Abstract
The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e., a collection of intervals, to a finite metric space. Because of the nature of the invariant, barcodes are not well adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes, and these outputs can then be used as input to standard algorithms. It is the purpose of this paper to identify tropical coordinates on the space of barcodes and prove that they are stable with respect to the bottleneck distance and Wasserstein distances.
Highlights
In the past two decades, with the emergence of ‘big data,’ topology started playing a more prominent role in data analysis [5,6]
The unusual structure of the invariant makes the method hard to combine with standard algorithms within machine learning
We are currently working on automating this step and using machine learning methods on this collection of coordinate functions to select their weights
Summary
In the past two decades, with the emergence of ‘big data,’ topology started playing a more prominent role in data analysis [5,6]. Adcock et al [1] identified an algebra of polynomials on the barcode space that can be used as coordinates The problem with these functions is that they are not stable (i.e., Lipschitz) with respect to the Bottleneck and Wasserstein p-distances usually used. Countable generating set (Theorem 6.7) that separates the barcodes and prove that each function in this set is stable with respect to the bottleneck and Wasserstein distances (Theorems 7.1 and 7.3). These functions and their sums, minima and maxima can be used by researchers interested in analyzing datasets of shapes. We are currently working on automating this step and using machine learning methods (for example, the Lasso method) on this collection of coordinate functions to select their weights
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