Persistent Homology Tutorial 4 of 4

(See also Part 1, Part 2, and Part 3.)

Topological feature vectors - persistent homology and machine learning

There are by now a wide variety of ways to incorporate persistent homology information into feature vectors for a machine learning task (Adcock et al 2016, Adams et al 2017, Atienza 2018, Bendich et al 2016, Bubenik 2015, Bubenik & Dlotko 2016, Carriere et al 2015, Carriere & Bauer 2018, Carriere et al 2018, Chazal & Divol 2018, Chen et al 2015, Chevyrev et al 2018, Chung et al 2009, Di Fabio & Ferri 2015, Hofer et al 2017, Kalisnik 2018, Reininghaus et al 2015, Skraba 2018, Topaz et al 2015, Zeppelzauer et al 2016). We tersely describe a few of these approaches.

For Henry's "from persistent homology to machine learning" survey slides from the ICERM TRIPODS bootcamp, please see here.

Persistence landscpaes

One example is persistence landscapes (Bubenik 2015). Bubenik & Dlotko (2017) have made their code available, and persistence landscapes are also implementable in R-TDA.

Persistence images

A second example is persistence images (Adams et al 2017), with Python code available from Nathaniel Saul (Persims) or Francis Motta, and with Matlab code available here.

We'll attempt to describe how to compute persistence images using the Jupyter notebooks accompanying Nathaniel Saul's Persim code. First, download the Persim code. Second, install Jupyter, perhaps using Anaconda as described in that link. Open a new terminal (which is often required after new installations) and in that terminal, try the following command.

jupyter notebook

This should open a window in an internet browser. In this browser window, change directories to persim/notebooks, and then open the Jupyter notebook Persistence Images.ipynb. You should then be able to run the code in the browser!

Topological feature vectors - coding challenges

6 shape classes

Change directory to the folder topological-data-analysis/topological-feature-vectors/data-6-shape-classes. NOTE: Folder missing--must fix! This subfolder contains persistence diagram data for 6 shape classes:

  1. A unit cube
  2. A circle of diameter one
  3. A sphere of diameter one
  4. Three clusters with centers randomly chosen in the unit cube
  5. Three clusters within three clusters
  6. A torus with a major diameter of one and a minor diameter of one half.

These shape classes are described in Section 6.1 of the paper Persistence Images: A Stable Vector Representation of Persistent Homology, and are shown below.

Illustrations of the six shape classes listed above

We produced 25 point clouds of 500 randomly sampled points from each shape class. We then added a level of Gaussian noise to each point, at a noise level neta=0.1 or neta=0.05. We then have already computed the persistent homology intervals in homological dimension i=0 and i=1. Should "neta" here be "eta" (Greek letter)? If so, we can then use MathJax on these equations.

For example, the file ToyData_PD_n05_23_6_0.txt corresponds to noise level neta=0.05, the 23rd point cloud randomly sampled from shape class 6., with persistent homology computed in dimension 0. Each row of this file has two entries: the birth and death time of a 0-dimensional persistent homology interval.

By contrast, the file ToyData_PD_n1_21_3_1.txt corresponds to noise level neta=1, the 21st point cloud randomly sampled from shape class 3., with persistent homology computed in dimension 1. Each row of this file has two entries: the birth and death time of a 1-dimensional persistent homology interval.

Your task is to use machine learning to distinguish these six classes from each other. In a K-medoids clustering test, some accuracies and computation times are displayed for bottleneck distances, Wasserstein distances, persistence landscapes, and persistence images in Table 1 of Persistence Images: A Stable Vector Representation of Persistent Homology. Do you have ideas for beating these accuracies or computation times?

Table comparing classification accuracy and time for persistence diagrams, persistence landscapes, and persistence images

In the above table, PD means the persistence diagram equipped with either the L^1 (1-Wasserstein), L^2 (2-Wasserstein), or sup (bottleneck) metrics. PL means the persistence landscape, equipped with either the L^1, L^2, or sup metric. PI means the persistence image, equipped with either the L^1, L^2, or sup metric. But you should develop your own techniques for turning barcodes into feature vectors, and see how they compare!

Some Matlab code for creating points from these shape classes is available here.

Bibliography

A. Adcock, E. Carlsson, and G. Carlsson. The ring of algebraic functions on persistence bar codes. Homology, Homotopy and Applications, 18:381-402, 2016

H. Adams, S. Chepushtanova, T. Emerson, E. Hanson, M. Kirby, F. Motta, R. Neville, C. Peterson, P. Shipman, and L. Ziegelmeier. Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18:1-35, 2017.

P. Bendich, J.S. Marron, E. Miller, A. Pieloch, and S. Skwerer, Persistent homology analysis of brain artery trees. Ann. Appl. Stat., 10:198-218, 2016.

M. A. Armstrong. Basic Topology. Springer, New York, Berlin, 1983.

N. Atienza, R. Gonzalez-Diaz, and M. Soriano-Trigueros. On the stability of persistent entropy and new summary functions for topological data analysis. arXiv:1803.08304, 2018.

U. Bauer. Ripser: A lean C++ code for the computation of Vietoris-Rips persistence barcodes. Software available at https://github.com/Ripser/ripser, 2015.

M. W. Brown, S. Martin, S. N. Pollock, E. A. Coutsias, and J. P. Watson. Algorithmic dimensionality reduction for molecular structure analysis. Journal of Chemical Physics, 129:064118, 2008.

P. Bubenik. Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16:77-10, 2015.

P. Bubenik and P. Dlotko, A persistence lanscapes toolbox for topological statistics. Journal of Symbolic Computation, 78:91-114, 2017.

G. Carlson, T. Ishkhanov, V. de Silva, and A. Zomorodian. On the local behavior of spaces of natural images. Int. J. Computer Vision, 76:1-12, 2008.

M. Carriere, U. Bauer. On the metric distortion of embedding persistence diagrams into reproducing kernel Hilbert spaces. arXiv:1806.06924, 2018

M. Carriere, M. Cuturi, and S. Oudot. Sliced Wasserstein kernel for persistence diagrams. arXiv:1706.03358, 2017.

M. Carriere, S. Oudot, and M. Ovsjanikov. Stable topological signatures for points on 3d shapes. In Computer Graphics Forum, 34:1-2, 2015.

Y.-C. Chen, D. Wong, A. Rinaldo, and L. Wasserman. Statistical analysis of persistence intensity functions. arXiv:1510.02502, 2015.

L. Crawford, A. Monod, A. X. Chen, S. Mukherjee, and R. Rabadan. Functional data analysis using a topological summary statistic: The smooth Euler characteristic transform. arXiv: https://arxiv.org/abs/1611.06818, 2017.

F. Chazal, V. de Silva, and S. Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, pages 1-22, 2013.

F. Chazal and V. Divol. The density of expected persistence diagrams and its kernel based estimation. arXiv:1802.10457, 2018.

I. Chevyrev, V. Nanda, H. Oberhauser. Persistence paths and signature features in topological data analysis. arXiv:1806.00381, 2018.

M.K. Chung, P. Bubenik, and P.T. Kim. Persistence diagrams of cortical surface data. In Information Processing in Medical Imaging, 386-297, Springer, 2009.

B. Di Fabio and M. Ferri. Comparing persistence diagrams through complex vectors. In International Conference on Image Analysis and Processing, LNCS 9279, pages 294-305, 2015.

H. Edelsbrunner and J. Harer. Computational Topology: An Introduction. American Mathematical Society, Providence, 2010.

H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511-533, 2002.

A. Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, 2002.

C. Hofer, R. Kwit, M. Niethammer, and A. Uhl. Deep learning with topological signatures. In Advances in Neural Information Processing Systems, pages 1634-1644, 2017.

S. Kalisnik. Tropical coordinates on the space of persistence barcodes. In Foundations of Computational Mathematics, pages 1-29, 2018.

A. B. Lee, K. S. Pedersen, and D. Mumford. The nonlinear statistics of high-contrast patches in natural images. Int. J. Comput. Vision, 54:83-103, 2003.

S. Martin and J. P. Watson. Non-manifold surface reconstruction from high-dimensional point cloud data. Computational Geometry, 44:427-441, 2011.

S. Martin, A. Thompson, E. A. Coutsias, and J. P. Watson. Topology of cyclo-octane energy landscape. Journal of Chemical Physics, 132: 234115, 2010.

J. Reininghaus, S. Huber, U. Bauer, and R. Kwitt. A stable multi-scale kernel for topological machine learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4741-4748, 2015.

P. Skraba. Persistent homology and machine learning. Informatica, 24(2), 2018.

C. Topaz, L. Ziegelmeier, and T. Halverson. Topological data analysis of biological aggregation models. PloS One, 10(5):e0126383, 2015.

M. Zeppelzauer, B. Zielinski, M. Juda, and M. Seidl. Topological descriptors for 3d surface analysis. In Computational Topology in Image Context: 6th International Workshop Proceedings, pages 77-87, 2016.

J. H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proc. R. Soc. Lond. B, 265:359-366, 1998.

A. Zomorodian. Advances in Applied and Computational Topology. American Mathematical Society, 2012.

A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput. Geom., 33:249-274, 2005.

(See also Part 1, Part 2, and Part 3.)