (Left) The figure shows the 2- 2-wrm centered at p with width d. (Right) The highlighted part denotes the boundary cap of the worm. The arrows in the figure denote the direction arrows in the zigzag filtration.
Refer to Figure 8 for an illustration of the zigzag filtration along the boundary cap of a 2-worm.
D EXPERIMENTAL SETUP
D.1 HOURGLASS DATASET
The two traversals Ti and T2 are designed as follows: Ti traverses G₁, then followed by Gz; Tr traverses upper halves GT C G1 and G C G2 sequentially first, then followed by the other halves GT S G₁ and G2 S G2. For cross edges, we randomly pick 2 pairs of nodes (with replacement) in G1 x G2 which we place cross edges. We don't place multiple edges the same pair of nodes. a similar way we place cross edges on G x G.2 Therefore,
has roughly 6|V| cross edges between G1 and G2- The (roughly) total number of edges: [E| 5 Vi. For methods based on persistence modules, we take two filtration functions fI, f2 : UUE + on as follows: let rrv) be the node attribute on given by the order index of the trace. Then
fi is given by Ve E (v) = x(t) and Ve = (v, w) E F, fi(e) = max (#()),
fa is given by fa(u)= = 0 and fz = (ee) where ((e) is a curvature value of Here we use version of discrete Ricci called Forman-Ria curvature (Forman 2003) computed by the code provided in (Ni et al. 2019).
We compute for all points in a uniform X grid the PERSGRIL values A(p, k. E) for general- ized rank 2, worm size é = 2, and homology of dimension 0 and 1. Therefore, for each graph our PERSGRIL generates 64-ddimensional vector as representation. For the method based on 1-parameter persistence modules with persistence image vectorization, we compute 1- 1-parameter persistence modules for homology dimension 0, 1 on fi and f2 independently. Each persistence grid. Therefore, it also produces a 644-dimensional vector as
module will vectorized on a representation. For graph neural networks models, we use 3-layer GCN and 3-layer GNN with fixed hidden dimension be 16, followed by sum pooling and one fully-connected layer. We use 3-layer multilayer perceptron (MLP) with fixed hidden dimension to be 16 as classifiers for all models. We train all the models 100 epochs with cross entropy loss and Adam optimizer (Kingma Ba 2015)) with learning rate fixed to be 1=00001. We do 5-fold cross validation and report the mean accuracy and standard deviation.
D.2 GRAPH EXPERIMENTS
We performed a series of experiments on graph classification using GRIL, We used standard datasets with node features such as PROTEINS, DHIRR, Cox2 and MUTAG (Morris al. 2020). Description of the graph classification tasks is given in Table The node features of all the nodes were treated points in higher dimensional space and we computed the Density-Alpha bi-filtration the nodes. We extended the filtration on the edges by considering the maximum of the values on the corresponding nodes.
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Under review as a conference paper at ICLR 2023
Dataset PROTHINS
Num Graphs
Num Classes
Avg. No. Nodes
Avg. No. Edges
1113
2
39.06
72.82
Cox2
467
41.22
43.45
DHER
756
42.43
44.54
MUTAG
188
17.93
19.79
Table 3: Description of Graph Datasets
The Density-Alpha bi-filtration and the Heat Kernel Signature- Ricci Curvature bi-filtration, as done in (Carrière & Blumberg 2020), values are normalized so that they lie between O and 1. For the experiments reported in 4 we fix the grid resolution p 0.01. Thus, the square [o, 1] [o. 1] has 100 x 100 many grid points. We sample 128 center points, p. out of these grid points uniformly. We fix for our experiments. We compute X(p, k; E) where p varies over the sampled 128 center points and & varies from 1 to 10. Each such computation is done for dimension 0 homology (Ho) and dimension homology (H1). We fix the value learning rate as 0,001 for the experiments.
Dataset MUTAG
== 0.02
0.01
0.005
8.1
87.8
8.8
87.9 + 8.1
Table 4: 10-fold cross-validated test accuracy for different grid resolutions.
In T 5 we provide study of the performance of GRIL on different values of & on MUTAG and Cox2 datasets. We compare the 10-fold cross-validated test accuracy of GRIL on Density-Alpha bi- bi-filtration. For this study, The columns in Table represent the values of k chosen. For instance, [1 2] represents that we computed A(p. 1, and A(p. 2, E) and concatenated these vectors before passing them the 1-layer MLP classifier. It seems that for datasets with smaller graphs such as MUTAG, using ranks higher than 6 are not very useful. However, for datasets with comparatively bigger graphs, using higher ranks seems to increase the performance of the model.
1-layer MLP classifier and fix the learning rate be 0.001.
Dataset MUTAG
[1-2]
[1-4]
[1-6]
[1-8]
[1-10]
83.1 + 9.0
86.8
1 7.5
88.4 1 8.4
87.8 1 8.8
87.8 + 8.8
Cox2 78.7 + 0.0
78.3
+ 1.2
79.6 + 3.0
80.4
+ 2.2
80.6 + 2.5
Table 5: 10-fold cross-validated test accuracy of Gril D-Alpha for different values of k.
We report the computation time for computing X(p, k. E) where / = 2, k E f1, 10) and p E P where |P| 128 Table The GRIL features were calculated on Density-Alpha bi-filtration. The computations were done on a Intel(R) Xeon(R) Gold 6248R CPU machine and the computation was carried out on 32 cores. We report total computation time per dataset average time takes for computation time (in seconds) per graph for each dataset.
Dataset
Total Computation Time (s)
Computation Time per graph(s)
PROTEINS
27851
25.02
Cox2
15878
34.00
DHFR
23121
30.58
MUTAG
4358
23.18
Table 6: Computation times for GRIL features on Density-Alpa bi-filtration for graph datasets
In Table 4 we report the accuracy of GRIL on Density-Alpha bi-filtration with different grid resolu- tions on the MUTAG dataset.
21


Reviews:
[{"review_id": "S_IoapSmcdq", "reviewer": "Reviewer_Jgx9", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper proposes a vector representation of a 2-parameter persistent homology. This is a very important problem. Application of multiparameter persistence can have significance impact in developing topology-based learning methods. \n\nThe proposed method is based on the generalized rank invariance by Kim and Memoli and is inspired by the classic persistence landscape on 1-parameter persistence by Bubenik. Theoretical results (stability and differentiability, etc) are provided, although they are not particularly surprising given the known nice properties of rank invariance and 1-parameter landscapes. An algorithm is proposed to compute the proposed representation. Experimental results on a synthetic dataset is used to show that the representation can be used as topological feature for undirected graphs, and can be used as a differentiable layer for end-to-end learning. \n\n", "summary_of_the_review": "Overall, I think the problem is important and the proposed idea is sufficiently significant. I am holding my scores at this moment because the experimental evaluation is not sufficient to support the main claim.", "strength_and_weaknesses": "Generally, I like the paper. It is well written considering how challenging the topic is. Multiparameter persistence is very important and can be quite useful in practice. The idea of landscape is well aligned with data analytics purpose. I tend to believe the idea will work in practice. However, I am holding my scores below the bar due to several concerns regarding the experiments. \n\nThe experiment section, in my opinion, should be showing that the representation is useful and delivers rich information than existing representations. But this is not thoroughly conveyed in the experiment section. The experiment has two parts: landscape as a static feature and as a differentiable layer. \n\nAs a static feature,\n1, I am not sure why only the HourGlass data is used when plenty of previous papers (Hofer et al, PersLay (Carrier et al), etc) used public graph classification benchmarks. The dataset is interesting. But to show the representation is useful, previously used graph classification benchmarks should be used. \n\n2, I am not really worried about comparing to SOTA GNN methods. But the proposed representation should outperform existing persistence-based representations. This includes not only persistence image, but also 1-parameter persistence kernels and some recent 2-parameter persistence kernels/representations. As long as these comparisons can be shown on datasets like REDDIT, IMDB, etc, used in Hofer et al. and PersLay, I would be convinced that the proposed landscape is indeed a better representation. \n\nThe differentiable layer experiment is an interesting proof-of-concept with regard to a specific manually chosen loss function. In this sense, emphasizing it as a main benefit (and thus the title) is a bit of an oversell. To me, the contribution is sufficiently important as long as the representation can be proven as a rich feature in the static experiments (with benchmarks and stronger baselines).\n\nMinor comments:\n1) end of the first paragraph - \"From the perspective of direct use of 2-parameter persistence modules into ML models, to the best of our knowledge, is the first of its kind\". I am not sure i understand this, isn't previous 2-parameter persistence kernels already doing this? Or you meant you are the first to use the non-slicing version of 2-parameter persistence in ML? Please elaborate.\n\n2) it would be helpful to make it more explicit the novel contribution of this paper compared to Kim and Memoli. \n\n3) Definition 2.1 could have been better illustrated with more examples considering this is one of the key contributions. The examples in Fig 2 and its caption is quite short.\n\n4) in complexity analysis, shouldn't $t$ be $n^2$ in the worst case? What is $\\omega$? Is it the exponent of the matrix multiplication complexity? If so, please also discuss the complexity of a practical implementation of the method (if I understand correctly, matrix multiplication complexity for persistence is only theoretical).\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "RrU8-UM-urF", "reviewer": "Reviewer_SJck", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper presents a new neural network layer for leveraging 2-parameter persistent homology, i.e. a topology-driven method that is capable of incorporating information from two independent filtration functions (serving as different \"perspectives\" under which to view data set, such as \"density\" versus \"distance\") of a point cloud data set. This is achieved by employing a novel vectorisation of topological features, based on the rank of 2-parameter persistence modules, a way to represent topological features algebraically. The proposed method can be used in a supervised setting, which is demonstrated by means of classifying graphs, as well as an unsupervised setting (serving as a loss term for adjusting the shape of point clouds according to their topological features.\n", "summary_of_the_review": "While I appreciate the novel and creative direction taken by the paper, I cannot endorse it for publication in its present form. The changes that are required to improve it cannot be done within a conference cycle and necessitate a major revision. I understand that this is not the desired outcome for the authors, but I want to stress that I believe that this paper has the potential make a substantial contribution to the field, provided the two issues mentioned above are addressed.\n", "strength_and_weaknesses": "The main strength of the paper lies in its **novel, original vectorisation strategy**. Bi-filtrations being a natural way of describing many data sets, the new method addresses a clear gap in the literature, and it has the potential to become a strong contribution to the topological data analysis literature. That being said, the paper in its current form suffers from two major weaknesses:\n\n1. *Lack of clarity*: the paper is currently not accessible for a non-expert reader. Given the high complexity of the topic, a more intuitive description of the required methods, as well as some more \"hand-holding\" of readers, would be necessary. In addition, the write-up is at times quite cursory, and will require a substantial revision.\n\n2. *Lack of experiments*: the experiments shown in the paper do not provide a sufficient depth to appreciate the contributions; the graph classification example, while interesting, would require more explanations concerning the task at hand, while the loss term example would require a comparison to other techniques.\n\nI will subsequently comment more on these two aspects.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "QgGwhffIHj", "reviewer": "Reviewer_nKKg", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "This article introduces a vectorization---called PersGril---of $2$-parameter-based persistence, akin to the well-known persistence landscapes routinely used for 1-persistent homology. They prove that their vectorization is stable (1-Lipschitz for the infinite norm) with respect to the inputs functions $\\mathcal{X} \\to \\mathbb{R}^2$, hence its differentiability (a.e.). \n\nThey derive a practical algorithm that rely on zigzag persistence (a sort of intermediate between 1 and 2 persistence, loosely) for which existing software already exists. \n\nThe showcase their approach on few numerical experiments ", "summary_of_the_review": "While I would like to see the development of more (computational) tools dedicated to multi-persistence---and in that regard I am quite positive about the proposed approach---I think that this work has some caveats in terms of (i) clarity of the presentation, (ii) experimental descriptions, that prevent me to support its publication for now. ", "strength_and_weaknesses": "## Strengths\n\n- Multi-parameter persistence is an extremely challenging topic (from both a computational and theoretical side), and there are very few practical tools developed to handle it (and most of them essentially rely on computing 1d-persistence). To that respect, any improvement in that direction---as the one proposed in this work---is worth of interest. \n\n## Weaknesses\n\n- The experimental section is pretty limited: the practical performance of PersGril is only showcased on a toy dataset, and while the results are reasonably convincing at showing that PersGril is capable of doing great on a dedicated dataset, I would not say that these experiments truly showcase the use of 2-parameter persistence in machine learning tasks. It is quite below, for instance, the experiments conducted in the work of Carrière and Blumberg, NeurIPS 2020, that also leverage 2-parameter persistence---and the current work does not compare to it (if the comparison is irrelevant, please discuss it). In addition, \n   - The presentation of the experiments lacks details. For instance, PersGril outputs some vector (one real number of each $p,k,\\ell$). How is the actual classification performed afterward? [note: ok I found the answer in Section C.0.1. It's a 3-layer perceptron. This should be explained in the main body directly.]\n   - There is no explicit running time reported. While Time Complexity is discussed, practical running time is also important (if not more) to give an idea of how practical the proposed method is. If PersGril is computationally expensive to run, this should be discussed. \n   - The influence of parameters (noting that there values are only discussed in Section C.0.1) is not discussed. For instance, we learn in C.0.1 that $p$ ranges on a $4 \\times 4$ grid. What's the influence of taking a $3 \\times 3$ grid, and a $5 \\times 5$ one in terms of both computational time and model accuracy? (Note : since the model achieves perfect accuracy on this dataset, this may not be very enlightening. Showcasing it on a harder dataset would be more interesting). \n\n- Despite some effort that have been made in the presentation (e.g. in the introduction), the paper is quite hard to understand in details. In particular the description of the algorithm, in Section 3, is hard to parse, especially for the reader that is not familiar with zigzag persistence (which, arguably, is likely to be a general case). I would think that a step-by-step example (at least in the appendix, if space constraints do not allow for it) would allow the reader to understand \"what is going on\". Similarly, \n   - The \"density-Rips filtration\" in Figure 4 has not been defined as far as I can tell. Rips filtration should be quickly defined at least in the appendix (saying it's the 1d filtration induced by the map $f : x \\mapsto \\mathrm{dist}(x, P)$ where $P$ denotes the point cloud should be enough). Also, being unfamiliar with 2-parameter persistence, I struggle to understand the second plot. Why do points organize along discrete vertical lines? It seems that some of the parameter (density I guess?) has only been taken at discrete steps (0.1), is that correct ? \n   - The distance $d_I$ used in the proof of Proposition 2.1 in the appendix has not been defined as far as I can tell. I guess it is some sort of interleaving distance, but I'm not sure. \n   - What is the exponent $\\omega$ in the Time Complexity paragraph? (I may have miss its definition, but a quick ctrl-F suggest that this symbol is only used there.)\n   - I don't think that section C.0.1 should belong to the appendix. At least some of it should belong to the description of the dataset. The only thing that the main body says about the two classes is that \"they essentially have the same structure\", which is not very instructive. What are we trying to classify here? Reading the appendix should not be necessary to understand such central points of the work. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Og61McjoC-u", "reviewer": "Reviewer_HJRc", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "The paper proposes an approach of vectorization of 2-parameter persistence modules, based on the generalized rank invariant, computed over so-called worm-shaped 2-intervals. The authors show that this construction is stable w.r.t. the interleaving distance, and that this construction can be differentiated w.r.t. the bi-filtration used to construct the 2-persistence module, when the latter is 1-critical. Finally they provide some experiments on synthetic data sets.", "summary_of_the_review": "Overall I think that this work is encouraging, but the lack of practical experiments makes the performance improvements uncertain.\n", "strength_and_weaknesses": "Strengths:\n---This vectorization uses more information from the module than most of the current ones, which only use the fibered barcodes.\n---This pipeline is stable and differentiable with respect to the input. \n---This generalizes the multiparameter persistence landscapes (MPL), when l=1.\n---This construction relies on the generalized rank invariant, which can be efficiently computed for 2-persistence modules. Furthermore, each component of this vectorization can be computed independently, which makes the computation easily parallelizable. In particular, it may be scalable to larger datasets (but it would require more experiments).\n\nWeaknesses:\nThe major weaknesses come from the numerical experiments. \n---The authors only provide synthetic data sets, with small numbers of simplices, which makes it difficult to assess how scalable and efficient the approach is in practice\n---As far as I understand, when l=1, the proposed approach is equal to the MPL, however there is no score comparison (running time / accuracy tradeoff wrt to the parameter l)\n---Although the classification performances seem to be positive, the authors only compare to the performance of the persistence image, which 1) is not always the best persistence diagram vectorization (it would be good to also try landscapes, silhouette and kernels such as the sliced Wasserstein kernel), and 2) is not a 2-parameter persistence vectorization (it would be nice to compare also to the multiparameter persistence image of Carriere and Blumberg and the multiparameter persistence landscape of Vipond).\n---Why are the accuracy provided with no variances? ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject