Field theoretical approach for signal detection in nearly continuous positive spectra III: Universal features
Résumé
This review paper aims to use the renormalization group technique to address the problem of signal detection in nearly continuous positive spectra. It is especially devoted to highlighting universal aspects of the analogue field theory approach. There are three underlying information in this paper. The first one is an extended and self-consistent construction of the analogue effective field theory framework for data, which can be viewed as a maximum entropy model. In particular, we justify the $\mathbb{Z}_2$-symmetry of the classical action exploiting universality arguments and stress out the existence of two regimes, a local regime valid for a large scale and a small scale regime exhibiting a specific non-locality. The second is based on a systematic investigation around standard models of noise: we point out the universal relation between phase transition and symmetry breaking in the vicinity of the detection threshold. Last but not least, we propose to tackle the open issue of a good definition of the covariance matrix for tensorial like data. Based on the "cutting graph" prescription and through a systematic investigation we stress out the superiority of definitions based on complete graphs of large size for data analysis.