Wavelet Scattering Transform. Mathematical Analysis and Applications to VIRGO Gravitational Waves Data

In this thesis we investigate the properties of the Wavelet Scattering Transform (WST), a mathematical technique recently proposed in the context of signal processing and ML. We explore the application of WST in distinguishing glitches that impact the accurate detection of gravitational waves (GW). This thesis is done in collaboration with the Istituto Nazionale di Fisica Nucleare (INFN), in the interTwin project framework. This work investigates the possibility of applying WST to extract discriminative features from glitches, i.e. noises captured by gravitational wave detectors not related to astrophysical phoenomena. Physical issues, e.g. non-homogeneity in centering the time-series over the energy spike, environmental noise and deformations caused by the high sensitivity of the interferometer are problems that critically plague the state-of-the-art techniques. The work is divided in three sections. The first section provides a formal definition of the wavelet scattering operator and a comparison to other standard signal representations, with particular regards to three fundamental properties: translation invariance, stability to additive perturbations, local invariance to continuous stretches. These properties are crucial for classification and discrimination tasks, as natural elements of the same class exhibit slight variations. However, standard methods like Q-transform based on Short Time Fourier Transform (STFT) lack these properties, leading to distorted representations that are less accurate in capturing essential features. The second part of the thesis focuses on the experiments and results on the Free Spoken Digits dataset, analyzing the effect on classification algorithms of WST, compared to the state-of-the-art representation technique based on STFT. On average the algorithms trained in the wavelet scattering domain outperformed the same classifiers trained in the Fourier domain, proving from an empirical point of view that WST enhances discriminative features across the different classes. The third part focuses on applying WST to GW data collected by VIRGO interferometer, and comparing the results with the state-of-the-art method, based on Q-transform, which, being strictly related to STFT, does not provide translation invariant features. This work is the first that applies WST to gravitational data in order to classify and discriminate different kinds of glitches. Trials are developed on the dataset provided by INFN, consisting of 855 samples of scattered-light glitches. Q-transform is known to perform bad on this class, since it is not able to discriminate the most significant features. An analysis of dispersion has been conducted to compare the behaviour of WST against Q-transform. We develop a technique to estimate dispersion for the two representations based on the Principal Component Analysis (PCA). The results show that in WST space scattered light glitches appear to be much more clustered. Promising future directions lie in understanding how to leverage WST patterns to whiten astrophysical signals and in developing efficient neural architectures with optimal parametrization to classify glitches belonging to different classes, exploiting invariance properties of WST. This study sets the stage for further advancements in the field, offering potential solutions to improve the accuracy and efficiency of gravitational glitch detection methods. Additionally lighter architectures are consistently preferred due to their lower training costs.