Download Persistence Theory: From Quiver Representations to Data by Steve Y. Oudot PDF

By Steve Y. Oudot

Patience conception emerged within the early 2000s as a brand new conception within the zone of utilized and computational topology. This e-book presents a large and smooth view of patience conception, together with its algebraic, topological, and algorithmic elements. It additionally elaborates on functions in information research. the extent of aspect of the exposition has been set in order to preserve a survey type, whereas delivering adequate insights into the proofs so the reader can comprehend the mechanisms at paintings. The e-book is prepared into 3 components. the 1st half is devoted to the principles of endurance and emphasizes its connection to quiver illustration concept. the second one half specializes in its connection to functions via a number of chosen issues. The 3rd half presents views for either the speculation and its functions. it may be used as a textual content for a path on utilized topology, on info research, or on utilized records.

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Extra info for Persistence Theory: From Quiver Representations to Data Analysis

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22 1. 1. The four decorated points corresponding to intervals b± j , dj . For simplicity we will also omit the index set T in the notation when it is irrelevant or obvious from the context. 12) ± I b± j , dj . j∈J ± b± j , dj The set of intervals , ordered by the lexicographical order on the decorated coordinates, is called the persistence barcode of V. Technically it is a multiset, as an interval may occur more than once. 1. This multiset of decorated points is called the decorated persistence diagram of V, noted Dgm(V).

10. g. [2, 3] in the previous example. 6 This rule has been applied implicitly until now. It is arbitrary and not motivated by mathematical considerations. Other rules can be applied as well, leading to different writings of the same interval S of T . 22 1. 1. The four decorated points corresponding to intervals b± j , dj . For simplicity we will also omit the index set T in the notation when it is irrelevant or obvious from the context. 12) ± I b± j , dj . j∈J ± b± j , dj The set of intervals , ordered by the lexicographical order on the decorated coordinates, is called the persistence barcode of V.

It provides a complete matching between the interval decompositions of the two x-monotone paths considered. 3] for the details. 40 2. TOPOLOGICAL PERSISTENCE that is, a simplex never appears in K before its faces. 7) K = {σ1 , σ2 , · · · , σm }, where i < j whenever t(σi ) < t(σj ) or σi is a proper face of σj . The sequence of simplices σ1 , σ2 , · · · , σm is the actual input taken in by persistence algorithms. It defines the following simplicial filtration of K, denoted Kσ : ∅ ⊆ {σ1 } ⊆ {σ1 , σ2 } ⊆ · · · ⊆ {σ1 , · · · , σm } = K.

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