Science Mathematics

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

By Steve Y. Oudot

Endurance concept emerged within the early 2000s as a brand new thought within the region of utilized and computational topology. This publication presents a huge and sleek view of patience thought, together with its algebraic, topological, and algorithmic points. It additionally elaborates on purposes in facts research. the extent of aspect of the exposition has been set for you to continue a survey variety, whereas offering 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 idea. the second one half makes a speciality of its connection to functions via a couple of chosen themes. The 3rd half offers views for either the idea and its purposes. it may be used as a textual content for a direction on utilized topology, on facts research, or on utilized records.

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Given a subset T ⊆ R, a filtration X over T is a family of topological spaces Xi , parametrized by i ∈ T , such that Xi ⊆ Xj whenever i ≤ j ∈ T. Thus, X is a special type of representation of the poset (T, ≤) in the category of topological spaces. X is called (finitely) simplicial if the spaces Xi are (finite) simplicial complexes and Xi is a subcomplex of Xj whenever i ≤ j. When the Xi 29 30 2. TOPOLOGICAL PERSISTENCE are subsets of a common topological space X, with i∈T Xi = ∅ and i∈T Xi = X, X is called a filtration of X, and the pair (X, X ) is called a filtered space.

6) ∅ ⊆ Kt1 ⊆ Kt2 ⊆ · · · ⊆ Ktn = K. Define the time of appearance of a simplex σ ∈ K to be t(σ) = min{ti | σ ∈ Kti }. The level sets of t are the Kti \ Kti−1 , where by convention we let Kt0 = ∅. The order induced by t on the simplices of K is only partial because it is unspecified within each level set. 4 of Appendix A. 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.

11] for the details. 4 applies. 2. CALCULATIONS 43 homomorphism is induced by the boundary operator of the simplicial filtration Kσ , whose coefficients are in k[t]. This provides us with an algorithm for computing persistence: build the matrix of the boundary operator of Kσ , then compute its Smith normal form over k[t] using Gaussian elimination on the rows and columns, then read off the interval decomposition. The algorithm is the same as for non-persistent homology, except the ring of coefficients is k[t] instead of Z.

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