Field Theoretical Approach for Signal Detection in Nearly Continuous Positive Spectra I: Matricial Data

Renormalization group techniques are widely used in modern physics to describe the relevant low energy aspects of systems involving a large number of degrees of freedom. Those techniques are thus expected to be a powerful tool to address open issues in data analysis when datasets are highly correlated. Signal detection and recognition for a covariance matrix having a nearly continuous spectra is currently one of these opened issues. First, investigations in this direction have been proposed in recent investigations from an analogy between coarse-graining and principal component analysis (PCA), regarding separation of sampling noise modes as a UV cut-off for small eigenvalues of the covariance matrix. The field theoretical framework proposed in this paper is a synthesis of these complementary point of views, aiming to be a general and operational framework, both for theoretical investigations and for experimental detection. Our investigations focus on signal detection. They exhibit numerical investigations in favor of a connection between symmetry breaking and the existence of an intrinsic detection threshold.