OOn C -persistent homology and trees Daniel Perez ∗ D´epartement de math´ematiques et applications, ´Ecole normale sup´erieure, CNRS, PSLUniversity, 75005 Paris, France Laboratoire de math´ematiques d’Orsay, Universit´e Paris-Saclay, CNRS, 91405 Orsay,France DataShape, Centre Inria Saclay, 91120 Palaiseau, France
December 8, 2020
Abstract
The study of the topology of the superlevel sets of stochastic pro-cesses on [0 ,
1] in probability led to the introduction of R -trees whichcharacterize the connected components of the superlevel-sets. We pro-vide a generalization of this construction to more general deterministiccontinuous functions on some path-connected, compact topological set X and reconcile the probabilistic approach with the traditional meth-ods of persistent homology. We provide an algorithm which functoriallylinks the tree T f associated to a function f and study some invariantsof these trees, which in 1D turn out to be linked to the regularity of f . A problem of interest in topological data analysis (TDA) is the study ofthe barcode of random functions. The easiest case-study of this problem isto look at some stochastic process S : [0 , → R , and study the barcodesof its sample paths. Studying the level sets and superlevel sets of stochasticprocesses in this particular setting has been widely studied by probabilistsfrom different approaches [3, 4, 13, 15, 16, 18, 24], although some hedgewayin understanding the allure of the barcode of Brownian motion has alsobeen made by the TDA community [10]. When looking at the probabilityliterature, one very quickly encounters trees which can be constructed froma deterministic function f : [0 , → R , and which exactly describe thetopology of the (super)level sets of the function. This construction wasoriginally thought of by Le Gall [15], who was able to describe fine propertiesof the sample paths of L´evy stochastic processes in [16] using this formalism. ∗ Email: [email protected] a r X i v : . [ m a t h . A T ] D ec his work aims at reconciling the tree approach taken by the probabilitytheory community with objects relevant to TDA, i.e. persistent diagramsor barcodes. Trees and barcodes turn out to be closely related. In fact,the link between the two is functorial. Establishing this connection hasthe merit to – among others – allow us to use the work of the probabilitytheory community to further our understanding of the allure of barcodesof random processes. More importantly, it enlarges the TDA toolbox bylooking at persistence diagrams as being themselves metric objects (trees),which yields new invariants for barcodes which are useful in a C -setting,such as the upper-box dimension. These trees and their construction have been extensively studied over[0 ,
1] [13, 15, 16, 24]. In particular, there are two results to keep in mind forthe rest of this paper:1. There is a link between the metric invariants of trees and the regularityof the function f from which they stem (theorem 3.7), this result wasestablished by Picard for functions f : [0 , → R [24].2. The L ∞ -stability of trees, with respect to the natural distance on thespace of compact trees : the Gromov-Hausdorff distance (theorem 4.5).This is a result of Le Gall for functions on [0 ,
1] [15].
Our contribution can be summed up along the following four points.1. A construction of trees which correctly computes the persistent homol-ogy of a function f : X → R , for any compact, connected and locallypath connected topological space X (section 2.1). This is the naturalgeneralization of the construction done by Le Gall on [0 , H -barcode of the function f and the tree constructed from this function, T f , and do so construc-tively by virtue of algorithm 1.3. An extension of the realm of validity of the known theorems for trees on[0 , (cid:96) p (denotedPers p in the TDA literature) of [2, 8, 14, 29]. These extensions aretheorems 3.7, 3.11 and 4.6.4. A constructive proof of existence for the inverse problem, that is :given some tree T (of finite upper-box dimension), can we construct afunction f : [0 , → R such that T f = T ?2 .3 Layout of the paper First, we construct the tree T f associated to a continuous function f : X → R , on a connected and locally path-connected, compact topologicalspace X by introducing a pseudo-distance d f on X and defining T f := X/ { d f = 0 } . We will then discuss the relation between barcodes and treesand give the functorial relation between the two. Then, we give the linkbetween the regularity of f : X → R and the metric properties of T f . Sub-sequently, we provide a generalization of Le Gall’s theorem on the stabilityof trees by relating d GH ( T f , T g ) with (cid:107) f − g (cid:107) L ∞ . Finally, we answer thequestion of the inverse problem positively and provide the construction ofa function f : [0 , → R given a tree of finite upper-box dimension T . Forthe rest of the paper, we take all homology groups with respect to the field Z / Z . For the rest of this paper, let X denote a path-connected, compact topo-logical space and let f : X → R be a continuous function. Let us denote( X r ) r ∈ R the filtration of X by the superlevels of f , that is X r := { x ∈ X | f ( x ) ≥ r } . (2.1) Remark . Notice that X r is closed. In particular, despite X being locallypath connected and connected, X r might not be locally path connected.However, the interior of X r − ε is for all ε > X r ⊂ Int( X r − ε ). It is inthis sense that the arguments of this paper are to be understood wheneverwe use the local path connectedness of a superlevel X r .There exists a pseudo-distance on X , denoted d f , given by: Definition 2.2.
Let X and f be defined as above. The H -distance , d f ,is the pseudo-distance d f ( x, y ) := f ( x ) + f ( y ) − γ : x (cid:55)→ y inf t ∈ [0 , f ( γ ( t )) , (2.2)where the supremum runs over every path γ linking x to y . Remark . Notice there are different ways of writing this distance. Inparticular, the sup above is also characterized bysup γ : x (cid:55)→ y inf t ∈ [0 , f ( γ ( t )) = sup { r | [ x ] H ( X r ) = [ y ] H ( X r ) } (2.3)= sup { r | ∃ γ ∈ C ( X r ) such that ∂γ = x − y } . (2.4)These equalities hold, since we take the coefficients of homology with respectto Z / Z , so we can interpret 1-cycles as sums of paths on X .3his pseudo-distance is a generalization of the distance introduced byCurien, Le Gall and Miermont in [13]. Note that d f has the following prop-erties:1. Idenfitication of the connected components of superlevel sets : d f ( x, y ) = 0 if and only if there exists t ∈ R such that x, y ∈ { f = t } and [ x ] H ( X t ) = [ y ] H ( X t ) . In words, that if two points x and y areon the same level-set of f and they both lie in the same connectedcomponent of the super-level set above the level-set, then they are adistance zero away from one another.2. Compatibility with the filtration induced by f : Let x, y ∈ X and suppose that f ( x ) < f ( y ), then if [ x ] H ( X f ( x ) ) = [ y ] H ( X f ( x ) ) , d f ( x, y ) := | f ( x ) − f ( y ) | . (2.5)Let us now prove that d f indeed satisfies all of the statements above. Proposition 2.4.
The function d f : X → R + of definition 2.2 is indeed apseudo-distance. Proof.
Checking symmetry and positivity is easy. The only non-obviouspoint is that the triangle inequality is satisfied by this expression, so we willfocus on this point. Let x, y, z ∈ X and denote[ x (cid:55)→ y ] := sup γ : x (cid:55)→ y inf t ∈ [0 , f ◦ γ ( t ) . (2.6)We will now show the following inequality[ x (cid:55)→ z ] + [ z (cid:55)→ y ] ≤ [ x (cid:55)→ y ] + f ( z ) , (2.7)which implies the triangle inequality. If we denote γ a path from x to z and η a path from z to y and by γ ∗ η the concatenation of these two path, bydefinition, we have that inf t ∈ [0 , f ◦ ( γ ∗ η )( t ) ≤ [ x (cid:55)→ y ] . (2.8)Since this holds for any such two paths γ and η , it follows that[ x (cid:55)→ z ] ∧ [ z (cid:55)→ y ] ≤ [ x (cid:55)→ y ] . (2.9)Without loss of generality, suppose that [ x (cid:55)→ z ] achieves the above mini-mum. Furthermore, note that [ z (cid:55)→ y ] ≤ f ( z ) (2.10)by definition of [ z (cid:55)→ y ]. Adding the two last inequalities together we getthat: [ x (cid:55)→ z ] + [ z (cid:55)→ y ] ≤ [ x (cid:55)→ y ] + f ( z ) , (2.11)as desired. (cid:4) f is immediate fromthe definition of d f , so let us verify that d f correctly identifies the connectedcomponents of superlevel sets. Proposition 2.5.
Let f be a continuous function as above, then d f identifiesthe connected components of the superlevel sets. Proof.
We must check that d f ( x, y ) = 0 if and only if there exists t ∈ R such that x, y ∈ { f = t } and [ x ] H ( X t ) = [ y ] H ( X t ) . The ( ⇐ ) direction isimmediate, so let us focus on ( ⇒ ).Suppose that d f ( x, y ) = 0 and that f ( x ) (cid:54) = f ( y ), then,sup γ : x (cid:55)→ y inf t ∈ [0 , f ( γ ( t )) = f ( x ) + f ( y )2 > f ( x ) ∧ f ( y ) . (2.12)However, sup γ : x (cid:55)→ y inf t ∈ [0 , f ( γ ( t )) ≤ f ( x ) ∧ f ( y ) , (2.13)which leads to a contradiction, so f ( x ) = f ( y ). The condition d f ( x, y ) = 0becomes: f ( x ) = sup γ : x (cid:55)→ y inf t ∈ [0 , f ( γ ( t )) . (2.14)This is only possible if a path achieving the supremum lies entirely above f ( x ). In other words, there is a path linking x and y contained within aconnected component of X f ( x ) , which implies the second condition. If thereis no path achieving this supremum, we can proceed by an approximationargument and conclude that for every ε > f ( x ) − ε , implying that : x, y ∈ (cid:92) ε> X f ( x ) − ε (2.15)but since all X f ( x ) − ε are closed and so is X f ( x ) , this is only possible if x, y ∈ X f ( x ) . (cid:4) With these technicalities out of the way, let us consider the metric space( T f , d f ) := ( X/ { d f = 0 } , d f ) , (2.16)where the quotient denotes the quotient of X where we identify all points x and y on X which satisfy d f ( x, y ) = 0. Slightly abusing the notation let d f denote the distance induced on the quotient by the pseudo-distance d f on X . The metric structure of T f is simple as T f is an R -tree. Recall that R -trees are defined as follows : Definition 2.6 (Chiswell, [11]) . An R -tree ( T, d ) is a connected metricspace such that any of the following equivalent conditions hold:5 T is a geodesic connected metric space and there is no subset of T which is homeomorphic to the circle, S ; • T is a geodesic connected metric space and the Gromov 4-point con-dition holds, i.e. : ∀ x, y, z, t ∈ T d ( x, y )+ d ( z, t ) ≤ max [ d ( x, z ) + d ( y, t ) , d ( x, t ) + d ( y, z )] ; • T is a geodesic connected 0-hyperbolic space.A rooted R -tree ( T, O, d ) is an R -tree along with a marked point O ∈ T .Note that T f is clearly connected, since X is connected. Our proofstrategy will be to use the first characterization above and to split the proofinto two parts. First, we will show that there are no subspaces of T f whichare homeomorphic to S , and then that T f is in fact a geodesic metric space.Before proving this, it is helpful to introduce some notation for somemaps and quantities which will become useful later on. Let π f : X → T f denote the canonical projection onto T f and let O denote the root of T f ( i.e. f ( O ) = min f ), let us define the following quantities: (cid:96) ( τ ) := inf X f + d f ( O, τ ) (2.17) h ( τ ) := sup x ∈ X τf ( τ ) f ( x ) − (cid:96) ( τ ) (2.18)where X τf ( τ ) denotes the connected component of the superlevel set X f ( τ ) containing a preimage of τ . These quantities are well-defined, by definitionof d f . Proposition 2.7.
The metric space T f := X/ { d f = 0 } equipped withdistance d f possesses no subspace homeomorphic to S . Proof.
We will reason by contradiction. Suppose that T f contains U ⊂ T f such that U is homeomorphic to the circle, S . Note that f descends to afunction on T f which is not locally constant anywhere by definition of d f .The restriction of f to U defines a function which admits some point x for which f is maximal on U . What we will now show is that, the filtrationof the topological space X by f and the fact that d f correctly identifies theconnected components of superlevel sets of f forbid that we can “branchdownwards” from x , thereby leading to a contradiction.An element x ∈ T f can be seen as a representative of a homology class[ x ] H ( X f ( x ) ) lying at a certain level { f = f ( x ) } . For all ε > U is homeomorphic to S and since f is not locally constant on T f ,two different points x ε + and x ε − satisfying: f ( x ε + ) = f ( x ε − ) = f ( x ) − ε (2.19)6xist on U .Notice that X x ε + f ( x ) − ε and X x ε − f ( x ) − ε are two connected components of X f ( x ) − ε ,so they are either equal or disjoint. If they are equal, then their image by π f is the same, but x ε + and x ε − were supposed to be distinct in T f , i.e. ata non-zero distance away from one another. But since all their preimageslie at the same level, and are in the same connected component of X f ( x ) − ε , x ε + and x ε − are in fact the same, leading to a contradiction. It follows that X x ε + f ( x ) − ε and X x ε − f ( x ) − ε must be disjoint.If these two connected components are disjoint, then their image by π f must be as well. Otherwise, there exists a point τ on T f lying in π f ( X x ε + f ( x ) − ε )and π f ( X x ε − f ( x ) − ε ) simultaneously. However, every preimage of τ in X lies onlevel f ( τ ) > f ( x ) − ε and they are all in the same connected component of X f ( τ ) . The inclusion X f ( τ ) (cid:44) −→ X f ( x ) − ε is injective, so every preimage of τ must lie in one and only one connected component of X f ( x ) − ε . It followsthat π f ( X x ε − f ( x ) − ε ) and π f ( X x ε − f ( x ) − ε ) are disjoint. Let us note: T r := { τ ∈ T f | f ( τ ) ≥ r } (2.20)These images are in fact nothing other than the connected component of T f ( x ) − ε containing x ε + or x ε − respectively, as if z / ∈ X x ε ± f ( x ) − ε then π f ( z ) / ∈ T x ε ± f ( x ) − ε . But, there is an arc linking x ε + and x ε − contained enitirely in T f ( x ) − ε ,so these connected components are the same, leading to a contradiction. (cid:4) Proposition 2.8.
The metric space ( T f , d f ) is an R -tree and we can choosea root for T f to be the point O the image in T f of a point x ∈ X for whichthe function f is minimal. Proof.
The only thing left to show is that T f is a geodesic space. Let x and y be two points of X . If f ( x ) = f ( y ) and x and y are in the sameconnected component, there is nothing to show, so suppose that f ( x ) < f ( y ).As before, note that f descends to the quotient and induces a non-locallyconstant function on T f . 7irst, suppose that x and y are in the same connected component of X f ( x ) and consider a path in X f ( x ) going from y to x , γ : [0 , → X , whichexists since X is path connected. The path γ can be modified into a path˜ γ ( t ) := π f (cid:32) γ (cid:32) arg min s ∈ [0 ,t ] f ◦ γ ( s ) (cid:33)(cid:33) . (2.21)On this modified path f is decreasing implying that it does not self-intersect,although it may be locally constant. The length of ˜ γ is defined as L (˜ γ ) = sup ( t i ) (cid:88) ( t i ) d f (˜ γ ( t i +1 ) , ˜ γ ( t i )) , (2.22)where the supremum is taken over all finite partitions of [0 , f ( y ) − f ( x ), since along ˜ γf (˜ γ ( t i )) ≥ f (˜ γ ( t i +1 )) = ⇒ d f (˜ γ ( t i +1 ) , ˜ γ ( t i )) = f (˜ γ ( t i )) − f (˜ γ ( t i +1 )) (2.23)by monotonicity of f along ˜ γ . This leads to pairwise cancelation of termsin the sum of equation 2.22. And so, L (˜ γ ) = d f ( x, y ) . (2.24)Now, suppose that x and y are two points on X , such that f ( x ) ≤ f ( y )but such that x and y no longer lie in the same connected component of X f ( x ) and pick a maximizer γ of the supremum ( cf. remark 2.9)sup γ : x (cid:55)→ y inf t ∈ [0 , f ◦ γ ( t ) . (2.25)Since y is not connected to x in X f ( x ), by continuity of f , the path γ musteventually go under the level f ( x ). Let us set t ∗ := sup (cid:40) arg min s ∈ [0 , f ◦ γ ( s ) (cid:41) (2.26)and note that γ ( t ∗ ) < f ( x ).On [0 , t ∗ ], the path γ lies entirely in X f ( γ ( t ∗ )) and similarly, entirely in X f ( γ ( t ∗ )) on [ t ∗ , , t ∗ ], we can define a modification of γ , ˜ γ : [0 , t ∗ ] → T f by ˜ γ ( t ) := π f (cid:32) γ (cid:32) arg min s ∈ [0 ,t ] f ◦ γ ( s ) (cid:33)(cid:33) . (2.27)Analogously, if we define η ( t ) := γ (1 − t ) – the reversed version of γ – it ispossible to define a modification of η , ˜ η : [0 , − t ∗ ] → T f , by˜ η ( s ) := π f (cid:32) η (cid:32) arg min r ∈ [0 ,s ] f ◦ η ( r ) (cid:33)(cid:33) . (2.28)8n particular, ˜ η (1 − t ∗ ) = ˜ γ ( t ∗ ). If ˜ η − denotes the reversed path along ˜ η , theconcatenation (without reparametrization), ζ := ˜ γ ∗ ˜ η − (2.29)is a path going from π f ( x ) to π f ( y ) monotone decreasing on [0 , t ∗ ] andmonotone increasing on ] t ∗ , ε > ζ ( t ∗ + ε ) does not lie in the same connected component of X f ( ζ ( t ∗ + ε )) as π f ( x ), but lies in the same connected component of X f ( ζ ( t ∗ + ε )) as π f ( y ). We are thus reduced to examine the length of the path along twodifferent sections of ζ , each lying in the same connected component as either π f ( x ) and π f ( y ). By the previous argument for points of T f lying in thesame connected component of a superlevel set, the length of ζ is L ( ζ ) = f ( x ) − f ( ζ ( t ∗ )) + f ( y ) − f ( ζ ( t ∗ )) = d f ( x, y ) (2.30)by definition of d f ( x, y ). Thus, T f is indeed geodesic and it is an R -tree, byvirtue of proposition 2.7.Finally, the tree is rooted since for any r < inf f , every single point of X r = X is identified in the quotient (since X was supposed to be connected),so we can identify the root with the point of T f achieving this infimum. (cid:4) Remark . If the suprema in the proof of proposition 2.8 are not achieved,it suffices to take a sequence of approximating paths ( γ n ) and apply the samereasoning as above. For all ε > d f ( x, y ) + ε , from which the statement follows. Remark . If X = [0 , x and y , so the definition above boils down to d f ( x, y ) := f ( x ) + f ( y ) − t ∈ [0 , f , (2.31)which is exactly the distance originally introduced by Le Gall et al. [15]. Given a tree stemming from a continuous function f : X → R , it ispossible to reconstruct the H -barcode of f by only using T f . If T f is finite,the relation between the barcode of H ( X, f ) with respect to the superlevelfiltration and the tree T f is given by algorithm 1.If T f is infinite, we can still give a correspondence between the barcodeand the tree proceeding by approximation. This approximation procedurerequires the introduction of so-called ε -trimmings of T f , of which we brieflyrecall the definition. Definition 2.11.
The ε -simplified tree of f , T εf or the ε -trimmed treeof f , is the subtree of T f defined as T εf := { τ ∈ T f | h ( τ ) ≥ ε } (2.32)9 lgorithm 1: A functorial relation between persistent modulesand R -trees Result: V F ← T ; V ← i ← while F (cid:54) = ∅ do Find γ the longest path in F starting from a root α and endingin a leaf β ; if i = 0 then V ← V ⊕ k [ (cid:96) ( α ) , ∞ [ ; else V ← V ⊕ k [ (cid:96) ( α ) , (cid:96) ( β )[ ; end F ← F \
Im( γ ); i ← i + 1 ; endreturn V Figure 1: The first four iterations of algorithm 1. For every step, in red is thelongest branch of the tree, which we use to progressively construct the per-sistent module V by associating an interval module whose ends correspondexactly to the values of the endpoints of the branches. Remark . The definition above can be extended to hold for any rootedtree (
T, d, O ) if we give define an equivalent for h ( τ ). This amounts todefining a suitable filtration of T by some function f . For any tree, we willtake this function f to be f ( τ ) = d ( O, τ ).10n ε -trimmed tree is always finite by virtue of the compactness of X .For a monotone decreasing sequence ( ε n ) n ∈ N such that ε n →
0, we have thefollowing chain of inclusions T ε (cid:44) −→ T ε (cid:44) −→ T ε (cid:44) −→ · · · . (2.33)Applying algorithm 1, we get a set of maps on the persistence modulesinduced by these inclusions. More precisely, denoting Alg( T ε n f ) the outputof the algorithm Alg( T ε ) → Alg( T ε ) → Alg( T ε ) → · · · . (2.34)where the morphisms are the maps induced at the level of the interval mod-ules generating Alg( T ε n ). Indeed, the interval modules k [ α, β n [ of Alg( T ε n )satisfy that there is exactly one interval module of Alg( T ε m ) ( m > n ) suchthat [ α, β n [ ⊂ [ α, β m [. A natural definition for infinite T is thusAlg( T ) := lim −→ Alg( T ε n f ) . (2.35)In categoric terms, the algorithm above in fact is a functorAlg : Tree → PersMod k , (2.36)where Tree is the category of rooted R -trees seen as metric spaces, whosemorphisms are isometric embeddings (which are not required to be surjec-tive) preserving the roots, and where PersMod k is the category of q-tamepersistence modules over a field k ( cf. Oudot’s book for details on the cate-gory of persistence modules [23]). The action of Alg on morphisms betweentwo trees ζ : T → T (cid:48) is defined as follows. If both T and T (cid:48) are finite, since ζ is an isometric embedding and it preserves the root, we can define Alg( ζ )to be Alg( ζ ) := (cid:77) i id k [ ζ ( α i ) ,ζ ( β i )[ , (2.37)where k [ α i , β i [ denotes the modules in the interval module decompositionof Alg( T ) (which is finite, since T is as well). If T is infinite, we extendthe above definition by taking successive ε n -simplifications of T and takingthe direct limit of the construction above. Note that this procedure is well-defined since ε n -simplifications only depend on the function h , which in turncan be taken to only depend on the distance to the root. Remark . To define Alg, we do not need the tree T to stem from afunction f , since the algorithm only depends on the function (cid:96) , which wecan define to be the distance from the root to a point τ ∈ T .Let us now consider a tree T f stemming from a function f and show thatAlg( T f ) = H ( X, f ). To do this, we will need the following proposition.11 roposition 2.14.
Let τ and η be elements of T f such that f ( τ ) < f ( η )and let x ∈ π − ( τ ) and y ∈ π − ( η ), then ∃ path γ : x (cid:55)→ y s.t. ∀ t, f ( γ ( t )) ≥ f ( τ ) ⇐⇒ h ( τ ) ≥ f ( η ) − f ( τ ) and x, y ∈ X τf ( τ ) . (2.38) Proof.
Since there exists γ connecting x and y and since γ always staysabove f ( τ ), we conclude naturally that Im( γ ) ⊂ X τf ( τ ) , which implies that h ( τ ) ≥ f ( η ) − f ( τ ) by definition of h ( τ ).The implication ( ⇐ ) is clear since if x, y ∈ X τf ( τ ) and X τf ( τ ) is connected,by path connexity of X there exists a path between x and y which staysabove f ( τ ). (cid:4) This proposition suffices to prove the following theorem on the validityof algorithm 1.
Theorem 2.15.
Let f : X → R be continuous. Then Alg( T f ) = H ( X, f ) .Proof. Suppose that T f is finite. If this is the case, then Alg( T f ) is a de-composable persistence module Alg( T f ) := V . The fact that V is pointwiseisomorphic to H ( X, f ) holds since d f correctly identifies the connected com-ponents of the superlevel sets. This guarantees the existence of a pointwiseisomorphism since both spaces have the same (finite) dimension.Let us now check that rank( V ( r → s )) = rank( H ( X r → X s )). Theinclusion X r (cid:44) −→ X s induces the following long exact sequence in homology · · · H ( X s ) H ( X s , X r ) H ( X r ) H ( X s ) H ( X s , X r ) 0Since this sequence is exactrank( H ( X r → X s )) = dim ker( H ( X s ) → H ( X s , X r )) . (2.39)For notational simplicity, let us denote φ : H ( X s ) → H ( X s , X r ). Notethat φ [ c ] = [0] if and only if there is a path γ between the representative c ∈ X s and an element b ∈ X r such that γ stays within X s . Without lossof generality, let us take c such that c ∈ { f = s } . Finding such a path γ is only possible if c and b lie in the same connected component of X r . Byproposition 2.14, this can happen if and only if h ([ c ] T f ) ≥ r − s . It followsthat dim ker φ = { τ ∈ T f | h ( τ ) ≥ r − s and f ( τ ) = s } , (2.40)which concludes the proof for the finite case.If T f is infinite, we consider a sequence of ε n -trimmings of T f such that ε n −−−→ n →∞
0. For any r > s , there exists n such that r − s > ε n . But T ε n f isfinite, so we are reduced to the previous case. (cid:4) Regularity of f and metric properties of T f From the general theory of persistence modules [9, 23], we already knowthat for all C -functions over a compact set X , H ( X, f ) is a q-tame per-sistence module. The previous section is a particular case of this generalfact in degree 0 in homology. Of course, if this was the only conclusionwe could draw from the tree construction, the theory we have just exposedabove would be completely redundant. However, the metric structure on T f is richer than the persistence diagram structure of f . This is because thetree construction gives us access to a new family of (metric) invariants forbarcodes.In this section, we will give an interpretation of some of the metric in-variants of T f . While these results originally stemmed from probabilitytheory [15, 24] they are completely deterministic. In 1D, the small bars ofthe barcode are related to the regularity of the function. The correct quan-tity to characterize this regularity in 1D is the true or total p -variation ofthe function f , of which we briefly recall the definition. Definition 3.1.
Let f : [0 , → R be a continuous function. The true p -variation of f is defined as (cid:107) f (cid:107) p − var := sup D (cid:88) t k ∈ D | f ( t k ) − f ( t k − ) | p /p , (3.41)where the supremum is taken over all finite partitions D of the interval [0 , Remark . We speak about true p -variation to make the distinction withanother notion of variation typically considered in the probabilistic context(more precisely, stochastic calculus), where instead of the supremum overall partitions, we have a limit as the mesh of the partition considered tendsto zero.This true p -variation has the nice property of being closely related to an (cid:96) p functional of the barcode, which has been used extensively by the TDAcommunity where it is typically denoted Pers p [8, 12, 14, 22, 29], it is definedas: Definition 3.3.
Let f : X → R be a continuous function. The (cid:96) p -lengthof the barcode B ( f ) is the following functional (cid:96) p ( f ) := (cid:88) I ∈B ( f ) µ ( I ∩ [inf( f ) , sup( f )]) p /p , (3.42)where µ is the Lebesgue measure on R and where B denotes the barcode of f obtained from the superlevel filtration.13t turns out that the (cid:96) p bounds the total p -variation of the function ascan be shown by adapting and slightly tweaking a result by Picard [24]: Theorem 3.4 (Picard, § . Let f : [0 , → R be a continuous function,then (cid:107) f (cid:107) p − var is finite as soon as (cid:96) p ( f ) is finite. In fact, for any p (cid:107) f (cid:107) pp − var ≤ (cid:96) p ( f ) p . (3.43) Furthermore, if (cid:107) f (cid:107) ( p − δ ) -var is finite for some δ > , (cid:96) p ( f ) is also finite. Intuition would have us think that generalizing this sort of statementrelating regularity to the (cid:96) p -length of B ( f ) beyond dimension 1 would bestraightforward, since this norm is sensitive to the oscillations of the func-tion. This turns out not to be the case, and in fact remains an open problem.One of the main difficulties is that it is not so clear how we should definean equivalent of the total variation on a more general topological space X ,which is compatible enough with the study of superlevel sets to be useful.For smooth f , there also seems to be a close link between the functionals (cid:96) p and different notions of regularity. For instance, on S , (cid:96) is the totalvariation of f and on T , Polterovitch et al. have given a result [26] relating (cid:96) with the Sobolev W , -norm. Unfortunately, the proof of this resultheavily relies on the geometry of the torus and other properties which holdonly in 2D. Theorem 3.5 (Polterovich, et al. [26]) . For every function f : T → R : (cid:96) ( f ) ≤ C (cid:107) f (cid:107) W , , (3.44) where (cid:107)·(cid:107) W , denotes the Sobolev (2 , -norm. These links between the functionals (cid:96) p and the regularity of f motivatethe study of this functional in more detail. This is also suggested by thefollowing result by Picard [24]: Theorem 3.6 (Picard, § . Let f : [0 , → R be a continuous functionand denote V ( f ) := inf { p | (cid:107) f (cid:107) p -var < ∞} and L ( f ) := inf { p | (cid:96) p ( f ) < ∞} . (3.45) Then, V ( f ) = L ( f ) = lim ε → log( λ ( T εf ) /ε )log(1 /ε ) = lim ε → log N ε log(1 /ε ) ∨ T f (3.46) where a ∨ b := max { a, b } , N ε is the number of leaves of the ε -trimmed tree T εf , λ ( T εf ) denotes the length of T εf and dim denotes the upper-box dimension. This result generalizes for more general topological spaces X and pro-vides a connection between the topology of T f and the (cid:96) p functional.14 heorem 3.7. With the same notation as above and supposing that dim T f is finite, the following chain of equalities hold L ( f ) = lim ε → log N ε log(1 /ε ) ∨ ε → log( λ ( T εf ) /ε )log(1 /ε ) = dim T f . (3.47) Furthermore, lim ε → log N ε log(1 /ε ) ∨ ≤ dim T f ≤ lim ε → log( λ ( T εf ) /ε )log(1 /ε ) , (3.48) where dim is the lower-box dimension. For dim T f > , these inequalitiesturn into equalities if either: lim ε → N ε N ε < or lim ε → λ ( T εf ) λ ( T εf ) < . (3.49) Remark . The study of N ε is in fact completely equivalent to the studyof (cid:96) pp ( f ). Indeed, (cid:96) pp ( f ) = p (cid:90) ∞ ε p − N ε dε , (3.50)which is finite as soon as p > L ( f ). This is nothing other than the Mellintransform of N ε . By the Mellin inversion theorem, for any c > L ( f ), wehave N ε = 12 πi (cid:90) c + i ∞ c − i ∞ (cid:96) pp ( f ) ε − p dpp . (3.51) Proof of theorem 3.7.
By the procedure detailed in section 5, since dim T f is finite we can construct a function ˆ f : [0 , → R such that T f and T ˆ f areisometric. Applying Picard’s theorem to T ˆ f and noting that L ( f ) dependsonly on the T f , we have that L ( f ) = lim ε → log( λ ( T εf ) /ε )log(1 /ε ) = lim ε → log N ε log(1 /ε ) ∨ T f . (3.52)Let us now show the inequalities for lim. First, λ ( T εf ) = (cid:90) ∞ ε N a da , (3.53)which implies that lim ε → log N ε log(1 /ε ) ∨ ≤ lim ε → log( λ ( T εf ) /ε )log(1 /ε ) . (3.54)Additionally, N ε ≤ N ( ε/
2) (3.55)15here N ( ε ) denotes the minimal number of balls of radius ε necessary tocover T f . This inequality holds as above each leaf of T εf , at least one ball ofradius ε is necessary to cover this section of the tree. It follows thatlim ε → log N ε log(1 /ε ) ∨ ≤ dim T f . (3.56)We can bound this minimal number of balls N ( ε ) by the following N ( ε ) ≤ N ε/ + λ ( T ε/ f ) ε/ ≤ N ε/ ∨ λ ( T ε/ f ) ε/ , (3.57)which holds since, at most N ε balls are needed to cover T f \ T εf . To cover T εf ,at most: (cid:108) λ ( T ε/ f ) / ( ε/ (cid:109) balls are needed, so the inequality above followsby further majoring the terms. This implies thatdim T f ≤ (cid:20) lim ε → log N ε log(1 /ε ) ∨ (cid:21) ∨ (cid:34) lim ε → log( λ ( T εf ) /ε )log(1 /ε ) (cid:35) , (3.58)but by inequality 3.56 this means thatdim T f ≤ lim ε → log( λ ( T εf ) /ε )log(1 /ε ) . (3.59)Finally, λ ( T εf ) − λ ( T εf ) ε = 1 ε (cid:20)(cid:90) ∞ ε N a da − (cid:90) ∞ ε N a da (cid:21) = 1 ε (cid:90) εε N a da ≤ N ε , (3.60)since N ε is monotone decreasing. This reasoning also gives a lower bound: N ε ≤ λ ( T εf ) − λ ( T εf ) ε ≤ N ε . (3.61)which entails that:lim ε → log N ε log(1 /ε ) = lim ε → log (cid:20) λ ( T εf ) − λ ( T εf ) ε (cid:21) log(1 /ε ) . (3.62)Suppose that this limit is larger than 1. Rearranging, we get: εN ε λ ( T εf ) ≤ − λ ( T εf ) λ ( T εf ) ≤ εN ε λ ( T εf ) . (3.63)16t follows that if any of these quantities admits a lim which is stricly greaterthan zero, we have lim ε → log N ε log(1 /ε ) = lim ε → log λ ( T εf )log(1 /ε ) . (3.64)Another equivalent condition for the validity of this equality is whetherlim ε → N ε N ε < , (3.65)which finishes the proof. (cid:4) Remark . If dim = dim, all the limits of the above theorem are well-defined, yielding exact asymptotics for λ ( T εf ) and N ε . This is in particularthe case if dim = dim H , where dim H denotes the Hausdorff dimension.The functional λ ( T εf ) is what some authors [25,26] refer to as the Banachindicatrix and its asymptotics have a topological interpretation as describedin the statement of the theorem. It is interesting to note that the study ofthe upper-box dimension is natural in the tree approach. Nonetheless, dimhas also been used in the context of persistent homology in degree 0, albeitunder a completely different setting by Schweinhart et al. in [20, 27] and byAdams et al. [1]. It is possible to further extend Picard’s theorem by some rudimentaryconsiderations and by imposing the so-called locally linearly connected con-dition X . Let us briefly recall its definition. Definition 3.10. A locally linearly connected (LLC) metric space ( X, d ), is a metric space such that for all r > z ∈ X , for all x, y ∈ B ( z, r ), there exists an arc connecting x and y such that the diameterof this arc is linear in d ( x, y ).With this extra assumption, it is possible to link the regularity of thefunction f to the quantities defined in theorem 3.7. Theorem 3.11 (The regularity of f bounds the upper-box dimension of T f ) . Let X be a compact LLC metric space. Keeping the same notations asin theorem 3.7, the following inequality holds dim T f ≤ H ( f ) dim X , (3.66) where: H ( f ) := inf (cid:26) α (cid:12)(cid:12)(cid:12) (cid:107) f (cid:107) C α < ∞ (cid:27) (3.67)17he proof of this theorem relies on two lemmas: Lemma 3.12.
Let X and Y be two metric spaces such that there is asurjective map π : X → Y such that π ∈ C α ( X, Y ), thendim Y ≤ α dim X . (3.68)
Lemma 3.13.
Let X be a compact locally linearly connected (LLC) metricspace ( cf. definition 3.10) and let f : X → R be a continuous function, then f ∈ C α ( X, R ) = ⇒ π f ∈ C α ( X, T f ) . (3.69)Let us show that these two lemmas imply the theorem. Proof of theorem 3.11. If f / ∈ C α ( X, R ) for any α , there is nothing to show,since the statement is vacuous. Otherwise, the projection onto the tree of f , π f : X → T f is in C α ( X, T f ) according to lemma 3.13. It follows fromlemma 3.12 that dim T f ≤ α dim X . (3.70)The statement of the theorem follows by taking the infimum over all such α . (cid:4) All that remains to show is the two lemmas.
Proof of lemma 3.12.
Since π : X → Y is surjective and C α ( X, Y ), for any x ∈ X π (cid:18) B X (cid:18) x, (cid:16) εK (cid:17) /α (cid:19)(cid:19) ⊂ B Y ( π ( x ) , ε ) (3.71)for some constant K . It follows that the minimal number of balls neededto cover X , N X dominates the minimal number of balls needed to cover Y , N Y . More precisely N Y ( ε ) ≤ N X (cid:18)(cid:16) εK (cid:17) /α (cid:19) ⇐⇒ α N Y ( ε )log(1 /ε ) + log( K ) ≤ N X (cid:16)(cid:0) εK (cid:1) /α (cid:17) log (cid:16)(cid:0) Kε (cid:1) /α (cid:17) . The statement of the lemma follows. (cid:4)
Proof of lemma 3.13.
Suppose that f : X → R is in C α ( X, R ) and let x, y ∈ X be two points inside a ball of radius r >
0. Without loss ofgenerality, suppose that f ( x ) < f ( y ). Since T f is a geodesic space, the dis-tance d f ( π f ( x ) , π f ( y )) is the length of the geodesic arc in T f linking π f ( x )and π f ( y ). By compactness of this geodesic path, there is a point τ ∈ T f where f achieves its minimum, thus d f ( π f ( x ) , π f ( y )) = | f ( y ) − f ( x ) | + 2 | f ( x ) − f ( τ ) | . (3.72)18his minimum f ( τ ) has the particularity that f ( τ ) = sup γ : x (cid:55)→ y inf t ∈ [0 , f ◦ γ , (3.73)where the supremum is taken over all paths on X linking x and y . Theequality holds, since it is always possible to find a path in X r whose imagein T f contains a path in T f lying entirely above level r . Since X is LLC,there is a path η linking x and y in X whose diameter we can control linearlyin terms of d X ( x, y ). Along η , we have that f ( τ ) = sup γ : x (cid:55)→ y inf t ∈ [0 , f ◦ γ ≥ inf t ∈ [0 , f ◦ η =: f ( z ) (3.74)for some z ∈ X along the path η . Combining the LLC condition and thefact that f is α -H¨older gives: f ( x ) − f ( τ ) ≤ f ( x ) − f ( z ) ≤ d ( x, z ) α ≤ C d ( x, y ) α (3.75)for some constant C , which is determined by quantitative LLC condition.Putting everything together we have that: d f ( π f ( x ) , π f ( y )) ≤ (2 C + 1) d X ( x, y ) α , (3.76)which finishes the proof. (cid:4) Theorem 3.11 is sharp. Indeed, Brownian sample paths almost surelysaturate this inequality. However, there is no hope to prove equality in allgenerality. Indeed note that for any f ∈ C ( T , R ) T f is a finite tree andhas upper-box dimension 1, but :dim T f = 1 < H ( f ) dim T . (3.77)An interesting problem would be to either prove that for irregular functionsequality holds, or alternatively, to find a counter example of an irregularfunction for which we don’t have equality. Of course, this can only be donein dimensions ≥
2. More precisely, we can phrase this question in the formof the following conjecture.
Conjecture 3.14.
Given an LLC space X , we have:dim X = sup f ∈ C ( X, R ) dim T f H ( f ) . (3.78)Furthermore, there may be instances of spaces X for which this supremumis achieved by some f ∈ C α ( X, R ) for some 0 < α ≤ Stability of trees with respect to the L ∞ -norm It is well known that the Gromov-Hausdorff distance is a natural notionof distance between metric spaces. Recall that this distance is defined as:
Definition 4.1.
Let X and Y be two compact metric spaces, the Gromov-Hausdorff distance, d GH ( X, Y ) between X and Y , is defined as d GH ( X, Y ) := inf f : X → Zg : Y → Z max (cid:40) sup x ∈ X inf y ∈ Y d Z ( f ( x ) , g ( y )) , sup y ∈ Y inf x ∈ X d Z ( f ( x ) , g ( y )) (cid:41) . (4.79)where the infimum is taken over all metric spaces Z and all isometric em-beddings f : X → Z and g : Y → Z .From this definition, we see that d GH quantifies how far away two metricspaces X and Y are from being isometric to each other. However, thisdefinition, while practical in theory is very difficult to compute in practice.To somewhat alleviate this, we will use the following characterization of theGromov-Hausdorff distance: Proposition 4.2 (Burago et al. , § . The Gromov-Hausdorf distance ischaracterized by the following equality: d GH ( X, Y ) = 12 inf R sup ( x,y ) ∈ R ( x (cid:48) ,y (cid:48) ) ∈ R (cid:12)(cid:12) d X ( x, x (cid:48) ) − d Y ( y, y (cid:48) ) (cid:12)(cid:12) , (4.80)where the infimum is taken over all correspondences , i.e. subsets R ⊂ X × Y such that for every x ∈ X there is at least one y ∈ Y such that ( x, y ) ∈ R and a symmetric condition for every y ∈ Y . Remark . Given two surjective maps π X : Z → X and π Y : Z → Y , it ispossible to build a correspondence between X and Y by considering the set { ( π X ( z ) , π Y ( z )) ∈ X × Y | z ∈ Z } .Recall that there is also a natural notion of distance in the space of bar-codes (or equivalently, persistence diagrams) called the bottleneck distance,which we will note d b . For the definition of this distance, we refer the readerto the books by Chazal and Oudot on persistence theory [9,23]. With respectto this distance, we have a “stability theorem” which we briefly state: Theorem 4.4 (Bottleneck stability with respect to L ∞ , Corollary 3.6 [23]) . Let f, g : X → R be two continuous functions, then d b ( B ( f ) , B ( g )) ≤ (cid:107) f − g (cid:107) L ∞ (4.81) where B ( f ) and B ( g ) denote the barcodes (or diagrams) of f and g respec-tively.
20 natural question is to ask whether we have an equivalent statementabout the stability of d GH with respect to (cid:107)·(cid:107) L ∞ and whether the two notionsof distances are in some sense “compatible”. We will positively answer thisfirst question. However, in general d b and d GH are not compatible, in thesense that no inequality between the two holds in all generality ( cf. remark4.7). Le Gall and Duquesne [15] gave a first stability result of d GH respectto the L ∞ -norm on continuous functions on [0 , Theorem 4.5 ( L ∞ -stability of trees, [15]) . Let f, g : [0 , → R be continu-ous functions. Then d GH ( T f , T g ) ≤ (cid:107) f − g (cid:107) L ∞ . (4.82)This result for functions on [0 ,
1] generalizes to more general topologicalspaces X . Theorem 4.6 (Stability theorem for trees) . Let f and g : X → R becontinuous, then d GH ( T f , T g ) ≤ (cid:107) f − g (cid:107) L ∞ . (4.83) Proof.
We will use the distortion characterization of the Gromov-Hausdorffdistance, which yields the following inequality d GH ( T f , T g ) ≤
12 sup x,y ∈ X | d f ( x, y ) − d g ( x, y ) | . (4.84)Following the logic of the proof of lemma 3.13, the distance between π f ( x )and π f ( y ) is of the form d f ( π f ( x ) , τ ) + d f ( τ, π f ( y )) = f ( x ) − f ( τ ) + f ( y ) − f ( τ ) (4.85)where τ is the lowest point of the geodesic path in T f between π f ( x ) and π f ( y ). This geodesic path on T f admits preimages by π f which are pathsconnecting x to y . These paths achieve the following supremumsup γ : x (cid:55)→ y inf t ∈ [0 , f ◦ γ = f ( τ ) ≤ f ( x ) ∧ f ( y ) (4.86)where a ∧ b := min { a, b } since by construction γ must always stay above f ( τ ) and since for r > f ( τ ), x and y lie in different connected componentsof X r . If ν is the analogous vertex to τ on T g , we have that d GH ( T f , T g ) ≤
12 sup x,y ∈ X | d f ( x, y ) − d g ( x, y ) | = 12 sup x,y ∈ X | f ( x ) − g ( x ) + f ( y ) − g ( y ) − f ( τ ) + 2 g ( ν ) |≤ (cid:107) f − g (cid:107) L ∞ + sup x,y ∈ X (cid:12)(cid:12)(cid:12)(cid:12) sup γ : x (cid:55)→ y inf t ∈ [0 , f ◦ γ − sup η : x (cid:55)→ y inf t ∈ [0 , g ◦ η (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) f − g (cid:107) L ∞ , (4.87)as desired. (cid:4) emark . One can be tempted to establish a general inequality between d GH and d b since both of these distances are bounded by the L ∞ -norm.However, this is not possible.Indeed, there is a simple counter-example to d GH ≥ d b . To illustrate thisconsider two barcodes k [ s, −∞ [ and k [ s + ε, −∞ [. The bottleneck distancebetween these two is clearly ≥ ε . But supposing that the functions f and g generating these barcodes are such that f = g + ε the trees T g and T f areisometric, so d GH ( T f , T g ) = 0 < ε ≤ d b ( B ( f ) , B ( g )).Conversely, there are also counter-examples to d b ≥ d GH , as this inequal-ity would imply that two trees which have the same barcode are isometric.This is clearly false, as one can “glue” the bars of a given barcode is manydifferent ways to give a tree, which generically will not be isometric. An interesting question is whether every (compact) tree stems from afunction f : X → R . We can positively answer this question under theassumptions that dim T < ∞ and that X = [0 ,
1] by constructing a function f : [0 , → R . The rest of this section will focus on proving the followingtheorem: Theorem 5.1.
Let T be a compact R -tree such that dim T < ∞ . Then, forany δ > it is possible to construct a continuous function f : [0 , → R of finite (dim T + δ ) -variation such that T = T f . In particular, up to areparametrization, f can be taken to be T + δ -H¨older continuous. The idea is to once again use ε -simplifications T ε for which we can con-struct a function by taking the contour of the tree. Such a construction isreferred to as the Dyck path in the terminology of [28].In what will follow, we will lay down notation which will simplify ourtask. In so doing, we will have shown the result for finite R -trees. We thenshow the result for infinite trees. We can regard a rooted discrete tree as being an operator with N inputs,where N is the number of leaves of the tree. There is a natural operationon the space of discrete trees which composes these operations by:22hese objects are called operads and originated in the study of iteratedloop spaces [5,6,21]. Since then, these objects have been studied in differentfields for a variety of purposes [17,19]. We will not give the explicit definitionof an operad here, as we don’t really need it, but we introduce this notionof composition of trees for notational simplicity.Figure 2: The Dyck path is the function f which assigns the height (thedistance from the root) of each vertex of the tree as we wrap around thetree following a clockwise contour around it. There is a map φ : T (cid:55)→ [0 , ζ ]where [0 , ζ ] is now marked at the points at which f achieves its local maxima.The figure is taken from [15].Given a discrete R -tree T , if we have an embedding of T in R , orequivalently, a partial order on its vertices, we can assign to T an interval I of a certain length with N marked points as well as a function f T : I → R ,where N is the number of leaves of T . Using the terminology of [28], a way todo this is by considering the so-called Dyck path or contour path wherethe path around T parametrized by arclength in T . The construction of theDyck path has been carefully detailed in [15, 28], but it is better understoodby looking at figure 2. By construction the equality: T f T = T holds for anydiscrete R -tree T . Here, equality is taken up to isometry.As per the description of figure 2, the construction of the Dyck pathyields a map φ which to T assigns an interval φ ( T ) with N marked points.An example of the action of φ is illustrated in figure 3.This operation φ is in fact a “morphism” with respect to a compositionoperation on the intervals, defined as follows. If we have an interval I with N marked points and N intervals J k each with M j marked points, the resultof the operation I ◦ ( J , · · · , J N ) is the insertion of the marked interval J k at the k th marked point of I . The length of I ◦ ( J , · · · , J N ) is | I ◦ ( J , · · · , J N ) | = | I | + n (cid:88) k =1 | J k | , (5.88)23igure 3: The action of φ on trees with two and three leaves respectively.The length of the intervals assigned is exactly the length of the contouraround the trees and the marked points are the points at which f T achievesits maxima.where |·| denotes the lengths of the intervals. The fact that φ is a “mor-phism” results from the definitons of compositions for trees and intervals.We can also define a variant of this morphism φ , which we will call φ λ , whichfor any tree T simply scales the (marked) interval φ ( T ) by a factor λ .Given a tree T the Dyck path f T : φ ( T ) → R can be transformed into afunction f λT : φ λ ( T ) → R by setting f λT ( x ) := f T ( x/λ ) . (5.89)This is a rescaling of the x -axis which means that T f λT = T f T = T still holds.These equalities are taken up to isometry. Remark . The definition of f λT is readily generalizable to forests. If F denotes a forest, then we define f λ F = (cid:70) T ∈F f λT .For discrete trees, there is an upper bound of the number of vertices ofthe tree given its number of leaves. Lemma 5.3.
Let T be a rooted discrete tree, N be its number of leavesand V be its number of vertices, then V ≤ N − . (5.90)In particular, if the edges of T all have length 1, the contour of the tree canbe done over an interval of length at most 4 N − Proof.
For binary trees, it is known that [15, 28] V = 2 N − . (5.91)Given a tree with N leaves, we can obtain a binary tree with N leaves bysimply blowing up the vertices which are non-binary. The inequality of thelemma follows. On a binary tree, the Dyck path passes through almost everypoint in T twice, so the length of the interval is exactly 4 N −
2. Since binarytrees are the extremal case, a bound for all trees with N leaves follows. (cid:4) The results above show the result of theorem 5.1 for finite trees, sincetheir upper-box dimension is equal to 1.24 .2 Infinite trees: a construction idea
The concatenation of trees can be defined for R -trees too in the obviousway. Given an infinite number of compositions, we can define a limit treeby defining it to be the limit of the partial compositions in the Gromov-Hausdorff sense. Ideally, we would like to have an equality of the followingtype T = T a ◦ ( T \ T a ) , (5.92)where T \ T a now denotes the rooted forest corresponding to the set T \ T a .This equality is desirable because by taking infinitely many compositions,we can eventually recover the original tree T , by composing successive ε n -simplifications with each other. However, this equality does not hold since T a might not have the right amount of leaves for this operation to be well-defined. Nonetheless, we can decide to count the vertices T a ∩ ( T \ T a ) asleaves, so that the equality above holds.In particular, the equality above would imply the following “proposi-tion”. Fictional proposition 5.4.
For any sequence ( ε n ) n ∈ N ∗ such that ε n → monotonously, we have that for any compact R -tree T = T ε ◦ ( T ε \ T ε ) ◦ ( T ε \ T ε ) ◦ · · · . (5.93)For an infinite compact tree with dim T < ∞ , the idea is to take someappropriate rapidly decreasing sequence ( ε n ) n ∈ N ∗ such that the interval I = φ ε ( T ε ) ◦ φ ε ( T ε \ T ε ) ◦ φ ε ( T ε \ T ε ) ◦ · · · (5.94)has finite length. On each φ ε k ( T ε k \ T ε k − ) we can consider the Dyck pathon the forest T ε k \ T ε k − . Defining a correct superposition of these Dyckpaths, we would be done ( cf. figure 4) Now that we have laid out the idea of the proof, we need to providea rigorous construction of the ideas above. For this, we need to introducemultiple definitions.
Definition 5.5.
Let I ⊂ R + be a marked interval with n marked points,which we will denote ( i k ) { ≤ k ≤ n } . Furthermore, let ( J k ) { ≤ k ≤ n } be a set of n marked intervals of R + , each with j k marked points. Define σ I : I → I ◦ ( J , · · · J n ) by σ I ( x ; J , · · · , J n ) := x + arg max k { i k Let f : I → R be a continuous function from an inter-val I with n marked points and let ( J , · · · , J n ) be intervals with eachwith j i marked points as before. Abusing the notation, we define anotherfunction σ ( − ; J , · · · , J n ) which assigns a function on I to a function on σ I ( I ; J , · · · , J n ) via the following formula σ I ( f ; J , · · · , J n )( x ) := (cid:40) f ( σ − I ( x ; J , · · · , J n )) x ∈ σ I ( I ; J , · · · , J n )Linearly extend elsewhere (5.96) Remark . By continuity of f : I → R , this linear extension on I ◦ ( J , · · · , J n ) is in fact constant everywhere outside σ I ( I ; J , · · · , J n ) (thisis the dotted region in figure 4). Note also that σ I ( f ; J , · · · , J n ) is contin-uous. Definition 5.9. Given a tree T f associated to a continuous function f , wedefine: • The projection onto the tree as the mapping π : X → T f = X/ { d f = 0 } ; (5.97) x (cid:55)→ [ f ( x )] (5.98) • Let τ ∈ T f , define the left preimage of τ , ←− τ and the right preim-age of τ by π , −→ τ as ←− τ := inf π − ( τ ) (5.99) −→ τ := sup π − ( τ ) . (5.100) Definition 5.10. Let T be a discrete rooted tree and T (cid:48) ⊂ T be a subtreesharing roots with T and suppose that we have chosen some embeddings of T and T (cid:48) on the plane such that these embeddings are consistent. Supposethere is a function f : I → R on a certain interval I such that T f = T (cid:48) .Then, the marking of I induced by T is the marking induced by markingthe preimage π − f ( T (cid:48) ∩ ( T \ T (cid:48) )) chosen in the following way: • If τ ∈ T (cid:48) ∩ ( T \ T (cid:48) ) admits a single preimage, choose this preimage; • Else, if the connected component of τ in T \ T (cid:48) is smaller (with respectto the partial order on the tree induced by the embedding of T ) thanevery vertex strictly greater than τ ∈ T (cid:48) , choose ←− τ . Otherwise, choose −→ τ . 26e will denote this marking operation by µ ( I ; T (cid:48) , T, f ).We can also define analogous maps to σ I , but this time on the intervals J k as follows. Definition 5.11. Let I ⊂ R + be a marked interval with n marked points,which we will denote ( i k ) { ≤ k ≤ n } . Furthermore, let ( J k ) { ≤ k ≤ n } be a set of n marked intervals of R + , each with j k marked points. Define η J k I : J k → I ◦ ( J , · · · , J n ) by η J k I ( x ; J , · · · , J n ) := x + i k . (5.101)These maps define a map η I = (cid:70) k η J k I on (cid:70) k J k and η I also induces a mapon the functions f : (cid:70) k J k → R , defined analogously to σ I , which we shallalso denote η I .With this notation, the construction is made in accordance to algorithm2. A depiction of the mechanism of algorithm 2 can be found in figure 4. Algorithm 2: Construction of approximants Output: A set of unions of intervals ( I i ) i ∈{ , ··· ,n } and a set offunctions on I n , ( f i : I n → R ) i ∈{ , ··· ,n } Input: An infinite tree T and a > I ← φ ( T a ) ; f ← f T a ; I ← I ; i ← while i ≤ n do I i +1 := I i ◦ φ λ i ( T a/ i +1 \ T a/ i ) ; f ← η I i +1 ( f λ i +1 T a/ i +1 \ T a/ i ; I , · · · I i ) ; I i ← µ ( I i ; T a/ i − , T a/ i , f i ) ; for j=1; j ≤ i do I j ← σ ( I j ; φ λ i ( T a/ i +1 \ T a/ i )) ; f j ← σ ( f j ; φ λ i ( T a/ i +1 \ T a/ i )) ; j ← j + 1 ; end f i +1 := f i + f ; i ← i + 1 ; endreturn ( I i ) i ∈{ , ··· ,n } , ( f i ) i ∈{ , ··· ,n } .For an infinite tree, it suffices to show that the sequence generated by thisalgorithm converges in the Gromov-Hausdorff sense to an interval of finitelength I and that ( f i ) i converge in L ∞ ( I ) to some function f . The firstthing we must show is thus: 27igure 4: Starting from a tree T a/ k (black) we construct the Dyck patharound it in the first step. Then, we look at T a/ k +1 which leads to theaddition of intervals (dotted), and a correction of the function at the k thstep f k (which is the function depicted in black, extended linearly over thenew intervals). We can further define a function which by pasting the Dyckpaths of the forest over the corresponding leaves, which leads to the functiondepicted in the second step (red and black). Lemma 5.12. If T is a compact R -tree of finite upper-box dimension, it ispossible to define such an interval of finite length I defined by the construc-tion above.We need to show the convergence of the corresponding functions ( f n ) n .This can be done by proving that the sequence is Cauchy. Lemma 5.13. Given the definition of functions f n above, then the sequence( f n ) n ∈ N ∗ is Cauchy in L ∞ ( I ), we have (cid:107) f n − f m (cid:107) L ∞ ≤ a − ( n ∧ m ) (5.102)for any n and m ∈ N ∗ .By completeness of L ∞ ( I ), the sequence ( f n ) n ∈ N ∗ uniformly convergesto a continuous function f . By virtue of stability theorem for trees (theorem4.6) it follows that T is isometric to T f . Using Picard’s theorem (theorem3.4) V ( f ) = dim T f = dim T (5.103)which concludes the proof of theorem 5.1.28 .4 Proofs of the key lemmas Proof of lemma 5.12. Recall that, according to the proof of theorem 3.7, thefollowing equality holds for any tree T lim ε → log N ε log(1 /ε ) ∨ ≤ dim T := α . (5.104)Unpacking the definition of the lim sup, for any δ > a > ε < a , we have that N ε < ε − α − δ . (5.105)Let us fix such a δ and pick a small enough so that the condition aboveholds. For any n ∈ N ∗ , the partial composition of intervals has length | I n | = | φ ( T a ) | + n (cid:88) k =1 (cid:12)(cid:12)(cid:12) φ λ k ( T a/ k \ T a/ k − ) (cid:12)(cid:12)(cid:12) . (5.106)However, we can bound (cid:12)(cid:12)(cid:12) φ λ k ( T a/ k \ T a/ k − ) (cid:12)(cid:12)(cid:12) by (cid:12)(cid:12)(cid:12) φ λ k ( T a/ k \ T a/ k − ) (cid:12)(cid:12)(cid:12) = λ k (cid:12)(cid:12)(cid:12) φ ( T a/ k \ T a/ k − ) (cid:12)(cid:12)(cid:12) ≤ λ k (cid:16) a k (cid:17) (4 N a/ k ) , (5.107)since on T a/ k \ T a/ k − the distances between the vertices of each tree areat most a/ k and there are at most 4 N a/ k such edges by virtue of lemma5.3. Thus, (cid:12)(cid:12)(cid:12) φ λ k ( T a/ k \ T a/ k − ) (cid:12)(cid:12)(cid:12) < λ k (cid:16) a k (cid:17) − α − δ = 4 a − α − δ (cid:16) α + δ − λ (cid:17) k . (5.108)Setting λ < − α − δ I n converges to some interval of finite length I , sincethe partial sums | I n | converge. (cid:4) Proof of lemma 5.13. Suppose that n < m . It is sufficient to show that on I m the equality holds, since in all further iterations of the algorithm, thefunctions f n and f m are locally constant over the intervals introduced. Bydefinition of f n , f n and f m agree on I n . Outside of this set, f n is constantand the difference in the L ∞ -norm depends only on what happens above T a/ n , thus we can write (cid:107) f n − f m (cid:107) L ∞ ≤ (cid:13)(cid:13)(cid:13) f T a/ m \ T a/ n (cid:13)(cid:13)(cid:13) L ∞ (5.109)by definition of f n . However, the Dyck path on T a/ m \ T a/ n can at mostreach a height of a (2 − n − − m ) < a − n , which finishes the proof. (cid:4) Limitations and prospects One of the clear limitations of the work we have done so far is that,in the way it has been presented, trees only give a valid description of the H -barcode. It would be interesting to extend these notions and build an H k -distance in general, which we expect would generate a forest instead of a tree.Furthermore, the metric invariants related to persistence diagrams in thiswork are only really useful in a C -setting. 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