Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations

We present a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction.

[1]  Jose A. Perea.  Persistent homology of toroidal sliding window embeddings , 2016, 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[2]  Gunnar Carlsson,et al.  Homological Coordinatization , 2011, ArXiv.

[3]  Mark Guzdial,et al.  From science to engineering , 2011, Commun. ACM.

[4]  G. Carlsson,et al.  Topology of viral evolution , 2013, Proceedings of the National Academy of Sciences.

[5]  P. Schönemann,et al.  A generalized solution of the orthogonal procrustes problem , 1966 .

[6]  Frédéric Chazal,et al.  A Sampling Theory for Compact Sets in Euclidean Space , 2006, SCG '06.

[7]  Dmitriy Morozov,et al.  Dualities in persistent (co)homology , 2011, ArXiv.

[8]  Grzegorz Jablonski,et al.  The Persistent Homology of a Self-Map , 2015, Found. Comput. Math..

[9]  Tamal K. Dey,et al.  Optimal homologous cycles, total unimodularity, and linear programming , 2010, STOC '10.

[10]  S. Weinberger.  Persistent Homology , 2019, Brain Network Analysis.

[11]  Gunnar E. Carlsson,et al.  Topological pattern recognition for point cloud data* , 2014, Acta Numerica.

[12]  Rick Miranda,et al.  Algebraic Curves and Riemann Surfaces , 1995 .

[13]  Facundo Mémoli,et al.  Characterization, Stability and Convergence of Hierarchical Clustering Methods , 2010, J. Mach. Learn. Res..

[14]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[15]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2005, Discret. Comput. Geom..

[16]  Lek-Heng Lim,et al.  Hodge Laplacians on graphs , 2015, SIAM Rev..

[17]  Nicholas J. Cavanna,et al.  A Geometric Perspective on Sparse Filtrations , 2015, CCCG.

[18]  Jose A. Perea,et al.  A Klein-Bottle-Based Dictionary for Texture Representation , 2014, International Journal of Computer Vision.

[19]  Loring W. Tu,et al.  Differential forms in algebraic topology , 1982, Graduate texts in mathematics.

[20]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[21]  Jean-Pierre Serre.  Faisceaux algébriques cohérents , 1955 .

[22]  Vin de Silva,et al.  Coverage in sensor networks via persistent homology , 2007 .

[23]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[24]  Tamal K. Dey,et al.  Approximating loops in a shortest homology basis from point data , 2010, SoCG '10.

[25]  J. Marron,et al.  Analysis of principal nested spheres. , 2012, Biometrika.

[26]  Jean-Paul Watson,et al.  Topology of cyclo-octane energy landscape. , 2010, The Journal of chemical physics.

[27]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[28]  J. Kruskal.  Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis , 1964 .

[29]  Facundo Mémoli,et al.  Classifying Clustering Schemes , 2010, Foundations of Computational Mathematics.

[30]  May,et al.  A Concise Course in Algebraic Topology , 1999 .

[31]  Jose A. Perea,et al.  Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis , 2013, Found. Comput. Math..

[32]  Chao Chen,et al.  Hardness Results for Homology Localization , 2010, SODA.

[33]  E. Coutsias,et al.  Topology of cyclo-octane energy landscape. , 2010, The Journal of chemical physics.

[34]  R. Ho.  Algebraic Topology , 2022 .

[35]  John Milnor,et al.  Characteristic Classes. (Am-76), Volume 76 , 1962 .

[36]  Vin de Silva,et al.  Persistent Cohomology and Circular Coordinates , 2009, SCG '09.

[37]  Primoz Skraba,et al.  Topological Analysis of Recurrent Systems , 2012, NIPS 2012.

[38]  Abubakr Muhammad,et al.  Coverage and hole-detection in sensor networks via homology , 2005, IPSN 2005. Fourth International Symposium on Information Processing in Sensor Networks, 2005..