Abstract
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.
Highlights
From the point of view of information theory, statistical physics looks like a consequence of a statistical inference based on the maximum entropy estimate, disregarding the specific aspects of the microscopic problem, and it is for this reason a general paradigm [2,3,4]
We show in the same figure that there are other renormalization group (RG) trajectories which do not allow a restoration of the symmetry
We showed that the field theory approximation works well up to some scale Λ0
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Anticipating the results of the section, we discuss the relevance of interactions and argue the existence of a “wall” in generalized momenta, defining a physical cut-off, below which the relevant sector diverges and the field theoretical embedding breaks down Working above this wall, we show that only a few number of local couplings are relevant, essentially quartic and sixtic couplings for small perturbations around MP law. Within this approximation scheme, we are able to identify the presence of a signal (materialized by a few number of discrete spikes disturbing the purely noisy data) as a lack of symmetry restoration in the deep IR, for k ∼ 1/N.
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