The Discrete Fourier Transform for Persistent Homology: Applications to Music Analysis
Victoria Callet is a PhD student at the Institut de Recherche Mathématiques Avancées de Strasbourg, under the supervision of Pierre Guillot and Moreno Andreatta. In her thesis, she studies some links between algebraic topology and music analysis. More precisely, she works on the topological analysis of musical processes and structures via a tool of simplicial theory called persistent homology.
Victoria Callet is hosted in December by the Musical Representations team of Ircam-STMS (Ircam, Sorbonne University, CNRS, Ministry of Culture). The Mercredis de STMS invited her to present her research.
If you couldn't attend the event or want to see it again, it is now on-line on https://youtu.be/AGNBWqHqBMs and on https://medias.ircam.fr/xb7b504_la-transformee-de-fourier-discrete-au-serv
Abstract:
Persistent homology is a tool of simplicial theory built at the end of the 20th century. It is used in particular in the field of Topological Data Analysis (TDA) and shape recognition: the principle is to extract a point cloud from an object to be studied, then to transform this cloud into a filtered simplicial complex using for example the Vietoris-Rips method. In general, the choice of the distance between the points is crucial, and it is at this point that the intervention of the Discrete Fourier Transform (DFT) is decided.
The goal is then to calculate the simplicial homology of a filtered complex at each moment of the filtration, and to look at the topological features that persist over time. This approach allows to encode the topological evolution of an object through a single algebraic structure.
This talk helped us to redefine persistent homology, and to understand how it can be used to recover topological information present in music scores; in particular by extracting point clouds and using DFT as a distance on them.