Landscapes of data sets and functoriality of persistent homology
Wojciech Chacholski, Alessandro De Gregorio, Nicola Quercioli, Francesca Tombari
LLandscapes of data sets and functoriality of persistenthomology
Wojciech Chach´olski a , Alessandro De Gregorio b , Nicola Quercioli c, ∗ , FrancescaTombari a a Mathematics Department, KTH, Lindstedtsvgen 25, Stockholm, 11428, Sweden b Mathematics Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino,10129, Italy c Mathematics Department, University of Bologna, Piazza di Porta S. Donato 5, Bologna,40126, Italy
Abstract
The aim of this article is to describe a new perspective on functoriality of per-sistent homology and explain its intrinsic symmetry that is often overlooked.A data set for us is a finite collection of functions, called measurements, witha finite domain. Such a data set might contain internal symmetries which areeffectively captured by the action of a set of the domain endomorphisms. Dif-ferent choices of the set of endomorphisms encode different symmetries of thedata set. We describe various category structures on such enriched data setsand prove some of their properties such as decompositions and morphism for-mations. We also describe a data structure, based on coloured directed graphs,which is convenient to encode the mentioned enrichment. We show that persis-tent homology preserves only some aspects of these landscapes of enriched datasets however not all. In other words persistent homology is not a functor onthe entire category of enriched data sets. Nevertheless we show that persistenthomology is functorial locally. We use the concept of equivariant operators tocapture some of the information missed by persistent homology.
Keywords: persistent homology, topological data analysis, equivariantoperators,
1. Introduction
In this article we give an answer to the question: what is persistent homologya functor of? ∗ Corresponding author.
Email addresses: [email protected] (Wojciech Chach´olski), [email protected] (Alessandro De Gregorio), [email protected] (Nicola Quercioli ), [email protected] (Francesca Tombari)
Submitted preprint a r X i v : . [ m a t h . A T ] M a y e will consider data sets given by finite sets of functions on a finite set X with real values. There are several important consequences of data setshaving this form. For example, they endow X with a pseudometric, enablingus to extract non-trivial homological information in form of persistent homol-ogy, one of the key invariants studied in Topological Data Analysis. A sin-gle measurement does not contain any higher non-trivial homological informa-tion. Sets of measurements however do. Thus it is essential that measure-ments, on a given set X , are grouped together to form various data sets. Inthis case persistent homology becomes a non-expansive (1-Lipschitz) function P H Φ d : Φ → Tame([0 , ∞ ) × R , Vect), assigning to each measurement in the dataset Φ a tame vector space parametrized by [0 , ∞ ) × R . It is important to noticethat the choice of a set of measurements on X affects the pseudometric definedon it. One can use this fact to change the metric on X in order to extractmore meaningful information from persistent homology. For example consider X to be a finite sample of points on a circle. If Φ consists of only one functiongiven by the x − coordinate, then the persistent homology of this measurementis trivial in degrees greater than 0. If we enlarge the data set by adding to the x − coordinate the function given by precomposing x with rotation by 90 degrees,then the persistent homology of the function x with respect to this bigger dataset gains a non-trivial homology in degree 1. This illustrates how our knowledgeof an object is affected by the number and the type of measurements done onit. Furthermore in this example we gain additional information by enlargingthe set of measurements through the action of some of the endomorphisms of X on the existing measurements. We can then take advantage of these actionsto inject geometrical features of our choice on a given data set. For exhibitingand extracting interesting homological features of data sets, such actions aretherefore important.A data set Φ is naturally equipped with an action of the monoid of itsoperations End Φ ( X ), which are endomorphisms of X preserving Φ. This actiongives the set Φ a structure of Grothendieck graph. Persistent homology turns outto be a functor indexed by this graph, rather than simply a function. Thus notonly persistent homology can be assigned to individual measurements in a dataset, but operations can be used to compare persistent homologies of differentmeasurements. That is what we call local functorial properties of persistenthomology.Persistent homology also has certain global functorial properties. Thereare various ways of representing data in the form of sets of measurements,we might choose different units or different parametrizations of a domain ofmeasurements, or we might need to focus only on certain operations such asrotations. Furthermore, the same measurements might be part of different datasets. These are some of the reasons why it is essential to be able to comparedata sets equipped with different structures. For that purpose we introducethe notion of incarnations of data sets to encode different actions, and SEOs tocompare incarnations. An incarnation of a data set Φ is an action of a subset M ⊂ End Φ ( X ). A SEO (set equivariant operator) between two incarnations(Φ , M ) and (Ψ , N ) is a pair consisting of a map T : M → N and an equivariant2with respect to T ) function α : Φ → Ψ. The use of this kind of operators forthe comparison of incarnations of data sets has been inspired by [1, 2] , whereGENEOs (group equivariant non-expansive operators) are introduced and usedfor applications to neural networks. If a SEO is geometric, then there is acomparison map between persistent homologies of the incarnations connectedby the SEO. However if a SEO is not geometric, such as the change of unitsSEO, there is no direct comparison of persistent homologies of the involvedincarnations. Such SEOs therefore exhibit diverse homological features of datasets enhancing the analysis. This suggests complementarity of these operatorsand persistent homology for a geometric analysis of a data set. Consider thechange of unit as an example. In general it is the SEO obtained by composingmeasurements in a data set by a given real valued function defined on the realnumbers. Multiplication by −
2. Data sets
For us a data set, which we regard as a point in the data landscape, isgiven by a finite set of real valued functions on some finite set X also calledmeasurements: Φ = { φ i : X → R | i = 1 , · · · , m } . We define dom(Φ), the domain of dataset Φ, to be the set X which is thedomain of the functions in Φ. The most fundamental aspect of a data set Φis that it is a set. All such data sets with different domains form a categorywith functions as morphisms. This is the most primitive landscape of datasets. The nature of our data sets however can be used to impose more intricatestructures and more meaningful landscapes. This is reminiscent of the case ofgroups. The most fundamental aspect of a group is that it is a set. However thecategory whose morphisms are group homomorphism is a much more meaning-ful landscape in which to study relationships between groups. To understandrelationships between topological groups, the category with continuous grouphomomorphisms provides an even more meaningful landscape.In this most primitive landscape however we can already perform productsand coproducts. Let φ : X → R and ψ : Y → R be functions. Define φ + ψ : X (cid:96) Y → R to be the function that maps x in X to φ ( x ) and y in Y to ψ ( y ). The coproduct of two data sets Φ and Ψ, denoted by Φ (cid:96) Ψ, is definedto be the data set given by the measurements { φ + 0 | φ ∈ Φ } ∪ { ψ | ψ ∈ Ψ } on X (cid:96) Y . Their product , denoted by Φ × Ψ, is defined to be the data set given3y the measurements { φ + ψ | φ ∈ Φ and ψ ∈ Ψ } on X (cid:96) Y . The functions:ΦΦ × Ψ Φ (cid:96) ΨΨ i n Φ φ (cid:55) → φ + p r Φ φ + ψ (cid:55) → φ p r Ψ φ + ψ (cid:55) → ψ i n Ψ ψ (cid:55) → + ψ satisfy the following universal properties, which justify the names coproductand product: • for any data set Π, and any two functions α : Φ → Π and β : Ψ → Π, thereis a unique function µ : Φ (cid:96) Ψ → Π for which µ in Φ = α and µ in Ψ = β ; • for any data set Π, and any two functions α : Π → Φ and β : Π → Ψ, thereis a unique function µ : Π → Φ × Ψ for which pr Φ µ = α and pr Ψ µ = β .Let f : R → R be a function. By composing with f , a data set Φ is trans-formed into a new data set f Φ := { f φ | φ ∈ Φ } . This operation is called changeof units along f . The symbol f − : Φ → f Φ denotes the function mapping φ to f φ . For example let f : R → R map { r ∈ R | r < } to − { r ∈ R | r ≥ } to 1. Consider X = { x , x } , two data sets { , } and {− , } given by the con-stant functions − , , X → R , and a function α : { , } → {− , } mapping1 to − f { , } = { } and f − : {− , } → f {− , } is theidentity. Thus there is no function f { , } → f {− , } making the followingdiagram commutative: { , } f { , } = { }{− , } f {− , } = {− , } f − α f − =id Consequently, for that f there is no functor F assigning to a data set Φ itschange of units f Φ along f for which f − : Φ → f Φ is a natural transformationbetween F and the identity functor. If f is invertible, then f − : Φ → f Φ isa bijection whose inverse is given by f − − . The association ( α : Φ → Ψ) (cid:55)→ (cid:0) ( f − ) α ( f − − ) : f Φ → f Ψ (cid:1) is a functor for which f − : Φ → f Φ is a naturaltransformation between this functor and the identity functor. Changing theunits along any function preserves products and coproducts i.e., f (Φ (cid:96) Ψ) isisomorphic to f (Φ) (cid:96) f (Ψ), and f (Φ × Ψ) is isomorphic to f (Φ) × f (Ψ). Asimilar reasoning is used in [7] to study brain data, in order to obtain resultsthat are invariant under transformations given by change of units with invertiblefunctions, and in [8] to study metric spaces that are isometric up to a rescalingof the distance functions.Let Φ be a data set with the domain X . By composing a function f : Y → X with the measurements in Φ, we obtain a new data set Φ f := { φf | φ ∈ Φ } withthe domain Y . This operation is called domain change along f . The symbol − f : Φ → Φ f denotes the function that maps φ to φf .4et f : Z → X and f : Z → Y be functions and f (cid:96) f : Z (cid:96) Z → X (cid:96) Y be their coproduct. For any datasets Φ and Ψ with dom(Φ) = X anddom(Ψ) = Y , the following equalities hold:(Φ (cid:97) Ψ)( f (cid:97) f ) = Φ f (cid:97) Ψ f , (Φ × Ψ)( f (cid:97) f ) = Φ f × Ψ f .
3. Metrics and persistent homology
We can think about a data set Φ as a subset Φ ⊂ R | X | . Via this inclusion Φinherits a metric induced by the infinity norm (cid:107) v (cid:107) ∞ = max {| v i |} on R | X | . Weuse the symbol (cid:107) φ − ψ (cid:107) ∞ to denote the distance between φ and ψ in Φ. Theconsidered data sets are not just sets anymore but metric spaces. Thereforenon-expansive (1-Lipschitz) functions between data sets play a special role. Forexample, let f : R → R be a function. If f is non-expansive, then so is thechange of units along f , f − : Φ → f Φ, that maps φ to f φ . The domain change − h : Φ → Φ h is non-expansive along any h . Non-expansiveness is an importantassumption to prove some stability results in [1] and it is also reasonable inapplications, since it is important that these functions between data sets do notalter the information too much.By taking all the measurements of Φ together, we can form a function[ φ · · · φ m ] : X → R m . Via this function, X inherits a pseudometric d Φ inducedby the infinity norm on R m . Explicitly d Φ ( x, y ) := max ≤ i ≤ m | φ i ( x ) − φ i ( y ) | .This metric plays a fundamental role as it permits us to extract persistent ho-mologies (see [3, 6]). In this article, persistent homology of a data set Φ withcoefficients in a field and in a given degree d assigns a vector space P H Φ d ( φ ) r,s to each measurement φ in Φ, for every ( r, s ) in [0 , ∞ ) × R , and it is defined as: P H Φ d ( φ ) r,s := H d (VR r ( φ ≤ s, d Φ )) , where: • φ ≤ s := φ − ( −∞ , s ]; • VR r ( φ ≤ s, d Φ ) is the Vietoris-Rips complex whose simplices are givenby the subsets σ ⊂ ( φ ≤ s ) of diameter not exceeding r with respect to d Φ ; • H d is the homology in degree d with coefficients in a given field.If s ≤ s (cid:48) and r ≤ r (cid:48) , then ( φ ≤ s ) ⊂ ( φ ≤ s (cid:48) ) and therefore VR r ( φ ≤ s ) ⊂ VR r (cid:48) ( φ ≤ s (cid:48) ). The linear function induced on homology by this inclusion isdenoted by: P H Φ d ( φ ) ( r,s ) ≤ ( r (cid:48) ,s (cid:48) ) : P H Φ d ( φ ) r,s → P H Φ d ( φ ) r (cid:48) ,s (cid:48) . These functions form a functor
P H Φ d ( φ ) indexed by the poset [0 , ∞ ) × R withvalues in the category of vector spaces. Since X is finite, P H Φ d ( φ ) is tame (see [11]). This means that values of P H Φ d ( φ ) are finite dimensional, and thereare two finite sequences 0 = r < r < · · · < r m in [0 , ∞ ) and s < s < · · < s l = ∞ in R such that P H Φ d ( φ ), restricted to subposets of the form[ r i , r i +1 ) × ( ∞ , s ) ⊂ [0 , ∞ ) × R and [ r i , r i +1 ) × [ s j , s j +1 ) ⊂ [0 , ∞ ) × R , isconstant. The category of such functors is denoted by Tame([0 , ∞ ) × R , Vect).Thus a data set Φ leads to a function assigning to each measurement φ itspersistent homology in a given degree: P H Φ d : Φ → Tame([0 , ∞ ) × R , Vect) . Next, recall a definition of the interleaving metric in the direction ofthe vector (0 ,
1) on Tame([0 , ∞ ) × R , Vect) (see [9]). Let P and Q be inTame([0 , ∞ ) × R , Vect). • P and Q are (cid:15) - interleaved if, for all ( r, s ) in [0 , ∞ ) × R , there are linearfunctions f s,r : P r,s → Q r,s + (cid:15) and g s,r : Q r,s → P r,s + (cid:15) making the followingdiagram commutative: P r,s P r,s +2 (cid:15) Q r,s − (cid:15) Q r,s + (cid:15) Q r,s +3 (cid:15)f s,r P ( r,s ) < ( r,s +2 (cid:15) ) f r,s +2 (cid:15) g r,s − (cid:15) Q ( r,s − (cid:15) ) < ( r,s + (cid:15) ) Q ( r,s + (cid:15) ) < ( r,s +3 (cid:15) ) g r,s + (cid:15) • d (cid:46)(cid:47) ( P, Q ) := inf { (cid:15) ∈ [0 , ∞ ) | P and Q are (cid:15) -interleaved } .The function P, Q (cid:55)→ d (cid:46)(cid:47) ( P, Q ) is an extended ( ∞ is allowed) metric on the setTame([0 , ∞ ) × R , Vect) called interleaving metric in the direction of the vector(0 , Proposition 1.
The function
P H Φ d : Φ → Tame([0 , ∞ ) × R , Vect) is non-expansive if the set Φ is equipped with ∞ -norm metric (cid:107) φ − ψ (cid:107) ∞ and the set Tame([0 , ∞ ) × R , Vect) is equipped with the interleaving metric in the directionof the vector (0 , .Proof. Let φ, ψ : X → R be measurements in Φ and (cid:15) = (cid:107) φ − ψ (cid:107) ∞ . For every s in R , the sublevel set φ ≤ s is a subset of ψ ≤ s + (cid:15) , and ψ ≤ s is a subset of φ ≤ s + (cid:15) . This translates into inclusions:VR r ( φ ≤ s, d Φ ) ⊂ VR r ( ψ ≤ s + (cid:15), d Φ ) VR r ( ψ ≤ s, d Φ ) ⊂ VR r ( φ ≤ s + (cid:15), d Φ )leading functions: f s,r : P H Φ d ( φ ) r,s → P H Φ d ( ψ ) r,s + (cid:15) g s,r : P H Φ d ( ψ ) r,s → P H Φ d ( φ ) r,s + (cid:15) . These functions provide (cid:15) interleaving between
P H Φ d ( φ ) and P H Φ d ( ψ ), giving (cid:107) φ − ψ (cid:107) ∞ ≥ d (cid:46)(cid:47) ( P H Φ d ( φ ) , P H Φ d ( ψ )).A measurement φ : X → R can be part of many data sets and its persistenthomology depends on what data set this function is part of. For example, let X = { x , x , x , x } and φ, ψ : X → R be measurements defined as follows: φ ( x ) = − φ ( x ) = φ ( x ) = 0 φ ( x ) = 1 ψ ( x ) = − ψ ( x ) = ψ ( x ) = 0 ψ ( x ) = 16he measurement φ is part of two data sets Φ = { φ } and Ψ = { φ, ψ } . Theinduced pseudometrics d Φ and d Ψ on X can be depicted by the following dia-grams where the continuous, dashed, and dotted lines indicate distance 0, 1 and2 respectively: d Φ x x x x d Ψ x x x x In this case
P H Φ1 ( φ ) r,s = 0 for all r and s , however:dim P H Ψ1 ( φ ) r,s = (cid:40) ≤ s and 1 ≤ r <
20 otherwiseTo understand persistent homology, it is therefore paramount to understandhow it changes when data sets change and here functoriality plays an essentialrole.Let Φ and Ψ be data sets consisting of measurements on X and Y re-spectively. A function α : Φ → Ψ is called geometric if there is a function f : Y → X , called a realization of α , making the following diagram commuta-tive for every φ in Φ: Y R X f α ( φ ) φ For example − f : Φ → Φ f is geometric, as it is realized by f .The commutativity of the triangle above has two consequences. First, f is non-expansive with respect to the pseudometrics d Φ on X and d Ψ on Y .Second, for s in R and φ in Φ, the subset ( α ( φ ) ≤ s ) ⊂ Y is mapped via f into( φ ≤ s ) ⊂ X , i.e., the following diagram commutes: α ( φ ) ≤ s Y R φ ≤ s X f f α ( φ ) φ The realization f induces therefore a map of Vietoris-Rips complexes and theirhomologies: f s,r : VR r ( α ( φ ) ≤ s, d Ψ ) → VR r ( φ ≤ s, d Φ ); P H Ψ d ( α ( φ )) r,s P H Φ d ( φ ) r,s H d (VR r ( α ( φ ) ≤ s, d Ψ )) H d (VR r ( φ ≤ s, d Φ )) . H d ( f r,s ) If f, f (cid:48) : Y → X are two realizations of α , then for y in Y , d Φ ( f ( y ) , f (cid:48) ( y )) = 0,hence they are points of the same simplex in the Vietoris-Rips complex, implying7hat f r,s and f (cid:48) r,s are homotopic for all r and s . Consequently, H d ( f r,s ) = H d ( f (cid:48) r,s ). The linear function H d ( f r,s ) depends therefore only on α and it isindependent on the choice of its realization f . We denote this function by: P H αd ( φ ) r,s : P H Ψ d ( α ( φ )) r,s → P H Φ d ( φ ) r,s . These functions are natural in r and s and induce a morphism in the categoryTame([0 , ∞ ) × R , Vect) between persistent homologies:
P H αd ( φ ) : P H Ψ d ( α ( φ )) → P H Φ d ( φ ) . If α : Φ → Ψ and β : Ψ → Ξ are geometric functions realized by f : Y → X and g : Z → Y , then the composition βα : Φ → Ξ is also geometric, andrealized by the composition f g : Z → X . Consequently, for every measurement φ in Φ, P H βαd ( φ ) = P H αd ( φ ) P H βd ( α ( φ )), that assures the commutativity of thediagram: P H Ξ d ( βα ( φ )) P H Ψ d ( α ( φ )) P H Φ d ( φ ) P H βd ( α ( φ )) P H βαd ( φ ) P H αd ( φ ) For any α : Φ → Ψ, taking persistent homology leads to two functions on Φ:Tame([0 , ∞ ) × R , Vect)Φ Ψ Tame([0 , ∞ ) × R , Vect)
P H Φ d α P H Ψ d These functions rarely coincide. However, when α is geometric, we can use themorphisms P H αd ( φ ) : P H Ψ d ( α ( φ )) → P H Φ d ( φ ) to compare the values of these twofunctions on Φ. For non-geometric α , we are not equipped with such comparisonmorphisms and there is no reason for such a comparison to even exist. Forexample, consider the change of unit along the function f : R → R , f ( x ) := − x .Then f − : Φ → f Φ is an isomorphism. In this case
P H Φ d ( φ ) r,s := H d (VR r ( φ ≤ s, d Φ )) ( f − ) P H f Φ d ( φ ) = H d (VR r ( φ ≥ − s, d Φ )) . Thus
P H Φ d encodes information about sub-level sets of the measurements in Φand ( f − ) P H f Φ d encodes information about super-level sets of the measurements.These persistent homologies encode therefore the same information as the socalled extended persistence (see [4, 10]).
4. Actions
To describe symmetries of a data set Φ with domain X , we consider op-erations on X that convert measurements into measurements. By definition a8- operation is a function g : X → X such that, for every measurement φ in Φ,the composition φg also belongs to Φ. If g : X → X is such an operation, then,for all φ and ψ in Φ: (cid:107) φ − ψ (cid:107) ∞ = max x ∈ X | φ ( x ) − ψ ( x ) | ≥ max x ∈ im( g ) | φ ( x ) − ψ ( x ) | = (cid:107) φg − ψg (cid:107) ∞ . Thus the function − g : Φ → Φ that maps φ to φg is non-expansive.The composition of Φ-operations is again a Φ-operation, and the identityfunction id X is also a Φ-operation. In this way the set of Φ-operations withthe composition becomes a unitary monoid, called the structure monoid ofΦ, and denoted by:End Φ ( X ) = { g : X → X | φg ∈ Φ for every φ ∈ Φ } ⊂ End( X ) . A Φ-operation g is invertible if there is a Φ-operation h such that gh = hg = id X .Since Φ is finite, a Φ-operation is invertible if and only if it is a bijection. Theircollection is denoted by:Aut Φ ( X ) = { g : X → X | g is a bijection, and φg ∈ Φ for every φ ∈ Φ } . With the composition operation, Aut Φ ( X ) becomes a group for which the in-clusion Aut Φ ( X ) ⊂ End Φ ( X ) is a monoid homomorphism.A data set Φ is equipped with an associative right action:Φ × End Φ ( X ) → Φ , ( φ, g ) (cid:55)→ φg. Thus Φ is not just a set, but a set with an action of the monoid End Φ ( X ). Toencode the symmetries of Φ induced by this action, we consider its incarnations.An incarnation of Φ is a choice of a subset M ⊂ End Φ ( X ) (in general, notnecessarily a submonoid). An incarnation is denoted as a pair (Φ , M ). We thinkabout M as an additional structure on Φ. An incarnation of the form (Φ , M )is called an M -incarnation. We also refer to an M -incarnation as an M -action.The choice of an M -action on Φ encodes certain symmetries of Φ. Differentchoices of M can encode different symmetries. This flexibility is important inapplications. For example in data sets that represent images, we might wantto focus on rotational symmetries, so we may use an appropriate action on thedata set to inject the corresponding geometry. The incarnation (Φ , End Φ ( X ))is an example of a incarnation called universal.An incarnation (Φ , M ) is called a monoid incarnation if M ⊂ End Φ isa submonoid, and our convention here is that all such submonoids contain theidentity element. If (Φ , M ) is an incarnation, we use the symbol (Φ , (cid:104) M (cid:105) ) to de-note the monoid incarnation where (cid:104) M (cid:105) ⊂ End Φ ( X ) is the submonid generatedby M .If a submonoid M ⊂ End Φ ( X ) is a group, then (Φ , M ) is called a group in-carnation . The incarnation (Φ , Aut Φ ( X )) is an example of a group incarnationcalled universal.Let (Φ , M ) be an incarnation for which any element g in M is a bijection.Such incarnations are called group-like . For group like incarnations (Φ , M )9he finiteness implies that the monoid (cid:104) M (cid:105) is in fact a subgroup of Aut Φ ( X ).Thus any group-like incarnation (Φ , M ) leads to a group incarnation (Φ , (cid:104) M (cid:105) ).Let (Φ , M ) be an incarnation. For a subset Ω ⊂ Φ, the symbol Ω M denotesthe set of all the measurements in Φ which either belong to Ω or are of theform ωg · · · g k , for some ω in Ω and some sequence of elements g , . . . g k in M .If Ω M = Φ, then Ω is said to generate the incarnation (Φ , M ). In the case(Φ , M ) is a monoid incarnation, then any element in Ω M is of the form ωg forsome ω in Ω and g in M . Note that Ω M = Ω (cid:104) M (cid:105) for every incarnation (Φ , M ).If ψ belongs to φM := { φ } M , then ψ is said to be a deformation of φ .If (Φ , M ) is a group incarnation, then the relation of being a deformation isan equivalence relation. For a general incarnation however being a deformationcan fail to be even a symmetric relation. Two measurements in Φ are said tobe connected if they are related by the equivalence relation generated by therelation of being a deformation. The symbol Φ /M denotes the partition of Φinduced by this equivalence relation. We refer to Φ /M as the quotient of theincarnation (Φ , M ). The partitions Φ /M and Φ / (cid:104) M (cid:105) coincide. If (Φ , M ) isa group incarnation, then Φ /M coincide with the orbit partition of the usualgroup action of M on Φ.Let (Φ , M ) be an incarnation. For a measurement ψ in Φ, the symbol [ ψ ]denotes the block in Φ /M containing ψ . Explicitly, [ ψ ] is the subset of Φconsisting of all the measurements connected to ψ . Note that, for all g in M , if φ is connected to ψ , then φg is also connected to ψ . We thus have the followinginclusions: M End Φ ( X )End [ ψ ] ( X ) End( X )The M incarnation ([ ψ ] , M ) of the block [ ψ ], given by the above inclusions M ⊂ End [ ψ ] , is called a block incarnation of (Φ , M ). In this way we canthink about [ ψ ] and ([ ψ ] , M ) as a new data set.An incarnation (Φ , M ) is called transitive if all the elements in Φ are con-nected to each other. For example, let M be a finite submonoid of End( X ). Fora given function φ : X → R , define a data set φM := { φg | g ∈ M } to consistof all functions of the form x (cid:55)→ φ ( g ( x )) for all g in M . Then every g : X → X in M is a φM -operation. The obtained incarnation ( φM, M ) is transitive. Anytransitive group incarnation is of such form. For all measurements φ in anyincarnation (Φ , M ), the block incarnation ([ φ ] , M ) is transitive. Any transitiveincarnation is of this form.Let (Φ , M ) be an incarnation. A subset Ω ⊂ Φ is called independent if noelement in Ω is a deformation of any other element in Ω, explicitly: ω (cid:54)∈ ω (cid:48) M for all ω (cid:54) = ω (cid:48) in Ω.A basis of (Φ , M ) is an independent subset Ω ⊂ Φ such that Ω M = Φ (Ωgenerates (Φ , M )).Two measurements ψ and φ are called indistinguishable if ψ is a defor-mation of φ and φ is a deformation of ψ . If (Φ , M ) is a group incarnation, then10 and φ are indistinguishable if and only if ψ = φg for some g in M , i.e., if ψ is a deformation of φ . Proposition 2. Every incarnation has a basis. Let Ω , Ω (cid:48) ⊂ Φ be two bases of an incarnation (Φ , M ) . Then there is abijection σ : Ω → Ω (cid:48) such that ω and σ ( ω ) are indistingishable for every ω in Ω .Proof. (1): Let (Φ , M ) be an incarnation. Choose Ω ⊂ Φ to be an independentsubset for which Ω M is maximal. Existence of Ω is guaranteed by finiteness ofΦ. We claim that Ω M = Φ and hence Ω is a basis. If this is not the case, let ψ be in Φ \ Ω M . Define Ω (cid:48) = { ψ } ∪ { ω ∈ Ω | ω (cid:54)∈ { ψ } M } . Then Ω (cid:48) M containsΩ and hence Ω M . It also contains ψ . Since Ω (cid:48) is independent, we would obtaina contradiction to the maximality assumption about Ω M , and thus the claimholds.(2): Let ω be in Ω. Since Ω M = Φ = Ω (cid:48) M , there is ω (cid:48) in Ω (cid:48) such that ω ∈ ω (cid:48) M .Let ω in Ω be such that ω (cid:48) ∈ ω M . Then ω ∈ ω (cid:48) M ⊂ ω M , and hence ω = ω by the independence of Ω. The desired bijection is then given by ω (cid:55)→ ω (cid:48) .According to Proposition 2, any two bases of an incarnation have the samenumber of elements. We define the dimension of an incarnation to be the car-dinality of its bases. For example a transitive group incarnation has dimension1. In fact for a transitive group incarnation any single measurement forms abasis. More generally, the dimension of a group incarnation (Φ , M ) equals thecardinality of Φ /M . In this case Ω ⊂ Φ is a basis if and only if, for every blockΨ in Φ /M , the intersection Ω ∩ Ψ has only one element. Since being a basisdepends only on the monoid (cid:104) M (cid:105) , the dimension of a group-like incarnation(Φ , M ) equals also the cardinality of Φ /M , and similarly a subset Ω ⊂ Φ is abasis if and only if, for every block Ψ in the partition Φ /M , the intersectionΩ ∩ Ψ has only one element.The dimension of a transitive monoid incarnation can be bigger than 1. Forexample, let X = { x , x , x } and consider functions φ , φ , φ : X → R and g , g , g : X → X defined as follows: φ ( x ) = 2 φ ( x ) = 2 φ ( x ) = 1 g ( x ) = x g ( x ) = x g ( x ) = x φ ( x ) = 2 φ ( x ) = 2 φ ( x ) = 2 g ( x ) = x g ( x ) = x g ( x ) = x φ ( x ) = 3 φ ( x ) = 2 φ ( x ) = 2 g ( x ) = x g ( x ) = x g ( x ) = x The compositions g i g j and φ i g j are described by the following tables: g g g g g g g g g g g g g g g g g g φ φ φ φ φ φ φ φ φ φ φ φ Thus the functions g , g , and g are Φ := { φ , φ , φ } -operations. Furthermorethe subset M := { id , g , g , g } ⊂ End Φ ( X ) is a submonoid. The incarnation11Φ , M ) is a transitive monoid incarnation. Since the set { φ , φ } is independentand generates (Φ , M ), it is a basis. Thus (Φ , M ) is an example of a transitivemonoid incarnation of dimension 2.
5. Nirvana
To compare incarnations of various data sets we are going to use SEOs (setequivariant operators). A
SEO from an incarnation (Φ , M ) to an incarnatiopn(Ψ , N ), denoted as ( α, T ) : (Φ , M ) → (Ψ , N ), is a pair of functions ( α : Φ → Ψ , T : M → N ) for which the following diagram commutes:Φ × M Φ × End Φ ( X ) ΦΨ × N Ψ × End Ψ ( Y ) Ψ α × T action α action Explicitly, for φ in Φ and g in M , it holds α ( φg ) = α ( φ ) T ( g ). This implies thatfor φ in Φ and every sequence of elements g , . . . , g k in M , it holds: α ( φg · · · g k ) = α ( φ ) T ( g ) · · · T ( g k ) . Be however aware that in general there may not be a homomorphism T : (cid:104) M (cid:105) →(cid:104) N (cid:105) of monoids which extends T : M → N and makes the following diagramcommutative:Φ × M Φ × (cid:104) M (cid:105) Φ × End Φ ( X ) ΦΨ × N Ψ × (cid:104) N (cid:105) Ψ × End Ψ ( Y ) Ψ α × T α × T action α action A SEO between monoid incarnations ( α, T ) : (Φ , M ) → (Ψ , N ) is called aMEO (monoid equivariant operators) if T : M → N is a monoid homomorphism.A MEO between group incarnations is also called a GEO (group equivariantoperators).Let ( α , T ) : (Φ , M ) → (Φ , M ) and ( α , T ) : (Φ , M ) → (Φ , M ) beSEOs. Then the compositions ( α α , T T ) form a SEO. Furthermore the pair(id Φ , id M ) : (Φ , M ) → (Φ , M ) is also a SEO. The composition of SEOs is anassociative operation and defines a category structure on the collection of dataset incarnations with SEOs as morphisms. This category is called Nirvana .A SEO ( α, T ) : (Φ , M ) → (Ψ , N ) is an isomorphism if and only if both ofthe functions α and T are bijections. Isomorphisms preserve independence andbeing a basis: Proposition 3. If ( α, T ) : (Φ , M ) → (Ψ , N ) is an isomorphism, then a subset Ω ⊂ Φ is independent or a basis if and only if its image α (Ω) ⊂ Ψ is independentor a basis. roof. Assume α and T are bijections. This assumption imply that φ belongsto φ M if and only if α ( φ ) belongs to α ( φ ) N . It follows that two elements inΦ are (in)dependant if and only if their images via α are (in)dependent in Ψ.By the same argument, Ω M = Φ if and only α (Ω) T ( M ) = α (Φ).According to Proposition 3 two isomorphic incarnations have the same di-mension.The universal incarnations (Φ , End Φ ( X )) and (Φ , Aut Φ ( X )) are special inthe category Nirvana. For any (Φ , M ), the pair (id , i : M (cid:44) → End Φ ( X )) de-fines a SEO (Φ , M ) → (Φ , End Φ ( X )) called canonical . If (Φ , M ) is a groupincarnation, then the pair (id , i : M (cid:44) → Aut φ ( X )) defines a GEO (Φ , M ) → (Φ , Aut φ ( X )) also called canonical.The rest of this section is devoted to present three ways of constructingSEOs. Change of units.
Choose a function f : R → R . For any incarnation (Φ , M ),consider the data set f Φ (see Section 2). If g is a Φ-operation, then it is alsoa f Φ-operation. Thus there is an inclusion End Φ ( X ) ⊂ End f Φ ( X ), which isan equality if f is invertible, therefore we have an incarnation ( f Φ , M ). If(Φ , M ) is a monoid or a group incarnation, then so is ( f Φ , M ). The pair( f − , id M ) : (Φ , M ) → ( f Φ , M ) is a SEO called the change of units along f .Assume f is invertible. If ( α, T ) : (Φ , M ) → (Ψ , N ) is a SEO, then the pair offunctions (cid:0) ( f − ) α ( f − − ) , T (cid:1) forms a SEO between ( f Φ , M ) and ( f Ψ , N ). Theassignment ( α, T ) (cid:55)→ (( f − ) α ( f − − ) , T ) is a self functor C( f ) of Nirvana alsocalled the change of units along f . It is an equivalence of categories. Indeed,C( f )C( f − )((Φ , M )) = C( f )( f − Φ , M ) = (Φ , M )C( f )C( f − )(( α, T )) = C( f )(( f − − ) α ( f − ) , T ))= (( f − )( f − − ) α ( f − )( f − − ) , T ) = ( α, T ) . The same holds for C( f − )C( f ), hence C( f ) is an equivalence of categories.The SEOs ( f − , id M ) : (Φ , M ) → ( f Φ , M ), for all incarnations (Φ , M ), form anatural transformation between the identity functor on Nirvana and the changeof units along f functor. Domain change.
Let (Φ , M ) and (Ψ , N ) be incarnations of data sets consist-ing of measurements on X and Y respectively. A SEO ( α, T ) : (Φ , M ) → (Ψ , N )is called geometric if there is a function f : Y → X , called a realization of( α, T ), making the following diagram commutative for every φ in Φ and g in M : Y Y R X X T ( g ) f α ( φ ) fg φ For example, let (Φ , M ) be an incarnation of a data set consisting of measure-ments on X . Then the SEO (id Φ , id M ) : (Φ , M ) → (Φ , M ) is geometric. Theidentity function id X : X → X is one of its realizations.13et Y ⊂ X be M -invariant: g ( y ) belongs to Y for all y in Y and g in M . Consider the data set Φ | Y given by the domain change along the inclusion Y ⊂ X . The restriction of g to Y is a Φ | Y -operation for every g in M . Weuse the symbol T Y : M → End Φ | Y ( Y ) to denote the function that maps g in M to the restriction of g to Y . The incarnation (Φ | Y , T Y ( M )) is called the restriction of (Φ , M ) to the invariant subset Y . The pair (Φ (cid:16) Φ | Y , T Y )forms a geometric SEO. The inclusion i Y : Y (cid:44) → X is one of its realizations.Let f : Y → X be a bijection. Consider the data set Φ f . For any g in M ,the function f − gf : Y → Y is a Φ f -operation. Define T : M → End Φ f ( Y )to map g in M to f − gf . The incarnation (Φ f, T ( M )) is called the domainchange of (Φ , M ) along f . The pair ( − f : Φ → Φ f, T ) forms a geometric SEOand f : Y → X is one of its realizations. Extending from a basis.
SEOs can be effectively constructed using bases.
Proposition 4.
Let (Φ , M ) and (Ψ , N ) be incarnations and Ω be a basis of (Φ , M ) . Then two SEOs ( α, T ) , ( α (cid:48) , T (cid:48) ) : (Φ , M ) → (Ψ , N ) are equal if and onlyif T = T (cid:48) and α ( ω ) = α (cid:48) ( ω ) for any ω in Ω .Proof. The only non trivial thing to prove in the statement of the propositionis that α = α (cid:48) when their restrictions to Ω are equal. Assume T = T (cid:48) and α ( ω ) = α (cid:48) ( ω ) for any ω in Ω. Since Ω generates (Φ , M ), any element in Φ is ofthe form φ = ωg · · · g k for some ω in Ω and a sequence of elements g , . . . , g k in M . The assumption and the fact that ( α, T ) and ( α (cid:48) , T ) are SEOs, imply: α ( φ ) = α ( ωg · · · g k ) = α ( ω ) T ( g ) · · · T ( g k ) == α (cid:48) ( ω ) T ( g ) · · · T ( g k ) = α (cid:48) ( ωg · · · g k ) = α (cid:48) ( φ ) . Consequently α = α (cid:48) .According to Proposition 4, a SEO is determined by what it does on a basisof the domain. This is analogous to a linear map between vector spaces beingdetermined by its values on a basis. However unlike for linear maps, we cannotfreely map elements of a basis of an incarnation to obtain a SEO. To obtaina SEO certain relations have to be preserved. Let (Φ , M ) be an incarnation.A relation between measurements φ and ψ in Φ is by definition a pair ofsequences (( g , . . . , g k ) , ( h , . . . , h l )) of elements in M for which the followingequality holds: φg · · · g k = ψh · · · h l . Proposition 5.
Let (Φ , M ) and (Ψ , N ) be incarnations, Ω be a basis of (Φ , M ) ,and ¯ α : Ω → Ψ and T : M → N be functions. Assume that for every relation (( g , . . . , g k ) , ( h , . . . , h l )) between any twoelements ω , ω (cid:48) in Ω , the pair (( T ( g ) , . . . , T ( g k )) , ( T ( h ) , . . . , T ( h l ))) isa relation between α ( ω ) and α ( ω (cid:48) ) in Ψ . Under this assumption, thereis a unique SEO ( α, T ) : (Φ , M ) → (Ψ , N ) for which the restriction of α : Φ → Ψ to Ω is ¯ α . Assume (Φ , M ) and (Ψ , N ) , are monoid incarnations, T is a monoid ho-momorphism, and if ωg = ω (cid:48) h for some ω, ω (cid:48) in Ω and g, h in M , then α ( ω ) T ( g ) = α ( ω (cid:48) ) T ( h ) . Under these assumptions, there is a unique MEO ( α, T ) : (Φ , M ) → (Ψ , N ) for which the restriction of α : Φ → Ψ to Ω is ¯ α . Assume (Φ , M ) and (Ψ , N ) are group incarnations, T is a group homomor-phism, and if ω = ωg , for some ω in Ω and g in M , then α ( ω ) = α ( ω ) T ( g ) .Under these assumptions, there is a unique GEO ( α, T ) : (Φ , M ) → (Ψ , N ) for which the restriction of α : Φ → Ψ to Ω is ¯ α .Proof. Since the proofs are analogous, we illustrate only how to show statement(2). For every φ in Φ, there exist (not necessarily unique) ω in Ω and g in M such that φ = ωg . The assumption implies that the expression α ( ω ) T ( g )depends on φ and not on the choices of ω and g for which φ = ωg . Thusby mapping φ in Φ to α ( ω ) T ( g ) in Ψ, we obtain a well defined function alsodenoted by α : Φ → Ψ. The pair ( α, T ) is the desired MEO. The uniqueness isa consequence of Proposition 4.For example assume (Φ , M ) is a transitive group incarnation and (Ψ , N ) isa group incarnation. Choose an element ω in Φ. Recall that any such elementis a basis of (Φ , M ). Fix a group homomorphism T : M → N . Then anyGEO ( α, T ) : (Φ , M ) → (Ψ , N ) is uniquely determined by the element α ( ω ) inΨ. Thus by choosing a basis element ω in Φ, we can identify the collection ofGEOs of the form ( α, T ) : (Φ , M ) → (Ψ , N ) with a subset of Ψ. To describethis subset explicitly, we apply Proposition 5.2. It states that there is a GEO( α, T ) : (Φ , M ) → (Ψ , N ) (necessarily unique) such that α ( ω ) = ψ if and onlyif the following implication holds: if ω = ωg , then ψ = ψT ( g ). The collection M ω := { g ∈ M | ω = ωg } is the isotropy subgroup of ω consisting of all theelements in M that fix ω . Thus GEOs of the form ( α, T ) : (Φ , M ) → (Ψ , N )can be identified with the subset of all the elements in Ψ whose isotropy groupcontains T ( M ω ).
6. Decomposition
Let (Φ , M ) be an incarnation of a data set Φ. Consider its quotient Φ /M ,which is a partition of Φ, and the block incarnations (Ψ , M ) for every block Ψin Φ /M (see Section 4). Let X be the domain of Φ. Recall that the domainof the data set (cid:96) Ψ ∈ Φ /M Ψ is given by the disjoint union (cid:96) Ψ ∈ Φ /M X , and thatthis data set consists of functions (cid:96) Ψ ∈ Φ /M X → R whose restrictions to allbut one summands X in (cid:96) Ψ ∈ Φ /M X is the 0 function and the restriction to theremaining summand belongs to the corresponding block of the partition Φ /M .Define: M (cid:48) = (cid:97) Ψ ∈ Φ /M g : (cid:97) Ψ ∈ Φ /M X → (cid:97) Ψ ∈ Φ /M X | g ∈ M . M (cid:48) ⊂ End (cid:96) Ψ ∈ Φ /M Ψ ( (cid:96) Ψ ∈ Φ /M X ). We call ( (cid:96) Ψ ∈ Φ /M Ψ , M (cid:48) ) the diagonalincarnation. Define T : M → M (cid:48) to map g : X → X in M to (cid:96) Ψ ∈ Φ /M g in M (cid:48) .Define α : Φ → (cid:96) Ψ ∈ Φ /M Ψ to map φ to the function (cid:96) Ψ ∈ Φ /M X → R whoserestriction to the summand X corresponding to the block [ φ ] is φ and that mapsall other summands to 0. Note that both of the functions α and T are bijections.Furthermore they form a SEO between (Φ , M ) and ( (cid:96) Ψ ∈ Φ /M Ψ , M (cid:48) ). Proposition 6.
The SEO ( α, T ) : (Φ , M ) → ( (cid:96) Ψ ∈ Φ /M Ψ , M (cid:48) ) is an isomor-phism.
7. Grothendieck graphs
In this section we explain a convenient data structure to encode incarnationsof data sets.A
Grothendieck graph is a triple (
V, M, E ) consisting of a finite set V whose elements are called vertices, a finite set M whose elements are calledcolors or operations, and a subset E ⊂ V × M × V whose elements are callededges, such that, for every vertex v in V , the following composition is a bijection:( { v } × M × V ) ∩ E E V × M × V M. pr M This condition assures that, for every v in V and g in M , there is a uniqueelement in V , denoted by vg , such that ( v, g, vg ) is an edge in E . For examplelet (Φ , M ) be an incarnation of a data set Φ. Define: E Φ ,M := { ( φ, g, ψ ) ∈ Φ × M × Φ | φg = ψ } . Then the triple (Φ , M, E Φ ,M ) is a Grothendieck graph. We think about thisgraph as a convenient data structure representing the incarnation (Φ , M ).Grothendieck graphs are also convenient to represent SEOs. Define a mor-phism between Grothendieck graphs ( V, M, E ) and (
W, N, F ) to be apair of functions α : V → W and T : M → N such that, if ( v, g, w ) belongsto E , then ( α ( v ) , T ( g ) , α ( w )) belongs to F . Such a morphism is denoted as( α, T ) : ( V, M, E ) → ( W, N, F ). Componentwise composition defines a cate-gory structure on the collection of Grothendieck graphs and we use the symbolGGraph to denote this category. If ( α, T ) : (Φ , M ) → (Ψ , N ) is a SEO, then( α, T ) : (Φ , M, E Φ ,M ) → (Ψ , N, E Ψ ,N ) is a morphism between the associatedGrothendieck graphs. By assigning to a SEO ( α, T ) the graph morphism givenby the same pair ( α, T ), we obtain a fully faithful functor from the categoryNirvana to GGraph.Grothendieck graphs can also be used to encode pseudometric informationon incarnations. A pseudometric on a Grothendieck graph ( V, M, E ) is a pseu-dometric d on V such that d ( v, w ) ≥ d ( vg, wg ) for all v and w in V , and g in M . For example, the pseudometric (cid:107) φ − ψ (cid:107) ∞ on Φ is a pseudometric on thegraph (Φ , M, E Φ ,M ). 16 Grothendieck graph ( V, M, E ) is said to be compatible with a monoidstructure on M if ( v, , v ) is in E , and whenever ( v , g , v ) and ( v , g , v )belong to E , then so does ( v , g g , v ). In this case the composition operationgiven by the association ( v , g , v )( v , g , v ) (cid:55)→ ( v , g g , v ) defines a categorystructure, denoted by Gr M V , with V as the set of objects and E as the set ofmorphisms. This category is a familiar Grothendieck construction [5, 12]. Forexample, the Grothendieck graph associated with a monoid incarnation (Φ , M )is compatible with the monoid structure on M . We think about Gr M Φ as anadditional structure on the data set Φ: objects are the measurements in Φ,morphisms are triples ( φ, g, φg ), where φ is in Φ, g is in M , and the compositionof ( φ, g, φg ) and ( φg, h, φgh ) is given by ( φ, gh, φgh ).A contravariant functor indexed by a Grothendieck graph ( V, M, E ) withvalues in a category C , denoted as P : ( V, M, E ) → C , is by definition a sequenceof objects { P ( v ) | v ∈ V } and a sequence of morphisms { P ( v , g, v ) : P ( v ) → P ( v ) | ( v , g, v ) ∈ E } in C subject to: if ( v , g , v ), ( v , g , v ), and ( v , h, v )are edges in E , then P ( v , h, v ) = P ( v , g , v ) P ( v , g , v ). If ( V, M, E ) iscompatible with a monoid structure on M , then a contravariant functor indexedby ( V, M, E ) is simply a contravariant functor indexed by the category Gr M V .Let (Φ , M ) be an incarnation of a data set Φ consisting of measurementson X , and (Φ , M, E Φ ,M ) be the associated Grothendieck graph. For every g in M , the function − g : Φ → Φ, mapping φ to φg , is geometric and realized by g : X → X (see Section 3). Persistent homology leads therefore to the followingcollections of objects and morphisms in Tame([0 , ∞ ) × R , Vect) as explained inSection 3: (cid:8)
P H Φ d ( φ ) | φ ∈ Φ (cid:9) , (cid:8) P H − gd ( φ ) : P H Φ d ( φg ) → P H Φ d ( φ ) | ( φ, g, φg ) ∈ E Φ ,M (cid:9) . These sequences form a functor
P H Φ d : (Φ , M, E Φ ,M ) → Tame([0 , ∞ ) × R , Vect)also referred to as the persistent homology functor of the incarnation (Φ , M ).Let ( α, T ) : (
W, N, F ) → ( V, M, E ) be a morphism and P : ( V, M, E ) → C be a functor. The following sequences of objects and morphisms in C form acontravariant functor denoted by P ( α, T ) : ( W, N, F ) → C and called the com-position of ( α, T ) with P : { P ( α ( v )) | v ∈ V } , { P ( w , g, w ) : P ( α ( w )) → P ( α ( w )) | ( w , g, w ) ∈ F } . For example, let (id Φ , i ) : (Φ , M ) → (Φ , End Φ ( X )) be the canonical SEO (seeSection 5). Consider the induced morphism of the associated Grothendieckgraphs: (id Φ , i M ) : (Φ , M, E Φ ,M ) → (Φ , End Φ ( X ) , E Φ , End Φ ( X ) ) . Consider also the persistent homology of the universal incarnation:
P H Φ d : (Φ , End Φ ( X ) , E Φ , End Φ ( X ) ) → Tame([0 , ∞ ) × R , Vect) . , M ): P H Φ d : (Φ , M, E Φ ,M ) → Tame([0 , ∞ ) × R , Vect) . In this way we obtain a commutative diagram:(Φ , End Φ ( X ) , E Φ , End Φ ( X ) )(Φ , M, E Φ ,M ) Tame([0 , ∞ ) × R , Vect)
P H Φ d (id Φ ,i M ) P H Φ d Such a commutativity does not hold for arbitrary SEOs. Consider a SEO( α, T ) : (Φ , M ) → (Ψ , N ). We can form two functors indexed by the graph(Φ , M, E Φ ,M ): Tame([0 , ∞ ) × R , Vect)(Φ , M, E Φ ,M ) (Ψ , N, E Ψ ,N ) Tame([0 , ∞ ) × R , Vect)
P H Φ d α P H Ψ d These functors rarely coincide. However, in the case ( α, T ) is geometric, themorphisms
P H αd ( φ ) : P H Ψ d ( α ( φ )) → P H Φ d ( φ ) (see Section 3), for all φ in Φ,form a natural transformation.
8. Conclusions
In the following figure we give a graphical representation of some of the con-cepts introduced in this article. Data sets can be equipped with three structures:a pseudometric, an incarnation describing an action, and a Grothendieck graph.We imagine Nirvana as the landscape of all possible incarnations of data sets,represented by the shaded region in the following figure. Each point in Nirvanahas a lot of internal structure allowing the extraction of persistent homology.In this landscape the black arrows represent geometric SEOs and the grey onesnon-geometric SEOs. Recall that geometric SEOs enable us to compare relevantpersistent homology. Non-geometric SEOs contain complementary information.
Acknowledgements
The research carried out by N.Q. was partially supported by GNSAGA-INdAM (Italy). A.D. has been supported by the SmartData@PoliTO center onBig Data and Data Science and by the Italian MIUR Award “Dipartimento diEccellenza 2018-202” - CUP: E11G18000350001. The work of W.C. and F.T.was partially supported by the Wallenberg AI, Autonomous System and Soft-ware Program (WASP) funded by Knut and Alice Wallenberg Foundation. Thework of W.C. was also in part funded by VR and G¨oran Gustafsson foundation.18 ( , k k )( , M )( , M , E ,M )
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