Next Article in Journal
Feature Selection from Lyme Disease Patient Survey Using Machine Learning
Previous Article in Journal
Applying Neural Networks in Aerial Vehicle Guidance to Simplify Navigation Systems
Previous Article in Special Issue
Lumáwig: An Efficient Algorithm for Dimension Zero Bottleneck Distance Computation in Topological Data Analysis
Article

From Trees to Barcodes and Back Again: Theoretical and Statistical Perspectives

1
Blue Brain Project, École Polytechnique Fédérale de Lausanne (EPFL), Campus Biotech, 1202 Geneva, Switzerland
2
Laboratory for Topology and Neuroscience, Brain Mind Institute, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
*
Author to whom correspondence should be addressed.
Algorithms 2020, 13(12), 335; https://doi.org/10.3390/a13120335
Received: 30 September 2020 / Revised: 3 December 2020 / Accepted: 4 December 2020 / Published: 11 December 2020
(This article belongs to the Special Issue Topological Data Analysis)
Methods of topological data analysis have been successfully applied in a wide range of fields to provide useful summaries of the structure of complex data sets in terms of topological descriptors, such as persistence diagrams. While there are many powerful techniques for computing topological descriptors, the inverse problem, i.e., recovering the input data from topological descriptors, has proved to be challenging. In this article, we study in detail the Topological Morphology Descriptor (TMD), which assigns a persistence diagram to any tree embedded in Euclidean space, and a sort of stochastic inverse to the TMD, the Topological Neuron Synthesis (TNS) algorithm, gaining both theoretical and computational insights into the relation between the two. We propose a new approach to classify barcodes using symmetric groups, which provides a concrete language to formulate our results. We investigate to what extent the TNS recovers a geometric tree from its TMD and describe the effect of different types of noise on the process of tree generation from persistence diagrams. We prove moreover that the TNS algorithm is stable with respect to specific types of noise. View Full-Text
Keywords: tree; topological data analysis; persistence barcode; symmetric group; inverse methods tree; topological data analysis; persistence barcode; symmetric group; inverse methods
Show Figures

Figure 1

MDPI and ACS Style

Kanari, L.; Garin, A.; Hess, K. From Trees to Barcodes and Back Again: Theoretical and Statistical Perspectives. Algorithms 2020, 13, 335. https://doi.org/10.3390/a13120335

AMA Style

Kanari L, Garin A, Hess K. From Trees to Barcodes and Back Again: Theoretical and Statistical Perspectives. Algorithms. 2020; 13(12):335. https://doi.org/10.3390/a13120335

Chicago/Turabian Style

Kanari, Lida, Adélie Garin, and Kathryn Hess. 2020. "From Trees to Barcodes and Back Again: Theoretical and Statistical Perspectives" Algorithms 13, no. 12: 335. https://doi.org/10.3390/a13120335

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop