  The large scale behavior of systems having a large number of interacting
degrees of freedom is suitably described using renormalization group, from
non-Gaussian distributions. Renormalization group techniques used in physics
are then expected to be helpful for issues when standard methods in data
analysis break down. Signal detection and recognition for covariance matrices
having nearly continuous spectra is currently an open issue in data science and
machine learning. Using the field theoretical embedding introduced in
arXiv:2011.02376 to reproduces experimental correlations, we show in this paper
that the presence of a signal may be characterized by a phase transition with
$\mathbb{Z}_2$-symmetry breaking. For our investigations, we use the
nonperturbative renormalization group formalism, using a local potential
approximation to construct an approximate solution of the flow. Moreover, we
focus on the nearly continuous signal build as a perturbation of the
Marchenko-Pastur law with many discrete spikes.
