Análisis de las sesiones realizadas en el aula

In the recent years, methods for computing multipersistence, limitedly to simplicial complex filtrations, have been proposed. Proposing new algorithms for multipersistence retrieval is a very active research branch for the computational topology community. Indeed, only some of the methods we present in this overview were available before we started the work of this thesis. As reported in Section 1.7, the invariants or, more in general, the representations available for multipersistence are not equivalent one another. So, we classify the available methods according to the invariant or the representation they provide:

• persistence module retrieval [33,94];

• slice-compatible invariants retrieval [17,27,133]; • other invariants retrieval [94,33,133].

Persistence module retrieval. The first algorithm for the persistence module retrieval of a simplicial filtered complex S of S is proposed in [33], where the three tasks of computing the k- boundaries, k-cycles and their quotients at each grade u are translated into submodule membership problems in computational commutative algebra. As drawbacks, the algorithm introduces an artefact dependency on the chosen basis for expressing the chain complex and implies high computational costs in terms of time: Op|S|4n3q, where n is the number of independent parameters in the filtration.

The actual persistence module can be retrieved by means of the algorithm proposed in [94]. The algorithm has been implemented in the Topcat library [154] and distributed in public domain. See Section6.2for a preliminary evaluation of our method as a preprocessing to the Topcat library performances. The algorithm acts on the filtration S at chain level rather than at homology level. First, k-cycles and k-boundaries are expressed in terms of the same basis along the filtration at the chain level. Then, the Smith Normal Form reduction is applied at each step u in the filtration leading to a worst time complexity of Op|S|3u¯nq, where ¯uis the maximum cardinality among the components uiin the poset of filtration grades, and n the number of dimensions in the filtration.

Slice-compatible invariants retrieval. In the time line, the first approach in this class is the foliation methodintroduced by Biasotti et al. in [17] for the retrieval of the persistence space limitedly to the case of 0th-homology. Then, in [27], the foliation method is extended to higher homology degrees. The foliation method is not properly an algorithm but rather a framework, particularly suited for shape comparison purposes in the context of Size Theory [89] but not strictly limited to it. The foliation method reduces the computations to multiple iterations of one-persistence computations over each possible filtration slice. The cost of the foliation method is the cost of each persistence computation times the number of possible slices. Encoding the persistence space retrieved by the foliation method has the cost of a single persistence diagram for each possible slice in the set of filtration grades. In Section6.3, we evaluate the impact on our implementation of the foliation method of the preprocessing described in Chapter4.

Here, the filtration is induced by sublevel sets with respect to a continuous filtering function φ : Y ´Ñ Rnwith Y a topological space. Each injective linear map l : R ´Ñ Rndetermining the filtration slice FlpX q is uniquely determined by two parameters

• m P Rn_{such that}řn

i“1mi“ 1 and mią 0, for all 1 ď i ď n;

• b P Rn_{such that}řn

i“1mi“ 0.

Indeed, if R is parametrized by a P R, then we define l : R ´Ñ Rnby

lpaq :“ a ¨ m ` b. (2.3)

φl : Y ´Ñ R such that, for each point x P Y ,

φlpxq :“ min

i“1,...,nmi¨ maxi“1,...,n

φipxq ´ bi

mi

.

By Proposition 1 in [17], each pair pu, vq such that u ă v or pu, 8q with 8 greater than all other elements in Rnuniquely determines a linearization l through pu, vq. Hence, there is a unique pair pa, bq P Rnˆ Rnsuch that lpaq “ u and lpbq “ v. Hence, Theorem 3 in [17], written equivalently in terms of rank invariant rather than size functions, ensures that, if M “ Hk˝ F and Ml“ M ˝ l,

then

rankMpu, vq “ rank_Mlpa, bq. (2.4)

Thus, Equation (2.4) ensures that one-parameter persistence can be performed independently over each slice. Moreover, this can be done by means of any available algorithm or tool.

In the context of Size Theory, the main purpose is that of comparing shapes. Existing algorithms implementing the foliation method have not as final goal that of retrieving the entire persistence space. The persistence space is a mean towards the computation of the matching distance between persistence spaces which quantifies the (dis)similarity between shapes. To this purpose, an approximate version of the persistence space is retrieved by the algorithm proposed in [47]. The space of all available filtration slices is uniformly sampled. Authors quantify the density of the sampling which is necessary for determining the matching distance up to a fixed threshold error. The same procedure is specialized and optimized for two-parameter filtrations in [16] in a hierarchical way. This method finds applications for shape comparison in the PHOG library [15] where authors use the approximate persistence space to deal with photometric attributes. The tool handles graph domains with multiple filtering functions. The foliation method is applied and then the persistence diagram of the graph is computed by applying the optimization by ∆˚_-reduction

introduced in [57].

For the two-parameter case, i.e., for n “ 2, a complete representation of a multipersistence slice-compatible invariant is the one proposed in [133] for the retrieval of the fibered barcode of M “ Hk˝ F pSq with S a simplicial complex. The algorithm is a variation of the foliation method

where slices are parametrized to exploit suitable duality properties for n “ 2. The algorithm is implemented in the Rivet visualization tool [133] which is available online at [197].

Rivet consists of three main steps:

• a preprocessing by multigraded Betti numbers to reduce the number of filtration grades; • introduction of the augmented arrangement data structure for handling a barcode template; • successive and efficient updatings of barcodes in the template from one slice to another.

In the preprocessing step, Rivet locates λiinteresting grades along each component in the grade

poset via bigraded Betti numbers [127,83]. This procedure is optimized for the n “ 2 case and requires time Op|S|3λq, where λ is the product of λ1λ2.

Secondly, the support of the bigraded Betti numbers is crucial for Rivet to introduce a specific data structure, called augmented arrangement based on the duality line-to-point for lines in the plane connecting points determined by grades in the support of bigraded Betti numbers. The barcode template is obtained in time Op|S|3λ` p|S| ` log λ qλ2q.

In the final step, the barcode retrieval is successively updated over each entity in the augmented arrangement. The final step is linear in |S| and the implementation is optimized to decide whether to modify the barcode from adjacent entities in the augmented arrangement or to compute it from scratch. Though limitedly to the two-parameter case, this strategy sensibly increases the size of the inputs which is possible to handle and, depending on the kinds of filtration, it has led to computations over simplicial complexes of up to 15M simplices. This is not comparable to one-parameter persistence standards but, up to now, it is way the highest size available for multipersistence.

Other invariants retrieval. The rank invariant value over a single pair pu, vq can be easily derived from a persistence module representation by an algorithm discussed in [33]. However, the full rank invariant computation requires the iteration of this simple procedure for all possible grades satisfying u ĺ v in the filtration which multiplies the complexity by approximately 1₂|F |2, where |F| is the, typically very large, cardinality of filtration grades. The Topcat library [154] implements the function for the rank invariant computation starting from the persistence module representation obtained through [94]. The implemented function computes the rank invariant in each pu, vq but not the full rank invariant.

The algorithm in [94] is not limited to the persitence module retrieval but it also allows for the retrieval of the feature counting map.

The preprocessing performed by Rivet allows to compute the multigraded Betti numbers, limitedly to the two-parameter persistence case.