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Persistent homology (PH) provides topological descriptors for geometric data,such as weighted graphs, which are interpretable, stable to perturbations, andinvariant under, e.g., relabeling. Most applications of PH focus on theone-parameter case -- where the descriptors summarize the changes in topologyof data as it is filtered by a single quantity of interest -- and there is nowa wide array of methods enabling the use of one-parameter PH descriptors indata science, which rely on the stable vectorization of these descriptors aselements of a Hilbert space. Although the multiparameter PH (MPH) of data thatis filtered by several quantities of interest encodes much richer informationthan its one-parameter counterpart, the scarceness of stability results for MPHdescriptors has so far limited the available options for the stablevectorization of MPH. In this paper, we aim to bring together the best of bothworlds by showing how the interpretation of signed barcodes -- a recent familyof MPH descriptors -- as signed measures leads to natural extensions ofvectorization strategies from one parameter to multiple parameters. Theresulting feature vectors are easy to define and to compute, and provablystable. While, as a proof of concept, we focus on simple choices of signedbarcodes and vectorizations, we already see notable performance improvementswhen comparing our feature vectors to state-of-the-art topology-based methodson various types of data.

Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures