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2008

Zigzag Persistence

Carlsson, Gunnar, de Silva, Vin

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We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence.

Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by that theory.
In this paper we develop theoretical and algorithmic foundations with a view towards applications in topological statistics.

Zigzag Persistence · Around

Built on

On the Krull–Schmidt theorem with application to sheaves
M. F. Atiyah · 1956
Earlier work this paper cites.
Unzerlegbare darstellungen I
P. Gabriel · 1972
Earlier work this paper cites.
Infinite root systems, representations of graphs and invariant theory
V. G. Kac · 1980
Earlier work this paper cites.
Three-dimensional alpha shapes
Herbert Edelsbrunner and Ernst P. Mücke · 1994
Earlier work this paper cites.

Similar

Topological persistence and simplification
Herbert Edelsbrunner, David Letscher, and Afra Zomorodian · 2002
Cited alongside, same era.
Topological estimation using witness complexes
Vin de Silva and Gunnar Carlsson · 2004
Cited alongside, same era.
Quiver representations
Harm Derksen and Jerzy Weyman · 2005
Cited alongside, same era.
Computing persistent homology
Afra Zomorodian and Gunnar Carlsson · 2005
Cited alongside, same era.

Then

Stability of persistence diagrams
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer · 2007
Later among the works it cites.
Zigzag persistent homology and real-valued functions
Gunnar Carlsson, Vin de Silva, and Dmitriy Morozov · 2008
Closest in time.
On the local behavior of spaces of natural images
Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, and Afra Zomorodian · 2008
Closest in time.
Extending persistence using Poincaré and Lefschetz duality
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer · 2008
Closest in time.

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