Persistence Modules

2016 ◽  
pp. 15-29
Author(s):  
Frédéric Chazal ◽  
Vin de Silva ◽  
Marc Glisse ◽  
Steve Oudot
Keyword(s):  
Persistence Modules

scholarly journals Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules

2015 ◽  
Vol 22 (1) ◽  
pp. 227-296 ◽  
Author(s):  
Leonid Polterovich ◽  
Egor Shelukhin
Keyword(s):  
Hamiltonian Flows ◽  
Persistence Modules

scholarly journals Higher Interpolation and Extension for Persistence Modules

2017 ◽  
Vol 1 (1) ◽  
pp. 272-284 ◽  
Author(s):  
Peter Bubenik ◽  
Vin de Silva ◽  
Vidit Nanda
Keyword(s):  
Persistence Modules

scholarly journals Computing the Interleaving Distance is NP-Hard

2019 ◽  
Vol 20 (5) ◽  
pp. 1237-1271 ◽  
Author(s):  
Håvard Bakke Bjerkevik ◽  
Magnus Bakke Botnan ◽  
Michael Kerber
Keyword(s):  
Computational Complexity ◽  
Complete Characterization ◽  
Np Hard ◽  
Slight Improvement ◽  
Approximation Factor ◽  
Distance Computation ◽  
Persistence Modules ◽  
Np Complete ◽  
Interleaving Distance

Abstract We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are 1-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving distance computation of a special type of persistence modules. We show that this matrix invertibility problem is NP-complete. We also give a slight improvement in the above reduction, showing that also the approximation of the interleaving distance is NP-hard for any approximation factor smaller than 3. Additionally, we obtain corresponding hardness results for the case that the modules are indecomposable, and in the setting of one-sided stability. Furthermore, we show that checking for injections (resp. surjections) between persistence modules is NP-hard. In conjunction with earlier results from computational algebra this gives a complete characterization of the computational complexity of one-sided stability. Lastly, we show that it is in general NP-hard to approximate distances induced by noise systems within a factor of 2.

scholarly journals Persistence Modules with Operators in Morse and Floer Theory

2017 ◽  
Vol 17 (4) ◽  
pp. 757-786 ◽  
Author(s):  
Leonid Polterovich ◽  
Egor Shelukhin ◽  
Vukašin Stojisavljević
Keyword(s):  
Floer Theory ◽  
Persistence Modules

scholarly journals The observable structure of persistence modules

2016 ◽  
Vol 18 (2) ◽  
pp. 247-265 ◽  
Author(s):  
Frédéric Chazal ◽  
William Crawley-Boevey ◽  
Vin de Silva
Keyword(s):  
Persistence Modules

scholarly journals Decomposition of pointwise finite-dimensional persistence modules

2015 ◽  
Vol 14 (05) ◽  
pp. 1550066 ◽  
Author(s):  
William Crawley-Boevey
Keyword(s):  
Chain Condition ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Persistence Modules ◽  
Descending Chain Condition ◽  
Persistence Module

We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on images and kernels.

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