Learning Hyperbolic Representations of Topological Features

1 INTRODUCTION . Persistent homology is a topological data analysis tool which tracks how topological features ( e.g . connected components , cycles , cavities ) appear and disappear as we analyze the data at different scales or in nested sequences of subspaces ( 1 ; 2 ) . A nested sequence of subspaces is known as a filtration . As an informal example of a filtration consider an image of variable brightness . As the brightness is increased , certain features ( edges , texture ) may become less or more prevalent . The birth of a topological feature refers to the `` time '' ( i.e. , the brightness value ) when it appears in the filtration and the death refers to the `` time '' when it disappears . The lifespan of the feature is called persistence . Persistent homology summarizes these topological characteristics in a form of multiset called persistence diagram , which is a highly robust and versatile descriptor of the data . Persistence diagrams enjoy the stability property , which ensures that the diagrams of two similar objects are similar ( 3 ) . Additionally , under some assumptions , one can approximately reconstruct the input space from a diagram ( which is known as solving the inverse problem ) ( 4 ) . However , despite their strengths , the space of persistence diagrams lacks structure as basic operations , such as addition and scalar multiplication , are not well defined . The only imposed structure is induced by the Bottleneck and Wasserstein metrics , which are notoriously hard to compute , thereby preventing us from leveraging them for machine learning tasks . Related Work . To address these issues , several vectorization methods have been proposed . Some of the earliest approaches are based on kernels , i.e. , generalized products that turn persistence diagrams into elements of a Hilbert space . Kusano et al . ( 5 ) propose a persistence weighted Gaussian kernel which allows them to explicitly control the effect of persistence . Alternatively , Carrière et al . ( 6 ) leverage the sliced Wasserstein distance to define a kernel that mimics the distance between diagrams . The approaches by Bubenik ( 7 ) based on persistent landscapes , by Reininghaus et al . ( 8 ) based on scale space theory and by Le et al . ( 9 ) based on the Fisher information metric are along the same line of work . The major drawback in utilizing kernel methods is that they suffer from scalability issues as the training scales poorly with the number of samples . In another line of work , researchers have constructed finite-dimensional embeddings , i.e. , transformations turning persistence diagrams into vectors in a Euclidean space . Adams et al . ( 10 ) map the diagrams to persistence images and discretize them to obtain the embedding vector . Carrière et al . ( 11 ) develop a stable vectorization method by computing pairwise distances between points in the persistence diagram . An approach based on interpreting the points in the diagram as roots of a complex polynomial is presented by Di Fabio ( 12 ) . Adcock et al . ( 13 ) identify an algebra of polynomials on the diagram space that can be used as coordinates and the approach is extended by Kališnik in ( 14 ) to tropical functions which guarantee stability . The common drawback of these embeddings is that the representation is pre-defined , i.e. , there exist no learnable parameters , therefore , it is agnostic to the specific learning task . This is clearly sub-optimal as the eminent success of deep learning has demonstrated that it is preferable to learn the representation . The more recent approaches aim at learning the representation of the persistence diagram in an end-to-end fashion . Hofer et al . ( 15 ) present the first input layer based on a parameterized family of Gaussian-like functionals , with the mean and variance learned during training . They extend their method in ( 16 ) allowing for a broader class of parameterized function families to be considered . It is quite common to have topological features of infinite persistence ( 1 ) , i.e. , features that never die . Such features are called essential and in practice are usually assigned a death time equal to the maximum filtration value . This may restrict their expressivity because it shrinks their importance relative to non-essential features . While we may be able to increase the scale sufficiently high and end up having only one trivial essential feature ( i.e. , the 0-th order persistent homology group that becomes a single connected component at a scale that is sufficiently large ) , the resulting persistence diagrams may not be the ones that best summarize the data in terms of performance on the underlying learning task . This is evident in the work by Hofer et al . ( 15 ) where the authors showed that essential features offer discriminative power . The work by Carrière et al . ( 17 ) , which introduces a network input layer the encompasses several vectorization methods , emphasizes the importance of essential features and is the first one to introduce a deep learning method incorporating extended persistence as a way to deal with them . In this paper , we approach the issue of essential features from the geometric viewpoint . We are motivated by the recent success of hyperbolic geometry and the interest in extending machine learning models to hyperbolic spaces or general manifolds . We refer the reader to the review paper by Bronstein et al . ( 18 ) for an overview of geometric deep learning . Here , we review the most relevant and pivotal contributions in the field . Nickel et al . ( 19 ; 20 ) propose Poincaré and Lorentz embeddings for learning hierarchical representations of symbolic data and show that the representational capacity and generalization ability outperform Euclidean embeddings . Sala et al . ( 21 ) propose low-dimensional hyperbolic embeddings of hierarchical data and show competitive performance on WorldNet . Ganea et al . ( 22 ) generalize neural networks to the hyperbolic space and show that hyperbolic sentence embeddings outperform their Euclidean counterparts on a range of tasks . Gulcherhe et al . ( 23 ) introduce hyperbolic attention networks which show improvements in terms of generalization on machine translation and graph learning while keeping a compact representation . In the context of graph representation learning , hyperbolic graph neural networks ( 24 ) and hyperbolic graph convolutional neural networks ( 25 ) have been developed and shown to lead to improvements on various benchmarks . However , despite this success of geometric deep learning , little work has been done in applying these methods to topological features , such as persistence diagrams . The main contribution of this paper is to bridge the gap between topological data analysis and hyperbolic representation learning . We introduce a method to represent persistence diagrams on a hyperbolic space , more specifically on the Poincare ball . We define a learnable parameterization of the Poincare ball and leverage the vectorial structure of the tangent space to combine ( in a manifoldpreserving manner ) the representations of individual points of the persistence diagram . Our method learns better task-specific representations than the state of the art because it does not shrink the relative importance of essential features . In fact , by allowing the representations of essential features to get infinitesimally close to the boundary of the Poincare ball , their distance to the representations of non-essential features approaches infinity , therefore preserving their relative importance . To the best of our knowledge , this is the first approach for learning representations of persistence diagrams in non-Euclidean spaces . 2 BACKGROUND . In this section , we provide a brief overview of persistent homology leading up to the definition of persistence diagrams . We refer the interested reader to the papers by Edelsbrunner et al . ( 1 ; 2 ) for a detailed overview of persistent homology . An overview of homology can be found in the Appendix . Persistent Homology . Let K be a simplicial complex . A filtration of K is a nested sequence of subcomplexes that starts with the empty complex and ends with K , ∅ = K0 ⊆ K1 ⊆ . . . ⊆ Kd = K. ( 1 ) A typical way to construct a filtration is to consider sublevel sets of a real valued function , f : K → R. Let a1 < · · · < ad be a sorted sequence of the values of f ( K ) . Then , we obtain a filtration by setting K0 = ∅ and Ki = f−1 ( ( −∞ , ai ] ) for 1 ≤ i ≤ d. ( 2 ) We can apply simplicial homology to each of the subcomplexes of the filtration . When 0 ≤ i ≤ j ≤ d , the inclusion Ki ⊆ Kj induces a homomorphism f i , jn : Hn ( Ki ) → Hn ( Kj ) ( 3 ) on the simplicial homology groups for each homology dimension n. We call the image of f i , jn a n-th persistent homology group and it consists of homology classes born before i that are still alive at j . A homology class α is born at Ki if it is not in the image of the map induced by the inclusion Ki−1 ⊆ Ki . Furthermore , if α is born at Ki , it dies entering Kj if the image of the map induced by Ki−1 ⊆ Kj−1 does not contain the image of α but the image of the map induced by Ki−1 ⊆ Kj does . The persistence of the homology class α is j − i . Since classes may be born at the same i and die at the same j , we can use inclusion-exclusion to determine the multiplicity of each ( i , j ) , µi , jn = β i , j−1 n − βi−1 , j−1n − βi , jn + βi−1 , jn , ( 4 ) where the n-th persistent Betti numbers βi , jn are the ranks of the images of the n-th persistent homology group , i.e. , βi , jn = rank ( im ( f i , j n ) ) , and capture the number of n-dimensional topological features that persist from i to j . By setting µi , ∞n = β i , d n − βi−1 , dn we can account for features that still persist at the end of the filtration ( j = d ) , which are known as essential features . Persistence Diagrams . Persistence diagrams are multisets supported by the upper diagonal part of the real plane and capture the birth/death of topological features ( i.e. , homology classes ) across the filtration . Definition 2.1 ( Persistence Diagram ) . Let ∆ = { x ∈ R∆ : mult ( x ) = ∞ } be the multiset of the diagonal R∆ = { ( x1 , x2 ) ∈ R2 : x1 = x2 } , where mult ( · ) denotes the multiplicity function and let R2∗ = { ( x1 , x2 ) ∈ R ∪ ( R ∪∞ ) : x2 > x1 } . Also , let n be a homology dimension and consider the sublevel set filtration induced by a function f : K → R over the complex K. Then , a persistence diagram , Dn ( f ) , is a multiset of the form Dn ( f ) = { x : x ∈ R2∗ } ∪∆ constructed by inserting each point ( ai , aj ) for i < j with multiplicity µi , jn ( or µ i , ∞ n if it is an essential feature ) . We denote the space of all persistence diagrams with D. Definition 2.2 ( Wasserstein distance and stability ) . Let Dn ( f ) , En ( g ) be two persistence diagrams generated by the filtration induced by the functions f , g : K → R , respectively . We define the Wasserstein distance wqp ( Dn ( f ) , Eg ( g ) ) = inf η ( ∑ x∈D ‖x− η ( x ) ‖pq ) 1/p , ( 5 ) where p , q ∈ N and the infimum is taken over all bijections η : Dn ( f ) → En ( g ) . The special case p = ∞ is known as Bottleneck distance . The persistence diagrams are stable with respect to the Wasserstein distance if and only if wqp ( Dn ( f ) , Eg ( g ) ) ≤ ‖f − g‖∞ . Note that a bijection η between persistence diagrams is guaranteed to exist because their cardinalities are equal , considering that , as per Def . 2.1 , the points on the diagonal are added with infinite multiplicity . The strength of persistent homology stems from the above stability definition , which essentially states that the map taking a sublevel function to the persistence diagram is Lipschitz continuous . This implies that if two objects are similar then their persistence diagrams are close .

Summary: In this paper, the authors proposed a new representation of persistence diagrams that can include `''essential features''. Essential features correspond to the intrinsic topology of the underlying space that will not die during the filtration. To include the fact that the essential features are infinitely far from other normal features in the diagram, the authors proposed to use a Poincare ball representation, which maps the diagram into a disk whose boundary is infinitely far from inside.