Field theoretical approach for signal detection in nearly continuous positive spectra I: Matricial data

TL;DR

This paper introduces a field theoretical approach to improve signal detection in large covariance matrices with nearly continuous spectra, linking symmetry breaking to detection thresholds and providing a unified framework for analysis.

Contribution

It develops a novel field theoretical framework combining previous ideas for analyzing large covariance matrices, enhancing theoretical understanding and practical detection methods.

Findings

Experimental evidence links symmetry breaking to detection thresholds.

The framework unifies coarse-graining and PCA concepts.

Improves detection in nearly continuous spectra scenarios.

Abstract

Renormalization group techniques are widely used in modern physics to describe the low energy relevant 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 data sets are very larges. Signal detection and recognition for covariance matrix having a nearly continuous spectra is currently one of these opened issues. First investigations in this direction has been proposed in [Journal of Statistical Physics, 167, Issue 3-4, pp 462-475, (2017)] and [arXiv:2002.10574], 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, and exhibit experimental evidences in favor of a connection between symmetry breaking and the existence of an intrinsic detection threshold.