Persistence paths and signature features in topological data analysis
This provides a method for improving topological data analysis in statistical learning, though it appears incremental as it builds on existing persistent homology techniques.
The paper tackles the problem of representing barcodes from persistent homology for machine learning by introducing a new feature map that converts barcodes to paths and then to tensor series, achieving state-of-the-art results on classification benchmarks.
We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations - barcode to path, path to tensor series - results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.