EMP: E FFECTIVE M ULTIDIMENSIONAL P ERSISTENCE FOR G RAPH R EPRESENTATION L EARNING
Anonymous authors
Paper under double-blind review
A BSTRACT
Topological data analysis (TDA) has become increasingly popular in a broad range of machine learning tasks, ranging from anomaly detection and manifold learning to graph classiﬁcation.
Persistent homology being the key approach in TDA provides a unique topological ﬁngerprint of the data by assessing the evolution of various hidden patterns in the data as we vary a scale parameter.
Current PH tools are limited to analyze the data by ﬁltering with single parameter while in many applications, several relevant parameters are equally important to get a much ﬁner information on the data.
In this paper, we overcome this problem by introducing Effective Multidimensional Persistence (EMP) framework which enables to investigate the data by varying multiple scale parameters simultaneously.
EMP framework provides a highly expressive summary of the data by integrating the multiple descriptor functions to the process successfully.
EMP naturally adapts to many known single PH summaries and converts them into multidimensional summaries, for example, EMP Landscapes, EMP Silhouettes, EMP Images, and
EMP Surfaces.
These summaries deliver a multidimensional ﬁngerprint of the data as matrices and arrays which are suitable for various machine learning models.
We apply EMP framework in graph classiﬁcation tasks and observe that EMP boosts the performances of various single PH descriptors, and outperforms the most state-of-the-art methods on benchmark datasets.
We further derive theoretical guarantees of the proposed EMP summary and prove the stability properties.
I NTRODUCTION
In the past decade, topological data analysis (TDA) proved to be a powerful machinery to discover many hidden patterns in various forms of data which are otherwise inaccessible with more traditional methods.
In particular, for graph machine learning tasks, while many traditional methods fail, TDA and, speciﬁcally, tools of persistent homology (PH), have demonstrated a high potential to detect local and global patterns and to produce a unique topological ﬁngerprint to be used in various machine learning tasks.
This makes PH particularly attractive for capturing various characteristics of the complex data which may play the key role behind the learning task performance.
In turn, multiparameter persistence, or multipersistence (MP) is a novel idea to further advance the
PH machinery by analysing the data in much ﬁner way simultaneously along multiple dimensions.
However, because of the technical problems related to commutative algebra because of its multidimensional structure, it has not been deﬁned for general settings yet (See Section 2.1).
In this paper, we develop an alternative approach to utilize multipersistence idea very efﬁciently for various types of data, with main focus on graph representation.
In particular, we bypass technical issues with the
MP by obtaining a very practical summaries by utilizing slicing idea in a structured way.
In turn, we obtain suitable multidimensional topological ﬁngerprints of the data as matrices and arrays where
ML models can easily detect the hidden patterns developed in the complex data.
Our contributions can be summarized as follows:
• We develop a new computationally efﬁcient and highly expressive EMP framework which provides multidimensional topological ﬁngerprints of the data.
EMP expands many popular summaries of single persistence to multidimensions by adapting an effective slicing
direction.
As such, our EMP framework provides a practical way to utilize the promising multipersistence approach in real-life applications.
• We illustrate the utility of EMP summaries in various settings and compare our results to state-of-the-art methods.
Our numerical experiments demonstrate that EMP summaries outperforms SOTA in several benchmark datasets for graph classiﬁcation tasks.
• We derive theoretical stability guarantees of the new topological summaries.
R ELATED W ORK 2. M ULTIPARAMETER P ERSISTENCE
Persistent homology (PH), being a key tool in topological data analysis (TDA), delivers invaluable and complementary information on the intrinsic properties of data that are inaccessible with conventional methods (Chazal & Michel, 2021; Hensel et al., 2021).
In the past decade, PH has become quite popular in various ML tasks, ranging from manifold learning to medical image analysis, material science to ﬁnance (TDA applications library (Giunti, 2022)).
One of the key beneﬁts of PH is that it allows us to extract the evolution of subtler patterns in the data shape dynamics at multiple resolution scales which are not accessible with more conventional, non-topological methods (Wasserman, 2018).
Multipersistence (MP) is a highly promising approach to signiﬁcantly improve the success of single parameter persistence (SP) in applied topological data analysis, but there are some issues to convert this novel idea into an effective feature extraction method as desired (See Appendix D.3).
Except for some special cases, MP theory suffers from the problem of the nonexistence of barcode decomposition because of the partially ordered structure of the index set { ( α i , β j ) } (Botnan & Lesnick, 2022).
Lesnick & Wright (2015) suggested to bypass this issue via slicing technique by studying onedimensional ﬁbers of the multiparameter domain where one restricts the multidimensional persistence module to a single direction (slice) and to use single persistence on this one dimensional slice.
Later, by using this novel idea, Carrière & Blumberg (2020) combined several slicing directions (vineyards) and obtained a vectorization by summarizing the persistence diagrams (PDs) in these directions.
There are several promising recent studies in this direction (Botnan et al., 2021; Vipond, 2020), but these approaches are not computationally feasible, and cannot provide the expected effectiveness of MP approach in real life applications.
Here we develop a highly efﬁcient way to use MP approach for various forms of data, and provide a multidimensional topological vectorization with EMP summaries.
2. G RAPH R EPRESENTATION L EARNING
After the success of convolutional neural networks (CNN) on image-based tasks, graph neural networks (GNNs) have emerged as a powerful tool for graph-level classiﬁcation and representation learning.
A wide variety of models are developed based on numerous theories (see Appendix A for further details).
However, to our best knowledge, most of existing approaches do not account for the important topological information on the shapes of the node neighborhoods.
While GNNs produce great performances in many graph learning tasks, they tend to suffer from over-smoothing problems and are vulnerable to graph perturbations.
To address these challenges, TDA provides a computationally efﬁcient alternative to GNNs, and can be used as an effective feature extractor to be combined with the deep learning methods Hofer et al. ; Bodnar et al.
; Horn et al. Recently, this direction has been explored in many works, and single persistent homology give highly competitive results with kernel-based methods Togninalli et al.
; Rieck et al.
Le & Yamada (2018); Zhao & Wang (2019); Kyriakis et al. (2021) and neural networks Hofer et al.
; Carrière et al.
(2020) in graph classiﬁcation tasks.
In this work, we apply MP approach for the ﬁrst time in this setting, and our EMP model outperforms most deep learning models in benchmark datasets.
B ACKGROUND
We start from providing the basic background for our framework.
Since we mainly focus on graph representation learning in this paper, we explain our construction on graph setting.
Note that our techniques can easily be adapted to other types of data, e.g., point clouds and images (See
Appendix D.1).
3. P ERSISTENT H OMOLOGY
Persistent Homology (PH) is a mathematical machinery to capture the hidden shape patterns in the data by using algebraic topology tools.
PH extracts this information by keeping track of evolution of the topological features (components, loops, cavities) created in the data while looking at it in different resolutions Dey & Wang (2022).
Appendix B provides additional background on classic PH for the avid reader.
For a given graph G , consider a nested sequence of subgraphs G 1 ⊆ . . .
⊆
G N = G .
For each
G i , deﬁne an abstract simplicial complex (cid:98) G i , 1 ≤ i ≤ N , yielding a ﬁltration of complexes (cid:98) G 1 ⊆ . . .
⊆ (cid:98) G N .
Here, clique complexes are among the most common ones, i.e., a clique complex (cid:98) G is obtained by assigning (ﬁlling with) a k -simplex to each complete ( k + 1) -complete subgraph in G , e.g., a 3 -clique, a complete 3 -subgraph, in G will be ﬁlled with a 2 -simplex (triangle).
Then, in this sequence of simplicial complexes, one can systematically keep track of the evolution of topological patterns mentioned above.
A k -dimensional topological feature (or k -hole) may represent connected components ( 0 -hole), loops ( 1 -hole) and cavities ( 2 -hole).
For each k -hole σ , PH records its ﬁrst appeareance in the ﬁltration sequence, say (cid:98) G b σ , and ﬁrst disappearence in later complexes, (cid:98) G d
σ
with a unique pair ( b σ , d σ ) , where 1 ≤ b σ < d σ ≤ N .
We call b σ the birth time of σ and d σ the death time of σ .
We call d σ − b σ the life span of σ .
PH records all these birth and death times of the topological features in persistence diagrams (PD).
Let 0 ≤ k ≤ D where D is the highest dimension in the simplicial complex (cid:98) G N .
Then the k th persistence diagram PD k ( G )
= { ( b σ , d σ ) | σ ∈ H k
( (cid:98) G i ) for b σ ≤ i < d σ } .
Here, H k ( (cid:98) G i ) represents the k th homology group of (cid:98) G i which keeps the information of the k -holes in the simplicial complex (cid:98) G i .
For sake of notation, we skip the dimension (subscript k ).
With the intuition that the topological features with long life span (persistent features) describe the hidden shape patterns in the data, these PDs provide a unique topological ﬁngerprint of G .
As one can easily notice, the most important step in the PH machinery is the construction of the nested sequence of subgraphs G 1 ⊆ . . .
⊆
G N = G .
For a given unweighted graph G =
( V , E ) with
V = { v 1 , . .
. , v N } the set of nodes and E ⊂ {{ v i , v j } ∈ V × V , i (cid:54) =
j } the set of edges, the most common technique is to use a ﬁltration function f :
V → R with a choice of thresholds
I = { α i } where α 1 = min v ∈V f ( v ) <
α 2 < . . .
< α N = max v ∈V f ( v ) .
For α i ∈ I , let
V i = { v r ∈ V | f ( v r ) ≤ α i } .
Let G i be the induced subgraph of G by V
i , i.e., G i =
( V i , E i ) where
E i = { e rs ∈ E | v r , v s ∈ V i } .
This process yields a nested sequence of subgraphs G 1 ⊂ G 2 ⊂ . . .
⊂
G N = G , called the sublevel ﬁltration induced by the ﬁltration function f .
We denote PDs obtained via sublevel ﬁltration for a ﬁltration function f as PD ( G , f ) .
The choice of f is crucial here, and in most cases, f is either an important function from the domain of the data, e.g., atomic number in protein or a function deﬁned from intrinsic properties of the graph, e.g., degree and betweenness.
Similarly, for a weighted graph, one can use sublevel ﬁltration on the weights of the edges and obtain a suitable ﬁltration reﬂecting the domain information stored in the edge weights.
(For further details on different ﬁltrations see Aktas et al.
; Hofer et al.
3. S INGLE P ERSISTENCE V ECTORIZATIONS
While PH extracts hidden shape patterns from data as persistence diagrams (PD), PDs being collection of points in R 2 by itself are not very practical for statistical and machine learning purposes.
Instead, the common techniques are by faithfully representing PDs as kernels Kriege et al.
or
vectorizations Hensel et al.
This provides a practical way to use the outputs of PH in real life applications.
Single Persistence Vectorizations transform obtained PH information (PDs) into a function or a feature vector form which are much more suitable for machine learning tools than PDs.
Common single persistence (SP) vectorization methods are Persistence Images Adams et al.
Persistence Landscapes Bubenik (2015), Silhouettes Chazal et al.
(2014), and various Persistence
Curves Chung & Lawson (2019).
These vectorizations deﬁnes a single variable or multivariable functions out of PDs, which can be used as ﬁxed size 1 D or 2 D vectors in applications, i.e 1 × n vectors or m × n vectors.
For example, a Betti curve for a PD with n thresholds can also be expressed as 1 × n size vectors.
Similarly, Persistence Images is an example of 2 D vectors with the chosen resolution (grid) size.
See the examples given in Section 4.2 for further details.
3. M ULTIDIMENSIONAL P ERSISTENCE
MultiPersistence (MP) is a novel idea to signiﬁcantly boost the performance of the single parameter persistence technique described above.
The reason for the term “single” is that we are ﬁltering the data in only one direction G 1 ⊂ · · · ⊂ G N = G .
The construction of the ﬁltration is the key to get ﬁne analysis of the data to ﬁnd the hidden patterns.
If one uses one function f :
V
→ R which has very valuable domain information (e.g., amount for blockchain networks, atomic number for protein networks), then this induces a single parameter ﬁltration as above.
However, various data have more than one very natural domain functions to analyze the data, and using them simultaneously would give a much better understanding of the hidden patterns.
With this intuition, multiparameter persistence (MP) theory is suggested as natural generalization of single persistence (SP).
In simple terms, if one uses only one ﬁltration function, sublevel sets induces a single parameter ﬁltration (cid:98) G 1 ⊂ · · · ⊂ (cid:98) G N = (cid:98) G .
Instead, if one uses two or more functions, then it would give a way to study the data in much ﬁner resolution.
For example, if we have two functions f :
V → R and g :
V → R with very valuable complementary information of the network, MP idea is presumed to produce a unique topological ﬁngerprint combining the information from both functions.
These pair of functions f, g induces a multivariate ﬁltration function F : V → R 2 with F ( v ) =
( f ( v ) , g ( v )) .
Again, one can deﬁne a set of nondecreasing thresholds { α i } m 1 and { β j } n 1 for f and g respectively.
Then, V
[Equation] = [Equation]
} , i.e., V ij = F − 1 (( −∞ , α i ] × ( −∞ , β j ]) .
Then, let G ij be the induced subgraph of G by V ij , i.e., the smallest subgraph of G generated by V ij .
Then, instead of a single ﬁltration of complexes, we get a biﬁltration of complexes { (cid:98) G ij | 1 ≤ i ≤ m, 1 ≤ j ≤ n } .
One can imagine { (cid:98) G ij } as a rectangular grid of size m × n such that for each 1 ≤ i 0 ≤ m , { (cid:98) G i 0 j } nj =1 gives a nested (horizontal) sequence of simplicial complexes.
Similarly, for each 1 ≤ j 0 ≤ n , { (cid:98) G ij 0 } mi =1 gives a nested (vertical) sequence of simplicial complexes (See Figure 2 in the Appendix).
By computing the homology groups of these complexes, { H k ( (cid:98) G ij ) } , we obtain the induced bigraded persistence module (a rectangular grid of size m × n ).
Again, the idea is to keep track of the k -dimensional topological features via the homology groups { H k ( (cid:98) G ij ) } in this grid.
As explained in Appendix D.3, because of the technical issues related to commutative algebra, there are still issues to deﬁne a mathematical output like "Multipersistence Diagram", and hence, to obtain an effective vectorization of the MP module (Section 3). These technical issues prevent this promising approach to reach its full potential in real life applications.
In this paper, we overcome this problem by producing very practical vectorizations by utilizing slicing idea in the multipersistence grid in a structured way.
The key point in our approach is that we produce multidimensional vectors (matrices and arrays) as unique topological multidimensional ﬁngerprints of the data.
In other words, by using MP approach, we extract multidimensional features of the data as same size matrices and arrays which are quite suitable for various machine learning models, and we can facilitate the appropriate ML method to analyze these unique ﬁngerprints and detect the patterns developing in these multi-resolution view of the data.
E FFECTIVE M ULTIDIMENSIONAL P ERSISTENCE S UMMARIES
We now introduce our Effective MultiPersistence (EMP) framework which describes a way to expand most single persistence vectorizations (Section 3.2) as multidimensional vectorizations by utilizing
MP approach.
In particular, by using the existing single parameter persistence vectorizations, we produce multidimensional vectorization by effectively using the one of the ﬁltering direction as “slicing direction” in the multipersistence module.
4. EMP F RAMEWORK
In simple terms, for d = 2 , for a given two ﬁltration functions f, g , the main idea is to use ﬁrst ﬁltration function f to get a single parameter ﬁltering of the data, i.e., G 1 ⊆ . . .
⊆
G m = G .
Then, use the second function in each subgraph to obtain persistence diagram PD ( G i , g ) for 1 ≤ i ≤ m .
In particular, we obtain m persistence diagrams { PD ( G i , g ) } mi =1 .
Then, by applying the chosen SP vectorization ϕ to each PD, we obtain m different same length vector (cid:126)ϕ ( PD (
G i , g )) , say 1 × k vector.
Then, by combining all m 1 D -vectors, we obtain EMP vectorization M ϕ such that M iϕ = (cid:126)ϕ ( PD ( G i , g )) where M iϕ represents the i th row of M ϕ which is a 2 D -vector (matrix) of size m × k (See Figure 3 in Appendix).
For d > 2 , the construction is similar and given below.
Here, we give the details for sublevel ﬁltration for two ﬁltration functions.
In Appendix D.2, we explain how to modify the construction for weight ﬁltrations or power ﬁltrations.
Let G =
( V , E ) be a graph, and let f, g :
V → R be two ﬁltration functions with threshold sets { α i } mi =1 and { β j } nj =1 respectively.
Let
V i = { v r ∈ V | f ( v r ) ≤ α i } .
Let G i be the induced subgraph of G by V i .
This gives a ﬁltering of the graph (nested subgraphs) as
G 1 ⊆ . . .
⊆
G m = G .
Recall that g : V → R is another ﬁltration function for
G .
Now, we ﬁx 1 ≤ i 0 ≤ m , and consider G i 0 .
By restricting g on G i 0 , we get persistence diagram PD (
G i 0 , g ) as follows.
Let V i
j = { v r ∈ V
i 0 | f ( v r ) ≤ β j } , and let G
i
j be the induced subgraph of
G i 0 by V i 0 j .
This deﬁnes a ﬁner ﬁltering of the graph G i 0 as G i 0 1 ⊆
G i 0 2 . . .
⊆
G i 0 n = G i 0 .
Corresponding clique complexes deﬁnes a ﬁltration (cid:98)
[Equation]
This ﬁltration gives the persistence diagram PD (
G i 0 , g ) .
Hence, for each 1 ≤ i ≤ m , we obtain a persistence diagram PD (
G i , g ) .
The next step is to use the vectorization on these m persistence diagrams.
Let ϕ be a single persistence vectorization, e.g., Persistence Landscape, Silhouette, Entropy, Betti, Persistence Image.
By applying the chosen SP vectorization ϕ to each PD, we obtain a function ϕ
i = ϕ ( PD ( G i , g )) where in most cases it is a single variable function on the threshold domain [ β 1 , β n ] , i.e., ϕ i : [ β 1 , β n ] → R .
For the multivariable case (e.g., Persistence Image), we give explicit description in the examples section below.
Most such vectorizations being induced from a discrete set of points PD ( G ) , they naturally can be expressed as a 1 D vector of length k .
In the examples below, we explain this conversion in detail.
Then, let (cid:126)ϕ i be the corresponding 1 × k vector for the function ϕ
i .
Now, we are ready to deﬁne our EMP summary M ϕ which is a 2 D -vector (a matrix)
[Equation]
where M iϕ is the i th -row of M ϕ .
Hence, M ϕ is a 2 D -vector of size m × k .
Each row M iϕ is the vectorization of the persistence diagram PD ( G i , g ) via the SP vectorization method ϕ .
We use the ﬁrst ﬁltration function f to get a ﬁner look at the graph as G 1 ⊆ . . .
⊆
G m = G .
Then, we use the second ﬁltration function g to obtain the persistence diagrams PD (
G i , g ) of these ﬁner pieces.
In a way, we look at G with a two dimensional resolution {G ij } and we keep track of the evolution of topological features in the induced biﬁltration { (cid:98) G ij } .
The main advantage of this technique is that the outputs are ﬁxed size matrices (or arrays) for each dataset which is highly suitable for various machine learning models.
Order of the Filtration Functions.
We need to note that for given two ﬁltration functions f, g , the order is quite important for our algorithm.
In particular, let M ϕ ( f, g ) represent the above construction where we ﬁrst apply f to get ﬁltering {G i } mi =1 , and then we obtain m different PD ( G i , g ) .
Hence, M ϕ ( f, g ) would be a m × k matrix.
On the other hand, if we apply g ﬁrst, we would get a ﬁltering {G j } nj =1 .
Then, by using sublevel ﬁltration with g for each G j , we would get n persistence diagrams
PD ( G j , f ) .
Assuming the produced vectors (cid:126)ϕ j for [ α 1 , α m ] domain has length k (cid:48) , then M ϕ ( g, f ) would give us n × k (cid:48) size matrices.
In particular, in the ﬁrst one M ϕ ( f, g ) we use “horizontal” slicing in the bipersistence module, while in the latter M ϕ ( g, f ) , we use “vertical” slicing.
For the question “Which function should be used ﬁrst” , the answer is that the function with more important domain information should be used as second function ( g in the original construction), as we get much ﬁner information via persistence diagram PD ( G i , g ) for the second function.
Higher Dimensional Multipersistence.
Similarly, for d = 3 , let f, g, h be ﬁltration functions and let {G ij } be the biﬁltering of the data, e.g., sublevel ﬁltration for two functions f, g .
Then, again by using the third function h , we ﬁnd m · n persistence diagrams { PD ( G ij , h ) } m,ni,j =1 .
Similarly, for a given SP vectorization ϕ , one obtains a 1 D -vector (cid:126)ϕ ( PD ( G ij , g )) of size 1 × k for each i, j .
This produces a 3 D -vector (array)
M ϕ of size m × n × k where M ijϕ = (cid:126)ϕ ( PD ( G ij , g )) .
For d > 3 , one could follow a similar route.
4. E XAMPLES OF EMP S UMMARIES
Here, we discuss explicit constructions of EMP summaries for most common SP vectorizations.
As noted above, the framework is very general and it can be applied to most SP vectorization methods.
In all the examples below, we use the following setup: Let G =
( V , E ) be a graph, and let f, g : V → R be two ﬁltration functions with threshold sets { α i } mi =1 and { β j } nj =1 respectively.
As explained above, we ﬁrst apply sublevel ﬁltering with f to get a sequence of nested subgraphs,
[Equation]
Then, for each
G i , we apply sublevel ﬁltration with g to get persistence diagram PD ( G i , g ) .
Therefore, we will have m PDs.
In the examples below, for a given SP vectorization ϕ , we explain how to obtain a vector (cid:126)ϕ ( PD ( G i , g )) , and deﬁne the corresponding EMP M ϕ .
Note that we skip the homology dimension (subscript k for PD k ( G ) ) in the discussion.
In particular, for each dimension k = 0 , 1 , . . .
, we will have one EMP M ϕ ( G ) (a matrix or array) corresponding to { (cid:126)ϕ ( PD k ( G i , g )) } .
The most common dimensions are k = 0 and k = 1 in applications.
EMP Landscapes.
Persistence Landscapes λ are one of the most common SP vectorizations introduced by Bubenik (2015).
For a given persistence diagram PD ( G ) = { ( b i , d i ) } , λ produces a function λ (
G ) by using generating functions Λ i for each ( b i , d i ) ∈ PD ( G ) , i.e., Λ i : [ b i , d i ] → R is a piecewise linear function obtained by two line segments starting from ( b i , 0) and ( d i , 0) connecting to the same point ( b i + d i 2 , b i − d i 2 ) .
Then, the Persistence Landscape function λ ( G ) :
[ (cid:15) 1 , (cid:15) q ] → R for t ∈ [ (cid:15) 1 , (cid:15) q ] is deﬁned as
[Equation]
where { (cid:15) k } q 1 are thresholds for the ﬁltration used.
Considering the piecewise linear structure of the function, λ ( G ) is completely determined by its values at 2 q − 1 points, i.e., b i ±
d i∈ { (cid:15) 1 , (cid:15) 1 . 5 , (cid:15) 2 , (cid:15) 2 . 5 , . .
, (cid:15) q } where (cid:15) k.
= ( (cid:15) k
+ (cid:15) k +1 ) / 2 .
Hence, a vector of size× (2 q − 1) whose entries the values of this function would sufﬁce to capture all the information needed, i.e. (cid:126)λ =
[ λ ( (cid:15) 1 ) λ ( (cid:15) 1 . 5 ) λ
( (cid:15) 2 ) λ ( (cid:15) 2 . 5 ) λ
( (cid:15) 3 ) . . .
λ ( (cid:15) q )]
Considering we have threshold set { β j } nj =1 for the second ﬁltration function g , (cid:126)λ i = (cid:126)λ ( PD ( G i , g )) will be a vector of size 1 × 2 n − 1 .
Then, as M iλ =
(cid:126)λ i for each 1 ≤ i ≤ m , EMP Landscape M λ ( G ) would be a 2 D -vector (matrix) of size m × (2 n − 1) .
EMP Silhouettes.
Silhouette Chazal et al.
(2014) is another very popular SP vectorizations method in machine learning applications.
The idea is similar to Persistence Landscapes, but this vectorization uses the life span of the topological features more effectively.
For PD ( G ) = { ( b i , d i ) } Ni =1 , let Λ i be the generating function for ( b i , d i ) as deﬁned in Landscapes above.
Then, Silhouette function ψ is deﬁned as
[Equation]
where the weight w i is mostly chosen as the life span d i − b i , and { (cid:15) k } qk =1 represents the thresholds for the ﬁltration used.
Again such a Silhouette function ψ ( G ) produces a 1 D -vector (cid:126) ψ ( G ) of size 1 × (2 q − 1) as in Persistence Landscapes case.
Similar to the EMP Landscapes, with the threshold set
{ β j } nj =1 for the second ﬁltration function g , (cid:126)ψ i = (cid:126)ψ ( PD ( G i , g ) will be a vector of size 1 × 2 n − 1 .
Then, as M iψ = (cid:126)ψ i for each 1 ≤ i ≤ m , EMP Landscape M ψ ( G ) would be again a 2 D -vector (matrix) of size m × (2 n − 1) (Figure 1).
EMP Persistence Images.
Next SP vectorization in our list is Persistence Images Adams et al.
Different than the most SP vectorizations, Persistence Images produces 2 D -vectors.
The idea is to capture the location of the points in the persistence diagrams with a multivariable function by using the 2 D Gaussian functions centered at these points.
For PD ( G ) = { ( b i , d i ) } , let φ i represent a 2 D -Gaussian centered at the point ( b i , d i )
∈
R 2 .
Then, one deﬁnes a multivariable function,
Persistence Surface , (cid:101) µ = (cid:80) i
w i φ i where w i is the weight, mostly a function of the life span d
i − b i .
To represent this multivariable function as a 2 D -vector, one deﬁnes a k × l grid (resolution size) on the domain of (cid:101) µ , i.e., threshold domain of PD ( G ) .
Then, one obtains the Persistence Image , a 2 D -vector (matrix) (cid:126)µ = [ µ rs ] of size k × l where µ rs = (cid:82) ∆ rs (cid:101) µ ( x, y ) dxdy and ∆ rs is the corresponding pixel (rectangle) in the k × l grid.
This time, the resolution size k × l is independent of the number of thresholds used in the ﬁltering, the choice of k and l is completely up to the user.
Recall that by applying the ﬁrst function f , we have the nested subgraphs {G i } m i =1 .
For each G i , the persistence diagram PD ( G i , g ) obtained by sublevel ﬁltration with g induces a 2 D vector (cid:126)µ i = (cid:126)µ ( PD (
G i , g )) of size k × l .
Then, deﬁne EMP Persistence Image as M iµ = (cid:126)µ i , where M iµ is the i th -ﬂoor of the array M µ .
Hence, M µ (
G ) would be a 3 D -vector (array) of size m × k × l where m is the number of thresholds for the ﬁrst function f and k × l is the chosen resolution size for the Persistence Image (cid:126)µ .
EMP Surfaces and EMP Betti Summaries.
Next, we give an important family of SP vectorizations, Persistence Curves Chung & Lawson (2019).
This is an umbrella term for several different
SP vectorizations, i.e., Betti Curves, Life Entropy, Landscapes, et al.
Our EMP framework naturally adapts to all Persistence Curves to produce multidimensional vectorizations.
As Persistence Curves produce a single variable function in general, they all can be represented as 1 D -vectors by choosing a suitable mesh size depending on the number of thresholds used.
Here, we describe one of the most common Persistence Curves in detail, i.e., Betti Curves.
It is straightforward to generalize the construction to other Persistence Curves.
Betti curves are one of the simplest SP vectorization as it gives the count of topological feature at a given threshold interval.
In particular, β k (∆) is the total count of k -dimensional topological feature in the simplicial complex ∆ , i.e., β k (∆)
= rank ( H k (∆))
(See Figure 2 in the Appendix).
In particular, β k ( G ) :
[ (cid:15) 1 , (cid:15) q +1 ]
→ R is a step function deﬁned as
[Equation]
for t ∈ [ (cid:15) i , (cid:15) i +1 ) , where { (cid:15) i } q 1 represents the thresholds for the ﬁltration used.
Considering this is a step function where the function is constant for each interval [ (cid:15) i , (cid:15) i +1 ) , it can be perfectly represented by a vector of size 1 × q as (cid:126)β ( G ) =
[ β (1) β (2) β (3) . . .
β ( q )] .
Then, with the threshold set { β j } n j =1 for the second ﬁltration function g , (cid:126)β i = (cid:126)β ( PD ( G i , g )) will be a vector of size 1 × n .
Then, as M iβ = (cid:126)β i for each 1 ≤ i ≤ m , EMP Betti Summary M β ( G ) would be a 2 D -vector (matrix) of size m × n (Figure 1).
In particular, each entry M β = [ m ij ] is just the
Betti number of the corresponding clique complex in the biﬁltration { (cid:98) G ij } , i.e., m ij = β ( (cid:98) G ij ) .
This matrix M β is also called bigraded Betti numbers in the literature, and computationally much faster than other vectorizations Lesnick & Wright (2022).
4. S TABILITY OF EMP S UMMARIES
We now show that when the source single parameter vectorization ϕ is stable, then so is its induced
EMP vectorization M ϕ .
(We give the details of stability notion in persistence theory and examples of stable SP vectorizations in Appendix C.1.)
Let G + =
( V + , E + ) and G − =
( V − , E − ) be two graphs.
Let ϕ be a stable SP vectorization with the stability equation
[Equation]
for some 1 ≤ p ϕ ≤ ∞ .
Here, ϕ ( G ± ) represent the corresponding vectorizations for PD ( G ± ) and
W p represents Wasserstein-
p distance as deﬁned in Appendix C.1.
Now, by taking d = 2 for EMP construction, let f, g : V ±
→ R be two ﬁltration functions with threshold sets { α i } mi =1 and { β j } nj =1 respectively.
Then, by deﬁning V ± i = { v r ∈ V ± | f ( v r ) ≤ α i } , their induced subgraphs {G ± i } give the ﬁltration (cid:98) G 1 ⊂ (cid:98) G 2 ⊂ . . .
(cid:98) G m as before.
For each 1 ≤ i ≤ m , we will have persistence diagram PD (
G i , g ) as detailed in Section 4.1.
We deﬁne the induced matching distance between the multiple persistence diagrams as
[Equation]
Now, we deﬁne the distance between induced EMP Summaries as
[Equation]
Theorem 4.1.
Let ϕ be a stable SP vectorization.
Then, the induced EMP Vectorization M ϕ is also stable, i.e., with the notation above, there exists (cid:98) C ϕ > 0 such that for any pair of graphs G + and G − , we have the following inequality.
[Equation]
The proof is given in Appendix C.2.
E XPERIMENTS 5. D ATASETS
To verify our EMP summaries on graph representation learning, we evaluate EMP summaries with random forest on nine widely used benchmark datasets, i.e., (i) three molecule graphs (Sutherland et al., 2003): BZR_MD, COX2_MD, and DHFR_MD; (ii) two biological graphs (Kriege & Mutzel, 2012; Borgwardt et al., 2005): MUTAG and PROTEINS; and (iii) four social graphs: IMDBBinary (IMDB-B), IMDB-Multi (IMDB-M), REDDIT-Binary (REDDIT-B), and REDDIT-Multi-5K (REDDIT-M-5K).
We also calculate (i) average node-level centrality measures: average degree centrality, average betweenness centrality, average closeness centrality, average clustering centrality, average eigenvector centrality, and average Katz centrality; and (ii) average edge-level centrality measures: average betweenness centrality, average Ricci curvature, and average Ricci positive curvature, and feed these node-/edge-level measures into random forest model.
Furthermore, it is worth noting that COX2_MD and DHFR_MD all have three node features, whereas PROTEINS has one node feature.
We use all these available features as input to a random forest model.
Further details on the datasets and experimental setup are provided in Appendix E.1.
5. E XPERIMENTAL S ETUP
We vectorize our proposed EMP representations and use random forest (RF) for the graph classiﬁcation task.
For RF, we set the number of trees in the forest as 1000, the minimum number of samples as 2, and the function to measure the split quality as Gini impurity.
We apply ﬁltrations based on the available information of each dataset, either using their graph structure or their provided node/edge features.
Our pool of ﬁltration functions include, but is not limited to: type of atom, closeness, edge-betweenness, weighted degree, katz centrality, ricci curvature, and distance between atoms.
In this experiments, ﬁrst we apply a sublevel ﬁltration on nodes, and secondly a power ﬁltration over edges.
To test our EMP framework we use three vectorizations: Betti curves, silhouette functions, and entropy summary functions, thus producing EMP matrix representations which can be embedded in classic and modern machine learning algorithms.
We compare EMP summary with two types of state-of-the-art baselines, covering six graph kernels, one graph neural networks, and one topological method.
Graph kernels: (i) comprised of the [Table] subgraph matching kernel (CSM) Kriege & Mutzel (2012), (ii) Shortest Path Hash Graph Kernel (HGK-SP) Morris et al.
(iii) Weisfeiler–Lehman Hash Graph Kernel (HGK-WL) Morris et al. (2016), (iv) Multiscale Laplacian Graph Kernel (MLG) Kondor & Pan (2016), (v) Weisfeiler–
Lehman (WL) Shervashidze et al.
(2011), and (vi) Weisfeiler-Lehman Optimal Assignment (WLOA) Kriege et al.
O’Bray al methods we report the average accuracy of 10 runs of 10-fold CV with standard deviation, and running time for computation of each EMP representation.
The source code is freely available 1 .
5. E XPERIMENTAL R ESULTS
Table 1 shows the results of different methods on nine graph datasets.
Out-of-time (OOT) results indicate that a method could not complete the classiﬁcation task within 12 hours, and OOM means “out-of-memory” (from an allocation of 128 GB RAM).
We observe the following phenomena:
• Compared with all baselines, out of 9 benchmark datasets, the proposed EMP summaries achieve the best performance on 5 datasets (BZR_MD, DFHR_MD, MUTAG, IMDB-B, REDDIT-B), and second best in 2 datasets (COX2_MD, REDDIT-5K).
• EMP summary consistently outperforms Filtration Curves on all datasets except COX2_MD and PROTEINS, indicating that multiparameter structure of the EMP summaries can better capture the complex structural properties and local topological information in heterogeneous graphs.
• Moreover, EMP Summaries consistently delivers the competitive results with GNNs in all benchmark datasets.
This indicates that EMP summary introduces powerful topological and geometric schemes for node features and graph representation learning.
To further investigate the effectiveness of the EMP vectorization function in graph representation learning, we have conducted ablation studies of various EMP summaries on the benchmark datasets, see Appendix G. Furthermore, Appendix F includes an analysis of computational complexity.
D ISCUSSION
We have proposed a new computationally efﬁcient summary for multidimensional persistence for various forms of data, and include focus-cases on graph machine learning.
The new Effective
Multidimensional Persistence (EMP) framework provides a practical way to employ the promising multidimensional persistence idea in real-life applications.
As such, EMP summaries are highly compatible with machine learning models and can boost the performances of popular single persistent summaries in a uniﬁed manner.
In a graph classiﬁcation task, EMP summaries outperform state-of-theart methods in several benchmark datasets.
Furthermore, we have shown that EMP maintain important stability guarantees.
Therefore, EMP makes an important step toward bringing the theoretical concepts of multipersistence to the machine learning community and advances the use of persistent homology in various settings.
In future work, we aim to enhance the EMP framework by using more than one slicing direction in the multipersistence grid and, thus, effectively combining the outputs of multiple slicing directions with deep learning approaches.
Appendix A A DDITIONAL L ITERATURE ON G RAPH R EPRESENTATION L EARNING
After the success of convolutional neural networks (CNN) on image-based tasks, graph neural networks (GNNs) have emerged as a powerful tool for graph classiﬁcation and representation learning.
Based on the spectral graph theory, Bruna et al.
(2014) introduced a graph-based convolution in
Fourier domain.
However, complexity of this model is very high since all Laplacian eigenvectors are needed.
To tackle this problem, ChebNet Defferrard et al.
(2016) integrated spectral graph convolution with Chebyshev polynomials.
Then, Graph Convolutional Networks (GCNs) of Kipf & Welling (2017) simpliﬁed the graph convolution with a localized ﬁrst-order approximation.
More recently, there have been proposed various approaches based on accumulation of the graph information from a wider neighborhood, using diffusion aggregation and random walks.
Such higher-order methods include approximate personalized propagation of neural predictions (APPNP)
Klicpera et al.
(2019), higher-order graph convolutional architectures (MixHop) Abu-El-Haija et al.
(2019), multi-scale graph convolution (N-GCN)
Abu-El-Haija et al.
(2020), and Lévy Flights Graph Convolutional
Networks (LFGCN)
Chen et al.
In addition to random walks, other recent approaches include GNNs on directed graphs (MotifNet)
Monti et al.
(2018), graph convolutional networks with attention mechanism (GAT, SPAGAN)
Veli ˇ ckovi ´ c et al.
; Yang et al.
(2019), and graph
Markov neural network (GMNN)
Qu et al.
Most recently, Liu et al.
Liu et al.
(2020) consider utilizing information on the node neighbors’ features in GNN, proposing Deep Adaptive Graph
Neural Network (DAGNN).
However, DAGNN, and other state-of-the-art approaches, does not account for the important information on the shapes of the node neighborhoods.
B F URTHER B ACKGROUND ON S INGLE P ERSISTENT H OMOLOGY
Here, we give further details on single parameter persistent homology.
To sum up, PH machinery is a 3 -step process.
The ﬁrst step is the ﬁltration step, where one can integrate the domain information into the process.
The second step is the persistence diagrams , where the machinery records the evolution of topological features (birth/death times) in the ﬁltration sequence of the simplicial complexes.
The ﬁnal step is the vectorization (ﬁngerprinting), where one can convert these records to a function or vector to be used in suitable ML models.
i. Constructing Filtrations: As PH is the machinery to keep track of the evolution of topological features in a sequence, the most important step is inducing this nested sequence of simplicial complexes, ∆ 0 ⊂ ∆ 1 ⊂ · · · ⊂ ∆ m .
This is the key step where one can inject valuable domain information into the PH process by using important domain functions.
Two most common methods are Sublevel/superlevel ﬁltration and Vietoris-Rips (VR) ﬁltration .
We already described sublevel ﬁltration in Section 3.1.
VR ﬁltration is another common method especially used for point clouds, where coarse geometry of the data set X play key role (Chazal & Michel, 2021).
Let X
= { x 1 , x 2 , ..., x N } be the given data set.
For a given threshold (cid:15) i , one forms a Vietoris-Rips complex ∆ i by adding a k -simplex to X for any subset { x n 0 , x n 1 , ..., x n k } , where the pairwise distances are all d ( x n r , x n s ) < (cid:15) i .
In particular, if a pair of points x n 0 , x n 1 has distance < (cid:15) i , then in the induced simplicial complex ∆ i , we add an edge e n 0 n 1 between the corresponding vertices x n 0 and x n 1 .
If three such points x n 0 , x n 1 , x n 2 have pairwise distances < (cid:15) i , then we ﬁll the triangle e n 0 n 1 ∪ e n 1 n 2 ∪ e n 2 n 0 with a 2 -simplex, and so on. For
each (cid:15) i , one obtains a simplicial complex ∆ i by using this procedure.
Changing threshold values (cid:15) < (cid:15) < . . .
< (cid:15) m results
in a
hierarchical nested
sequence of simplicial
complexes ∆ 1 ⊂ ∆ 2 ⊂ · · · ⊂ ∆ m that is termed the Vietoris-Rips ﬁltration of the data set X
Note that VR ﬁltration can also be considered as a sublevel ﬁltration for the distance
function to
X from the ambient space R d , i.e., f : R d
→ R with f y
= d y, X ) where X ⊂ R d . ii.
Persistence Diagrams:
The second step in PH process is to obtain persistence diagrams (PD) for the ﬁltration ∆
⊂ ∆ 1 ⊂ · · · ⊂ ∆ m .
As explained in Section 3.1, PDs are collection of 2 -tuples, marking the birth and death times of the topological features appearing in the ﬁltration, i.e. PD k ( X )
= { ( b σ , d σ ) | σ ∈ H k (∆ i ) for b σ ≤ i < d σ } .
Current algorithmic advancements make possible to efﬁciently compute persistence diagrams via modern software libraries (Otter et al., 2017).
PDs, a collection of points in R 2 by itself, are not very practical for statistical and ML purposes.
Instead, the common techniques are faithfully representing PDs as kernels (Kriege et al., 2020) or vectorizations (Hensel et al., 2021).
One can consider this step as converting PDs into a useful format to be used in the ML process as ﬁngerprints of the dataset.
This provides a practical way to use the outputs of PH in real-life applications.
Single Persistence Vectorizations transform obtained PH information (PDs) into a function or a feature vector form which is much more suitable for ML tools than PDs.
Common single persistence (SP) vectorization methods are Persistence Images (Adams et al., 2017), Persistence Landscapes (Bubenik, 2015), Silhouettes (Chazal et al., 2014), Betti Curves and various Persistence Curves (Chung & Lawson, 2019).
These vectorizations deﬁne a single variable or multivariable function out of PDs, which can be used as ﬁxed-size 1 D or 2 D vectors in applications, i.e., 1 × n vectors or m × n vectors.
For example, a Betti curve for a PD with n thresholds can also be expressed as 1 × n size vectors.
Similarly, Persistence Images is an example of 2 D vectors with the chosen resolution (grid) size.
See the examples given in Section 4.2 for further details.
C S TABILITY C. S TABILITY OF S INGLE P ERSISTENCE S UMMARIES
For a given PD vectorization, the stability is one of the most important properties for statistical purposes.
Intuitively, stability question is whether a small perturbation in PD cause a big change in the vectorization or not.
To make this question meaningful, one needs to deﬁne what "small" and “big" means in this context.
Therefore, we need to deﬁne distance notion, i.e., metric in the space of persistence diagrams.
The most common such metric is called Wasserstein distance (or matching distance) which is deﬁned as follows.
Let PD ( X + ) and PD ( X − ) be persistence diagrams two datasets X + and X −
(We omit the dimensions in PDs).
Let PD ( X + ) =
{ q + j } ∪ ∆ + and
PD ( X − ) = { q − l } ∪ ∆ − where ∆ ± represents the diagonal (representing trivial cycles) with inﬁnite multiplicity.
Here, q + j = ( b + j , d + j ) ∈ PD ( X + ) represents the birth and death times of a hole σ j in X + .
Let φ : PD ( X + ) → PD ( X − ) represent a bijection (matching).
With the existence of the diagonal ∆ ± in both sides, we make sure the existence of these bijections even if the cardinalities |{ q + j }| and |{ q − l }| are different.
Then, the p th Wasserstein distance W p deﬁned as
[Equation]
Then, a vectorization (function) ϕ ( PD ( X )) is called stable if d( ϕ + , ϕ − ) ≤
C ·
W p ( PD ( X + ) , PD ( X − ))
where ϕ ± =
ϕ ( PD ( X ± )) and d( ., . ) is a suitable metric on the space of vectorizations used.
Here, the constant C > 0 is independent of X ± .
This stability inequality interprets as the changes in the vectorizations are bounded by the changes in PDs.
Two nearby persistence diagrams are represented by nearby vectorizations.
If a given vectorization ϕ satisﬁes such a stability inequality for some d and W p , we call ϕ a stable vectorization Atienza et al.
Persistence Landscapes Bubenik (2015), Persistence Images Adams et al.
(2017), Stabilized Betti
Curves Johnson & Jung (2021) and several Persistence curves Chung & Lawson (2019) are among well-known examples of stable vectorizations.
C. P ROOF OF T HEOREM 4.1: S TABILITY
Proof.
As ϕ is a stable SP vectorization, for any 1 ≤ i ≤ m , we have d( ϕ ( G + i ) , ϕ ( G − i )) ≤
C ϕ · W p ϕ ( PD ( G + i ) , PD ( G − i )) for some C ϕ > 0 by Equation (1), where W p ϕ is Wasserstein-
p distance.
Notice that the constant C ϕ > 0 is independent of i .
Hence,
[Equation]
where the ﬁrst and last equalities are due to Equation (2) and Equation (3), while the inequality follows from Equation (1) which is true for any i .
This concludes the proof of the theorem.
D EMP F RAMEWORK D. EMP FOR O THER T YPES OF D ATA
So far, to keep the exposition simple, we described our construction in the graph setup.
However, our framework is suitable for various types of data.
Let X be a an image data or a point cloud.
Let f : X → R and g : X → R be two ﬁltration functions on X .
For example, it can be grayscale function for image data, or density function on point cloud data.
Let f : X → R be the ﬁltration function with threshold set { α i } m 1 .
Let X i =
f
− 1 (( −∞ , α i ]) .
Then, we get a ﬁltering of
X as nested subspaces X 1 ⊂ X 2 ⊂ · · · ⊂ X m = X .
By using the second ﬁltration function, we obtain ﬁner ﬁltrations for each subspace
X i where 1 ≤ i ≤ m .
In particular, ﬁx≤ i 0 ≤ m and let {
β j } nj =1 be the threshold set for the second ﬁltration function g .
Then, by restricting g to
X i 0 , we get a ﬁltration function on X i 0 , i.e., g : X i
→ R which produces ﬁltering X i 0 1 ⊂ X i 0 2 ⊂ · · · ⊂ X i 0 n = X i 0 .
By inducing a simplicial complex (cid:98) X i 0 j for each
X i
j , we get a ﬁltration (cid:98) X i 0
⊂ (cid:98) X i 0 2 ⊂ · · · ⊂ (cid:98) X i 0 n = (cid:98) X i 0 .
This ﬁltration results in a persistence diagram (PD) PD (
X i 0 , g ) .
For each 1 ≤ i ≤ m , we obtain PD ( X i , g ) .
Note that after getting {X i } mi =1 via f , instead of using second ﬁltration function g , one can apply power ﬁltration or Vietoris-Rips construction based on distance for each X i 0 in order to get a different ﬁltration (cid:98) X
[Equation]
By using m PDs, we follow a similar route to deﬁne our EMP summaries.
Let ϕ be a single persistence vectorization.
By applying the chosen SP vectorization ϕ to each PD, we obtain a function ϕ
i = ϕ ( PD ( X i , g )) on the threshold domain [ β 1 , β n ] , which can be expresses as a 1 D (or 2 D ) vector in most cases (Section 4.2).
Let (cid:126)ϕ i be the corresponding 1 × k vector for the function ϕ i .
Deﬁne the corresponding EMP M ϕ as M i ϕ
= (cid:126)ϕ i where M i ϕ is the i th row of M ϕ .
In particular, M ϕ is a 2 D -vector (a matrix) of size m × k where m is the number of thresholds for the ﬁrst ﬁltration function f , and k is the length of the vector (cid:126)ϕ .
D. EMP WITH O THER F ILTRATIONS
Weight ﬁltration For a given weighted graph G =
( V , E , W ) , it is common to use edge weights
[Equation]
} to describe ﬁltration.
By choosing the threshold set similarly
I = { α i } m 1 with α 1 = min { ω rs ∈ W} <
α 2 < . . .
<
α m = max { ω rs ∈ W} .
For α i ∈ I , let
E i = { e rs ∈ V | ω rs ≤ α i } .
Let G i be a subgraph of G induced by V i .
This induces a nested sequence of subgraphs
[Equation]
See top row in Figure 2)
[Equation]
In the case of weighted graphs, one can apply the EMP framework just by replacing the ﬁrst ﬁltering (via f ) with weight ﬁltering.
In particular, let g : V
→ R be a ﬁltration function with threshold set { β j } nj =1 .
Then, one can ﬁrst apply weight ﬁltering to get G 1 ⊂ · · · ⊂ G m =
G as above, and then apply f to each
G i to get a bilﬁltration {G ij } ( m × n resolution).
One gets m PDs as PD ( G i , g ) and induce the corresponding M ϕ .
Alternatively, one can change the order by applying g ﬁrst, and get a different ﬁltering G 1 ⊂ G 2 ⊂ · · · ⊂ G n = G induced by g .
Then, apply to edge weight ﬁltration
d e g r ee
[Table]
1 I 1 I 1 I 1 I
A
G
I
A
B
F 3 G
E
{+I  
+}
A
B
F 3 G
D
C
E
I  {+
+}
A
G I A G
I A G
I[-
-]
A
B
F 3 G {+
+}
E
I
A
B
F 3 G
E
I
A
B
F 3 G
E
I
A
B
F 3 G[-
-]
[-I   
-]
D
C
E
I  4
A
B
F 3 G
D
C
E
I  4 4
A
B
F 3 G
D
C
E
I  4 42  43
[Table]
A
B
F 3 G
D
C
E
I   42
Introduction to L A T E X
Author’s Name
June 4,
Abstract
The abstract text goes here.
Comparison with other multiparameter descriptors
B k ( C ) is the k th Betti number of C
[Equation]
[Equation] [Equation]
k
In our ﬁnal set of experiments, we test our MPGF method in time series classiﬁcation problems and compare results with other new multiparameter persistence approaches.
Here, we focus on 17 time series datasets from the UCR archive [1], using training/testing sizes as described in table 1.
We choose such benchmark problems because we aim to contrast our MPGF versus newly state- of-the-art results of multiparameter persistence approaches, as described in [2].
To keep all experiments under same conditions we parallelize the MPGF method and run on a AWS machine with Xeon Platinum 8175M.
Similarly, we apply Taken’s embeddings on time series, ﬁlter based on distance to the point cloud and density estimates, train using a XGBoost classiﬁer, and perform 5-fold cross validation; see further and other details in [2].
Table 2 shows the accuracy of other 6 methods along with our proposed
MPGF: Multiparameter Persistence Image (MP-I) [2], Multiparameter Persis- tence Kernel (MP-K) [3], Multiparameter Persistence Landscape (MP-L) [REF], 1D Persistence Landscape (P-L) [4], 1D Persistence Image (P-I) [5], Persistence Space Scale Kernel (PSS-K) [6].
Here, we focus on contrasting the amount of signiﬁcant information, for a machine learning method, contained in the topo- logical summaries.
The classiﬁcation results are consistent with our hypothesis that MPGF cap- tures signiﬁcant multidimensional information produced by ﬁltration functions.
We notice that the proposed method outperforms current state-of-the-art ap- proaches on
% of benchmark problems
Furthermore most of best results are split among
MP
K, MP-L and MPGF, leading us to conclude that these
Introduction to L A TEX
Author’s Name
June 4,
Abstract
The abstract text goes here.
Comparison with other multiparameter descriptors
B k ( C ) is the k th Betti number of C
B 1 =
[Equation]
In our ﬁnal set of experiments, we test our MPGF method in time series classiﬁcation problems and compare results with other new multiparameter persistence approaches.
Here, we focus on 17 time series datasets from the UCR archive [1], using training/testing sizes as described in table 1.
We choose such benchmark problems because we aim to contrast our MPGF versus newly state- of-the-art results of multiparameter persistence approaches, as described in [2].
To keep all experiments under same conditions, we parallelize the MPGF method and run on a AWS machine with Xeon Platinum 8175M.
Similarly, we apply Taken’s embeddings on time series, ﬁlter based on distance to the point cloud and density estimates, train using a XGBoost classiﬁer, and perform 5-fold cross validation; see further and other details in [2].
Table 2 shows the accuracy of other 6 methods along with our proposed
MPGF: Multiparameter Persistence Image (MP-I) [2], Multiparameter Persis- tence Kernel (MP-K) [3], Multiparameter Persistence Landscape (MP-L) [REF], 1D Persistence Landscape (P-L) [4], 1D Persistence Image (P-I) [5], Persistence Space Scale Kernel (PSS-K) [6].
Here, we focus on contrasting the amount of signiﬁcant information, for a machine learning method, contained in the topo- logical summaries.
The classiﬁcation results are consistent with our hypothesis that MPGF cap- tures signiﬁcant multidimensional information produced by ﬁltration functions.
We notice that the proposed method outperforms current state-of-the-art ap- proaches on 23.5% of benchmark problems.
Furthermore, most of best results are split among MP-K, MP-L and MPGF, leading us to conclude that these
[Figure]
G D C E
I
[Figure] [Figure]
[Figure] [Figure]
[Figure]
Interview Questions
Ignacio Segovia-Dominguez
The University of Texas at Dallas / NASA Jet Propulsion Lab, Caltech
Filtration:
B 0 = B 0 =
B 0 = B 0 =
B 1 = B 1 =
B 1 =
Second ﬁltration: V
V 2 V
V 4 V 9  1 V
V 9 + 1 V =
V = ...
First ﬁltration: U
U 2 U
U 4 U 8  1 U
U 8 + 1 U <
U < ...
Dimension: 38< 38< dim 1  dim 0 -dim 1 -dim0-dim 1-dim0-dim 1 - dim
[Table]
Ignacio Segovia-Dominguez
The University of Texas at Dallas / NASA Jet Propulsion Lab, Caltech
Filtration:
B 0 = B 0 =
B 0 = B 0 =
B 1 = B 1 =
B 1 =
Second ﬁltration: V
V 2 V
V 4 V 9  1 V
V 9 + 1 V =
V = ...
First
[Table]
: U
U 2 U
U 4 U 8  1 U
U 8 + 1 U <
U < ...
Dimension: 38< 38< dim 1  dim 0 -dim 1 -dim0-dim 1-dim 0 - dim 1 - dim
[Table]
Ignacio Segovia-Dominguez
The University of Texas at Dallas / NASA Jet Propulsion Lab, Caltech
Filtration:
B 0 =
[Table]
B
[Table]
B
[Table]
Second
[Table]
: V
V 2 V
[Table]
V 4 V
[Table]
1 V
V 9 + 1 V =
V = ...
First
[Table]
: U
U 2 U
U 4 U 8  1 U
U 8 + 1 U <
U < ...
Dimension:
[Table]
38<
[Table]
38<
[Table] [Table]
dim 1 -dim
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dim 1-dim 0 - dim 1 - dim
[Table]
Ignacio Segovia-Dominguez
The
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Ignacio
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dim 0 -dim 1 -dim0-dim 1-dim0-dim 1-dim
Interview Questions
Ignacio Segovia-Dominguez
The University of Texas at Dallas / NASA Jet Propulsion Lab, Caltech
Filtration:
B 0 = B 0 =
B 0 = B 0 =
B 1 = B 1 =
B 1 =
Second ﬁltration: V
V 2 V
V 4 V 9  1 V
V 9 + 1 V =
V = ...
First ﬁltration:
[Table]
[Table]
[Figure]
[Table] [Table] [Figure] [Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
to any G j , one gets a biﬁltration { (cid:98) G ji } ( n × m resolution) this time.
As a result, one gets n PDs as PD (
G i , ω ) and induce the corresponding M ϕ .
The difference is that in the ﬁrst case (ﬁrst apply weights, then g ), the ﬁltration function plays more important role as M ϕ uses PD ( G i , g ) while in the second case (ﬁrst apply g , then apply weights) weights have more important role as M ϕ is induced by
PD ( G j , ω ) .
Note also that there is a very different ﬁltration method for weighted graphs by applying the following the following VR-complexes method.
Power (Vietoris-Rips)
Filtration There is a highly different ﬁltration technique using distances between the data points in the dataset.
The technique is called power ﬁltration for unweighted graphs Aktas et al.
(2019), while it is called Vietoris-Rips ﬁltration for other types of data Edelsbrunner & Harer (2010).
The idea is for a point cloud X
= { x 1 , x 2 , . . .
, x N } , one uses the pairwise distances d ( x r , x s ) to construct the simplicial complexes in the ﬁltration.
In particular, for a threshold set (cid:15) 1 < (cid:15) 2 < · · · < (cid:15) n = diam ( X ) , one forms a Vietoris-Rips complex ∆ j by adding a k -simplex to
X for any subset { x r 0 , x r 1 , . . .
, x r k } , where the pairwise distances are all < (cid:15) j .
If a pair of points x r 1 , x r 2 has distance < (cid:15) j , then in the induced simplicial complex ∆ j , we add an edge between the corresponding vertices x r and x s .
If three such points x r 1 , x r 2 , x r 3 have pairwise distances < (cid:15) j , then we ﬁll the triangle e r 1 r 2 ∪ e r 2 r 3 ∪ e r 3 r 1 with a 2 -simplex, and so on.
This procedure induces in a hierarchical nested sequence of simplicial complexes ∆⊂ ∆ 2 ⊂ . . .
⊂ ∆ m that is termed Vietoris-Rips ﬁltration induced by the point cloud
X .
For unweighted graphs, one takes the vertex set
V as the point cloud, and deﬁnes the distances d ( v i , v j ) as the shortest distance in the graph where each edge has length 1 .
For weighted graphs, one can do the same by deﬁning edge lengths induced by the weights.
One can adapt Vietoris-Rips ﬁltrations to our EMP setting as follows.
Start with a ﬁltration function f : X → R with threshold set { α i } m 1 and obtain X 1 ⊂ X 2 ⊂ · · · ⊂ X m =
X where X i
=
f − 1 (( −∞ , α i ]) .
Then, apply Vietoris-Rips ﬁltration to each X i 0 for threshold set { (cid:15) j } nj =1 which produces a ﬁltration (cid:98) X i 0 1 ⊂ (cid:98)
X i 0 2 ⊂ · · · ⊂ (cid:98) X i 0 n where (cid:98) X i
j is the Vietoris-Rips complex of X i 0 for threshold (cid:15) j .
Construct PD (
X i , V R ) of these ﬁltrations for each 1 ≤ i ≤ m .
The following steps are same Section 4.2.
For a given SP vectorization ϕ , let (cid:126)ϕ i be the corresponding× k vector induced by ϕ ( PD (
X i , V R )) with domain [ (cid:15) 1 , (cid:15) n ] .
Then, deﬁne EMP M ϕ as M iϕ = (cid:126)ϕ i where M iϕ is the i th row of M ϕ .
Again, M ϕ is a 2 D -vector (a matrix) of size m × k where m is the number of thresholds for the ﬁltration function f , and k is the length of the vector (cid:126)ϕ .
D. M ULTIPARAMETER P ERSISTENCE T HEORY
Multipersistence theory is under intense research because of its promise to signiﬁcantly improve the performance and robustness properties of single persistence theory.
While single persistence theory obtains the topological ﬁngerprint of single ﬁltration, a multidimensional ﬁltration with more than one parameter should deliver a much ﬁner summary of the data to be used with ML models.
However, because of the technical issues in the theory, multipersistence has not reached to its potential yet and remains largely unexplored by the ML community.
Here, we provide a short summary of these technical issues.
For further details, Botnan & Lesnick (2022) gives a nice outline of current state of the theory and major obstacles.
In single persistence, the threshold space { α i } being a subset of R , is totally ordered, i.e., birth time < death time for any topological feature appearing in the ﬁltration sequence { ∆ i } .
By using this property, it was shown that “barcode decomposition” is well-deﬁned in single persistence theory in 1950s [Krull-Schmidt-Azumaya Theorem Botnan & Lesnick (2022)–Theorem 4.2].
This decomposition makes the persistence module M =
{ H k (∆ i ) } Ni =1 uniquely decomposable into barcodes.
This barcode decomposition is exactly what we call a Persistence Diagram .
However, when one goes to higher dimensions, i.e. d = 2 , then the threshold set { ( α i , β j ) } is no longer totally ordered, but becomes partially ordered (Poset).
In other words, some indices have ordering relation (1 , 2) <
(4 , 7) , while some do not, e.g., (2,3) vs. (1,5).
Hence, if one has a multipersistence grid { ∆ ij } , we no longer can talk about birth time or death time as there is no ordering anymore.
Furthermore, Krull-Schmidt-Azumaya Theorem is no longer true for upper dimensions Botnan & Lesnick (2022)–Section 4.2.
Hence, for general multipersistence modules barcode decomposition is not possible, and the direct generalization of single persistence to multipersistence fails.
On the other hand, even if the multipersistence module has a good barcode decomposition, because of partial ordering, representing these barcodes faithfully is another major problem.
Multipersistence modules are an important subject in commutative algebra, where one can ﬁnd the details of the topic in Eisenbud (2013).
While complete generalization is out of reach for now, several attempts have been tried to utilize MP idea (Lesnick, 2015). One of the ﬁrst such novel idea came from Lesnick & Wright (2015) where they suggest to use one dimensional slices in the MP grid, and to get the signature of the most dominant features. Later, Carrière & Blumberg (2020) combined several slicing directions (vineyards) and obtained a vectorization by summarizing several persistence diagrams (PDs) in these directions.
Slicing techniques use the persistence diagrams of predetermined one-dimensional slices in the multipersistence grid, and then combine (compress) them as one dimensional output Botnan &
Lesnick (2022).
In that respect, one major issue is that the topological summary highly depends on the predetermined slicing directions in this approach, and how to decide this direction.
The other problem is the loss of information when compressing the information in various persistence diagrams.
As explained above, MP approach has still technical problems to reach its full potential, and there are several attempts to utilize this idea.
In this paper, we do not claim to solve theoretical problems of multipersistence homology, but offer a novel, highly practical multidimensional topological summary by advancing the existing methods.
We use the grid directions in the multipersistence module as natural slicing directions and produce mutidimensional topological summary of the data.
As a result, these multidimensional topological ﬁngerprints are capable of capturing very ﬁne topological information hidden in the data.
Furthermore, in the case the data provides more than two very important ﬁltration functions, our framework easily accommodates these functions and induces corresponding substructures.
Then, our EMP model capture the evolving topological patterns of these substructures and summarize them in matrices and arrays which are highly practical output format to be used with various ML models.
Our model is highly different from previous work mostly because of its practicality and computational efﬁciency. Among these, the closest method to ours is Carrière & Blumberg (2020) which employs slicing techniques a different way.
Like us, they have predetermined slicing options (vineyards), and they compute the single persistence diagrams on these slices and combine them in a unique way by using weight functions induced from lifespans of the topological features in this collection of persistence diagram.
In our approach, we use only horizontal slices and do not compress the information.
First, choosing horizontal slices computationally very feasible to obtain persistence diagrams. Second, we offer a variety of options on how to vectorize these persistence diagrams.
N
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[Table] [Table]
D i m e n s i on : 0  3 8 < 1  3 8 <
d
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Secondﬁltration: V 1 V 2 V
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: U 1 U 2 U
U 4 U 8  1 U 8 U 8 + 1 U <  1 U < ...
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Hence, depending on the dataset, one can use vectorization methods which emphasize long barcodes (Silhouette with p > 1 , Entropy, Persistence Image) or the ones which considers all signals equally (Betti).
Our experiments proved that these variety of options can be quite useful as some EMP vectorizations give much better result than others in different datasets (Appendix G).
E F URTHER D ETAILS ON E XPERIMENTS E. B ENCHMARK D ATASETS AND E XPERIMENTAL S ETUP
In our experiments, we use 11 benchmark datasets for graph classiﬁcation tasks (see Table 2).
We have run our models for graph classiﬁcation tasks on an 8-core DO droplet machine with Intel Xeon
Scalable processors at a base frequency of 2.5 Ghz.
Table 2 summarize the statistics of the datasets in our experiments.
[Table]
The resolution of vectorization is the most signiﬁcant parameter [Table] which may impact the computational performance and results. As such
[Table]
we use a ﬁxed resolution to get consistent results in all experiments and consider time constraints on server usage.
We use resolution size of 50 × 50 for each summary function, and the standard parameters set by the Gudhi library in Python 2 .
The order of landscape summary function is set to 1 ( max ), whilst the power of weights is set to 1 for silhouette summaries.
Results in Table 1 comes from computing MP using ﬁltration functions as follows.
BZR_MD,
COX2_MD, and DHFR_MD use weighted node-degree and edge-power ﬁltrations.
PROTEINS uses node-closeness and edge-betweenness power ﬁltrations. MUTAG, IMDB-BINARY, and IMDBMULTI use node-katz centrality and edge-ricci curvature power ﬁltrations. REDDIT-BINARY, and REDDIT-MULTI-5K use node-katz centrality and edge-ricci curvature ﬁltrations.
We performed an empirical analysis to select previous ﬁlter functions for each graph network.
We use three types of vectorizations: Betti curves, silhouette functions and entropy summary functions.
For each case we compute both 0-dim and 1-dim MP topological features.
F C OMPUTATIONAL C OMPLEXITY
Computational complexity (CC) of persistence diagram PD k (∆) is O ( N 3 ) , where N is the number of k -simplices in ∆ (Otter et al., 2017).
CC of EMP summary M dϕ depends on the vectorization ϕ used and the number d of the ﬁltration functions one uses.
If r is the resolution size of the multipersistence grid, then one needs r ( d − 1) single persistence diagrams to obtain M d ϕ .
Therefore,
CC( M dϕ ) =
O ( r ( d − 1) · N 3 · C ϕ ( m )) where C ϕ ( m ) is CC for ϕ and m is the number of barcodes in PD k , e.g., if ϕ is persistence landscape, then C ϕ ( m ) =
m 2 and hence CC for EMP Landscape with d = 2 is O ( r · N 3 · m 2 ) .
In practice, r is a constant and m is small compared to N , hence the complexity is again reduced to
O ( N 3 ) .
On the other hand, as Betti numbers do not need PD k to be computed, it is possible to obtain much faster algorithms for EMP Betti Summary (Edelsbrunner &
Parsa, 2014).
Recently, Lesnick & Wright (2022) introduced a quite fast algorithm for EMP Betti summaries with O ( M 3 ) time where M is the rank of the multipersistence module with minimal representation.
G A BLATION S TUDY G. D IFFERENT TYPES OF EMP SUMMARIES
Furthermore, to evaluate the performance of different types of EMP summaries (i.e., EMP Silhouette (EMP-S), EMP Entropy (EMP-E), and EMP Betti (EMP-B) with different dimensions (i.e., H 0 : 0-dimensional topology, and H 1 : 1-dimensional topology) and graph-based features (i.e., f V : node features, and f E : edge features; see more details in Section 5.2 and Appendix E.1).
We include comparative statistics using all combinations of input variables, i.e. ablation study, for 3 datasets in our analysis: BZR-MD (Table 3), DHFR-MD (Table 4), and REDDIT-BINARY (Table 5).
Each cell shows the classiﬁcation accuracy (in % ± standard deviation) when using different combinations of variables.
The best accuracy result is highlighted using bold font.
A row/column named as ’none’ means that either graph-based features, or EMP summaries where not used to compute the experiments.
As such, the top row shows all the accuracy results without using any EMP topological features.
The ﬁrst/left column shows the results only using EMP topological features.
The cell in the bottom right corner, of each EMP group, contains the accuracy results using all graph-extracted/
EMP features available.
We can observe that: (i) best results contains 1-dimensional EMP summaries, demonstrating the necessity of capturing higher-order structures (e.g., triangles/cycles), (ii) the choice of the EMP summary can signiﬁcantly affect the performance, and (iii) the addition of EMP topological features generally improves accuracy of versions without EMP summaries, thus, demonstrating the importance of modeling global and topological graph structures. G. O RDER OF F ILTRATION With our novel approach, especially sublevel/superlevel ﬁltration, we give a computationally very efﬁcient way to extract information of substructures of the data induced by multiple functions. Since
[Table]
[Table]
[Table]
we only use horizontal slices, the ﬁrst function is only used for ﬁner ﬁltration, and the second function gives the persistence diagrams.
This makes our method a-symmetric (the order is important).
Hence, one can change the order and get different performance results for the model.
This asymmetry enriches our method as one can combine both feature vectors obtained by different order as they do not contain the same information about the multipersistence grid.
To observe the effect of changing the order of the ﬁltering functions, we run experiments on two benchmark datasets, BZR_MD and
REDDIT-BINARY.
Our experiments show that in some datasets, the order can be highly important, while in others, it has negligible effect on the performance.
Table 6 (BZR_MD) shows signiﬁcant changes in the performance for all our vectorizations (EMPBetti, EMP-Silhouette and EMP-Entropy).
In Figure 4a, we also see in half of the models, one order is better than the other, while in the other half of the models, the opposite is true.
On the other hand, we observe that for the REDDIT-BINARY dataset, the order is not as important for these functions (Table 7 & Figure 4b) as the change in performance is not as signiﬁcant as in BZR_MD.
[Table] [Table] ± 0. 88.250 ± 0.
87.410 ± 0.
88.590 ± 0.
{+H 
+}
+ H 1 W E (cid:32) S V 89.715 ± 0.279 ∗∗∗ 90.495 ± 0.248 ∗∗∗ 89.780 ± 0.337 ∗∗∗ 90.580 ± 0.217 ∗∗∗ [Figure] [Figure] [Figure]