Abstract
This chapter begins with a brief history of topological persistence from the early 1990s to the present day. This gives context for the contributions of the present monograph. The need for an effective theory of real-parameter persistence modules (the main topic of the monograph) is illustrated by two applications of persistence, to the machine-learning tasks of classification and clustering. We provide an extensive reading list, arranged by topic, to help acquaint the reader with the wider literature on persistence.
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Notes
- 1.
We do not give the details of this construction here; see [13, 16] for instance. What matters is that the signature \(\mathsf {dgm}(\mathrm{H}_*(\mathbb {R}\mathrm {ips}(P)))\) is easily computed, the distance between two signatures is easily computed, the distance is robust in the sense of Theorem 1.1, and that the lower bound in the theorem is sufficiently tight to solve the learning problem under consideration.
- 2.
Readers who wish to adopt our notation are invited to contact us for the LaTeX macros.
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Chazal, F., de Silva, V., Glisse, M., Oudot, S. (2016). Introduction. In: The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-42545-0_1
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DOI: https://doi.org/10.1007/978-3-319-42545-0_1
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