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.