Abstract
The large scale behavior of systems having a large number of interacting degrees of freedom is suitably described using the renormalization group from non-Gaussian distributions. Renormalization group techniques used in physics are then expected to provide a complementary point of view on standard methods used in data science, especially for open issues. 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 Entropy, 23(9), 1132 to reproduce experimental correlations, we show in this paper that the presence of a signal may be characterized by a phase transition with Z2-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.
Highlights
The renormalization group (RG) is one of the most important discoveries of the twentieth century in physics
This hierarchy is intrinsically related to the notion of scale; and RG aims to construct large scale effective theories integrating out microscopic degrees of freedom in such a way to preserve long-distance physics
The paper is organized as follows: In Section 2 we provide a short state of the art, allowing us to position our work in the existing literature, especially in regard to the continuous spectra issue
Summary
The renormalization group (RG) is one of the most important discoveries of the twentieth century in physics. The most universal formalization of the RG [5–13] is based on the existence of an intrinsic hierarchy of degrees of freedom in such a way that we can progressively ignore some of them, and “integrated” in a less fundamental effective description for the remaining ones For this reason, the RG is relevant in many-body physics, for all problems involving a very large number of interacting degrees of freedom. It seems reasonable to investigate the RG flow associated with the eigenvalue distribution of the covariance matrix through a suitable field theoretical embedding that is able to reproduce (at least partially) the data correlations and to extract the relevant features of the distributions Note that such a strategy follows the current point of view of field theory, understood as effective descriptions at a large scale of some partially understood microscopic physics [1,2,30].
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