Persistent Homology: Course Plan

Transcription

1 Persistent Homology: Course Plan Andrey Blinov 16 October 2017 Abstract This is a list of potential topics for each class of the prospective course, with references to the literature. Each class will have some basic material in it, and something more advanced if time allows. 1 Outline The class will last for weeks. It could be informally divided in three parts: theory, practice, and applications. There is a plan for the course, but it might change a little bit, depending on how fast we move and what applications want to investigate. The main sources used in this course are the following ones: the book by De Silva et al. [10]; the book by Oudot [21]; the summary of the course read by Carlsson [3]; the handbook by Edelsbrunner and Morozov [17]; the survey by Edelsbrunner and Morozov [18]; finally, the survey by Ghirst [19]. 1.1 Theory Week 0 and week 1. Introduction. Image recognition. Basics of topology: homeomorphisms. Basics of homology: simplicial complexes. Simplicial and singular homology. What H 0 and H 1 mean for planar sets. Literature: [19], [3, pp. 5-26], [18]. Week 2. Persistent modules: definiton, properties. Main classification theorem. Persistence diagrams. Integer coefficients and multivariable persistent homology. Quiver representations. Zigzag persistent homology. 1

2 Literature: [10, ch. 2], [3, pp , 46-48]; [6] for multidimensional persistence. Commentary: The plan is quite ambitious; just discussion of the decomposition theorem would be good enough. Week 3. Persistence diagrams and measure. Barcode spaces and persistence diagrams. Metrics: interleaving distance, bottleneck distance, Wasserstein distance. Literature: [10, ch. 3]; [3, pp ]. Commentary: The measure theory of persistence diagrams is quite strange, but we need it for the topic of interleaving. Week 4. Interleaving. Tameness. Isometry theorem. Stability theorems. Literature: [10, ch. 4, 5], [21, ch. 3].. Commentary: A good property of the persistence theory is that small perturbations give us close results. Week 5. Persistent relative homology and cohomology. Zigzag persistence. Extended persistence diagram. Literature: [14, 15], [21, pp ]. Commentary: These are more complex topics in the persistence theory, but they are still useful in applications. 1.2 Practice In practice, we usually have a huge data point sets in a high-dimensional space (or just a huge metric space). We want to compute persistence homology of a set as fast as possible, either by using faster algorithms, or by choosing better complexes that still represent the topology of the data point set well enough. This is also the moment where other students can start giving talks (but not necessary). Week 6. Comparison of complexes. Literature: [21, ch. 5]; [13], recent papers like [23] and [24]. Commentary: This topic is tightly connected with the next one, but has more theory in it. 2

3 Week 7. Simple algorithms used in computing persistent homology. Row algorithm, column algorithm. H 0 and clustering. Literature: [17, pp ]; perhaps something from the book on computational homology [20]; some of the first papers on the subject of persistent homology like [5] and [16]. For clustering, look at [21, ch. 6], also [11]. More advanced approach is in [1] and [12]. It also might have some sense to look at the libraries like PERSEUS ( perseus/index.html) or DIONYSUS ( dionysus/) and see how they work. Commentary: We should also talk about noise reduction at some point. Moreover, this topic might be too large for one meeting, so we can extend it to the next week. Finally, clustering (computing H 0 ) deserves a closer look, and could be classified as applications. 1.3 Applications It is supposed that this part will be presented mostly by students. There are many papers that talk about different applications of the persistence theory. Good places to start are [3, 50-67], [21, ch. 5-6], [10, pp. 5-10], and [18, pp ]. Also, the plan for this section is much less definite: previous topics might take additional time to cover, and your choice of papers might be different. Week 8. Applications. natural images, viral evolution, etc. Literature: [21, pp ], [3, pp ] and [4] for natural images, [3, pp ] [8] for viral evolution. Commentary: The first topic appears in almost every survey. The viral evolution paper is a biological one, and its main part does not contain a lot of mathematics; however, the appendix should. Week 9. Morse theory and persistence. Curvature and persistence. Literature: [19, p. 9], [2] for the first topic, and [7] for the second. Commentary: The first paper is quite technical and talks about general theory. The second one is more practical. 3

4 Week 10. Periodic functions and persistent homology. Literature: [3, pp ], [22]. Week 11. Stable descriptors for metric spaces. Literature: [10, pp. 5-7], [9]. 2 Final Commentary Of course, you are free to choose to read something not mentioned on the list, as far as it has some connection with the topic of persistence homology. It is also a good idea to observe the home pages by Carlsson, de Silva, Zomoridian, Oudot, Edelsbrunner, Morozov, Ghirst, and others the recent papers might contain something completely new. References [1] Ulrich Bauer, Michael Kerber, and Jan Reininghaus. Clear and compress: Computing persistent homology in chunks. In Topological Methods in Data Analysis and Visualization III, pages Springer, Cham, [2] Peter Bubenik and Peter T. Kim. A statistical approach to persistent homology. Homology, Homotopy and Applications, 9(2): , August [3] Gunnar Carlsson. Topological pattern recognition for point cloud data. Acta Numerica, 23: , May [4] Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, and Afra Zomorodian. On the local behavior of spaces of natural images. International Journal of Computer Vision, 76(1):1 12, January [5] Gunnar Carlsson and Afra Zomoridian. Computing persistent homology. Discrete and Computational Geometry, 33(2): , February [6] Gunnar Carlsson and Afra Zomoridian. The theory of multidimensional persistence. Discrete and Computational Geometry, 42(1):71 93, July

5 [7] Gunnar Carlsson, Afra Zomorodian, Anne Collins, and Leonidas Guibas. Persistence barcodes for shapes. International Journal of Shape Modeling, 11(2): , January [8] Joseph Minhow Chan, Gunnar Carlsson, and Raul Rabadan. The topology of viral evolution. Proceedings of the National Academy of Sciences of the United States of America (PNAS), 110(46): , November [9] Frédéric Chazal, David Cohen-Steiner, Leonidas J. Guibas, Facundo Mémoli, and Steve Y. Oudot. Gromov-hausdorff stable signatures for shapes using persistence. In SGP 09 Proceedings of the Symposium on Geometry Processing, pages Eurographics Association, [10] Frédéric Chazal, Vin de Silva, Marc Glisse, and Steven Oudot. The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics, [11] Frédéric Chazal, Leonidas J. Guibas, Steve Y. Oudot, and Primoz Skraba. Persistence-based clustering in riemannian manifolds. Journal of the ACM, 60(6), November [12] Chao Chen and Michael Kerber. Persistent homology computation with a twist. In Proceedings 27th European Workshop on Computational Geometry, [13] Vin de Silva and Robert Ghirst. Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology, 7(1): , April [14] Vin de Silva, Dmitriy Morozov, and Mikael Veldemo-Johansson. Dualities in persistent (co)homology. Inverse Problems, 27(12), November [15] Vin de Silva and Mikael Veldemo-Johansson. Persistent homology and circular coordinates. Discrete and Computational Geometry,, 45(4): , June [16] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simplification. Discrete and Computational Geometry, 28(4): , November

6 [17] Herbert Edelsbrunner and Dmitriy Morozov. Persistent homology. [18] Herbert Edelsbrunner and Dmitriy Morozov. Persistent homology: Theory and practice. [19] Robert Ghirst. Barcodes: The persistent topology of data. Bulletin of the American Mathematical Society, 45(1):61 75, January [20] Tomasz Kaczynski, Konstantin Mischaikow, and Marian Mrozek. Computational Homology. Springer-Verlag, New York, [21] Steve Y. Oudot. Persistence Theory: From Quiver Representations to Data Analysis. AMS Providence, Rhode Island, [22] Jose Perea and John Harer. Sliding windows and persistence: An application of topological methods to signal analysis. Foundations of Computational Mathematics, 15(3): , June [23] Donald R. Sheehy. Linear-size approximations to the vietoris-rips filtration. Discrete and Computational Geometry, 49(4): , June [24] Afra Zomorodian. Fast construction of the vietoris-rips complex. Computer and Graphics, 3(34): , June