Cover Filtration and Stable Paths in the Mapper

Keywords : cover and nerve , Jaccard distance , stable paths in filtration , Mapper , recommender systems , explainable machine learning . 1 INTRODUCTION AND MOTIVATION . The need to rigorously seed a solution with a notion of stability in topological data analysis ( TDA ) has been addressed primarily using topological persistence ( Carlsson , 2009 ; Ghrist , 2008 ) . Persistence arises when we work with a sequence of objects built on a data set , a filtration , rather than with a single object . One line of focus of this work has been on estimating the homology of the data set . This typically manifests itself as examining the persistent homology represented as a diagram or barcode , with interpretations of zeroth and first homology as capturing significant clusters and holes , respectively ( Adams & Carlsson , 2009 ; Edelsbrunner et al. , 2002 ; Edelsbrunner & Harer , 2009 ; Zomorodian , 2005 ) . In practice it is not always clear how to interpret higher dimensional homology ( even holes might not make obvious sense in certain cases ) . A growing focus is to use persistence diagrams as a form of feature engineering to help compare different data sets rather than interpret individual homology groups ( Adams et al. , 2017 ; Chazal et al. , 2009 ; Turner et al. , 2014 ) . The implicit assumption in most such TDA applications is that the data is endowed with a natural metric , e.g. , points exist in a high-dimensional space or pairwise distances are available . In certain applications , it is also not clear how one could assign a meaningful metric . For example , memberships of people in groups of interest is captured simply as sets specifying who belongs in each group . An instance of such data is that of recommendation systems , e.g. , as used in Netflix to recommend movies to the customer . Graph based recommendation systems have been an area of recent research . Usually these systems are modeled as a bipartite graph with one set of nodes representing recommendees and the other representing recommendations . In practice , these systems are augmented in bespoke ways to accommodate whichever type of data is available . It is highly desirable to analyze the structure directly using the membership information . Another distinct TDA approach for structure discovery and visualization of high-dimensional data is based on a construction called Mapper ( Singh et al. , 2007 ) . Defined as a dual construction to a cover of the data ( see Figure 1 ) , Mapper has found increasing use in diverse applications in the past several years ( Lum et al. , 2013 ) . Attention has recently focused on interpreting parts of the 1-skeleton of the Mapper , which is a simplicial complex , as significant features of the data . Paths , flares , and cycles have been investigated in this context ( Li et al. , 2015 ; Nicolau et al. , 2011 ; Torres et al. , 2016 ) . The framework of persistence has been applied to this construction to define a multi-scale Mapper , which permits one to derive results on stability of such features ( Dey et al. , 2016 ) . At the same time , the associated computational framework remains unwieldy and still most applications base their interpretations on a single Mapper object . sampled from the circle is represented by just four vertices and four edges here . More generally , we consider higher dimensional intervals covering a subspace of Rd . But in recommendation systems , the cover is just a collection of abstract sets providing membership info ( rather than intervals over the range of function values ) . Could we define a topological construction on such abstract covers that still reveals the topology of the dataset ? We could study paths in this construction , but as the topological constructions are noisy , we would want to define a notion of stability for such paths . With this goal in mind , could we define a filtration from the abstract cover ? But unlike in the setting of , e.g. , multiscale Mapper ( Dey et al. , 2016 ) , we do not have a sequence of covers ( called a tower of covers ) —we want to work with a single cover . How do we define a filtration on a single abstract cover ? Could we prove stability results for such a filtration ? Finally , could we demonstrate the usefulness of our construction on real data ? 1.1 OUR CONTRIBUTIONS We introduce a new type of filtration defined on a single abstract cover . Termed cover filtration , our construction uses Steinhaus distances between elements of the cover . We generalize the Steinhaus distance between two elements to those of multiple elements in the cover , and define a filtration on a single cover using the generalized Steinhaus distance as the filtration index . Working with a bottleneck distance on covers , we show a stability result on the cover filtration—the cover filtrations of two covers are α/m interleaved , where α is a bound on the bottleneck distance between the covers and m is the cardinality of the smallest element in either cover ( see Theorem 3.5 ) . We conjecture that in Euclidean space , the cover filtration is isomorphic to the standard Čech filtration built on the data set . We prove the conjecture holds in dimension 1 and independently that the VietorisRips filtration completely determines the cover filtration in arbitrary dimensions ( see Section 4 ) . This filtration is quite general , and enables the computation of persistent homology for data sets without requiring strong assumptions . With real life applications in mind , we study paths in the 1-skeleton of our construction . Paths provide intu- itive explanations of the relationships between the objects that the terminal vertices represent . Our perspective of path analysis is that shortest may not be more descriptive—see Figure 2 for an illustration . Instead , we define a notion of stability of paths in the cover filtration . Under this notion , a stable path is analogous to a highly persistent feature as identified by persistent homology . We demonstrate the utility of stable paths in cover filtrations on two real life applications : a problem in movie recommendation system and Mapper . We first show how recommendation systems can be modeled using the cover filtration , and then show how stable paths within this filtration suggest a sequence of movies that represent a “ smooth ” transition from one genre to another ( Appendix C.1 ) . We then define an extension of the traditional Mapper ( Singh et al. , 2007 ) termed the Steinhaus Mapper Filtration , and show how stable paths within this filtration can provide valuable explanations of populations in the Mapper , focusing on the case of explainable machine learning ( Appendix C.2 ) . 1.2 RELATED WORK . Cavanna & Sheehy ( 2017 ) developed theory for a cover filtration , built from a cover of a filtered simplicial complex . But we work from more general covers of arbitrary spaces . We are inspired by similar goals as those of Dey et al . ( 2016 ) and Carrière et al . ( 2018 ) , who addressed the question of stability in the Mapper construction . Our goal is to provide some consistency , and thus interpretability , to the Mapper . We incorporate ideas of persistence in a different manner into our construction using a single cover , which considerably reduces the effort to generate results . While stability of persistence diagrams is a well studied topic ( Edelsbrunner & Harer , 2009 ) , we may not get stability to simplices and cycles that generate the persistence homology classes ( Bendich et al. , 2019 ) . In contrast , our definition of path stability ( Section 5 ) aligns exactly with our result on stability for cover filtrations ( Section 3 ) . We believe this is a unique feature of our method and its stability result—paths are automatically stable with respect to perturbations of input data . The multi-scale Mapper defined by Dey et al . ( 2016 ) builds a filtration on the Mapper by varying the parameters of a cover . This construction yields nice stability properties , but is unwieldy in practice and difficult to interpret . Carrière et al . develop ways based on extended persistence to automatically select a single cover that best captures the topology of the data ( Carrière et al. , 2018 ) , producing one final Mapper that is easy to interpret Methods have been developed to track populations within the Mapper by identifying interesting paths ( Kalyanaraman et al. , 2019 ) and interesting flares ( Kamruzzaman et al. , 2018 ) . Interesting paths maximize an interestingness score , and are manifested in the Mapper as long paths that track particular populations that show trending behavior . Flares capture subpopulations that diverge , i.e. , show branching behavior . In our context , we are interested in shorter paths , under the assumption that they provide the most succinct explanations for relationships between subpopulations . Our work is similar to that of Parthasarathy et al . ( 2017 ) in that they use the Jaccard Index of an observed graph to estimate the geodesic distance of the underlying graph . We take an approach more akin to persistence and make fewer assumptions about underlying data . Hence we are unable to make rigorous estimates of distances and instead provide many possible representative paths . S-paths defined by Purvine et al . ( 2018 ) are similar to stable paths when we realize that covers can be modeled as hypergraphs , and vice versa . Stable paths incorporate the size of each cover elements ( or hyperedges ) , normalizing the weights by relative size . This perspective allows us to compare different parts of the resulting structure which may have wildly difference sizes of covers . In this context , a large overlap of small elements is considered more meaningful than a proportionally small intersection of large elements . 2 COVER FILTRATIONS . We introduce the notions of distance on covers required to construct our filtrations and then provide the general definition of the cover filtration . We begin with the definition of Steinhaus distance , a generalization of the standard Jaccard distance between two sets , and further generalize it to an arbitrary collection of subsets of a cover . These generalizations take a measure µ and assumes that all sets are taken modulo differences by sets of measure 0 . We will take µ as arbitrary unless otherwise stated . Definition 2.1 ( Steinhaus Distance ( Marczewski & Steinhaus , 1958 ) ) . Given a measure µ , the Steinhaus distance between two sets A , B is dSt ( A , B ) = 1− µ ( A ∩B ) µ ( A ∪B ) = µ ( A ∪B ) − µ ( A ∩B ) µ ( A ∪B ) . This distance is also bounded on [ 0 , 1 ] , i.e. , two sets have Steinhaus distance 0 when they are equal ( as sets differing by measure 0 are identified ) and distance 1 when they do not intersect . We extend this distance from an operator on a pair of elements to an operator on a set of elements . Definition 2.2 ( Generalized Steinhaus distance ) . Define the generalized Steinhaus distance of a collection of sets { Ui } as dSt ( { Ui } ) = 1− µ ( ⋂ Ui ) µ ( ⋃ Ui ) . Much of our results that consider paths used the standard version of Jaccard distance for finite sets . But we require the Steinhaus distance for general measures to present equivalence results to Čech and VR filtrations ( Section 4 ) . Hence we present all results for the generalized Steinhaus distance . We make use of this generalized distance to associate birth times to simplices in a nerve . Given a cover , we define the cover filtration as the filtration induced from sublevel sets of the generalized Steinhaus distance function . In other words , consider a cover of the space and the nerve of this cover . For each simplex in the nerve , we assign as birth time the value of its Steinhaus distance . This filtration captures information about similarity of cover elements , and the overall structure of the cover . Definition 2.3 ( Nerve ) . A nerve of a cover U = { Ui } i∈C is an abstract simplicial complex defined such that each subset { Uj } j∈J ⊆ U , i.e. , with J ⊆ C , defines a simplex if ⋂ j∈J { Uj } 6= ∅ . In this construction , each cover element Ui ∈ U defines a vertex . Definition 2.4 ( Steinhaus Nerve ) . The Steinhaus nerve of a cover U , denoted NrvSt ( U ) , is defined as the nerve of U with each simplex assigned their generalized Steinhaus distance as weight : wσ = dSt ( { Ui | i ∈ σ } ) ∀σ ∈ U . Note that wσ < 1 by definition for every simplex σ ∈ U . We will use NrvSt when the cover U is evident from context . This can be thought of as a weighted nerve , but the weighting scheme satisfies the conditions of a filtration ( see Page 14 for the proof ) . Theorem 2.5 . The Steinhaus nerve of a cover U is a filtered simplicial complex . Following this result , we refer to the construction as the Steinhaus filtration . The only cover filtrations we use in this paper is Steinhaus filtrations , so we will use the two terms interchangeably . We could study an adaptation of cover filtration to an analog of the VR complex by building a weighted clique rank filtration from the 1-skeleton of the cover filtration ( Petri et al. , 2013 ) . This adaptation drastically reduces the number of intersection and union checks required for the construction . The weight rank clique filtration is a way of generating a flag filtration from a weighted graph ( Petri et al. , 2013 ) . We can apply this technique to build a VR analog of the cover filtration .

Summary: The paper presents an interesting filtration method to find staple maps, which proves effective in the recommendation system and explainable machine learning. The new framework is built upon a new concept termed cover filtration based on the generalized Steinhaus distance derived from Jaccard distance. A staple path discovery mechanism is then developed using this filtration based on persistent homology. Experiments on Movielense and FashionMNIST has quantitatively and qualitatively demonstrated the effectiveness of the new model. It also showcases the explainable factors with visualized samples from FashionMNIST dataset. In addition, theoretical discussion in the appendix makes the theory part of this work solid and convincing.