A derived isometry theorem for sheaves

Berkouk, Nicolas; Ginot, Grégory

doi:10.1016/j.aim.2021.108033

Cited by 8 publications

(9 citation statements)

References 23 publications

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“…When V is a one-dimensional real vector space, the category D b Rc (k V ) enjoys a structure theorem which ensures the existence of a graded-barcode for its objects, and allow to derive explicit computations for the convolution distance. This was studied in detail in [16,1]. These results extends the key theorem of Crawley-Boevey [9] to constructible sheaves on the real line.…”

Section: The Dimension One Casementioning

confidence: 70%

“…Here, we strengthen this result by removing the properness assumption. We then recall some results specific to the dimension one case, where γ-sheaves have a gradedbarcode [1]. We end the section with the notion of fibered barcode which was introduced in [7] for persistence modules, that we translate in our γ-sheaf setting.…”

Section: γ-Sheavesmentioning

confidence: 99%

“…Implementing such an approach requires relating precisely the classical theory of persistent modules and sheaf theory. This has largely been achieved thanks to several works: [1,2,3,13,16,20]. Heuristically, this correspondence goes as follows.…”

Section: Introductionmentioning

confidence: 97%

“…Multi-parameter persistence is difficult both in theory (there cannot exist any notion of barcode after [5]), and in practice (the interleaving distance is NPhard to compute [4]). Using tools from sheaf theory, we develop new invariants and pseudo-metrics that can be efficiently computed on specific classes of multiparameter persistence modules thanks to [1,2]: sub-level and level sets multiparameter persistence modules.…”

Section: Introductionmentioning

confidence: 99%

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Projected distances for multi-parameter persistence modules

Berkouk¹,

Petit²

2022

Preprint

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We introduce the new notions of projected distances and projected barcodes for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto R. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). We conduct a systematic study of the stability of projected barcodes and show that the fibered barcode is a particular instance of projected barcodes. We prove that the ISM and the SCD provide lower bounds for the convolution distance. Furthermore, we show that the γ-linear ISM and the γ-linear SCD which are projected distances tailored for γ-sheaves can be computed using TDA software dedicated to one-parameter persistence modules. Moreover, the time and memory complexity required to compute these two metrics are advantageous, since our approach does not require computing nor storing an entire n-persistence module. N. B. was supported by the Swiss Innovation Agency (Innosuisse project 41665.1 IP-ICT). F. P. was supported by the

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Section: The Dimension One Casementioning

confidence: 70%

Section: γ-Sheavesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Projected distances for multi-parameter persistence modules

Berkouk¹,

Petit²

2022

Preprint

Self Cite

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“…If N is real analytic and F , G are constructible and satisfy γ g (F , G ) = 0, then F G (see [PSW] and also Proposition B.7 for limits of constructible sheaves). However γ g is only a pseudo-metric as shown by Berkouk and Ginot (see [BG18], proposition 6.9) and the following example.…”

Section: The γ G -Metric On Sheavesmentioning

confidence: 80%

The singular support of sheaves is $γ$-coisotropic

Guillermou¹,

Viterbo²

2022

Preprint

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Introduction 2. Notation, comments and acknowledgements 3. Reminders on the singular support of sheaves 4. Quantization of Hamiltonian isotopies 5. Spectral invariants for sheaves and Lagrangians 5.1. Definition of spectral invariants 5.2. The case of Hamiltonian maps 5.3. Defining γ-coisotropic subsets 6. The γ g -metric on sheaves 6.1. The γ g -topology 6.2. Link with spectral invariants 6.3. γ g -limits and colimits 7. Proof of Theorem 1.2 8. γ-coisotropic vs cone-coisotropic 8.1. A γ-coisotropic set is cone-coisotropic 8.2. A cone-coisotropic set that is not γ-coisotropic 9. Questions and comments Appendix A. Persistence modules and Barcodes Appendix B. Decomposition in the completion of D l c (R) B.1. Limits of constructible sheaves B.2. Decomposition in D l c (R) Appendix C. Comparing different interleaving metrics on D(N × R).

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Thickening of the diagonal and interleaving distance

Petit,

Schapira

2023

Sel. Math. New Ser.

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A derived isometry theorem for sheaves

Cited by 8 publications

References 23 publications

Projected distances for multi-parameter persistence modules

Projected distances for multi-parameter persistence modules

The singular support of sheaves is $γ$-coisotropic

Thickening of the diagonal and interleaving distance

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