We review the renormalization group framework for signal detection in high-dimensional data, tailored to the regime where the signal may be of extensive rank and does not separate from the noise bulk as isolated spikes. The framework provides a conceptually simple criterion for distinguishing signal from noise within a quasi-continuous spectral region near a random-matrix universality class. This scenario lies beyond the reach of standard methods such as the Baik-Ben Arous-Péché threshold, which requires eigenvalues to be cleanly separated from the bulk. The renormalization group approach, by contrast, directly tracks spectral deformations and consistently yields a lower limit of detection without relying on spike separation. We review results that identify the presence of a signal by testing the stability of the Gaussian fixed point of an effective field theory for the collective behaviour of the degrees of freedom in the spectral tail, where the signal resides. We also discuss how the scale dependence of the canonical dimension, induced by the signal, manifests as a dimensional phase transition.
We review the renormalization group framework for signal detection in high-dimensional data, tailored to the regime where the signal may be of extensive rank and does not separate from the noise bulk as isolated spikes. The framework provides a conceptually simple criterion for distinguishing signal from noise within a quasi-continuous spectral region near a random-matrix universality class. This scenario lies beyond the reach of standard methods such as the Baik-Ben Arous-Péché threshold, which requires eigenvalues to be cleanly separated from the bulk. The renormalization group approach, by contrast, directly tracks spectral deformations and consistently yields a lower limit of detection without relying on spike separation. We review results that identify the presence of a signal by testing the stability of the Gaussian fixed point of an effective field theory for the collective behaviour of the degrees of freedom in the spectral tail, where the signal resides. We also discuss how the scale dependence of the canonical dimension, induced by the signal, manifests as a dimensional phase transition.
This review paper utilizes renormalization group techniques for signal detection in nearly continuous positive spectra. We emphasize the universal aspects of the analogue field-theory approach. The primary objective is to present an extended self-consistent construction of the analogue effective field-theory framework for data, which can be interpreted as a maximum entropy model. In particular, we leverage universality arguments to justify the \mathbb{Z}_2 ℤ 2 symmetry of the classical action, highlighting the existence of both a large-scale (local) regime and a small-scale (nonlocal) regime. Secondly, in relation to noise models, we observe the universal relationship between phase transitions and symmetry breaking near the detection threshold. Finally, we address the challenge of defining the covariance matrix for tensor-like data. Based on the cutting graph prescription, we note the superiority of definitions that rely on complete graphs of large size for data analysis.
We establish a correspondence between anomaly detection in high-noise regimes and the renormalization group flow of non-equilibrium field theories. We provide a physical grounding for this framework by proving that the detection of phase transitions in interacting non-equilibrium systems maps to the study of an effective equilibrium field theory near its Gaussian fixed point, which we identify with the universal Marchenko-Pastur distribution. Applying the Functional Renormalization Group to the two-dimensional Model A, we demonstrate that the noise-to-signal ratio acts as a physical temperature, where the signal emerges as ordered domains within a thermalized background of fluctuations. Using the exact Onsager solution as a benchmark, we show that this approach identifies critical thresholds with an error below 4%, significantly outperforming standard information-theoretic metrics such as the Kullback-Leibler divergence. Our results provide a universal strategy for resolving structures in complex datasets near criticality, bridging the gap between statistical mechanics and statistical inference.
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.
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
The tensorial principal component analysis is a generalization of ordinary principal component analysis focusing on data which are suitably described by tensors rather than matrices. This paper aims at giving the nonperturbative renormalization group formalism, based on a slight generalization of the covariance matrix, to investigate signal detection for the difficult issue of nearly continuous spectra. Renormalization group allows constructing an effective description keeping only relevant features in the low "energy" (i.e., large eigenvalues) limit and thus providing universal descriptions allowing to associate the presence of the signal with objectives and computable quantities. Among them, in this paper, we focus on the vacuum expectation value. We exhibit experimental evidence in favor of a connection between symmetry breaking and the existence of an intrinsic detection threshold, in agreement with our conclusions for matrices, providing a new step in the direction of a universal statement.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].