The persistent homology of a sampled map: From a viewpoint of quiver representations

This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of correspondences using the framework of quiver representations. Our definition of homology induced maps is given by most persistent direct summands of representations, and the direct summands uniquely determine a persistent homology. We provide stability theorems of this process and show that the output persistent homology of the sampled map is the same as that of the underlying map if the sample is dense enough. Compared to existing methods using eigenspace functors, our filtration analysis has an advantage that no prior information on the eigenvalues of the underlying map is required. Some numerical examples are given to illustrate the effectiveness of our method.


Introduction
Consider the following problem.
Problem 1.1. Let X and Y be topological spaces, and f : X → Y be a continuous map. If we know only X, Y , and sampling data f S , which is a restriction of f on a finite subset S ⊂ X, then can we retrieve any information about the homology induced map f * : HX → HY ?
The map f S is called a sampled map of f . This paper is motivated by [Harker et al. 2016], which suggests the following analysis for sampled maps. A grid X of X is a finite collection of subsets of X with disjoint interiors such that X := X ∈X X = X. we divide the topological spaces X and Y into grids X and Y, and let F be the union of regions which have elements of the sample Gr(f S ) := { (s, f (s)) | s ∈ S }. That is, the purple regions in Figure 2. The set F is an approximation of the graph Gr(f ) from the sampled map by this subspace, which is called a correspondence.
Definition 1.3. For a correspondence F , let p : F → X and q : F → Y be canonical projections, and p * : HF → HX and q * : HF → HY be their homology induced maps. If p * and q * satisfy two properties • Im p * = HX (homologically complete) • q * (Ker p * ) = 0 (homologically consistent), then the induced map of F is defined by F * := q * • p −1 * : HX → HY , and is well-defined.
The graph Gr(f ) of f is a correspondence. Since f is continuous, p : Gr(f ) → X is homeomorphic. As a consequence, p * and q * for Gr(f ) satisfy homologically completeness and homologically consistency, hence Gr(f ) * is well-defined. We remark that this induced map Gr(f ) * coincides with f * . The following theorem guarantees that F * restores f * when the grid is fine and the sample is dense enough.
In Section 3, we give a new definition of induced maps of correspondences within the framework of quiver representations. We will see that the indecomposable decompositions of quiver representations give us an assignment among the bases of HX, HF , and HY , which defines the induced map from HX to HY .
The paper [Edelsbrunner et al. 2015] gives a way to analyze the eigenspaces of the homology induced map of a self-map (discrete dynamical system). In this analysis, the authors construct a filtration of simplicial maps from the sampled map and build its persistent homology by applying the homology functor and eigenspace functors. This construction of the filtration and the new definition of homology induced maps provide a further technique that enables another persistence analysis of sampled maps, shown in Section 4. Specifically, the assignment among the bases can compress the three persistent homologies generated by the finite sets S, f S , and f (S), yielding a persistent homology which describes the persistence of the topological mapping from the domain to the image of the induced map f * . Moreover, we can provide such a persistence analysis also in the above gridded setting, as mentioned in Subsection 4.2.
The main theorem of this paper is a stability theorem for these processes, Theorems 5.4 and 5.6 in Section 5, which state that these mappings from the input (sampled maps) to output (persistent homology or persistence diagrams) are non-expanding maps.
Finally, we approach 2-D persistence modules in Section 6 using the above ideas, and show some numerical results in Section 7.

Preliminaries
In this section, we introduce quiver representations and matrix notation for morphisms between A n type representations. For more details, the reader can refer to [Assem et al. 2006] and [Asashiba et al. 2019], respectively.

Quivers and their representations
Throughout this paper, scalars of vector spaces and coefficient rings of homology groups are a fixed field K. A quiver Q = (Q 0 , Q 1 , s, t) (or simply (Q 0 , Q 1 )) is a directed graph with a set of vertices Q 0 , a set of arrows Q 1 , and morphisms s, t : Q 1 → Q 0 identifying the source and the target vertex of an arrow. An arrow α ∈ Q 1 is denoted by α : s(α) → t(α). A representation of a quiver Q, denoted M = (M a , ϕ α ) a∈Q 0 ,α∈Q 1 (or simply (M a , ϕ α ) or (M, ϕ)), is a collection of a (finite dimensional) vector space M a for each vertex a ∈ Q 0 and a linear map ϕ α : The composition of morphisms These definitions determine an additive category of representations rep(Q). Specifically, rep(Q) has a zero representation, isomorphisms of representations, and direct sums of representations. One can see the concrete construction of these in [Assem et al. 2006].
A representation M is indecomposable if M ∼ = N ⊕ N implies N = 0 or N = 0. From the Krull-Remak-Schmidt theorem, every representation M can be uniquely decomposed into a direct sum of indecomposables M ∼ = N 1 ⊕ · · · ⊕ N s , unique up to isomorphism and permutations.
A quiver Q is of finite type if the number of distinct isomorphism classes of indecomposables is finite, and is of infinite type otherwise.
A n (τ n ) type quivers (or simply A n type quivers) are a class of quivers with the following shape: where ←→ denotes a forward arrow −→ or backward arrow ←−, and τ n is a sequence of n − 1 symbols f and b which determine the orientation of the arrows. From Gabriel's theorem [Gabriel 1972], every A n type representation can be uniquely decomposed into a direct sum of indecomposable interval representations In topological data analysis, a central role is played by persistent homology. The homology of a filtration of simplicial complexes HX : HX 1 → HX 2 → · · · → HX n can be regarded as a representation of an A n (f f · · · f ) type quiver in the framework of quiver representations. Each interval representation I[b, d] corresponds to a generator of a homology group which is born at HX b and dies at HX d+1 , and d − b is called its lifetime. The persistence diagram is a multiset determined by the unique decomposition of the persistent homology or an image made by plotting this on a plane. This description allows us to overview the generators of all homology groups, and hence this approach is frequently used for the application of persistent homology.
The framework of quiver representations has extended persistent homology to general representations of quivers. We call representations of quivers persistence modules. Zigzag persistence modules [Carlsson and de Silva 2010] are a typical example of the extension, which enable persistence analysis of time series data.
Consider deformations of topological spaces (X 1 , . . . , X T ), a sequence of topological spaces. The zigzag persistence of the sequence is the representation of an A 2T −1 (f bf b · · · f b) type quiver composed by the unions of neighboring spaces and their canonical inclusions. The decomposition of the zigzag persistence module as a representation yields a persistence diagram again, where each interval captures the persistence of a homology generator in the deformations of spaces.

Matrix notation for morphisms in rep(A n )
The paper [Asashiba et al. 2019] established a new matrix notation for morphisms in rep(A n ) in the following way. This notation will give a clear perspective when arguing the well-definedness of persistence analysis in Section 4.
Definition 2.1 ( [Asashiba et al. 2019, Definition 3]). The arrow category arr(rep(Q)) of the category rep(Q) is a category whose objects are all morphisms in rep(Q), where morphisms are defined as follows. For two objects f : M → N and f : M → N in this category, an morphism from f to g is a pair (F M We remark that every morphism ϕ : V → W between representations of rep(A n ) is isomorphic to a morphism between direct sums of interval representations We use the notation [a, b] := { a, a + 1, . . . , b } over Z. A candidate for the basis is In a similar way, each block Φ c:d a:b can also be written in matrix form with the entries in where each M c:d a:b is a m c,d × m a,b matrix with the entries in K.
Definition 2.4. Let ϕ be a morphism in rep(A n ). The block matrix form Φ(ϕ) of ϕ is The paper [Asashiba et al. 2019] shows that isomorphisms in the arrow category arr(rep(A n )) correspond to row and column operations in block matrix form. These operations are performed by matrix multiplication with the same restriction, that is, the block of I [a, b] I[c, d] must be always zero. The column and row operations are almost the same as that of K-matrices, however because of the restriction, addition from a block to certain blocks is not permissible.

6
Let us discuss column operations. We define a morphism Φ such that the following diagram commutes: where Θ is an isomorphism. Namely, Φ and Φ are isomorphic in the arrow category. The morphism Θ is also a morphism between direct sums of interval representations, hence in the same way, Θ can also be written in block matrix form C c: The following is an example of permissibility in the case of arr(rep(A 3 (bf ))).
Example 2.5. The following matrix is the block matrix form of arr(rep(A 3 (bf ))), where we use the symbols a:b to denote the rows and columns corresponding to the direct summands The prohibited additions for columns and rows, written as red arrows, correspond to the positions of the zero blocks ∅. Since column addition from lower to upper blocks and row addition from left to right blocks are always prohibited, we write down only the prohibited column additions from upper to lower, and prohibited row addition from right to left. A notable fact is that either f 1:3 a:b or f a:b 1:3 is 0 for arbitrary (a, b) = (1, 3). In other words, I[1, 3] I[a, b] I[1, 3] if and only if (a, b) = (1, 3). We will refer to this fact in the proof of Lemma 4.3.

The induced maps via quiver representations
In this section, we redefine the induced map of a correspondence by using quiver representations. Let H = H(−; K) be the homology functor with coefficient K. As a representation of an A 3 type quiver, the diagram HX p * ← HF q * → HY induced by a correspondence F ⊂ X ×Y can be decomposed into a direct sum of interval representations: This indecomposable decomposition can be written as the diagram The choice of bases gives us a relationship between bases of HX and HY , which can be regarded as a map from HX to HY . For example, an interval representation I[1, 2] assigns an element of the standard basis of K dim HX to 0 in K dim HY . Therefore, nontrivial assignment happens only on the interval representations . By regarding the other interval representations as 0 maps from HX to HY , we can F ⇤

HX
HF HY Figure 3: An overview of our definition of the induced map F * of a correspondence. The isomorphism from the first row to the second row is an indecomposable decomposition. The inclusion map on the right-hand side is the canonical injection of the vector space.
where the arrows are the canonical projections of the vector spaces, and the morphism ι Y is the canonical injection of the vector space. Composing the path of morphisms, we can define the induced map of F through h as HX → HY. This definition does not need the two assumptions mentioned in Definition 1.3. Although our definition depends on the choice of isomorphism of indecomposable decomposition, when the two assumptions are satisfied, our definition coincides with the original definition q * • p −1 * by the following theorem.
In addition, q * = q * • p −1 * • p * because of the homological consistency q * (Ker(p * )) = 0, hence what we should prove is By chasing the diagram of Figure 3, this equation results in The standard basis of K dim HF corresponds to the standard bases of the four intervals I[2, 3], I[2, 2], I[1, 2], and I[1, 3]. Here we remark that the homological consistency q * (Ker(p * )) = 0 is equivalent to m 2,3 = 0, namely I[2, 3] does not exist as a direct summand. Moreover, the basis corresponding to I[2, 2] and I[1, 2] is mapped to 0 by both

Persistence analysis for sampled maps
The ability to decompose and focus exclusively on the interval representation I[1, 3] provides persistence analysis for sampled maps. Let us consider the following problem, which is similar to Problem 1.1 but requires additional assumptions of embeddings.
Y , and f are unknown, and we know only a sampled map f S : S → f (S) which is a restriction of f on a finite subset S ⊂ X, then can we retrieve any information about the homology induced map f * : HX → HY ?
Note that the sampling S is a point cloud capturing topological features of X when S is dense enough. Originally the paper [Edelsbrunner et al. 2015] sets this problem with the more additional assumption that X = Y and constructs a persistent homology of eigenspaces of the sampled map, in order to analyze the eigenspaces of the discrete dynamical system f .
In this section, we explain how to construct another persistent homology of a sampled map, which captures the generator of HX and Hf (X) connected by f . We can utilize two types of filtrations in Subsections 4.1 and 4.2. The former filtration is generated using simplicial complexes, and we will prove a stability theorem (Theorem 5.4) of this construction in Section 5. The latter filtration is generated using grids as in Section 3, and also derives stability (Theorem 5.6). The stability theorem for the latter, however, requires more assumptions than for the former. Hence we explain in this order.

Construction using simplicial complexes
First, we generate a filtration of abstract simplicial complexes of S, each simplex of which has elements of S as its vertices, so that the filtration can capture the topology of the underlying space X. For example,Čech complexes or Vietoris-Rips complexes [Edelsbrunner and Harer 2010] are available.
Definition 4.2. Let P ⊂ R n be a finite set. TheČech complex Γ r for P with a radius r is an abstract simplicial complex defined as where B(p; r) is the closed ball of center p and radius r. Let d R n be the Euclidean metric on R n . The Vietoris-Rips complex V r for P with a radius r is an abstract simplicial complex defined as Similarly, we also generate a filtration of abstract simplicial complexes Using these filtrations, we attempt to build a filtration of maps from f S to analyze the persistence of the original map f , in analogy with the classical technique of persistent homology. Although we expect the sampled map to derive a simplicial map C i → D i on each i-th filter, in general, they can derive only a simplicial partial map 1 f i : Hence, the conventional technique computing topological persistence for simplicial maps [Dey et al. 2014] is not available in this setup.
We remark that the graph Gr(f S ) = { (s, f (s)) ∈ R n × R n | s ∈ S } of the sampled map is a point cloud in R n × R n . Let us define the i-th abstract simplicial complex G i of Gr(f S ) as One can show that { G i } forms a filtration. The sequence of the partial maps can be regarded as a sequence { C i . . .
Applying the homology functor to the sequence, we obtain a sequence of representations of the A 3 (bf ) type quiver: Remark 1. The earlier research [Edelsbrunner et al. 2015] used domains of partial maps to construct a similar filtration. For a partial map f i : ( By the definitions of G i and dom f i , it is straightforward to see that the induced representation (3) is isomorphic to our representation (2). For the sake of consistency from the viewpoint of graphs and correspondences, we adopt the simplicial complexes { G i }.
Here, decomposing each filter HC i ← HG i → HD i as a representation to the intervals Projecting to I[1, 3] again, we obtain a sequence of subrepresentations which is three copies of an A type representation as we will see later.
We should be careful in the construction of Λ[1, 3]. We write canonical projections and injections defined by direct sum as respectively, and the morphisms in Λ as is defined by Φ i 1:3 1:3 := π i+1 • Φ i • ι i , which is the submatrix at (1:3, 1:3) in block matrix form of Φ i .
At a glance, this construction seems natural, but "π : Λ → Λ[1, 3]" is not a morphism in the representation category. Namely, does not always commute. Consequently, the choice of isomorphism of indecomposable decomposition on each filter may make a difference in the output persistence diagram. In order to make sense of this analysis, the output persistence diagram should be uniquely determined and independent of the choice of isomorphism. The following theorem guarantees uniqueness and independence. The restriction to the block (1:3, 1:3) has the following functoriality. is uniquely determined and independent of the choice of the bases of Proof. Let Ψ i be a morphism isomorphic to Φ i , which is written as a commutative diagram for some isomorphisms C and R. Namely, Φ i = RΨ i C, and by Lemma 4.3, applying the restriction yields Φ i which is an A type representation. We call this representation persistent homology of the sampled map f S . Decomposing into intervals, we can draw a persistence diagram, which shows us the robustness of the generators of homology in both filtrations, which are assigned by f . Simultaneously we have constructed the filtration of complexes approximating the unknown spaces X and f (X).
In comparison with the earlier research This persistence diagram does not provide any information about eigenvectors, unlike [Edelsbrunner et al. 2015]; however, it can be widely applied. First, since our method does not use the eigenspace functor, we need not require both sides' spaces to be the same. Therefore, even in the case of sampled dynamical systems X = Y like the previous research, we can weaken the assumption f S : S → S to f S : S → f (S) and take another filtration on f (S). (If f (S) is not dense enough for sampling X, then we can take S ∪ f (S) instead.) Moreover, since the previous method needs to set an eigenvalue before analysis, they have to predict some behavior of f in advance, but our method does not need any prior information. The numerical experiments in Section 7 will emphasize this difference.

Constriction using a grid
Subsequently, we provide another construction of a filtration constructed by dividing the spaces. Suppose the spaces X and Y are embedded into Euclidean space R n , and both R n are divided by n-dimensional ε-cubes { [a 1 ε, (a 1 + 1)ε] × · · · × [a n ε, (a n + 1)ε] | a 1 , . . . , a n ∈ Z } .
To distinguish the two divisions, we write this set as X ε for the X side Euclidean space and Y ε for Y side. Let f S be a sampled map of a continuous map f : X → Y , and p and q be the canonical projections of R n × R n to the X side and Y side Euclidean spaces, respectively. First, we generate a correspondence where Gr(f S ) := { (s, f (s)) | s ∈ S } (see Figure 2). In this setup, we use the L ∞ metric d ∞ ((x i ), (y i )) := max i (|x i − y i |) for both spaces R n and R n ×R n . To construct a filtration along with the grids, let us define the filtration of a correspondence and morphisms p iε := p F iε and q iε := q F iε . Here we restrict i = 1, . . . , for sufficiently large . Then we have a similar diagram as before, allowing us to obtain the filtrations { p(F iε ) } and { q(F iε ) }, capturing the persistent topological features of X and f (X). Again, the homology functor derives the sequence of morphisms in rep(A 3 (bf )), therefore we can execute the same analysis as before, transforming it into block matrix form, restricting to the blocks (1:3, 1:3), identifying it with a representation of the A type quiver, and producing a persistence diagram.

Stability
In order for a tool in topological data analysis to be considered practical, the output persistence diagrams should behave continuously toward input data. Such a property is known as a stability theorem [Cohen-Steiner et al. 2007, Chazal et al. 2009] and has been proved for persistence modules on R.
Let vect be the category of finite dimensional vector spaces, R be the poset category of real numbers 3 . An object of the functor category vect R is also called a persistence module in some papers. To distinguish it from our definition, we call this an R-persistence module.
Specifically, for an R-persistence module M , we assign a vector space M t for t ∈ R and a linear map ϕ M (s, t) : for all r ≤ s ≤ t ∈ R. We call the linear maps ϕ M (s, t) transition maps. A morphism f : M → N of R-persistence modules is a natural transformation, that is a collection of for all s ≤ t ∈ R. We remark that every persistence module can be similarly regarded as a functor from a finite poset category to vect.
The fundamental objects of R-persistence modules are interval modules K I for intervals I ⊂ R, given by (K I ) t = K for t ∈ I and (K I ) t = 0 otherwise, and with the morphism corresponding to s ≤ t ∈ I is an identity map. As is the case with persistent homology, every R-persistence module can be decomposed into a direct sum of interval modules [Crawley-Boevey 2015].
We can define a distance between R-persistence modules, called the interleaving distance.
An often used distance between persistence diagrams is the bottleneck distance, which is defined by bijections between them. It is well-known that the interleaving distance of R-persistence modules is equal to the bottleneck distance of their persistence diagrams [Lesnick 2015, Bauer andLesnick 2014]. Hence, by showing that a distance between input data is greater than the interleaving distance of their R-persistence module, we can prove the stability of the persistence diagrams toward input data.
In analogy with [Edelsbrunner et al. 2015], stability theorems for some filtrations also hold on our analysis as follows. The discrete setting discussed in Section 4 is enough for implementation, but we extend it to a continuous analysis to prove its stability. Now we use the following filtrations for S and f (S). Let d R n ×R n be a distance on R n × R n defined by d R n ×R n ((x 1 , y 1 ), (x 2 , y 2 )) := max{d R n (x 1 , x 2 ), d R n (y 1 , y 2 )}, where d R n is the Euclidean metric on R n . For a subset U of R n , we define a function d U : R n → R ≥0 to be infimum distance to a point in U . In the same way, we define the function d U : R n × R n → R ≥0 for a subset U of R n × R n . We use the notation U r := d −1 U [0, r] to denote the sublevel sets. Let Top (bf) be the functor category from the A 3 (bf ) type quiver (· ← · → ·) as a poset category to the category of topological spaces. The sublevel sets S r , f (S) r , and commutes for every s ≤ r ∈ R ≥0 . In the same way as in the discrete analysis, applying the homology functor H to the filtration produces { HS r ← H Gr(f S ) r → Hf (S) r }, which is a family of objects in the representation category rep(A 3 (bf )) with the induced morphisms from Diagram (4).
Remark 2. We have constructed the different representations from the previous representations using complexes in Subsection 4.1, but these are isomorphic if we adopť Cech complexes. It is known by the Nerve Lemma [Borsuk 1948] that, if U is a finite subset in a metric space, then the sublevel set U r is homotopy equivalent to theČech complex of U with radius r. Therefore, letting C r , G r , and D r beČech complexes with radius r of the finite subsets S, Gr(f S ), and f (S), respectively, the induced family { HC r ← HG r → HD r } is isomorphic to the family { HS r ← H Gr(f S ) r → Hf (S) r }.

Since decomposing every representation into intervals is isomorphic in the functor category rep(A
b,d }, and the induced morphisms can be written in block matrix form again.
By Lemma 4.3 and Theorem 4.5, the family { I[1, 3] m r 1,3 } and the induced morphisms are uniquely determined up to isomorphism. This gives us three copies of the R-persistence module { K m r 1,3 = K m r 1,3 = K m r 1,3 }. Thus, we obtain an R-persistence module { K m r 1,3 }. We denote this R-persistence module of the sampled map f S as M f S and call it the R-persistence module of the sampled map.
Remark 3. The construction of an R-persistence module using graphs does not require the assumption that S is a finite set. Therefore, if we assume that dim HX r , dim H Gr(f ) r , and dim Hf (X) r are finite for an arbitrary r, then the same analysis can be executed on the filtration { X r ← Gr(f ) r → f (X) r }, deriving an R-persistence module M f in the same way. We call this the R-persistence module of the map f . The output persistence diagram portrays the robustness of the generators of the homology induced map f * .
After these setups, we can show the following stability theorem.
Theorem 5.4. Let d H be a Hausdorff distance induced by d R n ×R n . For two sampled maps h : S → R n and h : S → R n , let M h , M h be the R-persistence modules of the sampled maps. Then, Proof. Let ε := d H (Gr(h), Gr(h )), and r be an arbitrary real number. By the definition of Hausdorff distance, Gr(h) r ⊂ Gr(h ) r+ε and Gr(h ) r ⊂ Gr(h) r+ε . Moreover, ε = d H (Gr(h), Gr(h )) implies that d H (S, S ) ≤ ε and d H (h(S), h (S )) ≤ ε, hence S r ⊂ S r+ε , S r ⊂ S r+ε , h(S) r ⊂ h (S ) r+ε , and h (S ) r ⊂ h(S) r+ε as well. These inclusions induce the commutative diagrams: By those functoriality, it is straightforward that these inclusions induce ε-interleaving morphisms between M h and M h , and we have d Accordingly, the obtained persistence modules and persistence diagrams can only have as much noise as S or its evaluation by f .
Since the proof does not use the assumption that S is a finite set, a similar inequality holds for R-persistence modules of maps.
Corollary 5.5. Let U and U be subsets in R n . If R-persistence modules M h and M h of maps h : U → R n and h : U → R n are defined, then By Corollary 5.5, the error (bottleneck distance) between the persistence diagram of a sampled map f S and that of the original map f is bounded by the error d H (Gr(f ), Gr(f S )) of the sampled map. If the sampled map is dense enough, we can infer the persistent generators of f * from the persistence module of the sampled map.
Finally, let us provide the stability of the persistence analysis using grids. We regard the persistent homology constructed using a grid as an R-persistence module by the embedding induced by Theorem 5.6. Let d H be a Hausdorff distance induced by d ∞ . For two sampled maps h : S → R n and h : S → R n , we write the filtrations of correspondences as { F r } and { F r }, and let M h and M h be their output R-persistence modules. If .
It is clear that these inclusions induce the ε-interleaving morphisms between M h and M h .

Application of functoriality to 2-D persistence modules
The functoriality lemma, Lemma 4.3, can be generalized for the restriction to every "diagonal" block. Precisely, since the candidate of intervals I[c, d] satisfying relations holds for all I [a, b]. This result can be checked easily, not only on the orientation bf but also on every orientation of any length, as follows. Suppose I[c, d] = I[a, b], which can happen when a = c or when b = d. In the case that a = c, we may assume a < c without loss of generality. When (c − 1)-th orientation . Then, the commutative diagram of the morphism g from (c − 1) to c is gc .
It is obvious that g c−1 = 0, and the commutativity derives g c = 0. Since the commutativity of the diagram on g derives g i = 0 for the other vertices i, g = 0.  a, b] when (c−1)-th orientation is b in a similar discussion, using the commutative diagram gc .

Similar arguments also hold in the case that
That is why we can extend the statement on the orientation bf to general τ n as follows.
Proposition 6.1. Let and Ψ : 1≤a≤b≤n where every row has the same orientation τ n 2 . The 2-D persistence modules sometimes appear and cause problems in the context of persistence analysis for time series data. See [Carlsson and Zomorodian 2009] for details and higher dimensional persistence. In our context, the 2-D persistence module naturally appears when we consider iterations of a sampled map or compositions of sampled maps. Suppose we have a time series of some point clouds { S 1 , S 2 , . . . , S T } in the same Euclidean space, with their transition as maps { f i : S i → S i+1 }. We generate a filtration of abstract simplicial complexes C t 1 ⊂ · · · ⊂ C t n for each S t . As we saw in Section 4, the maps between points induce a filtration of partial maps f t i : by taking the i-th simplicial complex G t i of Gr(f i ) defined as Eq. (1). As a consequence, the homology functor induces the 2-D persistence module from the above diagram. In this case, we can observe the 2-D persistence module from the viewpoint that the horizontal (vertical) direction on the diagram describes persistence in time (space, respectively).
Let us go back to Diagram (5). In the same way as in the specific case τ n 2 = bf , Diagram (5) can be regarded as a sequence of morphisms in the category rep(A n 2 (τ n 2 )). By decomposing representations of A n 2 (τ n 2 ) in each row, we can deal with the morphisms as matrices in block matrix form. Restricting each matrix to the diagonal block (a:b, a:b) derives a sequence of matrices whose domains and codomains are direct sums of I[a, b]. As this sequence is b−a copies of nonzero representations of A n 1 and n 2 −(b−a) copies of zero representations, we can take one of the nonzero representations. Finally, we obtain the persistence diagram by decomposing. Proposition 6.1 ensures the uniqueness of the output persistence diagram.
In the case of 2-D persistence modules derived from Diagram (6), when we take the block (a:b, a:b) as (1:n 2 , 1:n 2 ), each generator of the output persistence module survives under all transitions, and its lifetime in the persistence diagram shows how robust it is in the Euclidean space. Although this process ignores much information stored in the other blocks, it is an approach to 2-D persistence analysis that can capture the rough topological structures.

22
The author has implemented the persistence analysis in Subsection 4.1. Here we fix the field for the coefficient of matrices and the homology functor as Z/1009Z.

Remark 4.
To implement the persistence analysis on computers, we must use finite fields as the coefficient. Here, every map is written as a matrix. If we choose the field Z/2Z as the coefficient, then every entry with the prime factor 2 in the matrix is regarded as 0. For example, a homology generator a mapped as f * (a) = 2a, such as the example discussed later and in Figure 4, is ignored. Therefore, it is better to choose a larger prime number p as the coefficient Z/pZ to retrieve more generators.
The implementation uses Vietoris-Rips complexes for simplicity, whileČech complexes are theoretically more satisfying (see [Edelsbrunner and Harer 2010, Section III.2] for details). Except for the construction of the persistence module from the sequence of the pairs of the maps { (p i * , q i * ) }, it basically follows the algorithm in [Edelsbrunner et al. 2015] (recall Remark 1).
First, we generate the boundary matrix induced by the filtration of the Vietoris-Rips complexes for each point cloud S and f (S), and then the boundary matrix of the filtration { G i }. We can use the original persistence algorithm [Edelsbrunner and Harer 2010] to compute the reduced boundary matrices and the bases of the persistent homology of the filtration.
Second, since we can obtain the homology bases for each filtration, we generate the maps p i * and q i * as matrices between the homology basis for each filter. In the same way, we compute the induced maps of the inclusions j * : HG i → HG i+1 as matrices. To obtain the basis of I[1, 3] for each matrix p i * and q i * , we execute the following elementary row and column operations.
1. Transform p i * to Smith normal form: where P 1 and Q 1 are regular matrices corresponding to elementary operations, and r 1 is the rank of p i * .
2. Since p i * and q i * share the same basis for the columns, the elementary column operations Q 1 are simultaneously performed on q i * where X 1 is the submatrix on the basis of columns corresponding to the above I r 1 , and X 2 is the submatrix corresponding to the 0 columns in P 1 p i * Q 1 .
3. Transform X 2 to Smith normal form with elementary row operations P 2 and elementary column operations Q 2 P 2 X 1 I r 2 0 0 0 = X 3 I r 2 0 X 4 0 0 where P 2 X 1 is divided into submatrices of appropriate sizes on the right-hand side. We remark that these column operations have no side effect on the p i * side matrix since every column corresponding to the basis is zero.
4. Zero out X 3 by I r 2 using column operations without any side effect: 5. Transform X 4 to Smith normal form with elementary row operations P 4 and elementary column operations Q 4 : Here the column operations have side effect on I r 1 transforming it to a matrix Q 4 , but Q 4 is regular, hence we can transform it to I r 1 again using only row operations.
6. Finally, we obtain the matrix transformations which are decomposed into intervals. The rows and columns corresponding to two I r 3 are pairs of identity maps, which are I[1, 3]. Therefore the basis in the columns of I r 3 is what we want.
Applying the change of basis of HG i during the above column operations and restricting to the basis corresponding to I r 3 , we finally obtain the persistent homology of the sampled map. At last, we decompose the persistent homology into intervals using the decomposition algorithm in [Edelsbrunner et al. 2015, Subsection 3.4] and plot a persistence diagram.
By tracking the inverse of the matrix transformations executed above, we can write down the cycles corresponding to the generators in the persistence diagram. As we remarked in Section 3, the cycles depend on the choice of the bases of I[1, 3]. Nevertheless, in the following numerical experiments, we succeed in reconstructing underlying maps.

Twice mapping on a circle
As an example for input data, let us consider the twice map on the unit circle f : S 1 → S 1 defined by f (z) := z 2 . Note that, in this case, the spaces in Problem 4.1 are given as X = Y = S 1 embedded in R 2 , and we regard R 2 as C. The sampled points of the unit circle are 100 points z j := cos(2π j 100 ) + √ −1 sin(2π j 100 ) for 0 ≤ j < 100, with added Gaussian noise with σ ∈ [0.00, 0.30]. Gaussian noise is at σ = 0.18. The black points are sampled points for the domain and the blue crosses are its image by f . As presented in Figure 5, the generator is unique. The corresponding generator in the domain side is described by the black edges approximating the unit circle. The generator in the image side is the blue dashed edges, and we can observe that it turns around the origin twice.
A computational result is presented in Figure 4, which portrays the sampled map at σ = 0.18 and its unique generator of the persistence diagram. The generator is the corresponding cycle in HG b at the birth radius b and is indeed approximating the unit circle, and we can see that its image is turning around the origin twice. Results for other noises are shown in Figure 6. Figure 5 presents the persistence diagrams under changing σ from 0 to 0.3. As expected, the lifetime of the unique generator decreases as the noise increases.

Inverse mapping on a circle
To emphasize the difference from the existing method using eigenspace functors, let us consider the inverse map on the unit circle g : S 1 → S 1 defined by f (z) := z −1 . The sampled points { z j } and the range of the parameter σ of Gaussian noises are the same as before. The computational results are presented in Figures 7 and 8.
The analysis using eigenspace functors can detect such a generator using the eigenspace functor with eigenvalue −1. In other words, prior knowledge about the eigenvalue is essential. On the other hand, our method can use the same construction both for the inverse mapping and twice mapping.