Abstract
This chapter examines the interleaving relation between persistence modules, and the associated interleaving metric. Interleavings are approximate isomorphisms, and in the first instance may be defined by a pair of ‘shifted’ homomorphisms between the two persistence modules being compared. More abstractly, an interleaving can be thought of as a solution to a functor extension problem. The Interpolation Lemma is the main result of this chapter: it asserts that a pair of interleaved persistence modules can be interpolated by a 1-Lipschitz 1-parameter family. We give three different explicit constructions of the interpolation; two of them are the left and right Kan extensions (in the functor extension point of view), while the third mediates between the two.
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Notes
- 1.
The image and the coimage are naturally isomorphic; the difference is whether we wish to think of \(\overline{{\mathbb {W}}}\) as a submodule of \(\mathbb {C}\oplus \mathbb {D}\) or as a quotient module of \(\mathbb {A}\oplus \mathbb {B}\). In the following pages, we will treat the two points of view with equal emphasis.
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Chazal, F., de Silva, V., Glisse, M., Oudot, S. (2016). Interleaving. In: The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-42545-0_4
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DOI: https://doi.org/10.1007/978-3-319-42545-0_4
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42543-6
Online ISBN: 978-3-319-42545-0
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