Topological data analysis of human vowels: Persistent homologies across representation spaces

Topological Data Analysis (TDA) has been successfully used for various tasks in signal/image processing, from visualization to supervised/unsupervised classification. Often, topological characteristics are obtained from persistent homology theory. The standard TDA pipeline starts from the raw signal data or a representation of it. Then, it consists in building a multiscale topological structure on the top of the data using a pre-specified filtration, and finally to compute the topological signature to be further exploited. The commonly used topological signature is a persistent diagram (or transformations of it). Current research discusses the consequences of the many ways to exploit topological signatures, much less often the choice of the filtration, but to the best of our knowledge, the choice of the representation of a signal has not been the subject of any study yet. This paper attempts to provide some answers on the latter problem. To this end, we collected real audio data and built a comparative study to assess the quality of the discriminant information of the topological signatures extracted from three different representation spaces. Each audio signal is represented as i) an embedding of observed data in a higher dimensional space using Taken's representation, ii) a spectrogram viewed as a surface in a 3D ambient space, iii) the set of spectrogram's zeroes. From vowel audio recordings, we use topological signature for three prediction problems: speaker gender, vowel type, and individual. We show that topologically-augmented random forest improves the Out-of-Bag Error (OOB) over solely based Mel-Frequency Cepstral Coefficients (MFCC) for the last two problems. Our results also suggest that the topological information extracted from different signal representations is complementary, and that spectrogram's zeros offers the best improvement for gender prediction.