Abstract
We present in this paper an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, we derive, from a 1-dimensional persistent cohomology computation, explicit formulas for circle-valued functions on data with nontrivial underlying topology. We show that the language of principal bundles leads to coordinates defined on an open neighborhood of the data, but computed using only a smaller subset of landmarks. It is in this sense that the coordinates are sparse. Several data examples are presented, as well as theoretical results underlying the construction.
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Notes
- 1.
So that partitions of unity always exist.
- 2.
That is, so that support(φ j) ⊂closure(U j) for all j ∈ J.
- 3.
That is, we use the harmonic cocycle representative for appropriate inner products on cochains.
References
Bauer., U.: Ripser: a lean C++ code for the computation of Vietoris-Rips persistence barcodes. 2017. Software: https://github.com/Ripser/ripser.
Belkin, M., Niyogi P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation, 15 (6) , 1373–1396 (2003)
Ben-Israel, A., Greville, T. N.: Generalized inverses: theory and applications, volume 15. Springer Science & Business Media, (2003)
Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. International journal of computer vision, 76 (1) , 1–12, (2008)
de Silva V., Carlsson, G. E.: Topological estimation using witness complexes. SPBG, 4 , 157–166, (2004)
de Silva, V., Morozov, D., Vejdemo-Johansson, M.: Persistent cohomology and circular coordinates. Discrete & Computational Geometry, 45 (4) , 737–759, (2011)
de Silva, V., Skraba, P., Vejdemo-Johansson, M.: Topological analysis of recurrent systems. In Workshop on Algebraic Topology and Machine Learning, NIPS, (2012)
v. Hateren, J. H., v. d. Schaaf, A.: Independent component filters of natural images compared with simple cells in primary visual cortex. In Proceedings: Biological Sciences, 265 (1394) , 359–366, (1998)
Husemoller, D.: Fibre bundles, volume 5. Springer, (1966)
Jolliffe, I.: Principal component analysis. Wiley Online Library, (2002)
Kruskal, J. B.: Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29 (1) , 1–27, (1964)
Lee, A. B., Pedersen, K. S., Mumford, D.: The nonlinear statistics of high-contrast patches in natural images. International Journal of Computer Vision, 54 (1–3) , 83–103, (2003)
Milnor, J.: Construction of universal bundles, ii. Annals of Mathematics, pages 430–436, (1956)
Miranda, R.: Algebraic curves and Riemann surfaces, volume 5. American Mathematical Soc., (1995)
Nene, S. A., Nayar, S. K., Murase, H., et al: Columbia object image library (coil-20). 1996. Data available at http://www.cs.columbia.edu/CAVE/software/softlib/coil-20.php.
Perea, J. A.: A brief history of persistence. Morfismos, 23(1), 1–16, (2019)
Perea, J. A.: Multiscale projective coordinates via persistent cohomology of sparse filtrations. Discrete & Computational Geometry, 59 (1) , 175–225, (2018)
Radovanović, M., Nanopoulos, A., Ivanović, M.: Hubs in space: Popular nearest neighbors in high-dimensional data. Journal of Machine Learning Research, 11 (Sep) , 2487–2531, (2010)
Roweis, S. T., Saul, L. K.: Nonlinear dimensionality reduction by locally linear embedding. Science, 290 (5500) , 2323–2326, (2000)
Rybakken, E., Baas, N., Dunn, B.: Decoding of neural data using cohomological learning. Neural computation 31(1), 68–93, (2019)
Singh, G., Mémoli, F., Carlsson, G. E.: Topological methods for the analysis of high dimensional data sets and 3d object recognition. In SPBG, pages 91–100, (2007)
Tenenbaum, J. B., de Silva, V., Langford, J. C.: A global geometric framework for nonlinear dimensionality reduction. Science, 290 (5500) , 2319–2323, (2000)
Tralie, C. J., Berger, M.: Topological Eulerian synthesis of slow motion periodic videos. In: 2018 25th IEEE International Conference on Image Processing (ICIP), pages 3573–3577, (2018)
Xu, B., Tralie, C. J., Antia, A., Lin, M., Perea, J. A.: Twisty Takens: A geometric characterization of good observations on dense trajectories. Journal of Applied and Computational Topology, (2019). https://doi.org/10.1007/s41468-019-00036-9
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This work was partially supported by the NSF under grant DMS-1622301 and DARPA under grant HR0011-16-2-003.
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Perea, J.A. (2020). Sparse Circular Coordinates via Principal \(\mathbb {Z}\)-Bundles. In: Baas, N., Carlsson, G., Quick, G., Szymik, M., Thaule, M. (eds) Topological Data Analysis. Abel Symposia, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-030-43408-3_17
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