Supervised Learning Using Homology Stable Rank Kernels

Exciting recent developments in Topological Data Analysis have aimed at combining homology-based invariants with Machine Learning. In this article, we use hierarchical stabilization to bridge between persistence and kernel-based methods by introducing the so-called stable rank kernels. A fundamental property of the stable rank kernels is that they depend on metrics to compare persistence modules. We illustrate their use on artificial and real-world datasets and show that by varying the metric we can improve accuracy in classification tasks.

[1]  Marian Gidea,et al.  Topological Data Analysis of Financial Time Series: Landscapes of Crashes , 2017, 1703.04385.

[2]  Sayan Mukherjee,et al.  Fréchet Means for Distributions of Persistence Diagrams , 2012, Discrete & Computational Geometry.

[3]  Adam R Ferguson,et al.  Topological data analysis for discovery in preclinical spinal cord injury and traumatic brain injury , 2015, Nature Communications.

[4]  Leonidas J. Guibas,et al.  Proximity of persistence modules and their diagrams , 2009, SCG '09.

[5]  S. Mukherjee,et al.  Probability measures on the space of persistence diagrams , 2011 .

[6]  Ulrich Bauer,et al.  Ripser: efficient computation of Vietoris–Rips persistence barcodes , 2019, Journal of Applied and Computational Topology.

[7]  Henri Riihimäki,et al.  Metrics and Stabilization in One Parameter Persistence , 2019, SIAM J. Appl. Algebra Geom..

[8]  Wojciech Chachólski,et al.  Multidimensional Persistence and Noise , 2015, Foundations of Computational Mathematics.

[9]  Hunter S. Snevily.  Combinatorics of finite sets , 1991 .

[10]  W. Massey.  A basic course in algebraic topology , 1991 .

[11]  Oliver Gafvert,et al.  Stable Invariants for Multidimensional Persistence , 2017, 1703.03632.

[12]  Qi Zhao,et al.  Learning metrics for persistence-based summaries and applications for graph classification , 2019, NeurIPS.

[13]  Steve Oudot,et al.  The Structure and Stability of Persistence Modules , 2012, Springer Briefs in Mathematics.

[14]  J. Hausmann.  On the Vietoris-Rips complexes and a Cohomology Theory for metric spaces , 1996 .

[15]  Nello Cristianini,et al.  Kernel Methods for Pattern Analysis , 2004 .

[16]  Afra Zomorodian,et al.  Computing Persistent Homology , 2005, Discret. Comput. Geom..

[17]  Nello Cristianini,et al.  Kernel Methods for Pattern Analysis , 2003, ICTAI.

[18]  Ulrich Bauer,et al.  A stable multi-scale kernel for topological machine learning , 2014, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[19]  Herbert Edelsbrunner,et al.  Topological Persistence and Simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.